diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlzbm" "b/data_all_eng_slimpj/shuffled/split2/finalzzlzbm" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlzbm" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn the present climate of fiscal austerity, research funding is falling in a number of countries \\citep{BIS2015}. Questions about the efficiency of funding mechanisms are therefore very much in the minds of policy makers and science policy analysts. Some advocate closer monitoring and measurement to optimise the `yield' from research \\citep{Chalmers2014}. Others suggest funders should consider the types of research they support, ensuring they maintain diversity \\citep{Wallace2015}. In this context, understanding the relationship between funding inputs and research outputs has important implications for the design and management of research at the level of grants, programmes, agencies' portfolios, and even at the national and international funding landscape level (e.g.\\ where interdependencies and complementarities between funders could be identified, and co-ordination can improve efforts to address particular diseases).\n\nFunders are investing in the development of data infrastructures capable of supporting such efforts as well as of increasing the transparency around funding decisions. These include the RePORTER database of the US National Institutes for Health (NIH), the Gateway to Research of the Research Councils UK (RCUK), and the OpenAIRE of the European Commission. Some funders have also combined efforts in order to compare their research portfolios and to track national trends in funding \\citep{UKCRC2014}. However, considering the diversity of funders that populate funding systems, these efforts, led by major funders, do not include data on the activities of numerous smaller organisations that also support research. \n\nIn order to consider the research being funded in a given field or country and to reflect on changes to funding policies, it is desirable to obtain a comprehensive overview of the funding landscape by tracing inputs and outputs from a wide range of funders and researchers that are active in a given area. One promising source of data for this purpose is the information included in the acknowledgement \\textit{paratext} of scientific publications where authors commonly give thanks for support \\citep[e.g.][]{Costas2012, Dawson1998, Rigby2011}. In particular, funding information in acknowledgements has the potential to establish a direct link between funding inputs and research outputs on a grand scale, without the need to gain direct access to data via individual funders or researchers. This paper explores a number of approaches to using funding data derived from in publication acknowledgements. In doing so, we address key questions about the coverage of these data and explore the utility of mapping funding landscapes using these data.\n\nMaking systematic use of the funding information included in the acknowledgment sections of publications is, however, difficult due to the lack of standardization in the structure and content of acknowledgements --- in contrast to references, for example. In order to support the use of acknowledgement \\textit{paratext} some databases, such as ISI Web of Science (WoS) and MEDLINE\/PubMed, provide ready-extracted and coded funding data. Yet, uncertainties remain about the coverage of such sources \\citep{Koier2014, Rigby2011}. The motivation for this paper is to resolve some of the uncertainties of working with funding data, in the hope that improving methods and resources for the use of these data can help to improve research policy.\n\nOur analysis focuses on 7,510 publications in cancer research published in the year 2011 and involving at least one author affiliated with a UK research host organisation \\citep[see][]{CRUK2014}. Cancer represents by far the largest single disease area studied in the UK, attracting an estimated 20\\% of funding invested by the government and major research charities \\citep{UKCRC2014}. For each publication in the sample, we extracted and coded funding data from the acknowledgment sections and used these data for three purposes: (i) to devise heuristics that can provide grounds for future systematic collection of funding data from acknowledgements; (ii) to provide one of the first assessments, in terms of \\textit{recall} and \\textit{precision}, of the coverage of funding data available from MEDLINE\/PubMed and WoS; (iii) to illustrate how bibliometric analysis of funding data taken from acknowledgment sections can be used to generate intelligence on funding systems to inform policy making in terms of most active funders, co-funding (i.e.\\ joint occurrence of funders in publication acknowledgements), and funders' portfolio profiles of funded research.\n\nThe paper proceeds as follows: The next section positions the contribution of the present paper within the growing bibliometric literature on the use of acknowledgment sections of scientific publications; we then describe the methodological approach used to collect and code the data and introduce a number of procedural points of guidance on how to recognise and isolate funding information in the acknowledgment sections, as well as on how to codify this information; in the Results section, we examine the extent to which authors acknowledge funders in their publications; we compare, in terms of \\textit{precision} and \\textit{recall}, our data with those extracted by queries of the same set of publications from MEDLINE\/PubMed and WoS; we then use the data to illustrate applications in funding portfolio and landscape analysis; we conclude the paper by discussing the main results of the approach we propose as well as the caveats and limitations associated with it.\n\n\n\n\n\n\\section{Literature review and emerging questions}\n\nThe breadth of information on scientific research activity that is included in the acknowledgment \\textit{paratext} of scientific publications has attracted the attention of bibliometric researchers since the 1970s. Pioneering work by \\cite{Mackintosh1972}, \\cite{Cronin1991}, and \\cite{McCain1991} focused on taxonomising the information contained in these sections. \\cite{Mackintosh1972}, for example, analysed the scientific articles published in the \\textit{American Sociological Review} from 1940 to 1962 and defined three types of acknowledgments, i.e.\\ facilities, access to data, and help of individuals. Cronin's (\\citeyear{Cronin1991}) taxonomy specifically included: (i) paymasters (i.e.\\ funding), (ii) moral support (e.g.\\ use of equipment, access to data), (iii) dogsbody (e.g.\\ secretarial support, editorial guidance, data collection and analysis), (iv) technical (e.g.\\ access to know-how, guidance on statistical procurers), (v) prime mover (e.g.\\ inspiration provided by principal investigator, project directors, mentor), and (vi) trusted assessor (e.g.\\ feedback, critical analysis and comments from peers).\n\n\nScholars have subsequently exploited publication acknowledgements for different bibliometric and evaluation purposes. These include the analysis of patterns of \\textit{subauthorship} or \\textit{peer interactive communication}, such as formal or informal assistance and acknowledgments to reviewers \\citep[e.g.][]{Cronin2006, Cronin1992, Cronin2004, Cronin2003, Tiew2002}, as well as the cataloguing of the funding authors received to produce their publications \\citep[e.g.][]{Harter1992, Lewison1995, Lewison1994}. The latter has received extensive attention because of the potential for linking research funding inputs with scientific outputs.\n\nA considerable effort in this direction was made by \\cite{Dawson1998} with the development of the Wellcome Trust's Research Output Database (ROD). The ROD included bibliographic information of about 214,000 UK biomedical publications between 1988 and 1995 and funding data extracted from the acknowledgement sections of these publications. This database enabled researchers to examine, for the first time and on a relatively large scale, the mix of funding sources in specific biomedical domains \\citep{Lewison1998} and the extent to which funded and unfunded publications differ in terms of type of research \\citep{Lewison1999} and impact as measured on the basis of citation-based indicators \\citep{Lewison1998a, Lewison2001, MacLean1998}. These studies provided seminal evidence, for example, that publications acknowledging funders are likely to receive more citations than those not reporting acknowledgment to funders. Yet, this relationship was also found to be dependent on the research domain and type of citation-based indicator used \\citep{Cronin1999, Lewison2003}.\n\nThese studies provided important insights on the extent to which authors acknowledged their funders. \\Tabref{review} reports a summary of these and subsequent studies. It is worth noting that substantial variation in funding acknowledgment frequencies exists --- varying with research focus, journal, funder, and country. For example, the ROD database provides an overview of funding in several biomedical fields, with around 64\\% of the publications acknowledging funders \\cite{Dawson1998}. However, this figure masks considerable variation with publications. For example, in gastroenterology, arthritis and malaria research funders were acknowledged in about 54\\%, 60\\%, and 80\\% of the cases, respectively \\cite[e.g][]{Lewison1999, Lewison1998, MacLean1998}. More recently, \\cite{Costas2012} found that more than 50\\% of the publications in natural sciences (e.g.\\ molecular biology and biochemistry, chemistry, physics and astronomy, and geosciences) reported funders in their acknowledgments, while publications in applied\/clinical sciences (engineering, clinical medicine), mathematics, social sciences applied to medicine, psychology, psychiatry, and behavioral sciences reported funders in only 20-50\\% of cases. Publications in economics, social sciences, and humanities and arts reported funding in even fewer cases, typically less than 10\\%. Similar patterns were found by \\cite{Diaz-Faes2014} on a sample of about 40,000 publications published in 2010 by Spanish researchers. On the one hand these figures highlight differences in the funding required to undertake research (e.g.\\ between humanities and natural sciences), but on the other hand differences may also reflect norms of reporting at different times and in different places. \n\nAcknowledgement patterns in journals might be expected to follow the rates of acknowledgements in the field they cover. For example, \\cite{Zhao2010} showed that articles published in library and information science journals (e.g.\\ \\textit{Journal of the American Society for Information Science}, \\textit{Information Processing and Management}, \\textit{Journal of Documentation}, and \\textit{College and Research Libraries}) include funding acknowledgements from about 6\\% to 35\\% of the cases (see \\Tabref{review}), while Rigby's (\\citeyear{Rigby2011, Rigby2013}) analysis of articles included in Cell revealed much higher funding acknowledgement rates (94\\%). Great variations were found across journals in social sciences and humanities, such as psychology, sociology, history, and philosophy \\cite{Cronin1993}: articles in \\textit{Psychological Review}, \\textit{American Sociological Review}, \\textit{American Historical Review}, and \\textit{Mind} acknowledged funders in 70.3\\%, 53.5\\%, 29.7\\%, and 3.0\\% of the cases, respectively.\n\n\n\\begin{spacing}{1}\\scriptsize\n\\begin{longtable}{p{3.5cm}p{6.5cm}cc}\n\\caption{\\label{tab:review}Reporting of funding data in publication acknowledgements in terms of journals, research domains, funders, and countries.} \\\\\n\\hline\\hline\n\\textbf{Studies by main}&\t\\textbf{Data source}&\t\t\t\\textbf{Sample} &\t \\textbf{\\% of records reporting} \\\\\n\\textbf{analytical focus}&\t\\textbf{(period of observation)}&\t\\textbf{ } & \t\t\\textbf{funding data} \\\\\n\\hline\n\\endfirsthead\n\\caption{Reporting of funding data in publication acknowledgements in terms of journals, research domains, funders, and countries\\textit{(continued)}.} \\\\\n\\hline\\hline\n\\textbf{Studies by main}&\t\\textbf{Data source}&\t\t\t\\textbf{Sample} &\t \\textbf{\\% of records reporting} \\\\\n\\textbf{analytical focus}&\t\\textbf{(period of observation)}&\t\\textbf{ } & \t\t\\textbf{funding data} \\\\\n\\hline\n\\endhead\n\\\\\n\\textbf{Journal(s)} & & & \\\\\n\\\\\n\\cite{Cronin1991} & \\textit{J. of the Am. Soc. for Inf. Science} (1970-1990) \t\t\t& 938 & 29.1\\% \\\\\n\\\\\n\\cite{Harter1992} & \\textit{J. of the Am. Soc. for Inf. Science} (1972-1974, 1982-1984, 1988-1990)\t& 391 & 32.2\\% \\\\\n\\\\\n\\cite{Cronin1993} \t& Sample of journals (1971-1990): & & \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{Psychological Review} \t\t\t& 629 & 70.3\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{Am. Sociological Review} & 1,186 & 53.5\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{Am. Historical Review} & 464 & 29.7\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{Mind} & 1,027 & 3.0\\% \\\\\n\\\\\n\\cite{Cronin1999} \t\t& Four information science journals (1989-1993) & 716 & 26.5\\% \\\\\n\\\\\n\\cite{Tiew2002} & \\textit{J. of Natural Rubber Research} (1986-1997) & 310 & 20.3\\% \\\\\n\\\\\n\\cite{Cronin2003,Cronin2004} \t\t& Sample of journals (1900-1990): & & \\\\\n \t\t\t\t\t\t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{Psychological Review} & 2,707 & 31.9\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{Mind} & 1,850 & 3.4\\% \\\\\n\\\\\n\\cite{Cronin2006} & Cells (1975, 1985, 1995, 2004) & 1,106 & 86.8\\%-96.6\\% \\\\\n\\\\\n\\cite{Zhao2010} & Sample of journals (1998): & & \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{J. of the Am. Soc. for Inf. Science} & 99 & 35.4\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{Inf. Processing and Management} & 46 & 34.8\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{J. of Documentation} & 24 & 33.3\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{Library and Inf. Science Research} & 17 & 29.4\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{Library Trends} & 33 & 12.1\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{Library Quarterly} & 12 & 16.7\\% \\\\\n \t& \\addtolength{\\leftskip}{1em}\t\\textit{College and Research Libraries} & 35 & 5.7\\% \\\\\n\\\\\n\\cite{Rigby2011} & Sample of journals (-): & & \\\\\n & \\addtolength{\\leftskip}{1em}\t\\textit{Cell} & 301 & 94.0\\% \\\\\n & \\addtolength{\\leftskip}{1em}\t\\textit{Physical Review Letters} & 3,414 & 83.0\\% \\\\\n\\\\\n\\cite{Rigby2013} & \\textit{J. of Biological Chemistry} (2009) & 3,596 & 89.9\\% \\\\\n\\\\\n\\\\\n\\textbf{Research domain(s)} & & & \\\\\n\\\\\n\\cite{Lewison1995} \t\t& Sample of UK biomedical articles (1988-1992) \t\t& $\\sim$122,000 & 61.0\\% \\\\\n\\\\\n\\cite{Dawson1998} \t& Wellcome Trust's Research Outputs Database (ROD) (1988-1995) \t\t& 214,364 & 63.6\\% \\\\\n\\\\\n\\cite{Lewison1998a, Lewison1998, Lewison2001} \t& Sample of gastroenterology articles (1988-1994) \t\t& 12,925 & 54.4\\% \\\\\n\\\\\n\\cite{MacLean1998} \t& Sample of malaria articles (1989) \t\t\t& 758 & 80.5\\% \\\\\n\\\\\n\\cite{Lewison1999} \t\t& UK articles in arthritis research (1988-1995) \t\t\t& 6,672 & 60.3\\% \\\\\n\\\\\n\\cite{Lewison2003} \t\t& Articles in language therapy research (1991-2000) \t\t& 1,048 & 51.0\\% \\\\\n\\\\\n\\cite{Giles2004} \t\t\t& Articles in CiteSeer (1990-2004) \t\t\t\t& 335,000 & 56.1\\% \\\\\n\\\\\n\\cite{Shapira2010a, Wang2011} \t\t& Sample of nanotechnology articles (2008-2009) \t\t& 91,164 & 66.9\\% \\\\\n\\\\\n\\cite{Costas2012} \t\t& WoS articles (2009) \t\t\t\t\t& 1,253,909 & 43.2\\%\\\\\n\\\\\n\\hline\\hline\n\\\\\n\\\\\n\\\\\n\\textbf{Funder(s)} \t& & & \\\\\n\\\\\n\\cite{Lewison1994} \t& Sample of articles supported by the EU BAP as reported by surveyed authors (1986-1992) \t& 584 & 72.9\\% \\\\\n\\\\\n\\cite{Butler2001} \t& Articles funded by the Australian NHMRC (1994-1995) \t\t& 2,962 & 75.2\\% \\\\\n\\\\\n\\cite{Koier2014} \t\t\t& Articles funded by the Dutch climate programmes (2009-2012) \t\t\t& 221 & 52.9\\% \\\\\n\\\\\n\\\\\n\\textbf{Countries} \t& \t& & \\\\\n\\\\\n\\cite{Salager-Meyer2009} \t& Sample of medical articles (2005-2007): \t& & \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{Spain} \t& 50 & 4.0\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{United States} \t& 50 & 26.0\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{Venezuela} \t& 50 & 72.0\\% \\\\\n\\\\\n\\cite{Wang2012a} \t\t& Articles from 10 countries (2009): & & \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{United States} \t& 379,321 \t& 44.1\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{China} \t& 126,931 \t& 70.3\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{Germany} \t& 102,927 \t& 40.8\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{United Kingdom} & 99,832 \t& 42.6\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{Japan} \t& 87,582 \t\t& 43.0\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{France} \t& 73,186 \t\t& 38.1\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{Italy} \t\t& 62,609 \t\t& 33.1\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{Canada} \t& 60,723 \t\t& 49.1\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{Spain} \t& 48,245 \t\t& 51.6\\% \\\\\n \t\t\t& \\addtolength{\\leftskip}{1em}\t\\textit{Australia} \t& 18,590 \t\t& 44.9\\% \\\\\n\\\\\n\\cite{Diaz-Faes2014} \t\t& Articles produced by Spanish researchers (2010) & 38,257 \t\t& 72.6\\% \\\\\n\\\\\n\\cite{Gok2015} \t\t& Articles from 6 countries (2008-2011) \t& 242,406 & \t$\\sim$44\\% \\\\\n\\\\\n\\hline\\hline\n\\multicolumn{4}{l}{\\footnotesize \\textit{Source: Authors' elaboration.}}\\\\\n\\\\\n\\end{longtable}\n\\end{spacing}\n\n\n\n\n\n\n\n\nCountries with very different funding systems also can be expected to have markedly different acknowledgment rates. \\cite{Butler2001} found that researchers funded by the Australian National Health and Medical Research Council (NHMRC) acknowledge their funder in 75\\% of papers. \\cite{Costas2012} found that articles authored by Chinese researchers acknowledge funders in about 65\\% of cases. Publications authored by researchers in South Korea, Taiwan, Sweden, Finland, Denmark, Spain, and Canada acknowledged funders in more than 50\\% of the publications, while funding was acknowledged in less than 40\\% of the publications of authors based in Turkey, Greece, Iran, Poland, India, Italy, and Israel. Similarly, \\cite{Wang2012a} has shown in a study of 10 countries that 70\\% of publications from China, but only 33\\% of those from Italy, carry funding acknowledgements. \n\nBeyond variation in the occurrence of funding acknowledgments, the analysis of the relationship between funding and impact has been a key application of funding data in publication acknowledgments. Evidence of the positive correlation between the presence of funding acknowledgments in a publication and the citations the publication received were found by \\cite{Zhao2010} on a sample of articles published in library and information science journals. \\cite{Boyack2011} examined over 2.5 million publication records from 1980 to 2009 in MEDLINE\/PubMed and found that publications acknowledging funding from the US Public Health Service (including the NIH) were cited twice as much as publications authored by US researchers acknowledging no funding source. However, as discussed above, the relationship between funding and impact varies across journals, research domains, and countries. For example, \\cite{Rigby2011, Rigby2013} found that there is a weak positive relationship between the number of acknowledged funding sources and the citation impact of the publications included in \\textit{Physical Review Letters} and the \\textit{Journal of Biological Chemistry} and that this relationship becomes statistically non-significant in the case of publications included in \\textit{Cell}. Furthermore, a recent study by \\cite{Gok2015} provides evidence that the variety of the acknowledged funding sources (i.e.\\ the number of distinct funding sources) is positively related to the total number of citations a publication receives and to the top percentile citations. Yet, the funding intensity (i.e.\\ the number of unique funders acknowledged in a publication over the number of authors listed in the publication) was found to negatively affect citation impact. \n\nWith important findings beginning to accumulate from the study of funding acknowledgements, the question arises about the `quality' of the underlying data. Early work on the manual coding of acknowledgments data pointed towards the difficulty of using these data, attributed to the lack of structure in unstandardized free-form formats. There are challenges in terms of finding acknowledgement data in the full-text as well as in extracting funding data from text in a consistent manner. Acknowledgements are not solely found in specific sections called `acknowledgements'. Some journals have separate sections on funding, or include acknowledgements at the beginning or the end of the documents or even in small print without a section indicator. Building samples of data has typically relied on intensive manual work both to retrieve publications, for which the electronic access was not available, and to identify and classify the text of the acknowledgment sections of these publications (e.g.\\ identification and characterisation of funders by type and nationality). Only a few attempts were made to develop algorithms capable of extracting publication acknowledgments and the funding data included in these \\citep[e.g.][]{Giles2004}.\\footnote{\\cite{Giles2004} developed an algorithm to extract acknowledgements (and funding data included in these) of about 335,000 research documents within the CiteSeer computer science archive. This algorithm searched for lines of text that included acknowledgement information achieving a level \\textit{recall} of about 90\\% --- i.e.\\ about 10\\% of acknowledgment data were not retrieved.} \n\nFor this reason, research on the use of acknowledgment funding data for large-scale bibliometric analyses was limited until in more recent years a new wave of studies has emerged with much higher sample sizes \\citep[e.g.][]{Costas2012, Diaz-Faes2014, Gok2015, Shapira2010a}. These studies have mostly been facilitated by the increasing availability of ready-classified information. WoS, for example, has recently provided access to the funding text of acknowledgments and to the list of funders and grant codes mentioned in these --- this information is reported in the ``FX'' and ``FU'' fields, respectively --- for publications from August 2008. \n\nNonetheless, we have limited knowledge on the \\textit{recall} and \\textit{precision} of acknowledgement funding data gathered through automated routines and provided by publication databases such as MEDLINE\/PubMed and WoS. For example, on a sample of 117 publications supported by ``Dutch climate programmes'', \\cite{Koier2014} found that WoS did not correctly report the list of funders in about 51\\% of the cases. This therefore raises questions of the extent to which these limitations of WoS funding data also occurs in other domains and on how much additional benefit remains to be gained from manual coding of funding data in acknowledgments as compared to relying on data from major databases. A second problem that emerges when working with acknowledgement data concerns the standardisation and disambiguation of funders' names. Algorithms can support this process, yet ambiguity remains with regard to the selection of the level of analysis and on the treatment of funders that have changed their names over time \\citep{Rigby2011}.\n\nThis paper aims to address some of these concerns and to contribute to the extant literature in three ways. Firstly, we develop a series of `heuristics' to guide the collection and coding of acknowledgment funding data, and provide some cautionary advice on their use. This feeds into ongoing debates over a greater standardisation of acknowledgment sections, which can facilitate guidelines for authors of publications, and for analysts extracting and coding of information there included. Secondly, we examine the \\textit{recall} and \\textit{precision} of MEDLINE\/PubMed and WoS in retrieving publications that reported acknowledgements to funders as well as the accuracy of the list of funders provided by these databases as compared to a sample of manually collected funding data on UK cancer research. Finally, we provide further elaboration on the value of carefully gathered funding data for profiling funders' portfolios and national funding landscapes.\n\n\n\n\n\\section{Data and methods}\nOur empirical analysis is focused on the UK cancer research domain, which is particularly suitable for the purpose of this paper for two main reasons. First, cancer research has been characterised by intense funding activity over the last few decades. This has involved a large variety of funders --- from industry to government and philanthropic organisations \\citep{Eckhouse2008}. Second, cancer research falls within the broader domain of biomedical research where the rates of reporting funding data in the acknowledgement sections of publications are relatively high \\citep{Dawson1998}.\n\nThe search set out to capture all publications in the domain of cancer research, involving at least one author from a UK research host organisation, and published in the year 2011. Delineating a broad topic such as cancer is, however, a complex task from a bibliometric perspective due to the breadth of the `cancer' field. We therefore preferred to rely on the Medical Subject Heading (MeSH) classification of MEDLINE\/PubMed, an extensively developed and evolving vocabulary of medical terms \\citep{Leydesdorff2012, Petersen2016}. The MeSH classification is composed of several thousands of descriptor terms, representing topics in medical research --- 26,404 descriptors composed the 2011 MeSH classification. These descriptors are organised in a tree-like structure and assigned, through a standardised indexing process performed by examiners at the US National Library of Medicine (NLM), to MEDLINE\/PubMed publications in order to classify the content of these at different levels of specificity.\\footnote{A descriptor may belong to more than one branch of the MeSH tree and may be complemented by qualifiers that further specify a publication's content in relation to the assigned descriptor. For more details on the classification see \\url{www.nlm.nih.gov\/pubs\/factsheets\/mesh.html}.}\n\nWe defined cancer research using the MESH descriptor ``Neoplasms'' (which captures any of its children descriptors, such as different types of cancer). The ``Neoplasms'' descriptor is formally defined as: ``\\textit{New abnormal growth of tissue. Malignant neoplasms show a greater degree of anaplasia and have the properties of invasion and metastasis, compared to benign neoplasms}''. We queried MEDLINE\/PubMed for 2011 data in early 2013, by using the following search string: ``Neoplasms''[MeSH Terms].\\footnote{Retrieving publications related to cancer by using the MeSH descriptor ``Neoplasms'' inevitably misses some of those publications that might be deemed by researchers or funders to be associated with `neoplasms'. For example, a funder or researcher may consider studies of angiogenesis in healthy tissue to be relevant for understanding how tumours develop a blood supply, but if these papers are not considered to actually study cancerous tissue they may not be understood as within the study of neoplasms per se, and they will not be coded using the ``Neoplasms'' descriptor (or its children descriptors). The publications collected by the search employed in this study are therefore referred to as a `sample', reflecting the fact that the search is exhaustive within the neoplasms field (including cancerous and pre-cancerous growths, but also non-cancerous growths, which we accept as a minor limitation), but not comprehensive of neoplasms by other definitions.} This returned 115,101 documents published globally in that year.\\footnote{We used the electronic date, i.e.\\ the earliest date when a document is made publicly available.}\n\nMEDLINE\/PubMed data were matched with SCOPUS data to retrieve bibliographic data on all authors' affiliation addresses as MEDLINE\/PubMed provides affiliation data only for the first listed author in the given publication. Records between the two databases were matched on the basis of the ``PubMed unique identifier'' (PMID) field of MEDLINE\/PubMed that is also included in SCOPUS records. SCOPUS-MEDLINE\/PubMed match was, however, obtained for 98.1\\% of the records. The full match was not achieved because of time lag in publication indexing between MEDLINE\/PubMed and SCOPUS (even though data collection began in early 2013, a full year after the close of the target year).\n\n\\begin{figure}\n\\includegraphics[height=7cm]{journals}\n\\centering\n\\caption{Journal-publication distribution (a) and top-10 most frequent journals (b). \\newline\\textit{Source: Authors' elaboration.}}\n\\label{fig:journals}\n\\end{figure}\n\nThis match enabled us to identify UK publications as those involving at least one author based in a UK research host organisation --- unmatched records were manually screened. We sought to include as many records as possible --- including all publication types (from editorials to reviews and all forms of research papers).\n\nThis process returned 7,922 publications, indicating that UK authors are represented in 6.9\\% of the global production of publications on cancer for 2011. Publications were distributed across 1,449 journals. As depicted in \\Figref{journals}a, the journal-publication distribution is skewed --- the \\textit{British Journal of Cancer} is the journal in which UK publications on cancer were most frequently published in 2011 (see \\Figref{journals}b). We were not able to electronically access 130 journals (712 records) due to access restrictions. E-mails to authors provided access for about about 42\\% of publications included in these journals. As a result, we obtained access to the full-text of 7,510 publications (94.8\\% of the initial sample of publications). For each of these, we collected and coded acknowledgements to funders that authors made, as detailed below.\n\n\n\n\\section{Extracting funding data}\nThe automatic extraction of funding information from acknowledgment sections in publications is challenging. As discussed above, authors use acknowledgments for a variety of purposes. Our focus is on financial support (e.g.\\ external grants) that authors have received in order to undertake research that is then reported in a given publication. Funding data must therefore be separated from the text reporting other non-financial forms of support. For this reason, for each publication in our sample, we read the text of acknowledgments (including other sections that may report funding data), interpreted it, and extracted funding data manually. These data constitute the `reference data' on which we will assess the \\textit{recall} and \\textit{precision} of funding data provided by MEDLINE\/PubMed and WoS. This section sets out steps taken to assure consistency in the data collection process, including the heuristics that were used to guide the coding of sample data (together with illustrative examples). These heuristics may be of assistance for future studies adopting similar approaches or aimed at developing routines capable of extracting funding information from acknowledgements in an automatic manner.\n\n\n\\setlength{\\tabcolsep}{10pt}\n\\renewcommand{\\arraystretch}{1.0}\n\\begin{table}\\footnotesize\n\t\\caption{\\label{tab:cases}Cases of acknowledgment text and reporting of funding data (extracted funding information is represented with bold font).}\n\t\\centering\n{\\begin{tabular}{cp{14cm}}\n\\hline\\hline\n\\textbf{Case} & \\textbf{Acknowledgement text}\\\\\n\\hline\n1 & \t\\textit{Acknowledgements} \\newline \n\t\\textit{The authors thank our colleagues who have suggested or contributed data for current or previous versions of the database. They also would like to thank the swift and helpful advice from the ICGC DCC and BioMart team.}\\newline\n\t\n\t\\textit{Funding} \\newline \n\t\\textit{\\textbf{Cancer Research UK} (programme grant C355\/A6253) and \\textbf{FW6 EU project} MolDiag-Paca. R.C. is funded by \\textbf{Breast Cancer Campaign}. Funding for open access charge: Cancer Research UK.}\\newline\n\t\\citep{Cutts2011}\\\\\t\n\\\\\t\n2 &\t\\textit{Acknowledgments}\\newline\n\t\\textit{We are grateful to Ms. Haruka Sawada and Ms. Noriko Ikawa for technical assistance. Our bio-repository is supported by funding from the \\textbf{National Institute for Health Research} (UK) and the \\textbf{Cambridge Biomedical Research Centre}. This work was supported by a Grant-in-Aid for Young Scientists (A) (22681030) from the \\textbf{Japan Society for the Promotion of Science}.} \\newline\n\n\t\\textit{Disclosure Statement} \\newline \n\t\\textit{The research was funded by \\textbf{OncoTherapy Science, Inc.} YI, YY, and KM are employees of OncoTherapy Science, Inc. YD, YN, and RH are scientific advisors of OncoTherapy Science, Inc.}\\newline\n\t\\citep{Takawa2011}\\\\\n\\\\\n3 &\t\\textit{Funding} \\newline \n\t\\textit{\\textbf{Grant Agency of Czech Republic} (303\/09\/0472 and 305\/09\/H008); \\textbf{Ministry of Education of Czech Republic} (MSM0021620808 and 1M0505). Work at the Institute of Cancer Research is supported by \\textbf{Cancer Research UK}.} Portions of this work were funded by \\textbf{National Institutes of Health} (R01 ES014403 and P30 ES006096 to D.W.N. and Z.S.).\\newline \n\n\t\\textit{Acknowledgments}\\newline \n\t\\textit{The authors would like to thank the Grant Agency of the Czech Republic and the Ministry of Education of the Czech Republic.}\\newline \n\t\\citep{Stiborova2012}\\\\\n\\\\\n4 &\t\\textit{Acknowledgments}\\newline\n\t\\textit{The authors wish to thank colleagues in the Histology Laboratory and the Electron Microscopy Unit, Veterinary Laboratory Services, School of Veterinary Science, University of Liverpool, for excellent technical support. T. Soare was supported by a scholarship from the Agency for Credits and Studies, \\textbf{Romanian Ministry of Education, Research and Innovation}.}\\newline\n\t\\citep{Soare2012}\\\\\n\\\\\n5 &\t\\textit{Acknowledgments}\\newline\n\t\\textit{MG was supported by the grant Gu 1170\/1-1 of the \\textbf{German Research Foundation}. The support for using the research version of the Pinnacle treatment planning software from Philips Radiation Oncology Systems, Fitchburg, WI, USA is acknowledged. This work was partially supported by research Grant C46\/A2131 from \\textbf{Cancer Research UK}. We acknowledge \\textbf{NIHR} funding to the NHS Biomedical Research Centre. The fruitful discussions and thoughtful proofreading of the manuscript by Kevin Brown and Joel Goldwein (Elekta) is acknowledged.}\\newline\n\t\\citep{Guckenberger2011}\\\\\n\\\\\n6 & \t\\textit{Acknowledgments}\\newline\n\t\\textit{The study was supported by the grant MA1659\/6-1\/2 of the \\textbf{Deutsche Forschungsgemeinschaft (DFG)}. The recombinant topoisomerase II was a generous gift of Fritz Boege, Institute of Clinical Chemistry and Laboratory Diagnostics, Heinrich Heine University (Duesseldorf, Germany). We thank Dr. Antonella Riva and Dr. Paolo Morazzoni (Indena SpA, Milan, Italy) for provision of test material. The authors have declared no conflict of interest.}\\newline\n\t\\citep{Esselen2011}\\\\\n\n\\hline\\hline\n\\multicolumn{2}{p{14cm}}{\\footnotesize \\textit{Notes: Publications for cases 3 and 4 were electronically published in 2011, but included in a journal volume and issue in 2012.\\newline\nSource: Authors' elaboration.}}\n\\end{tabular}\n}\n\\end{table}\n\n\n\nThe first step of data collection was the identification of the portions of publication full-text where funding data are reported. These data, as expected, are often included in the acknowledgment sections, which are usually positioned before or after references, or, in a few cases, at the beginning or in the footnotes of the publication. However, there are cases where funding information is reported in dedicated sections called, for example, ``Financial Support'', ``Financial Information'', or ``Funding''. These are often located next to the acknowledgment sections. Case 1 in \\Tabref{cases}, for example, presented two separate sub-sections: ``Acknowledgements'' and ``Funding''. There are also cases of publications where funding information is reported in multiple sections. For example, in Case 2, the analysis of the acknowledgments section only would have inevitably missed funding to this publication coming from the industry, as accounted for in the disclosure of conflict section. \n\nCase 2 also provides indication of the importance of interpreting the wording used by authors to declare financial support. For example, it is crucial to distinguish between past and current financial support made available to the authors (e.g.\\ historic honoraria as opposed to direct funding for research leading to a given publication). In addition, Case 2 reveals that the authors acknowledged their employer as funding the study. However, in many cases authors do not do so, and where they do not, it is important to note that we do not infer that their employer was the funder of the research. We have only extracted funding information that is explicitly declared by authors.\\footnote{Previous studies on the use of funding data reported in publication acknowledgments have adopted two different coding approaches. 8 of the 30 studies listed in \\Tabref{review} (notably those by Lewison and colleagues) have complemented the information on funding reported by authors in acknowledgments with the information on authors' affiliations (e.g.