diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbuwt" "b/data_all_eng_slimpj/shuffled/split2/finalzzbuwt" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbuwt" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{s:introduction}\n\n This work is a continuation of the survey search for bright\ncompact radio sources. Several major applications require an\nextended list of sources with positions known at the nanoradian\nlevel: geodetic observations including space navigation; very\nlong baseline interferometry (VLBI) phase-referencing of weak\ntargets, and differential astrometry. For satisfying the needs of\nthese applications, 878 sources were observed under various\ngeodetic and astrometric programs from 1979 thorough 2002, and\nover 80\\,\\% of them were detected. Results of these observations\nwere presented in the catalog ICRF-Ext.2\n\\citep{icrf-ext2-2004} that contains positions of 776 sources.\n Additionally, 2952 flat-spectrum sources were observed in nineteen\n24-hour sessions from 1994 through 2005 in the Very Long Baseline Array\n(VLBA) Calibrator Survey (VCS) program. The positions of 2505 sources\nwere determined from the observations of the VCS project: VCS1\n\\citep{VCS1}, VCS2 \\citep{VCS2}, VCS3 \\citep{VCS3}, and VCS4\n\\citep{VCS4}. Since 364 sources are listed in both the ICRF-Ext.2 and\nthe VCS catalogs, the total number of sources for which positions were\ndetermined with VLBI in IVS (International VLBI Service for astrometry\nand geodesy) and VCS1 to VCS4 experiments is 2917. Among them, 2468\nsources, or 85\\,\\%, are considered acceptable calibrators: having at\nleast eight successful observations at both X band (central frequency\n8.6~GHz) and S band (central frequency 2.3~GHz), and the\nsemi-major axis of the error ellipse of their coordinates being less\nthan 25~nrad ($\\approx$~5~mas). When observations from both the geodetic\nprograms and VCS1--4 are combined, the overall catalog provides fairly\ngood sky coverage. The probability of finding a calibrator within\n4\\degr\\ of any target north of declination $-40\\degr$ is 98.1\\,\\%.\n\nIn this paper we present an extension to the VCS catalogs, called the\nVCS5 catalog. It concentrates on the brightest flat-spectrum sources\nnorth of declination $-30\\degr$ not previously observed with VLBI under\ngeodetic and absolute astrometry programs. VCS5 is different from the\nprevious campaigns since its prime goal is to collect data needed for\nastrophysical analysis of active galactic nuclei (see\n\\S\\ref{s:objectives}).\n\nSince the observations, calibration, astrometric solutions and imaging\nare similar to that of VCS1--4, most of the details are described by\n\\cite{VCS1} and \\cite{VCS3} and will not be repeated here.\nIn \\S\\ref{s:objectives} we discuss\nscientific objectives for the VCS5 survey. In \\S\\ref{s:selection} we\ndescribe the strategy for selecting the 675 candidate sources observed in\nthree 24-hour VCS5 sessions with the VLBA based on analysis of the\navailable multi-frequency non-VLBI continuum radio measurements. The\nsame strategy was successfully applied by us earlier to select one\nhundred objects with the strongest estimated flux density at 8.6~GHz in\nthe framework of the VCS4 survey. Sixty seven out of these one hundred\nVCS4 candidates showed X~band correlated flux density greater than\n0.2~Jy \\citep{VCS4}. In \\S\\ref{s:obs_anal} we briefly outline the\nobservations and data processing. We present the VCS5 catalog in\n\\S\\ref{s:catalog}, and summarize our results in \\S\\ref{s:summary}.\n\n\\section{Scientific objectives for the VCS5 survey}\n\\label{s:objectives}\n\n There are two main scientific objectives for the VCS5 survey. We\nwould like to perform statistical analysis of physical properties of a\ndeep sample of compact AGNs on the basis of milliarcsecond scale images\nmeasured simultaneously at S~band and X~band with the VLBA. A cursory\nanalysis of the sample of the 2917 VCS\/ICRF sources observed with the\nVLBA in the S\/X mode revealed that it is nearly complete down to\n0.5~Jy but becomes incomplete at lower flux densities. As a result,\npossible usage of this largest collection of VLBI data for statistical\nanalysis of properties of active galactic nuclei at milliarcsecond\nscales is limited. The first goal of the current VCS5 project is to\nobserve the remaining bright sources with expected correlated flux\ndensities in the range 200~mJy to 600~mJy to create a statistically\ncomplete sample of extragalactic flat-spectrum radio sources with\nintegrated flux density at milliarcsecond scales greater than 200~mJy\nat X~band. This will make results of the VCS survey significantly more\nuseful for astrophysical applications. The uniformity of VCS data\nreduction as well as the completeness and homogeneity of the source\nsample will guarantee robust results from further statistical studies.\n\nThe second goal is to find more compact sources and to measure their\npositions precisely for use in geodetic applications including space\nnavigation, VLBI phase referencing of weak targets, and differential\nastrometry. Many applications prefer a more distant bright calibrator to\na near-by but weaker calibrator, since more time can be spent on the\ntarget. The VCS5 observations cover almost all remaining {\\em bright}\ncalibrators with correlated flux density at or greater that 200~mJy.\n\n\n\\section{Source selection}\n\\label{s:selection}\n\nOur source selection goal was to find all flat-spectrum radio sources\nbrighter than 0.2~Jy at 8.6~GHz that were missing from the VCS1 to VCS4 and\nICRF-Ext2 catalogs. We define flat radio spectrum as having a spectral\nindex $\\alpha>-0.5$ ($S\\sim\\nu^\\alpha$). To compile a list of missing\nobjects, we first selected all sources from the NVSS catalog\n\\citep{NVSS} with flux density at 1.4 GHz $S>50$~mJy, declination\n$\\delta>-30\\degr$, a Galactic latitude $|b|>1\\fdg5$, and not identified\nwith Galactic objects. Since the NVSS catalog is more than 99\\,\\%\ncomplete for flux density $S>50$~mJy, it is unlikely that sources with\nhighly inverted spectra and flux density $S>200$~mJy at 8.6~GHz will\nhave been missed.\n\n\\begin{figure}[t]\n\\begin{center}\n\\resizebox{1.0\\hsize}{!}\n{\n \\includegraphics[trim=0cm -0.5cm 0cm 0cm]{f1a.eps}\n \\includegraphics[trim=0cm -0.5cm 0cm 0cm]{f1b.eps}\n}\n\\resizebox{1.0\\hsize}{!}\n{\n \\includegraphics[trim=0cm 0cm 0cm 0cm]{f1c.eps}\n \\includegraphics[trim=0cm 0cm 0cm 0cm]{f1d.eps}\n}\n\\end{center}\n\\caption\nIllustration of the candidates selection procedure. The data from all\navailable publications were located through the CATS database.\nThe dotted falling lines represent $\\alpha=-0.5$.\nJ1109+3738 was not selected since its integrated spectral\nindex was less than $-0.5$. \nJ0446+5802 was not selected since its flux density extrapolated to\n8.6~GHz was less than 170~mJy. J1250+0216 and J2210+2013 satisfied all\nthe selection criteria mentioned in \\S~\\ref{s:selection} and were\nselected. Their $uv$ and image data are presented in\nFigure~\\ref{f:images}.\n\\label{f:selection}\n}\n\\end{figure}\n\nWe then searched the CATS database \\citep{CATS} containing almost all\nradio catalogs to find flux density measurements at other radio\nfrequencies for the selected NVSS sources.\nThese data were supplemented by results of the 1--22 GHz\ninstantaneous broad-band spectra measurements of $\\sim$3000\nextragalactic flat-spectrum radio sources which we performed at the\ntransit mode 600~m ring radio telescope \\mbox{RATAN-600} of the Russian\nAcademy of Sciences \\citep[see, e.g.,][]{Kovalev_etal99}. The collected\ndata were then analyzed semi-automatically, and bad data points, wrong\nidentifications, multiple data points corresponding to different\ncomponents of the same extended object were flagged. We found\nthat we could compile a complete sample of sources with total flux\ndensity spectrum\nflatter than $\\alpha=-0.5$, and with estimated total flux density of\n$S>170$~mJy at 8.6~GHz. In this complete sample were 675 candidates\nnot previously observed in geodetic VLBI mode, and these are the sources\nselected for VCS5 observations. Figure~\\ref{f:selection} presents\nexamples of plots of the total flux density spectra collected by the\nCATS database which we used for source selection.\n\nOur analysis of the multi-frequency catalogs and RATAN observations used\nfor selection indicates that we have found almost all of the sources\nwith spectral index greater than $-0.5$ and estimated total flux density\nat 8.6~GHz $S>170$~mJy. It is based on the fact that many used catalogs\nincluding NVSS \\citep{NVSS}, FIRST \\citep{FIRST}, 87GB \\citep{87GB}, GB6\n\\citep{GB6}, CLASS \\citep{Myers_etal03}, JVAS \\citep{JVAS1,JVAS2,JVAS3},\nPMN \\citep{Wright_etal94,Griffith_etal95,Wright_etal96}, and PKSCAT90\n\\citep{pkscat90}, are complete down to 150--250~mJy and below. This\nshould provide us with a sample of the same completeness\ncharacteristics. However, it is well known that flat spectrum sources\nare variable \\citep[e.g.,][]{KelPau_ARAA68}; consequently, the\nvariability corrupts at some level our estimations of spectral\nindex and total flux density. The membership of a source in the\ncompleteness sample is also changeable and depends on the observation\nepochs of the various compilation surveys. The quantitative analysis of\ncompleteness of the resulting correlated flux density limited sample of\nthe sources from the combined ICRF-Ext.2 and VCS1 to VCS5 catalogs will have\nto take into account the frequency dependent variability properties\n\\citep[e.g.,][]{KKNB02} as well as the compactness characteristics of flat\nspectrum sources \\citep[e.g.,][]{PopovKovalev99,2cmPaperIV}. This is\nbeyond the scope of the present paper and is deferred to another\npublication. We expect the present sample to be sufficiently complete,\nrobust, and unbiased for most statistical studies of flat-spectrum radio\nsources.\n\n\\section{Observations and data processing}\n\\label{s:obs_anal}\n\n The VCS5 observations were carried out in three 24-hour observing\nsessions with the VLBA on 2005 July 8, July 9, and July 20. Each of the\n675 target sources was observed in two scans of 120 seconds each. The\ntarget sources were observed in a sequence designed to minimize loss of\ntime from antenna slewing. In addition to these objects, 97 strong\nsources were taken from the GSFC astrometric catalog\n\\mbox{\\tt{}2004f\\_astro}\\footnote{\\url{http:\/\/vlbi.gsfc.nasa.gov\/solutions\/astro}}.\nObservations of three or four strong sources from this list were made every\n1\\,h to 1.5\\,h, 70\\,s to 80\\,s seconds per scan. These observations were scheduled\nin such a way that at each VLBA station at least one of these sources\nwas observed at an elevation angle less than 20\\degr, and at least one\nat an elevation angle greater than 50\\degr. The purpose of these\nobservations was to provide calibration for mis-modeled atmospheric path\ndelays and to tie the VCS5 source positions to the ICRF catalog\n\\citep{icrf98}. The list of tropospheric\ncalibrators\\footnote{\\url{http:\/\/vlbi.gsfc.nasa.gov\/vcs\/tropo\\_cal.html}}\nwas selected from the sources that, according to the 2~cm VLBA survey\nresults \\citep{2cmPaperIV}, showed the greatest compactness index, i.e.\\\nthe ratio of the correlated flux density measured at long VLBA spacings\nto the flux density integrated over the VLBA image. In total, 772\ntargets and calibrators were observed. The antennas were on-source about\n65\\,\\% of the time.\n\n Similar to the previous VLBA Calibrator Survey observing campaigns\n\\citep[e.g.,][]{VCS4}, we used the VLBA dual-frequency geodetic mode,\nobserving simultaneously at S~band and X~band. Each band was separated into\nfour 8~MHz channels (IFs) which spanned 140 MHz around 2.3~GHz and 490\nMHz around 8.6~GHz to provide precise measurements of group\ndelays for astrometric processing. Since the a~priori coordinates of\ncandidates were expected to have errors of up to 30\\arcsec, the data\nwere correlated with an accumulation period of 1~second in 64~spectral\nchannels in order to provide an extra-wide window for fringe search.\n\n Processing of the VLBA correlator output was done in three steps. In\nthe first step the data were calibrated using the Astronomical Image\nProcessing System (AIPS) \\citep{aips}. In the second step data were\nimported to the Caltech DIFMAP package~\\citep{difmap}, $uv$ data\nflagged, and maps were produced using an automated procedure of hybrid\nimaging developed by Greg Taylor~\\citep{difmap-script} which we adopted\nfor our needs. We were able to reach the VLBA image thermal noise level\nfor most of our CLEAN images \\citep{VLBA_summ}. Errors on our estimates\nof correlated flux density values for sources stronger than\n$\\sim$100~mJy were dominated by the accuracy of amplitude\ncalibration, which for the VLBA, according to \\citet{VLBA_summ}, is at\nthe level of 5\\,\\% at 1~GHz to 10~GHz. An additional error is introduced by\nthe fact that our frequency channels are widely spread over receiver\nbands while the VLBA S~band and X~band gain-curve parameters are measured\naround 2275~GHz and 8425~MHz respectively \\citep{VLBA_summ}, and the noise\ndiode spectrum is not ideally flat. However, this should not add more\nthan a few percent to the total resulting error. Our error estimate was\nconfirmed by comparison of the flux densities integrated over the VLBA\nimages with the single-dish flux densities which we measured with\n\\mbox{RATAN--600} in June and August 2005 for slowly varying sources\nwithout extended structure. The methods of single-dish observations and\ndata processing can be found in \\citet{Kovalev_etal99}. In the third\nstep, the data were imported to the Calc\/Solve program, group delays\nambiguities were resolved, outliers eliminated, and coordinates of new\nsources were adjusted using ionosphere-free combinations of X~band and\nS~band group delay observables of the three VCS5 sessions, 19 VCS1 to\nVCS4 experiments and 3976 24-hour International VLBI Service for\nastrometry and geodesy (IVS)\nexperiments\\footnote{\\url{http:\/\/vlbi.gsfc.nasa.gov\/solutions\/2005c}} in\na single least square solution. Positions of 3486 sources were estimated\nincluding all detected VCS5 sources: 590 targets and 97 tropospheric\ncalibrators. Boundary conditions were imposed requiring zero\nnet-rotation of position adjustments of the 212 sources listed as\ndefining sources in the ICRF catalog with respect to their coordinates\nfrom that catalog.\n\n In a separate solution, coordinates of the 97 well known tropospheric\ncalibrators were estimated from the VCS5 observing sessions only.\nComparison of these estimates with coordinates derived from the 3976 IVS\ngeodetic\/astrometric sessions provided us a measure of the accuracy of\nthe coordinates from the VCS5 observing campaign. The differences in\ncoordinate estimates were used for computation of parameters $a$ and\n$b(\\delta)$ of an error inflation model in the form $ \\sqrt{ (a\\,\n\\sigma)^2 + b(\\delta)^2}$, where $\\sigma$ is an uncertainty derived from\nthe fringe amplitude signal to noise ratio using the error propagation\nlaw and $\\delta$ is declination. More details about the analysis and\nimaging procedures can be found in \\cite{VCS1} and \\cite{VCS3}. The\nhistogram of source position errors is presented in\nFigure~\\ref{f:errhist}.\n\n\\begin{figure}[b]\n\\begin{center}\n\\resizebox{\\hsize}{!}{\n\\includegraphics[trim=0cm 0cm 0cm 0cm]{f2.eps}\n}\n\\end{center}\n\\caption{Histogram of semi-major error ellipse of position errors.\nThe last bin shows errors exceeding 9~mas.\nSee explanation of different assigned source classes in\n\\S\\,\\ref{s:obs_anal}, \\S\\,\\ref{s:catalog}.\n\\label{f:errhist}\n}\n\\end{figure}\n\n\\begin{figure*}[p]\n\\begin{center}\n\\resizebox{1.0\\hsize}{!}\n{\n \\includegraphics[trim=-0.5cm 5cm 0cm 0cm]{f3a.eps}\n \\includegraphics[trim=-1.0cm 5cm 0cm 0cm]{f3b.eps}\n \\includegraphics[trim=-1.0cm 5cm 0cm 0cm]{f3c.eps}\n}\n\n\\resizebox{1.0\\hsize}{!}\n{\n \\includegraphics[trim=0cm 0.0cm 0cm 0cm,angle=270]{f3d.eps}\n \\includegraphics[trim=0cm -0.5cm 0cm 0cm,angle=270]{f3e.eps}\n \\includegraphics[trim=0cm -0.5cm 0cm 0cm,angle=270]{f3f.eps}\n}\n\n\\resizebox{1.0\\hsize}{!}\n{\n \\includegraphics[trim=-0.5cm 5cm 0cm 0cm]{f3g.eps}\n \\includegraphics[trim=-1.0cm 5cm 0cm 0cm]{f3h.eps}\n \\includegraphics[trim=-1.0cm 5cm 0cm 0cm]{f3i.eps}\n}\n\n\\resizebox{1.0\\hsize}{!}\n{\n \\includegraphics[trim=0cm 0.0cm 0cm 0cm,angle=270]{f3j.eps}\n \\includegraphics[trim=0cm -0.5cm 0cm 0cm,angle=270]{f3k.eps}\n \\includegraphics[trim=0cm -0.5cm 0cm 0cm,angle=270]{f3l.eps}\n}\n\\end{center}\n\\figcaption{\\footnotesize \nFrom top to bottom.\n{\\em Row~1:}\nNaturally weighted CLEAN images at S~band (2.3~GHz). The lowest contour\nis plotted at the leven given by ``clev'' in each panel title (Jy\/beam),\nthe peak brightness is given by ``max'' (Jy\/beam). The contour levels\nincrease by factors of two. The dashed contours indicate negative flux.\nThe beam is shown in the bottom left corner of the images.\n{\\em Row~2:}\nDependence of the correlated flux density at S~band on projected\nspacing. Each point represents a coherent\naverage over one 2~min observation on an individual interferometer\nbaseline. The error bars represent only the statistical errors.\n{\\em Row~3:} Naturally\nweighted CLEAN images at X~band (8.6~GHz).\n{\\em Row~4:} Dependence of the\ncorrelated flux density at X~band on projected spacing.\n\\label{f:images}\n}\n\\end{figure*}\n\n In total, 590 out of 675 sources were detected and yielded at least\ntwo good points for position determination. This 87\\,\\% detection rate \nconfirms the validity of the applied candidate selection procedure\n(\\S\\ref{s:selection}). It should be noted that, due to an omission, the\nlist of target sources contained 21 objects previously observed and\ndetected in the VCS4 campaign.