\\ when an author of a given publication was found to be affiliated to a firm, the firm was included in the list of funders of the publication). The remaining studies in \\Tabref{review} (22 out of 30) have instead exclusively focused on the funding information included in the acknowledgments. Our study adopts the latter approach. This, in turn, enables us to make a broader comparison with the results of previous studies as well as to conduct a more precise comparative analysis on \\textit{recall}and \\textit{precision} of WoS data, which are based on the text of publication acknowledgments only.}\n\nOnce the relevant funding information for a given publication is identified, the information included in this section requires interpretation before being coded, i.e.\\ before the list of funders can be defined. In certain cases, identifying the list of organisations that financially supported a publication is relatively straightforward. There are, however, instances where this distinction is ambiguous because of the general usage of the term `support' in the English language. As discussed, authors often use this term to give credits to a variety of types of support such as financial, material, or moral \\citep[e.g.][]{Cronin1991}. For this reasons, we included an organisation in the list of funders if we could infer, with a certain degree of confidence, from the acknowledgment text that the `support' from a given acknowledged organisation was financial. For example, in the Case 3, we interpreted that \\textit{``supported by Cancer Research UK''} referred to financial support because this funder was acknowledged in a dedicated funding section. In those cases where there was no clear way to detect the financial nature of the support acknowledged, we adopted a conservative approach and assumed that the support was not financial --- the acknowledged organisation was therefore not included in the list of funders.\\footnote{`Travel support', `in-kind donations' and some sorts of support (technical, lab access, access to journals and the like) are often acknowledged by authors. To choose what constitute acknowledgment for funding and what does not, we follow our principle to codify as funding support only when we conclude there is a direct link between the publication and monetary support. So of the three items mentioned, only travel support has to be codified, given some money were given to pay for travel during the course of the research.}\n\nAfter identifying the relevant funding information, the last step was to codify this information in a consistent manner. To do so, we first extracted the full name of an organisation as stated in the acknowledgment text and added the ISO 2-digit code of the country where the organisation is located as we could infer from the acknowledgment text or authors' addresses. Funders that were acknowledged more than once in the same publication were counted only once. Harmonisation of names was undertaken using the highest organisation level of a funder in most cases. For example, in Case 4, the highest organisational level is the ministerial level, i.e.\\ \\textit{Ministry of Education, Research and Innovation (RO)}.\n\nThe language in which the name of the funder is reported can also be an issue. A funder based in a non-English speaking country can have its name reported in the acknowledgement either in the native language (so expressed in a language other than English) or in its translated English form. The acknowledgments of Case 5 and 6 in \\Tabref{cases} are a clear example of this, given that the \\textit{German Research Foundation} and the \\textit{Deutsche Forschungsgemeinschaft} are the same organisation.\n\nFinally, also the task of adding next to each funder name the ISO 2-digit code of the country where it is located was not unambiguous. A funder's country of origin is not always clearly identifiable by its name or via a web search. In ambiguous cases, we assumed that the organisation acknowledged was from the same country as the author who acknowledged it if no further information was available. In case of multinational companies, we used the location of their headquarters as the home country.\n\nUsing the above heuristics helps to ensure consistency in the data collection and codification, but some degree of ambiguity persists, especially in cases where the wording of the acknowledgment was not clear. Consistent coding was ensured through use of a protocol containing illustrative examples (as above) and overseeing the coding process through periodic cross-checking of entries among the coding team. \n\n\n\\section{Results}\nThe sections below report our findings in relation to the extent to which authors acknowledge their funders in the field of cancer research, and on the \\textit{precision} and \\textit{recall} of MEDLINE\/PubMed and WoS in retrieving UK cancer publications that report funding information in their acknowledgments. We also examine the accuracy of the list of funders (acknowledged in publications) as provided by these databases. We then use the collected data to explore the UK cancer research funding system and to provide examples of the strategic intelligence that such data can provide to analysts and policy makers.\n\n\n\n\n\\subsection{Do UK authors of cancer research acknowledge their funders?}\nThe analysis revealed that 52.1\\% (3,914 out 7,510 publications) of the sample disclosed at least one funder.\\footnote{The percentage of publications reporting funding data increases to 57.0\\% (3,741 out 6,560 publications) when considering only articles and reviews as classified by SCOPUS.} We further examined our sample of publications below to explore whether authors might not be reporting funding information. We first distinguished publications in our sample in three sub-samples: (i) publications with funding data in acknowledgments (52.1\\%), (ii) publications with no funding data in acknowledgments (30.4\\%), and (iii) publications with no acknowledgement sections (17.4\\%).\n\nPublications in each of these sub-samples were then classified by type on the basis of the classification of publication records provided by SCOPUS, i.e.\\ articles, reviews, conference papers, editorials, errata, notes, letters, and short surveys. The results of this analysis, which are depicted in \\Figref{coverage}, provide evidence that the sub-samples of publications with no acknowledgment sections and with acknowledgements but no named funders are composed to a much greater extent of `less cost-intensive' publications (conference papers, editorials, errata, notes, letters, and short surveys) than within the sub-sample of publications with acknowledgements, i.e. 45.5\\% and 36.7\\% against 16.8\\%. These findings are also in line with previous research \\citep[e.g.][]{Salager-Meyer2009}. Also, about 22\\% of the publications with no acknowledgment to funders are represented by `less cost-intensive' research outputs, while this proportion significantly reduces --- it ranges from 0\\% to 5\\% --- when considering publications with acknowledgment to at least one funder (see \\Figref{publication_types}). \n\n\\begin{figure}[h]\n\\includegraphics[width=\\textwidth]{coverage}\n\\centering\n\\caption{Reporting of funding data by type of publication.\n\\newline\\textit{Source: Authors' elaboration of the basis of MEDLINE\/PubMed and SCOPUS data.}}\n\\label{fig:coverage}\n\\end{figure}\n\nWe then explored two additional explanations for the lack of named funders in publication acknowledgments: (i) the publication required research funding, but relevant information was omitted from the publication full-text either by the authors or the publisher; or (ii) the publication was supported by the authors' employers, which are indirectly acknowledged through the authors' affiliations. To determine how these works were funded, we examined the two sub-samples of publications with no funding data as follows. First, we randomly selected more than 10\\% (208) of the publications within the sub-sample of publications with no acknowledgment sections. A closer examination of this sub-sample revealed that about 38\\% (79\/208) of publications could be classified as case reports. These are likely to require no external funding other than the support of authors' employers. We therefore did not further examine this category of outputs. For the remaining publications (129\/208) an e-mail query was sent to the corresponding authors. We obtained a response rate of about 38\\% (49 replies). In about 18\\% of responses, a funding contribution to the publication other than the authors' employers was revealed. In 61\\% of responses, authors instead revealed a contribution by their employer, while in the remaining 21\\% of the cases, authors indicated that the publications did not require any funding. In summary, we can estimate that 11.2\\% (0.18*0.62) of the publications with no acknowledgment sections were actually supported by an external funder, i.e.\\ less than 2\\% (0.11*1310\/7510) of the entire sample of publications.\n\n\n\\begin{figure}\n\\includegraphics[height=7cm]{publication_types}\n\\centering\n\\caption{Number of acknowledged funders by type of publication (``Other outputs'' include: books, conference papers, editorials, errata, letters, notes, short surveys).\n\\newline\\textit{Source: Authors' elaboration of the basis of MEDLINE\/PubMed and SCOPUS data.}}\n\\label{fig:publication_types}\n\\end{figure}\n\nWe then performed a similar analysis for the sub-sample of publications with acknowledgements but no stated funders. 69\\% of these publications explicitly stated that they did not benefit from financial support. Of the remaining 31\\% of publications, we randomly selected a sub-sample of more than 10\\% (89) publications for further investigation. On reading the papers we found that about 16\\% (14\/89) of the publications could be deemed to be case reports and therefore it was assumed no funding (beyond the authors' employers) was necessary to produce the publication. Financial support for the remaining publications (75) was queried via e-mail to the corresponding author, and a response rate of 37\\% (28 responses) was obtained. About 14\\% of responders revealed contribution by funders other than authors' employers; 50\\% of the responses suggested that the research was supported by their employers; authors indicated the remaining publications did not require any funding. We can therefore estimate that 3.6\\% (0.31*0.84*0.14) of the publications with acknowledgements but no stated funders were externally funded, i.e. about 1\\% (0.04*2286\/7510) of the entire sample of publications.\n\nOn the basis of the findings above, we concluded that when a UK cancer publication reports no funding data, it is likely that the publication was `less cost-intensive' (e.g.\\ editorials, notes, letters, case reports) and therefore did not require any additional funding or that the publication was supported by authors' employers. From a review of the text of acknowledgements and survey of a sample of the authors, it seems that the omission of all funding information is unlikely --- we estimate overall less than 3\\% of the entire sample. However, it is clear that sometimes authors do not think it is relevant to acknowledge their employers for funding the publication --- the disclosure of their address somewhat fulfils this purpose.\n\n\n\n\n\\subsection{\\textit{Precision} and \\textit{recall} of MEDLINE\/PubMed and WoS}\nTo enable the comparison of the funding data extracted through the process described above with those available from MEDLINE\/PubMed and WoS, we first harmonised the names of the funders listed in the acknowledgements of the publications included in our sample.\\footnote{The Vantage Point software package aided the harmonisation of organisation names} This led to the identification of 2,549 distinct funding organisations of which 663 organisations in the public or charitable sector were based in the UK. A further 1,579 public or charitable sector organisations were identified outside the UK. Industry (e.g. pharmaceutical companies) were classified separately, with 307 firms being identified. About 64\\% of the publications reporting funding information recognised support from two or more funders, with authors of cancer research publications acknowledging an average of 3.3 funders per publication. We compared these data, namely the `reference data', with those available in MEDLINE\/PubMed and WoS across few descriptive indicators and in terms of \\textit{precision} and \\textit{recall}.\\footnote{For the sake of clarity, we defined \\textit{recall} as the ratio between \\textit{true positives} (i.e.\\ the number of publications for which both MEDLINE\/PubMed or WoS and the `reference data' reported at least one funder) and the sum of \\textit{true positives} and \\textit{false negatives} (i.e.\\ the number of publications for which these databases did not report funding data, but we found funding information in publications' full-text). \\textit{Precision} was instead defined as the ratio between \\textit{true positives} and the sum of \\textit{true positives} and \\textit{false positives} (i.e.\\ the number of publications for which MEDLINE\/PubMed or WoS reported funding data, but our analysis revealed no funding information in publications' full-text).} This comparison is reported in \\Tabref{comparison}.\n\n\n\\setlength{\\tabcolsep}{7pt}\n\\renewcommand{\\arraystretch}{1.0}\n\\begin{table}\\footnotesize\n\t\\caption{\\label{tab:comparison}\\textit{Recall} and \\textit{precision} of MEDLINE\/PubMed and WoS funding data.}\n\t\\centering\n{\\begin{tabular}{lcccc}\n\\hline\\hline\n\t& \\multicolumn{4}{c}{\\textbf{Database}}\\\\\n\t\\cline{2-5}\n\t&\t\\textbf{Reference data}&\t\\textbf{MEDLINE\/PubMed}& \\multicolumn{2}{c}{\\textbf{ISI Wed of Science (WoS)}}\\\\\n\\hline\nNumber of publications &\t\t\t\t7,510&\t\t7,510&\t\t\\multicolumn{2}{c}{7,082}\\\\\n\\\\\nPublications reporting&\t\t3,914&\t\t1,712&\t\t\\multicolumn{2}{c}{3,736}\\\\\nfunding data\t\t\t\t\t\t&\t\t(52.1\\%)&\t\t(22.8\\%)&\t\t\\multicolumn{2}{c}{(52.7\\%)}\\\\\n\\\\\n\\textit{Recall}&\t\t\t\t\t\t-&\t\t\t41.9\\%&\t\t\\multicolumn{2}{c}{92.8\\%}\\\\\n\\\\\n\\textit{Precision}&\t\t\t\t\t-&\t\t\t95.7\\%&\t\t\\multicolumn{2}{c}{94.3\\%}\\\\\n\\\\\n\t\t\t&\t\t\t\t\t&\t\t\t\t&\t\t\\textit{Before}& \\textit{After}\\\\\n\t\t\t&\t\t\t\t\t&\t\t\t\t&\t\t\\textit{harmonization}& \\textit{harmonization}\\\\\nNumber of distinct& \t\t\t2,549& \t\t17& \t\t\t6,714& \t\t\t\t\t\t3,541\\\\\nfunders\\\\\n\\\\\nNumber of funders\\\\\nper publication \\\\\n\\addtolength{\\leftskip}{1em}\t\\textit{Mean}& \t\t\t\t\t\t3.3& \t\t\t1.4 & \t\t4.1& \t\t\t\t\t\t\t3.7\\\\\n\\addtolength{\\leftskip}{1em}\t\\textit{Std. Dev.} & \t\t\t\t\t(5.0)& \t\t(0.8)& \t\t(6.1)& \t\t\t\t\t\t(5.3)\\\\\n\\addtolength{\\leftskip}{1em}\t\\textit{Max} & \t\t\t\t\t\t78 & \t\t6 & \t\t92 & \t\t\t\t\t\t76 \\\\\n\\\\\n\\textit{Accuracy} of the list & \t\t- & \t\t24.8\\%& \t\t- & \t\t\t\t\t\t68.0\\%\\\\\n of funders\\\\\n\\hline\\hline\n\\multicolumn{5}{l}{\\footnotesize \\textit{Source: Authors' elaboration.}}\n\\end{tabular}\n}\n\\end{table}\n\n\n\nMEDLINE\/PubMed reported funding information only for 23\\% of publications (1,712 records). These publications overlapped with those in the `reference data' in about 96\\% of the case (1,639 records), while the remaining 4\\% (73 records) reported funding data even though we could not identify funding sources in the publications' full-text (\\textit{false positives}). In other words, MEDLINE\/PubMed did not report funding data for 2,275 publications (\\textit{false negatives}). Accordingly, MEDLINE\/PubMed funding data have very low \\textit{recall}, i.e. 41.9\\% (1639\/(1639+2275)) and relatively high \\textit{precision} of 95.7\\% (1639\/(1639+73)). Yet, MEDLINE\/PubMed publications with funding data identified only 17 distinct funders and 1.4 funders per publication against the 2,549 funders and the 3.3 funders per publication reported in the `reference data'. This reflects the focus of MEDLINE\/PubMed on major US funders and few large non-US funding organisations.\\footnote{See \\url{www.nlm.nih.gov\/bsd\/grant_acronym.html}} As a result, funders are not correctly listed in 75.2\\% of the 1,639 \\textit{true positives}. About 73\\% of these cases miss at least one funder, while less than 2\\% of them report at least one funder not acknowledged in publications' full-text.\n\nIn the case of WoS, we first matched our data with WoS data.\\footnote{The matching was performed by using the \\textit{medlineR} routine \\cite{Rotolo2015a}.} 7,082 out of 7,510 publications (about 94.3\\%) included in the `reference data' were also found in WoS. We then harmonised funders' names listed in the ``FU'' field of WoS data in order to make WoS data comparable with the `reference data'. The proportion of publications reporting funding data is 52.7\\% (3,736 records), i.e.\\ similar in proportion to the `reference data'. 3,524 publications of these are also included in the `reference data' (\\textit{true positives}), while WoS did not report funding data for 274 publications for which the `reference data' included funding information (\\textit{false negatives}). 212 publications with no funding information in the `reference data' were instead found to contain funding data in WoS (\\textit{false positives}). This indicates that within the UK cancer research domain WoS data \\textit{recall} is of 92.8\\% (3524\/(3524+274)) and its \\textit{precision} is 94.3\\% (3524\/(3524+212)). These findings also suggest that the \\textit{recall} and \\textit{precision} of WoS funding data vary across research domains --- \\cite{Koier2014} found that WoS did not correctly recognise and retrieve acknowledgement sections in about 24\\% of cases of research publications sponsored by the ``Dutch climate programmes'' in the field of climate change research.\n\nThe number of distinct funders listed in WoS data (after the harmonisation of their names) is much higher than those identified in the `reference data': 3,541 against 2,549. This is also evident from the average number of funders per publications, i.e.\\ 3.7 in WoS data and 3.3 in the `reference data'. To further investigate these differences, we focused on the sample of publications for which both the `reference data' and WoS reported funding information, i.e. 3,524 publications. Within this sample, WoS did not report the same number of funders as in the `reference data' in 32.0\\% of the cases: in 10.5\\% of the records WoS missed at least one funder, while at least one funder more than in the `reference data' was included by WoS in 21.5\\% of the sample. This includes cases where organisations were acknowledged, but under the guidance provided here (in Section 4) would not be classified as providing financial support. For example, the acknowledgments of a publication in the sample stated: \n\n\\bigskip\n\n``\\textit{[...] \\textbf{This study was not funded}. GPRD operates within the MHRA. GPRD has received funding from the MHRA, Wellcome Trust, Medical Research Council, NIHR Health Technology Assessment programme, Innovative Medicine Initiative, UK Department of Health, Technology Strategy Board, Seventh Framework Programme EU, various universities, contract research organisations and pharmaceutical companies. The Department of Pharmacoepidemiology and Pharmacotherapy, Utrecht Institute for Pharmaceutical Sciences, has received unrestricted funding for pharmacoepidemiological research from GlaxoSmithKline, Novo Nordisk, the private-public funded Top Institute Pharma (www. tipharma.nl, includes co-funding from universities, government, and industry), the Dutch Medicines Evaluation Board, and the Dutch Ministry of Health [...]''} \\citep{vanStaa2012}.\\footnote{The publication was electronically published in 2011.}\n\n\\bigskip\n\nDespite the authors? explicit declaration that the focal study did not receive any external funding, WoS data listed all the organisations, subsequently mentioned in the acknowledgments, as funders of the publication. This suggests that the ``FU'' field of WoS requires careful cleaning to remove cases where organisations are reported as funders despite the clear indication that they are listed as a declaration of potential conflict of interests (e.g.\\, in the case of authors paid by companies for unrelated work) and for previous funding, rather than funding of the work contained in that particular paper.\n\nIn summary, the comparison provides evidence that in a very substantial number of cases, manual coding of the data reveals a different outcome to that provided by MEDLINE\/ PubMed and WoS.\n\n\n\n\n\\subsection{Exploring UK cancer funding}\nThis section demonstrates how carefully gathered funding acknowledgements data, although labour intensive to prepare, can be used to provide research portfolio profiles for individual organisations as well as providing landscape overviews of wider funding environments. This is an important approach because alternatives based on self-reporting of funding inputs by research funders have a number of limitations including, \\textit{inter alia}, reliance on funders to provide data and different data collection and coding conventions amongst funders \\citep{Hopkins2013}. Although efforts are underway to bring together and classify funding inputs data (e.g.\\ the UK Clinical Research Collaboration's coverage of 60 funders' self-reporting of funding allocations --- mentioned in the introduction) as the data explored in this section show, biomedical research funding involves a very large number of national and international funders, including governments, charities and firms, and funding landscape methodologies need to be able to take this diversity into account.\n\n\n\n\n\n\\subsubsection{The contribution of different types of funders}\nAn advantage of collecting funding data from publications is the breadth of funders that it is possible to reveal without a priori knowledge of the field or privileged access to funders' data. As noted above, 2,549 funders were associated with at least one publication by a UK author in 2011, although, as reported in \\Figref{funders}a, the distribution funder-publication is highly skewed. 75\\% (1,921) of funders were associated with just a single publication in the sample. More than 296 public and charitable sector funders supported two or more publications. If we define `major UK funders' as those acknowledged in at least 2\\% of the publications acknowledging funding in our sample, these supported about 50\\% (1978\/3914) of the publications. A further 37\\% (1440\/3914) of publications acknowledging funding were supported by `minor UK funders' (that is, funders that supported less than 2\\% of funded publications individually). About 22\\% (849\/3914) of the sample acknowledged at least one major and one minor UK funder. Although clearly a relatively small number of funders fund the majority of UK cancer research, these descriptive statistics highlight the important role that the myriad of small funders play in the overall funding system.\n\n\\begin{figure}[h]\n\\includegraphics[height=14cm]{funders}\n\\centering\n\\caption{Funder-publication distribution (a) and top-10 UK (b), industry (c) and non-UK (d) acknowledged funders.\n\\newline\\textit{Source: Authors' elaboration.}}\n\\label{fig:funders}\n\\end{figure}\n\n\\Figref{funders} reports the top-10 most acknowledged funders in the categories of UK public sector and charitable funders (\\Figref{funders}b), industry (\\Figref{funders}c), and non-UK public sector and charitable funders (\\Figref{funders}d). The analysis suggests that international funders make a strong contribution to UK scientific output. The US National Institutes of Health (NIH) was, for example, acknowledged in 11.5\\% of publications in the sample, i.e.\\ in a similar proportion of publications that acknowledged UK Medical Research Council (MRC). The nature of this contribution is, however, more likely to be via support to the foreign collaborators of UK authors than to UK authors directly --- with unstructured acknowledgements it is impossible to tell which authors are supported by which funders most of the time. The European Commission (EC) funding (acknowledged in 9.8\\% of publications in the sample) is more likely to be provided as direct support to UK researchers, but much of this will also be linked to international collaborators. Nonetheless, given that 43\\% of the sample of 7,510 publications involved an international collaboration, it is clear that the UK cancer research output is substantially supported by funding coming from countries like the US, Italy, Sweden and France (and the relationship goes both ways, clearly). Specifically, about 47\\% (1853\/3914) of the publications acknowledging funders mention at least a non-UK public sector or charitable funder. Of the 307 firms acknowledged for their funding in UK cancer publications, large pharmaceutical companies are the industrial actors most acknowledged in publications, led by Pfizer, AstraZeneca, and Novartis. Collectively these firms supported about 18\\% (699\/3914) of the publications acknowledging funders. About 20\\% (141 publications) of these acknowledged at least two industrial actors for funding, thus providing some indication on the extent to which firms jointly support pre-competitive research. \n\nFunding acknowledgements also enable the exploration of the extent to which funders are jointly acknowledged in publications (co-funding). Use of the term co-funding for publications does not imply funders were aware of collaborative work \\textit{ex-ante}. Rather, it is anticipated that funded researchers work together in ad-hoc combinations in order to drive their research forward. In some cases, co-funding will be as a result of individual researchers accessing multiple sources of funding in order to undertake their work. However, again, due to the unstructured nature of funding acknowledgments it is often difficult to tell how particular funders contributed to a given publication. Given that our data are focused on the UK context, we only examined the co-occurrence of UK funders in funding acknowledgements.\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{cofunding}\n\\centering\n\\caption{Co-occurrence of UK funders in publication acknowledgement sections: nodes represent UK funding organisations (296) acknowledged in at least in two publications (the size of a node is proportional to the number of publications that acknowledged the funder the node represents, while the width of a tie is proportional to the number of times the two funders connected by the tie were jointly acknowledged in publications). The high-definition version of this figure is available at \\url{https:\/\/dx.doi.org\/10.6084\/m9.figshare.2064897.v1}.\n\\newline\\textit{Source: Authors' elaboration.}}\n\\label{fig:cofunding}\n\\end{figure}\n\n\n\nResults are depicted in \\Figref{cofunding}, where each node is a UK funder and the width of the tie between two nodes indicates the extent to which the two funders are jointly acknowledged in publications. For the sake of clarity in the visualisation, we reported UK funders that were acknowledged in at least two publications in our sample (296 funders). The analysis revealed that the UK funders most frequently jointly acknowledged in cancer research publications are the MRC, Cancer Research UK, and the UK Departments of Health (including the NHS in England and the devolved regions and NIHR). The majority of funders shown in \\Figref{cofunding} are external to the research host organisation undertaking the research. Nonetheless, it is worth noting that a small proportion of funders are research host organisations that also appear to support cancer research through internal funds.\\footnote{Although some authors did acknowledge such contributions, generally, as discussed, authors did not acknowledge their employers as a funder. However, it is not possible to exclude these where they do occur. } \\Tabref{co-occurence} also reports the co-occurrence matrix for the top-10 UK funders. \n\n\n\\setlength{\\tabcolsep}{6pt}\n\\renewcommand{\\arraystretch}{1.0}\n\\begin{table}\\footnotesize\n\t\\caption{\\label{tab:co-occurence}Co-occurrence matrix (joint acknowledgements) of the top-10 UK funders.}\n\t\\centering\n{\\begin{tabular}{lcccccccccc}\n\\hline\\hline\n\\textbf{UK funder} \t\t& \\textbf{1} & \\textbf{2} & \\textbf{3} & \\textbf{4} & \\textbf{5} & \\textbf{6} & \\textbf{7} & \\textbf{8} & \\textbf{9} & \\textbf{10} \\\\\n\\hline\n1.\\ Cancer Research UK \t\t\t& - & & & & & & & & & \\\\\n2.\\ Departments of Health (NHS, NIHR) \t\t& 315 & - & & & & & & & & \\\\\n3.\\ Medical Research Council \t\t\t& 216 & 183 & - & & & & & & & \\\\\n4.\\ Wellcome Trust \t\t\t\t& 83 & 75 & 86 & - & & & & & & \\\\\n5.\\ Leukaemia and Lymphoma Research \t\t& 42 & 24 & 41 & 9 & - & & & & & \\\\\n6.\\ Breakthrough Breast Cancer \t\t\t& 51 & 52 & 15 & 10 & 3 & - & & & & \\\\\n7.\\ Breast Cancer Campaign \t\t\t& 54 & 19 & 7 & 3 & 0 & 7 & - & & & \\\\\n8.\\ Engineering and Physical Science Research Council \t& 44 & 46 & 43 & 7 & 2 & 4 & 2 & - & & \\\\\n9.\\ Biotechnology and Biology Sciences Research Council & 21 & 10 & 25 & 15 & 9 & 1 & 1 & 15 & - & \\\\\n10.\\ Yorkshire Cancer Research \t\t& 19 & 14 & 8 & 1 & 3 & 4 & 9 & 0 & 5 & - \\\\\n\n\n\\hline\\hline\n\\multicolumn{11}{l}{\\footnotesize \\textit{Source: Authors' elaboration.}}\n\\end{tabular}\n}\n\\end{table}\n\n\n\\subsubsection{Funders' research portfolios}\nThe MeSH classification enables us to link funders acknowledged in publications and the content of publications in terms of medical topics. We focused on the third level of the MeSH classification and specifically on the `children' of ``Neoplasms by Site'' and ``Neoplasms by Histologic Type'' descriptors, which hold 68\\% and 43\\% of the publications in our sample, respectively. Overall, these descriptors classify 6,174 publications (82.2\\%).\\footnote{The publication count includes also descriptors at lower levels of the MeSH classification. 1,126 publications (15\\%) are classified as ``Neoplasms'' at the first level of the MeSH tree, i.e.\\ no additional MeSH descriptors at second or lower levels are reported. Other second-level descriptors under the ``Neoplasms'' are assigned to 934 publications.} The selected descriptors with the associated number of publications in our sample are reported in \\Tabref{mesh}. The three areas with most publications are glandular and epithelial, digestive system, and urogenital neoplasms.\n\n\\setlength{\\tabcolsep}{5pt}\n\\renewcommand{\\arraystretch}{1.0}\n\\begin{table}[h]\\footnotesize\n\t\\caption{\\label{tab:mesh}Cancer research areas.}\n\t\\centering\n{\\begin{tabular}{lllc}\n\\hline\\hline\n\\textbf{Descriptor} & \\textbf{Abbreviation} & \\textbf{Tree number} & \\textbf{Number of} \\\\\n\\textbf{ } & \\textbf{ } & \\textbf{} & \\textbf{publications} \\\\\n\n\\hline\n\\textbf{Neoplasms by Site} & - & \\textbf{C04.588} & \\textbf{5,137} \\\\\n\\addtolength{\\leftskip}{1em} Digestive System Neoplasms & Digestive System & C04.588.274 & 1,253 \\\\\n\\addtolength{\\leftskip}{1em} Urogenital Neoplasms & Urogenital & C04.588.945 & 1,017 \\\\\n\\addtolength{\\leftskip}{1em} Breast Neoplasms & Breast & C04.588.180 & 1,008 \\\\\n\\addtolength{\\leftskip}{1em} Head and Neck Neoplasms & Head \\& Neck & C04.588.443 & 663 \\\\\n\\addtolength{\\leftskip}{1em} Endocrine Gland Neoplasms & Endocrine Gland & C04.588.322 & 565 \\\\\n\\addtolength{\\leftskip}{1em} Thoracic Neoplasms & Thoracic & C04.588.894 & 533 \\\\\n\\addtolength{\\leftskip}{1em} Nervous System Neoplasms & Nervous System & C04.588.614 & 373 \\\\\n\\addtolength{\\leftskip}{1em} Skin Neoplasms & Skin & C04.588.805 & 287 \\\\\n\\addtolength{\\leftskip}{1em} Bone Neoplasms & Bone & C04.588.149 & 236 \\\\\n\\addtolength{\\leftskip}{1em} Soft Tissue Neoplasms & Soft Tissue & C04.588.839 & 62 \\\\\n\\addtolength{\\leftskip}{1em} Eye Neoplasms & Eye & C04.588.364 & 61 \\\\\n\\addtolength{\\leftskip}{1em} Hematologic Neoplasms & Hematologic & C04.588.448 & 60 \\\\\n\\addtolength{\\leftskip}{1em} Abdominal Neoplasms & Abdominal & C04.588.33 & 46 \\\\\n\\addtolength{\\leftskip}{1em} Mammary Neoplasms, Animal & Mammary (Animal) & C04.588.531 & 23 \\\\\n\\addtolength{\\leftskip}{1em} Pelvic Neoplasms & Pelvic & C04.588.699 & 13 \\\\\n\\addtolength{\\leftskip}{1em} Splenic Neoplasms & Splenic & C04.588.842 & 5 \\\\\n\\addtolength{\\leftskip}{1em} Anal Gland Neoplasms & Anal Gland & C04.588.83 & 1 \\\\\n & & & \\\\\n\\textbf{Neoplasms by Histologic Type} & - & \\textbf{C04.557} & \\textbf{3,252} \\\\\n\\addtolength{\\leftskip}{1em} Neoplasms, Glandular and Epithelial & Glandular \\& Epithelial & C04.557.470 & 1,633 \\\\\n\\addtolength{\\leftskip}{1em} Neoplasms, Germ Cell and Embryonal & Germ Cell \\& Embryonal & C04.557.465 & 652 \\\\\n\\addtolength{\\leftskip}{1em} Neoplasms, Nerve Tissue & Nerve Tissue & C04.557.580 & 614 \\\\\n\\addtolength{\\leftskip}{1em} Leukemia & Leukemia & C04.557.337 & 435 \\\\\n\\addtolength{\\leftskip}{1em} Neoplasms, Connective and Soft Tissue & Connective \\& Soft Tissue & C04.557.450 & 328 \\\\\n\\addtolength{\\leftskip}{1em} Lymphoma & Lymphoma & C04.557.386 & 302 \\\\\n\\addtolength{\\leftskip}{1em} Nevi and Melanomas & Nevi \\& Melanomas & C04.557.665 & 229 \\\\\n\\addtolength{\\leftskip}{1em} Neoplasms, Plasma Cell & Plasma Cell & C04.557.595 & 132 \\\\\n\\addtolength{\\leftskip}{1em} Neoplasms, Vascular Tissue & Vascular Tissue & C04.557.645 & 115 \\\\\n\\addtolength{\\leftskip}{1em} Neoplasms, Complex and Mixed & Complex \\& Mixed & C04.557.435 & 62 \\\\\n\\addtolength{\\leftskip}{1em} Neoplasms, Gonadal Tissue & Gonadal Tissue & C04.557.475 & 11 \\\\\n\\addtolength{\\leftskip}{1em} Lymphatic Vessel Tumors & Lymphatic Vessel Tumors & C04.557.375 & 10 \\\\\n\\addtolength{\\leftskip}{1em} Histiocytic Disorders, Malignant & Histiocytic Disorders, Malignant & C04.557.227 & 5 \\\\\n\\addtolength{\\leftskip}{1em} Odontogenic Tumors & Odontogenic Tumors & C04.557.695 & 1 \\\\ \n\n\n\\hline\\hline\n\\multicolumn{4}{l}{\\footnotesize \\textit{Source: Authors' elaboration on the basis of MEDLINE\/PubMed data.}}\n\\end{tabular}\n}\n\\end{table}\n\nWe used this classification to profile cancer research supported by four UK funders: Cancer Research UK, the Medical Research Council (MRC), the Engineering and Physical Science Research Council (EPSRC), and the Biotechnology and Biology Sciences Research Council (BBRSC) --- a broader set of cases is available elsewhere \\citep{CRUK2014}. These funders were acknowledged in 24.2\\%, 12.2\\%, 2.4\\%, and 2.3\\% of the sub-sample of publications reporting funding data, respectively --- the low percentage for the last two cases is expected given that cancer research is not the main focus of these funders (indeed their strategic priorities extend beyond healthcare). \\Figref{profile1, profile2} profiles the four funders in terms of the proportion of research that they supported in a given cancer domain, relative to overall number of publications that they supported in cancer.\n\n\\begin{figure}\n\\includegraphics[width=13cm]{profile1}\n\\centering\n\\caption{Profile of four UK funders' research portfolios by cancer types (as based on the `abbreviated' MeSH descriptors assigned to publications): Cancer Research UK (a) and MRC (b).\n\\newline\\textit{Source: Authors' elaboration.}}\n\\label{fig:profile1}\n\\end{figure}\n\n\\Figref{profile1}a shows that Cancer Research UK's support is mostly focused on breast, digestive system, glandular and epithelial, and urogenital areas. A similar profile can be observed for the case of MRC (\\Figref{profile1}b) except for breast neoplasm, where CR-UK has a much more pronounced focus in its portfolio than MRC. EPSRC (\\Figref{profile2}a) has also a similar portfolio in its cancer-related funded research (despite cancer not being the main focus of this funder). In the case of BBSRC (\\Figref{profile2}b), the analysis shows that this funder tends to support research on germ cell and embryonal, thoracic, and nerve tissue neoplasms as well as on leukaemia and lymphoma. It is worth noting that about 15\\% of the BBSRC-funded publications are within the thoracic neoplasm area, while less than 10\\% of the publications supported by Cancer Research UK, EPSRC, and MRC fall within this area. Thoracic neoplasms include lung neoplasms, which account for the biggest cancer burden \\citep{Stewart2014}.\n\n\\begin{figure}\n\\includegraphics[width=13cm]{profile2}\n\\centering\n\\caption{Profile of four UK funders' research portfolios by cancer types (as based on the `abbreviated' MeSH descriptors assigned to publications): EPSRC (a) and BBSRC (b).\n\\newline\\textit{Source: Authors' elaboration.}}\n\\label{fig:profile2}\n\\end{figure}\n\n\n\n\n\n\n\\section{Discussion and conclusions}\nAt the present time there is an increasing pressure on research funding. At the same time, there is increasing access to data, facilitated by the Internet and improvements in computing power. These trends have combined to give new impetus for the use of funding data to analyse relationships between funding inputs and research outputs, such as publications. Funders have started to invest in large data infrastructures. For example, the US NIH's grants have been linked to their outcomes (publications and patents) in the RePORTER database, which is also made publicly available. Similarly, an increasing number of UK and international funders are relying on Researchfish to capture outcomes generated from their funded research, while the UK Clinical Research Collaboration has focused on the collection and analysis of data on the portfolios of scores of charitable and public sector funders. However, these `top-down' initiatives allow analysis only of research that funders have shared. Many funders that also contribute to funding systems are excluded. The analysis of funding data included in the acknowledgment \\textit{paratext} of scientific publications this paper has focused is instead a `bottom-up' approach, which, in turn, can provide a more comprehensive perspective on the variety of funders involved in funding systems. This does not rely on a priori knowledge of the population of funders nor does it require funders' support to access the data. Nevertheless, significant challenges exist in the collection process (extraction and coding) of these data.\n\nThis paper has addressed a series of questions related to the use of this `bottom-up' approach to gathering funding data. To do so, we focused on a sample of 7,510 publications published in 2011 in cancer research by UK authors. First, the paper sets out a number of heuristics for guiding the collection and coding of funding data included in acknowledgements \\textit{paratext}. These can inform future studies that aim to build datasets of funding data from publications as well as to develop natural language algorithms capable of extracting funding information from the acknowledgements \\textit{paratext} automatically.\n\nSecond, we have examined the extent to which authors acknowledge funders in their publications. We specifically reviewed extant literature on the use funding data. These studies suggest that the reporting of funding data in publications is relatively limited in certain domains, such as some social sciences, due to their lower reliance on grant funding than the natural sciences, in which authors tend to acknowledge external funders more frequently. On the basis of reading and interpreting acknowledgements sections and surveying UK (corresponding) authors of a sample of publications that did not contain acknowledgements to funders, we estimated that in less than 3\\% of the cases funding acknowledgements were entirely omitted when at least one external funder should have been disclosed.\n\nThird, we provide evidence on the `quality' of funding data available in existing publication databases. Specifically, we compare MEDLINE\/PubMed and ISI Web of Science (WoS), with those data that we manually collected and extracted following the heuristics we set out. On the one hand, we found that MEDLINE\/PubMed has a very low \\textit{recall} (about 42\\%), thus many publications that include funding information are not identified as such by this database. Also, given the focus of this databases on few funders, those MEDLINE\/PubMed records including funding information correctly listed funders in only about 25\\% of the cases. On the other hand, WoS reported values of \\textit{recall} and \\textit{precision} well above 90\\%, but the set of funders acknowledged in a publications were not correctly listed in about 32\\% of the cases (i.e.\\ the list of funders provided by the ``FU'' field could not be reconciled with the authors' reading of the acknowledgements \\textit{paratext}). \n\nFourth, through the application of bibliometric analysis to these data, we illustrated how funding data can be used to inform policy making on research funding. We used the data to provide a number of illustrative examples of major (charitable, governmental and private) funders supporting cancer research in the UK, showing the extent to which funders are co-funding research publications (i.e.\\ being jointly acknowledged in the same publications), and profiling the research portfolios of these funders by cancer type.\n\nImplications for development and use of funding data can be derived on the basis of these findings. Our review of extant literature on the proportion of publications with funding acknowledgements found a lack of studies on similar fields and regions over time. As a result, it remains difficult to determine whether the propensity to acknowledge funders has increased or not in recent years --- although one could argue that funders' efforts to link their funding to outputs and journals' efforts to boost transparency should yield an improvement, particularly in the biomedical field. \n\nFunders, journal editors and publishers have an important role to play in ensuring better reporting and transparency. Standardisation of the reporting modes of funding (e.g.\\ consistent linking of particular funders to authors) would be a major contribution to the utility of funding data. It is often not possible to identify how much funding is linked to a given author or even to identify which author is acknowledging a funder linked to a publication. Also, authors can strategically (or mistakenly) acknowledge funders that have not supported the specific publications in which they are mentioned. It is generally not possible to link publications with the \\textit{quanta} of research funding using funding acknowledgements \\textit{paratext}. \n\nIt is desirable that the contributions of all kinds of funders should be recognised. There is a clear tendency to acknowledge external funders, but not to acknowledge funding support from authors' employers and funders that provide block funding (e.g. the UK Higher Education Funding Council was acknowledged in about 0.4\\% of the publications included in our sample). Such omissions are a substantial hindrance to the accurate description of funding landscapes. Improvements in transparency and completeness of funding data would be to the benefit of analysts of funding systems, as the potential for comprehensiveness of the bottom-up approach are attractive.\n\nThe findings reported here also have important implications for future research on funding data. We provided evidence that the ``FU'' field (list of funders included in publications' acknowledgements) of WoS has some major limitations, with around one third of records in our sample not being correctly coded. These limitations may, however, be less prominent in areas of science that have lower funding intensity, for example with fewer funders per publication. It is worth noting that, the ``FX'' field (including the full-text of publications' acknowledgements) provided by WoS can be a helpful starting point to build datasets matching funding sources with scientific publications. Nonetheless, these data also require substantial cleaning and aggregation to ensure that all of the publications related to a given funder are appropriately linked. MEDLINE\/PubMed can provide indications of funding activity for only a limited number of major US and non-US funders and therefore is not a suitable starting point for analysis of funding landscapes owing to the large number of funders that are not captured in the data at present.\n\nMuch of this paper has focused on cautions of the limitations of funding data, and naturally the analysis provided here suffers from many of the shortcomings we have identified, particularly in relation to data coverage. Those working with data outside biomedical sciences will likely find these problems too. Those seeking to provide longitudinal analysis on the dynamics of funding systems, which we have not explored here, will likely find the difficulties in applying our approach will multiply. Yet such efforts are worthwhile as bottom-up efforts to study funding systems have much to offer in terms of strategic intelligence for policy-making \\citep{Rotolo2016}. This includes intelligence on active funders, joint-funding (as in the extent to which funders are jointly acknowledged in publications), and profiling of funders' research portfolios (e.g.\\ which areas are more intensively supported by certain funders and where funders' interests overlap). Dynamic data can show how interaction between types of funder may be causally linked to certain benefits for particular types of research. This can aid the design of funding programmes and systems as well as improving our understanding of how funders may be complementary or interdependent.\n\n\n\n\n\n\\section*{Acknowledgements}\nAll the authors acknowledge the support of Cancer Research UK for the research project ``Exploring the Interdependencies of Research Funders in the UK''. MH and DR also acknowledge support from the UK Economic and Social Research Council for the award ``Mapping the Dynamics of Emergent Technologies'' (RES-360-25-0076) during which approaches used for this study were developed. DR further acknowledges the support of the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7\/2007-2013) (award PIOF-GA-2012-331107 - \\href{http:\/\/www.danielerotolo.com\/netgenesis}{\"{\\color{blue}NET-GENESIS: Network Micro-Dynamics in Emerging Technologies}}\"). During the course of this research MH has received funding from the Higher Education Funding Council for England QR funding stream (as distributed through the UK's REF funding allocation system). All authors thank the University of Sussex for bearing residual costs of this research not met by the above named funders. The findings and observations contained in this paper are those of the authors and do not necessarily reflect the funders' views. We are thankful to the research assistance provided by Philippa Crane, Christopher Farrell, Abigail Mawer, Chelsea Pateman, and Tammy-Ann Sharp. We also are grateful to Aoife Regan, Emma Greenwood, Daniel Bridge, Jon Sussex, Ismael Rafols, Ben Martin, Paul Nightingale, Richard Sullivan, Virginia Acha, Michael O'Neill, Kevin Dolby, Shemila Nebhrajani, the two anonymous referees of the SPRU Working Paper Series (SWPS), and the two anonymous referees of the Journal of the Association for Information Science and Technology for their comments, criticisms and suggestions. We are also grateful for helpful feedback from audiences at the 2014 Eu-SPRI conference, the 2014 Global TechMining Conference, the 2015 Atlanta Conference, the 2015 Manchester Data Forum, and the 2015 British Council Newton Workshop on ``Science, Technology and Innovation in Neglected Diseases: Policies, Funding and Knowledge Creation'' in Belo Horizonte (Brazil). All remaining errors in the paper are those of the authors. We offer our kind regards to anyone using this piece of acknowledgement \\textit{paratext} for analytical purposes. \n\n\n\n\n\\newpage\n\\singlespace\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOne of the most interesting aspects of graphene is the tight relation between its morphological and\nelectronic properties. Although this issue has been explored at length in the theoretical literature\n \\cite{NGetal09,VKG10}, and there is a fair amount of related experiments\n\\cite{BMetal09,TLetal09,DJetal10,HYetal10,YTetal10,TKetal11,GBetal11}, recent results\n\\cite{DWetal11,GMetal12,KZetal12} have given an extra push to the subject that will be explored in\nthis work. \n\nIn the continuum limit of the standard tight binding (TB) approach, lattice deformations couple to\nthe electronic excitations in the form of effective gauge fields and scalar potentials\n\\cite{SA02,VKG10}. In particular, the so called pseudomagnetic fields have acquired a physical\nreality after the observation\nof Landau levels from strain in graphene samples \\cite{LBetal10} predicted theoretically in\nref. \\onlinecite{GKG10,GGetal10}.\nThese deformation gauge fields are at the basis of the proposed strain modifications of the \nelectronic properties (strain engineering) of graphene \\cite{GLZ08,PC09,GKG10,LGK11,JCetal11,CAH12}\nand have been used in the design and modeling of recent experiments exploring the physics of lattice\nsystems either with cold atoms \\cite{JBetal08,WGS08} or with artificial lattices made with CO\nmolecules in a Cu surface\\cite{GMetal12}.\n\nHence it is very important to establish the accuracy and completeness of the TB description,\nto ascertain if there are modifications to the model and, if so, how they will affect the experiments.\n\nIn the standard approach the parameter that links the TB electronics with the continuum elasticity theory,\n$\\beta$, is related to the electron--phonon coupling and appears in the definition of the\nstrain--induced effective magnetic fields. $\\beta$ reflects the changes in the hopping parameter $t$ of the TB model\nwith the changes of the relative distances between atomic nearest neighbors due to the lattice\ndeformations. \nIn a recent work \\cite{KPetal12}\nit was claimed that extra $\\beta$-independent pseudomagnetic fields arise in the standard TB\ndescription coming from the displacements of the atomic positions of the lattice. Following this work\nthere have been attempts to correct the previous calculations leading \nto ``strain Landau levels\" \\cite{ON12}. Moreover, inspired by a geometric approach to curved graphene\n\\cite{GGV92,GGV93,CV07a,CV07b,JCV07}, the continuum TB Hamiltonian was supplemented in\nref. \\onlinecite{JSV12} with additional\n$\\beta$-dependent terms arising from a higher order derivative expansion, which can be interpreted\nas a position-dependent Fermi velocity and an new vector field. \n\n\nIn what follows we will show that no $\\beta$-independent pseudomagnetic fields exist in\nstrained graphene: the only pseudogauge fields are the well-known $\\beta$-{\\it dependent} fields in\nEq.~\\eqref{gaugefield}. We will indeed identify all the new terms arising from ``frame effects\" \n(i.e., due to the actual atomic positions) needed to complete the TB description whenever the\nsystem is coupled to external probes. But we will see that they only modify the coefficients of the\nposition-dependent Fermi velocity and new vector field obtained in ref. \\onlinecite{JSV12}. We will\nfurther clarify the\nnature of the new vector field and show that it does not act as a pseudogauge field, although it\nmay have interesting physical effects, such as pseudospin precession. We will also show that the\nextra gauge fields suggested in ref. \\onlinecite{KPetal12} can be completely eliminated by a gauge\ntransformation and have no physical consequences. Finally\nwe will discuss the experimental context in which the newly derived terms might lead to observable effects.\n\n\n \n\\section{Frame effects}\n\nWe will assume for simplicity that there are no short range interactions or disorder\nconnecting the two Fermi points of graphene, so that the low energy description around each point remains valid.\nAs is well known in the TB-elasticity approach \\cite{GHD08,VKG10}, elastic deformations of the\nlattice give rise, in the continuum limit, to vector potentials that mimic the coupling of real magnetic fields to the electronic current. The standard TB Hamiltonian in the continuum limit is\n\\begin{equation}\nH_{TB}=-iv_0\\int d^2 x\\psi^{\\dagger}\\sigma_j(\\partial_j + i A_j)\\psi.\n\\label{HTB}\n\\end{equation}\nwhere $v_0=3\\, t a\/2 $ is the Fermi velocity for the perfect lattice, with $t$ the hopping\nparameter for nearest neighbors and $a$ the lattice constant; j=1,2 (summation over a repeated index is understood over\nthe article), and $\\sigma_j$ are the Pauli matrices. The potential $A_j$ is related to\nthe strain tensor by \n\\begin{equation}\nA_1 = \\frac{\\beta }{2 a} (u_{xx}-u_{yy})\\;\\;,\\;\\;\nA_2 = \\frac{\\beta }{2 a} (-2u_{xy}),\n\\label{gaugefield}\n\\end{equation}\nwhere $\\beta\\!=\\! \\vert\\partial \\log t \/ \\partial \\log a\\vert$. The strain\ntensor is defined as $u_{ij} = \\frac{1}{2} \\left( \\partial_i u_j + \\partial_j u_i +\\partial_i h\n\\partial_j\nh \\right)$, where $u_i$ and $h$ are in- and out-of-plane displacements respectively. Note that one usually assumes that crystal deformations are small and~\\eqref{HTB} is valid only up to $O(u_{ij}^2)$ corrections. We will follow this practice for the rest of the paper.\n\n\nAs shown in ref. \\onlinecite{JSV12}, if one uses the TB approach to go one order higher in the\nderivative expansion, the Hamiltonian \\eqref{HTB} becomes \n\\begin{equation}\nH_{TB} = -i\\int d^2 x\\psi^{\\dagger} [v_{ij}({ x}) \\sigma_i \\partial_j + v_0 \\sigma_i\n\\Gamma_i + i v_0 \\sigma_i A_i]\\psi,\n\\label{HTBcomplete}\n\\end{equation}\nwhere the field $A_i$ is the one given in \\eqref{gaugefield}, $v_{ij}$ is the tensorial and space\ndependent Fermi velocity, \n$v_{ij} = v_0 \\left[\\eta_{ij} - {\\frac{ \\beta}{4}}(2 u_{ij} + \\eta_{ij} u_{kk})\\right],$\nand $\\Gamma_i$ is a new vector field given by\n\\begin{equation}\n\\Gamma_i = \\frac{1}{2v_0}\\partial_j v_{ij}=-\\frac{\\beta}{4} \\left(\\partial_j u_{ij}+ \\frac{1}{2}\\partial_i u_{jj}\\right).\n\\label{Aprima}\n\\end{equation}\n\n\n The key observation of the present work is that TB Hamiltonians describing strained graphene~\\cite{GHD08,VKG10}, and~\\eqref{HTBcomplete} in particular, are commonly derived in a specific reference system, the ``crystal frame\". \nThe reason is that the Bloch waves $a_k\\!=\\!\\sum_x e^{-i\\vec k\\cdot\\vec x}a_x$ used to diagonalize the TB hamiltonian are written using the atomic \\textit{equilibrium} positions\n$\\{ x\\}$, which are regularly spaced and independent of the crystal deformation. On the other hand, in the presence of strain the\npositions measured in the ``lab frame\" are the actual positions of the atoms $y_i$. The two sets of coordinates are related by $y_i=x_i+u_i(x)$, where $u_i$ is the in-plane horizontal displacement vector. Note that the vertical displacements $h$ are identical in both systems. In the classical theory of elasticity, crystal (lab) frame coordinates are usually referred to as Lagrangian (Eulerian) coordinates \\cite{CL95}.\n\nThus, the TB hamiltonian~\\eqref{HTBcomplete} is actually the crystal frame hamiltonian $H_c(x)$. In order to describe the interaction of \nelectrons with external probes or fields, we must use the lab frame hamiltonian $H_{Lab}(y)$, i.e., \nthe TB Hamiltonian has to be rewritten in lab frame coordinates. The TB hamiltonian is the sum of the Dirac hamiltonian $H_0$ plus the terms induced by the lattice\ndeformations. As these are already $O(u_{ij})$, we have to compute change-of-frame corrections \nonly for the $u_{ij}$-independent piece $(H_{0})_c$ of the crystal hamiltonian. The computation is simplified by using the symmetric convention for the derivatives of the fermion fields \n\\begin{equation}\n(H_{0})_c=-iv_0\\int d^2 x\\psi^{\\dagger}_c(x)\\sigma_i\\overleftrightarrow{\\partial_i}\\psi_c(x)\n\\label{Hcrystal}\n\\end{equation}\nwhere $\\psi^{\\dagger}\\overleftrightarrow{\\partial_i}\\psi\\equiv\n1\/2(\\psi^{\\dagger}\\partial_i\\psi-(\\partial_i\\psi^{\\dagger})\\psi$ and the subscript in $\\psi_c$ indicates that this is the fermion field operator in the crystal frame.\nThe derivatives transform according to\n\\begin{align}\n\\frac{\\partial}{\\partial x_i}=\\frac{\\partial y_k}{\\partial x_i}\\frac{\\partial}{\\partial\ny_k} =(\\delta_{ik}+\\partial_iu_k)\\partial_k =(\\delta_{ik}+\\tilde u_{ik}+\\omega\\varepsilon_{ik})\\partial_k,\n\\label{1}\n\\end{align}\nwhere $\\tilde u_{ik}=(\\partial_iu_k+\\partial_k u_i)\/2$ is the linear piece of the strain tensor and \n$\\omega\\varepsilon_{ik}=(\\partial_i u_k-\\partial_k u_i)\/2$. We also have to\ntransform the integration measure\n\\begin{align}\nd^2x&=\\left|\\det\\left( \\frac{\\partial x_k}{\\partial\ny_i}\\right)\\right|d^2y =\n\\left|\\det(\\delta_{ik}-\\tilde u_{ik}-\\omega\\varepsilon_{ik})\\right|d^2y \\nonumber \\\\\n&\\simeq(1-\\tilde u_{ii}\n)d^2y.\n\\label{2}\n\\end{align}\nOn the other hand, $\\psi_c^\\dagger \\psi_c$ is the particle density operator in the crystal frame. As the number of fermions in any region should be frame independent, we must impose $\\psi_c^\\dagger \\psi_c\\, d^2 x = \\psi^\\dagger \\psi\\, d^2 y$, where $\\psi(y)$ is the lab frame field operator. This implies\n\\begin{equation}\n\\psi_c(x)=\\left|\\det\\left( \\frac{\\partial x_k}{\\partial\ny_i}\\right)\\right|^{-1\/2} \\psi(y),\\label{rescaling}\n\\end{equation}\nwhich exactly cancels the Jacobian in~\\eqref{2}. The net result is\n\\begin{align}\n-&i v_0\\int d^2 x\\psi^{\\dagger}_c(x)\\sigma_i\\overleftrightarrow{\\partial_i}\\psi_c(x)\n\\simeq \\nonumber\\\\ -iv_0\\int d^2 y & \\left[ \\psi^{\\dagger}(y)\\sigma_i\n\\overleftrightarrow{\\partial_i}\\psi(y)+(\\tilde u_{kl}+\\omega\\varepsilon_{kl})(\\psi^{\\dagger}\\sigma_k\\overleftrightarrow{\n\\partial_l}\\psi) \\right],\n\\label{trans}\n\\end{align}\nwhere the derivatives in the last term act only on the fermion fields. \nFinally, the dependence on the antisymmetric piece $\\omega\\varepsilon_{ij}$ may be eliminated by a\nlocal rotation of the spinors \n\\begin{equation}\n\\psi(y)\\rightarrow e^{-\\frac{i}{2}\\omega\\sigma_3}\\psi(y)\\simeq\\psi(y)-\\frac{i}{2}\\omega\\sigma_3\\psi(y).\n\\label{3}\n\\end{equation}\nIndeed, the identity $i\\sigma_k\\sigma_3=\\varepsilon_{kl}\\sigma_l$ shows that this rotation cancels the term proportional to $\\omega$ in~\\eqref{trans}. A contribution proportional to $\\partial_k\\omega$ vanishes as well due to the anticommutation relation $\\{\\sigma_3,\\sigma_k\\}=0$ for $k=1,2$. \nThis yields\n\\begin{equation}\\label{main1}\nH_{Lab}=H_{TB}+H_{Geom}\n\\end{equation}\nwhere $H_{TB}$ is given by~\\eqref{HTBcomplete} and\n\\begin{align}\nH_{Geom}&=-iv_0\\int d^2 x\\,\\tilde u_{kl}(\\psi^{\\dagger}\\sigma_k\\overleftrightarrow{\n\\partial_l}\\psi)\\nonumber\\\\\n&=-iv_0\\int d^2 x\\,\\psi^{\\dagger}\\left[\\tilde u_{kl}\\sigma_k\n\\partial_l +\\frac{1}{2}(\\partial_l \\tilde u_{kl})\\sigma_k\\right]\\psi .\n\\label{main2}\n\\end{align}\nIn the last line we have used integration by parts to revert to the asymmetric derivative convention. \nNote that, to first order in the strain, $\\beta$-dependent terms are the same in both frames. \nEqs.~\\eqref{main1} and \\eqref{main2} are the main results in this paper.\n\nAs $\\beta\\simeq 2$, \nthe new $\\beta$--independent terms in $H_{Geom}$ are of the same order of magnitude\nas those in the standard TB hamiltonian~\\eqref{HTBcomplete}. In particular, the space--dependent Fermi velocity\nderived in the TB formalism in ref. \\onlinecite{JSV12} will become\n\\begin{equation}\nv_{ij}=v_0 \\left[\\delta_{ij} - {\\frac{ \\beta}{4}}(2 u_{ij} + \\delta_{ij} u_{kk})+\\tilde u_{ij}\\right]\n\\end{equation}\nwith the corresponding correction for the vector field\n\\begin{equation} \n\\Gamma_i=\\frac{1}{2v_0}\\partial_j v_{ij}=-\\frac{\\beta}{4} \\left(\\partial_j u_{ij}+ \n\\frac{1}{2}\\partial_i u_{jj}\\right)+\\frac{1}{2}\\partial_j \\tilde u_{ij}.\n\\end{equation}\n\n\nThe hamiltonian $H_{Lab}$ can also be obtained by performing the TB calculation directly in the lab\nframe. This derivation is explicitly given in the Supplemental Material, where we also show that\nthe additional pseudogauge field found in ref. \\onlinecite{KPetal12} has zero curl everywhere and\ncan be eliminated by a gauge transformation of the electronic wave function.\nAs $\\Gamma_i$ is the only ``new\" vector field in strained or curved graphene, \nin what follows we will comment briefly on its physical significance and compare \nit with the well known pseudogauge field $A_i$ in Eq.~\\eqref{gaugefield}.\nFirst of all, note that, unlike $A_i$, $\\Gamma_i$ is not a functionally independent field. \nThe reason is that the hamiltonian~\\eqref{HTBcomplete} is hermitian only \nfor $\\Gamma_i \\!=\\! \\frac{1}{2v_0}\\partial_j v_{ij}$. Thus, a position \ndependent Fermi velocity requires the existence of the new vector field $\\Gamma_i$.\n\nA look at~\\eqref{HTBcomplete} might suggest that $\\Gamma_i$ is some sort of purely \nimaginary~\\footnote{As a consequence of the extra $i$, the vector field $\\Gamma_i$ is odd under time reversal and couples with equal signs at the two Fermi points.} counterpart to $A_i$. However, this is obviously wrong, as \ngauge potentials have to be real (hermitian). The true nature of $\\Gamma_i$ \nis made apparent if we use the identity $i\\sigma_k\\sigma_3=\\varepsilon_{kl}\\sigma_l$ to rewrite the relevant term as \n$-i v_0\\sigma_i\\Gamma_i = v_0\\sigma_i \\tilde\\Gamma_i $, with\n\\begin{equation}\\label{redef}\n\\tilde \\Gamma_1 = \\Gamma_2 \\sigma_3\\;\\; , \\;\\; \\tilde \\Gamma_2 = -\\Gamma_1 \\sigma_3 .\n\\end{equation}\nNote that $\\tilde \\Gamma_i$ is matrix-valued and hermitian. This shows that the vector field $\\Gamma_i$ plays the role a hermitian connection for the $SO(2)$ group of local pseudospin rotations~\\eqref{3} generated by $\\sigma_3$.\nAs a consequence, a position dependent Fermi velocity will be accompanied by pseudospin rotation\n(``pseudospin precession\"), i.e., by electronic transitions between the two sublattices. In more\nphysical terms, whereas electrons propagating in a (pseudo)gauge field acquire a path-dependent\ncomplex phase, the new vector field induces pseudospin rotation, very much like an optically active\nmedium turns the polarization plane of light. Thus $\\Gamma_i$ is not a gauge field \nand can not give rise to the characteristic Landau levels of real or pseudo-magnetic fields: \nthe only pseudogauge field in strained graphene is the well known $A_i$ given by~\\eqref{gaugefield}.\nNote also that,\n in general, observable effects will not be associated to the field $\\tilde \\Gamma_i$ itself but to\nits curl, which by \\eqref{redef} is proportional to the divergence of $\\Gamma_i$. This is even more\nobvious in the covariant model\\cite{JSV12}, where $\\Gamma_i$ appears as the spin connection\nassociated to fermions propagating in a curved background and its divergence is proportional to the\nscalar curvature $R$. \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=8cm]{fig2.pdf}\n\\includegraphics[width=9cm]{fig3.pdf}\n\\caption{Pictorial view of the strain field discussed in the text and the changes it produces in\nthe density of states. The dotted (green) line represents the contribution from $\\beta$ dependent\nterms alone, while the thick (blue) line represents the total correction including frame effects.\nThe black, dashed line represents the density of states of the perfect lattice. \nThe three plots correspond to $x=-L,0,L$ for the displacement discussed in the text with\n$u_{max}=0.2$.}\n\\label{stmfig2}\n\\end{center}\n\\end{figure}\n\\section{Some physical examples}\nTo see the physical implications of this work to actual measurements we now work out some practical\nexamples. Consider first a density of states measurement. The frame effects discussed are rather\ntrivial in this case but enough to exemplify the issue. The effect of the coordinate change will\naffect STM measurements when the tip resolution is large in units of the lattice constant (no\natomic resolution). The local density of states (LDOS) in the lab frame can be computed\napproximately in the local limit, for a sufficiently smooth $u_{ij}$. To do this, $\\rho(E,u_{ij})$\nis computed assuming $u_{ij}$ is constant, and then its dependence on the position is restored in the\nfinal expression $\\rho(E,x) \\equiv \\rho(E,u_{ij}(x))$. The LDOS can be computed in momentum space\n\\begin{equation}\n\\rho (E) = \\int dq_x d q_y tr (E-H(q_x,q_y))^{-1}\n\\end{equation}\nwith the Hamiltonian \\eqref{main1} by diagonalizing $v_{ij}$, which amounts to a change of integration\nvariables \n\\begin{equation}\n\\rho (E) = \\int \\frac{dq_+ dq_-}{v_+v_-}tr (E-H_0)^{-1},\n\\end{equation}\nwith $H_0$ the unperturbed Hamiltonian and $v_{\\pm}$ the velocity eigenvalues. This yields\n\\begin{equation}\n\\rho (E) = \\frac{4}{2\\pi} \\frac{E}{v_+v_-} = \\rho_0 (E) \\frac{v_0^2}{v_+v_-}\n\\end{equation}\n(the factor of $4$ is due spin and valley degeneracy) which to first order in strain can be computed to\ngive\n\\begin{equation}\n\\rho(E,x) = \\rho_0(E) \\left[(1+\\beta\\,\\text{tr}\\; u-\\text{tr}\\;\\tilde u)\\right] .\n\\end{equation}\n\nA simple but interesting example is provided by in-plane strains that are quadratic in the position, such as\nthose associated to the triangular bumps that led to the observation of pseudo--Landau levels in STM\n\\cite{DWetal11} and that have been explicitly produced in artificial graphene\\cite{GMetal12}.\nRemember that the TB\ngauge field associated to a strain tensor $u_{ij}$ is $\\vec A \\propto (u_{xx}-u_{yy},-2u_{xy})$.\nConsider first a deformation vector given by $\\vec u=(x^2-y^2,2xy)u_{max}\/4L$ shown in the\nupper part of\nFig.~\\ref{stmfig2}. It is easy to see that the associated pseudomagnetic field will be zero in this\ncase. \nThe trace of the strain tensor is tr u = $u_{max}x\/L$,\nhence a line scan along the y direction will give a perfect constant V shape ($\\rho(E, x)\\sim \\vert\nE\\vert$), while along the x direction there will be a dilatation effect such that $\\rho(E, x)\\sim\n(1+u_{max}(\\beta-1)x\/L)\\vert\nE\\vert$, as depicted in the lower part of Fig.~\\ref{stmfig2} for different values of $x$. Due to the frame effects discussed in this work there is an additional, material\nindependent change in the magnitude of the LDOS that adds on top of the $\\beta$ dependent\ncontributions. This is important to consider if one wants to measure the space-dependent Fermi\nvelocity from a local probe with resolution larger than the lattice constant. \n\nAn interesting thing happens if we now consider the same deformation vector but exchange $u_x$ and\n$u_y$, i.e., $\\vec u \\propto (2xy,x^2-y^2)$.\nIn this case\nthere will be no volume effect (tr u=0) and the strain will give rise to a constant pseudomagnetic\nfield whose associated density of states will show similar Landau levels oscillations along any\nscanline.\nA $90$ degree rotation of the strain deformation\nwill\ngive the same V shape with a Fermi velocity increasing this time along x=const. Finally, for the\nstrain $\\vec u \\propto (x^2-y^2,-2xy)$\nboth the trace and the pseudo-magnetic field\nwill be zero and there will be no effect altogether. It can be shown that the geometric vector field\ncoming from the frame change does not affect the DOS at the linear order in $u_{ij}$ considered in this\nwork.\n\nOn the other hand, these examples are a simple illustration of the fact that the Honeycomb lattice is\nvery anisotropic and, of course, does not have full rotational symmetry \\footnote{See for\nexample ref. \\onlinecite{MF12} where the dependence of the pseudomagnetic field on the lattice\norientation was explored}; hence similar looking\ndeformations give rise to very different effects in the STM images. The important point is\nthat, in the case of general strain, the frame effects discussed in this work will be responsible\nfor additional spatial modulation of the intensity of the LDOS while preserving its energy\ndependence. \n\nFrame effects will also be important when the absolute orientation of the lattice changes locally.\nAn example of this effect can be observed in the polarization dependence of ARPES signal\n\\cite{HPetal11}. \nThe usual ARPES pictures of\nDirac cones see only one half of the cones, due to the form of the matrix element of the lattice electron at\nthe K point with the free electron that comes out. This effect sees the absolute orientation of the\nlattice: if the lattice is rotated with respect to the polarization of light, the part of the Dirac\ncone that is observed also rotates. \nAs before, in order to describe the physics in the lab frame, vectors in the\ncrystal frame have to be rotated to the lab frame. This is again a $\\beta$-independent\ncontribution. Note, however, that the suppression of part of the observed Dirac cones in ARPES is\ndue to the interference between photoelectrons emitted from the two sublattices and, as such, goes\nbeyond the continuum limit considered in this paper. Effects of local lattice rotations in ARPES\nhave been reported recently in\n\\onlinecite{WBetal12}. The frame effects associated to lattice rotations could also be observed in\nref.\noptical experiments like those described in ref. \\onlinecite{PRetal11}.\n\\section{Conclusions}\nAs a summary, we have shown that the TB description of general crystal systems on distorted lattices\nmust be supplemented with geometric terms originating in the change of coordinates needed to describe interactions with external probes. These are of course always present in the experiments. The\ncorrect Hamiltonian to use when trying to fit experiments is $H_{lab}=H_{TB}+H_{Geom}$.\nThe new terms are material independent and different from the usual gauge fields arising from\ndeformation induced changes in the hopping parameter. We have worked out in detail the case of\nstrained graphene and tried to clarify some confusions in the literature. We have seen that the\nextra terms \nare of the same form as those already\npresent in the complete TB Hamiltonian~\\eqref{HTBcomplete}, but come with $\\beta$--independent coefficients. Moreover, aside from the well known pseudogauge fields in Eq.~\\eqref{gaugefield}, the only\n vector field in strained graphene is the connection $\\Gamma_i$ (also present in the geometric\nformalism~\\cite{JSV12}), which is compatible with the symmetry analysis\\cite{M07,WZ10,Lin11,JMSV12b}\nand required by the hermiticity of the hamiltonian whenever we have a position dependent Fermi\nvelocity. We have clarified that $\\Gamma_i$ is not\na gauge field and will not give rise to the standard Landau levels in the density of states, \nalthough it may have other physical effects, such as pseudospin precession. We have also shown that\nthe extra gauge fields claimed in ref. \\onlinecite{KPetal12} can be gauged away and do not lead to\nphysical consequences. \nThe frame effects described in this work will be\nrelevant to local experiments with resolution $\\lambda \\gg a$, for which a continuum limit is\nappropriate.\n\n\\begin{acknowledgments}\nWe specially thank M. Sturla for very useful conversations. Discussions with B. Amorim, A. Cortijo,\nD. Faria, A. G. Grushin, F. Guinea, H. Ochoa, A. Salas, and N. Sandler are also acknowledged.\nThis research was supported in part by the Spanish MECD grants FIS2008-00124, FIS2011-23713,\nPIB2010BZ-00512, FPA2009-10612, the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007- 00042)\nand by the Basque Government grant IT559-10. F. de J. acknowledges support from the ``Programa\nNacional de Movilidad de Recursos Humanos\" (Spanish MECD).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRecent cosmological observations \\cite {recent} continue to\nprovide strong evidence that the $\\Lambda$CDM model is a\nreasonable approximation to the recent history of our universe.\nHowever, the true nature of the dominant term - the dark energy -\nremains one of the foremost problems in all of science today\n\\cite{dark}. With the emergence of at least three major players in\ncurrent cosmology; the cosmological constant $\\Lambda$ (or,\nperhaps, a refinement), spatial curvature $k$ and the matter $M$\n(dominated by the yet to be understood dark matter), the view that\nthe history of our universe is best represented by a trajectory in\nphase space has become commonplace. Indeed, many (but notably not\nall \\cite{john}) analyses of the observations are now filtered\nthrough the $\\Lambda$CDM model and reported by way of our current\nposition in the $\\Omega_{\\Lambda}$ - $\\Omega_{M}$ plane. Here we\nuse this phase space approach to completely solve the central\nproblem in classical theoretical cosmology, the integration of the\nFriedmann equation. This program is completed for all models\nconsisting of an arbitrary number of non-interacting species and\nis accomplished through the algorithmic construction of a complete\nset of constants of the motion. These constants are to be\nconsidered as the fundamental characteristics of the model\nuniverses. In this approach the universe is viewed as a whole\nwithout reference to any particular class of observer. The\nquestion then naturally arises as to whether or not events in a\ngiven universe can be ordered globally without reference to any\nparticular class of observer. This question is also explored by\nway of the construction of a gravitational epoch function. It is\nshown that such a function need not exist, but all observations\nstrongly suggest that we live in a universe for which it does.\n\n\\section{The Friedmann equation}\nWe use the following notation: $a(t)$ is the (dimensionless) scale\nfactor and $t$ the proper time of comoving streamlines. $H \\equiv\n\\dot{a}\/a$, $q\\equiv -\\ddot{a}a\/\\dot{a}^2$, $^{.} \\equiv d\/dt$,\n$k$ is the spatial curvature (scaled to $\\pm 1$ for $k\\neq0$ and\nin units of $L^{-2}$ ($L$ signifies length)), $\\rho$ is the energy\ndensity and $p$ the isotropic pressure. $\\Lambda$ is the (fixed)\neffective cosmological constant and $_{0}$ signifies current\nvalues. In currently popular notation, the Friedmann equation is\ngiven by\n\\begin{equation}\n\\Omega+\\Omega_{\\Lambda}+\\Omega_{k}=1 \\label{omegatotal}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Omega_{\\Lambda} \\equiv \\frac{c^2 \\Lambda}{3 H^2},\\;\\;\\;\\Omega_{k}\n\\equiv -\\frac{c^2 k}{H^2 a^2} \\label{omegas}\n\\end{equation}\nand\n\\begin{equation}\n\\Omega\\equiv\\frac{8 \\pi G \\rho}{3 H^2 c^2}. \\label{omegarho}\n\\end{equation}\n\nConsider an arbitrary number of non-interacting separately\nconserved species so that\n\\begin{equation}\n\\rho=\\sum_{i}\\rho_{i} \\label{rho}\n\\end{equation}\nwith each species characterized by\n\\begin{equation}\np_{i}=w_{i}\\rho_{i} \\label{barotropic}\n\\end{equation}\nwhere $w_{i}$ is a constant and distinct for each species. (In the\nreal universe different species interact. The principal\nassumption made here is that these interactions are not an\nimportant effect.) The conservation equation now gives us\n\\begin{equation}\n\\frac{8 \\pi G \\rho_{i}}{3 c^2}\n=\\frac{\\mathcal{C}_{i}}{a^{3(1+w_{i})}} \\equiv\nH^2\\Omega_{i}\\label{Cconstant}\n\\end{equation}\nwhere for each species $\\mathcal{C}_{i}$ is a constant. We can\nformally incorporate $\\Lambda$ and $k$ into the sum (\\ref{rho}) by\nwriting $w_{\\Lambda}=-1$ so that $\\mathcal{C}_{\\Lambda}=c^2\n\\Lambda\/3$ and $w_{k}=-1\/3$ so that $\\mathcal{C}_{k}= -c^2 k$ and\nwrite (\\ref{omegatotal}) in the form\n\\begin{equation}\n\\sum_{i}\\Omega_{i}=1 \\label{omegasum}\n\\end{equation}\nwhere the sum it is now over all ``species\". Since $w_{i}$ is\nassumed distinct for each species, there is no way to distinguish,\nsay, Lovelock \\cite{lovelock} (\\textit{i.e.} geometric) and vacuum\ncontributions to $\\Lambda$. However, one can introduce an\narbitrary number of separate species about the ``phantom divide\"\nat $w=-1$.