\n\nHowever, not all of these 590 sources are suitable as phase reference\ncalibrators or as targets for geodetic observations. Following\n\\citet{VCS3} we consider a source suitable as a calibrator if 1) the\nnumber of good X\/S pairs of observations is eight or greater in order to\nrule out the possibility of a group delay ambiguity resolution error,\nand 2) the position error before re-weighting is less than 5~mas\nfollowing the strategy adopted in processing VCS observations. Only 433\nsources satisfy this calibrator criteria. Other detected sources were\nsomewhat resolved and\/or below the detection limit of these observations\nof 60 mJy. Some of these may become suitable phase calibrators for\nfuture experiments with higher data rates and more sensitivity than the\nVCS surveys. Among the 157 non-calibrators, 53 sources had less than\neight observations at X~band and less than eight observations S~band.\nPositions of these sources we consider as unreliable, because we \ncannot rule out errors in group delay ambiguity resolution. The other 104\nsources had eight or more observations at one of the bands, so we can rule\nout the possibility of group delay ambiguity resolution errors, and\ntherefore, we consider that our estimates of positions of these sources\nare reliable.\n\n\\section{The VCS5 catalog}\n\\label{s:catalog}\n\n\\begin{figure}[t]\n\\begin{center}\n\\resizebox{1.0\\hsize}{!}\n{\n \\includegraphics[trim= 0.00cm 0cm 0cm 0cm]{f4a.eps}\n \\includegraphics[trim= 1.5cm 0cm 0cm 0cm,clip]{f4b.eps}\n}\n\\end{center}\n\\figcaption{\nDistributions of flux density integrated over VLBA image for all\ndetected VCS5 sources (columns 10 and 12 of the Table~\\ref{t:cat}).\n\\label{f:hist_s_spi}\n}\n\\end{figure}\n\nThe VCS5 catalog of 590 detected target sources is listed in\nTable~\\ref{t:cat}. The first column gives source class: ``C'' if the\nsource can be used as a calibrator, ``N'' if it cannot but determined\npositions are reliable, ``U''---non-calibrator, unreliable positions.\nThe second and third columns give IVS source name (B1950 notation), and\nIAU name (J2000 notation). The fourth and fifth columns give measured\nsource coordinates at the J2000.0 epoch. Columns 6 and 7 give inflated\nsource position uncertainties in right ascension (without $\\cos\\delta$\nfactor) and declination in mas, and column 8 gives the correlation\ncoefficient between the errors in right ascension and declination. The\nnumber of group delays used for position determination is listed in\ncolumn 9. Columns 10 and 12 give the estimate of the flux density\nintegrated over the entire map in janskies at X~band and S~band\nrespectively. This estimate was computed as a sum of all CLEAN\ncomponents if a CLEAN image was produced. If we did not have enough\ndetections of the visibility function to produce a reliable image, the\nintegrated flux density was estimated as the median of the correlated\nflux density measured at projected spacings less than 25~M$\\lambda$ and\n7~M$\\lambda$ for X~band and S~bands respectively. The integrated flux\ndensity means the total flux density with spatial frequencies less than\n4~M$\\lambda$ at X~band and 1~M$\\lambda$ at S~band filtered out, or in\nother words, the flux density from all components within a region about\nor less than 50~mas at X~band and 200~mas at S~band. Columns 11 and 13\ngive the flux density of unresolved components estimated as the median\nof correlated flux density values measured at projected spacings greater\nthan 170~M$\\lambda$ for X~band and greater than 45~M$\\lambda$ for\nS~band. For some sources no estimates of the integrated and\/or\nunresolved flux density are presented, because either no data were\ncollected on the baselines used in the calculations, or these data were\nunreliable. Column 14 gives the data type used for position estimation:\nX\/S stands for ionosphere-free linear combination of X and S wide-band\ngroup delays; X stands for X-band-only group delays; and S stands for\nS-band-only group delays. Some sources for which less than eight pairs\nof X~band and S~band group delay observables were available had two or\nmore observations at X~band and\/or S~band. For these sources either\nX~band or S~band only estimates of coordinates are listed in the VCS5\ncatalog.\n\n\\begin{figure}[t]\n\\begin{center}\n\\resizebox{1.0\\hsize}{!}\n{\n \\includegraphics[trim=0cm 0cm 0cm 0cm,angle=270]{f5.eps}\n}\n\\end{center}\n\\figcaption{\nThe probability (filled circles) of finding a calibrator in any given\ndirection within a circle of a given radius, north of declination\n$-30\\degr$. All sources from 3976 IVS geodetic\/astrometric sessions and\n22 VCS1 to VCS5 VLBA sessions that are classified as calibrators are taken\ninto account.\n\\label{f:prob}\n}\n\\end{figure}\n\n\n In addition to this table, the html version of the catalog is posted\non the Web\\footnote{\\url{http:\/\/vlbi.gsfc.nasa.gov\/vcs5}}. For each\nsource there are eight links: to a pair of postscript images of the\nsource at X~band and S~band; a pair of plots of correlated flux\ndensity as a function of baseline length projected to the source plane;\na pair of FITS files of CLEAN components of naturally weighted source\nimages; and to a pair of FITS files with calibrated $uv$ data. This\ndataset is also accessible from the NRAO\narchive\\footnote{\\url{http:\/\/archive.nrao.edu}} which links the files to\nthe Virtual Observatory. The positions and the plots are also accessible\nfrom the updated NRAO VLBA Calibrator Search\nweb-page\\footnote{\\url{http:\/\/www.vlba.nrao.edu\/astro\/calib}}.\n\n\\begin{deluxetable*}{c l l l r r r r r r r r r c}\n\\tablewidth{0pt}\n\\tablecaption{\\rm The VCS5 catalog \\label{t:cat}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n \\colhead{} &\n \\multicolumn{2}{c}{Source name} &\n \\multicolumn{2}{c}{J2000.0 Coordinates} &\n \\multicolumn{2}{c}{Errors (mas)} &\n \\colhead{} &\n \\colhead{} &\n \\multicolumn{4}{c}{Correlated flux density (Jy)} &\n \\colhead{}\n \\vspace{0.5ex} \\\\\n \\multicolumn{9}{c}{} &\n \\multicolumn{2}{c}{8.6 GHz} &\n \\multicolumn{2}{c}{2.3 GHz} \\vspace{0.5ex} \\\\\n \n \\colhead{Class} &\n \\colhead{IVS} &\n \\colhead{IAU} &\n \\colhead{Right ascension} &\n \\colhead{Declination} &\n \\colhead{$\\Delta \\alpha$} &\n \\colhead{$\\Delta \\delta$} &\n \\colhead{Corr} &\n \\colhead{\\# Obs} &\n \\colhead{Total } &\n \\colhead{Unres } &\n \\colhead{Total } &\n \\colhead{Unres } &\n \\colhead{Band} \n \\vspace{0.5ex} \\\\\n \\colhead{(1)} &\n \\colhead{(2)} &\n \\colhead{(3)} &\n \\colhead{(4)} &\n \\colhead{(5)} &\n \\colhead{(6)} &\n \\colhead{(7)} &\n \\colhead{(8)} &\n \\colhead{(9)} &\n \\colhead{(10)} &\n \\colhead{(11)} &\n \\colhead{(12)} &\n \\colhead{(13)} &\n \\colhead{(14)} \n }\n\\startdata\nC & \\objectname{0008$+$006} & \\objectname{J0011$+$0057} & 00 11 30.403309 & $+$00 57 51.87984 & 1.02 & 2.00 & 0.114 & 25 & 0.09 & 0.07 & \\nodata & \\nodata & X \\\\\nC & \\objectname{0009$+$467} & \\objectname{J0012$+$4704} & 00 12 29.302900 & $+$47 04 34.73946 & 0.77 & 1.08 & $-$0.371 & 35 & 0.13 & 0.12 & 0.10 & $<$0.06 & X\/S \\\\\nN & \\objectname{0013$-$240} & \\objectname{J0016$-$2343} & 00 16 05.738818 & $-$23 43 52.18956 & 30.58 & 16.85 & $-$0.808 & 17 & \\nodata & \\nodata & 0.15 & 0.08 & S \\\\\nC & \\objectname{0015$-$054} & \\objectname{J0017$-$0512} & 00 17 35.817204 & $-$05 12 41.76727 & 0.46 & 0.92 & $-$0.278 & 54 & 0.20 & 0.12 & 0.14 & 0.09 & X\/S \\\\\nC & \\objectname{0015$-$280} & \\objectname{J0017$-$2748} & 00 17 59.006128 & $-$27 48 21.57153 & 1.75 & 3.51 & 0.712 & 32 & 0.24 & 0.16 & 0.20 & 0.06 & X\/S \\\\\nC & \\objectname{0034$+$078} & \\objectname{J0037$+$0808} & 00 37 32.197173 & $+$08 08 13.05750 & 0.38 & 0.50 & $-$0.225 & 76 & 0.25 & 0.14 & 0.19 & 0.10 & X\/S \\\\\nC & \\objectname{0035$-$037} & \\objectname{J0038$-$0329} & 00 38 20.794340 & $-$03 29 58.96178 & 0.32 & 0.63 & $-$0.347 & 77 & 0.20 & 0.16 & 0.31 & 0.21 & X\/S \\\\\nN & \\objectname{0036$-$191} & \\objectname{J0039$-$1854} & 00 39 16.924431 & $-$18 54 05.61863 & 4.97 & 9.83 & 0.749 & 8 & 0.12 & \\nodata & 0.18 & 0.10 & X\/S \\\\\nC & \\objectname{0037$+$011} & \\objectname{J0040$+$0125} & 00 40 13.525489 & $+$01 25 46.35014 & 0.99 & 1.94 & $-$0.124 & 29 & 0.11 & $<$0.06 & $<$0.06 & \\nodata & X \\\\\nC & \\objectname{0041$+$677} & \\objectname{J0044$+$6803} & 00 44 50.759589 & $+$68 03 02.68607 & 0.91 & 0.58 & $-$0.130 & 59 & 0.28 & 0.22 & 0.14 & $<$0.06 & X\/S \\\\\n\\enddata\n\\tablecomments{\nTable~\\ref{t:cat} is presented in its entirety in the electronic edition\nof the Astronomical Journal. A portion is shown here for guidance\nregarding its form and contents. Assigned source class in (1) is `C' for\ncalibrator, `N' for non-calibrator with reliable coordinates,\n`U' for non-calibrator with unreliable coordinates; see\n\\S\\,\\ref{s:obs_anal}, \\S\\,\\ref{s:catalog} for details. Units of right ascension\nare hours, minutes and seconds; units of declination are degrees,\nminutes and seconds.\n}\n\\end{deluxetable*}\n\n\\begin{deluxetable}{l l c c}\n\\tablewidth{0pt}\n\\tablecaption{\\rm Sources not detected in VCS5 VLBA observations\\label{t:nondetected}}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n \\multicolumn{2}{c}{Source name} &\n \\multicolumn{2}{c}{J2000.0 Coordinates} \n \\vspace{0.5ex} \\\\\n \\colhead{IVS} &\n \\colhead{IAU} &\n \\colhead{Right ascension} &\n \\colhead{Declination} \n \\vspace{0.5ex} \\\\\n \\colhead{(1)} &\n \\colhead{(2)} &\n \\colhead{(3)} &\n \\colhead{(4)} \n }\n\\startdata\n\\objectname{0032$-$011} & \\objectname{J0034$-$0054} & 00 34 43.93 & $-$00 54 13.0 \\\\\n\\objectname{0039$+$211} & \\objectname{J0041$+$2123} & 00 41 45.10 & $+$21 23 41.1 \\\\\n\\objectname{0104$+$650} & \\objectname{J0107$+$6521} & 01 07 51.35 & $+$65 21 20.5 \\\\\n\\objectname{0137$-$273} & \\objectname{J0139$-$2705} & 01 39 55.42 & $-$27 05 29.4 \\\\\n\\objectname{0212$-$214} & \\objectname{J0214$-$2113} & 02 14 40.73 & $-$21 13 28.2 \\\\\n\\objectname{0258$+$356} & \\objectname{J0301$+$3551} & 03 01 47.96 & $+$35 51 24.5 \\\\\n\\objectname{0426$+$351} & \\objectname{J0430$+$3516} & 04 30 14.41 & $+$35 16 23.1 \\\\\n\\objectname{0434$-$225} & \\objectname{J0436$-$2226} & 04 36 34.31 & $-$22 26 18.6 \\\\\n\\objectname{0506$-$196} & \\objectname{J0508$-$1935} & 05 08 19.03 & $-$19 35 56.4 \\\\\n\\objectname{0512$-$129} & \\objectname{J0515$-$1255} & 05 15 17.51 & $-$12 55 27.8 \\\\\n\\enddata\n\\tablecomments{\nTable~\\ref{t:nondetected} is presented in its entirety in the electronic\nedition of the Astronomical Journal. A portion is shown here for\nguidance regarding its form and contents. Units of right ascension are\nhours, minutes and seconds; units of declination are degrees, minutes\nand seconds. The 85 sources presented in this electronic table include\nthe gravitation lens \\objectname{1422+231} which was detected, but not\nprocessed in VCS5. The J2000 source positions are taken from the NVSS\nsurvey \\citep{NVSS}, they were used for VCS5 VLBA observing and\ncorrelation.\n}\n\\end{deluxetable}\n\nTable~\\ref{t:nondetected} presents 85 sources not detected in VCS5 VLBA\nobservations. Source positions used for observations and correlation are\npresented. They are taken from \\cite{NVSS}. The correlated flux density\nfor the non-detected sources is estimated to be less than 60~mJy at 2.3~GHz\nand 8.6~GHz.\n\nFigure~\\ref{f:images} presents examples of naturally weighted contour\nCLEAN images as well as correlated flux density versus projected spacing\ndependence for three sources: the strongest VCS5 source at X~band,\n\\objectname{J1250$+$0216}, with two bright components resolved at X~band and\nnot resolved at S~band; a steep spectrum source with extended structure,\n\\objectname{J0932$+$6507}; and the source with the most inverted\nspectrum and very compact structure at the milliarcsecond scale,\n\\objectname{J2210$+$2013}.\n\nFigure~\\ref{f:hist_s_spi} presents histograms of the 2.3~GHz and 8.6~GHz\nintegrated flux density for 590 detected VCS5 sources, 132 out of which\nhave integrated flux density $S\\ge 200$~mJy at 8.6~GHz. Their addition\nto the previously observed sources will form the statistically complete\nsample north of declination $-30\\degr$. It is interesting to note that\nmany of the discovered VCS5 sources have inverted radio spectra. The\n50~mJy cutoff for the original selection of sources from the NVSS\ncatalog allowed us to add inverted-spectrum objects to the list of\ncandidates. A few tens of new compact VCS5 objects with high flux\ndensity on VLBA baselines will be useful for geodetic applications. \n\n The sky calibrator density for different radii of a search circle for\ndeclination $\\delta>-30\\degr$ is presented in Figure~\\ref{f:prob}. As\ndiscussed in \\cite{VCS4}, the addition of these sources to the VLBA\nCalibrator list did not affect significantly the density for the search\nradius of 4\\degr, but increases it for smaller search circles, e.g.,\nthe probability of finding a calibrator within $2\\fdg5$ is now 83\\,\\%. This\nis beneficial for many applications requiring bright compact calibrator\nwithin 2\\degr to 3\\degr of a target.\n\n\\section{Summary}\n\\label{s:summary}\n\nThe VCS5 Survey has made a significant step towards constructing a\nhomogeneous statistically complete sample of flat-spectrum compact\nextragalactic radio sources north of declination $-30\\degr$ with\nintegrated VLBA flux density greater than about 200~mJy at 8~GHz. The\nVCS5 Survey has added 569 new sources, not previously observed with VLBI\nunder geodesy and absolute astrometry programs. Among them, 433 sources\nare suitable as phase referencing calibrators and as target sources for\ngeodetic applications. After processing the VCS5 experiments, the total\nnumber of sources with positions known at the nanoradian level is 3486,\nand the number of VLBI calibrators has grown from 2472 to 2905. This\npool of calibrators was formed from analysis of 22 VLBA Calibrator\nSurvey and 3976 24-hour IVS astrometry and geodesy observing\nsessions.\n\nIn the present paper we do not yet provide quantitative estimates of\ncompleteness of our list of compact flat-spectrum sources. In order to\nget these estimates we are going to (i) complete the homogeneous imaging\nof all of 3486 sources and get estimates of their integrated flux\ndensities at milliarcsecond scales in the X~band and S~band, (ii) complete\nprocessing of instantaneous single-dish multi-frequency, multi-epoch flux\ndensity measurements with \\mbox{RATAN--600} for this sample, (iii)\nobserve with the VLBA a total-flux-density limited sample of all sources\nregardless of their spectral index over a relatively large portion of\nthe sky complemented with simultaneous multi-frequency single-dish\nmeasurements. The latter will allow us to assess whether conclusions\ndrawn from the VLBI flat-spectrum source samples can be extended to the\nwhole population of extragalactic objects regardless their continuum\nspectrum characteristics.\n\n\n\n\\acknowledgments\n\n We would like to thank A.~Roy and the anonymous referee for useful\ncomments which helped to improve the manuscript.\n \\facility[NRAO(VLBA)]{The National Radio Astronomy Observatory is a\nfacility of the National Science Foundation operated under cooperative\nagreement by Associated Universities, Inc.} We thank the staff of the\nVLBA for carrying these observations in their usual efficient manner.\n Y.~Y.~Kovalev is a Research Fellow of the Alexander von Humboldt\nFoundation. This work was done while D.~Gordon worked for Raytheon,\nunder NASA contract NAS5--01127. \\mbox{RATAN--600} observations were\npartly supported by the Russian Ministry of Education and Science, the\nNASA JURRISS Program (project W--19611), and the Russian Foundation for\nBasic Research (projects 01--02--16812 and 05--02--17377). The authors\nmade use of the database CATS \\citep{CATS} of the Special Astrophysical\nObservatory. This research has made use of the NASA\/IPAC Extragalactic\nDatabase (NED) which is operated by the Jet Propulsion Laboratory,\nCalifornia Institute of Technology, under contract with the National\nAeronautics and Space Administration.\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{What is {\\sc Cinderella}?}\n\n{\\sc Cinderella} is an abbreviation of ``{\\bf C}omparison of {\\bf INDE}pendent {\\bf REL}ative {\\bf L}east-squares {\\bf A}mplitudes''. It provides a quantitative comparison between the DFT amplitude spectra of time-resolved astronomical measurements.\n\nThe {\\sc SigSpec} technique (Reegen~2005, 2007) allows to determine probabilities for coincident peaks in the DFT amplitude spectra of different datasets quantitatively and in a statistically unbiased way. The theoretical background of this procedure is introduced by Reegen et al.~(2008).\n\n{\\sc Cinderella} uses the standard output of the program {\\sc SigSpec}, which represents the results of a cascade of consecutive prewhitenings employing least-squares fits to obtain amplitudes and phases of signal components. Following the {\\sc SigSpec} terminology, {\\sc Cinderella} returns a spectral significances (hereafter abbreviated by `sig') rather than a probability. This manual uses cross-references to {\\sc SigSpec} frequently. In these cases, the reader is referred to the {\\sc SigSpec} manual (Reegen 2009).\n\n\\section{Projects}\\label{CINDERELLA_Projects}\n\n{\\sc Cinderella} is called by the command line\n\n\\begin{scriptsize}\\begin{verbatim}\nCinderella \n\\end{verbatim}\\end{scriptsize}\n\n\\noindent where {\\tt } is the name (or path, if desired) of the {\\sc Cinderella} project. The project structure is strictly consistent with {\\sc SigSpec}.