\n\nIf $\\mathcal{C}_{i}>0$ for species $i$ it follows that all the\nclassical energy conditions hold \\cite{energy} for that species as\nlong as\n\\begin{equation}\n-\\frac{1}{3} \\leq w_{i} \\leq 1.\\label{energy}\n\\end{equation}\nThe situation is summarized for convenience in Fig \\ref{fig1}.\n\n\\begin{figure}[ht]\n\\epsfig{file=fig1.eps,height=0.5in,width=3in,angle=0}\n\\caption{\\label{fig1}The range in $w_{i}$ is shown (in the box)\nfor which the species satisfies the classical energy conditions.\nHere $M$ stands for dust ($w_{M}=0$) and $R$ for radiation\n($w_{R}=1\/3$).}\n\\end{figure}\n\nIt follows from (\\ref{Cconstant}) that\n\\begin{equation}\n2q-1=3\\sum_{i}w_{i}\\Omega_{i}\\label{q}\n\\end{equation}\nand writing $1+z \\equiv \\frac{a_{0}}{a}$ \\cite{nophotons}, it\nalso follows that\n\\begin{equation}\n\\Omega_{i}=\\frac{(1+z)^{3(1+w_{i})}\\Omega_{io}}{\\sum_{j}\\Omega_{jo}(1+z)^{3(1+w_{j})}}.\\label{OmegaZ}\n\\end{equation}\nAs $1+z \\rightarrow \\infty$ (Big Bang or Big Crunch) it follows\nthat $\\Omega_{i} \\rightarrow 0$ for all species except that with\nthe largest $w_{i}>-1$. Then $\\Omega_{i} \\rightarrow 1$. As $1+z\n\\rightarrow 0$, $\\Omega_{i} \\rightarrow 0$ for all species except\nthat with the smallest $w_{i}\\leq-1$. Then $\\Omega_{i} \\rightarrow\n1$. From (\\ref{q}) and (\\ref{OmegaZ}) it follows that the\nevolution of the universes considered are governed by the\nautonomous (but non-linear) system \\cite{dynamical}\n\\begin{equation}\n\\Omega_{i}^{'}=\\Omega_{i}((2q-1)-3w_{i})\\label{system}\n\\end{equation}\nwhere $^{'}\\equiv -(1+z)\\frac{d}{d(1+z)}$.\n\n\\section{Solving the Friedmann equation}\n\nWe solve the system (\\ref{system}) algorithmically by way of the\nconstruction of a complete set of constants of the motion.\nConsider all species or any subset thereof. To distinguish this\nlatter possibility we replace $i$ by $\\alpha$. A constant of the\nmotion, defined globally over the history of a universe, and not\nwith respect to any particular observer, is defined here to be a\nfunction\n\\begin{equation}\n\\mathcal{F}=\\mathcal{F}(\\Omega_{\\alpha})\\label{function}\n\\end{equation}\nfor which\n\\begin{equation}\n\\mathcal{F}^{\\;'}=0. \\label{constant}\n\\end{equation}\nConsider the product\n\\begin{equation}\n\\mathcal{F}=\n\\prod_{\\alpha}\\Omega_{\\alpha}^{v_{\\alpha}}\\label{product}\n\\end{equation}\nwhere the exponents $v_{\\alpha}$ are constants. Note that we do\nnot make use of (\\ref{omegasum}) at this stage. For\n(\\ref{constant}) to hold for (\\ref{product}), it follows from\n(\\ref{system}) that\n\\begin{equation}\n(2q-1)V=3W\\label{constantmotion}\n\\end{equation}\nwhere\n\\begin{equation}\nV\\equiv\\sum_{\\alpha}v_{\\alpha},\\;\\;\\;\nW\\equiv\\sum_{\\alpha}w_{\\alpha}v_{\\alpha}.\\label{VW}\n\\end{equation}\nIf $V \\neq 0$ then according to (\\ref{constantmotion}) $q$ itself\nis a constant and so from (\\ref{system}) for each species the\nevolution is given by\n\\begin{equation}\n\\Omega_{i}= \\Omega_{io} (1+z)^{1-2q+3w_{i}}. \\label{qconst}\n\\end{equation}\nWe do not consider this case further here. If $q$ is not constant\nthen for $\\mathcal{F}$ of the form (\\ref{product}) to be a\nconstant of the motion it follows from (\\ref{constantmotion}) that\n\\begin{equation}\nV=W=0 \\label{vwconst}\n\\end{equation}\nand the constant itself then reduces to\n\\begin{equation}\n\\mathcal{F}=\\prod_{\\alpha} \\mathcal{C}_{\\alpha}^{v_{\\alpha}}.\n\\label{constantC}\n\\end{equation}\n\nSums, differences, products and quotients of constants of the\nmotion are of course constants as well and so the range in\n$\\alpha$ must be examined. First consider two species, say\n$\\alpha=a, b$. It then follows form (\\ref{vwconst}) that\n$w_{a}=w_{b}$ which is not possible as we require that the species\nbe distinguished by $w$. Next suppose that $\\alpha$ ranges over\nmore than three species. It now follows from (\\ref{vwconst}) that\nany such constant can be reduced to products and quotients of\nconstants involving only three distinct species. As a result we\nneed consider only those constants $\\mathcal{F}$ constructed from\nthree distinct species which we now do.\n\nThe trivial case is that of a universe for which there are only\nthree species, say $a, b,$ and $c$. Then\n$\\mathcal{F}=\\Omega_{a}^{v_{a}}\\Omega_{b}^{v_{b}}\\Omega_{c}^{v_{c}}$\nand since we can always choose $v_{a}$, (\\ref{vwconst}) reduces to\ntwo equations in two unknowns. The constant of motion is then\ndetermined up to the choice in $v_{a}$. For example, with dust we\nhave the species $\\Lambda, M$ and $k$ (assumed $\\neq 0$\n\\cite{two}, the $\\Lambda$CDM model) and it follows immediately\nthat \\cite{alpha}\n\\begin{equation}\n\\mathcal{F}=\\frac{\\Omega_{\\Lambda}\\Omega_{M}^2}{\\Omega_{k}^3}.\n\\label{dust}\n\\end{equation}\nWe can now substitute for $\\Omega_{k}$ from (\\ref{omegasum}).\n\nMore generally, consider $n$ species, $n>3$. If, say, species $a$\nand $b$ are chosen to construct $\\mathcal{F}$ then there are $n-2$\nchoices for the remaining species, the number of independent\nconstants of the motion. Further, for example, there are\n$(n-2)(n-3)\/2$ dependent constants of motion that follow which\ninvolve $a$ but not $b$. In any event, as with the case of only\nthree species, since we can always specify one exponent,\n(\\ref{vwconst}) always reduces to two equations in two unknowns.\n\nAs the foregoing algorithm makes clear, if species are added to a\nmodel, but none taken away, the constants of motion are\n\\textit{inherited} from the simpler model. For example,\nirrespective of what we add to the $\\Lambda$CDM model,\n$\\mathcal{F}$, given by (\\ref{dust}), remains a constant of the\nmotion.\n\nNow consider evolution in a three dimensional subspace, say\n$\\Omega_{a}, \\Omega_{b}, \\Omega_{c}$. We can construct the\nconstant $\\mathcal{F}_{1}=\\Omega_{a}\\Omega_{b}^{v_{b}}\n\\Omega_{c}^{v_{c}}$, and the constant\n$\\mathcal{F}_{2}=\\Omega_{a}^{\\bar{v}_{a}}\\Omega_{b}^{\\bar{v}_{b}}\n\\Omega_{d}$ where $d$ is any species other than $a, b$ or $c$.\nThen with (\\ref{omegasum}) we replace $\\Omega_{d}$. All species\nother than $a,b$ or $c$ that now enter are replace with the aide\nof constants constructed from that species and any two of $a,b$\nand $c$. In the subspace then we have two independent relations\nthat relate $\\Omega_{a}, \\Omega_{b}, \\Omega_{c}$. The intersection\nof these surfaces in the subspace defines the evolution trajectory\nof the associated universe in that subspace.\n\nTo amplify the foregoing, consider, for example, dust and\nradiation. We have the species $\\Lambda, M, R$ and $k$ (assumed\n$\\neq 0$). Following the algorithm outlined above we immediately\nobtain the constants \\cite{rindler}\n\\begin{equation}\n\\mathcal{F}_{1}=\\frac{\\Omega_{\\Lambda}\\Omega_{M}^2}{\\Omega_{k}^3},\\;\\;\\;\\mathcal{F}_{2}=\\frac{\\Omega_{\\Lambda}\\Omega_{R}^3}{\\Omega_{M}^4},\n\\label{rad12}\n\\end{equation}\n\n\\begin{equation}\n\\mathcal{F}_{3}=\\frac{\\Omega_{\\Lambda}\\Omega_{R}}{\\Omega_{k}^2},\\;\\;\\;\\mathcal{F}_{4}=\\frac{\\Omega_{k}\\Omega_{R}}{\\Omega_{M}^2}.\n\\label{rad34}\n\\end{equation}\n\nFirst note that the constant $\\mathcal{F}_{1}$ is\n\\textit{inherited} from the $\\Lambda$CDM model as discussed above.\nMoreover, only two of the above constants are independent. For\nexample, $\\mathcal{F}_{3}=\\mathcal{F}_{1}\\mathcal{F}_{4}$ and\n$\\mathcal{F}_{2}=\\mathcal{F}_{1}\\mathcal{F}_{4}^3$. In, say, the\n$\\Lambda, M, R$ subspace the history of a universe is obtained by\nthe trajectory defined by the intersection of $\\mathcal{F}_{2}$\nwith one of the other constants and with $\\Omega_{k}$ replaced by\n$1-\\Omega_{\\Lambda}-\\Omega_{M}-\\Omega_{R}$ in that constant\n\\cite{further}.\n\nWe end our discussion of constants of the motion here by noting\nthat there are an infinite number of universes, of arbitrary\ncomplexity (but containing the species $\\Lambda, M$ and $k$), for\nwhich\n\\begin{equation}\n\\frac{\\Omega_{k}^3}{\\Omega_{\\Lambda}\\Omega_{M}^2} \\sim 0\n\\label{flatness}\n\\end{equation}\nthroughout the entire history of the universe. The fact that we\nwould appear to live in such a universe is a problem for some\ncosmologists - the ``flatness problem\". We take the view that this\n``problem\" arises only when too few species are considered, in the\nlimit a two species model with only $M$ and $k$, in order to put\nthe issue in perspective.\n\nCentral to the discussion given above is the system\n(\\ref{system}). We now discuss how this system can be used to make\na fundamental distinction between model universes.\n\n\\section{The gravitational epoch function}\n\nIn rigorous texts on general relativity, for example\n\\cite{O'Neill} and \\cite{Sachs}, spacetime is defined (in part) as\na time-oriented manifold. As stated in \\cite{O'Neill},\ntime-oriented is often weakened simply to time-orientable, and as\npointed out in \\cite{Sachs}, the local time orientation is, with\nguesswork, extrapolated to the universe as a whole. Here we take a\ngravitational epoch function to be defined as a scalar field\nconstructed from dimensionless ratios of invariants, each\nderivable from the Riemann tensor without differentiation, such\nthat the function is monotone throughout the history of the\nuniverse. The purpose of an epoch function is to allow the\nordering of events without reference to any specific class of\nobservers or, of course, coordinates. The existence of an epoch\nfunction, when unique, allows the rigorous time-orientation of an\nentire manifold without guesswork. This is why the existence of an\nepoch function is interesting.\n\nIn a conformally flat four dimensional spacetime the maximum\nnumber of independent scalar invariants derivable from the Riemann\ntensor without differentiation is four. These are usually taken to\nbe the Ricci scalar $R$ at degree $d = 1$ and the Ricci invariants\n$r(d-1)$ for $d=2,3,4$. (Degree ($d$) here means the number of\ntensors that go into the construction of the scalar. Let $R_{a b c\nd}$ signify the Riemann tensor, $R_{a}^{b}$ the Ricci tensor, $R$\nthe Ricci scalar ($\\equiv R_{a}^{a}$) and $S_{a}^{b}$ the\ntrace-free Ricci tensor ($ \\equiv R_{a}^{b}-\\frac{R}{4\n}\\delta_{a}^{b} $). The Ricci invariants are defined by (the\ncoefficients are of no physical consequence as they derrive from\nthe spinor forms of the invariants): $r(1) \\equiv \\frac{1}{2^2}\nS_{a}^{b} S^{a}_{b}$, $r(2) \\equiv -\\frac{1}{2^3} S_{a}^{b}\nS_{b}^{c} S_{c}^{a}$, and $r(3) \\equiv \\frac{1}{2^4} S_{a}^{b}\nS_{b}^{c} S_{c}^{d}S_{d}^{a}$. The Kretschmann invariant, for\nexample, is not used since $6R_{a b c d}R^{a b c d} = R^2+48 r(1)$\nin the conformally flat case.) The dimension of these invariants\nis $L^{-2d}$. In a Robertson - Walker spacetime only two of these\ninvariants are independent and in this spacetime it follows that\n$r(1)^2\/r(3)=12\/7, \\,r(1)^3\/r(2)^2=3, \\, R^4\/r(3)=12f^2\/7$ and\n$(R^3\/r(2))^2=3f^3$ where\n\\begin{equation}\nf \\equiv \\frac{R^2}{r(1)}. \\label{ratio}\n\\end{equation}\n It is clear that $f$ given by (\\ref{ratio}), or any\nmonotone function thereof, is the only epoch function that can be\nconstructed in a Robertson - Walker spacetime. Universes can then\nbe fundamentally categorized as to whether or not they admit\nmonotone $f$.\n\nFor convenience consider $\\mathcal{T} \\equiv f\/48$. It follows\ndirectly that in any Robertson - Walker spacetime\n\\begin{equation}\n\\mathcal{T} = (\\frac{\\Omega_{k}+q-1}{\\Omega_{k}-q-1})^2.\n\\label{epoch}\n\\end{equation}\nWhereas $\\mathcal{T}$ is defined through a bounce, its\nrepresentation in the $\\Omega$ notation is not. This is of no\nconcern here since we are concerned with the behaviour of\n$\\mathcal{T}$ strictly prior to any bounce should one exist. With\nthe aide of (\\ref{system}), applied only to the species $k$, it\nalso follows that\n\\begin{equation}\n\\mathcal{T}^{\\;'} = \\frac{4(\\Omega_{k}+q-1)(-2q^2\n\\Omega_{k}+q^{'}(\\Omega_{k}-1) )}{(\\Omega_{k}-q-1)^3}.\n\\label{epochprime}\n\\end{equation}\nIf we now apply the restriction (\\ref{barotropic}) and use the\nfull set of equations (\\ref{system}) it follows that in addition\nto (\\ref{q})\n\\begin{equation}\nq^{\\;'}=\\frac{9}{2}((\\sum_{i}w_{i}\\Omega_{i})^2-\\sum_{i}w_{i}^2\\Omega_{i}).\n\\label{qprime}\n\\end{equation}\n\nLet us again consider the $\\Lambda$CDM model \\cite{rad}. It\nfollows from (\\ref{epoch}) that\n\\begin{equation}\n\\mathcal{T} = (\\frac{\\Omega_{M}+4\n\\Omega_{\\Lambda}}{3\\Omega_{M}})^2 \\label{epochdust}\n\\end{equation}\nand from (\\ref{epochprime}) and (\\ref{qprime}) that\n\\begin{equation}\n\\mathcal{T}^{\\;'} = 8 \\Omega_{\\Lambda}(\\frac{\\Omega_{M}+4\n\\Omega_{\\Lambda}}{3\\Omega_{M}^2}) \\label{epochdustprime}\n\\end{equation}\nand finally from (\\ref{system}) applied to $M$ and $\\Lambda$ that\n\\begin{equation}\n\\mathcal{T}^{\\;''} = 8 \\Omega_{\\Lambda}(\\frac{\\Omega_{M}+8\n\\Omega_{\\Lambda}}{\\Omega_{M}^2}). \\label{epochdustprimeprime}\n\\end{equation}\nAs a result, $\\mathcal{T}$ has a global minimum value of $0$ when\n\\begin{equation}\n\\Omega_{\\Lambda}=-\\frac{\\Omega_{M}}{4}. \\label{epochdustminimum}\n\\end{equation}\nThe question then is, what $\\Lambda$CDM models intersect the locus\n(\\ref{epochdustminimum}) and therefore fail to have $\\mathcal{T}$\nmonotone \\cite{degenerate}?\n\nSince $\\Omega_{M} > 0$ clearly all $\\Lambda$CDM models with\n$\\Lambda > 0$ admit the epoch function (\\ref{epochdust}). Now\nconsider $\\Lambda <0$. For $k=0$ condition\n(\\ref{epochdustminimum}) holds at\n$(\\Omega_{M},\\Omega_{\\Lambda})=(4\/3,-1\/3)$. For $k \\neq 0$ use\n(\\ref{epochdustminimum}) in (\\ref{dust}) to define\n\\begin{equation}\n\\mathcal{F}_{int}=\\frac{16 \\Omega_{M}^3}{(3\\Omega_{M}-4)^3}.\n\\label{dustintersection}\n\\end{equation}\nWith $k=-1$ choose a value of $\\Omega_{M}$ in the range $0 <\n\\Omega_{M} < 4\/3$ and insert this value into\n(\\ref{dustintersection}). The resultant $\\mathcal{F}_{int}$ (which\nis negative) is the constant of motion associated with the\nintersection of the associated integral curve with the locus\n(\\ref{epochdustminimum}) with the intersection taking place at the\nchosen value of $\\Omega_{M}$. For $k=1$ choose a value of\n$\\Omega_{M}$ in the range $4\/3 < \\Omega_{M} < \\infty$ and follow\nthe foregoing procedure ($\\mathcal{F}_{int}$ is now positive).\nHowever, with $k=1$, as $\\Omega_{M} \\rightarrow \\infty$ the\nconstant $\\mathcal{F}_{int}$ reaches the lower bound of $16\/27$.\nThere is then the range $0 < \\mathcal{F} < 16\/27$ for which the\nassociated integral curves do not intersect the locus\n(\\ref{epochdustminimum}). We conclude then that for the\n$\\Lambda$CDM models, all models with $\\Lambda > 0$ admit the epoch\nfunction (\\ref{epochdust}), but for $\\Lambda < 0$ only those with\n$k=1$ and $0 < \\mathcal{F} < 16\/27$ do. There is no evidence that\nour universe lies in this latter category. Indeed, all the current\nevidence would suggest that we are very far away from it.\n\nWe now make some general observations on the monotonicity of\n$\\mathcal{T}$, but restricted to the case $\\Omega_{k}=0$\n\\cite{furthert}. In this case it follows from (\\ref{epochprime})\nthat the monotonicity of $\\mathcal{T}$ is closely associated with\nthe monotonicity of $q$. For example, if we assume that $q$ is\nmonotone decreasing, then we can put a limit on the maximum $w$\nallowed for a monotone $\\mathcal{T}$. To do this note that for\nmodels with a Big Bang the initial value of $q$ is $(3\n\\tilde{w}+1)\/2$ where $\\tilde{w}$ signifies the largest value of\n$w$ (assumed $>-1$) for all species. It follows from\n(\\ref{epochprime}) then that $\\mathcal{T}$ can be monotone only\nfor $\\tilde{w}\\leq 1\/3$. In view of (\\ref{energy}) this is a\nrather surprising restriction, but one which is in accord with all\nobservations.\n\n\\section{Summary}\n\nA complete set of constants of the motion has been constructed for\nall FLRW models consisting of an arbitrary number of separately\nconserved species, each with a constant ratio of pressure to\ndensity. These constants of the motion are to be considered as the\nfundamental characteristics of the model universes. The unique\ncandidate for a gravitational epoch function has been constructed\nfor all FLRW models. In the simplest of all models, the\n$\\Lambda$CDM model, it has been shown that the epoch function\nexists for all models with $\\Lambda > 0$, for no models with\n$\\Lambda = 0$, and for almost no models with $\\Lambda < 0$. This\nfunction allows the global ordering of events without reference to\nany particular class of observers or, of course, coordinates.\n\n\\bigskip\n\n\\begin{acknowledgments}\nIt is a pleasure to thank Nicos Pelavas for comments. This work\nwas supported by a grant from the Natural Sciences and Engineering\nResearch Council of Canada.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe photons that Penzias \\& Wilson (1965) detected coming from all\ndirections with a temperature of about $3\\,$K have traveled freely since their\nlast Thomson scattering, when the Universe became cool enough for the\nions and electrons to form neutral atoms.\nDuring this recombination epoch, the\nopacity dropped precipitously, matter and radiation were decoupled,\nand the anisotropies of the Cosmic Microwave Background radiation (CMB) were\nessentially\nfrozen in. These anisotropies of the CMB have now been detected \non a range of scales (e.g.~White, Scott, \\& Silk 1994;\nSmoot \\& Scott 1998) and developments in the field have been very rapid.\nRecently, two post-{\\sl COBE\\\/}\\ missions, the Microwave Anisotropy Probe\n({\\sl MAP\\\/}) and the {\\sl Planck\\\/}\\ satellite, were approved with the major science goal of\ndetermining\nthe shape of the power spectrum of anisotropies with experimental\nprecision at a level similar to current theoretical predictions.\n\nDetailed understanding of the recombination process is crucial for\nmodeling the power spectrum of CMB anisotropies. Since the seminal work\nof the late 1960s (Peebles 1968; Zel'dovich et al.~1968), several\nrefinements have been introduced by, for example,\nMatsuda, Sato, \\& Takeda (1971), Zabotin \\& Nasel'skii (1982),\nLyubarsky \\& Sunyaev (1983), Jones \\& Wyse (1985), Krolik (1990),\nand others, but in fact little has changed (a fairly comprehensive overview\nof earlier work on recombination is to be found in Section~IIIC of\nHu et al.~1995, hereafter HSSW).\nMore recently, refinements have been made independently in\nthe radiative transfer to calculate secondary spectral distortions\n(Dell'Antonio \\& Rybicki 1993; Rybicki \\& Dell'Antonio 1994),\nand in the chemistry (Stancil, Lepp, \\& Dalgarno 1996b). These\nimprovements may have noticeable effects (at the 1\\% level) on the\ncalculated shapes of the power spectrum of anisotropies. Given the\npotential to measure important cosmological parameters\nwith {\\sl MAP\\\/}\\ and {\\sl Planck\\\/}\\\n(e.g.~Jungman et al.~1996; Bond, Efstathiou, \\& Tegmark~1997;\nZaldarriaga, Spergel, \\& Seljak~1997; Eisenstein, Hu, \\& Tegmark~1998;\nBond \\& Efstathiou~1998) it is of\ngreat interest to make a complete and detailed calculation of the\nprocess of recombination. Our view is that this is in principle such a\nsimple process that it should be so well understood that it could never\naffect the parameter estimation endeavor.\n\nOur motivation is to carry out a `modern' calculation of the cosmic\nrecombination process. All of the physics is well understood, and so it\nis surprising that cosmologists have not moved much beyond the solution of a\nsingle ODE, as introduced in the late 1960s. With today's computing power,\nthere is no need to make the sweeping approximations that were implemented\n30 years ago. We therefore attempt to calculate to as great an extent\nas possible the {\\it full\\\/} recombination problem. The other difference compared with\nthree decades ago is that we are now concerned with high precision\ncalculations, because of the imminent prospect of high fidelity data.\n\nIt is our intention to present here a coupled treatment of the\nnon-equilibrium radiative transfer and the detailed chemistry.\nThe present investigation was motivated by indications (HSSW) that multi-level\nnon-equilibrium effects in H and He, as well as in some molecular\nspecies, may have measurable effects on the power spectrum of CMB\nanisotropies by affecting the low-$z$ and high-$z$ tails\nof the visibility function \n${\\rm e}^{-\\tau}\\slantfrac{{d\\tau}}{{dz}}$ (where $\\tau$ is the optical depth).\n\nTo that effect this paper presents a study of the recombination era by\nevolving neutral and ionized species of H and He, and molecular species of H\nsimultaneously with the matter temperature. We believe our work represents\nthe most accurate picture to date of how exactly the Universe as a whole\nbecame neutral.\n\n\\section{Basic Theory}\n\\subsection{The Cosmological Picture}\n\nWe will assume that we live in a homogeneous, expanding Universe within\nthe context of the canonical Hot Big Bang paradigm. The general picture\nis that at some sufficiently early time the Universe can be regarded as\nan expanding plasma of hydrogen plus some helium, with\naround $10^9$ photons per\nbaryon and perhaps some non-baryonic matter. As it expanded\nand cooled there came a time when the protons were able to keep hold of the\nelectrons and the Universe became neutral. This is the period of cosmic\nrecombination.\n\nIn cosmology it is standard to use redshift $z$ as a time coordinate, so\nthat high redshift represents earlier times. Explicitly, $a(t)=1\/(1+z)$\nis the scale factor of the Universe, normalized to be unity today, and\nwith the relationship between scale factor and time depending on the\nparticular cosmological model. It is sufficient to consider only hydrogen\nand helium recombination, since the other elements exist in minute amounts.\nThe relevant range of redshift is then $\\lesssim10{,}000$, during which the\ndensities are typical of those astrophysicists deal with every day, and\nthe temperatures are low enough that there are no relativistic effects.\n\n\\subsection{The Radiation Field}\n\\label{sec-radfield}\nIn describing the radiation field and its interaction with matter,\nwe must use a specific form of the radiative transfer equation. It\nshould describe how radiation is absorbed, emitted and scattered as it\npasses through matter in a medium which is homogeneous, isotropic,\ninfinite, and expanding. The basic time-dependent form\nof the equation of transfer is\n\\begin{eqnarray} \n\\lefteqn{\\frac{1}{c}\\frac{\\partial{I(\\vec{r},\\hat{n},{\\nu},t)}}{\\partial{t}}\n + \\frac{\\partial{I(\\vec{r},\\hat{n},{\\nu},t)}}{\\partial{l}}=}\\nonumber \\\\\n & & j(\\vec{r},\\hat{n},{\\nu},t) - \\kappa(\\vec{r},\\hat{n},{\\nu},t)\n I(\\vec{r},\\hat{n},{\\nu},t).\n\\label{eq:radtrans}\n\\end{eqnarray} \nHere the symbols are:\n$I(\\vec{r},\\hat{n},{\\nu},t)$ is the specific intensity of\nradiation at\nposition $\\vec{r}$, traveling in direction $\\hat{n}$ (the unit\ndirection vector), with\nfrequency $\\nu$\nat a time $t$ (in units of ergs s$^{-1}$cm$^{-2}$Hz$^{-1}$sr$^{-1}$);\n$l$ is the path length along the ray (and is a coordinate-independent\npath length); $j$ is the emissivity,\nwhich is calculated by summing products of upper excitation state\npopulations and transition probabilities over all relevant processes\nthat can release a photon at frequency $\\nu$, including electron\nscattering; and $\\kappa$ is the absorption coefficient, which is the\nproduct of an atomic absorption cross section and the number density\nof absorbers summed over all states that can interact with photons of\nfrequency $\\nu$.\n \nIn the homogeneous and isotropic medium of the early Universe, we\ncan integrate equation~(\\ref{eq:radtrans}) over all solid angles\n$\\omega$ (i.e.~integrating over the unit direction vector $\\hat{n}$):\n\\begin{eqnarray} \n\\lefteqn{\\frac{4\\pi}{c}\\frac{\\partial{J(\\vec{r},\\nu,t)}}{\\partial{t}} +\n\\vec{\\nabla}{\\cdot}\\vec{F}(\\vec{r},\\nu,t) =} \\nonumber\\\\\n& & \\mbox{} - \\oint\\Big[j(\\vec{r},\\hat{n},{\\nu},t) -\n\\kappa(\\vec{r},\\hat{n},{\\nu},t)I(\\vec{r},\\hat{n},{\\nu},t)\\Big]\\, d\\omega.\n\\label{eq:angleav}\n\\end{eqnarray}\nHere $J(\\nu,t)$ is the mean intensity, the zeroth order moment of\nthe specific intensity over all angles (in units of\nergs ${\\rm s}^{-1}{\\rm cm}^{-2}{\\rm Hz}^{-1}$), $\\vec{F}$ is the flux of\nradiation, which is the net rate of radiant energy flow across an\narbitrarily oriented surface per unit time and frequency interval, and\n$c$ is the speed of light. If\nthe radiation field is isotropic there is a ray-by-ray cancellation\nin the net energy transport across a surface and the net flux is\nzero.\nAlso, because of the isotropy of the radiation field and the medium\nbeing static, we can drop the dependence upon angle of $j$ and\n$\\kappa$ in equation~(\\ref{eq:angleav}).\nWith the definition of $J$,\n$\\oint Idw = 4{\\pi}J$, this simplifies equation~(\\ref{eq:angleav}) to:\n\\begin{equation}\n\\label{eq:rt-2}\n\\frac{1}{c}\\frac{\\partial{J(\\nu,t)}}{\\partial{t}} =\nj(\\nu,t) - \\kappa(\\nu,t)J(\\nu,t).\n\\end{equation}\n\nThe above equation is for a static medium. An isotropically expanding\nmedium would reduce the number\ndensity of photons due to the expanding volume, and reduce their\nfrequencies due to redshifting. The term due to the density change\nwill be simply a $3\\frac{\\dot{a}(t)}{a(t)}$ factor, while the redshifting\nterm will involve the frequency derivative of $J(\\nu,t)$ and hence a \n$\\nu\\frac{\\dot{a}(t)}{a(t)}$ factor.\n\nThen the equation for the evolution\nof the radiation field as affected by the expansion and the sources and\nsinks of radiation becomes\n\\begin{eqnarray}\n\\lefteqn{\\frac{dJ(\\nu,t)}{dt} =\n \\frac{\\partial{J(\\nu,t)}}{\\partial{t}} -\n \\nu H(t)\\frac{\\partial{J(\\nu,t)}}{\\partial{\\nu}}}\\nonumber \\\\\n & & \\quad = -3H(t)J(\\nu,t) + c\\left[j(\\nu,t) - \\kappa(\\nu,t)J(\\nu,t)\\right],\n\\label{eq:radfield}\n\\end{eqnarray}\nwhere $H(t)\\equiv {\\dot{a}}\/a$.\n\nThis equation is in its most general form and difficult to solve; \nfortunately we can make two significant\nsimplifications, because the primary spectral\ndistortions{\\footnote{Not to be confused here with the power spectrum of\nspatial anisotropies.}} are of\nnegligible intensity (Dell'Antonio \\& Rybicki 1993) and the quasi-static\nsolution for spectral line profiles is valid (Rybicki \\& Dell'Antonio 1994).\nThe first simplification is that \nfor the purposes of this paper (in which we do not study secondary spectral\ndistortions),\nwe set $J(\\nu,t)= B(\\nu,t)$, the Planck function, which is observed to\napproximate $J(\\nu)$ to at least 1 part in 10$^4$ (Fixsen et\nal.~1996). Thus\nwe eliminate explicit frequency integration from the simultaneous integration\nof all equations (\\S2.6). The validity of this assumption is shown in\n\\S\\ref{sec-Distortions} where we follow the dominant secondary\ndistortions of H Ly$\\,\\alpha$ and H\n2-photon by including their feedback on the recombination process, and find\nthat the secondary spectral distortions from the other Lyman lines and He are\nnot strong enough to feed back on the recombination process. \n\nThe second simplification is in the treatment of the evolution of the\nresonance lines\n(Ly$\\,\\alpha$, etc.) which must still be treated explicitly -- because\nof cosmological redshifting they cause $J(\\nu,t) \\neq B(\\nu,t)$ in the\nlines. These we call the primary distortions. We use escape \nprobability methods for moving (expanding) media (\\S\\ref{sec-Sobolev} and\n\\ref{sec-3level}).\nThis simplification is not an approximation, but is an exact\ntreatment -- a simple solution to the multi-level radiative transfer\nproblem afforded by the physics of the expanding early Universe.\n\nNot only does using $B(\\nu,t)$ with the escape probability method\ninstead of $J(\\nu,t)$ simplify the \ncalculation and reduce computing time\nenormously, but the effects from following the actual radiation\nfield will be small compared to the main improvements of our\nrecombination calculation which are the level-by-level treatment of H,\n\\ion{He}{1} and \\ion{He}{2},\ncalculating recombination directly, and the correct treatment of\n\\ion{He}{1} triplet and singlet states.\n\n\\subsection{The Rate Equations}\n\\label{sec-species}\nThe species we evolve in the expanding Universe are \\ion{H}{1}, \\ion{H}{2}, \n \\ion{He}{1} , \\ion{He}{2}, \\ion{He}{3}, e${^-}$, H${^-}$, H$_{2}$,\nand H$_{2}{^+}$. \nThe chemistry of the early\nUniverse involves the reactions of association and dissociation among\nthese species, facilitated by interactions with the radiation field,\n$J(\\nu,t)$.\nThe rate equations\nfor an atomic system with $N$ energy levels can be described as\n\\begin{eqnarray}\na(t)^{-3}\\frac{d\\left(n_{i}(t)a(t)^{3}\\right)}{dt}\n \\,{=}&\\Big[n_{\\rm e}(t) n_{\\rm c}(t) P_{{\\rm c}i}\n - n_i(t) P_{i{\\rm c}}\\Big] \\nonumber \\\\\n {+}&\\!\\! \\sum_{j=1}^N \\Big[n_j(t) P_{ji} - n_i(t) P_{ij}\\Big],\n\\end{eqnarray}\nwhere the $P_{ij}$ are the rate coefficients between bound levels $i$ and\n$j$, and the $P_{i{\\rm c}}$ are the rate coefficients between bound levels and\nthe continuum ${\\rm c}$:\n$P_{ij} = R_{ij} + n_{\\rm e} C_{ij}$ and $P_{i{\\rm c}}\n = R_{i{\\rm c}} + n_{\\rm e} C_{i{\\rm c}}$,\nwhere $R$ refers to radiative rates and $C$ to collisional rates.\nHere the $n$s are physical (as opposed to comoving) number\ndensities: $n_i$ refers to number density of the $i$th excited atomic\nstate, $n_{\\rm e}$ to the number density of electrons, and\n$n_{\\rm c}$ to the number density of a continuum particle such as a proton,\n\\ion{He}{2} or \\ion{He}{3}.\n$a(t)$ is the cosmological scale factor. \nThe rate equations for molecules take\na slightly different form because their formation and destruction\ndepends on the rate coefficients for the reactions discussed in\n\\S\\ref{sec-Chem}, \nand molecular bound states are not included.\n\n\\subsubsection{Photoionization and Photorecombination}\n\\label{sec-Recomb}\nBy calculating photorecombination rates $R_{{\\rm c}i}$ directly to each level\nfor multi-level H,\n\\ion{He}{1}, and \\ion{He}{2} atoms, we avoid the problem of finding an accurate\nrecombination coefficient, the choice of which has a large effect on\nthe power spectra (HSSW).\n\nPhotoionization rates are calculated by\nintegrals of the incident radiation field $J(\\nu,t)$ and the bound-free cross\nsection $\\sigma_{i{\\rm c}}(\\nu)$. The photoionization rate \nin s$^{-1}$ is\n\\begin{equation}\n\\label{eq:defRic}\n R_{i{\\rm c}} = 4\\pi\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}J(\\nu,t)d\\nu.\n\\end{equation}\nHere $i$ refers to the $i$th excited state and\n$c$ refers to the continuum. $\\nu_0$ is the threshold frequency for\nionization from the $i$th excited state. The radiation field\n$J(\\nu,t)$ depends on frequency $\\nu$ and time $t$. With $n_i$ as the\nnumber density of the $i$th excited state, the number of\nphotoionizations per unit volume per unit time (hereafter\nphotoionization rate) is $n_i R_{i{\\rm c}}$. \n\nBy using the principle of detailed balance\nin the case of local thermodynamic equilibrium (LTE), the radiative\nrecombination rate\ncan be calculated from the photoionization rate.\nThen, as described below, the photorecombination\nrate can be generalized to the non-LTE case by scaling the LTE\npopulations with the actual populations and substituting the actual\nradiation field for the LTE radiation field.\nIn LTE the radiation field $J(\\nu,t)$ is the Planck\nfunction $B(\\nu,t)$.\n$B(\\nu,t)$ is a function of time $t$ during recombination because\n$T_{\\rm R} = 2.728 (1 + z(t))\\,$K. We\nwill call the LTE temperature $T$ ($T = T_{\\rm R} = T_{\\rm M}$ at\nearly times), where $T_{\\rm R}$ is the radiation temperature, and\n$T_{\\rm M}$ the matter temperature.\nTo emphasize the Planck function's dependence on temperature,\nwe will use $B(\\nu,T)$ where $T$ is a function of time.\n\nBy detailed balance in LTE we have \n\\begin{displaymath}\n\\label{eq:db1}\n \\left[n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}\\right]^{\\rm LTE} =\n \\left[n_i R_{i{\\rm c}}\\right]^{\\rm LTE}\\!.\n\\end{displaymath}\n\nRadiative recombination includes spontaneous and stimulated\nrecombination, so we must rewrite the above equation as\n\\begin{eqnarray}\n\\label{eq:db2}\n\\lefteqn{\\left[n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}\\right]^{\\rm LTE} =\n \\left[n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}^{\\rm spon}\\right]^{\\rm LTE}\n + \\left[n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}^{\\rm stim}\\right]^{\\rm LTE}} \\\\\n & & = \\left( \\left[n_i R_{i{\\rm c}}\\right]^{\\rm LTE}\n - \\left[n_i R_{i{\\rm c}}^{\\rm stim}\\right]^{\\rm LTE}\\right)\n + \\left[n_i R_{i{\\rm c}}^{\\rm stim}\\right]^{\\rm LTE}. \\nonumber\n\\end{eqnarray}\nUsing the definition of $R_{i{\\rm c}}$ in equation~(\\ref{eq:defRic}), \n\\begin{displaymath}\n \\left[n_{\\rm e} n_{c}R_{{\\rm c}i}\\right]^{\\rm LTE}\\! =\n 4\\pi n_{i}^{\\rm LTE}\\!\\!\\int_{\\nu_{0}}^{\\infty}\\!\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T)\\!\n \\left(1 - {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T}\\right)\\!d\\nu \\nonumber\n\\end{displaymath}\n\\vspace{-3mm}\n\\begin{equation}\n \\qquad\\qquad {+}\\ 4\\pi n_{i}^{\\rm LTE}\\!\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T)\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T}d\\nu.\n\\label{eq:recombLTE}\n\\end{equation}\nThe first term on the right hand side is the spontaneous recombination rate and the second term on the right hand side is the stimulated recombination rate.\nHere $h_{\\rm P}$ is Planck's constant and $k_{\\rm B}$ is Boltzmann's constant.\nThe factor $(1 - {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T})$ is the correction for\nstimulated recombination (see Mihalas 1978, $\\S$4--3 for a derivation of this\nfactor). \nStimulated recombination can be treated as either negative ionization or as\npositive recombination; the physics is the same (see Seager \\& Sasselov,\nin preparation, for some subtleties). \nWith the LTE expression for recombination (equation~(\\ref{eq:recombLTE})),\nit is easy\nto generalize to the non-LTE case, considering spontaneous\nand stimulated recombination separately. Because the matter\ntemperature $T_{\\rm M}$ and the radiation temperature $T_{\\rm R}$\ndiffer at low $z$, it is important to understand how recombination\ndepends on each of these separately.\n\nSpontaneous recombination involves a free electron but its calculation\nrequires no knowledge of the local radiation field,\nbecause the photon energy is derived from the electron's kinetic energy. In\nother words, whether or not LTE is valid, the LTE spontaneous recombination\nrate holds per ion, as long as the velocity distribution is Maxwellian. The\nlocal Planck function (as representing the Maxwell distribution) depends on\n$T_{\\rm M}$, because the Maxwell distribution describes a collisional process.\nFurthermore, since the \nMaxwellian distribution depends on $T_{\\rm M}$, so does the\nspontaneous rate. To get the non-LTE rate, we only have\nto rescale the LTE ion density to the actual ion density:\n\\begin{eqnarray}\n\\lefteqn{n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}^{\\rm spon} =\n 4\\pi \\frac{n_{\\rm e} n_{\\rm c}}{(n_{\\rm e} n_{\\rm c})^{\\rm LTE}}n_{i}^{\\rm LTE}\n\\ \\times}\\nonumber \\\\\n & & \\qquad\\int_{\\nu_{0}}^{\\infty}\\frac{\\sigma_{i{\\rm c}}(\\nu)}\n {h_{\\rm P}\\nu}B(\\nu,T_{\\rm M})\n \\left(1 - {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}\\right)d\\nu\n\\end{eqnarray}\n\\vspace{-3mm}\n\\begin{equation}\n{\\!{=}\\, 4\\pi n_{\\rm e} n_{\\rm c}\n \\left(\\frac{n_{i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE}\\!\\!\n \\int_{\\nu_{0}}^{\\infty}\\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}\n \\frac{2h_{\\rm P}\\nu^{3}}{c^{2}}\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu.}\n\\label{eq:spon}\n\\end{equation}\n\nTo generalize the stimulated recombination rate from the LTE rate to the\nnon-LTE rate, we rescale the LTE ion density to the\nactual ion density, and replace the LTE\nradiation field by the actual radiation field\n$J(\\nu,t)$ because that is what is `stimulating' the recombination.\nThe correction for stimulated recombination depends on $T_{\\rm M}$,\nbecause the recombination process is collisional; the term always remains\nin the LTE form because equation~(\\ref{eq:recombLTE}) was derived from\ndetailed balance. So we have\n\\begin{eqnarray}\n\\lefteqn{n_{\\rm e} n_{\\rm c} R_{{\\rm c}i}^{\\rm stim} =\n 4\\pi \\frac{n_{\\rm e} n_{\\rm c}}\n{(n_{\\rm e} n_{\\rm c})^{\\rm LTE}}n_{i}^{\\rm LTE} \\times} \\nonumber \\\\\n & & {\\displaystyle \\int_{\\nu_{0}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}J(\\nu,t)\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu.}\n\\end{eqnarray}\nTherefore, the total non-LTE recombination rate $(R_{{\\rm c}i}^{\\rm spon} +\nR_{{\\rm c}i}^{\\rm stim})$ is\n\\begin{eqnarray}\n\\lefteqn{n_{\\rm e} n_{\\rm c} R_{{\\rm c}i} = n_{\\rm e} n_{\\rm c}\n \\left(\\frac{n{_i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE} \\times}\\nonumber \\\\\n & {\\displaystyle 4\\pi\\int_{\\nu_{i}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}\n \\left[\\frac{2h_{\\rm P}\\nu^{3}}{c^{2}}\n + J(\\nu,t)\\right]{\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu.}\n\\end{eqnarray}\nThe LTE population ratios\n$\\left( n_i \/ {n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE}$\ndepend only on $T_{\\rm M}$ through the Saha relation:\n\\begin{equation}\n\\label{eq:saha}\n\\left( \\frac{n_{i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE} =\n \\left( \\frac{h^{2}}{2\\pi m_{\\rm e}k_{\\rm B}T_{\\rm M}} \\right)^{3\/2}\n \\frac{g_i}{2g_{\\rm c}} {\\rm e}^{E_{i}\/k_{\\rm B}T_{\\rm M}}.\n\\end{equation}\nHere $m_{\\rm e}$ is the electron mass,\nthe atomic parameter $g$ is the degeneracy of the\nenergy level, and $E_{i}$ is the ionization energy of level $i$.\nIn the recombination calculation presented in this paper we \nuse the Planck function $B(\\nu,T_{\\rm R})$ instead of the radiation field\n$J(\\nu,T)$ as described earlier.