\n\nBefore running the program, the user has to provide\n\n\\begin{enumerate}\n\\item at least two time series input files consistent with the {\\sc SigSpec} MultiFile mode (see {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_MultiFile Mode}, and ``Time series input files'', p.\\,\\pageref{CINDERELLA_Time series input files}), and\n\\item a directory {\\tt } containing the {\\sc SigSpec} result files corresponding to the time series input files (see ``{\\sc SigSpec} result files'', p.\\,\\pageref{CINDERELLA_SigSpec result files}).\n\\end{enumerate}\n\nFurthermore, the user may pass a set of specifications to {\\sc Cinderella} by means of a file {\\tt .cnd} (see ``The {\\tt .cnd} file'', p.\\,\\pageref{CINDERELLA_The .cnd file}). This file is expected in the same folder as the time series input files and the project directory. For specifications not given by the user, defaults are used.\n\n{\\sc Cinderella} is designed to answer two different questions, depending on the problem it is applied to. By default, the program runs through both modes simultaneously and provides both conditional (p.\\,\\pageref{CINDERELLA_Conditional Mode}) and composed sigs (p.\\,\\pageref{CINDERELLA_Composed Mode}).\n\n\\section{Input}\\label{CINDERELLA_Input}\n\n\\subsection{Time series input files}\\label{CINDERELLA_Time series input files}\n\n{\\sc Cinderella} expects at least two time series input files named according to {\\tt \\#index\\#..dat}. In this context, {\\tt \\#index\\#} is a placeholder for a six-digit index of the file. Note that only files with consecutive indices are appropriately recognised by {\\sc Cinderella}. The conventions are compatible with the {\\sc SigSpec} MultiFile mode (see {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_MultiFile Mode}), whence the most convenient preparation of data for {\\sc Cinderella} is the {\\sc SigSpec} MultiFile computation.\n\nThe only restrictions to the format of the time series input files are that the number of items per row has to be constant for all rows in the file and that the columns have to be seperated by at least one whitespace character or tab. Dataset entries need not to be numeric, except for the columns specified as time, observable, and weights. The conventions for specifying these three column types are consistent with {\\sc SigSpec}. See {\\sc SigSpec} manual, pp.\\,\\pageref{SIGSPEC_Time series columns representing time and observable}, \\pageref{SIGSPEC_Time series columns containing statistical weights} for details.\n\n\\figureDSSN{f1.eps}{{\\sc SigSpec} result files used as input for the sample project {\\tt CinderellaNative}. The bottom panel refers to the file {\\tt CinderellaNative\/000000.result.dat}, which is used as the comparison dataset. Above, the files {\\tt CinderellaNative\/000001.result.dat} to {\\tt CinderellaNative\/000008.result.dat} are displayed from bottom to top. The underlying time series represent Gaussian noise plus a single sinusoidal signal at a frequency of $3.125\\,\\mathrm{d}^{-1}$ (grey line) with different amplitudes.}{CINDERELLA_normalrun.res}{!htb}{clip,angle=0,width=110mm}\n\n\\subsection{{\\sc SigSpec} result files}\\label{CINDERELLA_SigSpec result files}\n\nThe {\\sc SigSpec} result files {\\tt \\#index\\#.result.dat} are located in the project directory and contain a list of all sig maxima associated to each of the input time series {\\tt \\#index\\#..dat}. A {\\sc SigSpec} result file consists of seven columns. A full description is provided by the {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_Result files}. {\\sc Cinderella} uses only five columns:\n\n\\begin{itemize}\n\\item the frequency [inverse time units] (column 1),\n\\item the sig (column 2),\n\\item the amplitude [units of observable] (column 3),\n\\item rms scatter of the time series before prewhitening (column 5), and\n\\item point-to-point scatter of the time series before prewhitening (column 6).\n\\end{itemize}\n\nThe last line of a result file contains only two non-zero values in columns 5 and 6. These represent the rms and point-to-point scatter of the time series after the last prewhitening step, correspondingly, and are also used by {\\sc Cinderella}.\n\n\\vspace{12pt}\\noindent{\\bf Example.}\\label{CINDERELLA_EXnormalrun} {\\it The sample project {\\tt CinderellaNative} provides a run without any additional options by typing {\\tt Cinderella CinderellaNative}. The files {\\tt 000000.CinderellaNative.dat}, ..., {\\tt 000008.CinderellaNative.dat} are the same as {\\tt 000038.diffsig.dat} to {\\tt 000046.diffsig.dat} in the {\\sc SigSpec} example {\\tt diffsig} (Reegen 2009, p.\\,\\pageref{SIGSPEC_EXdiffsig}), correspondingly.\n\nThe {\\sc SigSpec} result files {\\tt 000000.result.dat} to {\\tt 000008.result.dat} are provided in the project directory {\\tt CinderellaNative}. They contain all signal components found with sig $>$ 2 and are displayed in Fig.\\,\\ref{CINDERELLA_normalrun.res}\\it .\n\nThe screen output at runtime starts with a standard header.}\n\n\\begin{scriptsize}\\begin{verbatim}\n CCCCC ii dd ll ll\nCC CC dd ll ll\nCC ii n nnn ddddd eeee r rrr eeee ll ll aaaa\nCC ii nn nn dd dd ee ee rr rr ee ee ll ll aa aa\nCC ii nn nn dd dd ee ee rr ee ee ll ll aa\nCC ii nn nn dd dd eeeeee rr eeeeee ll ll aaaaa\nCC ii nn nn dd dd ee rr ee ll ll aa aa\nCC CC ii nn nn dd dd ee ee rr ee ee ll ll aa aa\n CCCCC ii nn nn ddd d eeee rr eeee ll ll aaa a\n\n\nComparison of INDEpendent RELative Least-squares Amplitudes\nVersion 1.0\n************************************************************\nby Piet Reegen\nInstitute of Astronomy\nUniversity of Vienna\nTuerkenschanzstrasse 17\n1180 Vienna, Austria\nRelease date: April 29, 2008\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The program starts with processing the command line, checking if a present directory {\\tt CinderellaNative} is present, and searching for a configuration file {\\tt CinderellaNative.cnd} (see ``The {\\tt .cnd} file'', p.\\,\\pageref{CINDERELLA_The .cnd file}). Since there is no such file present, {\\sc Cinderella} produces a warning message.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** start **************************************************\n\nChecking availability of project directory CinderellaNative...\nproject directory CinderellaNative ok.\n\nWarning: CndFile_LoadCnd 001\n Failed to open .cnd file.\n\\end{verbatim}\\end{scriptsize}\n\n{\\it Now {\\sc Cinderella} counts the time series input files and checks for corresponding {\\sc SigSpec} result files.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** MultiFile count ****************************************\n\nNumber of files 9\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The next step is to count the rows in each {\\sc SigSpec} result file.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** count rows in SigSpec result files *********************\n\nCinderellaNative\/000000.result.dat: 106 rows\nCinderellaNative\/000001.result.dat: 21 rows\nCinderellaNative\/000002.result.dat: 21 rows\nCinderellaNative\/000003.result.dat: 21 rows\nCinderellaNative\/000004.result.dat: 21 rows\nCinderellaNative\/000005.result.dat: 21 rows\nCinderellaNative\/000006.result.dat: 21 rows\nCinderellaNative\/000007.result.dat: 21 rows\nCinderellaNative\/000008.result.dat: 21 rows\n\\end{verbatim}\\end{scriptsize}\n\n{\\it Before reading the input files, {\\sc Cinderella} performs the assignment of {\\tt target}, {\\tt comp} and {\\tt skip} tags to the datasets (see ``Dataset Types'', p.\\,\\pageref{CINDERELLA_Dataset Types}). Since there is no file {\\tt CinderellaNative.cnd} available, the defaults are used: the first file, {\\tt 000000.CinderellaNative.dat} is considered a comparison dataset, the eight remaining files are interpreted as targets.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** dataset type assignment ********************************\n\n\nWarning: CndFile_Cind 001\n Failed to open .cnd file.\n Assigning default types.\n\n000000.CinderellaNative.dat: comparison (default)\n000001.CinderellaNative.dat: target (default)\n000002.CinderellaNative.dat: target (default)\n000003.CinderellaNative.dat: target (default)\n000004.CinderellaNative.dat: target (default)\n000005.CinderellaNative.dat: target (default)\n000006.CinderellaNative.dat: target (default)\n000007.CinderellaNative.dat: target (default)\n000008.CinderellaNative.dat: target (default)\n\nnumber of target datasets 8\nnumber of comparison datasets 1\nnumber of datasets to skip 0\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The time series and {\\sc SigSpec} result files are read, and {\\sc Cinderella} displays the frequency resolution and the mean observable for each time series.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** read input files ***************************************\n\n000000.CinderellaNative.dat: Rayleigh resolution 0.1089880382935977\n000000.CinderellaNative.dat: mean observable 0.0074422789800987\nCinderellaNative\/000000.result.dat\n000001.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000001.CinderellaNative.dat: mean observable -0.0704869463068650\nCinderellaNative\/000001.result.dat\n000002.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000002.CinderellaNative.dat: mean observable -0.0875484339175066\nCinderellaNative\/000002.result.dat\n000003.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000003.CinderellaNative.dat: mean observable -0.1046099215281657\nCinderellaNative\/000003.result.dat\n000004.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000004.CinderellaNative.dat: mean observable -0.1216714091388107\nCinderellaNative\/000004.result.dat\n000005.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000005.CinderellaNative.dat: mean observable -0.1387328967494604\nCinderellaNative\/000005.result.dat\n000006.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000006.CinderellaNative.dat: mean observable -0.1557943843601256\nCinderellaNative\/000006.result.dat\n000007.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000007.CinderellaNative.dat: mean observable -0.1728558719707690\nCinderellaNative\/000007.result.dat\n000008.CinderellaNative.dat: Rayleigh resolution 0.0914470160931467\n000008.CinderellaNative.dat: mean observable -0.1899173595814251\nCinderellaNative\/000008.result.dat\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The amplitudes in the {\\sc SigSpec} result file of the comparison dataset (file {\\tt CinderellaNative\/000000.result.dat} have to be transformed in order to be comparable to the target datasets (see ``Amplitude transformation'', p.\\,\\pageref{CINDERELLA_Amplitude transformation}). {\\sc Cinderella} uses the rms residual as a measure for this transformation. This is the default setting.}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** amplitude transformation *******************************\n\nby rms error: file 0\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The core of {\\sc Cinderella} consists of three different analyses. The first part of these is the computation of conditional sigs for each pair of target vs.~comparison datasets. In this case we have eight target datasets and one comparison dataset, which results in eight operations. In general, the total number of operations performed in this step is the product of the number of target datasets times the number of comparison datasets. Detailed information on the computation of pairwise conditional sigs is found in ``Single-comparison output'' (p.\\,\\pageref{CINDERELLA_Single-comparison output}).}\\pagebreak\n\n\\begin{scriptsize}\\begin{verbatim}\n*** pairwise Cinderella analysis ***************************\n\n 1 vs. 0: conditional sig\n 2 vs. 0: conditional sig\n 3 vs. 0: conditional sig\n 4 vs. 0: conditional sig\n 5 vs. 0: conditional sig\n 6 vs. 0: conditional sig\n 7 vs. 0: conditional sig\n 8 vs. 0: conditional sig\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The second part of the {\\sc Cinderella} analysis is a computation of mean conditional sigs for each target dataset over all comparison datasets (see ``Multi-comparison output'', p.\\,\\pageref{CINDERELLA_Multi-comparison output}). Since there is only one comparison dataset available in this example, the output of this procedure is the same as of the previous pairwise analysis. Finally, {\\sc Cinderella} evaluates composed sigs for each target dataset. This calculation is applied to the ``raw'' {\\sc SigSpec} results and also to the conditional sigs obtained by the previous operation. Details on the computation of composed sigs are found in ``Composed Mode'' (p.\\,\\pageref{CINDERELLA_Composed Mode}).}\n\n\\begin{scriptsize}\\begin{verbatim}\n*** total Cinderella analysis ******************************\n\n 1: conditional sig\n 2: conditional sig\n 3: conditional sig\n 4: conditional sig\n 5: conditional sig\n 6: conditional sig\n 7: conditional sig\n 8: conditional sig\ncomposed sig\ncomposed sig of conditional sigs\n\\end{verbatim}\\end{scriptsize}\n\n{\\it On exit, {\\sc Cinderella} displays a goodbye message.}\n\n\\begin{scriptsize}\\begin{verbatim}\nFinished.\n\n************************************************************\n\nThank you for using Cinderella!\nQuestions or comments?\nPlease contact Piet Reegen (reegen@astro.univie.ac.at)\nBye!\n\\end{verbatim}\\end{scriptsize}\n\n{\\it The {\\sc Cinderella} output for this example is discussed in the subsequent chapters.}\n\n\\subsection{The {\\tt .cnd} file}\\label{CINDERELLA_The .cnd file}\n\nAn optional file {\\tt .cnd} consists of a set of keywords and arguments defining project-specific parameters for {\\sc Cinderella}. If this file is not present in the same folder as the time series input files, {\\sc Cinderella} uses a set of default parameters.\n\n\\vspace{12pt}\n{\\bf Caution: {\\sc Cinderella} demands a carriage-return character at the end of the file {\\tt .cnd}, otherwise the program hangs!}\n\\vspace{12pt}\n\nLines in the {\\tt .cnd} file starting with a {\\tt \\#} character are ignored by {\\sc Cinderella}. This provides the possibility to write comments into the file.\n\n\\section{Indexing}\\label{CINDERELLA_Indexing}\n\nBy default, {\\sc Cinderella} expects the file index of time series and {\\sc SigSpec} results to start at zero. If the user wants the program to start at a different index, the keyword {\\tt mfstart} may be given in the {\\tt .cnd} file. This keyword is followed by an integer representing the desired start index. Furthermore, the software takes into account as many files with consecutive indices as available. The number of indices to process may be restricted by means of the keyword {\\tt multifile}, followed by an integer representing the last index to take into account. The use of the keywords {\\tt mfstart} and {\\tt multifile} is strictly consistent with the {\\sc SigSpec} conventions ({\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_MultiFile Mode}).\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt index} contains the same input as the project {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}), but both the time series input files and the {\\sc SigSpec} result files are now enumerated from {\\tt 004847} to {\\tt 004855} rather than from {\\tt 000000} to {\\tt 000008}. The file {\\tt index.cnd} consists of a single line,\n\n\\begin{scriptsize}\\begin{verbatim}\nmfstart 4847\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent Since the keyword {\\tt multifile} is not specified in the file {\\tt index.cnd}, {\\sc Cinderella} uses all input files available through consecutive indices.\n\nThe computations are entirely the same as for {\\tt CinderellaNative}, but all indices are consistently incorporated by {\\sc Cinderella}. E.\\,g., the screen output for the single-comparison computations is now:\n\n\\begin{scriptsize}\\begin{verbatim}\n*** pairwise Cinderella analysis ***************************\n\n 4848 vs. 4847: conditional sig\n 4849 vs. 4847: conditional sig\n 4850 vs. 4847: conditional sig\n 4851 vs. 4847: conditional sig\n 4852 vs. 4847: conditional sig\n 4853 vs. 4847: conditional sig\n 4854 vs. 4847: conditional sig\n 4855 vs. 4847: conditional sig\n\\end{verbatim}\\end{scriptsize}\\sf\n\n\\section{Dataset Types}\\label{CINDERELLA_Dataset Types}\n\nIn order to avoid unnessecarily high computational effort and redundant output, if comparing all possible pairs of time series input files, {\\sc Cinderella} provides the possibility to specify which pairs of target\/comparison datasets to take into account. Moreover, the user has the opportunity to identify datasets to be ignored.\n\n{\\sc Cinderella} will produce one so-called single-comparison output file (see ``Single-comparison output'', p.\\,\\pageref{CINDERELLA_Single-comparison output}) for each target-comparison pair. If there is more than one comparison dataset available, additional files are generated for each target. They contain summaries concerning all comparison datasets examined for the target (see ``Multi-comparison output'', p.\\,\\pageref{CINDERELLA_Multi-comparison output}) and ``Output for composed mode'', p.\\,\\pageref{CINDERELLA_Output for composed mode}).\n\nContrary to the file nomenclature, the six-digit format is not required for file indices specified in the {\\tt .cnd} file.\n\n\\subsection{Target datasets}\\label{CINDERELLA_Target datasets}\n\nThe keyword {\\tt target}\\label{CINDERELLA_keyword.target} in the {\\tt .cnd} file is used for the specification of a target dataset. The keyword is followed by an integer value referring to the six-digit index of the desired time series input file. Multiple declarations of {\\tt target} are supported. If no {\\tt .cnd} file is available, {\\sc Cinderella} uses the first time series input file (i.\\,e. the one with the start index) as the only target dataset. \n\n\\subsection{Comparison datasets}\\label{CINDERELLA_Comparison datasets}\n\nThe keyword {\\tt comp}\\label{CINDERELLA_keyword.comp} in the {\\tt .cnd} file is used for the specification of a comparison dataset. The keyword is followed by an integer value referring to the six-digit index of the desired time series input file. Contrary to the file nomenclature, the six-digit format is not required for file indices specified in the {\\tt .cnd} file. If no {\\tt .cnd} file is available, {\\sc Cinderella} uses all time series input files -- except for the first one, which is considered target data -- as comparison datasets.\n\n\\subsection{Datasets to ignore}\\label{CINDERELLA_Datasets to ignore}\n\nThe keyword {\\tt skip}\\label{CINDERELLA_keyword.skip} in the {\\tt .cnd} file is used for the specification of a dataset not to be taken into account for computation. The keyword is followed by an integer value referring to the six-digit index of the desired time series input file. Contrary to the file nomenclature, the six-digit format is not required for file indices specified in the {\\tt .cnd} file.\n\n\\subsection{Default type}\\label{CINDERELLA_Default type}\n\nTo enhance the convenience for the user, not all files need to be specified by the keywords {\\tt target}, {\\tt comp} and {\\tt skip}. The keyword {\\tt deftype}\\label{CINDERELLA_keyword.deftype} may be used to assign a default dataset type.\n\\begin{enumerate}\n\\item Use {\\tt deftype target} to assign the target attribute by default. If no {\\tt deftype} keyword is provided, this setting is activated.\n\\item Use {\\tt deftype comp} to assign the comp attribute by default.\n\\item Use {\\tt deftype skip} to assign the skip attribute by default.\n\\end{enumerate}\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt types} contains the same input as the project {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}), and the file {\\tt types.cnd} contains the two lines\n\n\\begin{scriptsize}\\begin{verbatim}\ndeftype target\ncomp 0\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent This reproduces the default assignment of data types, and {\\sc Cinderella} performs the same calculations as for the project {\\tt CinderellaNative}. The only difference is the screen output:\n\n\\begin{scriptsize}\\begin{verbatim}\n*** dataset type assignment ********************************\n\n000000.types.dat: comparison\n000001.types.dat: target\n000002.types.dat: target\n000003.types.dat: target\n000004.types.dat: target\n000005.types.dat: target\n000006.types.dat: target\n000007.types.dat: target\n000008.types.dat: target\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent The fact that {\\tt (default)} is not attached to the file list indicates that {\\sc Cinderella} uses the specifications given in the file {\\tt types.cnd} rather than the standard settings applied to the project {\\tt CinderellaNative}.\\sf\n\n\\section{Conditional Mode}\\label{CINDERELLA_Conditional Mode}\n\nThe conditional sig is a measure of the probability that a signal component in a target star occurs, although a coincident signal component is found in a comparison star or sky background. It provides an answer to the question, ``What is the probability that a signal component with given amplitude and sig in the target data is not due to the same process that produces a coincident signal component with given amplitude and sig in the comparison data?''\n\nThe conditional {\\sc Cinderella} mode is comparable to the differential sig ({\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_Differential significance spectra}), although the numerical results are not the same.\n\n\\begin{itemize}\n\\item For the differential mode of {\\sc SigSpec}, the full spectral information is available. Thus {\\sc SigSpec} handles the DFT spectra as continuous functions. {\\sc Cinderella} accesses only a list of peaks detected by {\\sc SigSpec}. Deviations of corresponding peak frequencies in comparison and target spectra cannot be handled as accurately as in the case of differential sig computation.\n\\item The differential mode of {\\sc SigSpec} compares power integrals over the entire frequency range under consideration for the transformation of amplitudes from comparison into target data. Since {\\sc Cinderella} deals with a list of frequencies rather than the entire spectra, different strategies to transform amplitudes have to be employed. See ``Amplitude transformation'', p.\\,\\pageref{CINDERELLA_Amplitude transformation}.\n\\end{itemize}\n\n\\subsubsection{Single-comparison output}\\label{CINDERELLA_Single-comparison output}\n\nThe single-comparison output files contain three columns:\n\n\\begin{enumerate}\n\\item target frequency [inverse time units],\n\\item conditional sig,\n\\item conditional csig.\n\\end{enumerate}\n\nEach file refers to the analysis of a single target-comparison dataset pair. Correspondingly, two six-digit indices {\\tt \\#target\\#}, {\\tt \\#comparison\\#} are used to form the filenames, {\\tt \\#target\\#.cd.\\#comparison\\#.dat} (conditional mode) and {\\tt \\#target\\#.cd.\\#comparison\\#.dat} (composed mode).\n\n\\vspace{12pt}\\noindent{\\bf Example.} {\\it The output file {\\tt 000004.cd.000026.dat} contains conditional sigs for {\\tt 000004..dat} as target data and {\\tt 000026..dat} as comparison data.}\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it In the sample {\\tt CinderellaNative}, there are 8 single-compari\\-son output files: {\\tt 000001.cd.000000.dat} to {\\tt 000008.cd.000000.dat}. Since there is only one comparison dataset available, these files are redundant, because {\\tt \\#target\\#.cd.dat} $=$ {\\tt \\#target\\#.cd.000000.dat}.\\sf\n\n\\subsubsection{Multi-comparison output}\\label{CINDERELLA_Multi-comparison output}\n\nFor each target dataset, a multi-comparison output file {\\tt \\#target\\#.cd.dat} is generated for each target dataset. The three columns represent\n\n\\begin{enumerate}\n\\item target frequency [inverse time units],\n\\item mean conditional sig for all comparison datasets,\n\\item mean conditional csig for all comparison datasets.\n\\end{enumerate}\n\nIf there is only one comparison dataset available, the single-comparison and multi-comparison output files are identical.\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample {\\tt CinderellaNative} contains 8 multi-comparison output files in the project directory: {\\tt 000001.cd.dat} to {\\tt 000008.cd.dat}.\\sf\n\n\\subsection{Candidate selection}\\label{CINDERELLA_Candidate selection}\n\nFor each target dataset, {\\sc Cinderella} scans all comparison datasets, searching for coincident signal components. A pair of signal components is considered coincident, if the frequencies match to an accuracy that may be specified by the user, who may also define what {\\sc Cinderella} shall do if a comparison dataset does not contain a match for a given target frequency.\n\nIf more than one coincident frequency associated to a given target frequency is found, {\\sc Cinderella} chooses the candidate with the lowest conditional or composed sig in order to obtain the most conservative solution.\n\n\\subsubsection{Frequency resolution}\\label{CINDERELLA_Frequency resolution}\n\nThere are mainly two interpretations of the frequency resolution. It is either calculated as the inverse time interval width (Rayleigh frequency resolution),\n\\begin{equation}\\label{CINDERELLA_EQ Rayleigh}\n\\delta f := \\frac{1}{T}\\: ,\n\\end{equation}\nor as\n\\begin{equation}\\label{CINDERELLA_EQ Kallinger}\n\\delta f := \\frac{1}{T\\sqrt{\\mathrm{sig}\\left( A\\right)}}\\: ,\n\\end{equation}\nwhere $\\mathrm{sig}\\left( A\\right)$ denotes the sig of an amplitude $A$. This definition is called Kallinger resolution (Kallinger, Reegen \\& Weiss 2008).\n\nIn order to enhance the flexibility of {\\sc Cinderella}, the frequency resolution is computed as \\begin{equation}\\label{CINDERELLA_EQ fres}\n\\delta f := \\frac{1}{T\\sqrt{\\mathrm{sig}\\left( A\\right)^{\\tau}}}\\: ,\n\\end{equation}\nwhere the floating-point number $\\tau\\in\\left[ 0,1\\right]$ may be defined using the keyword {\\tt tol}\\label{CINDERELLA_keyword.tol} in the {\\tt .cnd} file. The special value $\\tau = 0$ transforms eq.\\,\\ref{CINDERELLA_EQ fres} into eq.\\,\\ref{CINDERELLA_EQ Rayleigh}, setting $\\tau = 1$ provides eq.\\,\\ref{CINDERELLA_EQ Kallinger}.\nThe default value is $\\tau = 0$.\n\n{\\sc Cinderella} checks for frequencies in the comparison datasets are within the frequency resolution around each frequency in the target dataset.\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample {\\tt cand} contains the same input as {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}), and the file {\\tt cand.cnd} contains the line\n\n\\begin{scriptsize}\\begin{verbatim}\ntol 2\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent The frequency tolerance parameter is increased compared to the default value 0, which means that the intervals taken into account to search for corresponding signal components are tendentially narrower. This setting is for demonstration only; in normal applications, only frequency tolerance parameters ranging from 0 to 1 will make sense.\n\nThe effect of this modification is visible, e.\\,g., comparing the output files {\\tt 000001.cd.000000.dat} of the project {\\tt CinderellaNative} to the project {\\tt cand}. In the project {\\tt CinderellaNative}, this file contains the line\n\n\\begin{scriptsize}\\begin{verbatim}\n58.3815412948909298 -8.8063413553403507 -5.6290446018702553\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent whereas the corresponding line in the project {\\tt cand} is\n\n\\begin{scriptsize}\\begin{verbatim}\n58.3815412948909298 2.1858505500938747 0.3972034177439077\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent In the project {\\tt CinderellaNative}, the 14th component in the {\\sc SigSpec} result file {\\tt 000001.result.dat} in the project directory is related to the 47th component in the file {\\tt 000000.result.dat}. The two frequencies differ by 0.054, and the Rayleigh frequency resolution of the target dataset is $0.091$, which is sufficient for a correspondence. In the project {\\tt cand}, the target sig of 2.387 becomes relevant. Eq.\\,\\ref{CINDERELLA_EQ fres} \\it yields a frequency resolution 0.042 for this component, which is now too small for a coincidence. The corresponding line in the file {\\tt 000001.cd.000000.dat} consistently indicates that no coincident peak is found for this signal component. In this case, {\\sc Cinderella} uses a default sig threshold for the comparison data, see ``Spectral significance threshold'', below.\\sf\n\n\\subsubsection{Spectral significance threshold}\\label{CINDERELLA_Spectral significance threshold}\n\nIf no coincidence in a comparison dataset is detected for a given target frequency, i.\\,e., if {\\sc Cinderella} does not find a valid candidate for this target frequency, a default value is used for the sig in the comparison dataset. The user may specify this {\\sc Cinderella} threshold by means of the keyword {\\tt defsig}\\label{CINDERELLA_keyword.defsig} in the {\\tt .cnd} file. The same specification may be set for the default csig\\footnote{abbreviation for cumulative sig} using the keyword {\\tt defcsig}\\label{CINDERELLA_keyword.defcsig}. If one of these keywords is not provided, $\\frac{\\pi}{4}\\log\\mathrm{e} \\approx 0.341$ is used correspondingly by default. According to Reegen~(2007), this is the expected value of the sig for white noise. The underlying assumption is that the residuals after prewhitening of all significant signal components in the comparison dataset represent pure noise, i.\\,e.~do not contain any further unresolved signal.\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample {\\tt cand} contains the same input as {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}), and the file {\\tt cand.cnd} contains the two lines\n\n\\begin{scriptsize}\\begin{verbatim}\ndefsig 0\ndefcsig 1\n\\end{verbatim}\\end{scriptsize}\n\nThe second row in the output file {\\tt 000001.cd.000000.dat} of the project {\\tt CinderellaNative}\n\n\\begin{scriptsize}\\begin{verbatim}\n30.7091991449662061 3.8506295783390758 3.6090130812823817\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent using the default sig and csig threshold $\\frac{\\pi}{4}\\log\\mathrm{e}$, because the comparison dataset does not contain a coincident signal component. The second row in the corresponding file of the project {\\tt cand} is\n\n\\begin{scriptsize}\\begin{verbatim}\n30.7091991449662061 4.1917236667995361 2.9501071697428420\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent Since the default sig is lower in the project {\\tt cand}, the resulting conditional sig is higher. On the other hand, the default csig is higher, which causes the resulting conditional csig to drop down.\\sf\n\n\\subsection{Amplitude transformation}\\label{CINDERELLA_Amplitude transformation}\n\nThe assumption that instrumental and environmental artifacts use to be additive in terms of intensity may create needs to transform amplitudes in mag from the comparison into the target spectra, if the conditional {\\sc Cinderella} mode is applied. The amplitude transformation is only performed to obtain conditional sigs.\n\nThree different strategies to adjust comparison amplitudes are offered, according to the specifications in the {\\tt .cnd} file.\n\n\\begin{itemize}\n\\item In photometry, the photon statistics may be employed to transform comparison into target amplitudes, if the mean magnitudes of the stars are known (Reegen et al.~2008). If the keyword {\\tt transam:mean}\\label{CINDERELLA_keyword.transam:mean} is provided in the {\\tt .cnd} file, {\\sc Cinderella} uses the mean observables $\\left< m_C\\right>$, $\\left< m_T\\right>$ of the comparison and target time series, respectively, to transform the comparison amplitude $A_C$ into a target amplitude $A_T$ according to\n\\begin{equation}\\label{CINDERELLA_EQtransam_mean}\nA_T = 2.5\\log\\left[ 1 + \\frac{10^{-0.4\\left(\\left< m_C\\right> - A_C\\right)} - 10^{-0.4\\left< m_C\\right>}}{10^{-0.4\\left< m_T\\right>}}\\right]\\: .\n\\end{equation}\n\\item If the keyword {\\tt transam:rms}\\label{CINDERELLA_keyword.transam:rms} is specified in the {\\tt .cnd} file, {\\sc Cinderella} interprets the residual rms errors $\\sigma _C$, $\\sigma _T$ of the comparison and target time series\\footnote{with all significant signal prewhitened}, respectively, as measures of the photon noise levels and evaluates the transformed amplitude according to\n\\begin{equation}\\label{CINDERELLA_EQtransam_rms}\nA_T = 2.5\\log\\left[ 1 + \\frac{\\sigma _C^2}{\\sigma _T^2}\\left( 10^{0.4\\,A_C} - 1\\right)\\right]\\: .\n\\end{equation}\n\\item The keyword {\\tt transam:ppsc}\\label{CINDERELLA_keyword.transam:ppsc} in the {\\tt .cnd} file causes {\\sc Cinderella} to use Eq.\\,\\ref{CINDERELLA_EQtransam_rms} employing residual point-to-point scatter instead of residual rms error for both $\\sigma _C$ and $\\sigma _T$.\n\\item If the keyword {\\tt transam:none}\\label{CINDERELLA_keyword.transam:none} is specified in the {\\tt .cnd} file, no amplitude transformation is performed at all, i.\\,e., {\\sc Cinderella} assumes $A_T = A_C$.\n\\end{itemize}\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt transam-mean} contains the same input as the project {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}). The time series data are considered to represent millimag photometry. The comparison dataset is assumed to refer to a 5 mag star, whereas the target datasets shall correspond to a 15 mag star. The resulting time series input files are {\\tt 000000.transam-mean.dat} {\\tt 000001.transam-mean.dat} to {\\tt 000008.transam-mean.dat}. The keyword\n\n\\begin{scriptsize}\\begin{verbatim}\ntransam:mean\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent in the file {\\tt transam-mean.cnd} forces {\\sc Cinderella} to employ the mean magnitudes of the datasets for the amplitude transformation.\\sf\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt CinderellaNative} contains an amplitude transformation based on the rms residual, which is the default method.\\sf\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt transam-ppsc} contains the same input as {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}). The line\n\n\\begin{scriptsize}\\begin{verbatim}\ntransam:ppsc\n\\end{verbatim}\\end{scriptsize}\n\nin the file {\\tt transam-ppsc.cnd} forces {\\sc Cinderella} to employ the residual point-to-point scatters of the datasets for the amplitude transformation.\\sf\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it The sample project {\\tt transam-none} contains the same input as {\\tt CinderellaNative} (p.\\,\\pageref{CINDERELLA_EXnormalrun}). The line\n\n\\begin{scriptsize}\\begin{verbatim}\ntransam:none\n\\end{verbatim}\\end{scriptsize}\n\n\\noindent in the file {\\tt transam-none.cnd} switches off the amplitude transformation.\\sf\n\n\\section{Composed Mode}\\label{CINDERELLA_Composed Mode}\n\nThe composed sig is a measure of the probability that two coincident signal components occur in two different datasets. This implements a logical `and', providing an answer to the question, ``What is the probability that two different datasets show coincident signal components with given amplitudes and sigs?''\n\nThe composed mode is useful for, e.\\,g., photometry of the same star in different filters, or if two short datasets of the same object obtained in different years are examined.\n\nNote that the composed sig in the {\\sc SigSpec} result files (see {\\sc SigSpec} manual) is consistently defined, but applies to the set of significant signal components displayed in the file, whereas {\\sc Cinderella} refers to the composed sig of signal components found in two or more different datasets.