\n\nIn the early Universe\nCase B recombination is used. This excludes recombinations to the\nground state and considers the Lyman lines to be\noptically thick. An implied assumption necessary to compute the\nphotoionization rate is that the excited states (n~$\\ge 2$) are in equilibrium\nwith the radiation. Our approach is more general than Case B, because\nwe don't consider the Lyman lines to be optically thick and\ndon't assume equilibrium among the excited states. For more details on the\nvalidity of Case B recombination see \\S\\ref{sec-CaseB}.\nTo get the total recombination coefficient, we sum\nover captures to all\nexcited levels above the ground state.\n\nTo summarize, the form of the total photoionization rate is\n\\begin{equation}\n\\label{eq:integral1}\n\\sum_{i>1}^N n_{i}R_{i{\\rm c}} = \\sum_{i>1}^N n_{i}4\\pi\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T_{\\rm R})d\\nu,\n\\end{equation}\nand the total recombination rate is\n\\begin{eqnarray}\n\\lefteqn{\\sum_{i>1}^N n_{\\rm e} n_{\\rm c} R_{{\\rm c}i}\n = n_{\\rm e}n_{\\rm c}\\sum_{i>1}^N\n \\left(\\frac{n_{i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE} \\times}\\nonumber \\\\\n & \\!\\!\\!{\\displaystyle 4\\pi\\!\\int_{\\nu_{i}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}\n \\left[\\frac{2h_{\\rm P}\\nu^{3}}{c^{2}} + B(\\nu,T_{\\rm R})\\right]\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu.}\n\\label{eq:integral2}\n\\end{eqnarray}\n\n\\subsubsection{Comparison With the `Standard' Recombination\n\\label{sec-StandardRecomb}\nCalculation of Hydrogen}\nThe `standard' recombination calculation refers to the calculation\nwidely used today and first derived by Peebles (1968,\n1993) and Zel'dovich and collaborators (1968, 1983),\nupdated with the most recent parameters and recombination\ncoefficient (HSSW). See also \\S\\ref{sec-3level}.\n\nFor a 300-level H atom in our new recombination calculation, the\nexpressions (\\ref{eq:integral1}) and (\\ref{eq:integral2}) include\n300 integrals at each redshift step.\nThe standard recombination calculation does not go through this\ntime-consuming task but avoids it entirely by using a\npre-calculated recombination coefficient that is a single expression dependent on $T_{\\rm M}$ only.\nThe recombination coefficient to each excited state $i$ is defined by\n\\begin{equation}\n\\alpha_{i}(T_{\\rm M}) = R_{{\\rm c}i}^{\\rm spon}.\n\\end{equation}\nHere $\\alpha$ is a function of $T_{\\rm M}$, because spontaneous recombination\nis a collisional process, as described previously.\nThe total Case B recombination coefficient ($\\alpha_{\\rm B}$) is obtained from\n\\begin{equation}\n\\alpha_{\\rm B}(T_{\\rm M}) = \\sum_{i>1}^{N} \\alpha_{i}(T_{\\rm M}).\n\\end{equation}\nWe will refer to $\\alpha_{\\rm B}(T_{\\rm M})$ as the `pre-calculated\nrecombination coefficient' because the recombination to each atomic level $i$\nand the summation over $i$ are pre-calculated for LTE conditions.\nSee Hummer (1994) for an example of how these recombination\ncoefficients are calculated. \nSome more elaborate derivations of $\\alpha(T_{\\rm M}) = f(T_{\\rm M}, n)$\nhave also been tried (e.g.~Boschan \\& Biltzinger 1998).\n\nThe standard recombination calculation uses a photoionization\ncoefficient $\\beta_{\\rm B}(T_{\\rm M})$ which is derived\nfrom detailed balance using the recombination rate: \n\\begin{equation}\n{(n_i \\beta_i)}^{\\rm LTE} = ({n_{\\rm e} n_{\\rm p} \\alpha_i})^{\\rm LTE}.\n\\end{equation}\nTo get the non-LTE rate, one uses the actual populations $n_i$,\n\\begin{equation}\nn_i \\beta_i(T_{\\rm M}) =\n n_i{\\left(\\frac{n_{\\rm e} n_{\\rm p}}{n_i}\\right)}^{\\rm LTE}\n \\alpha_i(T_{\\rm M}),\n\\end{equation}\nor with the Saha relation (equation~(\\ref{eq:saha})),\n\\begin{equation}\n\\label{eq:beta_saha}\nn_i \\beta_i(T_{\\rm M}) = n_i \\left( \\frac{2\\pi m_{\\rm e}k_{\\rm\nB}T_{\\rm M}}{h_{\\rm P}^2} \\right)^{3\/2}\n\\frac{2}{g_{i}} {\\rm e}^{-E_{i}\/k_{\\rm B}T_{\\rm M}} \\alpha_i(T_{\\rm M}).\n\\end{equation}\nConstants and variables are as described before.\nTo get the total photoionization rate, $\\beta_i$ is summed over all\nexcited levels. Because \nthe `standard' calculation avoids use of all levels $i$\nexplicitly, the $n_i$ are assumed to be in equilibrium with the\nradiation, and thus can be related to the first excited state number\ndensity $n_{2s}$ by the Boltzmann relation,\n\\begin{equation}\n\\label{eq:boltz}\nn_i = n_{2s} \\frac{g_i}{g_{2s}} {\\rm e}^{-(E_2-E_i)\/k_{\\rm B}T_{\\rm M}}.\n\\end{equation}\nWith this relation, the total photoionization rate is\n\\begin{equation}\n\\sum_{i>1}^N n_i \\beta_i = n_{2s} \\alpha_{\\rm B}\n {\\rm e}^{-E_{2s}\/k_{\\rm B}T_{\\rm M}}\n \\left( \\frac{2\\pi m_{\\rm e}k_{\\rm B}T_{\\rm M}}{h_{\\rm P}^2} \\right)^{3\/2} \n \\equiv n_{2s} \\beta_{\\rm B}.\n\\end{equation}\nIn this expression for the total photoionization rate, the excited\nstates are populated according to a Boltzmann distribution. $T_{\\rm M}$\nis used instead of $T_{\\rm R}$, because the Saha and Boltzmann\nequilibrium used in the derivation are collisional processes. The\nexpression says nothing about the excited levels being in equilibrium \nwith the continuum, because the actual values of $n_{\\rm e}$, $n_1$ and\n$n_{2s}$ are used, and the $n_i$ are proportional to $n_{2s}$.\n\nTo summarize, the standard calculation uses a single expression for \neach of the total recombination rate and the total photoionization rate\nthat is dependent on\n$T_{\\rm M}$ only. The\ntotal photoionization rate is\n\\begin{eqnarray}\n\\lefteqn{\\sum_{i>1}^N n_i R_{i{\\rm c}} = n_{2s} \\beta_{\\rm B}(T_{\\rm M})}\n \\nonumber \\\\\n & = n_{2s} \\alpha_{\\rm B}(T_{\\rm M}) {\\rm e}^{-E_{2s}\/k_{\\rm B}T_{\\rm M}}\n \\left(2\\pi m_{\\rm e} k_{\\rm B} T_{\\rm M} \\right)^{3\/2}\/h_{\\rm P}^3,\n\\label{eq:Betatot}\n\\end{eqnarray}\nand the total recombination rate is\n\\begin{equation}\n\\sum_{i>1}^N n_{\\rm e} n_{\\rm p} R_{{\\rm c}i} = n_{\\rm e} n_{\\rm p}\n \\alpha_{\\rm B}(T_{\\rm M}).\n\\end{equation}\nComparing the right hand side of equation~(\\ref{eq:Betatot}) to our\nlevel-by-level total photoionization rate (equation~(\\ref{eq:integral1})),\nthe main improvement in our method over the standard one is clear:\nwe use the actual excited level populations $n_i$,\n{\\it assuming no equilibrium distribution among them}. In this way\nwe can test the validity of the equilibrium assumption.\nFar less important is that the standard recombination treatment cannot\ndistinguish\nbetween $T_{\\rm R}$ and $T_{\\rm M}$, even though photoionization and\nstimulated recombination are functions of \n$T_{\\rm R}$ while spontaneous recombination is a function of $T_{\\rm\nM}$, as shown in\nequations~(\\ref{eq:integral1}) and~(\\ref{eq:integral2}). The\nnon-equilibrium of excited states is important at the \n$10\\%$ level in the residual ionization fraction for $z\\lesssim800$, while using $T_{\\rm R}$ in\nphotoionization and photoexcitation is only important at the few percent\nlevel for $z\\lesssim300$ (for typical cosmological models). \nNote that although the pre-calculated recombination coefficient\nincludes spontaneous recombination only,\nstimulated recombination (as a function of $T_{\\rm M}$)\nis still included as negative\nphotoionization via detailed balance (see\nequation~(\\ref{eq:db2})). \n\n\\subsubsection{Photoexcitation}\n\\label{sec-Sobolev}\n\nIn the expanding Universe, redshifting of the photons must be taken\ninto account (see equation~(\\ref{eq:radfield})). Line photons emitted\nat one position may be redshifted \nout of\ninteraction frequency (redshifted more than the width of the line) by\nthe time they reach another position in the \nflow.\nWe use the Sobolev escape probability to account for this, a method\nwhich was first used for the expanding Universe by Dell'Antonio and\nRybicki (1993). \nThe Sobolev escape probability (Sobolev 1946), also sometimes called the\nlarge-velocity gradient approximation, is not an approximation but is\nan exact, simple solution to the multi-level radiative transfer in the\ncase of a large velocity gradient. It is this solution which allows\nthe explicit inclusion of the line distortions to the radiation field\n-- without it our detailed approach to the recombination problem would\nbe intractable. \nWe will call the net bound-bound rate for each line transition $\\Delta\nR_{ji}$, where $j$ is the upper level and $i$ is the lower level: \n\\begin{equation}\n\\label{eq:delrp}\n\\Delta R_{ji} = p_{ij} \\left\\{n_j\\left[A_{ji} + B_{ji} B(\\nu_{ij}, t)\\right]\n - n_i B_{ij} B(\\nu_{ij},t)\\right\\}.\n\\end{equation}\nHere the terms $A_{ji}, B_{ji}, B_{ij}$ are the Einstein coefficients;\nthe escape probability $p_{ij}$ is the probability that photons associated\nwith this transition will `escape' without being further scattered or absorbed.\nIf $p_{ij} = 1$ the photons produced in the line transition escape to\ninfinity -- they contribute no\ndistortion to the radiation field. If $p_{ij} = 0$ no photons\nescape to infinity; all of them get reabsorbed and the line is\noptically thick. This is the case of primary distortions to the radiation\nfield, and the Planck function cannot be used for the line radiation.\nIn general $p_{ij} \\ll 1$ for the Lyman lines and\n$p_{ij} = 1$ for all other line transitions. With this method we have\ndescribed the redshifting of photons through the resonance lines and\nfound a simple solution to the radiative transfer problem for all\nbound-bound transitions. The rest of this section is devoted to deriving\n$p_{ij}$. \n\nFor the case of no cosmological redshifting, the radiative rates per\ncm$^{3}$ for transitions between\nexcited states of an atom are\n\\begin{eqnarray}\n\\label{eq:bbup}\nn_{i}R_{ij} &=& n_{i}B_{ij} \\overline{J} \\\\\n\\label{eq:bbdown}\n{\\rm and}\\quad n_{j}R_{ji} &=& n_{j}A_{ji} + n_{j}B_{ji}\\overline{J},\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{eq:Jbar}\n\\overline{J} = \\int_{0}^{\\infty}J(\\nu,t)\\phi(\\nu)d\\nu,\n\\end{equation}\nand $\\phi(\\nu)$ is the line profile function with its area normalized by\n\\begin{equation}\n\\label{eq:normalize}\n \\int_{0}^{\\infty}\\phi(\\nu)d\\nu = 1.\n\\end{equation}\nThe line profile $\\phi(\\nu)$ is taken to be a Voigt function that includes\nnatural and Doppler broadening. In principle, equation~(\\ref{eq:Jbar})\nis the correct approach. In practise we take $\\phi(\\nu)$ as a delta\nfunction, and use $J(\\nu,t)$ instead of $\\overline{J}$. The smooth\nradiation field is essentially constant over the width of the line and \nso the line shape is not important; we get the same results using\n$\\overline{J}$ or $J(\\nu,t).$\n\nThe Sobolev escape probability considers\nthe distance over which the expansion of the medium\ninduces a velocity difference equal to the thermal velocity (for the\ncase of a Doppler width): $L = v_{\\rm th}\/|{v^{\\prime}}|$,\nwhere $v_{\\rm th}$ is the\nthermal velocity width and $v^{\\prime}$ the velocity gradient. The\ntheory is valid when this distance $L$ is much smaller than typical\nscales of macroscopic variation of other quantities.\n\nWe follow Rybicki (1984) in the derivation of the\nSobolev escape probability. The general definition of escape\nprobability is given by the exponential extinction law,\n\\begin{equation}\np_{ij} = \\exp[-\\tau(\\nu_{ij})],\n\\end{equation}\nwhere $\\nu_{ij}$ is the frequency for a given line transition, and\n$\\tau(\\nu_{ij})$ is the monochromatic optical depth forward along a\nray from a given point to the boundary of the medium. Here $\\tau(\\nu_{ij})$\nis defined by\n\\begin{equation}\nd\\tau(\\nu_{ij}) = -{\\tilde k} \\phi(\\nu_{ij})dl,\n\\end{equation}\nwhere ${\\tilde k}$ is the integrated line absorption coefficient, so that the\nmonochromatic absorption coefficient or opacity is\n$\\kappa={\\tilde k}\\phi(\\nu_{ij})$, and $l$ is the distance along the ray\nfrom the emission point ($l=0$).\nRewriting the optical depth for a line profile function\n(which has units of inverse frequency) of the dimensionless\nfrequency variable $x = (\\nu - \\nu_{ij}) \/ \\Delta $,\nwith $\\Delta$ the width of the line in Doppler units,\nand $\\nu_{ij}$ the central line frequency,\nwe have\n\\begin{equation}\nd\\tau(\\nu_{ij}) = -\\frac{\\tilde k}{\\Delta} \\phi(x)dl.\n\\end{equation}\nHere $\\kappa$ is the absorption coefficient (defined in\nequation~(\\ref{eq:radtrans})),\nwith ${\\tilde k}$ just dividing out the line profile\nfunction, and\n\\begin{equation}\n{\\tilde k} = \\frac{h_{\\rm P}\\nu}{4\\pi}(n_iB_{ij} - n_jB_{ji}).\n\\end{equation}\nUsing the Einstein relations\n$g_i B_{ij} = g_j B_{ji}$ and \n$A_{ji} = (2 h \\nu^3\/c^2) B_{ji}$\nthe absorption coefficient can also be written as\n\\begin{equation}\n{\\tilde k} = \\frac{A_{ji}\\lambda_{ij}^2}{8\\pi}\n \\left(n_i\\frac{g_j}{g_i} - n_j\\right).\n\\end{equation}\n\nNote that distances along a ray $l$ correspond to shifts in\nfrequency $x$. This is due to the Doppler effect induced by the velocity\ngradient and is the essence of the Sobolev escape probability approach.\nFor example, the Ly$\\,\\alpha$ photons cannot be reabsorbed in the\nLy$\\,\\alpha$ line if they redshift out of the frequency interaction range.\nThis case will happen at some frequency $x$, or at some distance $l$\nfrom the photon emission point, where because of the expansion the\nphotons have redshifted out of the frequency interaction range.\nFor an expanding medium with a constant velocity gradient $v^\\prime = dv\/dl$,\nthe escape probability along a ray is then\n\\begin{equation}\np_{ij} = \\exp\\left[-\\frac{\\tilde k}{\\Delta} \\int_{0}^{\\infty}\\!\n \\phi(x - l\/L)dl \\right] \\equiv \\exp\n \\left[-\\tau_{\\rm S}\\! \\int_{-\\infty}^{x}\\!\\phi({\\tilde x})d{\\tilde x} \\right].\n\\end{equation}\nThe velocity field has in effect introduced an intrinsic escape mechanism\nfor photons; beyond the interaction limit with a given atomic transition, the photons can no longer be absorbed\nby the material, even if it is of infinite extent, but escape\nfreely to infinity (Mihalas 1978).\nHere the Sobolev optical thickness along the ray is defined by\n\\begin{equation}\n\\label{eq:Tau}\n\\tau_{\\rm S} \\equiv \\frac{\\tilde k}{\\Delta}L,\n\\end{equation}\nwhere L is the Sobolev length,\n\\begin{equation}\nL = v_{\\rm th}\/|{v^{\\prime}}| =\n \\left.\\sqrt{\\frac{3k_{\\rm B} T_{\\rm M}}{m_{\\rm atom}}}\\right\/|{v^{\\prime}}|,\n\\end{equation}\nand $\\Delta$ is the width of the line, which in the case of Doppler\nbroadening is\n\\begin{equation}\n\\Delta = \\frac{\\nu_0}{c}\\sqrt{\\frac{3k_{\\rm B} T_{\\rm M}}{m_{\\rm atom}}}.\n\\end{equation}\nWith these definitions, equation~(\\ref{eq:Tau}) becomes\n\\begin{equation}\n\\tau_{\\rm S} = \\frac{\\lambda_{ij} {\\tilde k}}{|{v^{\\prime}}|}.\n\\end{equation}\nIn the expanding Universe, the velocity gradient $v^{\\prime}$ is given by the\nHubble expansion rate $H(z)$, and using the above definition for ${\\tilde k}$,\n\\begin{equation}\n\\label{eq:sobtau}\n\\tau_{\\rm S} = \\frac{A_{ji} {\\lambda_{ij}}^3 \\left[n_i (g_j\/g_i) - n_j\\right]}\n {8\\pi H(z)}.\n\\end{equation}\n \nTo find the Sobolev escape probability for the ray, we average over the\ninitial frequencies $x$, using the line profile function $\\phi(x)$\nfrom equation~(\\ref{eq:normalize}),\n\\begin{displaymath}\np_{ij} =\n\\int_{-\\infty}^{\\infty}\\!dx\\phi(x)\\exp\\left[-\\tau_{\\rm S}\\!\n\\int_{-\\infty}^{x}\\!\\phi({\\tilde x})d{\\tilde x} \\right]\\! =\\!\n\\int_{0}^{1}\\!d\\zeta\\,\\exp(-\\tau_{\\rm S}\\zeta), \\\\\n\\end{displaymath}\n\\begin{equation}\n\\label{eq:indofphi}\n{\\rm i.e.}\\quad p_{ij} = \\frac{1 - \\exp(-\\tau_{\\rm S})}{\\tau_{\\rm S}}.\n\\end{equation}\nNote that this expression is independent of the line profile shape $\\phi(x)$.\nThe escape probability $p_{ij}$ is defined as a frequency average at a\nsingle point.\nFinally, we must average over angle, but in the case of the isotropically \nexpanding Universe, the angle-averaged Sobolev escape probability\ntakes the same form as $p_{ij}$ above.\nFor further details on the Sobolev escape probability see Rybicki\n(1984) or Mihalas (1978), \\S14.2. \n \nHow does the Sobolev optical depth relate to the usual meaning of\noptical depth? The optical depth for a specific\nline at a specific redshift point is equivalent to the Sobolev optical\ndepth,\n\\begin{equation}\nd\\tau(\\nu_{ij}) = -\\tau_{\\rm S} \\phi(x)\\frac{dl}{L}.\n\\end{equation}\nIf no other line or continua photons are redshifted into that\nfrequency range before or after the redshift point, then the optical\ndepth at a given frequency \ntoday (i.e.~summed over all redshift points) will be equivalent to the\nSobolev optical depth at that past point. Generally, behaviors in frequency\nand space are interchangeable in a medium with a velocity gradient.\n \nIn order to derive an expression for the bound-bound rate\nequations, we must consider the mean radiation field $\\overline{J}$ in\nthe line. For the case of spectral distortions $\\overline{J}$ does not\nequal the Planck function at the line frequency. We \nuse the core saturation method (Rybicki 1984) to get $\\overline{J}$ using\n$p_{ij}$; from this we get\nthe net rate of deexcitations in that transition ($j\\rightarrow i$), \ngiven in equation~(\\ref{eq:delrp}).\nIn general, only the Lyman lines of H and \\ion{He}{2}, and the\n\\ion{He}{1} n$^1$p--1$^1$s lines have $p_{ij} < 1$.\nWith this solution we have accomplished two things: described the\nredshifting of photons through the resonance lines, and found a simple\nsolution to the radiative transfer problem for all bound-bound lines.\n\nPeculiar velocities during the recombination era may cause line\nbroadening of the same order of magnitude as thermal broadening over\ncertain scales (A.~Loeb, private communication).\nBecause we use $J(\\nu) = B(\\nu,T)$, the\nradiation field is essentially constant over the width of the line and\nso the line shape is \nnot important. Similarly, the peculiar velocities will not affect the\nSobolev escape probability because it is independent of line shape\n(equation~(\\ref{eq:indofphi})). If \npeculiar velocities were angle dependent there would be an effect on the\nescape probability which is an angle-averaged function.\nOnly for computing spectral distortions to the CMB, where the line\nshape is important, should\nline broadening from peculiar velocities be included.\n\n\\subsubsection{Chemistry}\n\\label{sec-Chem}\nHydrogen molecular chemistry has been included because\nit may affect the residual electron densities at low redshift ($z<200$).\nDuring the recombination epoch, the H$_2$ formation reactions include\nthe H$^-$ processes\n\\begin{equation}\n{\\rm H} + {\\rm e}^{-} \\longleftrightarrow {\\rm H}{^-} + \\gamma,\n\\end{equation}\n\\begin{equation}\n{\\rm H}^{-} + {\\rm H} \\longleftrightarrow {\\rm H}_{2} + {\\rm e}^{-},\n\\end{equation}\nand the $\\mathrm{H_2}^+$ processes\n\\begin{equation}\n{\\rm H} + {\\rm H}^{+} \\longleftrightarrow {{\\rm H}_2}^{+} + \\gamma,\n\\end{equation}\n\\begin{equation}\n{{\\rm H}_2}^{+} + {\\rm H} \\longleftrightarrow {\\rm H}_2 + {\\rm H}^{+},\n\\end{equation}\ntogether with\n\\begin{equation}\n{\\rm H} + {\\rm H}_2 \\longleftrightarrow {\\rm H} + {\\rm H} + {\\rm H},\n\\end{equation}\nand\n\\begin{equation}\n{\\rm H}_2 + {\\rm e}^{-} \\longleftrightarrow {\\rm H} + {\\rm H} + {\\rm e}^{-}.\n\\end{equation}\n(see~Lepp \\& Shull 1984). The direct three-body process for H$_2$ formation\nis significant only at much higher densities.\nThe most recent rate coefficients and cross sections are listed with\ntheir references in Appendix A (see also Cen~1992; Puy et al.~1993;\nTegmark et al.~1997; Abel et al.~1997; Galli \\& Palla~1998). \n\nWe have not included the molecular chemistry of Li, He, or D. \nIn general, molecular chemistry only becomes important\nat values of $z < 200$. Recent detailed analyses of H, D, and He chemistry\n(Stancil et al.~1996a) and of Li and H chemistry (Stancil et al.~1996b;\nStancil \\& Dalgarno~1997)\npresented improved relative abundances\nof all atomic, ionic and molecular species. The values are certainly too small\nto have any significant effect on the CMB power spectrum.\n\nThere are two reasons for this. The visibility function is\n${\\rm e}^{-\\tau}\\slantfrac{{d\\tau}}{{dz}}$, where $\\tau$ is the optical depth.\nThe main component of the optical depth is Thomson scattering by\nfree electrons. Because the populations of Li, LiH, HeH$^+$, HD, and\nthe other species are so small relative to the electron density,\nthey do not affect the\ncontributions from Thomson scattering. Secondly, these atomic species\nand their molecules themselves make no contribution to the visibility\nfunction because they have no strong opacities. HD has\nonly a weak dipole moment. And while LiH has a very strong dipole moment,\nits opacity during this epoch is expected to be\nnegligible because of its tiny ($< 10^{-18}$) fractional abundance (Stancil,\net al.~1996b). Similarly, the fractional abundance of\nH$_2\\mathrm{D}^+$ ($< 10^{-22}$) is too small to have an effect on\nthe CMB spectrum (Stancil et al.~1996a). For more details,\nsee Palla et al.~(1995) and Galli \\& Palla (1998).\nAn interesting additional point is that \nbecause of the smaller energy gap between n=2 and the continuum\nin \\ion{Li}{1}, it actually recombines at a slightly lower redshift than\nhydrogen does (see e.g.~Galli \\& Palla 1998). Of course this has\nno significant cosmological effects.\n\nWe have also excluded atomic D from the calculation. D, like H,\nhas an atomic opacity much lower\nthan Thomson scattering for the recombination era conditions. D parallels H\nin its reactions with electrons and protons, and recombines in the same\nway and at the same time as H (see Stancil et al.~1996a). Although\nthe abundance of D is small ([D\/H] $\\simeq 10^{-5}$), its Lyman\nphotons are still trapped because\nthey are shared with hydrogen. This is seen, for example, by the\nratio of the isotopic shift of D Ly$\\,\\alpha$ to the width of the H\nLy$\\,\\alpha$ line, on\nthe order of $10^{-2}$. Therefore by excluding D we expect no change in the\nionization fraction, and hence none in the visibility function.\n\nWhile the non-hydrogen chemistry is still extremely important for\ncooling and triggering the collapse of primordial gas clouds, it is\nnot relevant for CMB power spectrum observations at the level measurable by\n{\\sl MAP\\\/}\\ and {\\sl Planck\\\/}.\n\n\\subsection{Expansion of the Universe}\nThe differential equations in time for the number densities and matter\ntemperature must be\nconverted to differential equations in redshift by multiplying by a\nfactor of $dz\/dt$ . The redshift $z$ is related to time by the\nexpression\n\\begin{equation}\n \\frac{dz}{dt} = -(1+z) H(z),\n\\end{equation}\nwith scale factor\n\\begin{equation}\n a(t) = \\frac{1}{1 + z}.\n\\end{equation}\nHere $H(z)={\\dot a}\/a$ is the Hubble factor\n\\begin{eqnarray}\n\\lefteqn{H(z)^2 = H_0^2 \\Big[\\Omega_0(1+z)^4\/(1+z_{\\rm eq}) }\n \\nonumber \\\\\n & & \\qquad \\mbox{} + \\Omega_0(1+z)^3 + \\Omega_K(1+z)^2 + \\Omega_{\\Lambda}\\Big],\n\\end{eqnarray}\nwhere $\\Omega_0$ is the density contribution, $\\Omega_K$ is the\ncurvature contribution and $\\Omega_{\\Lambda}$ is\nthe contribution associated with the cosmological constant,\nwith $\\Omega_0 + \\Omega_{\\rm R} + \\Omega_K + \\Omega_{\\Lambda} = 1$,\nand $\\Omega_{\\rm R}=\\Omega_0\/(1+z_{\\rm eq})$.\nHere $z_{\\rm eq}$ is the redshift of matter-radiation equality,\n\\begin{equation}\n 1+z_{\\rm eq} = \\Omega_0\\frac{3(cH_{0}){^2}}{8\\pi G(1+f_{\\nu})U},\n\\end{equation}\nwith $f_{\\nu}$ the\nneutrino contribution to the energy density in relativistic species\n($f_\\nu\\simeq0.68$ for three massless neutrino types),\n$G$ the gravitational constant, $U$ the photon energy density, and\n$H_{0}$ the Hubble constant today, which will be written as\n$100\\,h\\,{\\rm km}\\,{\\rm s}^{-1}{\\rm Mpc}^{-1}$.\nSince we are interested in redshifts $z\\sim z_{\\rm eq}$,\nit is crucial to include the radiation contribution explicitly (see HSSW).\n \n\\subsection{Matter Temperature}\nThe important processes that are considered in following the matter\ntemperature are Compton cooling, adiabatic cooling, and Bremsstrahlung\ncooling. Less important but also included are photoionization\nheating, photorecombination cooling, radiative and collisional line\ncooling, collisional ionization cooling, and collisional recombination\ncooling.\nNote that throughout the relevant time period, collisions and Coulomb\nscattering hold all the matter species at very nearly the same temperature.\n`Matter' here means protons (and other nuclei), plus electrons, plus\nneutral atoms; dark matter is assumed to be decoupled.\n\nCompton cooling is a major source of energy\ntransfer between electrons and photons. It is described by\nthe rate of transfer of energy per unit volume between photons and\nfree electrons when the electrons are near thermal equilibrium with\nthe photons:\n\\begin{equation}\n\\label{eq:E1}\n \\frac{dE_{{\\rm e},\\gamma}}{dt} = \n \\frac{4{\\sigma_{\\rm T}}Un_{\\rm e}k_{\\rm B}}{m_{\\rm e}{c}}\n {(T_{\\rm R} - T_{\\rm M})},\n\\end{equation}\n\\begin{equation}\n\\label{eq:cool1}\n{\\rm or}\\qquad \\frac{dT_{\\rm M}}{dt} =\n \\frac{8}{3}\\frac{\\sigma_{\\rm T} U n_{\\rm e}}{m_{\\rm e}{c}n_{\\rm tot}}\n (T_{\\rm R} - T_{\\rm M}),\n\\end{equation}\n(Weymann 1965), \nwhere $E_{{\\rm e},\\gamma}$ is the electron energy density, $k_{\\rm\nB}$, $m_{\\rm e}$ and $c$ are constants as before,\n$\\sigma_{\\rm T}$ is the Thomson scattering cross section, $T_{\\rm R}$ is the\nradiation temperature and $T_{\\rm M}$ is the electron or matter temperature.\nTo get from equation~(\\ref{eq:E1}) to equation~(\\ref{eq:cool1}) we use\nthe energy of all particles; collisions among all particles keep them\nat the same temperature.\nHere $n_{\\rm tot}$ represents the total\nnumber density of particles, which includes all of the species mentioned in\n\\S\\ref{sec-species}, while $U$ represents the radiation energy density\n(integrated over all frequencies) in units of ${\\rm ergs}\\,{\\rm cm}^{-3}$:\n\\begin{equation}\n U = \\int_{0}^{\\infty}u(\\nu)d\\nu,\n\\end{equation}\nwhere $u(\\nu,t)= 4\\pi{J(\\nu,t)}\/{c}$.\nIn thermal equilibrium the radiation field has a frequency distribution\ngiven by the Planck function, $J(\\nu,t)= B(\\nu,T_{\\rm R})$, and\nthus, in thermal equilibrium the energy density is\n\\begin{equation}\n u(\\nu,t) = \\frac{4\\pi}{c}B_{\\nu}(T_{\\rm R}),\n\\end{equation}\nand the total energy density $U$ is given by Stefan's law\n\\begin{equation}\n U = \\frac{8\\pi{h_{\\rm P}}}{c^3}\\int_{0}^{\\infty}\n ({\\rm e}^{h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm R}} - 1)^{-1}\n \\nu^{3}d\\nu = a_{\\rm R}T_{\\rm R}^{4}.\n\\end{equation}\nThe spectrum of the CMB remains close to blackbody because the heat\ncapacity of the radiation is very much larger than that of the matter\n(Peebles 1993), i.e.~there are vastly more photons than baryons.\n\nAdiabatic cooling due to the expansion of the Universe is described by\n\\begin{equation} \n\\label{eq:adiabaticcooling}\n\\frac{dT_{\\rm M}}{dt} = -2H(t)T_{\\rm M},\n\\end{equation}\nsince $\\gamma=\\slantfrac{5}{3}$ for an ideal gas implies\n$T_{\\rm M}\\propto(1+z)^2$. \\newline\nThe following cooling and heating\nprocesses are often represented by approximate expressions.\nWe used the exact forms, with the exception of Bremsstrahlung\ncooling, and the negligible collisional cooling. \\newline\nBremsstrahlung, or free-free cooling:\n\\begin{equation}\n\\Lambda_{\\rm brem} = \\frac{2^5\\pi e^6 Z^2}\n {3^{3\/2}h_{\\rm P}m_{\\rm e} c^3}\n \\left(\\frac{2\\pi k_{\\rm B}T}{m_{\\rm e}}\\right)^{1\/2}\n \\!\\!\\!\\!g_{\\rm ff}n_{\\rm e}(n_{\\rm p}+n_{\\rm He II} + 4 n_{\\rm He III}),\n\\end{equation}\nwhere $g_{\\rm ff}$ is the free-free Gaunt factor (Seaton 1960), $n_{\\rm\np}$ is the number density of protons, $n_{\\rm He II}$ and $n_{\\rm He\nIII}$ the number density of singly and double ionized helium\nrespectively, and other\nsymbols are as previously described. \\newline\nPhotoionization heating:\n\\begin{equation}\n\\Pi_{\\rm p\\!-\\!i} = \\sum_{i=1}^N n_{i}4\\pi\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\alpha_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T_{\\rm R})\n h_{\\rm P}(\\nu - \\nu_0)d\\nu.\n\\end{equation}\nPhotorecombination cooling:\n\\begin{eqnarray}\n\\lefteqn{\\Lambda_{\\rm p\\!-\\!r} = \\sum_{i=1}^N 4\\pi n_{\\rm e} n_{\\rm c}\n \\left(\\frac{n_{i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE} \\times}\\nonumber \\\\\n & \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\\displaystyle\n \\int_{\\nu_{i}}^{\\infty}\\!\\!\\frac{\\alpha_{i}(\\nu)}{h_{\\rm P}\\nu}\\!\\!\n \\left[\\frac{2h_{\\rm P}\\nu^{3}}{c^{2}}\\,{+}\\,B(\\nu,T_{\\rm R})\\!\\right]\\!\\!\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}h_{\\rm P}(\\nu\\,{-}\\,\\nu_0)d\\nu}.\n\\end{eqnarray}\nLine cooling:\n\\begin{equation}\n\\Lambda_{\\rm line} = h_{\\rm P}\\nu_0[n_jR_{ji} - n_iR_{ij}].\n\\end{equation}\nCollisional ionization cooling:\n\\begin{equation}\n\\Lambda_{\\rm c\\!-\\!i} = h_{\\rm P}\\nu_0C_{i{\\rm c}}.\n\\end{equation}\nCollisional recombination heating:\n\\begin{equation}\n\\Lambda_{\\rm c\\!-\\!r} = h_{\\rm P}\\nu_0C_{{\\rm c}i}.\n\\end{equation}\nHere $\\nu_0$ is the frequency at the ionization edge.\nWe used approximations for collisional ionization and recombination\ncooling because these collisional processes are essentially\nnegligible during the recombination\nera. $C_{i{\\rm c}}$ and $C_{{\\rm c}i}$ are the collisional ionization and\nrecombination rates respectively, computed as in e.g.~Mihalas (1978),~\\S5.4.\n\nThus, with \n\\begin{equation}\n \\frac{dT_{\\rm M}}{dz} = \\frac{dt}{dz}\\frac{dT_{\\rm M}}{dt},\n\\end{equation}\nthe total rate of change of matter temperature\nwith respect to redshift becomes\n\\begin{eqnarray}\n\\lefteqn{(1+z)\\frac{dT_{\\rm M}}{dz} = \\frac{8\\sigma_{\\rm T}U}{3H(z)m_{\\rm e}c}\\,\n \\frac{n_{\\rm e}}{n_{\\rm e}+n_{\\rm H}+n_{\\rm He}}\\,(T_{\\rm M} - T_{\\rm R})}\n \\nonumber \\\\\n & \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\mbox{}+ 2T_{\\rm M} + {\\displaystyle\n \\frac{2\\!\\left(\\Lambda_{\\rm brem}\\,{-}\\,\\Pi_{\\rm p\\!-\\!i}\n \\,{+}\\,\\Lambda_{\\rm p\\!-\\!r}\\,{+}\\,\\Lambda_{\\rm c\\!-\\!i}\n \\,{+}\\,\\Lambda_{\\rm c\\!-\\!r}\n \\,{+}\\,\\Lambda_{\\rm line}\\right)}{3 k_{\\rm B} n_{\\rm tot} H(z)}.}\n\\label{eq:cooling}\n\\end{eqnarray}\nHere $n_{\\rm He}$ is the total number density of helium, and \nthe denominator $n_{\\rm e}+n_{\\rm H}+n_{\\rm He}$ ($n_{\\rm tot}$ from\nequation~(\\ref{eq:cool1}))\ntakes into account the fact that the energy is shared among all the\navailable matter particles. All the terms except adiabatic cooling\nin equation~(\\ref{eq:cooling}) involve matter energy conversion into\nphotons. In particular, Compton and Bremsstrahlung cooling are the\nmost important, and they\ncan be thought of as keeping $T_{\\rm M}$ very close to $T_{\\rm R}$\nuntil their time scales\nbecome long compared with the Hubble time, and\nthereafter the matter cools as $T_{\\rm M}\\propto(1+z)^2$.\nPrevious recombination calculations only included Compton and adiabatic\ncooling, however the additional terms add improvements only at the\n$10^{-3}\\%$ level in the ionization fraction.\nThe reason for the negligible improvement is that it \nmakes little difference which mechanism keeps $T_{\\rm M}$ close to\n$T_{\\rm R}$ early on,\nand adiabatic cooling still becomes important at the same time.\n\n\\subsection{Summary of Equations}\nThe system of equations to be simultaneously integrated in redshift are:\n\\begin{eqnarray}\n(1+z)\\frac{dn_{i}(z)}{dz} = \n-&\\!\\!\\! {\\displaystyle\\frac{1}{H(z)}}\n\\Big\\{\\left[n_{\\rm e}(z) n_{\\rm c}(z) P_{{\\rm c}i} - n_i(z) P_{ic}\\right]\n \\nonumber \\\\\n & \\quad \\mbox{} + {\\rm \\sum_{{\\it j}=1}^{N}} \\Delta R_{ji}\\Big\\} + 3 n_i(z),\n\\label{eq:sum1}\n\\end{eqnarray}\n\\begin{displaymath}\n{\\displaystyle (1+z)\\frac{dT_{\\rm M}}{dz} =\n \\frac{8\\sigma_{\\rm T}U(J_{\\nu,z})}{3H(z)m_{\\rm e}c}\\,\n \\frac{n_{\\rm e}}{n_{\\rm e} + n_{\\rm H} + n_{\\rm He}}\\,(T_{\\rm M} - T_{\\rm R})}\n\\end{displaymath}\n\\vspace{-3mm}\n\\begin{equation}\n\\!\\!\\mbox{}+ 2T_{\\rm M}\n - \\frac{2\\left(\\Lambda_{\\rm brem} +\n \\Pi_{\\rm p\\!-\\!i} + \\Lambda_{\\rm p\\!-\\!r} + \\Lambda_{\\rm c\\!-\\!i}\n + \\Lambda_{\\rm c\\!-\\!r} + \\Lambda_{\\rm line}\\right)}\n {3 k_{\\rm B}n_{\\rm tot}H(z)},\n\\label{eq:sum2}\n\\end{equation}\nand\n\\begin{equation}\n (1+z)\\frac{dJ(\\nu,z)}{dz} =\n 3J(\\nu,z)\n -\\frac{c}{H(z)}\\Big[j(\\nu,z) - \\kappa(\\nu,z)J(\\nu,z)\\Big].\n\\label{eq:sum3}\n\\end{equation}\n\nFor $J(\\nu,z) = B(\\nu,z) = B(\\nu,T_{\\rm R})$ (see~\\S\\ref{sec-radfield}),\nequation~(\\ref{eq:sum3}) can\nbe omitted because the expansion of the Universe preserves the thermal\nspectrum of non-interacting radiation, and we can use the Sobolev escape\nprobability method for the primary spectral line distortions.\nThe system of coupled equations~(\\ref{eq:sum1}) that we use contains up to 609\nseparate equations, 300 for H (one for each of a maximum of 300 levels we\nconsidered), 200 for \\ion{He}{1}, 100 for \\ion{He}{2}, 1 for \\ion{He}{3},\n1 for electrons, 1 for protons, and 1 for each of the 5 molecular or ionic H\nspecies.\nThis system of equations, along with (\\ref{eq:sum2}), is extremely stiff,\nthat is\nthe dependent variables are changing on very different time scales.\nWe used the Bader-Deuflhard semi-implicit numerical integration scheme,\nwhich is described in Press et al.~(1992).\nTo test the numerical integration we checked at each time step that the\ntotal charge and total number of particles are conserved to one part\nin $10^{7}$.\n\n\\section{Results and Discussion}\nBy an `effective 3-level' H atom we mean a hydrogen atom that includes\nthe ground state,\nfirst excited state, and continuum. In an effective 3-level atom, the\nenergy levels between n=2 and \nthe continuum are accounted for by a recombination coefficient which\nincludes recombinations to those levels.\nThis should be distinguished from an {\\it actual\\\/} 3-level atom, which would\ncompletely neglect all levels above n=2, and would be a hopeless\napproximation. Good accuracy is obtained by considering an n-level atom,\nwhere n is large enough. In practice we find that a 300-level atom is\nmore than adequate.\nWe do not explicitly include angular momentum states $\\ell$, whose\neffect we expect to be negligible.\nIn contrast to the effective 3-level H atom, the 300-level H\natom has no\nrecombination coefficient with `extra' levels.\nThe `standard' recombination calculation refers to the calculation\nwith the effective 3-level atom that is\nwidely used today and first derived by Peebles (1968) and Zel'dovich\net al.~(1968), updated with the most recent parameters and recombination\ncoefficient (HSSW). \n\nThe primordial He abundance was taken to be $Y_{\\rm P}=0.24$ by mass\n(Schramm \\& Turner 1998). The present-day CMB temperature $T_{0}$ was\ntaken to be $2.728\\,$K, the central value determined by the FIRAS\nexperiment (Fixsen et al.~1996).\n\n\\subsection{The `Effective 3-level' Hydrogen Atom}\n\\label{sec-3level}\nFor comparison with the standard recombination calculation that only\nincludes hydrogen (see~Peebles 1968, 1993; Scott 1988), we reduce our chemical\nreaction network to an effective 3-level atom, i.e.~a two-level\nhydrogen atom plus continuum. \nThe higher atomic energy levels are included by way of the recombination\ncoefficient, which can effectively include recombination to hundreds of levels.\nThe following reactions are included:\n\\begin{eqnarray}\n{\\rm H}_{\\rm n=2,\\ell=2s} + \\gamma &\\longleftrightarrow&\n {\\rm e}{^-} + {\\rm H}{^+} \\nonumber\\\\\n{\\rm H}_{\\rm n=1} + \\gamma &\\longleftrightarrow&\n {\\rm H}_{\\rm n=2,\\ell=2p} \\nonumber\\\\\n{\\rm H}_{\\rm n=1} + 2\\gamma &\\longleftrightarrow&\n {\\rm H}_{\\rm n=2,\\ell=2s}.\\nonumber\n\\end{eqnarray}\n\nAs described in Peebles (1993), we omit the recombinations and\nphotoionizations to the ground state because any recombination directly\nto the ground state will emit a photon with energy $>13.6\\,$eV, where\nthere are few blackbody photons, and this will immediately re-ionize a\nneighboring H atom.\nWe include the two-photon rate from the $2s$ state with the rate\n$\\Lambda_{2s{-}1s} = 8.22458\\,$s$^{-1}$ (Goldman 1989).\nThe most accurate total Case B recombination coefficient is\nby Hummer (1994) and is fitted by the function\n\\begin{equation}\n\\alpha_{\\rm B} = 10^{-13}\\frac{at^{b}}{1 + ct^{d}} \\, \\mathrm{cm^{3}s^{-1}},\n\\end{equation}\nwhere $a=4.309$, $b=-0.6166$, $c=0.6703$, $d=0.5300$ and\n$t= T_{\\rm M}\/10^{4}\\,$K (P{\\'e}quignot et al.~1991; see also Verner\n\\& Ferland~1996).\n\nConsideration of detailed balance in the effective 3-level atom leads to\na single ordinary differential equation for the ionization fraction:\n\\begin{eqnarray}\n\\lefteqn{{dx_{\\rm e}\\over dz} =\n \\frac{\\big[x_{\\rm e}^2 n_{\\rm H} \\alpha_{\\rm B}\n - \\beta_{\\rm B} (1-x_{\\rm e})\n {\\rm e}^{-h_{\\rm P}\\nu_{2s}\/k_{\\rm B}T_{\\rm M}}\\big]} {H(z)(1+z)}\\,\\times}\n \\nonumber \\\\\n & & \\frac{\\big[1 + K \\Lambda_{2s{-}1s} n_{\\rm H}(1-x_{\\rm e})\\big]}\n {\\big[1+K \\Lambda_{2s{-}1s} n_{\\rm H} (1-x_{\\rm e})\n + K \\beta_{\\rm B} n_{\\rm H}(1-x_{\\rm e})\\big]}\n\\label{eq:standard_xe}\n\\end{eqnarray}\n(see e.g.~Peebles~1968; extra terms included in Jones \\& Wyse 1985,\nfor example, are negligible). Here $x_{\\rm e}$ is the residual\nionization fraction, that is the\nnumber of electrons compared to the total number of hydrogen nuclei\n($n_{\\rm H}$). Here the Case B recombination coefficient\n$\\alpha_{\\rm B}=\\alpha_{\\rm B}(T_{\\rm M})$, the total photoionization rate\n$\\beta_{\\rm B}=\\alpha_{\\rm B} (2\\pi m_{\\rm e} k_{\\rm B}\nT_{\\rm M}\/h_{\\rm P}^2)^{3\/2} \\exp(-E_{2s}\/k_{\\rm B}T_{\\rm M})$ as described in\n\\S\\ref{sec-StandardRecomb}, $\\nu_{2s}$ is the frequency of the $2s$ level from\nthe ground state,\nand the redshifting rate $K\\equiv \\lambda_\\alpha^3\/(8\\pi H(z))$, where\n$\\lambda_\\alpha$ is the Ly$\\,\\alpha$ rest wavelength.