\n\nContrary to the candidate selection procedure in conditional mode (p.\\,\\pageref{CINDERELLA_Candidate selection}), the frequency interval between the lowest and the highest frequency found in all target datasets is scanned in steps defined by half the frequency resolution (p.\\,\\pageref{CINDERELLA_Frequency resolution}). For each of the frequencies under consideration, {\\sc Cinderella} computes a composed sig, basically following the introduction by Reegen et al.~(2008). Since {\\sc Cinderella}'s composed mode takes into account all signal components in all datasets, statistical weights have to be introduced that put more emphasis to signal frequencies closer to the frequency under consideration. Hence the composed sig $\\mathrm{csig}\\left( A_n\\right)$ (annotation by Reegen et al.~2008) assigned to an arbitrary frequency $f$ is evaluated according to\n\\begin{equation}\n\\log\\left[ 1-10^{\\mathrm{csig}\\left( A_n\\right)}\\right] = \\frac{1}{N}\\sum _{n=1}^{N}\\mathrm{e} ^{-\\frac{1}{2}\\left[\\frac{f-f_n}{\\min\\left(\\delta f_n\\right)}\\right]^2}\\log\\left[ 1 - 10^{-\\mathrm{sig}\\,\\left( A_n\\right)}\\right]\\: .\n\\end{equation}\nIn this context, the total number of signal components in all target datasets is denoted $N$, $f_n$ referring to the frequency of one of these signal components. The minimum frequency resolution $\\min\\left(\\delta f_n\\right)$ incorporates the definition by Eq.\\,\\ref{CINDERELLA_EQ fres}.\n\nAn interpolation loop is used to exactly identify the maxima of this composed sig, which are written to the output file.\n\n\\subsection{Output for composed mode}\\label{CINDERELLA_Output for composed mode}\n\nThe calculation of the composed sig is applied to all target datasets at once. A file {\\tt cp.dat} is generated, which contains the composed sigs of all target datasets. The three columns refer to:\n\n\\begin{enumerate}\n\\item target frequency [inverse time units],\n\\item composed sig for all target datasets,\n\\item composed csig for all target datasets.\n\\end{enumerate}\n\nThe composed sigs are also calculated for the conditional sigs, i.\\,e., the files {\\tt \\#target\\#.cd.dat} (see ``Multi-comparison output'', p.\\,\\pageref{CINDERELLA_Single-comparison output}), and written to a file {\\tt cc.dat}. The column format is the same as for the file {\\tt cp.dat}.\n\n\\vspace{12pt}\\noindent{\\bf Example.} \\it For the sample project {\\tt CinderellaNative}, the project directory contains a file {\\tt cp.dat} with the composed sigs (and csigs) for all target datasets (provided by the {\\sc SigSpec} result files {\\tt 000001.result.dat} to {\\tt 000008.result.dat}). Furthermore, a file {\\tt cc.dat} is found in the project directory. It contains the composed sigs (and csigs) applied to the conditional ones for all target datasets, i.\\,e., the composed sigs are evaluated using the multi-comparison output files generated by the conditional mode, {\\tt 000001.cd.dat} to {\\tt 000008.cd.dat}.\\sf\n\n\\section{Keywords Reference}\\label{CINDERELLA_Keywords Reference}\n\nThis section is a compilation of all keywords accepted by {\\sc Cinderella}. A brief description of arguments and default values is given. The type of argument is provided by either {\\tt }, {\\tt }, or {\\tt }, and default values are given in parentheses, e.\\,g.~{\\tt (2)}. Empty parentheses indicate that there is no default setting.\n\n\\subsubsection{\\tt col:obs (2)}\n\nobservable column index (unique), starting with $1$, {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_keyword.col:obs}\n\n\\subsubsection{\\tt col:time (1)}\n\ntime column index (unique), starting with $1$, {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_keyword.col:time}\n\n\\subsubsection{\\tt col:weights }\n\nweights column index (also multiple), starting with $1$, {\\sc SigSpec} manual, p.\\,\\pageref{SIGSPEC_keyword.col:weights}\n\n\\subsubsection{\\tt comp (all except start index)}\n\nspecification of time series input files to be regarded as comparison datasets, p.\\,\\pageref{CINDERELLA_keyword.comp}\n\n\\subsubsection{\\tt defcsig ($\\frac{\\pi}{4}\\log\\mathrm{e} \\approx 0.341$)}\n\nthreshold to be used for the csig, if no coincidence is found in a comparison dataset, p.\\,\\pageref{CINDERELLA_keyword.defcsig}\n\n\\subsubsection{\\tt defsig ($\\frac{\\pi}{4}\\log\\mathrm{e} \\approx 0.341$)}\n\nthreshold to be used for the sig, if no coincidence is found in a comparison dataset, p.\\,\\pageref{CINDERELLA_keyword.defsig}\n\n\\subsubsection{\\tt deftype (target)}\n\nspecifies the type of dataset to be assigned to a time series by default, p.\\,\\pageref{CINDERELLA_keyword.deftype}\n\n\\subsubsection{\\tt skip ()}\n\nspecification of time series input files not to be taken into consideration, p.\\,\\pageref{CINDERELLA_keyword.skip}\n\n\\subsubsection{\\tt target (start index)}\n\nspecification of time series input files to be regarded as target datasets, p.\\,\\pageref{CINDERELLA_keyword.target}\n\n\\subsubsection{\\tt tol (0)}\n\n{\\sc Cinderella} frequency tolerance parameter, p.\\,\\pageref{CINDERELLA_keyword.tol}\n\n\\subsubsection{\\tt transam:mean}\n\namplitude transformation using the mean observable for photon statistics, p.\\,\\pageref{CINDERELLA_keyword.transam:mean}\n\n\\subsubsection{\\tt transam:none}\n\nno amplitude transformation at all, p.\\,\\pageref{CINDERELLA_keyword.transam:none}\n\n\\subsubsection{\\tt transam:ppsc}\n\namplitude transformation using the point-to-point scatter of the residual observable for photon statistics, p.\\,\\pageref{CINDERELLA_keyword.transam:ppsc}\n\n\\subsubsection{{\\tt transam:rms} (default)}\n\namplitude transformation using the rms residual observable for photon statistics, p.\\,\\pageref{CINDERELLA_keyword.transam:rms}\n\n\\section{Online availability}\n\nThe ANSI-C code is available online at {\\tt http:\/\/www.sigspec.org}. For further information, please contact P.~Reegen, {\\tt peter.reegen@univie.ac.at}.\n\n\\acknowledgments{\nPR received financial support from the Fonds zur F\\\"or\\-de\\-rung der wis\\-sen\\-schaft\\-li\\-chen Forschung (FWF, projects P14546-PHY, P17580-N2) and the BM:BWK (project COROT). Furthermore, it is a pleasure to thank M.~Gruberbauer (Univ.~of Vienna), D.\\,B.~Guenther (St.~Mary's Univ., Halifax), M.~Hareter, D.~Huber, T.~Kallinger (Univ.~of Vienna), R.~Kusch\\-nig (UBC, Vancouver), J.\\,M. Matthews (UBC, Vancouver), A.\\,F.\\,J.~Moffat (Univ.~de Montreal), D.~Punz (Univ.~of Vienna), S.\\,M. Rucinski (D.~Dunlap Obs., Toronto), D.~Sasselov (Harvard-Smithsonian Center, Cambridge, MA), L.~Schneider (Univ.~of Vienna), G.\\,A.\\,H.~Walker (UBC, Vancouver), W.\\,W. Weiss, and K.~Zwintz (Univ.~of Vienna) for valuable discussion and support with extensive software tests. I acknowledge the anonymous referee for a detailed examination of both this publication and the corresponding software, as well as for the constructive feedback that helped to improve the overall quality a lot. Finally, I address my very special thanks to J.\\,D.~Scargle for his valuable support.\n}\n\n\\References{\nKallinger, T., Reegen, P., Weiss, W.\\,W.~2008, A\\&A, 481, 571\\\\\n\nReegen, P.~2005, in {\\it The A-Star Puzzle}, Proceedings of IAUS 224, eds. J. Zverko, J. Ziznovsky, S.J. Adelman, W.W. Weiss (Cambridge: Cambridge Univ.~Press), p.~791\\\\\n\nReegen, P.~2007, A\\&A, 467, 1353\\\\\n\nReegen, P.~2009, CoAst, submitted\\\\\n\nReegen, P., Gruberbauer, M., Schneider, L., Weiss, W.\\,W.~2008, A\\&A, 484, 601\\\\\n\nZwintz, K., Marconi, M., Kallinger, T., Weiss, W.\\,W.~2004, in {\\it The A-Star Puzzle}, Proceedings of IAUS 224, eds. J.~Zverko, J.~Ziznovsky, S.\\,J.~Adelman, W.\\,W.~Weiss (Cambridge: Cambridge Univ.~Press), p.\\,353\\\\\n\nZwintz, K., Weiss, W.\\,W.~2006, A\\&A, 457, 237\\\\\n\n}\n\n\\end{document}\n\n\\section*{Abstract}\\normalsize\\sf}{}\n\\newcommand{\\References}[1]{\\begin{flushleft}{\\large References\\\\}\\vspace*{2mm}\\small #1 \\end{flushleft}}\n\n\\newcommand{\\chapterDSSN}[2]{\\chapter[\\sf\\normalsize #1\\\\ \\footnotesize \\hspace*{5mm}by #2 \\sf\\normalsize][]{#1\\\\}\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\headDSSN}[2]{\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\chapterdssn}[2]{\\chapter[\\sf\\normalsize #1\\\\ \\footnotesize \\hspace*{5mm}by #2 \\sf\\normalsize][]{#1\\\\}\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\headdssn}[2]{\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\figureDSSN}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand{\\figuredssn}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand {\\strich} {\\begin{center} \\vspace{0.5cm} \\rule{1.5cm}{0.3mm} \\end{center}}\n\\renewcommand{\\chaptername}{}\n\\newcommand{\\acknowledgments}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\acknowledgements}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\Acknowledgments}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\Acknowledgements}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\n\\section*{Affiliations of the authors} \\smallskip\\par\\noindent\\hangindent=4.25truemm $^1$ ~\\,LESIA-Observatoire de Paris-CNRS (UMR~8109)-Univ. Paris~6-Univ. Paris~7,\\\\pl. J. Janssen, F-92195 Meudon, France\n\n\n\n\n\n\\newcommand{\\Authors}[1]{\\begin{center}\\normalsize\\bf\\sf #1 \\end{center}}\n\\newcommand{\\authors}[1]{\\begin{center}\\normalsize\\bf\\sf #1 \\end{center}}\n\\newcommand{\\Author}[1]{\\begin{center}\\normalsize\\bf\\sf #1 \\end{center}}\n\\renewcommand{\\author}[1]{\\begin{center}\\normalsize\\bf\\sf #1 \\end{center}}\n\\newcommand{\\Address}[1]{\\begin{center}\\small\\sf #1 \\end{center}}\n\\newcommand{\\address}[1]{\\begin{center}\\small\\sf #1 \\end{center}}\n\n\\newcommand{\\Accepted}[1]{{\\vspace{2mm}\\small \\noindent \\hspace*{0mm}Accepted: } \\small #1 \\normalsize}\n\n\\newcommand{\\Objects}[1]{{\\vspace{2mm}\\small \\noindent \\hspace*{0mm}Individual Objects: } \\small #1 \\normalsize}\n\n\\newcommand{\\Multiobjects}[1]{{\\vspace{3mm}\\small \\noindent Individual Objects: }\\small\\sf \\hangindent=7truemm \\hangafter=1 #1}\t\n\n\\newcommand{\\discussion}[2]{{\\textbf{\\small#1:}}{ \\small #2 \\newline \\normalsize}}\n\n\\newcommand{\\listofparticipants}[2]\n{\\chapter[\\sf\\normalsize #\n \\footnotesize \\hspace*{10mm}\n \n\\sf\\normalsize][]\n{#1\\\\}\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]\n{\\fancyplain{}{\\sffamily \\thepage\n}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}\n{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize\n}}}\n\n\n\\newcommand{\\participant}[5]{ \\noindent\n{#1} {\\textbf{#2}}, {#3}, {#4}, {\\textit{#5}\\normalsize} \\\\ \\hangindent=0truemm}\n\n\\renewenvironment{abstract}{\\section*{Abstract}\\normalsize\\sf}{}\n\\newcommand{\\References}[1]{\\begin{flushleft}{\\large References\\\\}\\vspace*{2mm}\\small #1 \\end{flushleft}}\n\n\n\n\\newcommand{\\chapterCoAst}[2]\n{\\chapter[\\sf\\normalsize #1\\\\ \n\\footnotesize \\hspace*{5mm}by #2 \\sf\\normalsize][]\n{#1\\\\\n\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\n\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\n\n\n\n\\newcommand{\\sectionheading}[1]\n{\\chapter[\\sf\\large\\textbf{#1} \\footnotesize \n\\sf\\normalsize][]\n{ \\huge \\vspace{20mm} #1\\\\}\n}\n\n\n\n\n\\newcommand{\\headDSSN}[2]{\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\chapterdssn}[2]{\\chapter[\\sf\\normalsize #1\\\\ \\footnotesize \\hspace*{5mm}by #2 \\sf\\normalsize][]{#1\\\\}\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\headdssn}[2]{\\rhead[\\fancyplain{}{\\sf\\footnotesize \\center{#1}}]{\\fancyplain{}{\\sffamily\\thepage}}\\lhead[\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sffamily\\thepage}]{\\fancyplain{\\small\\itshape Comm. in Asteroseismology \\\\ Vol. 160, 2009}{\\sf\\footnotesize \\center{#2}}}}\n\n\\newcommand{\\figureCoAst}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand{\\figureDSSN}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand{\\figuredssn}[5]{\\begin{figure}[#4]\n\\centering\n\\includegraphics*[#5]{#1}\n\\caption{#2}\n\\label{#3}\n\\end{figure}}\n\\newcommand {\\strich} {\\begin{center} \\vspace{0.5cm} \\rule{1.5cm}{0.3mm} \\end{center}}\n\\renewcommand{\\chaptername}{}\n\n\\newcommand{\\acknowledgments}[1]{\\vspace*{5mm}\\noindent \\textbf{Acknowledgments.} #1}\n\\newcommand{\\acknowledgements}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\Acknowledgments}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\\newcommand{\\Acknowledgements}[1]{\\vspace*{5mm}\\noindent\\begin{bf}Acknowledgments. \\end{bf} #1}\n\n\n\\section*{Editorial statement for the following paper (entitled \\\\ {\\it MOST found evidence for solar-type oscillations in the K2 giant star HD\\,20884)} }\n{\\it \\small The main message of the following paper is to report the detection of nonradial pulsation in a K giant. This is an important and controversial topic.\n\nA key issue for this investigation is the reliability of the extracted frequencies. Your editor has spent his life with Fourier amplitude spectra of high-precision ground-based photometry. From that experience, Figure 3 of the paper\nrepresents the natural result of instrumental instabilities and atmospheric transparency changes and does not deserve a second look. However, these are satellite observations without the typical ground-based difficulties -- but there may be different problems.\n\nThe statistical analyses can only help us in a limited way. It does not matter much whether one applies my favorite criterion of amplitude signal\/noise ratio < 4, the SigSpec values, or another method. All these statistical analyses rely on assumptions of the nature of noise present in the data. Again, in ground-based photometry, we have found that the statistics are not reliable for low-frequency peaks. The obvious method to verify that certain peaks are not noise is to look at a comparison sample: the authors provide the results for another star with a simpler light curve. In my opinion, the issue is not yet solved.\\normalsize}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\bigskip\n\nFor $a,b>0$ with $a\\neq b$, the Schwab-Borchardt mean $SB(a,b)$ [1-3] of $a$ and $b$ is\ngiven by\n\\begin{equation*}\nSB(a,b)=\\begin{cases}\n\\frac{\\sqrt{b^2-a^2}}{\\cos^{-1}{(a\/b)}}, &\\quad ab,\n\\end{cases}\n\\end{equation*}\nwhere $\\cos^{-1}(x)$ and $\\cosh^{-1}(x)=\\log(x+\\sqrt{x^{2}-1})$ are the inverse cosine and inverse hyperbolic cosine functions, respectively. Recently, the Schwab-Borchardt mean has been the subject of intensive research. In particular, many remarkable inequalities for Schwab-Borchardt mean can be found in the literature [1-7]. Very recently, the Neuman mean $N(a,b)=(a+b^{2}\/SB(a,b))\/2$ derived from the Schwab-Borchardt was introduced and researched by Neuman in [8].\n\n\nLet $N_{AG}(a,b)=N(A(a,b), G(a,b))$, $N_{GA}(a,b)=N(G(a,b), A(a,b))$, $N_{QA}(a,b)=N(Q(a,b), A(a,b))$ and $N_{AQ}(a,b)=N(A(a,b), Q(a,b))$ be the Neuman means, where $G(a,b)=\\sqrt{ab}$, $A(a,b)=(a+b)\/2$ and $Q(a,b)=\\sqrt{(a^{2}+b^{2})\/2}$ are the classical geometric, arithmetic and quadratic means of $a$ and $b$, respectively. Then Neuman [8] proved that the inequalities\n$$G(a,b)0$ with $a\\neq b$.\n\nLet $a>b>0$ and $v=(a-b)\/(a+b)\\in (0, 1)$. Then we clearly see that\n\\begin{equation}\nG(a,b)=A(a,b)\\sqrt{1-v^{2}}, \\quad Q(a,b)=A(a,b)\\sqrt{1+v^{2}},\n\\end{equation}\nand the following explicit formulas for $N_{AG}(a,b)$, $N_{GA}(a,b)$, $N_{QA}(a,b)$ and\n$N_{AQ}(a, b)$ are given in [8]\n\\begin{equation}\nN_{AG}(a,b)=\\frac{1}{2}A(a,b)\\left[1+(1-v^{2})\\frac{\\tanh^{-1}v}{v}\\right],\n\\end{equation}\n\\begin{equation}\nN_{GA}(a,b)=\\frac{1}{2}A(a,b)\\left[\\sqrt{1-v^{2}}+\\frac{\\sin^{-1}v}{v}\\right],\n\\end{equation}\n\\begin{equation}\nN_{QA}(a,b)=\\frac{1}{2}A(a,b)\\left[\\sqrt{1+v^{2}}+\\frac{\\sinh^{-1}v}{v}\\right],\n\\end{equation}\n\\begin{equation}\nN_{AQ}(a,b)=\\frac{1}{2}A(a,b)\\left[1+(1+v^{2})\\frac{\\tan^{-1}v}{v}\\right],\n\\end{equation}\nwhere $\\tanh^{-1}(x)=\\log[(1+x)\/(1-x)]\/2$, $\\sin^{-1}(x)$, $\\sinh^{-1}(x)=\\log(x+\\sqrt{1+x^{2}})$ and $\\tan^{-1}(x)$ are the inverse hyperbolic tangent, inverse sine, inverse hyperbolic sine and inverse tangent functions, respectively.\n\nIn [8], Neuman also proved that the double inequalities\n$$\\alpha_{1} A(a,b)+(1-\\alpha_{1})G(a,b)0$ with $a\\neq b$ if and only if $\\alpha_{1}\\leq 2\/3$, $\\beta_{1}\\geq \\pi\/4$,\n$\\alpha_{2}\\leq 2\/3$, $\\beta_{2}\\geq (\\pi-2)\/[4(\\sqrt{2}-1)]=0.689\\ldots$, $\\alpha_{3}\\leq 1\/3$, $\\beta_{3}\\geq 1\/2$,\n$\\alpha_{4}\\leq 1\/3$ and $\\beta_{4}\\geq [\\log(1+\\sqrt{2})+\\sqrt{2}-2]\/[2(\\sqrt{2}-1)]=0.356\\ldots$.\n\nIn [9], the authors presented the best possible parameters $\\alpha_{1}, \\alpha_{2}, \\beta_{1}, \\beta_{2}\\in [0, 1\/2]$ and\n$\\alpha_{3}, \\alpha_{4}, \\beta_{3}, \\beta_{4}\\in [1\/2, 1]$ such that the double inequalities\n$$G(\\alpha_{1}a+(1-\\alpha_{1})b, \\alpha_{1}b+(1-\\alpha_{1})a)0$ with $a\\neq b$.\n\nThe main purpose of this paper is to find the greatest values $\\alpha_{1}$, $\\alpha_{2}$, $\\alpha_{3}$, $\\alpha_{4}$, $\\alpha_{5}$, $\\alpha_{6}$, $\\alpha_{7}$, $\\alpha_{8}$ and the least values $\\beta_{1}$, $\\beta_{2}$, $\\beta_{3}$, $\\beta_{4}$, $\\beta_{5}$, $\\beta_{6}$, $\\beta_{7}$, $\\beta_{8}$ such that the double inequalities\n$$A^{\\alpha_{1}}(a,b)G^{1-\\alpha_{1}}(a,b)0$ with $a\\neq b$ .\n\n\n\\bigskip\n\\section {Lemmas}\n\\bigskip\n\nIn order to prove our main results we need several lemmas, which we present in this section.\n\n\\setcounter{equation}{0}\n\n\\begin{lemma} (See [10, Theorem 1.25]) Let $-\\infty0$ and $b_{n}>0$ for all $n\\geq 0$. If the sequence $\\{a_{n}\/b_{n}\\}$ is (strictly) increasing (decreasing) for all $n\\geq 0$, then the function $f(x)\/g(x)$ is also (strictly) increasing (decreasing) on $(0, r)$.\n\\end{lemma}\n\n\\begin{lemma} The function\n\\begin{equation}\nf_{1}(x)=\\frac{\\log [\\sin(2x)]-\\log[2x+\\sin(2x)]+\\log 2}{\\log(\\cos x)}\n\\end{equation}\nis strictly increasing from $(0, \\pi\/2)$ onto $(2\/3, 1)$.\n\\end{lemma}\n{\\em Proof.} It follows from (3.1) that\n\\begin{equation}\nf_{1}(0)=\\frac{2}{3},\n\\end{equation}\n\\begin{equation}\nf_{1}\\left(\\frac{\\pi}{2}^{-}\\right)=1.\n\\end{equation}\n\nLet $g_{1}(x)=\\log [\\sin(2x)]-\\log[2x+\\sin(2x)]+\\log 2$, $h_{1}(x)=\\log(\\cos x)$, $g_{2}(x)=\\sin(2x)-2x\\cos(2x)$ and $h_{2}(x)=[2x+\\sin(2x)]\\sin^{2}x$. Then simple computations lead to\n\\begin{equation}\ng_{1}(0^{+})=h_{1}(0)=g_{2}(0)=h_{2}(0)=0,\n\\end{equation}\n\\begin{equation}\nf_{1}(x)=\\frac{g_{1}(x)}{h_{1}(x)}, \\quad \\frac{g'_{1}(x)}{h'_{1}(x)}=\\frac{g_{2}(x)}{h_{2}(x)},\n\\end{equation}\n\\begin{equation}\n\\frac{g'_{2}(x)}{h'_{2}(x)}=\\frac{1}{\\frac{1}{2}+\\frac{\\sin(2x)}{2x}}.\n\\end{equation}\n\nIt is well known that the function $\\sin x\/x$ is strictly decreasing on $(0, \\pi)$, hence equation (2.6) leads to the conclusion that the function\n$g'_{2}(x)\/h'_{2}(x)$ is strictly increasing on $(0, \\pi\/2)$. Therefore, Lemma 2.3 follows from Lemma 2.1 and (2.2)-(2.5) together with the monotonicity of $g'_{2}(x)\/h'_{2}(x)$.\n\n\\bigskip\n\\begin{lemma} The function\n\\begin{equation}\nf_{2}(x)=\\frac{\\log [2x+\\sinh(2x)]-\\log[\\sinh(x)]-2\\log 2}{\\log[\\cosh(x)]}\n\\end{equation}\nis strictly increasing from $(0, \\infty)$ onto $(1\/3, 1)$.\n\\end{lemma}\n{\\em Proof.} It follows from (2.7) that\n\\begin{equation}\nf_{2}(0^{+})=\\frac{1}{3},\n\\end{equation}\n\\begin{equation}\n\\lim_{x\\rightarrow \\infty}f_{2}(x)=1.\n\\end{equation}\n\nLet $g_{3}(x)=\\log [2x+\\sinh(2x)]-\\log[\\sinh(x)]-2\\log 2$ and $h_{3}(x)=\\log[\\cosh(x)]$. Then simple computations lead to\n\\begin{equation}\nf_{2}(x)=\\frac{g_{3}(x)}{h_{3}(x)}, \\quad g_{3}(0^{+})=h_{3}(0)=0,\n\\end{equation}\n\\begin{equation}\n\\frac{g'_{3}(x)}{h'_{3}(x)}=\\frac{\\sinh(4x)-4x\\cosh(2x)+2\\sinh(2x)-4x}{\\sinh(4x)+4x\\cosh(2x)-2\\sinh(2x)-4x}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\sum_{n=1}^{\\infty}\\frac{\\left(2^{2n}-2n\\right)2^{2n+2}}{(2n+1)!}x^{2n+1}}\n{\\sum_{n=1}^{\\infty}\\frac{\\left(2^{2n}+2n\\right)2^{2n+2}}{(2n+1)!}x^{2n+1}}\n\\end{equation*}\n\\begin{equation*}\n=\\frac{\\sum_{n=0}^{\\infty}\\frac{\\left(2^{2n+2}-2n-2\\right)2^{2n+4}}{(2n+3)!}x^{2n}}\n{\\sum_{n=0}^{\\infty}\\frac{\\left(2^{2n+2}+2n+2\\right)2^{2n+4}}{(2n+3)!}x^{2n}}.\n\\end{equation*}\nLet\n\\begin{equation}\na_{n}=\\frac{\\left(2^{2n+2}-2n-2\\right)2^{2n+4}}{(2n+3)!}, \\quad b_{n}=\\frac{\\left(2^{2n+2}+2n+2\\right)2^{2n+4}}{(2n+3)!}.\n\\end{equation}\nThen\n\\begin{equation}\nb_{n}>0, \\quad \\frac{a_{n+1}}{b_{n+1}}-\\frac{a_{n}}{b_{n}}=\\frac{(3n+2)2^{2n+2}}{\\left(2^{2n+3}+n+2\\right)\\left(2^{2n+1}+n+1\\right)}>0\n\\end{equation}\nfor all $n\\geq 0$.\n\nIt follows from Lemma 2.2 and (2.11)-(2.13) that the function $g'_{3}(x)\/h'_{3}(x)$ is strictly increasing on $(0, \\infty)$. Therefore, Lemma 2.4 follows from Lemma 2.1 and (2.8)-(2.10) together with the monotonicity of $g'_{3}(x)\/h'_{3}(x)$.\n\n\\bigskip\n\\begin{lemma} The function\n\\begin{equation}\nf_{3}(x)=\\frac{2x-\\sin(2x)}{(1-\\cos x)[2x+\\sin(2x)]}\n\\end{equation}\nis strictly increasing from $(0, \\pi\/2)$ onto $(2\/3, 1)$.\n\\end{lemma}\n{\\em Proof.} It follows from (2.14) that\n\\begin{equation}\nf_{3}\\left(0^{+}\\right)=\\frac{2}{3},\n\\end{equation}\n\\begin{equation}\nf_{3}\\left(\\pi\/2\\right)=1.\n\\end{equation}\n\nLet $g_{4}(x)=2x-\\sin(2x)$ and $h_{4}(x)=(1-\\cos x)[2x+\\sin(2x)]$. Then simple computations lead to\n\\begin{equation}\nf_{3}(x)=\\frac{g_{4}(x)}{h_{4}(x)}, \\quad g_{4}(0)=h_{4}(0)=0,\n\\end{equation}\n\\begin{equation*}\ng'_{4}(x)=4\\sin^{2}x,\n\\end{equation*}\n\\begin{equation*}\nh'_{4}(x)=2\\sin^{2}x\\cos x-4\\cos^{3}x+4\\cos^{2}x+2x\\sin x,\n\\end{equation*}\n\\begin{equation}\ng'_{4}(0)=h'_{4}(0)=0,\n\\end{equation}\n\\begin{equation}\n\\frac{g''_{4}(x)}{h''_{4}(x)}=\\frac{4}{9\\cos x+\\frac{x}{\\sin x}-4},\n\\end{equation}\n\\begin{equation}\n\\left(9\\cos x+\\frac{x}{\\sin x}\\right)'=-8\\sin x-\\frac{[2x-\\sin(2x)]\\cos x}{2\\sin^{2}x}<0\n\\end{equation}\nfor $x\\in (0, \\pi\/2)$.\n\nTherefore, Lemma 2.5 follows easily from Lemma 2.1 and (2.15)-(2.20).\n\n\\bigskip\n\\begin{lemma} The function\n\\begin{equation}\nf_{4}(x)=\\frac{\\sinh(x)\\cosh^{2}(x)-2\\sinh(x)\\cosh(x)+x\\cosh(x)}{\\sinh(x)\\cosh^{2}(x)-\\sinh(x)\\cosh(x)+x\\cosh(x)-x}\n\\end{equation}\nis strictly increasing from $(0, \\infty)$ onto $(1\/3, 1)$.\n\\end{lemma}\n{\\em Proof.} It follows from (2.21) that\n\\begin{equation}\nf_{4}(x)=\\frac{\\frac{1}{4}\\sinh(3x)+\\frac{1}{4}\\sinh(x)-\\sinh(2x)+x\\cosh(x)}{\\frac{1}{4}\\sinh(3x)+\\frac{1}{4}\\sinh(x)-\\frac{1}{2}\\sinh(2x)+x\\cosh(x)-x}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\sum_{n=1}^{\\infty}\\frac{3^{2n+1}-2^{2n+3}+8n+5}{4[(2n+1)!]}x^{2n+1}}{\\sum_{n=1}^{\\infty}\\frac{3^{2n+1}-2^{2n+2}+8n+5}{4[(2n+1)!]}x^{2n+1}}\n\\end{equation*}\n\\begin{equation*}\n=\\frac{\\sum_{n=0}^{\\infty}\\frac{3^{2n+3}-2^{2n+5}+8n+13}{4[(2n+3)!]}x^{2n}}{\\sum_{n=0}^{\\infty}\\frac{3^{2n+3}-2^{2n+4}+8n+13}{4[(2n+3)!]}x^{2n}}.\n\\end{equation*}\nLet\n\\begin{equation}\na_{n}=\\frac{3^{2n+3}-2^{2n+5}+8n+13}{4[(2n+3)!]}, \\quad b_{n}=\\frac{3^{2n+3}-2^{2n+4}+8n+13}{4[(2n+3)!]}.\n\\end{equation}\nThen simple computations lead to\n\\begin{equation}\nb_{n}>\\frac{3^{2n+3}-2^{2n+4}}{4[(2n+3)!]}=\\frac{2^{2n+3}\\left[\\left(\\frac{3}{2}\\right)^{2n+3}-2\\right]}{4[(2n+3)!]}>0\n\\end{equation}\n\\begin{equation}\n\\frac{a_{n+1}}{b_{n+1}}-\\frac{a_{n}}{b_{n}}=\\frac{\\left(135\\times 3^{2n}-24n-31\\right)2^{2n+4}}{\\left(3^{2n+3}-2^{2n+4}+8n+13\\right)\\left(3^{2n+5}-2^{2n+6}+8n+21\\right)}>0\n\\end{equation}\nfor all $n\\geq 0$.\n\nNote that\n\\begin{equation}\nf_{4}(0^{+})=\\frac{1}{3}, \\quad \\lim_{x\\rightarrow \\infty}f_{4}(x)=\\lim_{n\\rightarrow \\infty}\\frac{a_{n}}{b_{n}}=1.\n\\end{equation}\n\nTherefore, Lemma 2.6 follows easily from Lemma 2.2 and (2.22)-(2.26).\n\n\n\\bigskip\n\\section{Main Results}\n\\bigskip\n\\begin{theorem} The double inequalities\n\\begin{equation}\nA^{\\alpha_{1}}(a,b)G^{1-\\alpha_{1}}(a,b)0$ with $a\\neq b$ if and only if $\\alpha_{1}\\leq 2\/3$, $\\beta_{1}\\geq 1$, $\\alpha_{2}\\leq 0$ and\n$\\beta_{2}\\geq 1\/3$.\n\\end{theorem}\n{\\em Proof.} We clearly see that inequalities (3.1) and (3.2) can be rewritten as\n\\begin{equation}\n\\left(\\frac{A(a,b)}{G(a,b)}\\right)^{\\alpha_{1}}<\\frac{N_{GA}(a,b)}{G(a,b)}<\\left(\\frac{A(a,b)}{G(a,b)}\\right)^{\\beta_{1}}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{2}<\\frac{\\frac{1}{G(a,b)}-\\frac{1}{N_{GA}(a,b)}}{\\frac{1}{G(a,b)}-\\frac{1}{A(a,b)}}<1-\\alpha_{2},\n\\end{equation}\nrespectively.\n\nSince both the geometric mean $G(a,b)$ and arithmetic mean $A(a,b)$ are symmetric and homogeneous of degree 1, without loss of generality, we assume that $a>b$. Let $v=(a-b)\/(a+b)\\in (0, 1)$. Then from (1.1) and (1.3) we know that inequalities (3.3) and (3.4) are equivalent to\n\\begin{equation}\n\\alpha_{1}<\\frac{\\log\\left[\\frac{1}{2}\\left(1+\\frac{\\sin^{-1}(v)}{v\\sqrt{1-v^{2}}}\\right)\\right]}{\\log\\frac{1}{\\sqrt{1-v^{2}}}}<\\beta_{1}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{2}<\\frac{\\sin^{-1}v-v\\sqrt{1-v^{2}}}{(1-\\sqrt{1-v^{2}})(v\\sqrt{1-v^{2}}+\\sin^{-1}v)}<1-\\alpha_{2},\n\\end{equation}\nrespectively.\n\nLet $x=\\sin^{-1}(v)$. Then $x\\in (0, \\pi\/2)$,\n\\begin{equation}\n\\frac{\\log\\left[\\frac{1}{2}\\left(1+\\frac{\\sin^{-1}(v)}{v\\sqrt{1-v^{2}}}\\right)\\right]}{\\log\\frac{1}{\\sqrt{1-v^{2}}}}\n=\\frac{\\log[\\sin(2x)]-\\log[2x+\\sin(2x)]+\\log 2}{\\log(\\cos x)},\n\\end{equation}\n\\begin{equation}\n\\frac{\\sin^{-1}v-v\\sqrt{1-v^{2}}}{(1-\\sqrt{1-v^{2}})(v\\sqrt{1-v^{2}}+\\sin^{-1}v)}=\\frac{2x-\\sin(2x)}{(1-\\cos x)[2x+\\sin(2x)]}.\n\\end{equation}\n\n\nTherefore, inequality (3.1) holds for all $a,b>0$ with $a\\neq b$ follows from (3.5) and (3.7) together with Lemma 2.3,\nand inequality (3.2) holds for all $a,b>0$ with $a\\neq b$ follows from (3.6) and (3.8) together with Lemma 2.5.\n\n\\medskip\n\\begin{theorem} The double inequalities\n\\begin{equation}\nA^{\\alpha_{3}}(a,b)G^{1-\\alpha_{3}}(a,b)0$ with $a\\neq b$ if and only if $\\alpha_{3}\\leq 1\/3$, $\\beta_{3}\\geq 1$, $\\alpha_{4}\\leq 0$ and $\\beta_{4}\\geq 2\/3$.\n\\end{theorem}\n\n{\\em Proof.} We clearly see that inequalities (3.9) and (3.10) can be rewritten as\n\\begin{equation}\n\\left(\\frac{A(a,b)}{G(a,b)}\\right)^{\\alpha_{3}}<\\frac{N_{AG}(a,b)}{G(a,b)}<\\left(\\frac{A(a,b)}{G(a,b)}\\right)^{\\beta_{3}}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{4}<\\frac{\\frac{1}{G(a,b)}-\\frac{1}{N_{AG}(a,b)}}{\\frac{1}{G(a,b)}-\\frac{1}{A(a,b)}}<1-\\alpha_{4},\n\\end{equation}\nrespectively.\n\nWithout loss of generality, we assume that $a>b$. Let $v=(a-b)\/(a+b)\\in (0, 1)$. Then it follows from (1.1) and (1.2) that inequalities (3.11) and (3.12) are equivalent to\n\\begin{equation}\n\\alpha_{3}<\\frac{\\log\\left[\\frac{1}{\\sqrt{1-v^{2}}}+\\frac{\\sqrt{1-v^{2}}}{v}\\tanh^{-1}(v)\\right]-\\log 2}{\\log\\frac{1}{\\sqrt{1-v^{2}}}}<\\beta_{3}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{4}<\\frac{v+(1-v^{2})\\tanh^{-1}(v)-2v\\sqrt{1-v^{2}}}{(1-\\sqrt{1-v^{2}})[v+(1-v^{2})\\tanh^{-1}(v)]}<1-\\beta_{4},\n\\end{equation}\nrespectively.\nLet $x=\\tanh^{-1}(v)\\in (0, \\infty)$. Then simple computations lead to\n\\begin{equation}\n\\frac{\\log\\left[\\frac{1}{\\sqrt{1-v^{2}}}+\\frac{\\sqrt{1-v^{2}}}{v}\\tanh^{-1}(v)\\right]-\\log 2}{\\log\\frac{1}{\\sqrt{1-v^{2}}}}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\log[2x+\\sinh(2x)]-\\log[\\sinh(x)]-2\\log 2}{\\log[\\cosh(x)]}\n\\end{equation*}\nand\n\\begin{equation}\n\\frac{v+(1-v^{2})\\tanh^{-1}(v)-2v\\sqrt{1-v^{2}}}{(1-\\sqrt{1-v^{2}})[v+(1-v^{2})\\tanh^{-1}(v)]}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\sinh(x)\\cosh^{2}(x)-2\\sinh(x)\\cosh(x)+x\\cosh(x)}{\\sinh(x)\\cosh^{2}(x)-\\sinh(x)\\cosh(x)+x\\cosh(x)-x}.\n\\end{equation*}\n\nTherefore, inequality (3.9) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{3}\\leq 1\/3$ and $\\beta_{3}\\geq 1$ follows from (3.13) and (3.15) together with Lemma 2.4, and inequality (3.10) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{4}\\leq 0$ and $\\beta_{4}\\geq 2\/3$ follows from (3.14) and (3.16) together with Lemma 2.6.\n\n\\medskip\n\\begin{theorem} The double inequalities\n\\begin{equation}\nQ^{\\alpha_{5}}(a,b)A^{1-\\alpha_{5}}(a,b)0$ with $a\\neq b$ if and only if $\\alpha_{5}\\leq 2\/3$, $\\beta_{5}\\geq 2\\log(\\pi+2)\/\\log 2-4=0.7244\\ldots$, $\\alpha_{6}\\leq[6+2\\sqrt{2}-(1+\\sqrt{2})\\pi]\/(\\pi+2)=0.2419\\ldots$ and $\\beta_{6}\\geq 1\/3$.\n\\end{theorem}\n\n{\\em Proof.} We clearly see that inequalities (3.17) and (3.18) can be rewritten as\n\\begin{equation}\n\\left(\\frac{Q(a,b)}{A(a,b)}\\right)^{\\alpha_{5}}<\\frac{N_{AQ}(a,b)}{A(a,b)}<\\left(\\frac{Q(a,b)}{A(a,b)}\\right)^{\\beta_{5}}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{6}<\\frac{\\frac{1}{A(a,b)}-\\frac{1}{N_{AQ}(a,b)}}{\\frac{1}{A(a,b)}-\\frac{1}{Q(a,b)}}<1-\\alpha_{6},\n\\end{equation}\nrespectively.\n\nWithout loss of generality, we assume that $a>b$. Let $v=(a-b)\/(a+b)\\in (0, 1)$. Then from (1.1) and (1.5) we clearly see that inequalities (3.19) and (3.20) are equivalent to\n\\begin{equation}\n\\alpha_{5}<\\frac{2\\log(1+\\frac{1+v^{2}}{v}\\tan^{-1}(v))-2\\log 2}{\\log(1+v^{2})}<\\beta_{5}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{6}<\\frac{\\left[\\left(1+v^{2}\\right)\\tan^{-1}(v)-v\\right]\\sqrt{1+v^{2}}}{\\left[\\left(1+v^{2}\\right)\\tan^{-1}(v)+v\\right](\\sqrt{1+v^{2}}-1)}<1-\\alpha_{6},\n\\end{equation}\nrespectively.\n\nLet $x=\\tan^{-1}(v)$. Then $x\\in (0, \\pi\/4)$,\n\\begin{equation}\n\\frac{2\\log(1+\\frac{1+v^{2}}{v}\\tan^{-1}(v))-2\\log 2}{\\log(1+v^{2})}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\log[\\sin(2x)]-\\log[2x+\\sin(2x)]+\\log 2}{\\log(\\cos x)}=f_{1}(x)\n\\end{equation*}\nand\n\\begin{equation}\n\\frac{\\left[\\left(1+v^{2}\\right)\\tan^{-1}(v)-v\\right]\\sqrt{1+v^{2}}}{\\left[\\left(1+v^{2}\\right)\\tan^{-1}(v)+v\\right](\\sqrt{1+v^{2}}-1)}\n\\end{equation}\n\\begin{equation*}\n\\frac{2x-\\sin(2x)}{(1-\\cos x)[2x+\\sin(2x)]}=f_{3}(x).\n\\end{equation*}\nNote that\n\\begin{equation}\nf_{1}\\left(\\frac{\\pi}{4}\\right)=\\frac{2\\log(\\pi+2)}{\\log 2}-4,\n\\end{equation}\n\\begin{equation}\nf_{3}\\left(\\frac{\\pi}{4}\\right)=\\frac{(2+\\sqrt{2})(\\pi-2)}{\\pi+2}=1-\\frac{6+2\\sqrt{2}-(1+\\sqrt{2})\\pi}{\\pi+2}.\n\\end{equation}\n\nTherefore, inequality (3.17) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{5}\\leq 2\/3$ and $\\beta_{5}\\geq 2\\log(\\pi+2)\/\\log 2-4$ follows from (3.21), (3.23), (3.25) and Lemma 2.3, and inequality (3.18) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{6}\\leq[6+2\\sqrt{2}-(1+\\sqrt{2})\\pi]\/(\\pi+2)$ and $\\beta_{6}\\geq 1\/3$ follows from (3.22), (3.24), (3.26) and Lemma 2.5.\n\n\\medskip\n\\begin{theorem} The double inequalities\n\\begin{equation}\nQ^{\\alpha_{7}}(a,b)A^{1-\\alpha_{7}}(a,b)0$ with $a\\neq b$ if and only if $\\alpha_{7}\\leq 1\/3$, $\\beta_{7}\\geq 2\\log[\\sqrt{2}+\\log(1+\\sqrt{2})]\/\\log 2-2=0.3977\\ldots$, $\\alpha_{8}\\leq[2+\\sqrt{2}-(1+\\sqrt{2})\\log(1+\\sqrt{2})]\/[\\sqrt{2}+\\log(1+\\sqrt{2})]=0.5603\\ldots$ and $\\beta_{8}\\geq 2\/3$.\n\\end{theorem}\n\n{\\em Proof.} We clearly see that inequalities (3.27) and (3.28) can be rewritten as\n\\begin{equation}\n\\left(\\frac{Q(a,b)}{A(a,b)}\\right)^{\\alpha_{7}}<\\frac{N_{QA}(a,b)}{A(a,b)}<\\left(\\frac{Q(a,b)}{A(a,b)}\\right)^{\\beta_{7}}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{8}<\\frac{\\frac{1}{A(a,b)}-\\frac{1}{N_{QA}(a,b)}}{\\frac{1}{A(a,b)}-\\frac{1}{Q(a,b)}}<1-\\alpha_{8},\n\\end{equation}\nrespectively.\n\nWithout loss of generality, we assume that $a>b$. Let $v=(a-b)\/(a+b)\\in (0, 1)$. Then from (1.1) and (1.4) we clearly see that inequalities (3.29) and (3.30) are equivalent to\n\\begin{equation}\n\\alpha_{7}<\\frac{2\\log\\left[\\sqrt{1+v^{2}}+\\frac{\\sinh^{-1}(v)}{v}\\right]-2\\log 2}{\\log\\left(1+v^{2}\\right)}<\\beta_{7}\n\\end{equation}\nand\n\\begin{equation}\n1-\\beta_{8}<\\frac{\\left[v\\left(1+v^{2}\\right)+\\sqrt{1+v^{2}}\\sinh^{-1}(v)\\right]-2v\\sqrt{1+v^{2}}}{(\\sqrt{1+v^{2}}-1)\\left[v\\sqrt{1+v^{2}}+\\sinh^{-1}(v)\\right]}<1-\\alpha_{8},\n\\end{equation}\nrespectively.\n\nLet $x=\\sinh^{-1}(v)$. Then $x\\in (0, \\log(1+\\sqrt{2}))$,\n\\begin{equation}\n\\frac{2\\log\\left[\\sqrt{1+v^{2}}+\\frac{\\sinh^{-1}(v)}{v}\\right]-2\\log 2}{\\log\\left(1+v^{2}\\right)}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\log[2x+\\sinh(2x)]-\\log[\\sinh(x)]-2\\log 2}{\\log[\\cosh(x)]}=f_{2}(x),\n\\end{equation*}\n\\begin{equation}\n\\frac{\\left[v\\left(1+v^{2}\\right)+\\sqrt{1+v^{2}}\\sinh^{-1}(v)\\right]-2v\\sqrt{1+v^{2}}}{(\\sqrt{1+v^{2}}-1)\\left[v\\sqrt{1+v^{2}}+\\sinh^{-1}(v)\\right]}\n\\end{equation}\n\\begin{equation*}\n=\\frac{\\sinh(x)\\cosh^{2}(x)-2\\sinh(x)\\cosh(x)+x\\cosh(x)}{\\sinh(x)\\cosh^{2}(x)-\\sinh(x)\\cosh(x)+x\\cosh(x)-x}=f_{4}(x).\n\\end{equation*}\nNote that\n\\begin{equation}\nf_{2}[\\log(1+\\sqrt{2})]=\\frac{2\\log[\\sqrt{2}+\\log(1+\\sqrt{2})]}{\\log 2}-2,\n\\end{equation}\n\\begin{equation}\nf_{4}[\\log(1+\\sqrt{2})]=\\frac{(2+\\sqrt{2})\\log(1+\\sqrt{2})-2}{\\sqrt{2}+\\log(1+\\sqrt{2})}\n\\end{equation}\n\\begin{equation*}\n=1-\\frac{2+\\sqrt{2}-(1+\\sqrt{2})\\log(1+\\sqrt{2})}{\\sqrt{2}+\\log(1+\\sqrt{2})}.\n\\end{equation*}\n\nTherefore, inequality (3.27) holds for all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{7}\\leq 1\/3$ and $\\beta_{7}\\geq 2\\log[\\sqrt{2}+\\log(1+\\sqrt{2})]\/\\log 2-2$ follows from (3.31), (3.33), (3.35) and Lemma 2.4, and inequality (3.28) holds\nfor all $a, b>0$ with $a\\neq b$ if and only if $\\alpha_{8}\\leq[2+\\sqrt{2}-(1+\\sqrt{2})\\log(1+\\sqrt{2})]\/[\\sqrt{2}+\\log(1+\\sqrt{2})]$ and $\\beta_{8}\\geq 2\/3$ follows from (3.32), (3.34), (3.36) and Lemma 2.6.\n\n\n\n\\medskip\n\n\\noindent{\\bf Conflict of Interests}\n\n\\noindent{The authors declare that there is no conflict of interests regarding the publication of this paper.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\n\nThe two-dimensional toy models of quantum gravity are a very useful play ground for understanding the quantum nature of geometry quantitatively. \nThis is because many of them are simple enough to be dealt with analytically and complex enough to observe non-trivial quantum effects. \nLattice regularizations in particular are known to be quite powerful tools for investigating non-perturbative quantum effects analytically. \nAmongst these, two-dimensional Euclidean dynamical triangulations ($2$d EDT) \\cite{Ambjorn:1985az,Ambjorn:1985dn,David:1984tx,Billoire:1985ur,Kazakov:1985ea,Boulatov:1986jd} \n(see a pedagogical textbook \\cite{Ambjorn:1997di}) and two-dimensional causal dynamical triangulations ($2$d CDT) \\cite{Ambjorn:1998xu} (see a detailed review \\cite{Ambjorn:2022btk}) are good practical examples. \nThe former and the latter, respectively, are Euclidean and Lorentzian lattice models \nbased on Regge's discretization of geometries \\cite{Regge:1961px}. \n\n$2$d EDT discretizes Euclidean geometries by equilateral triangles and defines a regularized quantum amplitude as a sum over distinct triangulated geometries. \nMatrix models and combinatorics can be used for calculating such a statistical sum analytically (whenever possible). \nBy virtue of analytic treatments, one can explicitly remove the regularization through the continuum limit to calculate physical observables. \nWhat is remarkable is that exactly the same value of observables can be reproduced from a genuine continuum field theory called the Liouville quantum gravity \\cite{Polyakov:1981rd, Knizhnik:1988ak, David:1988hj, Distler:1988jt}. \nThis means that $2$d EDT serves as a well-defined regularization of the Liouville quantum gravity. \n\n \n$2$d CDT is a Lorentzian lattice model of quantum geometry which respects a global time foliation \nand prohibits the creation of so-called baby universes. \nOne can calculate the sum over such Lorentzian triangulated geometries using simple combinatorics, \nand take the continuum limit to remove the regularization. \nAll these processes can be performed analytically at least for the plain model without coupling to a matter. \nIt has been shown in Ref.