\nNote that $T_{\\rm M}$ is used in equation~(\\ref{eq:standard_xe}) and\nin $\\beta_{\\rm B}$, because the temperature terms come from detailed balance\nderivations that use Boltzmann and Saha equilibrium distributions, which are\ncollisional descriptions.\nIn the past, this equation has been solved (for $x_{\\rm e}(z)$)\nsimultaneously with\na form of equation~(\\ref{eq:cooling}) containing only adiabatic and\nCompton cooling. We refer to the approach of\nequation~(\\ref{eq:standard_xe}) as the `standard calculation.'\n\nFor the comparison test with the standard recombination calculation,\nwe also use an effective recombination coefficient, but three equations to\ndescribe the three reactions listed above. That is, we\nsimplified equation~(\\ref{eq:sum1}) to three equations, one for the\nground state population ($n_1$), one for the first excited state\npopulation ($n_2$), and one for the\nelectrons (for H recombination $n_e = n_p$).\n\\begin{equation}\n(1+z)\\frac{dn_{1}(z)}{dz} =\n-\\frac{1}{H(z)}\n\\left[\\Delta R_{2p{-}1s} + \\Delta R_{2s{-}1s}\\right] + 3 n_{1}\n\\end{equation}\n\\vspace{-3mm}\n\\begin{eqnarray}\n(1+z)\\frac{dn_{2}(z)}{dz} =& \\!\\!\\!\\!\\mbox{}-{\\displaystyle\\frac{1}{H(z)}}\n\\Big[(n_{\\rm e}(z) n_{\\rm p}(z) \\alpha_{\\rm B}\n - n_{2s}(z) \\beta_{\\rm B} \\nonumber \\\\\n & \\ \\mbox{} - \\Delta R_{2p{-}1s} - \\Delta R_{2s{-}1s}\\Big] + 3 n_2\n\\end{eqnarray}\n\\vspace{-3mm}\n\\begin{equation}\n(1+z)\\frac{dn_{\\rm e}(z)}{dz} =\n-\\frac{1}{H(z)}\n\\big[(n_{2s}(z) \\beta_{\\rm B} - n_{\\rm e}(z) n_{\\rm p}(z) \\alpha_{\\rm B}\n\\big] + 3 n_{\\rm e}\n\\end{equation}\nThe remaining\nphysical difference between our effective 3-level atom approach and that of\nthe standard calculation is the treatment of the redshifting of H Ly$\\,\\alpha$\nphotons (included in the $\\Delta R_{2p{-}1s}$ terms). In our calculation the redshifting is accounted for by \nthe Sobolev escape probability (see~\\S\\ref{sec-Sobolev}).\nFollowing Peebles (1968, 1993), the standard calculation accounts for the\nredshifting by\napproximating the intensity distribution as a step, and in effect takes the\nratio of the redshifting of the photons through the line to the expansion\nscale that produces the same amount of redshifting.\nIt can be shown that Peebles' step method considered as an escape\nprobability scales as $1\/ \\tau_{\\rm S}$, where $\\tau_{\\rm S}$ is the\nSobolev optical depth.\nFor high Sobolev optical depth, which holds for H Ly$\\,\\alpha$ during\nrecombination for any cosmological model (see\nFig.~7),\nthe Sobolev escape probability also scales as $1\/\\tau_{\\rm S}$:\n\\begin{equation} \n \\lim_{\\tau_{\\rm S} \\gg 1} p_{ij} = \\lim_{\\tau_{\\rm S} \\gg 1}\n \\frac{1}{\\tau_{\\rm S}}(1 - {\\rm e}^{-\\tau_{\\rm S}}) = \\frac{1}{\\tau_{\\rm S}}.\n\\end{equation}\nTherefore the two approximations are equivalent for Ly$\\,\\alpha$,\nalthough we would expect differences for lines with $\\tau_{\\rm S}\\lesssim1$,\nwhere $p\\to1$.\nBecause we treat recombination in the same way as Peebles, \nno individual\ntreatment of other lines is permitted, and therefore there are no other\ndifferences between the two calculations for this simple case. \nNote that with Peebles' step method to compute $\\Delta R_{2p{-}1s}$,\nand the assumption that $n_1 = n_{\\rm H} - n_{\\rm p}$, the above equations will\nreduce to the single ODE equation~(\\ref{eq:standard_xe}).\n\nThe results from our effective 3-level recombination calculation are\nshown in\nFig.~1,\nplotted along with values from a\nseparate code as used in HSSW, which represents the standard\nrecombination calculation updated with the most recent parameters.\nThe resulting ionization fractions are equal, which shows that our new approach\ngives exactly the standard result when reduced to an effective 3-level atom.\nTwo other results are plotted for comparison, namely values of $x_{\\rm e}$\ntaken from\nPeebles (1968) and Jones \\& Wyse (1985). Their differences can be largely\naccounted for by\nthe use of an inaccurate recombination coefficient with\n$\\alpha_{\\rm B}(T_{\\rm M}) \\propto T_{\\rm M}^{-1\/2}$.\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig1.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~1.---}\nComparison of effective three-level hydrogen recombination for\nthe parameters\n$\\Omega_{\\rm tot} = 1.0, \\Omega_{\\rm B} = 1.0, h=1.0$.\nNote that the Jones \\& Wyse (1985) and Peebles (1968) curves overlap as does\nour curve with the HSSW one.\n}\n\\label{fig:Hrec3level}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nAs an aside, we note the behavior for $z\\lesssim50$ in our curve and the HSSW\none. This is caused by inaccuracy in the recombination coefficient for very\nlow temperatures. The down-turn is entirely artificial and could be removed\nby using an expression for $\\alpha_{\\rm B}(T)$ which is more physical\nat small temperatures. The results of our detailed calculations are\nnot believable\nat these redshifts either, since accurate modeling becomes increasingly\ndifficult due to numerical precision as $T$ approaches zero.\nBut in any case the optical\ndepth back to such redshifts is negligible, and the real Universe is\nreionized at a similar epoch (between $z=5$ and 50 certainly).\n\n\\subsection{Multi-level Hydrogen Atom}\n\\label{sec-Multi}\nThe purpose of a multi-level hydrogen atom is to improve the\nrecombination calculation, by following the population of each atomic\nenergy level with redshift and by including all bound-bound and bound-free\ntransitions. This includes recombination to, and photoionization from,\nall levels {\\it directly\\\/} as a function of time,\nin place of a parameterized recombination and photoionization coefficient. The individual\ntreatment of all levels in a coupled manner allows for the\ndevelopment of departures from equilibrium among the states with time,\nand feedback on the rate of recombination. Since the accuracy of the\nrecombination coefficient is probably the single most important effect\nin obtaining accurate power spectra (HSSW), it makes sense to\nfollow the level populations as accurately as possible.\n\nIn the multi-level H atom recombination calculation, we do not consider\nindividual $\\ell$\nstates (with the exception of $2s$ and $2p$), but assume the $\\ell$ sublevels have populations proportional to\n(2$\\ell$ + 1). The $\\ell$ sublevels only deviate from this distribution\nin extreme non-equilibrium conditions (such as planetary nebulae).\nIn their H recombination calculation, Dell'Antonio \\& Rybicki (1993)\nlooked for such $\\ell$ level deviations for n$\\,{\\leq}\\,10$ and found none.\nFor n$\\,{>}\\,10$, the $\\ell$ states are even less likely to differ from an\nequilibrium distribution, because the energy gaps\nbetween the $\\ell$ sublevels are increasingly smaller as n increases.\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig2.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~2.---}\nMulti-level hydrogen recombination for the standard CDM\nparameters $\\Omega_{\\rm tot} = 1.0, \\Omega_{\\rm B} = 0.05, h = 0.5$\n(top), and for $\\Omega_{\\rm tot} = 1.0, \\Omega_{\\rm B} = 1.0, h = 1.0$\n(bottom), both with \n$Y_{\\rm P} = 0.24, T_0 = 2.728\\,$K. The `effective 3-level' calculation\nis essentially the same as in HSSW, and uses a recombination coefficient\nwhich attempts to account for the net effect of all relevant levels. We find\nthat we require a model which considers close to 300 levels for full\naccuracy. Note also that although we plot all the way to $z=0$, we know\nthat the Universe becomes reionized at $z>5$, and that our calculations\n(due to numerical precision at low $T$) become unreliable\nfor $z\\lesssim50$ in the upper two curves, and for $z\\lesssim20$ in\nthe other curves.\n\\label{fig:HrecMulti}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsubsection{Results From a Multi-level H Atom}\n\\label{sec-resultsmulti}\nFig.~2\nshows the ionization fraction $x_{\\rm e}$ from\nrecombination of a 2-, 10-, 50-, 100- and 300-level H atom, compared with\nthe standard effective 3-level results. The $x_{\\rm e}$ converges\nfor the highest n-level atom calculations. The effective\n3-level atom actually includes about 800 energy levels via the recombination\ncoefficient (e.g.~Hummer 1994).\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig3.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~3.---}\nEnergy separations between various hydrogen atomic levels and the\ncontinuum. The solid curve shows the energy width of the same energy\nlevels due to thermal\nbroadening. Thermal broadening was calculated using\n$\\nu (2 k_{\\rm B} T_{\\rm M} \/ m_{\\rm H} c^2)^{1\/2}$. The\nfrequency of the highest atomic energy level (n=300, with an energy\nfrom the ground state of 109676.547 cm$^{-1}$; the continuum energy\nlevel is 109677.766 cm$^{-1}$) was used for $\\nu$, but a\nthermal broadening value for any atomic energy level would overlap on\nthis graph.\n\\label{fig:thermbroad}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nFig.~2\nshows that the more levels that are included in the\nhydrogen atom, the lower\nthe residual $x_{\\rm e}$. The simple explanation is that the\nprobability for electron capture increases with more energy levels per\natom. Once captured, the electron can cascade\ndownwards before being reionized. Together this means \nadding more higher energy levels per atom\nincreases the rate of recombination.\nEventually $x_{\\rm e}$\nconverges as the atom becomes complete in terms of electron energy\nlevels, i.e.~when there is no gap between the highest energy level and\nthe continuum (see\nFig.~3).\nUltimately the uppermost levels\nwill have gaps to the continuum which are smaller than the thermal\nbroadening of those levels, and so\nenergy levels higher than about n=300 do not need to be considered,\nexcept perhaps at the very lowest redshifts.\n \nFor other reasons entirely, our complete (300-level) H atom recombination\ncalculation gives an $x_{\\rm e}$ lower than that of the\neffective 3-level atom calculation. The faster production of\nhydrogen atoms is due to non-equilibrium processes in the excited states\nof H, made obvious by our new, level-by-level treatment of recombination. The\ndetails are described in \\S\\ref{sec-noneq} below.\n\n\\subsubsection{Faster H Recombination in our Level-by-level\nRecombination Calculation}\n\\label{sec-noneq}\n\nThe lower $x_{\\rm e}$ in our calculation compared to the standard calculation\nis caused by the strong but cool radiation field. Specifically, \nboth a faster downward cascade rate and a lower total photoionization rate\ncontribute to a faster net recombination rate.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig4.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~4.---}\nHow the bound-bound rates of large energy separation\n(e.g. n=70 to n=20) go out of\nequilibrium at low $T$, illustrated using an upper level of\nn=70 for definiteness. The case of equilibrium corresponds to net\nbound-bound rates of zero. Each upward and downward bound-bound rate for\na given transition is represented by the same curve; non-equilibrium occurs\nwhere a single curve separates into two as redshift decreases.\nThe rates shown are for the sCDM model.\n\\label{fig:bbrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nBy following the population of each\natomic energy level with redshift, we relax the assumption used in the standard\ncalculation that the excited states are in equilibrium. In addition, we\ncalculate all bound-bound rates which control equilibrium among\nthe bound states.\nIn the standard calculation, equilibrium among the excited states\nn~$\\geq 2$ is\nassumed, meaning that the net bound-bound rates are\nzero.\nFig.~4\nshows that the net bound-bound\nrates are actually different from zero at $z\\lesssim1000$.\nThe reason for this is that at low temperatures, the strong but cool\nradiation field means that high energy transitions are rare due to few high\nenergy photons. More specifically,\nphotoexcitation and stimulated photodeexcitation for high energy transitions\nbecome rare (e.g.~70--10, 50--4 etc.). \nIn this case\nspontaneous deexcitation dominates, causing a\nfaster downward cascade to the n=2 state. \nIn addition, the faster downward cascade rate is faster than the\nphotoionization rate from the upper state, and one might view this as\nradiative decay stealing some of the depopulation `flux' from\nphotoionization. Both the faster downward cascade and the lower\nphotoionization rate contribute to the faster net recombination rate.\n\nThe cool radiation field is strong, so photoexcitations and \nphotodeexcitations are rapid among nearby energy\nlevels (e.g.~70--65, etc.;\nsee Fig.~4). What we see in Fig.~4 is that with time, after $z\\lesssim1000$,\nthe n=70 energy level becomes progressively decoupled from the distant\nlower energy levels (n=2,3,...20,...), but remains tightly coupled to its\nnearby `neighbors' (n=60,65, etc.). This explains the departures from an\nequilibrium Boltzmann distribution (in the excited states) as seen in the shape of the curves in\nFig.~5.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig5a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig5b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~5.---}\nRatio of the actual number densities of the excited\nstates ($n_i$) and\nnumber densities for a Boltzmann distribution of excited states\n($n_i^*$) at\ndifferent redshifts for recombination within the standard CDM model.\nSee section~\\ref{sec-noneq} for details. We give two different plots with\nlinear and logarithmic y-axes and different redshift values, to show\na wider range of out-of-equilibrium conditions. The ratio approaches\na constant for high n at a given $z$, because the high energy (Rydberg)\nstates are all very close in energy, and thus have similar behavior\n(i.e. remain coupled to the radiation field and each other).\n\\label{fig:Boltz}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nFig.~5\nillustrates the non-equilibrium of the excited\nstates by showing the ratio of populations of the excited states\ncompared to a Boltzmann equilibrium distribution with respect to n=2. \nWe find that the\nupper levels of the hydrogen atom are not in thermal equilibrium with\nthe radiation, i.e.~the excited levels are not populated according to\na Boltzmann distribution. The excited states are in fact overpopulated\nrelative to a\nBoltzmann distribution. This is not a surprise for the population of the\nn=2 state, which is strongly overpopulated compared to the n=1 ground\nstate, and so should be all n~$\\geq 2$ states because all Lyman lines remain\noptically thick during recombination. What {\\em is\\\/} surprising is that\nall excited states develop a further overpopulation {\\em with respect to\\\/}\nn=2 and each other. Note that this is not a population inversion.\nThe recombination rate to a given high level is faster than the downward\ncascade rate, and this causes a `bottleneck' creating the\noverpopulation.\nFig.~5\nshows that all states\nare in equilibrium at high redshifts, with the highest states going\nout of equilibrium first, followed by lower and lower states as the\nredshift decreases. The factor by which the excited states are\nover-populated approaches a constant at high n for a given redshift,\nwith this factor increasing as $z$ decreases. The ratio is constant\nbecause the high energy level Rydberg states have very similar energy\nlevels to each other, with a relatively large energy separation from the n=2\nstate (i.e.~the exponential term in equation~(\\ref{eq:boltz}) dominates\nover the $g_i$ ratios, and the exponential term is similar for all of\nthe Rydberg states.) \n\nFig.~5\nalso shows an enormous ratio at low\nredshift ($z < 500$) for number densities\nof the actual excited states to the number densities of a Boltzmann\ndistribution of excited states, on the order of $10^6$. At such a low\nredshift, there are almost no electrons in\nthe excited states ($\\sim 10^{-20}$ cm$^{-3}$), and so\nunlike at higher redshifts, the ratio is only an\nillustration of the strong departure from an equilibrium distribution;\nthe actual populations are very low in any case.\n\nIn comparison with the standard\nequilibrium capture-cascade calculation for $\\alpha_{\\rm B}$, the unusual\nsituation described above (caused by the strong but cool radiation field)\nleads to higher effective recombination\nrates for the majority of excited states without increasing photoionization\nproportionally. This results in a higher net rate of production of neutral\nhydrogen atoms, i.e.~a lower $x_{\\rm e}$.\n\n\\subsubsection{Accurate Recombination vs Recombination Coefficient}\nTo demonstrate why the non-equilibrium in the excited states of H affects\nthe recombination rate, we must consider the\ndifference in our new treatment of recombination compared to the\nstandard treatment.\nAn important new benefit of our level-by-level calculation\nlies in replacing the recombination coefficient with\na direct calculation of recombination to and photoionization from each level at each redshift step.\nIn other words, we calculate the recombination rate and the\nphotoionization rate using\nindividual level populations and parameters of the excited states $i$,\n\\begin{eqnarray}\n\\lefteqn{\\sum_{i=1}^N n_{\\rm e} n_{\\rm p} R_{{\\rm c}i}\n = n_{\\rm e} n_{\\rm p} \\sum_{i=1}^N\n {\\left(\\frac{n_{i}}{n_{\\rm e} n_{\\rm p}}\\right)}^{\\rm LTE}\\times}\\nonumber \\\\\n& \\!\\!{\\displaystyle 4\\pi\\!\\int_{\\nu_{i}}^{\\infty}\n \\frac{\\alpha_{i}(\\nu)}{h_{\\rm P}\\nu}\n \\left[\\frac{2h_{\\rm P}\\nu^{3}}{c^{2}} + B(\\nu,T_{\\rm R})\\right]\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu},\n\\label{eq:feedback2}\n\\end{eqnarray}\nand similarly for the photoionization rate,\n\\begin{equation}\n\\label{eq:feedback}\n\\sum_{i=1}^N n_{i}R_{i{\\rm c}} = \\sum_{i=1}^N\n n_{i}4\\pi\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\alpha_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T_{\\rm R})d\\nu.\n\\end{equation}\nFor the standard recombination calculation, the recombination rate is\n\\begin{equation}\n\\label{eq:standardrate}\n\\sum_{i=1}^N n_{\\rm e} n_{\\rm p} R_{{\\rm c}i} = n_{\\rm e} n_{\\rm p}\n \\alpha_{\\rm B}(T_{\\rm M}),\n\\end{equation}\nand the photoionization rate is\n\\begin{eqnarray}\n\\lefteqn{\\Sigma_i n_i R_{i{\\rm c}} \\equiv n_{2s} \\beta_{\\rm B}(T_{\\rm M})}\n \\nonumber \\\\\n& = n_{2s} \\alpha_{\\rm B}(T_{\\rm M}) {\\rm e}^{-E_{2s}\/k_{\\rm B}T_{\\rm M}}\n \\left(2\\pi m_{\\rm e} k_{\\rm B} T_{\\rm M} \\right)^{3\/2}\/h_{\\rm P}^3.\n\\label{eq:nofeedback}\n\\end{eqnarray}\nIn this last equation, the excited state populations are hidden by the\nBoltzmann relation with $n_{2s}$ (see~\\S\\ref{sec-StandardRecomb}).\nThe important point here is that our method allows redistribution of the\nH level populations over all 300 levels at each redshift step, which feeds\nback on the recombination process via\nequation~(\\ref{eq:feedback}), and leads to the lower $x_{\\rm e}$ shown in\nFig.~2.\nThis\nredistribution of the level populations is not possible in the standard \ncalculation's equation~(\\ref{eq:nofeedback}) which only considers the\npopulations $n_{\\rm e}$, $n_{1}$, and $n_{2s}$, and considers the\nexcited level populations $n\\,{>}\\,2s$\nto be proportional to $n_{2s}$ in an equilibrium distribution.\n\nA small improvement in our new recombination treatment over\nthe standard treatment is in our distinguishing the various temperature\ndependencies of recombination. Photoionization and\nstimulated recombination are radiative, so they should depend on\n$T_{\\rm R}$. Spontaneous recombination is collisional and depends on\n$T_{\\rm M}$ (see~\\S\\ref{sec-Recomb}). In the standard calculation, the radiative nature of \nrecombination and photoionization is\noverlooked because both the recombination\ncoefficient and photoionization coefficient are a function $T_{\\rm M}$ only\n(equations~(\\ref{eq:standardrate}) and (\\ref{eq:nofeedback})).\nAlthough adiabatic cooling (equation~(\\ref{eq:adiabaticcooling}))\ndoes not dominate until quite low redshifts ($z\\lesssim100$), it still\ncontributes partially to matter cooling\nthroughout recombination. The resulting difference between\n$T_{\\rm M}$ and\n$T_{\\rm R}$ in the net recombination rate affects $x_{\\rm e}$ at the\nfew percent level at $z\\lesssim300$ for the popular cosmologies,\nand has an even larger effect for high $\\Omega_{\\rm B}$ models.\n\n\\subsubsection{Collisions}\nThe standard recombination calculation omits collisional excitation\nand ionization because at the relevant temperatures and densities they\nare negligible for a two-level hydrogen\natom (Matsuda et al.~1971).\nWe have found that the collisional processes are also not important for \nthe higher levels, even though those electrons are bound with little\nenergy. In high $\\Omega_{\\rm B}$ models collisional ionization and\ncollisional recombination rates for \nthe highest energy levels are of the same order of\nmagnitude as the photoionization and recombination rates, though not\ngreater than them (see\nFig.~6).\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig6a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig6b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~6.---}\nComparison of ionization rates. The upper curves are\nphotoionization rates, the lower curves are collisional ionization rates.\nThe left panel shows a model with the standard CDM parameters, while\nthe right panel shows an extreme baryon cosmology.\n\\label{fig:Collrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsubsection{Departures From Case B}\n\\label{sec-CaseB}\nCase B recombination excludes recombination to the ground state and considers\nthe Lyman lines to be optically thick (i.e.~photons associated with all\npermitted radiative transitions to\nn=1 are assumed to be instantly reabsorbed).\nAn implied assumption necessary to compute the\nphotoionization rate is that the excited states (n~$\\ge 2$) are in\nequilibrium with the radiation.\nUnlike the standard recombination calculation, our method allows departures\nfrom Case B because the Lyman lines are\ntreated by the Sobolev escape probability method which is valid\nfor any optical thickness, and our\nmethod allows departures from equilibrium of the excited states\n(\\S\\ref{sec-resultsmulti}).\nWe find that the excited states depart from equilibrium at redshifts\n$\\lesssim800$, so Case B does not hold then.\nHowever, our calculations show that for hydrogen all Lyman lines\nare indeed\noptically thick during all of hydrogen recombination, so\nCase B holds for H recombination above redshifts $\\simeq 800$.\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig7.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~7.---}\nSobolev optical depth ($\\tau_{\\rm S}$) for the H Lyman lines for\ntwo different\ncosmological models. From upper to lower the curves represent the\noptical depth in the Lyman transitions from n=2 (i.e.~Ly$\\,\\alpha$), n=10,\nn=50, n=100, and n=270. The curves show that all the Lyman transitions are\noptically thick during H recombination ($z\\lesssim2000$)\nbut some are optically thin ($\\tau_{\\rm S} < 1$)\nearlier, during He recombination ($z \\gtrsim$ 2000).\n\\label{fig:SobolevTau}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nFig.~7\nshows that the Lyman lines are not\noptically thick\nat earlier times, e.g.~during helium recombination, where we\nfind some optically thin H\nLyman lines. The Sobolev escape probability treats this consistently,\nwhich is necessary because we evolve H, \\ion{He}{1}, and \\ion{He}{2}\nsimultaneously.\nwhich is\nnecessary because we evolve H, \\ion{He}{1}, and \\ion{He}{2} simultaneously.\n\n\\subsubsection{Other Recent Studies}\n\nThe previous study that was closest in approach to our own was that of\nDell'Antonio \\& Rybicki (1993), who calculated recombination\nfor a ten-level hydrogen atom in order to estimate the spectral distortions\nto the CMB blackbody radiation spectrum. Ten levels are insufficient to\ncalculate recombination accurately, because the higher energy levels of the\natom are completely ignored (see\nFig.~3).\nHowever the accuracy of the ionization\nfraction ($x_{\\rm e}$) was sufficient to determine the magnitude\nof the spectral distortions. Their recombination model treated individual\nlevels, but used a recombination coefficient to each level of the form\n${T_{\\rm M}}^{-1\/2}$. Because the form of the recombination\ncoefficient dominates the H recombination process, our models are not\nequivalent, and so there is little use in comparing the results.\n\nMore recently, Boschan \\& Biltzinger (1998) derived a new\nparameterized recombination coefficient to solve the recombination\nequation of the standard calculation, and to\ngenerate spectral distortions in the CMB. \nTheir calculation differs from ours in that their\nrecombination coefficient is pre-calculated. Hence it is not an interactive\npart of the calculation, and does not allow the advantages that our\ncalculation does, mainly the feedback of the non-equilibrium in the excited\nstates on the net recombination rate. While they include pressure\nbroadening for a cutoff in the partition function, they neglect\nthermal broadening. A more serious problem is their method\nof inclusion of stimulated recombination, as originally suggested by\nSasaki \\& Takahara (1993), who included stimulated recombination as\npositive recombination instead of negative ionization.\nThe physics (as described in \\S\\ref{sec-Recomb}) and our\ncomputational results are the same regardless of whether stimulated\nrecombination is treated as positive recombination or negative ionization.\nHowever, this may not be the case computationally for the standard\ncalculation, if it is\nnot treated with care. We defer a full discussion of these matters to a\nseparate paper (Seager \\& Sasselov, in preparation).\n\nWe have also investigated how we can approximate our calculations, so\nthat other researchers can obtain approximately accurate results without\nthe need to follow 300 levels in a hydrogen atom. Because the net effect \nof our new H calculation is a faster recombination (a lower freeze-out\nionization fraction), our results can be reproduced by artificially\nspeeding up recombination in the standard calculation. Further\ndetails are described in Seager, Sasselov, \\& Scott (1999).\n\n\\subsection{Helium}\n\\label{sec-Helium}\nWe compute helium and hydrogen recombination simultaneously.\nThe recombination of \\ion{He}{3} into \\ion{He}{2} and \\ion{He}{2} \ninto \\ion{He}{1}\nis calculated in much the same way as hydrogen, with recombination,\nphotoionization,\nredshifting of the n$^1p$--$1^1s$ lines (in H these are the Lyman\nlines), inclusion of the $2^1s$--$1^1s$ two-photon rates, collisional\nexcitation, collisional deexcitation, collisional ionization, and\ncollisional recombination, as described in \n\\S\\ref{sec-Recomb}--\\ref{sec-Sobolev}. The multi-level \\ion{He}{1} atom\nincludes the first 4 angular momentum states up to the level n=20, above which\nonly the principal quantum number energy levels and transitions are used.\nFig.~8\n(which shows the levels up to n=4 only)\nindicates how much more complicated the\n\\ion{He}{1} atom is compared with H or the hydrogenic \\ion{He}{2}.\nOur multi-level \\ion{He}{2} atom includes the first 4 angular momentum states\nup to the level n=4, above which only the principal quantum number energy\nlevels and transitions are used.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig8.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~8.---}\nGrotrian diagram for \\ion{He}{1}, showing the states with\nn~$\\le4$ and the continuum. In practice our model atom explicitly\ncontains the first 4 angular momentum states up to n=20, and\n120 principal quantum number energy levels beyond.\n\\label{fig:Hegrotrian}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nPhotoionizations from\nany \\ion{He}{1} excited state are allowed only into the ground state\nof \\ion{He}{2}, because there are few photons energetic enough ($>40\\,$eV)\nto do more than that. Two electron transitions in \\ion{He}{1} are\nnegligible at recombination era temperatures.\n\nCosmological helium recombination was discussed explicitly in\nMatsuda et al.~(1969, 1971, hereafter MST), and Sato, Matsuda \\& Takeda (1971),\nand to a lesser extent in Lyubarsky \\& Sunyaev (1983),\nwhile several\nother papers give results, but no details (e.g.~Lepp \\& Shull~1984;\nFahr \\& Loch~1991; Galli \\& Palla~1998).\nThe main improvement in our calculation over previous treatments of helium is\nthat we use a multi-level \\ion{He}{2} atom, a multi-level \\ion{He}{1}\natom with triplets and singlets treated correctly, and evolve the\npopulation of each energy level with redshift by including all\nbound-bound and bound-free transitions. This is not possible for the\nstandard recombination calculation method\n(equation~(\\ref{eq:standard_xe})) extended to \\ion{He}{1}, using an\neffective three-level \\ion{He}{1} atom with only a singlet ground\nstate, singlet first excited state and continuum.\n\n\\subsubsection{Results From \\ion{He}{1} Recombination}\n\nFig.~9,\nshows the ionization fraction $x_{\\rm e}$\nthrough \\ion{He}{2}, \\ion{He}{1} and H recombination, plotted against the\nstandard H calculation that includes \\ion{He}{2} and \\ion{He}{1}\nrecombination via the Saha equation. For completeness we give the\nhelium Saha equations here: \\\\\nfor \\ion{He}{1}$\\,\\leftrightarrow\\,$\\ion{He}{2}\n\\begin{equation}\n\\label{eq:heonesaha}\n{(x_{\\rm e}-1)x_{\\rm e}\\over 1+f_{\\rm He}-x_{\\rm e}} = \n 4 {(2\\pi m_{\\rm e} k_{\\rm B} T)^{3\/2}\\over h_{\\rm P}^3 n_{\\rm H}}\n {\\rm e}^{-\\chi_{\\rm He I}\/k_{\\rm B}T},\n\\end{equation}\nand for \\ion{He}{2}$\\,\\leftrightarrow\\,$\\ion{He}{3}\n\\begin{equation}\n\\label{eq:hetwosaha}\n{(x_{\\rm e}-1-f_{\\rm He})x_{\\rm e}\\over 1+2f_{\\rm He}-x_{\\rm e}} = \n {(2\\pi m_{\\rm e} k_{\\rm B} T)^{3\/2}\\over h_{\\rm P}^3 n_{\\rm H}}\n {\\rm e}^{-\\chi_{\\rm He II}\/k_{\\rm B}T}.\n\\end{equation}\nHere the $\\chi$s are ionization potentials, $n_{\\rm H}$ is the total\nnumber density of hydrogen, $f_{\\rm He}$ is the total number fraction of\nhelium to hydrogen $f_{\\rm He}=n_{\\rm He}\/n_{\\rm H}=Y_{\\rm\nP}\/4(1-Y_{\\rm P})$, and our definition of $x_{\\rm e}\\equiv n_{\\rm e}\/n_{\\rm H}$\nresults in the complicated-looking left hand sides. The extra factor of\n4 on the right hand side for \\ion{He}{1}$\\,\\leftrightarrow\\,$\\ion{He}{2}\narises from the statistical weights factor. \n \n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig9.eps}}}}\n\\baselineskip=8pt\n\\vspace{6mm}\n{\\footnotesize {\\sc Fig.~9.---}\nHelium and hydrogen recombination for two cosmological models with\n$Y_{\\rm P} = 0.24$ and $T_0 = 2.728\\,$K. The first step from\nright to left is recombination of \\ion{He}{3} to \\ion{He}{2}, the second\nstep is \\ion{He}{2} to \\ion{He}{1}, and the third step is H recombination.\n\\label{fig:Herec}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig10.eps}}}}\n\\baselineskip=8pt\n\\vspace{3mm}\n{\\footnotesize {\\sc Fig.~10.---}\nDetails of helium recombination for the standard CDM\ncosmology (top figure) and the high $\\Omega_{\\rm B}$ cosmology (bottom\nfigure). The dashed lines show our new results, and the dotted lines \nshow the results assuming the `standard calculation' (equivalent to\nSaha equilibrium).\n\\label{fig:Herecdetails}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nWhile our improved $x_{\\rm e}$ agrees fairly closely with Saha recombination\nfor \\ion{He}{2} (see~\\S\\ref{sec-HeII}), the \ndifference in $x_{\\rm e}$ from Saha recombination during \\ion{He}{1}\nrecombination is dramatic.\nOur new detailed treatment of \\ion{He}{1} shows \\ion{He}{1}\nrecombination finishing just after the start of H recombination (see\nFig.~9),\ni.e.~significantly delayed compared with the\nSaha equilibrium case. This\nis different from the earlier calculations (e.g.~MST), in which\n\\ion{He}{1} recombination is\nfinished well before H recombination begins. In this previous case,\n\\ion{He}{1} recombination still affected the CMB anisotropy power\nspectrum on small angular scales because the diffusion damping length\ngrows continuously and is\nsensitive to the full thermal history (HSSW). In our new case,\nparticularly for our low\n$\\Omega_{\\rm B}$ models \\ion{He}{1} recombination is still finishing at the\nvery beginning of H recombination, which further affects the power spectrum\nat large angular scales (see~\\S\\ref{sec-Powerspectrum}).\nWe show a `blow-up' of the two helium recombination epochs in\nFig.~10.\n\n\\subsubsection{Physics of \\ion{He}{1} Recombination}\nThe physics of \\ion{He}{1} recombination can be summarized as follows. There\nare three major aspects to it: (1) the \\ion{He}{1} has excited states which\nare able to retain charge; but (2) being very close to the continuum, the\nhighly excited states are easily photoionized by the radiation field at\n$z \\simeq 3000$; then (3) we have a standard hydrogenic-like Case B\nrecombination, which is unaffected by neutral H removing \\ion{He}{1}\n$2^1p$--$1^1s$ (resonance line) photons.\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig11.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~11.---}:\nThis figure shows why the Saha equilibrium recombination rate\n($R_{\\rm Saha}$) for\n\\ion{He}{1} is not valid. Comparing the dotted and dashed lines,\nthe photoexcitation rate (i.e.~photoabsorption rate) for He 2$^1p\\rm{-}1^1s$\n($R_{\\rm He}$) is orders\nof magnitude greater than the photoionization rate for H ($R_{\\rm H}$)\nfrom the same \\ion{He}{1} 2$^1p \\rm{-} 1^1s$\nphoton pool; there is no possibility for H to `steal' the\nphotons to speed up \\ion{He}{1} recombination. For\nSaha recombination to be valid, $R_{\\rm H} > R_{\\rm He}$, as well as\n$R_{\\rm H} \\geq R_{\\rm Saha}$. For the sCDM model shown here, \\ion{He}{1}\nrecombination begins around $z=3000$.\n\\label{fig:fracnu}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nThe \\ion{He}{1} atom has a metastable, i.e.~very slow, set of\nstates -- the triplets (e.g.~n$^3p$--n$^1s$). Therefore, overall the\nexcited states of \\ion{He}{1} can naturally retain more charge than a simple\nhydrogenic system under Boltzmann equilibrium. The situation would resemble\nwhat we found for H recombination with the enhanced populations of the\nhigher states, and would lead to faster reduction of $x_{\\rm e}$. However, the\nhigh excited states of \\ion{He}{1} are much more strongly `packed' towards the\ncontinuum compared to those of H;\nthe energy difference between the $3p$ levels and the continuum is\n$1.6\\,$eV for \\ion{He}{1} versus $1.5\\,$eV for H, \ncompared to $24.6\\,$eV versus $13.6\\,$eV for the ground state--continuum energy difference.\nThis is\nenough to depopulate the triplets (whose `ground state' is n=$2^3s$),\ngiven the much higher radiation temperature during \\ion{He}{1} recombination.\nLeft on its own under these circumstances, \\ion{He}{1} would recombine much\nlike the standard Case~B effective 3-level H atom, i.e.~slower than Saha\nrecombination. There is one possible obstacle -- it is the existence of some\nneutral H, which could `steal' \\ion{He}{1} resonance line photons,\ninvalidate the effective Case~B and make it a Saha recombination instead.\nHowever, our detailed calculation \nshows that neutral H during \\ion{He}{1} recombination is not able to\naccomplish that, and the process is {\\it not\\\/} described by Saha equilibrium.\n\nFig.~11\nshows that Saha equilibrium recombination is\ninvalid for \\ion{He}{1}, by comparing the three possible destruction\nprocesses of the \\ion{He}{1} $2^1p$--$1^1s$ (in H this is Ly$\\,\\alpha$)\nphotons: (1) cosmological redshift; (2) the $2^1s$--$1^1s$ two-photon rate;\nand (3) the photoionization rate of the ground state of H by the same\n\\ion{He}{1} $2^1p$--$1^1s$ photons.\nFig.~11\nclearly shows that process (3) is negligible\n(in contradiction to the discussion in HSSW).\nTo be doubly sure that\nabsorption of these photons by hydrogen is negligible we explicitly\nincluded the relevant rate in our models and found no discernible effects.\nIn order for \\ion{He}{1} recombination to be approximated by Saha equilibrium,\none of the 3 processes described above would have to be \nfaster than or equal to the Saha equilibrium rate, which we do not find to be\nthe case.\n\nThe recombination of \\ion{He}{1} is slow for the same reasons that H\nrecombination is,\nnamely because of the optically thick n$^1p$--$1^1s$ transitions which\nslow cascades to the ground state, and the \nexclusion of recombinations to the ground state.\nIn other words \\ion{He}{1} follows a Case B recombination. Because the\n`bottleneck' at n=2 controls recombination, it is not surprising\nthat \\ion{He}{1} and H recombination occur at a similar redshift; the\nionization energy of n=2 is similar in both. \\ion{He}{1}\nrecombination is slower than H recombination because of its different\natomic structure. The excited states of \\ion{He}{1} are more tightly packed,\nand the $2^1p$--$1^1s$ energy difference greater than that of H.\nThe strong radiation field keeps the ratio of photoionization rate\/downward\ncascade rate higher than in the H case, resulting in a slower recombination.\n\nWe find the strong radiation field also causes the triplet states to\nbe virtually unpopulated. \nThe lack of electrons in triplet states is easily understood by\nconsidering the blackbody radiation spectrum.\nAt \\ion{He}{1} recombination (z $\\simeq$ 3000), the blackbody\nradiation\npeak is around $2\\,$eV, so there are around 11 orders of magnitude\nmore photons that can\nionize the lowest triplet state 2$^3$s ($4.8\\,$eV),\nthan the singlet ground state ($24.6\\,$eV), since both are on the\nsteeply decreasing Wien tail.\nIt is interesting to note that in planetary nebulae where the young,\nhot ionizing star produces most of its energy in the UV, the opposite\noccurs: the \\ion{He}{1} atoms have few electrons in the singlet\nstates; instead most of them are in the triplet states.\n \nThere is one more possible method to speed up \\ion{He}{1}\nrecombination, and that is collisional rates between the triplets and\nsinglet states. If fast enough, the collisional rates would provide\nanother channel to keep hold of captured electrons -- by pumping them into the\ntriplet states faster than they can be reionized. The triplets are 3\ntimes as populated as the singlets due to the statistical weight factors.\nBy forcing the collisional\nrates to be greater than the recombination rates and the bound-bound\nradiative rates, we find an extremely fast He I recombination --\napproximated by the Saha equilibrium. Essentially we force electrons\nfrom the singlets into the triplets faster than they can cascade\ndownwards, and faster than they can be photoionized out of the\ntriplets. In reality, the collisions are\nnegligible, a few orders of magnitude less than the radiative rates.\nIt is important to note that\napart from collisions, the singlet and triplet states are {\\it only\\\/}\nconnected via the n$^3p$--n$^1s$ transitions, which are orders of\nmagnitude slower than the $2^3s$--$1^1s$ rate.\nWe note here that MST stated that the collisional rates were high\nenough to cause equilibrium between the triplet and singlet states.\nOne must be careful\nto compare all relevant rates, and we keep all of them in our code.\nWe find the allowed radiative rates\n(e.g.~photoexcitation and photodeexcitation) are greater than the\ncollisional rates. Therefore the allowed radiative rates control the\nexcited states' population distribution, {\\it not\\\/} the\ncollisional rates. In other words, electrons in the singlet states\nare jumping between bound singlet states faster than\nthe collisional rates can send them into the triplet states.\n\n\\subsubsection{Effective 3-level calculation for \\ion{He}{1}}\n\nWe note here that MST used an effective 3-level \\ion{He}{1} singlet atom\nand calculated\n\\ion{He}{1} recombination in the same way as the standard H calculation\n(equation~(\\ref{eq:standard_xe})) \nwith the appropriate \\ion{He}{1} parameters. When we follow their\ntreatment, we get essentially the same result as our multi-level \\ion{He}{1} calculation.\nWe are not sure why MST obtained such a fast \\ion{He}{1} recombination. \n\nAs with hydrogen, we have also investigated what is required to\nachieve an accurate solution for helium, without modeling the full\nsuite of atomic processes. We have found that the use of the\n`effective 3-level' equations for helium (as described in MST),\ntogether with an appropriate recombination coefficient for singlets only\n(equation~(\\ref{eq:hecoefficient})), results in a very accurate\ntreatment of $x_{\\rm e}(z)$ during the time of helium recombination.\nIn detail it is necessary to follow hydrogen and\nhelium recombination simultaneously, increasing the number of\ndifferential equations to solve. However, little accuracy is in fact lost\nby treating them independently -- since recombination is governed by dramatic\nchanges in time scales through Boltzmann factors and the like, and is\naffected little by small changes in the number of free electrons at a given\ntime. Further details are discussed in Seager, Sasselov, \\& Scott (1999).\n\nAlthough our model does not explicitly use a recombination coefficient,\nit does allow us to calculate one easily. \nTo aid other researchers it\nis worth presenting a fit for the singlet-only Case~B recombination\ncoefficient for \\ion{He}{1} (including recombinations to all states except the ground state) from the data in Hummer \\& Storey (1998). \nHummer \\& Storey (1998) compute photoionization cross sections that are more\naccurate than the ones we use (Hofsaess 1979), but are not publicly available. Following the functional forms used in the\nfits of Verner \\& Ferland (1996) we find\n\\begin{equation}\n\\label{eq:hecoefficient}\n\\alpha_{\\rm He}=a\\!\\left[\\sqrt{T_{\\rm M}\\over T_2}\n \\left(1+\\sqrt{T_{\\rm M}\\over T_2}\\right)^{1-b}\\!\\!\n \\left(1+\\sqrt{T_{\\rm M}\\over T_1}\\right)^{1+b}\\right]^{-1}\\!\\!\\! m^3 s^{-1},\n\\end{equation}\nwith\n$a=10^{-16.744}$m$^3$s$^{-1}$, $b=0.711$, $T_1=10^{5.114}\\,$K, and $T_2$ fixed\narbitrarily at $3\\,$K. This fit is good to $<0.1\\%$\nover the relevant temperature range (4{,}000--10{,}000\\,K), and still\nfairly accurate over a much wider range of temperatures.\n\n\\subsubsection{\\ion{He}{2} Recombination}\n\\label{sec-HeII}\n\\ion{He}{2} recombination occurs too early to affect the power spectrum\nof CMB anisotropies. For completeness, we mention it briefly\nhere. \\ion{He}{2} recombination is fast because of the very fast two\nphoton rate.\nFig.~14\nshows that for most cosmologies the two-photon\nrate is faster than the net recombination rate, meaning that as fast\nas electrons are captured from the continuum they can cascade down to\nthe ground state. Because of this, there is essentially no `bottleneck' at the\nn=2 level. In high baryon models, \\ion{He}{2} recombination can be\napproximated using the Saha recombination. As shown in\nFig.~9,\n\\ion{He}{2} recombination is slightly slower than the Saha recombination for\nlow baryon models. \n\n\\vspace{0.2in}\n\\centerline{{\\vbox{\\epsfxsize=8.25cm\\epsfbox{recombination_fig12.eps}}}}\n\\baselineskip=8pt\n\\vspace{0.3in}\n{\\footnotesize {\\sc Fig.~12.---}\nWhat controls H recombination? The net $2p$--$1s$ rate\n(dashed) compared to the $2s$--$1s$ two-photon rate (dotted) and the\nnet recombination rate (solid) for 4 different cosmologies. Except for\nlow $\\Omega_{\\rm B}$ and low $h$ models (e.g.~the sCDM model),\nthe $2s$--$1s$ rate dominates. The solid vertical line represents where\n5\\% of the atoms have recombined.\n\\label{fig:Hrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsubsection{What Controls Recombination?}\nH recombination is largely controlled by the $2s$--$1s$\ntwo-photon rate, which except for low-baryon cases, is much faster than\nthe H Ly$\\,\\alpha$ rate. The net recombination rate, net $2s$--$1s$\nrate, and net Ly$\\,\\alpha$ rate are compared for different cosmologies\nin\nFig.~12.\nFigs.~13\nand~14\nshow the same rate\ncomparison for \\ion{He}{1} and \\ion{He}{2}.\nThe three figures all have the same scale on the $x$ and $y$ axes, for\neasy comparison. \n\\ion{He}{1} recombination is controlled by the $2^1p$--$1^1s$ rate\nrather than the $2^1s$--$1^1s$ rate as previously stated (e.g.~MST).\nFig.~13\n(for \\ion{He}{1}) also illustrates the slow net\nrecombination rate, which is the primary factor in the slow Case B\n\\ion{He}{1} recombination.\nFig.~14\nalso illustrates that \\ion{He}{2} in the\nhigh $\\Omega_{\\rm B}$ and $h$ models has a $2s$--$1s$ rate faster\nthan the net recombination rate, meaning that there is no slowdown\nof recombination due to n$=2$, and the Saha equilibrium approximation\nis valid. \n\nThe rates change with cosmological model. Physically this is because\nall of the rates are very sensitive to the baryon density. The\n$2^1p$--$1^1s$ rates are further affected by the Hubble factor\nbecause the Sobolev approximation\n(equations~(\\ref{eq:indofphi}) and~(\\ref{eq:sobtau})), depends on the\nvelocity gradient. Whether most of the atoms in the Universe recombined\nvia a $2p$--$1s$ or a $2s$--$1s$ two-photon transition depends on the\nprecise values of the cosmological parameters. A confident answer to that\nquestion is still not known, given today's parameter uncertainties.\n\n\\vspace{0.2in}\n\\centerline{{\\vbox{\\epsfxsize=8.25cm\\epsfbox{recombination_fig13.eps}}}}\n\\baselineskip=8pt\n\\vspace{0.3in}\n{\\footnotesize {\\sc Fig.~13.---}\nWhat controls \\ion{He}{1} recombination? The net\n$2^1p$--$1^1s$ rate (dashed) compared to $2^1s$--$1^1s$\ntwo-photon rate (dotted)\nand the net recombination rate (solid) for 4\ndifferent cosmologies. Except for high $\\Omega_{\\rm B}$ and high $h$\nmodels, the $2^1p$--$1^1s$ rate dominates, in contrast to H. The solid\nvertical line represents where 5\\% of the atoms have recombined.\n\\label{fig:HeIrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\vspace{0.2in}\n\\centerline{{\\vbox{\\epsfxsize=8.5cm\\epsfbox{recombination_fig14.eps}}}}\n\\baselineskip=8pt\n\\vspace{0.25in}\n{\\footnotesize {\\sc Fig.~14.---}\nWhat controls \\ion{He}{2} recombination? The net $2p$--$1s$\nrate (dashed) compared to the $2s$--$1s$ two-photon rate (dotted) and\nthe net recombination rate (solid) for 4 different cosmologies. Except\nfor low $\\Omega_{\\rm B}$ and low $h$ \nmodels, the $2s$--$1s$ rate dominates during recombination, and the\n$2p$--$1s$ at the start of recombination. The solid vertical line\nrepresents where 5\\% of the atoms have recombined.\n\\label{fig:HeIIrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsection{Atomic Data and Estimate of Uncertainties}\n\\label{sec-AtomicData} \nOur approach in this work has been to include all relevant degrees of freedom\nof the recombining matter in a consistent and coupled manner. This requires\nspecial attention to the quality of the atomic data used. The challenge\nlies in building a consistent model for $all$ energy levels and transitions,\nnot just for the low-lying ones, which are often better known experimentally\nand theoretically.\n\n\\subsubsection{H and \\ion{He}{2}}\nHydrogen (and the hydrogenic ion of helium) have exactly known rate\ncoefficients for radiative processes from precise quantum-mechanical\ncalculations (uncertainties below 1\\%). We use exact values for the\nbound-bound radiative transitions and for radiative recombination,\nas in e.g.~Hummer (1994). For more details see Hummer \\& Storey (1987),\nbut also Brocklehurst (1970) and Johnson (1972). In particular, the\nrate of radiative recombination to level n of a hydrogenic ion can \nbe evaluated from the photoionization (bound-free) cross section for\nlevel n, ${\\sigma}_{{\\rm nc}}(\\nu)$, with the standard assumption of detailed\nbalance (see~\\S\\ref{sec-Recomb}).\nFor hydrogenic bound-free cross sections, we follow in essence\nSeaton's work (Seaton 1959) with its \nasymptotic expansion for the Gaunt factor (see Brocklehurst 1970).\nNote that the weak dependence of the Gaunt factor on wavelength has\na noticeable effect in our final recombination rate calculation.\nGiven our application, we do not require the resolution of resonances,\nas achieved for a few transitions by the Opacity Project (TOPbase,\nCanto et al.~1993).\nLike Hummer (1994), we work with the n-levels assuming that the\n$\\ell$-sublevels have populations proportional to (2$\\ell$+1). The\nresulting uncertainties for hydrogenic radiative rates at low\ntemperatures (T~$\\leq~$10$^5$K) certainly do not exceed the 1\\% level.\n\nCollisional rate coefficients cannot be calculated exactly.\nSo, compared to the hydrogenic radiative rate coefficients, the\nsituation for the bound-bound collisional rates\nand collisional ionization is poor, with errors typically about 6\\%,\nand as high as 20\\% in some cases (Percival \\& Richards 1978).\nA number of methods are used to evaluate\nelectron-impact excitation cross sections of hydrogen-like ions (Fisher\net al.~1997). These most recent values compare well to the older\nsources (Johnson 1972; Percival \\& Richards 1978). The helium ion, \\ion{He}{2},\nis hydrogenic and was treated accordingly. We basically\nfollowed Hummer \\& Storey (1987) and Hummer (1994) in building the model\natom. For our application,\ncollisional processes are negligible, so that the large\nuncertainties that still persist for the collisional rates have no\nimpact on our results.\n\n\\subsubsection{\\ion{He}{1}}\nHelium, in its neutral state, poses a challenge for building a multi-level\natomic model of high precision. Unlike atomic hydrogen, no exact solutions\nto the Schr\\\"odinger equation are available for helium. However, very high\nprecision approximations are now available (Drake 1992, 1994)\nwhich we have used. These approximations are essentially exact for all\npractical purposes. The largest \nrelativistic correction comes from singlet-triplet mixing between states\nwith the same n, $L$, and $J$, but is still small.\nTransition rates were calculated following\nthe recent comprehensive \\ion{He}{1} model built by Smits (1996) and some\nvalues in Theodosiou (1987). The source of our photoionization cross\nsections was TOPbase (Cunto et al.~1993) and Hofsaess (1979) for small n;\nabove n=10 we used scaled hydrogenic values.\nNew detailed calculations (Hummer \\& Storey 1998)\nshow that the \\ion{He}{1} \nphotoionization cross sections become strictly hydrogenic at about n~$>20$.\nThe uncertainties in the \\ion{He}{1} radiative rates are at the 5\\%\nlevel and below.\n\nThe situation with the collisional rates for \\ion{He}{1} is predictably much\nworse than for \\ion{He}{1} radiative rates, with good R-matrix\ncalculations existing only for n~$\\leq 5$ (Sawey \\&\nBerrington 1993). The collisional rates at large n are a crucial ingredient\nin determining the amount of singlet-triplet mixing, but fortunately\ncollisions are not very important for the low density conditions in the early\nUniverse, so the large uncertainty in these rates does not effect\nour calculation. The Born approximation,\nwhich assumes proportionality to the radiative transition rates, is used\n(see Smits 1996) to calculate the collisional cross sections for large n.\n\nFor the $2^1s$--$1^1s$ two-photon rate for \\ion{He}{1} we used the value\n$\\Lambda_{\\rm He I}=51.3\\,{\\rm s}^{-1}$ (Drake et al. 1969)\nwhich differs from a previously used value (Dalgarno~1966) by\n$\\sim10\\%$. An uncertainty even of this magnitude\nwould still make little difference in the final results.\nFor the \\ion{He}{2} $2s$--$1s$ two-photon rate we used the value\n$\\Lambda_{\\rm He I}=526.5\\,{\\rm s}^{-1}$ (for hydrogenic ions this is\nessentially $Z^6$ times the value for H) from Lipeles et al.~(1965).\nDielectronic recombination for \\ion{He}{1} is not at all\nimportant during \\ion{He}{1} recombination. While dielectronic\nrecombination dominates at temperatures above\n$6\\times10^4$K, for the range of temperatures relevant here it\nis at least 10 orders of magnitude below the radiative recombination\nrate (using the fit referred to in Abel et al.~1997).\n\n\\subsubsection{Combined Error From Atomic Data}\nWe have gathered together the uncertainties in the atomic data in\norder to estimate the resulting uncertainty in our derivation of $x_{\\rm e}$.\nThe atomic data with the dominant effect on our calculation are the set\nof bound-free cross sections for all H and \\ion{He}{1} levels -- not so much \nany individual values, but the overall consistency of the sets (which\nare taken from different sources). The differences between our\nmodel atom and Hummer's (1994) reflect the uncertainty in the atomic\ndata. To test the effect on our hydrogenic results, we compared the $x_{\\rm e}$\nresults of an effective 3-level atom using Hummer's (1994)\nrecombination coefficient with the\nresults using a recombination coefficient calculated with our own model H\natom. We find maximum\ndifferences of 1\\% at $z=300$, which corresponds to measurable effects on\nCMB anisotropies of much less than 1\\%.\n\nThe error in $x_{\\rm e}$ arising from \\ion{He}{1} is more difficult to\ncalculate. We estimate it to be considerably less than 1\\%, because the low\nlevel (n~$\\leq$~4)\nbound-bound and bound-free radiative rates dominate \\ion{He}{1}\nrecombination, and as described above, those data are accurate.\n\n\\subsection{Secondary Distortions in the Radiation Field}\n\\label{sec-Distortions}\nIn our recombination calculation we follow `secondary' distortions in the radiation\nfield that could affect the recombination process at a later time. The\nsecondary distortions are caused by the primary distortions that are\nfrozen into the radiation field. At a later time they are redshifted\ninto interaction frequency with other atomic transitions.\nExplicitly, we follow: \\newline\n(1) H Ly$\\,\\alpha$ photons; \\newline\n(2) H $2s$--$1s$ photons. \\newline\nBy the time of H recombination these photons have been redshifted\ninto an energy range where they\ncould photoionize H(n=2). In addition we follow: \\newline\n(3) \\ion{He}{1} $2^1p$--$1^1s$; \\newline \n(4) \\ion{He}{1} $2^1s$--$1^1s$. \\newline\nBy the time of H recombination these photons have been\nredshifted into an energy range which could photoionize H(n=1).\nAnd finally we also follow: \\newline\n(5) \\ion{He}{2} Ly$\\,\\alpha$ photons; \\newline\n(6) \\ion{He}{2} $2s$--$1s$ photons. \\newline\nBy the time of H recombination these photons have been redshifted\ninto an energy range\nwhich could photoionize H(n=1). These \\ion{He}{2} photons bypass\n\\ion{He}{1} because the photons have not been\nredshifted into a suitable energy range for interaction.\n\nHere we only attempt to investigate the maximum effects of secondary spectral\ndistortions. To that end we do not include additional distortions\nwhich are smaller. For example,\nLyman lines other than (1), (3), and (5), whose distortions are smaller \nthan Ly$\\,\\alpha$,\nwill produce a comparably smaller feedback on photoionization.\nThe \\ion{He}{1} singlet recombination photons could theoretically\nphotoionize \\ion{He}{1} triplet states, but as previously discussed\nthere are virtually no electrons in the triplet states, so this\nprocess is also negligible. Another possible effect is due to the\nsimilar energy levels of H and \\ion{He}{2}: $\\Delta E_{\\rm He II} =\n4\\Delta E_{\\rm H}$. For example, the transition from\n\\ion{He}{2} (n=4) to (n=2) produces the same frequency photons as the\ntransition from H~(n=2) to (n=1). These transitions are\ntheoretically competing for photons, and this effect can be important for other\nastrophysical situations (e.g.~planetary nebulae) where H and \\ion{He}{2}\nsimultaneously exist. However, any such effect is negligible for primeval\nrecombination because during \\ion{He}{2} recombination the amount of\nneutral H is very small ([H\/\\ion{He}{2}] $< 10^{-8}$), and during\nH recombination, there is almost no \\ion{He}{2}\n([\\ion{He}{2}\/H]$< 10^{-10}$).\n\nBecause we are only investigating maximum effects we assume the photons\nwere emitted at line center and are redshifted undisturbed until their\ninteraction with H(n=1) or H(n=2) as described above. We also \nassume two photons at half the energy for the $2s$--$1s$ transitions,\ncompared to the $2p$--$1s$ transitions.\nThe distorting photons emitted at a time $z_{\\rm em}$ are\nabsorbed at a later time $z$, where\n\\begin{equation}\nz = z_{\\rm em} \\nu_{\\rm edge}\/\\nu_{\\rm em}.\n\\end{equation}\nHere $\\nu_{\\rm edge}$ is the photoionization edge frequency where the\nphotons are being absorbed, and $\\nu_{\\rm em}$ is the photon's\nfrequency at emission.\nThe distortions are calculated as\n\\begin{eqnarray}\n\\lefteqn{J(\\nu,z) = {h \\nu(z) c}\\,\np_{ij}(z_{\\rm em})\\,\\times}\\nonumber \\\\\n& & \\qquad \\bigg\\{ n_j(z_{\\rm em})\\Big[A_{ji}\n+ B_{ji}B\\big(\\nu_{\\rm em},T_{\\rm R}(z_{\\rm em})\\big)\\Big]\\nonumber \\\\\n& & \\qquad\\qquad\\quad \\mbox{}- n_i(z_{\\rm em})B_{ij}\n B\\big(\\nu_{\\rm em},T_{\\rm R}(z_{\\rm em})\\big)\\bigg\\} ,\n\\end{eqnarray}\nwhere: $B(\\nu_{\\rm em},T_{\\rm R}(z_{\\rm em}))$ is the Planck function at the\ntime of emission; $A_{ji}, B_{ji}$ and $B_{ij}$ are the Einstein\ncoefficients; $p_{ij}$ is the Sobolev escape probability for the\nline; and the other variables are as described previously.\n\nThe distortions (1) and (2) were previously discussed by Rybicki\n\\& Dell'Antonio (1992). They pointed out that the effect from (1) should\nbe small, because the Ly$\\,\\alpha$ distortion must be redshifted by at\nleast a factor of 3 to have any effect. This means that the Ly$\\,\\alpha$\nphotons produced at $z\\lesssim2500$\nwill only affect the Balmer continuum at $z\\lesssim800$ when the\nrecombination process (and any possibility of photoionization) is\nalmost entirely over.\n\nWe find that including the distortions (1) through (6) improves\n$x_{\\rm e}$ during H\nrecombination at\nless than the $0.01\\%$ level.\nThis difference is far too small to make a\nsignificant change in the power spectrum, and it is negligible\ncompared to the major improvements in this paper, which are \nthe level-by-level treatment of H, \\ion{He}{1}, and \\ion{He}{2},\nallowing departures of the excited state populations from\nan equilibrium distribution, calculating recombination directly to\neach excited state, and the correct treatment of\n\\ion{He}{1} triplet and singlet states.\nHowever, the removal of these distorting photons by photoionization\nmust be taken into account when calculating spectral distortions to\nthe CMB blackbody, which we plan to study in a later paper.\n\n\\subsection{Chemistry}\nIncluding the detailed hydrogen chemistry (see~\\S\\ref{sec-Chem})\nmarginally affects the fractional abundances of protons and electrons at\nlow $z$. However, the correction is of the order\n$10^{-2}x_{\\rm e}$ at $z<150$.\nThis change in the electron density\nwould change the Thomson scattering optical depth by $\\sim 10^{-5}$,\ntoo little to make a difference in the CMB power spectrum.\n\n\\vspace{3mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig15a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig15b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~15.---}\nThe effect of the improved treatment of recombination on H\nchemistry. Shown is the standard CDM model. Solid lines are values\nfrom the standard calculation, dashed and dotted lines from our\nimproved results.\n\\label{fig:chem}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nOn the other hand, as shown in\nFig.~15,\nthe different\n$x_{\\rm e}(z)$ that we find will lead to \nfractional changes of similar size in molecular abundances at low $z$, since\nH$_2$ for example is formed via H$^{-}$ which is affected by the residual\nfree electron density. The delay in\n\\ion{He}{1} recombination compared to previous studies causes a\nsimilar delay in formation of He molecules (P.~Stancil, private\ncommunication). However, with the exception of He$_2^+$, no\nchanges are greater than those caused by the residual\nfree electron density at freeze-out. Since molecules can be important\nfor the cooling of primordial gas clouds and the formation of the first\nobjects in the Universe, the precise determination of molecular abundances is\nan important issue (e.g.~Lepp \\& Shull 1984, Tegmark et al.~1997, Abel\net al.~1997, Galli \\& Palla~1998). However, the roughly 10-20\\% change\nin the abundance of some chemical species is probably less than other\nuncertainties in the reaction rates (A.~Dalgarno, private communication).\nWith this in mind, we suspect no drastic implications for theories of\nstructure formation. \n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig16a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig16b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~16.---}\nDifferences in CMB power spectra arising from the improved treatment\nof hydrogen for: (a) the standard CDM parameters\n$\\Omega_{\\rm tot} = 1.0$, $\\Omega_{\\rm B} = 0.05$, $h =0.5$,\n$Y_{\\rm P} = 0.24$, $T_0 = 2.728\\,$K; and (b) an extreme baryon model\nwith $\\Omega_{\\rm B}=1$, $h=1.0$ (for which there is relatively little\neffect). The fractional\ndifference plotted is between our new hydrogen recombination calculation and\nthe standard hydrogen recombination calculation (e.g.~in HSSW), with the\nsense of $C_\\ell^{\\rm new}-C_\\ell^{\\rm old}$, and with the two calculations\nnormalized to have the same amplitude for the initial conditions.\nThe solid lines are for\ntemperature, while the dashed lines are for the (`E'-mode)\npolarization power spectrum.\n\\label{fig:h_powerspectrum}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsection{Power Spectrum}\n\\label{sec-Powerspectrum}\nEven relatively small differences in the recombination history of the\nUniverse can have potentially measurable effects on the CMB\nanisotropies. And so we might expect our two main changes (one in H\nand one in He) to be noticeable in the power spectrum.\nAs a first example\nFig.~16\ncompares the difference in the anisotropy power spectrum derived\nfrom our new $x_{\\rm e}(z)$ to that derived from the standard\nrecombination $x_{\\rm e}$ (essentially identical to that described in\nHSSW), for hydrogen recombination only. Here the $C_{\\ell}$s are\nsquares of the amplitudes in a spherical \nharmonic decomposition of anisotropies on the sky (the azimuthal index\n$m$ depends on the choice of axis, and so is irrelevant for an\nisotropic Universe). They represent the power and angular scale of\nthe CMB anisotropies by describing the rms temperatures at fixed\nangular separations averaged over the whole sky (see e.g.~White, Scott,\n\\& Silk~1994). These $C_{\\ell}$s\ndepend on the ionization fraction $x_{\\rm e}$\nthrough the precise shape of the thickness of the photon last\nscattering surface (i.e.~the visibility function).\nSince the detailed shape of the power spectrum may allow\ndetermination of fundamental cosmological parameters,\nthe significance of the change in $x_{\\rm e}$ is evident.\nTo determine the effect of the change in $x_{\\rm e}$ we have\nused the code {\\tt cmbfast} written and made available by Seljak\n\\& Zaldarriaga (1996), with a slight modification to allow for the input\nof an arbitrary recombination history.\n\nThe dominant physical affect arising from the new H calculation comes\nfrom the change in $x_{\\rm e}$ at low $z$. A process seldom mentioned in\ndiscussions of CMB anisotropy physics (which are otherwise quite\ncomprehensive, e.g.~Hu, Silk, \\& Sugiyama~1997) is that the low-$z$\ntail of the visibility function results in {\\it partial erasure\\\/} of the\nanisotropies produced at $z\\sim1000$. The optical depth in Thomson\nscattering back to, for example, $z=800$\n($\\tau=c\\sigma_{\\rm T}\\int n_{\\rm e}(dt\/dz)\\,dz$)\ncan be several percent. This partial rescattering of the photons\nleads to partial erasure of the $C_\\ell$s by an amount ${\\rm e}^{-2\\tau}$. Let\nus look at the standard CDM calculation first\n(Fig.~16(a)).\nOur change in the optical depth back to $z\\simeq800$ (see\nFig.~2)\nis around 1\\% less than that obtained\nusing the standard\ncalculation, and so we find that the anisotropies suffer less partial erasure\nby about 2\\%.\nThere is no effect on angular scales larger than the horizon at the\nscattering epoch (here redshifts of several hundred), so that all multipoles\nare effected except for the lowest hundred or so $\\ell$s. Hence this\neffect is largely a change in the overall normalization of the power spectrum,\nwith some additional differences at low $\\ell$ which will be masked by the\n`cosmic variance'. In addition there are smaller effects due to changes in\nthe {\\it generation\\\/} of anisotropies in the low-$z$ tail, giving small\nchanges in the acoustic peaks, which can be seen as wiggles in the figure.\nSince the partial erasing effect is essentially unchanged in the case of the\n$\\Omega_{\\rm B}=h=1.0$ model, these otherwise sub-dominant effects are more\nobvious in\nFig.~16(b).\n\nDifferences in the power spectra are rather small in absolute terms,\nso\nFig.~16\nplots the relative difference. We\nhave shown this for our two chosen models, one being standard Cold Dark Matter\n(a), which we will refer to as sCDM,\nand the other being an extreme baryon-only model (b). These models are\nmeant to be representative only, and changes in cosmological parameters will\nresult in curves which differ in detail. We describe how to calculate an\napproximately correct recombination history for arbitrary models in a\nseparate paper (Seager, Sasselov, \\& Scott~1999). Since the main effect is\nsimilar to an overall amplitude change, we normalized our CMB power spectra\nto have\nthe same large-scale matter power spectrum, which is equivalent to normalizing\nto the same amplitude for the initial conditions. The amplitude of the\neffect of our new H calculation clearly depends on the cosmology. For the high\n$\\Omega_{\\rm B}$ and $h$ case, the freeze-out value of $x_{\\rm e}$ is\nmuch smaller (around $10^{-5}$),\nand since the fractional change in $x_{\\rm e}$ is similarly $\\sim10$\\%,\nthe absolute change in ionization fraction is much lower than for the sCDM\nmodel. The integrated optical depth is directly proportional to\n$\\Omega_{\\rm B}h\\Delta x_{\\rm e}$, which is small, despite the increase in\n$\\Omega_{\\rm B}$ and $h$. Hence we see a much smaller increase from our\nhydrogen improvement in\nFig.~16(b).\nThe normalization\nchange is rather difficult to see, since it is masked by relatively small\nchanges around the power spectrum peaks, giving wiggles in the difference\nspectrum.\n\nThe dashed lines in\nFig.~16\nshow the effect\non the power spectrum for CMB polarization. In standard models polarization\nis typically at the level of a few percent of the anisotropy signal, and\nso will be difficult to measure in detail (see Hu \\& White~1997b for a\ndiscussion of CMB polarization).\nWe show the results here to indicate that there\nare further observational consequences of our improved recombination\ncalculation (explicitly we have plotted the `E' mode of polarization, see\ne.g.~Seljak 1997). The effect of our improvements on the polarization can be\nunderstood similarly through the visibility function. Since the polarization\npower spectrum tends to have sharper acoustic peaks, the wiggles in the\ndifference spectrum are more pronounced than for the temperature anisotropies.\nNote that the large relative differences at low $\\ell$\nare actually very small in absolute terms, since the polarization\nsignal is so small there. The polarization-temperature correlation power\nspectrum and the `B' mode of polarization (for models with gravity\nwaves) could also be plotted, but little extra insight is gained, and\nso we avoid this for the sake of clarity.\n\nThe other major difference we find compared with previous treatments\nis in the delayed recombination of \\ion{He}{1}.\nIn\nFig.~17\nwe show the effect of our new \\ion{He}{1} calculation, again as a fractional\nchange in the CMB anisotropy power spectrum versus multipole\n$\\ell$. The change in the recombination of \\ion{He}{1} affects the density\nof free electrons just before\nhydrogen recombination, which in turn affects the diffusion of the\nphotons and baryons, and hence the damping scale for the acoustic\noscillations which give rise to the peaks in the power spectrum.\nThe phases of the acoustic oscillations will also be affected\nsomewhat, which shows up in the wiggles in the difference spectrum.\nFor CDM-like models the\nmain effect is the change in the damping scale, since we now think there are\nmore free electrons at $z\\sim1500$--2000. The resulting change in the\n$C_\\ell$s is essentially the same as assuming the wrong angular scale for the\ndamping of the anisotropies (see Hu \\& White~1997a), which is\nthe same physical effect that HSSW found in arguing\nfor the need to include \\ion{He}{1} recombination {\\it at all\\\/} for obtaining\npercent accuracy in the $C_\\ell$s. The effect of this improved \\ion{He}{1} on\nthe power spectrum will depend on the background cosmology through the\nbaryon density ($\\propto\\Omega_{\\rm B}h^2$) and the horizon size at\nlast scattering through $\\Omega_0h^2$ -- hence there is no simple\nfitting formula, and it is necessary to calculate the effect on the\nanisotropy damping tail for each cosmological model considered.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig17a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig17b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~17.---}\nThe effect of the improved treatment of helium on the CMB power\nspectra for: (a) the standard CDM model; and (b) the extreme baryonic model\n(for which there is essentially no change). The fractional\ndifference plotted is between our new helium recombination calculation and\nthe assumption that helium follows Saha equilibrium (as in HSSW), with\nthe same `effective 3-level' hydrogen recombination used in both cases.\nAgain the sense is $C_\\ell^{\\rm new}-C_\\ell^{\\rm old}$, solid lines are\ntemperature, and dashed lines are polarization.\n\\label{fig:he_powerspectrum}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nThere are really two parts to the \\ion{He}{1} effect. Firstly the extra\n$x_{\\rm e}$ makes the tight coupling regime tighter, so that the\nphoton mean free path is shorter, and the length scale for diffusion\nis smaller. Secondly, the effective damping scale comes from an\naverage over the visibility function, so an increase in the high-$z$\ntail also leads to a smaller damping scale. The CMB anisotropies can\nbe thought of as a series of acoustic peaks multiplied by a roughly\nexponential damping envelope, with the characteristic multipole of the\ncut-off being determined by the damping length scale. As a result of\nthis smaller damping scale, the high $\\ell$ part of the power\nspectrum is {\\it less\\\/} suppressed, and so we see an increase in\nFig.~17(a)\ntowards high $\\ell$. For the\n$\\Omega_{\\rm B}=h=1$ model\n(Fig.~17(b))\nwe\nsee only a very small effect at the highest $\\ell$s.\nThis is easily understood by examining\nFig.~10,\nwhere we see that \\ion{He}{1} recombination is\npushed back to higher redshifts than for the sCDM case, and also that H\nrecombination happens earlier, which, together with the higher\n$\\Omega_{\\rm B}$ and $h$,\nshifts the peak of the visibility function to lower redshifts relative\nto the recombination curve. Hence the the high-$z$ tail of the\nvisibility function is much less affected in this case, and our\nimproved \\ion{He}{1} calculation has essentially negligible effect. However,\nfor less extreme models we find that the \\ion{He}{1} effect is always at least\nmarginally significant.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig18a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig18b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~18.---}\nThe total effect of our improvements on the CMB power\nspectra for: (a) the standard CDM model; and (b) an extreme baryonic model.\nThese plots are essentially the sum of the separate effects of hydrogen\nand helium.\n\\label{fig:tot_powerspectrum}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nTaking the two main effects together, for the sCDM model,\nwe find that they have essentially the\nsame sign, so that the total effect of our new calculation is more\ndramatic (see\nFig.~18(a)).\nBoth effects lead to a slight increase in the anisotropies,\nparticularly at small angular scales (high $\\ell$). Although the\nexact details will depend on the underlying cosmological model, we see\nthe same general trend for most parameters we have looked at. The\nchange can be\nhigher than 5\\% for the smallest scales, although it should be\nremembered that the absolute amplitude of the anisotropies at such scales is\nactually quite low. Nevertheless, changes of this size are well above the\nlevel which is relevant for future determinations of the power\nspectrum. As a rough measure, the cosmic variance at $\\ell\\sim1000$\nis about 3\\%. Hence\na 1\\% change over a range of say 1000 multipoles is something like a\n$10\\sigma$ effect for a cosmic variance limited experiment.\nWhat we can see is that although the effects are far from astonishing,\nthey are at a level which is potentially measurable. Hence our improvements\nare significant in terms of using future CMB data-sets to infer the values\nof cosmological parameters; if not properly taken into account\nthese subtle effects in the\natomic physics of hydrogen and helium might introduce biases in the\ndetermination of fundamental parameters.\n\n\\subsection{Spectral Distortions to the CMB}\n\\label{sec-Future}\nWith the model described in this paper we plan to calculate spectral\ndistortions to the blackbody radiation of the CMB today (see also\nDubrovich~1975; Lyubarsky \\& Sunyaev 1983;\nFahr \\& Loch 1991; Dell'Antonio \\& Rybicki~1993; Burdyuzha \\& Chekmezov 1994;\nDubrovich \\& Stolyarov 1995, 1997; Boschan \\& Biltzinger~1998).\nThe main emission from Ly$\\,\\alpha$ and the two-photon process will\nbe in the far-infrared part of the spectrum.\nTransitions among the very high energy levels, which have very\nsmall energy separations, may produce spectral distortions in the\nradio. Although far weaker than distortions by the lower Lyman lines,\nthey will be in a spectral region less contaminated by background\nsources.\n\nDetecting such distortions will not be easy, since they are generally swamped\nby Galactic infra-red or radio emission and other foregrounds.\nOur new calculation\ndoes not yield any vast improvement in the prospects for detection. However,\nconfirmation of the presence of these recombination lines would be a\ndefinitive piece of supporting evidence for the whole Big Bang paradigm.\nMoreover, detailed measurement of the lines, if ever possible to carry out,\nwould be a direct diagnostic of the recombination process. For these\nreasons we will present spectral results elsewhere.\n\n\\section{Conclusions}\n\nOne point we would like to stress is that\nour detailed calculation agrees very well\nwith the results of the effective 3-level atom. This underscores the\ntremendous achievement of Peebles, Zel'dovich and colleagues in so fully\nunderstanding cosmic recombination 30 years ago. However, the great goal\nof modern cosmology is to determine the cosmological parameters to an\nunprecedented level of precision, and in order to do so it is now\nnecessary to understand very basic things, like recombination, much more\naccurately. \n\nWe have shown that improvements upon previous recombination calculations\nresult in a roughly 10\\% change in $x_{\\rm e}$ at low redshift for most\ncosmological models, plus a substantial delay in \\ion{He}{1}\nrecombination, resulting in a few percent change in the CMB\npower spectrum at small angular scales.