~\\cite{Ambjorn:2013joa} that the resulting continuum theory is known to be in the same universality class of the projectable Ho\\v rava-Lifshitz quantum gravity (projectable HL QG) \\cite{horava} in two dimensions, which is different from the Liouville quantum gravity \\footnote{In fact, it has been shown that a direct lattice discretization of the $2$d projectable HL gravity, \nwhich has a lattice action different from that of the $2$d CDT, reproduces the same large-scale physics in the continuum limit \\cite{Glaser:2016smx}.}. \n \n \n\n$2$d HL QG is a quantum field theory in two dimensions, \nwhich has a preferred foliation structure. \nThis model is invariant only under the subclass of diffeomorphisms that respects the foliation, \nknown as the foliation-preserving diffeomorphisms\\footnote{At the cost of full diffeomorphism invariance, \nHL QG has been designed originally as a model of quantum gravity in higher dimensions such that it has a good convergence at UV in keeping with unitarity, \nand would approximately recover the diffeomorphism invariance at IR \\cite{horava}}. \n$2$d projectable HL QG is a certain version of HL QG where the time-time component of the metric called the lapse function is projectable, i.e. a function only of time. \nIn this chapter, we wish to explain in detail the relation between projectable HL QG and CDT in two dimensions\\footnote{The relation between HL QG and CDT in four dimension has been pointed out first in Ref.~\\cite{Horava:2009if} by looking at an observable called the spectral dimension, and the resemblance of the CDT phase diagram to a Lifshitz phase diagram has been shown in Ref.~\\cite{Ambjorn:2010hu}.}. \n \n \nIn fact one can generalize $2$d CDT in such a way that the creation and annihilation of (a finite number of) baby universes, and the formation of wormholes (handles) are allowed to occur in keeping with the foliation structure. \nThis model is called the generalized CDT (GCDT) introduced first as a continuum theory \\cite{Ambjorn:2007jm, Ambjorn:2008ta} and later defined at the discrete level \\cite{Ambjorn:2008gk, Ambjorn:2013csx}. \nThe full continuum description of GCDT is given by the so-called string field theory for CDT \\cite{Ambjorn:2008ta} in which the string means the one-dimensional closed spatial universe, \nand the baby universes and wormholes can be realized in terms of the splitting and joining interactions of strings. \n\n\nOne of remarkable facts is that focusing on a certain amplitude, i.e. loop-to-loop amplitude, one can read off a one-dimensional effective theory \nthat takes in all possible baby universe and wormhole contributions in an effective manner \\cite{Ambjorn:2009wi, Ambjorn:2009fm}: \nThe $1$d effective theory is a one-body quantum theory even though GCDT is a many-body theory that allows both creation and annihilation of strings. \nIt is known that one can correctly reproduce the $1$d effective theory if quantizing the projectable HL gravity with a certain bi-local interaction term \\cite{Ambjorn:2021wou, Ambjorn:2021ysb}. \nThis topic will be treated in this chapter. \n \n\nFurthermore, the $1$d effective theory that includes all contributions of baby universes and wormholes is known to be reproduced \nif one assumes that the cosmological constant of the continuum limit of $2$d CDT is not really a constant but fluctuates in time \\cite{Ambjorn:2021wdm}. \nThis idea leads to a certain realization of Coleman's mechanism about the cosmological constant \\cite{Coleman:1988tj} in the context of CDT \\cite{Ambjorn:2021wdm}, \nwhich will be also explained in this chapter. \n\n\nThe rest of this chapter is organized as follows. \nIn Sec.~\\ref{sec:2dcdt}, a self-contained introduction to $2$d CDT is presented. \nWe show that taking the continuum limit the physics of $2$d CDT can be described as a one-dimensional quantum system with a Hamiltonian. \n$2$d projectable HL QG is explained in Sec.~\\ref{sec:2dHLQG}. Through the path-integral quantization we read off the quantum Hamiltonian that is equivalent to the one obtained in the continuum limit of $2$d CDT. \nThereby one can confirm that the continuum limit of $2$d CDT is $2$d projectable HL QG. \nIn Sec.~\\ref{sec:sumoverallgeneraingenera}, we introduce GCDT that is a generalization of $2$d CDT such that baby universes and wormholes are introduced so as to be compatible with the foliation, \nand determine the $1$d effective theory obtained through the sum over all genera. \nIn particular, we explain in detail that quantizing $2$d projectable HL gravity with a bi-local interaction yields the $1$d effective theory, and discuss Coleman's mechanism in $2$d CDT. \nSec.~\\ref{sec:summary} is devoted to summary. \n\n \n\n \n\n\n\\section{$2$d causal dynamical triangulations}\n\\label{sec:2dcdt}\nTwo-dimensional causal dynamical triangulations ($2$d CDT) \\cite{Ambjorn:1998xu} is a lattice model of quantum geometries based on Regge's discretization \\cite{Regge:1961px}. \nIn this section, we give an overview of $2$d CDT, and in particular explain how to obtain the quantum Hamiltonian through the continuum limit. \n\nWe start with a two-dimensional globally hyperbolic manifold equipped with a global time foliation: \n\\[\n\\mathcal{M} = \\bigcup_{t\\in \\mathbb{R}} \\Sigma_t\\ , \n\\label{eq:m}\n\\]\nwhere each leaf $\\Sigma_t$ is a one-dimensional Cauchy ``surface'' (line). \nOne approximates the manifold with a foliation in such a way that the continuous label $t$ is discretized by integers, i.e. $t\\in \\mathbb{Z}$; \neach leaf (line) is partitioned by vertices connected by isometric edges; \nvertices among neighboring time steps are connected by isometric edges to form a triangulation of strip (see Fig.~\\ref{fig:triangulationofstrip}). \n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=.6]{triangulationofstrip.png}\n\\caption{A triangulation of a strip: Thick and thin lines are space-like and time-like edges, respectively.}\n\\label{fig:triangulationofstrip} \n\\end{figure}\nThe edges at a given time step and those connecting vertices in different time steps are respectively space-like and time-like edges \nsince the squared edge lengths of the space-like edge $a^2_s$ and the time-like edge $a^2_t$ are given by \n\\[\na^2_s = \\varepsilon^2\\ , \\ \\ \\ a^2_t = - \\alpha \\varepsilon^2\\ ,\n\\label{eq:latticespacing}\n\\] \nwhere $\\alpha$ is a positive number and $\\varepsilon$ is a lattice spacing that serves as a UV cutoff. \n\n\n$2$d CDT deals with a set of restricted class of Lorentzian triangulations as discussed above. \nIn particular, we consider that the topology of the one-dimensional universe (a graph consisting of vertices and edges at a given time) is either $S^1$ or $[0,1]$, \nand the topology will not change during (discrete) time propagation. \nSince the topology is fixed the curvature term plays no role in two dimensions and only the discrete analogue of the cosmological constant term, $\\Lambda_0 \\int d^2x \\sqrt{-g}$ is used as the lattice action of $2$d CDT: \n\\[\nS_T [\\lambda, \\alpha] \n= - \\frac{\\lambda}{\\varepsilon^2} \n\\left( \n\\frac{\\sqrt{4\\alpha +1}}{4}\\varepsilon^2\\ n(T) \n\\right)\\ , \n\\label{eq:action}\n\\] \nwhere $\\lambda\/\\varepsilon^2$ is the bare cosmological constant with the dimensionless number $\\lambda$, \n$n(T)$ the number of triangles in a triangulation $T$, \nand the term inside the parentheses denotes the total area of the triangulation. \nIt is useful to rotate to the Euclidean signature which can be performed by changing $\\alpha \\to - \\alpha -i0$. \nAccordingly the lattice action (\\ref{eq:action}) changes as follows:\n\\[\niS_T [\\lambda, \\alpha] \n\\to \niS_T [\\lambda, -\\alpha -i0] = - \\lambda \\frac{\\sqrt{4\\alpha -1}}{4} n(T)\n\\equiv - \\lambda n(T)\n\\ , \n\\label{eq:rotation}\n\\] \nwhere $\\alpha$ is chosen to be greater than $1\/4$ otherwise the triangle inequalities will not be satisfied after the rotation. \nIn any case we have absorbed the parameter $\\alpha$ by the redefinition of the dimensionless cosmological constant $\\lambda$. \n\nThe amplitude of the one-dimensional universe that starts with $\\ell_1$ edges and end up with $\\ell_2$ edges after the discrete time step $t$ is \ngiven by the sum over all allowed triangulations: \n\\[\nG^{(a)}_{\\lambda} (\\ell_1, \\ell_2; t) \n= \\sum_{T\\in \\mathcal{T}^{(a)}(\\ell_1,\\ell_2,t)} e^{-\\lambda n(T)}\n= \\sum_n e^{-\\lambda n} \\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n)\n\\ , \n\\label{eq:amplitude}\n\\] \nwhere $\\mathcal{T}^{(a)}$ is a set of triangulations whose topology is $[0,1] \\times [0,1]$ for $a=0$ and $S^1 \\times [0,1]$ for $a=1$, \nand \n\\[\n\\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n) \n= \\# \\left\\{ T \\in \\mathcal{T}^{(a)} \\ \\bigl{|}\\ n(T)=n \\right\\}\\ . \n\\label{eq:numberoftriangulations}\n\\] \nWhen defining the amplitude (\\ref{eq:amplitude}), we do not allow the one-dimensional universe to vanish during the discrete time propagation. \nFor later convenience, we introduce a marked amplitude: \n\\[\nG^{(-1)}_{\\lambda} (\\ell_1, \\ell_2; t) \n= \\ell_1 G^{(1)}_{\\lambda} (\\ell_1, \\ell_2; t)\\ , \n\\label{eq:markedamplitude}\n\\]\nwhere one of the edges in the initial one-dimensional universe is marked. This is because there exist $\\ell_1$ possible ways of marking the edges. \nThe three kinds of amplitude should satisfy the composition law: \n\\[\nG^{(1)}_{\\lambda} (\\ell_1, \\ell_2; t_1+t_2) \n&= \\sum^{\\infty}_{\\ell=1} G^{(1)}_{\\lambda} (\\ell_1, \\ell; t_1)\\ \\ell\\ G^{(1)}_{\\lambda} (\\ell, \\ell_2; t_2)\\ , \\\\\nG^{(0)}_{\\lambda} (\\ell_1, \\ell_2; t_1+t_2) \n&= \\sum^{\\infty}_{\\ell=1} G^{(0)}_{\\lambda} (\\ell_1, \\ell; t_1)\\ G^{(0)}_{\\lambda} (\\ell, \\ell_2; t_2)\\ , \\\\\nG^{(-1)}_{\\lambda} (\\ell_1, \\ell_2; t_1+t_2) \n&= \\sum^{\\infty}_{\\ell=1} G^{(-1)}_{\\lambda} (\\ell_1, \\ell; t_1)\\ G^{(-1)}_{\\lambda} (\\ell, \\ell_2; t_2)\\ , \n\\label{eq:compositionlaw}\n\\]\nwhere for $a=1$ one need to multiply the amputated amplitude by $\\ell$ since there exist $\\ell$ possible ways of gluing to recover the whole amplitude. \n\n\nIt is convenient to introduce the generating function of the number of triangulations. \nUsing the notation \n\\[\ng=e^{-\\lambda}\\ , \n\\label{eq:g}\n\\]\nwe define the generating function: \n\\[\n\\widetilde{G}^{(a)} (g,x,y;t) \n&= \\sum^{\\infty}_{\\ell_1=1} \\sum^{\\infty}_{\\ell_2=1} x^{\\ell_1} y^{\\ell_2} G^{(a)}_{\\lambda} (\\ell_1, \\ell_2; t)\\notag \\\\\n&= \\sum^{\\infty}_{\\ell_1=1} \\sum^{\\infty}_{\\ell_2=1} \\sum_n x^{\\ell_1} y^{\\ell_2} g^{n} \\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n)\\ ,\n\\label{eq:generatingfunction} \n\\] \nwhere in the context of quantum gravity, $x$ and $y$ are related to the boundary cosmological constants, $\\lambda_1$ and $\\lambda_2$, that control the size of the boundaries: \n\\[\nx=e^{-\\lambda_1}\\ , \\ \\ \\ y=e^{-\\lambda_2}\\ . \n\\label{eq:xy}\n\\]\nOne can reconstruct the amplitude from the generating function through the following relation:\n\\[\nG^{(a)}_{\\lambda} (\\ell_1, \\ell_2; t) \n= \\oint_{\\mathcal{C}_1} \\frac{dx}{2\\pi i x^{\\ell_1 +1}} \\oint_{\\mathcal{C}_2} \\frac{dy}{2\\pi i y^{\\ell_2 +1}}\\ \\widetilde{G}^{(a)} (g,x,y;t)\\ , \n\\label{eq:amplitudefromgeneratingfunction}\n\\]\nwhere the contour $\\mathcal{C}_1$ ($\\mathcal{C}_2$) is chosen to enclose $x=0$ ($y=0$) and to ensure the convergence of $\\widetilde{G}^{(a)} (g,x,y;t)$. \nOne can derive the relation (\\ref{eq:amplitudefromgeneratingfunction}) using the identity:\n\\[\n\\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i z^{n +1}} = \\delta_{n,0}\\ , \\ \\ \\ (n \\in \\mathbb{Z})\\ ,\n\\] \nwhere the contour $\\mathcal{C}$ encloses $z=0$. \nWe provide the composition law for the generating function when $a=0,-1$ in preparation for later calculations:\n\\[\n\\widetilde{G}^{(a)} (g,x,y;t_1+t_2) \n= \\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i z} \n\\widetilde{G}^{(a)} (g,x,z^{-1};t_1)\\ \n\\widetilde{G}^{(a)} (g,z,y;t_2)\\ ,\\ \\ (a=0,-1)\\ , \n\\label{eq:compositionlaw2}\n\\]\nwhere the contour encloses $z=0$, and for fixed $g$, $x$ and $y$ lies inside the radius of convergence \nfor $\\widetilde{G}^{(a)} (g,x,z^{-1};t_1)$ as the series in $1\/z$ and for $\\widetilde{G}^{(a)} (g,z,y;t_2)$ as the series in $z$, \nwhich is possible as we will see. \n\nIn what follows, we will discuss the one-step amplitude $G^{(a)}_{\\lambda} (\\ell_1, \\ell_2; 1)$. \nThis is because it becomes an important object when computing the whole amplitude. \n\n\n\n\n\\subsection{Counting triangulations}\n\\label{subsec:countingtriangulations}\n\nIn this section we focus on the one-step amplitude $G^{(a)}_{\\lambda} (\\ell_1, \\ell_2; 1)$ which is the sum over triangulations of a strip as shown in Fig.~\\ref{fig:triangulationofstrip}: \n\\[\nG^{(a)}_{\\lambda} (\\ell_1, \\ell_2; 1) \n= e^{-\\lambda (\\ell_1 + \\ell_2)} \\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2)\\ , \n\\label{eqonesteoamplitude} \n\\]\nand count the number of triangulations $\\mathcal{N}^{(a)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2)$. \nBased on simple combinatorics, one can calculate the case of $a=1$ which has the $S^1 \\times [0,1]$ topology:\n\\[\n\\mathcal{N}^{(1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \n= \\frac{1}{\\ell_1 + \\ell_2} \n\\begin{pmatrix}\n\\ell_1 + \\ell_2 \\\\\n\\ell_1\n\\end{pmatrix}\n= \\frac{(\\ell_1 + \\ell_2-1)!}{\\ell_1 ! \\ell_2 !}\\ .\n\\label{eq:n1}\n\\]\nBecause of the property (\\ref{eq:markedamplitude}), \none can easily compute the case of $a=-1$ that the topology is $S^1 \\times [0,1]$ and one of the edges in the initial one-dimensional universe is marked: \n\\[\n\\mathcal{N}^{(-1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \n= \\ell_1 \\mathcal{N}^{(1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \n= \\frac{\\ell_1 (\\ell_1 + \\ell_2-1)!}{\\ell_1 ! \\ell_2 !}\\ .\n\\label{eq:n-1}\n\\] \nConcerning the case of $a=0$ whose topology is $[0,1] \\times [0,1]$, there exist several possibilities depending on the restriction on the leftmost and rightmost triangles. \nIf the rightmost triangle is the upward triangle (downward triangle) and the leftmost triangles is the downward triangle (upward triangle), then the counting of triangulations yields \n\\[\n\\mathcal{N}^{(0)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \n= \\begin{pmatrix}\n\\ell_1 + \\ell_2 -2 \\\\\n\\ell_1 - 1\n\\end{pmatrix}\n= \\frac{(\\ell_1 + \\ell_2-2)!}{(\\ell_1 -1) ! (\\ell_2 -1) !}\\ .\n\\label{eq:n0}\n\\] \nIn the following, we will use eq.~(\\ref{eq:n0}) in the case of $a=0$ for computational simplicity\\footnote{\nIf we do not impose any restriction on the leftmost and rightmost triangles, the number of triangulations becomes \n$\\mathcal{N}^{(0)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) = \\ell_1 ! ( \\ell_1 + \\ell_2 )!\/(\\ell_1! \\ell_2!)$. \n}. \n\nThe one-step generating functions can be derived inserting eqs.~(\\ref{eq:n1}), (\\ref{eq:n-1}) and (\\ref{eq:n0}) into eq.~(\\ref{eq:generatingfunction}): \n\\[\n\\widetilde{G}^{(1)} (g,x,y;1) \n&= \\sum^{\\infty}_{\\ell_1 = 1} \\sum^{\\infty}_{\\ell_2=1} x^{\\ell_1} y^{\\ell_2} g^{\\ell_1+\\ell_2} \\mathcal{N}^{(1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \\notag \\\\\n&= - \\ln \\left( \n\\frac{1-gx-gy}{(1-gx)(1-gy)}\n\\right)\\ ; \n\\label{eq:generatingfunction1}\\\\\n\\widetilde{G}^{(-1)} (g,x,y;1) \n&= \\sum^{\\infty}_{\\ell_1 = 1} \\sum^{\\infty}_{\\ell_2=1} x^{\\ell_1} y^{\\ell_2} g^{\\ell_1+\\ell_2} \\mathcal{N}^{(-1)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \\notag \\\\\n&= \\frac{g^2xy}{(1-gx)(1-gx-gy)} \\ ; \n\\label{eq:generatingfunction-1}\\\\\n\\widetilde{G}^{(0)} (g,x,y;1) \n&= \\sum^{\\infty}_{\\ell_1 = 1} \\sum^{\\infty}_{\\ell_2=1} x^{\\ell_1} y^{\\ell_2} g^{\\ell_1+\\ell_2} \\mathcal{N}^{(0)}(\\ell_1,\\ell_2,n=\\ell_1+\\ell_2) \\notag \\\\\n&= \\frac{g^2xy}{1-gx-gy} \\ .\n\\label{eq:generatingfunction0}\n\\]\nIn fact, one can also obtain eq.~(\\ref{eq:generatingfunction-1}) through $\\widetilde{G}^{(-1)} (g,x,y;1) = x \\frac{\\partial}{\\partial x} \\widetilde{G}^{(1)} (g,x,y;1)$. \n\nAlternatively, it is possible to compute the one-step generating functions directly by simple combinatorics: \n\\[\n\\widetilde{G}^{(1)} (g,x,y;1) \n&= \\sum^{\\infty}_{s=1} \\frac{1}{s} \\left( \\sum^{\\infty}_{k=1} (gx)^k \\sum^{\\infty}_{l=1} (gy)^l \\right)^s \n= - \\ln \\left( \n\\frac{1-gx-gy}{(1-gx)(1-gy)}\n\\right)\\ ; \n\\label{eq:generatingfunction1v2}\\\\\n\\widetilde{G}^{(-1)} (g,x,y;1) \n&= \\sum^{\\infty}_{k=0} \\left( gx \\sum^{\\infty}_{l=0} (gy)^l \\right)^k - \\sum^{\\infty}_{k=0} (gx)^k \n= \\frac{g^2xy}{(1-gx)(1-gx-gy)} \\ ; \n\\label{eq:generatingfunction-1v2}\\\\\n\\widetilde{G}^{(0)} (g,x,y;1) \n&= \\sum^{\\infty}_{s=1} \\frac{1}{s} \\left( \\sum^{\\infty}_{k=1} (gx)^k \\sum^{\\infty}_{l=1} (gy)^l \\right)^s \n= \\frac{g^2xy}{1-gx-gy} \\ .