\nSpecifically, the low redshift difference in $x_{\\rm e}$ is due to the\nH excited states' departure from an equilibrium distribution. This in turn\ncomes from the level-by-level treatment of a 300-level H atom, which\nincludes all bound-bound radiative rates, and which allows\nfeedback of the disequilibrium of the excited states on\nthe recombination process. The large improvement in\n$x_{\\rm e}$ during \\ion{He}{1} recombination comes from the\ncorrect treatment of the atomic levels, including\ntriplet and singlet states. While\nit was already understood that \\ion{He}{1} recombination would\naffect the power spectrum at high multipoles (HSSW), our improved\n\\ion{He}{1} recombination affects even the start of H recombination for\ntraditional low $\\Omega_{\\rm B}$ models. There is thus a substantially\nbigger change in the $C_\\ell$s, reaching to larger angular scales.\n\nCareful use of $T_{\\rm M}$ rather than $T_{\\rm R}$ can also have\nnoticeable consequences, as to a lesser extent can the treatment of\nLy$\\,\\alpha$ redshifting using the Sobolev escape probability.\nOur other new contributions to the recombination calculation produce negligible \ndifferences in $x_{\\rm e}$. Collisional excitation and ionization for\nH, \\ion{He}{1} and for \\ion{He}{2} are of little importance. Inclusion\nof additional cooling and heating terms in the evolution of $T_{\\rm M}$ also\nproduce little change in $x_{\\rm e}$. The largest spectral distortions do not\nfeed back on the recombination process to a level greater than\n$0.01\\%$ in $x_{\\rm e}$. Finally, the H chemistry occurs too low in\nredshift to make any noticeable difference in the CMB power spectrum. \n\nAlthough we have tried to be careful to consider every process we can\nthink of, it is certainly possible that other subtle effects remain to\nbe uncovered. We hope that we do not have to wait another 30\nyears for the next piece of substantial progress in understanding how\nthe Universe became neutral.\n\n\\acknowledgments\nThe program {\\tt recfast}, which performs approximate calculations of the\nrecombination history is available at\n{\\tt http:\/\/www.astro.ubc.ca\/people\/scott\/recfast.html} (FORTRAN version) and\nat {\\tt http:\/\/cfa-www.harvard.edu\/ ${\\sim}$sasselov\/rec\/} (C version).\nWe would like to thank George Rybicki, Ian Dell'Antonio, Avi Loeb, and\nHan Uitenbroek for many useful conversations. Also David Hummer and\nAlex Dalgarno for discussions on the atomic physics, Alex Dalgarno and\nPhil Stancil for discussions on the chemistry, Martin White, Wayne Hu\nand Uro{\\v s} Seljak for discussion on the cosmology, and Jim Peebles\nfor discussions on several aspects of this work. We thank the referee\nfor a careful reading of the manuscript. Our study of effects\non CMB anisotropies was made much easier through the availability of\nMatias Zaldarriaga and Uro{\\v s} Seljak's code {\\tt cmbfast}. DS is\nsupported by the Canadian Natural Sciences and Engineering Research\nCouncil.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe study of centrality in networks goes back to the late forties. Since then,\nseveral measures of centrality with different properties have been published (see~\\cite{BoVAC} for a survey).\nTo sort out which measures are more apt for a specific application, one can try\nto classify them by means of some axiom that they might satisfy or not. \n\nIn a previous paper~\\cite{BLVRMCM}, two of the authors have studied in particular\n\\emph{score monotonicity}~\\cite{BoVAC} and \\emph{rank monotonicity} on directed graphs.\nThe first property says that when an arc\n$x\\to y$ is added to the graph, the score of $y$ strictly increases.\nRank monotonicity~\\cite{CDKLEAA}\nstates that after adding an arc $x\\to y$, all nodes with a score\nsmaller than or equal to $y$ have still a score smaller than or equal to $y$.\nScore and rank monotonicity complement themselves. Score\nmonotonicity tells us that ``something good happens''. Rank monotonicity\nthat ``nothing bad happens''. \n\nOnce we move to undirected graphs, however, previous definitions and results are no longer\napplicable. Note that adding a single edge to an undirected graph is equivalent to adding \\emph{two} opposite arcs\nin a directed graph, which may suggest why the situation is so different. In this paper, we\npropose definitions that are natural extensions of the directed case, and prove\nresults about classical types of spectral ranking~\\cite{VigSR}---eigenvector\ncentrality~\\cite{LanZRWT,BerTGA}, Seeley's\nindex~\\cite{SeeNRI}, and PageRank~\\cite{PBMPCR}. With minor restrictions,\nall these measures of centrality have been proven to be score and rank monotone in the directed case~\\cite{BLVRMCM}.\nHowever, we will prove that, surprisingly, this is no longer true in the undirected\ncase: in the case of eigenvector centrality and PageRank, at least one of the extremes of the edge might\nlower both its score and its rank.\n\nTo prove general results in the case of PageRank, we use the theory of graph fibrations~\\cite{BoVGF},\nwhich makes us able to reduce the computation of a spectral ranking of a graph of variable size to a similar\ncomputation on a finite graph.\nThis approach to proofs, which we believe is of independent interest,\nmakes it possible to use analytic techniques to control the PageRank values.\n\nWe conclude the paper with some anecdotal evidence from a medium-sized real-world\nnetwork, showing that violations of rank monotonicity do happen.\n\n\\section{Graph-theoretical preliminaries}\n\\label{sec:defs}\n\nWhile we will focus on simple undirected graphs, we are going to make use of\nsome proof techniques that require handling more general types of graphs.\n\nA \\emph{(directed multi)graph} $G$ is defined by a set $N_G$ of nodes, a set\n$A_G$ of arcs, and by two functions $s_G,t_G:A_G\\to N_G$ that specify the\nsource and the target of each arc (we shall drop the subscripts whenever no\nconfusion is possible); a \\emph{loop} is an arc with the same source and target.\nWe use $G(i,j)$ for denoting the set of\narcs from node $i$ to node $j$, that is, the set of arcs $a\\in A_G$ such that $s(a)=i$ and\n$t(a)=j$; the arcs in $G(i,j)$ are said to be \\emph{parallel} to one\nanother. \nSimilarly, we denote with $G(-} % in alternative to... \\newcommand{\\noarg}{\\:\\cdot\\:,i)$ the set of arcs coming\ninto $i$, that is, the set of arcs $a\\in A_G$ such that $t(a)=i$, and analogously with\n$G(i,-} % in alternative to... \\newcommand{\\noarg}{\\:\\cdot\\:)$ the set of arcs going out of $i$. \nFinally, we write $d_G^+(i)=|G(i,-} % in alternative to... \\newcommand{\\noarg}{\\:\\cdot\\:)|$ for the \\emph{outdegree}\nof $i$ in $G$ and $d_G^-(i)=|G(-} % in alternative to... \\newcommand{\\noarg}{\\:\\cdot\\:,i)|$ for the \\emph{indegree} of $i$ in\n$G$.\n\nThe main difference between this definition and the standard definition of a\ndirected graph is that we allow for the presence of multiple arcs between any\npair of nodes.\nSince we do not need to distinguish between graphs that only differ because of node names, we will\nalways assume that $N_G=\\{\\,0,1,\\dots,n_G-1\\,\\}$ where $n_G$ is the number of nodes of $G$.\nEvery graph $G$ \nhas an associated $n \\times n$ \\emph{adjacency matrix}, also denoted by $G$, where $G_{ij}=|G(i,j)|$. \n\nA \\emph{(simple) undirected graph} is a loopless\\footnote{Note that our negative results are \\emph{a fortiori} true if we consider\nundirected graphs with loops. Our positive results are still valid in the same case\nusing the standard convention that loops increase the degree by two.} graph $G$ such that for all $i,j \\in N$,\n$|G(i,j)|=|G(j,i)|\\leq 1$. In other words, there are no parallel arcs and if there is an\narc from $i$ to $j$ there is also an arc in the opposite direction.\nIn an undirected graph, an \\emph{edge} is an unordered set of nodes $\\{\\,i,j\\,\\}$ (simply denoted by $i \\scalebox{0.5}[1.0]{-} j$) such that $|G(i,j)|=1$; \nthe set of all edges will be denoted by $E_G$; obviously, the number of edges is exactly half of the number of arcs. \nFor undirected graphs,\nwe prefer to use the word ``vertex'' instead of ``node'', and use $V$ (instead of $N$) for the set of vertices\nand $d(x)$ for the degree of a vertex $x$.\n\n\n\n\\section{Score and rank monotonicity axioms on undirected graphs}\n\nOne of the most important notions that researchers have been trying to capture\nin various types of graphs is ``node centrality'':\nideally, every node (often representing an\nindividual) has some degree of influence or importance within the social domain\nunder consideration, and one expects such importance to be reflected in the\nstructure of the social network; centrality is a quantitative measure that\naims at revealing the importance of a node.\n\nFormally, a \\emph{centrality} (measure or index) is any function $c$ that, given a graph $G$, assigns a \nreal number $c_G(x)$ to every node $x$ \tof $G$; countless notions of centrality have been proposed over time, for\ndifferent purposes and with different aims; each of them was originally defined only for a specific category of graphs. Later some of\nthese notions of centrality have been extended to more general classes; in this paper, we shall only consider centralities \nthat can be defined\nproperly on all undirected graphs (even disconnected ones).\n\nAxioms are useful to isolate properties of different centrality measures and make it possible to compare them. One\nof the oldest papers to propose this approach is Sabidussi's paper~\\cite{SabCIG}, and many other proposals have appeared in the\nlast two decades.\n\nIn this paper we will be dealing with two properties of centrality measures:\n\n\\begin{definition}[Score monotonicity]\nGiven an undirected graph $G$, \na centrality $c$ is said to be \\emph{score monotone on $G$} iff for every pair of non-adjacent vertices $x$ and $y$ we have that\n\\[\n\tc_{G'}(x) > c_G(x) \\text{\\ and\\ } c_{G'}(y) > c_G(y),\n\\]\nwhere $G'$ is the graph obtained adding the new edge $x \\scalebox{0.5}[1.0]{-} y$ to $G$.\nIt is said to be \\emph{weakly score monotone on $G$} iff the same property holds, with $\\geq$ instead of $>$.\nWe say that $c$ is \\emph{(weakly) score monotone on undirected graphs} iff it is (weakly) score monotone on all \nundirected graphs $G$.\n\\end{definition}\n\n\\begin{definition}[Rank monotonicity]\nGiven an undirected graph $G$, \na centrality $c$ is said to be \\emph{rank monotone on $G$} iff for every pair of non-adjacent vertices $x$ and $y$ we have that for all vertices $z\\neq x,y$\n\\[\n\tc_{G}(x) \\geq c_G(z) \\Rightarrow c_{G'}(x) \\geq c_{G'}(z) \\text{\\ and \\ }\n\tc_{G}(y) \\geq c_G(z) \\Rightarrow c_{G'}(y) \\geq c_{G'}(z),\\\\\n\\]\nwhere $G'$ is the graph obtained adding the new edge $x \\scalebox{0.5}[1.0]{-} y$ to $G$.\nIt is said to be \\emph{strictly rank monotone\\footnote{Note that the published version of this paper~\\cite{BFVSRMUN}\ncontains a slightly different (and mistaken) definition which does not extend correctly the definition given in~\\cite{BLVRMCM}.} on $G$} if instead\n\\[\n\tc_{G}(x) \\geq c_G(z) \\Rightarrow c_{G'}(x) > c_{G'}(z) \\text{\\ and \\ }\n\tc_{G}(y) \\geq c_G(z) \\Rightarrow c_{G'}(y) > c_{G'}(z).\\\\\n\\]\nWe say that $c$ is \\emph{(strictly) rank monotone on undirected graphs} iff it is (strictly) rank monotone on all \nundirected graphs $G$.\n\\end{definition}\n\n\nThese four properties\\footnote{The asymmetric use of strict\/weak in the two definitions is for consistency with the previous literature on this topic.} can be studied on the class of all undirected graphs or only on the connected class, giving rise to eight possible ``degrees\nof monotonicity'' that every given centrality may satisfy or not. This paper studies these different degrees of monotonicity for three popular \nspectral centrality measures, also comparing the result obtained with the corresponding properties in the directed case. As we shall see, the\nundirected situation is quite different.\n\n\n\\section{Eigenvector centrality}\n\nEigenvector centrality is probably the oldest attempt at deriving a centrality from matrix information: a first\nversion was proposed by Landau in 1895 for matrices representing the results of chess tournaments~\\cite{LanZRWT}, and\nit was stated in full generality in 1958 by Berge~\\cite{BerG}; it has been rediscovered many times since then.\nOne considers the adjacency matrix of the graph and computes its left or right dominant eigenvector, which in our case coincide: the result\nis thus defined modulo a scaling factor, and if the graph is strongly\nconnected, the result is unique (modulo the scaling factor).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics{albiro-203-mps.eps}\n\\caption{\\label{fig:ec}A counterexample to score monotonicity for eigenvector centrality. After adding the edge between $0$ and $1$, the score of $0$ decreases:\nin norm $\\ell_1$, from $0.30656$ to $0.29914$; in norm $\\ell_2$, from $0.65328$ to $0.63586$; and when projecting the\nconstant vector $\\mathbf1$ onto the dominant eigenspace, from $1.39213$ to $1.35159$.}\n\\end{figure}\n\nDiscussing score monotonicity requires some form of normalization, due to the presence of the scaling factor. In Figure~\\ref{fig:ec} we show a very simple graph violating the property.\nIn particular, node $0$ score decreases after adding the arc $0\\adj1$ both\nin norm $\\ell_1$ and norm $\\ell_2$, and when projecting the constant vector $\\mathbf1$ onto the dominant eigenspace, which is an alternative way of circumventing the\nscaling factor~\\cite{VigSR}. The intuition \nis that initially node $0$ has a high score because of its largest degree (three). However, once we close the triangle\nwe create a loop that absorbs a large amount of rank, effectively decreasing the score of $0$. We conclude that\n\\begin{theorem}\nEigenvector centrality does not satisfy weak score monotonicity, even on connected undirected graphs (using norm $\\ell_1$, $\\ell_2$, or projection onto the dominant eigenspace).\n\\end{theorem}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics{albiro-204-mps.eps}\n\\caption{\\label{fig:ec2}A counterexample to rank monotonicity for eigenvector centrality.\nBefore adding the edge between $0$ and $1$, the score of $1$ is greater than the score of $3$; after, it is smaller.}\n\\end{figure}\n\n\nA similar counterexample, shown in Figure~\\ref{fig:ec2}, shows that eigenvector centrality does not satisfy rank monotonicity.\nThe scores of nodes $3$ and $1$ go from strictly increasing (without the edge $0\\adj1$) to strictly decreasing (with the edge $0\\adj1$);\nthus, $1$ loses rank.\nNote that in this case we do not have to choose a normalization, as the order of the two values does not change upon normalization. We conclude that \n\\begin{theorem}\nEigenvector centrality does not satisfy weak rank monotonicity, even on connected undirected graphs.\n\\end{theorem}\n\n\n\\section{Seeley's index}\n\nSeeley's index~\\cite{SeeNRI} is simply the steady state of the natural (uniform)\nrandom walk on the graph (for more details, see~\\cite{BoVAC}). It is a\nwell-known fact that if the graph is connected the steady-state probability of\nnode $x$ is simply $d(x)\/2m$---essentially, the index is just the $\\ell_1$-normalized degree. We\nwill thus use this definition for all graphs. As a consequence:\n\n\\begin{theorem}\nSeeley's index ($\\ell_1$-normalized degree) is strictly rank monotone on undirected graphs.\n\\end{theorem}\n\nThe situation is slightly different for score monotonicity:\n\\begin{theorem}\nSeeley's index ($\\ell_1$-normalized degree) is score monotone on undirected graphs, except in the case of a disconnected graph formed by a star graph and by one or more additional isolated vertices,\nin which case it is just weakly score monotone.\n\\end{theorem} \n\\begin{proof}\nWhen we add an edge between $x$ and $y$ in a graph with $m$ edges, the score of $x$ changes from $d(x)\/2m$ to $(d(x)+1)\/(2m+2)$. If we require\n\\[\n\\frac{d(x)+1}{2m+2}> \\frac{d(x)}{2m}\n\\]\nwe obtain $d(x)< m$. Since obviously $d(x)\\leq m$, the condition is always true except when $d(x)=m$, which corresponds to\nthe case of a disconnected graph formed by a star graph and by additional isolated vertices.\nIndeed, in that case adding an edge between an isolated vertex and the center of the star will not change the center's Seeley's index. \\ifmmode\\mbox{ }\\fi\\rule[-.05em]{.3em}{.7em}\\setcounter{noqed}{0}\n\\end{proof}\n\n\n\n\\section{Graph fibrations and spectral ranking}\n\\label{sec:fib}\n\nIt is known since seminal works from the '50s in the theory of \\emph{graph divisors}~\\cite{CDSSG} that fibrations~\\cite{BoVGF}, defined below,\nhave an important relationship with eigenvalues and eigenvectors: if there is a fibration $f:G\\to B$, the eigenvalues\nof $G$ and $B$ are the same, modulo multiplicity, and eigenvectors of $G$ can be obtained from the eigenvectors of $B$. The\nresults extend to weighted graphs, too. In this section, we are going to extend such results to \\emph{damped spectral rankings}~\\cite{VigSR} of the form\n\\[\n\\bm v\\sum_{i=0}^\\infty \\beta^iM^i = \\bm v( 1 - \\beta M)^{-1},\n\\]\nwhere $M$ is the weighted adjacency matrix of a graph, $\\beta$ is a parameter satisfying the condition\n$0\\leq\\beta<1\/\\rho(M)$, $\\rho(M)$ is the spectral radius of $M$, and $\\bm v$ is a \\emph{preference vector}: \nKatz's index~\\cite{KatNSIDSA}, Hubbell's index~\\cite{HubIOACI} and PageRank~\\cite{PBMPCR} are all examples of damped spectral rankings.\n\nWhile determining a damped spectral ranking for a \\emph{specific} graph essentially requires solving a system of linear equations,\npossibly approximating its solution with an iterative method,\ndoing that for \\emph{parametric families} of graphs is tricky and often requires\n\\emph{ad hoc} approaches. Nonetheless, when the graphs under consideration are sufficiently symmetric, one can try to reduce\nthe computation using a technique based on fibrations. The idea was introduced in~\\cite{BLSGFGIP} for random walks\nwith restart, and in this section we will extend it to general damped spectral rankings, providing\nthus a self-contained (and, in fact, simpler) proof.\n\nLet us start with some additional definitions.\nA \\emph{path} (of length $n\\geq 0$) is a sequence $\\pi=\\langle i_0 a_1 i_1 \\cdots\ni_{n-1} a_n i_n\\rangle$, where $i_k\\in N_G$, $a_k\\in A_G$, $s(a_k)=i_{k-1}$ and\n$t(a_k)=i_k$. \nWe define $s(\\pi)=i_0$ (the \\emph{source} of $\\pi$), $t(\\pi)=i_n$ (the \\emph{target} of $\\pi$), \n$|\\pi|=n$ (the \\emph{length} of $\\pi$) and let $G^*(i,j) = \\{\\, \\pi\n\\mid s(\\pi)=i, t(\\pi)=j\\,\\}$ (the\nset of paths from $i$ to $j$). \n\n\nA \\emph{(graph) morphism} $f:G\\to H$ is given by a pair of functions\n$f_N:N_G\\to N_H$ and $f_A:A_G\\to A_H$ commuting with the source and\ntarget maps, that is, $s_H(f_A(a))=f_N(s_G(a))$ and $t_H(f_A(a))=f_N(t_G(a))$ for all\n$a \\in A_G$ (again, we shall drop the subscripts whenever no confusion is possible). In other\nwords, a morphism maps nodes to nodes and arcs to arcs in such a way to\npreserve the incidence relation. \nThe definition of morphism we give here is the obvious extension to the case of multigraphs of the standard notion the\nreader may have met elsewhere.\nAn \\emph{epimorphism} is a morphism $f$ such that both $f_N$ and $f_A$ are surjective.\n\nA \\emph{fibration}~\\cite{BoVGF} between the graphs $G$ and $B$ is a morphism $f: G\\to B$ such\nthat for each arc $a\\in A_B$ and each node $i\\in N_G$ satisfying\n$f(i)=t(a)$ there is a unique arc $\\lift ai\\in A_G$ (called the \\emph{lifting of\n$a$ at $i$}) such that $f(\\lift ai)=a$ and $t(\\lift ai)=i$.\nIf $f:G\\to B$ is a fibration, $G$\nis called the \\emph{total graph} and $B$ the \\emph{base} of $f$. \nWe shall also say that $G$ is \\emph{fibered (over $B$)}. The \\emph{fiber over a\nnode $h\\in N_B$} is the set of nodes of $G$ that are mapped to $h$, and shall\nbe denoted by $f^{-1}(h)$. \n\nA more geometric way of interpreting the definition of\nfibration is that given a node $h$ of $B$ and a path $\\pi$ terminating at $h$,\nfor each node $i$ of $G$ in the fiber of $h$ there is a unique path terminating\nat $i$ that is mapped to $\\pi$ by the fibration; this path is called the\n\\emph{lifting of $\\pi$ at $i$}. \n\nIn Figure~\\ref{fig:exfib}, we show two graph morphisms; the morphisms are\nimplicitly described by the colors on the nodes. The morphism displayed on the\nleft is not a fibration, as the loop\non the base has no counterimage ending at the lower gray node, and\nmoreover the other arc has two counterimages with the same target. The\nmorphism displayed on the\nright, on the contrary, is a fibration. Observe that loops are not necessarily\nlifted to\nloops.\n\n\\begin{figure}[htbp]\n \\begin{center}\n\t\\includegraphics{albiro-5-mps.eps}\\qquad\\qquad\\qquad\\qquad\\includegraphics{albiro-6-mps.eps}\n \\end{center}\n \\caption{\\label{fig:exfib}On the left, an example of graph morphism that is\nnot a fibration; on the right, a fibration. Colors on the nodes are used to\nimplicitly specify the morphisms.}\n\\end{figure}\n\nWe will now show how fibrations can be of help in the computation of a damped spectral ranking.\nFirst of all, we are now going to consider \\emph{weighted} graphs, in which each arc is assigned\na real weight, given by a weighting function $w:A_G\\to \\mathbf R$: the adjacency matrix associated to\na weighted graph $G$ is defined by letting\n\\[\n\tG_{ij}=\\sum_{a \\in G(i,j)} w(a)\n\\]\nand we obtain the unweighted case when $w$ is the constant $a \\mapsto 1$ function.\nAll the morphisms (especially, fibrations) between weighted graphs are assumed to preserve weights. \n\nNote that every morphism $f: G \\to B$ extends to a mapping $f^*$ between paths of $G$ and paths of $B$\nin an obvious way. This map $f^*$ preserves not only path lengths, but also weight sequences if $f$ does.\n\nFor fibrations we can say more; using the lifting property, one can prove by induction that:\n\\begin{theorem}\n\\label{thm:bij}\nIf $f: G \\to B$ is an epimorphic fibration between weighted graphs,\nthen for every two nodes $j \\in N_G$ and $k \\in N_B$ the map $f^*$ is a bijection \nbetween $\\cup_{i \\in f^{-1}(k)} G^*(i,j)$ and $B^*(k,f(j))$.\n\\end{theorem}\n\n\nNow, for every $t\\geq 0$, $G^t$ is the matrix whose $ij$ entry contains a summation of contributions, one for each path $\\pi \\in G^*(i,j)$, \nand the contribution is given by the product of the arc weights found along the way;\nhence, by Theorem~\\ref{thm:bij}, under convergence assumptions we have that for all $\\beta$ and all $i \\in N_G$ and $k \\in N_B$\n\\[\n\t\\sum_{i \\in f^{-1}(k)}\\left(\\sum_{t \\geq 0} \\beta^t G^t\\right)_{ij} = \\left(\\sum_{t \\geq 0} \\beta^t B^t\\right)_{kf(j)},\n\\]\nor equivalently\n\\[\t\n\\sum_{i \\in f^{-1}(k)}\\left((1-\\beta G)^{-1}\\right)_{ij} = \\left((1-\\beta B)^{-1}\\right)_{kf(j)}.\n\\]\nNow, for every vector\\footnote{All vectors in this paper are row vectors.} $\\bm u$ of size $n_B$, define its \\emph{lifting along $f$} as\nthe vector $\\bm u^f$ of size $n_G$ given by\n\\[\n\t\\left(u^f\\right)_i=u_{f(i)}.\n\\]\nFor every $j$, we have\n\\begin{multline*}\n\t\\left(\\bm u^f (1-\\beta G)^{-1}\\right)_j\n\t=\\sum_{i\\in N_G} u^f_i \\left((1-\\beta G)^{-1}\\right)_{ij}\n\t=\\\\\n\t=\\sum_{k\\in N_B}\\sum_{i \\in f^{-1}(k)} u_{f(i)} \\left((1-\\beta G)^{-1}\\right)_{ij}\n\t=\\sum_{k\\in N_B}u_k\\left(\\sum_{i \\in f^{-1}(k)} \\left((1-\\beta G)^{-1}\\right)\\right)_{ij}\n\t=\\\\\n\t=\\sum_{k\\in N_B}u_k\\left((1-\\beta B)^{-1}\\right)_{kf(j)}\n\t=\\left(\\bm u (1-\\beta B)^{-1}\\right)_{f(j)}\n\\end{multline*}\nwhich can be more compactly written as\n\\begin{equation}\n\\label{eqn:resumedsp}\n\t\\bm u^f (1-\\beta G)^{-1}=\\left(\\bm u (1-\\beta B)^{-1}\\right)^f.\n\\end{equation}\n\n\nEquation (\\ref{eqn:resumedsp}) essentially states that if we want to compute the damped spectral ranking of the weighted graph $G$,\nfor a preference vector that is\nconstant along the fibers of an epimorphic fibration $f: G \\to B$, and thus of the form $\\bm u^f$, we can compute the damped spectral ranking of the weighted base $B$ \nusing $\\bm u$ as preference vector, and then lift along $f$ the result. For example, in the\ncase of Katz's index a simple fibration between graphs is sufficient, as in that case there are no weights to deal with.\n\n\n\\paragraph{Implications for PageRank.}\nPageRank~\\cite{PBMPCR} can be defined as\n\\[\n(1-\\alpha)\\bm v\\sum_{i=0}^\\infty \\alpha^i\\bar G^i = (1-\\alpha) \\bm v(1-\\alpha \\bar G)^{-1},\n\\]\nwhere $\\alpha\\in[0\\..1)$ is the damping factor, \n$\\bm v$ is a non-negative preference vector with unit $\\ell_1$-norm, \nand $\\bar G$ is the row-normalized version\\footnote{Here we are assuming that $G$ has no \\emph{dangling nodes} (i.e., nodes with outdegree $0$).\nIf dangling nodes are present, you can still use this definition (null rows are left untouched in $\\bar G$), but then to obtain PageRank you \nneed to normalize the resulting vector~\\cite{BSVPFD,DCGRFPCSLS}. So all our discussion can also be applied\nto graphs with dangling nodes, up to $\\ell_1$-normalization.} of $G$.\n\nThen $\\bar G$ is just the (adjacency matrix of the) weighted version of $G$ defined by letting $w(a)=1\/d^+_G(s_G(a))$.\nHence, if you have a weighted graph $B$,\nan epimorphic weight-preserving fibration $f: \\bar G \\to B$, and a vector $\\bm u$ of size $n_B$ such that \n$\\bm u^f$ has unit $\\ell_1$-norm, you can deduce from (\\ref{eqn:resumedsp}) that\n\\begin{equation}\n\\label{eqn:resumepr}\n (1-\\alpha) \\bm u^f (1-\\alpha \\bar G)^{-1}=\\left(\\bm (1-\\alpha) \\bm u (1-\\alpha B)^{-1}\\right)^f.\n\\end{equation}\n\nOn the left-hand side you have the actual PageRank of $G$ for a preference vector that is fiberwise constant;\non the right-hand side you have a spectral ranking of $B$ for the projected preference vector.\nNote that $B$ is not row-stochastic, and $\\bm u$ has not unit $\\ell_1$-norm, \nso technically the right-hand side of equation (\\ref{eqn:resumepr}) is\nnot PageRank anymore, but it is still a damped spectral ranking.\n\n\\section{PageRank}\n\nArmed with the results of the previous section, we attack the case of PageRank, which is the most interesting. The first observation\nis that\n\\begin{theorem}\nGiven an undirected graph $G$ there is a value of $\\alpha$ for which PageRank is strictly rank monotone on $G$.\nThe same is true for score monotonicity, except when $G$ is formed by a star graph and by one or more additional isolated vertices.\n\\end{theorem}\n\\begin{proof}\nWe know that for $\\alpha\\to 1$, PageRank tends to Seeley's index~\\cite{BSVPFDF}. Since Seeley's index is strictly rank monotone, for each non-adjacent pair \nof vertices $x$ and $y$ there is a value $\\alpha_{xy}$ such that for $\\alpha\\geq\\alpha_{xy}$ adding the edge $x\\scalebox{0.5}[1.0]{-} y$ is strictly\nrank monotone. The proof is completed by taking $\\alpha$ larger than all $\\alpha_{xy}$'s. \nThe result for score monotonicity is similar. \\ifmmode\\mbox{ }\\fi\\rule[-.05em]{.3em}{.7em}\\setcounter{noqed}{0} \n\\end{proof}\n\nOn the other hand, we will now show that \n\\emph{for every possible value of the damping factor $\\alpha$} there is a graph on which PageRank violates rank and score monotonicity.\n\nThe basic intuition of our proof is that when you connect a high-degree node $x$ with a low-degree node $y$, $y$ will pass to $x$\na much greater fraction of its score than in the opposite direction. This phenomenon is caused by the stochastic normalization of the\nadjacency matrix: the arc from $x$ to $y$ will have a low coefficient, due to the high degree of $x$, whereas the arc from $y$ to $x$\nwill have a high coefficient, due to the low degree of $y$.\n\nWe are interested in a parametric example, so that we can tune it for different values of $\\alpha$. At the same time, we want to \nmake the example analytic, and avoid resorting to numerical computations, as that approach would make it impossible to prove a result\nvalid for every $\\alpha$---we would just, for example, prove it for a set of samples in the unit interval.\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\n\\raisebox{.5cm}{$G_k$\\qquad}&\\includegraphics{albiro-201-mps.eps}\\\\\n\\raisebox{.5cm}{$B_k$\\qquad}&\\includegraphics{albiro-202-mps.eps}\n\\end{tabular}\n\\caption{\\label{fig:pr}The parametric counterexample graph for PageRank. The two $k$-cliques are represented here as $5$-cliques for simplicity. Arc\nlabels represent multiplicity; weights are induced by the uniform distribution on the upper graph.}\n\\end{figure}\n\nWe thus resort to fibrations, using equation (\\ref{eqn:resumepr}). In Figure~\\ref{fig:pr} we show a parametric\ngraph $G_k$ comprising two $k$-cliques (in the figure, $k=5$). \nBelow, we show the graph $B_k$ onto which $G_k$ can be fibred by mapping\nnodes following their labels. The dashed edge is the addition that we will study:\nthe fibration exists whether the edge exists or not (in both graphs).\n\nWhile $G_k$ has $2k+4$ vertices, $B_k$ has $9$ vertices, independently of $k$, and\nthus its PageRank can be computed analytically as rational functions of $\\alpha$ whose coefficients are rational functions\nin $k$ (as the number of arcs of each $B_k$ is different). The adjacency matrix of $B_k$ without the dashed arc, considering multiplicities, is\\footnote{Note that in the published version of this paper~\\cite{BFVSRMUN}\nthe denominators of the second row are $k-1$, mistakenly, instead of $k+1$.}\n\\[\n\\left(\\begin{matrix}\n \\frac{k-2}{k-1}& \\frac{k-1}{k-1}& 0& 0& 0& 0& 0& 0& 0\\\\\n \\frac1{k+1}& 0& \\frac1{k+1}& 0& \\frac1{k+1}& 0& 0& 0& 0\\\\\n 0& \\frac12& 0& \\frac12& 0& 0& 0& 0& 0\\\\\n 0& 0& \\frac12& 0& 0& 0& \\frac12& 0& 0\\\\\n 0& \\frac12& 0& 0& 0& \\frac12& 0& 0& 0\\\\\n 0& 0& 0& 0& 1& 0& 0 &\\fcolorbox{gray}{gray}{0}& 0\\\\\n 0& 0& 0& \\frac1k& 0& 0& 0& \\frac1k& \\frac1k\\\\\n 0& 0& 0& 0& 0& \\fcolorbox{gray}{gray}{0}& \\frac1{k-1}& 0& \\frac1{k-1}\\\\\n 0& 0& 0& 0& 0& 0& \\frac{k-2}{k-1}& \\frac{k-2}{k-1}& \\frac{k-3}{k-1}\\\\\n\\end{matrix}\\right)\n\\]\nAfter adding the edge between $5$ and $7$ we must modify the matrix by setting the two grayed \nentries to one and fix normalization accordingly. We will denote with $\\operatorname{pre}_\\alpha(x)$ the rational function returning the PageRank of\nnode $x$ with damping factor $\\alpha$ before the addition of the dashed arc, and \nwith $\\operatorname{post}_\\alpha(x)$ the rational function returning the PageRank of\nnode $x$ with damping factor $\\alpha$ after the addition of the dashed arc. \n\nWe use the Sage computational engine~\\cite{Sage} to perform all computations, as\nthe resulting rational functions are quite formidable.\\footnote{The Sage worksheet can be found at \\url{https:\/\/vigna.di.unimi.it\/pagerank.ipynb}.} \nWe start by considering\nnode $5$: evaluating $\\operatorname{post}_\\alpha(5)-\\operatorname{pre}_\\alpha(5)$ in $\\alpha=2\/3$ we obtain\na negative value for all $k\\geq 12$, showing\nthere is always a value of $\\alpha$ for which node $5$ violates weak score monotonicity, as long as $k\\geq 12$. \n\nTo strengthen our results, we are now going to show that for \\emph{every} $\\alpha$ there is a $k$ such that\nweak score monotonicity is violated. We use Sturm polynomials~\\cite{RaSATP} to compute the number of sign changes of the \nnumerator $p(\\alpha)$ of $\\operatorname{post}_\\alpha(5)-\\operatorname{pre}_\\alpha(5)$ \nfor $\\alpha\\in[0\\..1]$, as the denominator cannot have zeros. Sage\nreports that there are two sign changes for $k\\geq 12$, which means that $p(\\alpha)$ is initially positive; then, somewhere before $2\/3$\nit becomes negative; and finally it returns positive again somewhere after $2\/3$.\n\nDetermining the behavior of the points at which $p(\\alpha)$ changes sign is impossible due to the high degree of the polynomials\ninvolved. However, we can take two suitable parametric points in the unit interval that sandwich $2\/3$, such as \n\\[\na = \\frac34 - \\frac{3k}{4k+ 1000} \\leq \\frac23 \\leq \\frac12+\\frac{k}{2k+1000} = b,\n\\]\nand use again Sturm polynomials to count the number of sign changes in $[0\\..a]$ and $[b\\..1]$. In both cases, if $k\\geq 15$\nthere is exactly one sign change in the interval, and since $a\\to0$, $b\\to 1$ as $k\\to \\infty$, we conclude that as $k$ grows\nthe size of the interval of $\\alpha$'s in which $p(\\alpha)<0$ grows, approaching $[0\\..1]$ in the limit. Thus,\n\\begin{theorem}\nFor every value of $\\alpha\\in[0\\..1)$, there is an undirected graph for which PageRank violates weak score monotonicity\nwhen $\\alpha$ is chosen as damping factor.\n\\end{theorem}\n\nWe now use the same example to prove the lack of rank monotonicity. In this case, we study in a similar way\n$\\operatorname{pre}_\\alpha(5)-\\operatorname{pre}_\\alpha(2)$, which is positive in $\\alpha=2\/3$\nif $k\\geq 14$. Its numerator has two sign changes in the unit interval,\nwhich means that initially $5$ has a smaller PageRank than $2$; then, somewhere before $2\/3$\n$5$ starts having a larger PageRank than $2$; finally, we return to the initial condition.\n\n Once again, we sandwich $2\/3$ using\n\\[\n\\frac34 - \\frac{3k}{4k+200}\\leq \\frac23\\leq \\frac12 + \\frac{k}{2k+200},\n\\]\nand with an argument analogous to the case of score monotonicity we conclude that as $k$ grows the subinterval\nof values of $\\alpha$ in $[0\\..1)$ for which the score of $5$ is greater than the score of $2$ grows up to the whole interval.\n\nFinally, we study $r(\\alpha)=\\operatorname{post}_\\alpha(5)-\\operatorname{post}_\\alpha(2)$ which is negative in $\\alpha=2\/3$\nif $k\\geq 6$, and whose numerator has three sign changes in the unit interval. Once again, we sandwich $2\/3$ using\n\\[\na = \\frac1{10} - \\frac{k}{10k+2000}\\leq \\frac23\\leq \\frac12 + \\frac{k}{2k+200} = b.\n\\]\nIn this case, there are always two sign changes in $[0\\..a]$ and one sign change in $[b\\..1]$ for $k\\geq 25$, so \nthere is a subinterval of values of $\\alpha$ in $[0\\..1)$ for which the score of $5$ is smaller than the score of $2$ after adding the\nedge $5\\scalebox{0.5}[1.0]{-} 2$, and this subinterval grows in size up to the whole unit interval as $k$ grows. All in all, we proved that:\n\\begin{theorem}\n\\label{th:prrank}\nFor every value of $\\alpha\\in[0\\..1)$, there is an undirected graph for which PageRank violates rank monotonicity\nwhen $\\alpha$ is chosen as damping factor.\n\\end{theorem}\n\n\\section{Experiments on IMDB}\n\n\n\\begin{table}[t]\n\\renewcommand{\\arraystretch}{1.2}\n\\renewcommand{\\tabcolsep}{1ex}\n\\begin{tabular}{lll}\nScore increase & Score decrease & Violations of rank monotonicity\\\\\n\\hline\nMeryl Streep & Yasuhiro Tsushima & Anne--Mary Brown, Jill Corso,~\\ldots\\\\\nDenzel Washington & Corrie Glass & Patrice Fombelle, John Neiderhauser,~\\ldots\\\\\nSharon Stone & Mary Margaret (V) & Dolores Edwards, Colette Hamilton,~\\ldots\\\\\nJohn Newcomb & Robert Kirkham & Brandon Matsui, Evis Trebicka,~\\ldots \n\\end{tabular}\n\\vspace*{.5em}\n\\caption{\\label{tab:rank}A few examples of violations of score monotonicity and rank monotonicity in the Hollywood co-starship graph\n\\texttt{hollywood-2011}. If we add an edge between the actors in the first and second column, the first actor has a score increase, the second actor has a score decrease,\nand the actors in the third column, which were less important than the second actor, become more important after the edge addition.}\n\\end{table}\n\n\nTo show that our results are not only theoretical, we provide a few interesting anecdotal examples from\nthe PageRank scores ($\\alpha=0.85$) of the Hollywood co-starship graph,\nwhose vertices are actors\/actresses in the Internet Movie Database, with an edge connecting them if played in the same movie.\nIn particular, we used the \\texttt{hollywood-2011} dataset from the Laboratory for Web Algorithmics,\\footnote{\\url{http:\/\/law.di.unimi.it\/}}\nwhich contains approximately two million vertices and $230$ million edges.\n\nTo generate our examples, we picked two actors either at random, or considering\nthe top $1\/10000$ of the actors of the graph in PageRank order and the bottom\nquartile, looking for a collaboration that would hurt either actor (or\nboth).\\footnote{Note that for this to happen, the collaboration should be a\ntwo-person production. A production with more people would actually add more\nedges.} About $4$\\% of our samples yielded a violation of monotonicity, and in\nTable~\\ref{tab:rank} we report a few funny examples.\n\nIt is interesting to observe that in the first three cases it is the less-known actor that loses score (and rank) by the collaboration\nwith the star, and not the other way round, which is counterintuitive.\nIn the last case, instead, a collaboration would damage the most important vertex, and\nit is an open problem to prove a result analogous to Theorem~\\ref{th:prrank} for this case.\nWe found no case in which both actors would be hurt by the collaboration.\n\n\n\\section{Conclusions}\n\nWe have studied score and rank monotonicity for three fundamental kinds of spectral ranking---eigenvector\ncentrality, Seeley's index, and PageRank. Our results show that except for\nSeeley's index on connected graphs, there are always cases in which score and rank monotonicity fail,\ncontrarily to the directed case, and these failures can be found in real-world graphs.\nIn particular, for PageRank we can find a counterexample for every value of the damping factor.\nFinding such a class of counterexamples for Katz's index~\\cite{KatNSIDSA} is an interesting open problem.\nAnother valuable contribution would be to find another class of counterexamples for PageRank that is amenable to a simpler analytic proof without having to rely on computer algebra. \n\nOur results suggest that common knowledge about the behavior of PageRank in the directed case cannot\nbe applied automatically to the undirected case.\n\n\n\\vspace*{-.7em}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}