\n\\label{eq:generatingfunction0v2}\n\\] \n\n\n\\subsection{Continuum limit}\n\\label{subsec:continuumlimit}\n \nAll is now set for computing the amplitude in the continuum limit. \nIn this section, however, instead of directly computing the amplitude in the continuum limit, \nwe will derive the differential equation that the continuum amplitude satisfies. \n\nBefore going into details any further, let us explain some basic facts of the continuum limit. \nIn order to remove the cutoff $\\varepsilon$ through the continuum limit, one has to tune the bare coupling constants $(g,x,y)$ to their critical values $(g_c,x_c,y_c)$. \nAt the critical values, the generating function hits the radii of convergence and therefore becomes non-analytic. \nApproaching such a critical point, infinitely many triangles and boundary edges become important in the summation of the generating function, \ni.e. essentially the average number of triangles and boundary edges become infinity at the critical point. \nHaving this in mind, one may intuitively understand that the continuous surface would be obtained if $(g,x,y) \\to (g_c,x_c,y_c)$ and $\\varepsilon \\to 0$ in a correlated manner. \n\nIntroducing $\\lambda_c = - \\ln [g_c]$, $\\lambda_{1c} = - \\ln [x_c]$ and $\\lambda_{2c} = - \\ln [y_c]$, \none can transmute the dimension of the lattice spacing $\\varepsilon$ into the dimension of the renormalized bulk and boundary cosmological constants through the continuum limit: \n\\[\n\\Lambda = \\lim_{\\substack{\\lambda \\to \\lambda_c \\\\ \\varepsilon \\to 0}} \\frac{\\lambda - \\lambda_c}{\\varepsilon^2}\\ , \\ \\ \\ \nX = \\lim_{\\substack{\\lambda_1 \\to \\lambda_{1c} \\\\ \\varepsilon \\to 0}} \\frac{\\lambda_1 - \\lambda_{1c}}{\\varepsilon}\\ , \\ \\ \\ \nY = \\lim_{\\substack{\\lambda_2 \\to \\lambda_{2c} \\\\ \\varepsilon \\to 0}} \\frac{\\lambda_2 - \\lambda_{2c}}{\\varepsilon}\\ ,\n\\label{eq:renormalizedcc}\n\\] \nwhere $\\Lambda$ is the renormalized bulk cosmological constant, and $X$ and $Y$ are the renormalized boundary cosmological constants. \nTherefore, the divergent bare cosmological constants get additive renormalizations so as to obtain the finite renormalized cosmological constants that set the scale at IR. \n\nIn the following, we discuss the continuum limit in detail with respect to each topology of spacetime. \n\n\\subsubsection{$S^1 \\times [0,1]$ topology}\n\\label{subsubsec:s1times01}\n\n\nWe consider the case of $a=-1$, i.e. $S^1 \\times [0,1]$ topology with a marked boundary. \nUsing the composition law (\\ref{eq:compositionlaw2}) and the one-step generating function (\\ref{eq:generatingfunction-1v2}), \none obtains\n\\[\n\\widetilde{G}^{(-1)} (g,x,y;t+1) \n&= \\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i z} \n\\widetilde{G}^{(-1)} (g,x,z^{-1};1)\\ \n\\widetilde{G}^{(-1)} (g,z,y;t)\\notag \\\\\n&=\\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i} \n\\frac{g^2x}{(1-gx)^2 (z-g\/(1-gx))} \\frac{\\widetilde{G}^{(-1)} (g,z,y;t)}{z} \\notag\\\\\n&= \\frac{gx}{1-gx}\\ \\widetilde{G}^{(-1)} \\left( g, \\frac{g}{1-gx} ,y;t \\right)\\ . \n\\label{eq:-1relation}\n\\]\nIn the last equality, we have picked up a pole at $z=g\/(1-gx)$, and there exists no pole at $z=0$ since $\\frac{\\widetilde{G}^{(-1)} (g,z,y;t)}{z}$ is regular. \nThrough iterative use of eq.~(\\ref{eq:-1relation}), one can analytically compute the generating function, and extract the information of the critical point \\cite{Ambjorn:1998xu}. \nHowever, we do not compute the generating function directly to obtain the critical point. \nInstead, we follow the procedure shown in \\cite{Ambjorn:2022btk}: \nOne assumes the existence of the critical point, and determines the value of the critical coupling constants from the consistency. \n\nWe assume the critical point characterized by the critical coupling constants $(g_c,x_c,y_c)$ \nand use the following parametrization:\n\\[\ng=g_c e^{-\\varepsilon^2 \\Lambda}, \\ \\ \\ \nx=x_c e^{-\\varepsilon X}\\ , \\ \\ \\ \ny=y_c e^{-\\varepsilon Y}\\ . \n\\label{eq:parametrization}\n\\]\nAssuming the scalings \n\\[\nT = \\varepsilon t\\ , \\ \\ \\ L_1 = \\varepsilon \\ell_1\\ , \\ \\ \\ L_2 = \\varepsilon \\ell_2\\ , \n\\label{eq:continuumtandl}\n\\]\nwe introduce the renormalized amplitude and the renormalized generating function at the critical point by the multiplicative renormalizations:\n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2; T) \n&= \\lim_{\\varepsilon \\to 0} C_{\\varepsilon}\\ G^{(-1)}_{\\lambda} (\\ell_1,\\ell_2; t)\\ , \\label{eq:continuumg}\\\\\n\\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T) \n&= \\lim_{\\varepsilon \\to 0} \\widetilde{C}_{\\varepsilon}\\ \\widetilde{G}^{(-1)} (g, x, y; t)\\ , \\label{eq:continuumgtilde}\n\\]\nwhere $C_{\\varepsilon}$ and $\\widetilde{C}_{\\varepsilon}$ are real functions of $\\varepsilon$ that will be fixed below. \nThe function $C_{\\varepsilon}$ can be determined in such a way that \nthe composition law (\\ref{eq:compositionlaw}) holds in the continuum limit as \n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2; T_1 + T_2) \n= \\int^{\\infty}_{0}dL\\ \nG^{(-1)}_{\\Lambda} (L_1,L; T_1) G^{(-1)}_{\\Lambda} (L,L_2; T_2)\\ , \n\\label{eq:compositionlawcontinuum} \n\\]\nwhich is possible if $C_{\\varepsilon} = \\varepsilon^{-1}$. \nThe function $\\widetilde{C}_{\\varepsilon}$ can be determined in such way that eq.~(\\ref{eq:generatingfunction}) makes sense in the continuum limit, i.e. \n \\[\n\\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T) \n= \\int^{\\infty}_{0} dL_1 \\int^{\\infty}_{0}dL_2\\ e^{-XL_1} e^{-YL_2} \nG^{(-1)}_{\\Lambda} (L_1,L_2; T)\\ , \n\\label{eq:continuumlaplacetr}\n\\]\nwhich is possible if $\\widetilde{C}_{\\varepsilon} = \\varepsilon\/(x_c y_c)$. \n\nUsing the scaling behavior (\\ref{eq:continuumgtilde}), eq.~(\\ref{eq:-1relation}) can yield the sensible continuum limit if the critical coupling constants satisfy \n\\[\n\\frac{g_c x_c}{1-g_c x_c} = 1\\ , \\ \\ \\ \\frac{g_c}{1-g_c x_c} = x_c \\ \\ \\ \n\\Rightarrow \\ \\ \\ \ng_c = \\frac{1}{2}\\ , \\ \\ \\ x_c = 1\\ .\n\\] \nNow we wish to take the continuum limit of eq.~(\\ref{eq:-1relation}). \nFor notational convenience, we redefine the renormalized coupling constants as follows:\n\\[\ng= \\frac{1}{2} e^{-\\varepsilon^2 \\Lambda} \\equiv \\frac{1}{2} \\left( 1 - \\frac{1}{2}\\varepsilon^2 \\Lambda \\right), \\ \\ \\ \nx= e^{-\\varepsilon X} \\equiv 1 - \\varepsilon X \\ , \\ \\ \\ \ny= e^{-\\varepsilon Y} \\equiv 1 - \\varepsilon Y \\ . \n\\label{eq:parametrization2}\n\\]\nPlugging eqs.~(\\ref{eq:continuumtandl}), (\\ref{eq:parametrization2}) into eq.~(\\ref{eq:-1relation}), \none obtains the differential equation: \n\\[\n\\frac{\\partial}{\\partial T} \\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T)\n= - \\frac{\\partial}{\\partial X} \\left[ (X^2-\\Lambda) \\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T) \\right]\\ . \n\\label{eq:differentialeq-1}\n\\]\nDoing a little math, one can also derive the continuum description of eq.~(\\ref{eq:amplitudefromgeneratingfunction}):\n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2; T) \n= \\int^{c+i\\infty}_{c-i\\infty} \\frac{dX}{2\\pi i} \\int^{c+i\\infty}_{c-i\\infty} \\frac{dY}{2\\pi i}\\\ne^{L_1 X} e^{L_2 Y}\\ \\widetilde{G}^{(-1)}_{\\Lambda} (X,Y; T)\\ ,\n\\label{eq:inverselaplacetr} \n\\]\nwhere $c$ is a suitable real number. \nUsing the inverse Laplace transform (\\ref{eq:inverselaplacetr}) and eq.~(\\ref{eq:differentialeq-1}), \none obtains the differential equation that the continuum amplitude satisfies:\n\\[\n\\frac{\\partial}{\\partial T} G^{(-1)}_{\\Lambda} (L_1,L_2; T) \n= - \\hat{H}^{(-1)} (L_1)\\ G^{(-1)}_{\\Lambda} (L_1,L_2; T)\\ , \n\\label{eq:differentialeq-1v2}\n\\]\nwhere \n\\[\n\\hat{H}^{(-1)}(L) = - L \\frac{\\partial^2}{\\partial L^2} + \\Lambda L\\ . \n\\label{hamiltonian-1}\n\\] \nAs a result, one can interpret the continuum limit of $2$d CDT as a quantum system of the one-dimensional universe with length $L$ \nthat propagates in time $T$ following the quantum Hamiltonian (\\ref{hamiltonian-1}). \nThe quantum Hamiltonian (\\ref{hamiltonian-1}) is hermitian with respect to the inner product:\n\\[ \n\\int^{\\infty}_{0} \\frac{dL}{L} \\phi^* (L)\\ (\\hat{H}^{(-1)} \\psi) (L)\n=\n\\int^{\\infty}_{0} \\frac{dL}{L} (\\hat{H}^{(-1)} \\phi)^* (L)\\ \\psi (L)\\ . \n\\label{eq:innerproduct-1} \n\\] \n \nThe differential equation for the un-marked amplitude can be easily read off inserting the continuum limit of eq.~(\\ref{eq:markedamplitude})\n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2; T) \n= L_1 G^{(1)}_{\\Lambda} (L_1,L_2; T)\\ , \n\\label{eq:continuummarkedamplitude} \n\\]\ninto eq.~(\\ref{eq:differentialeq-1v2}): \n\\[\n\\frac{\\partial}{\\partial T} G^{(1)}_{\\Lambda} (L_1,L_2; T) \n= - \\hat{H}^{(1)} (L_1)\\ G^{(1)}_{\\Lambda} (L_1,L_2; T)\\ , \n\\label{eq:differentialeq1v2}\n\\]\nwhere \n\\[\n\\hat{H}^{(1)}(L) = - \\frac{\\partial^2}{\\partial L^2} L + \\Lambda L\\ . \n\\label{hamiltonian1}\n\\] \nThe quantum Hamiltonian (\\ref{hamiltonian1}) is hermitian with respect to the inner product:\n\\[\n\\int^{\\infty}_{0} LdL \\phi^* (L)\\ (\\hat{H}^{(1)} \\psi) (L)\n=\n\\int^{\\infty}_{0} LdL (\\hat{H}^{(1)} \\phi)^* (L)\\ \\psi (L)\n\\ . \n\\label{eq:innerproduct1} \n\\] \n\n \n \n\n\\subsubsection{$[0,1] \\times [0,1]$ topology}\n\\label{subsubsec:01times01}\n\nLet us consider the case of $a=0$, i.e. $[0,1] \\times [0,1]$ topology. \nWe basically follow the procedure shown in Sect.~\\ref{subsubsec:s1times01}. \nUsing the composition law (\\ref{eq:compositionlaw2}) and the one-step generating function (\\ref{eq:generatingfunction0v2}), \none obtains\n\\[\n\\widetilde{G}^{(0)} (g,x,y;t+1) \n&= \\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i z} \n\\widetilde{G}^{(0)} (g,x,z^{-1};1)\\ \n\\widetilde{G}^{(0)} (g,z,y;t)\\notag \\\\\n&=\\oint_{\\mathcal{C}} \\frac{dz}{2\\pi i} \n\\frac{g^2x}{(1-gx)(z-g\/(1-gx))} \n\\frac{\\widetilde{G}^{(0)} (g,z,y;t)}{z} \\notag\\\\\n&= gx\\ \\widetilde{G}^{(0)} \\left( g, \\frac{g}{1-gx} ,y;t \\right)\\ . \n\\label{eq:0relation}\n\\]\nFrom eq.~(\\ref{eq:0relation}), one may obtain a sensible continuum limit \nif the critical coupling constants are the same as before, i.e. $(g_c,x_c,y_c)=(1\/2,1,1)$, and if the multiplicative renormalization is treated carefully: \n\\[\n\\widetilde{G}^{(0)}_{\\Lambda} (X,Y; T) \n= \\lim_{\\varepsilon \\to 0} \\frac{\\varepsilon}{2^t} \\widetilde{G}^{(0)} (g, x, y; t)\\ . \n\\label{eq:continuumgtilde0}\n\\]\nIn fact, this assumption yields the correct continuum limit. \nPlugging eqs.~(\\ref{eq:continuumtandl}), (\\ref{eq:parametrization2}) into eq.~(\\ref{eq:0relation}) and using eq.~(\\ref{eq:continuumgtilde0}), \none obtains the differential equation:\n\\[\n\\frac{\\partial}{\\partial T} \\widetilde{G}^{(0)}_{\\Lambda} (X,Y; T)\n= - \\left( X + (X^2 - \\Lambda) \\frac{\\partial}{\\partial X} \\right) \\widetilde{G}^{(0)}_{\\Lambda} (X,Y; T) \\ . \n\\label{eq:differentialeq0}\n\\] \nDefining the continuum amplitude in such a way that the inverse Laplace transform (\\ref{eq:inverselaplacetr}) holds in the case of $a=0$ as well, i.e. \n\\[\nG^{(0)}_{\\Lambda} (L_1,L_2; T) \n= \\int^{c+i\\infty}_{c-i\\infty} \\frac{dX}{2\\pi i} \\int^{c+i\\infty}_{c-i\\infty} \\frac{dY}{2\\pi i}\\\ne^{L_1 X} e^{L_2 Y}\\ \\widetilde{G}^{(0)}_{\\Lambda} (X,Y; T)\\ ,\n\\label{eq:inverselaplacetr0} \n\\]\nand using eq.~(\\ref{eq:inverselaplacetr0}), the differential equation (\\ref{eq:differentialeq0}) becomes \n\\[\n\\frac{\\partial}{\\partial T} G^{(0)}_{\\Lambda} (L_1,L_2; T) \n= - \\hat{H}^{(0)} (L_1)\\ G^{(0)}_{\\Lambda} (L_1,L_2; T)\\ , \n\\label{eq:differentialeq0v2}\n\\]\nwhere $\\hat{H}^{(0)}$ is the quantum Hamiltonian obtained in Refs.~\\cite{DiFrancesco:2000nn, Durhuus:2001sp}: \n\\[\n\\hat{H}^{(0)}(L) = - \\frac{\\partial}{\\partial L} L \\frac{\\partial}{\\partial L} + \\Lambda L\\ . \n\\label{hamiltonian0}\n\\] \nThe quantum Hamiltonian (\\ref{hamiltonian0}) is hermitian with respect to the inner product:\n\\[\n\\int^{\\infty}_{0} dL \\phi^* (L)\\ (\\hat{H}^{(0)}\\psi) (L)\n=\n\\int^{\\infty}_{0} dL (\\hat{H}^{(0)} \\phi)^* (L)\\ \\psi (L)\n\\ . \n\\label{eq:innerproduct0} \n\\] \n\n\n\n\\subsection{Short summary of $2$d CDT}\n\\label{subsec:shortsummaryof2dcdt}\nAs discussed in Sect.~\\ref{subsec:continuumlimit}, the continuum limit of $2$d CDT is described by the quantum mechanics of a one-dimensional universe with length $L$ that propagates in time $T$ \nbased on the Hamiltonian $\\hat{H}^{(a)}$: \n\\[\n\\hat{H}^{(-1)} = - L \\frac{\\partial^2}{\\partial L^2} + \\Lambda L\\ , \\ \\ \\ \n\\hat{H}^{(1)} = - \\frac{\\partial^2}{\\partial L^2} L + \\Lambda L\\ , \\ \\ \\\n\\hat{H}^{(0)} = - \\frac{\\partial}{\\partial L} L \\frac{\\partial}{\\partial L} + \\Lambda L\\ ,\n\\label{eq:hamiltonians}\n\\] \nwhere the label $a$ classifies the topology of the one-dimensional universe: \n$S^1$ and $[0,1]$ for $a=1$ and $a=0$, respectively. \nWhen $a=-1$, the closed one-dimensional universe is marked. \nLet us define the eigenstates of $L$ as $|L\\rangle_{a}$ that satisfy \nthe completeness relation: \n\\[\n1 = \\int^{\\infty}_{0}L^{a}dL\\ |L\\rangle_{a} {}_{a}\\langle L | \\ \\ \\ \n\\Leftrightarrow \n\\ \\ \\ \n{}_a \\langle L' | L \\rangle_{a} = \\frac{1}{L^a} \\delta (L - L')\\ . \n\\label{eq:completenessrelations}\n\\] \nNote that $|L \\rangle_{-1} = L | L \\rangle_1$. \nOne can then express the amplitudes as matrix elements:\n\\[\nG^{(1)}_{\\Lambda} (L_1, L_2; T) \n&= {}_{1}\\langle L_2 | e^{-T \\hat{H}^{(1)}} |L_1 \\rangle_{1}\\ , \\\\\nG^{(-1)}_{\\Lambda} (L_1, L_2; T) \n&= {}_{1}\\langle L_2 | e^{-T \\hat{H}^{(-1)}} |L_1 \\rangle_{-1}\\ , \\\\\nG^{(0)}_{\\Lambda} (L_1, L_2; T) \n&= {}_{0}\\langle L_2 | e^{-T \\hat{H}^{(0)}} |L_1 \\rangle_{0}\\ , \n\\label{eq:matrixelement2}\n\\] \nUsing eq.~(\\ref{eq:completenessrelations}), one can show that the composition laws hold: \nFor $a=-1,0$, \n\\[\nG^{(a)}_{\\Lambda} (L_1,L_2; T_1 + T_2) \n= \\int^{\\infty}_{0}dL\\ \nG^{(a)}_{\\Lambda} (L_1,L; T_1) G^{(a)}_{\\Lambda} (L,L_2; T_2)\\ ,\n\\label{eq:compositionlawcontinuum_a-10} \n\\]\nand for $a=1$, \n\\[\nG^{(1)}_{\\Lambda} (L_1,L_2; T_1 + T_2) \n&= \\int^{\\infty}_{0}dL\\ \n^{(1)}_{\\Lambda} (L_1,L; T_1) L G^{(1)}_{\\Lambda} (L,L_2; T_2)\\ . \n\\label{eq:compositionlawcontinuum_a1}\n\\]\n\n\n\n\n\\section{$2$d projectable Ho\\v rava-Lifshitz quantum gravity}\n\\label{sec:2dHLQG}\n\nWe wish to introduce the classical field theory that reproduces the continuum limit of $2$d CDT once it is quantized. \nThe field theory is a certain version of the two-dimensional Ho\\v rava-Lifshitz gravity ($2$d HL gravity). \n\n\nThe starting point is the same class of manifold with a foliation (\\ref{eq:m}) where $\\Sigma_t$ is a one-dimensional space labelled by $t$: \n\\[\n\\Sigma_t = \\{ x^{\\mu} \\in \\mathcal{M}\\ |\\ f(x^{\\mu}) = t \\}\\ , \\ \\ \\ \\text{with} \\ \\ \\ \\mu = 0,1\\ . \n\\label{eq:sigma} \n\\] \nChoosing that $f(x^{\\mu}) =x^0$, the time direction can be decomposed into the two directions, i.e. the normal and the tangential to $\\Sigma_t$: \n\\[\n\\left( \\partial_t \\right)^{\\mu} \n= \\frac{\\partial x^{\\mu}}{\\partial t} \n= N n^{\\mu} + N^1 E^{\\mu}_{1}\\ , \n\\label{eq:timedirection}\n\\] \nwhere $n^{\\mu}$ and $E^{\\mu}_1$ are respectively the unit normal vector and the tangent vector defined as \n\\[\nn^{\\mu} = \n\\left(\n\\frac{1}{N}, - \\frac{N^1}{N}\n\\right)\\ , \\ \\ \\ \nE^{\\mu}_1 = \\delta^{\\mu}_1\\ . \n\\label{eq:normalandtangent}\n\\] \nHere $N$ and $N^1$ are the Lapse function and the shift vector. \nThrough the use of eq.~(\\ref{eq:timedirection}), \none can parametrize the metric $g_{\\mu \\nu}$ on $\\mathcal{M}$ as follows:\n\\[\nds^2 \n= g_{\\mu \\nu} dx^{\\mu} dx^{\\nu} \n= - N^2 dt^2 + h_{11} \\left(dx +N^1 dt \\right) \\left( dx +N^1 dt \\right) \\ , \n\\label{eq:admmetric}\n\\] \nwhere $t=x^0$ and $x=x^1$; $h_{11}$ is the spatial metric on $\\Sigma_t$ defined as $h_{11} = E^{\\mu}_{1} E^{\\nu}_1 g_{\\mu \\nu}$. \n\n$2$d HL gravity is a field theory that preserves the structure of the time foliation, or in other words, it is invariant under \nthe foliation-preserving diffeomorphisms (FPD): \n\\[\nt \\to t + \\xi^0 (t)\\ , \\ \\ \\ x \\to x + \\xi^1 (t,x)\\ . \n\\label{eq:fpd}\n\\]\nThe fields transform under FPD as follows:\n\\[\n\\delta_{\\xi} h_{11} &= \\xi^0 \\partial_0 h_{11} + \\xi^1 \\partial_1 h_{11} + 2h_{11}\\partial_1 \\xi^1\\ , \\label{eq:deltah11}\\\\\n\\delta_{\\xi} N_1 &= \\xi^{\\mu} \\partial_{\\mu} N_1 + N_1 \\partial_{\\mu} \\xi^{\\mu} + h_{11} \\partial_0 \\xi^1\\ , \\label{eq:deltan1}\\\\\n\\delta_{\\xi} N &= \\xi^{\\mu}\\partial_{\\mu} N + N \\partial_0 \\xi^0\\ . \\label{eq:deltan}\n\\] \nwhere $N_1 = h_{11}N^1$. \nHere if a function is a constant on each foliation $\\Sigma_t$, such a function is called projectable. \nIn fact, implementing FPD the projectable Lapse function, i.e. $N=N(t)$, stays as a function only of time. \nThe HL gravity with the projectable Lapse function is dubbed the projectable HL gravity. \nSince it is $2$d projectable HL gravity that reproduces the continuum limit of $2$d CDT once it is quantized, \nfrom now we focus on this special version of HL gravity. \n\nThe action of $2$d projectable HL gravity is given by \n\\[\nI =\n\\int dt\\ \\mathcal{L}\n=\\frac{1}{\\kappa} \n\\int dtdx\\ \nN(t) \\sqrt{h(t,x)} \n\\left(\n(1-\\eta) K^2(t,x) - 2 \\widetilde{\\Lambda}\n\\right)\\ , \n\\label{eq:hlaction}\n\\]\nwhere $\\mathcal{L}$ is the Lagrangian; \n$\\eta$, $\\widetilde{\\Lambda}$ and $\\kappa$ are a dimensionless parameter, \nthe cosmological constant and the (dimensionless) gravitational coupling constant, respectively; \n$h$ is the determinant of the spatial metric $h_{11}$, i.e. $h=h_{11}$; \n$K$ is the trace of the extrinsic curvature $K_{11}$ given by \n\\[\nK_{11} = \\frac{1}{2N} \\left( \\partial_0 - 2\\nabla_1 N_1 \\right)\\ , \\ \\ \\ \n\\text{with}\\ \\ \\ \n\\nabla_1 N_1 = \\partial_1 N_1 - \\Gamma^1_{11} N_1\\ .\n\\label{eq:extrinsiccurvature}\n\\] \nHere $\\Gamma^1_{11}$ is the spatial Christoffel symbol: \n\\[\n\\Gamma^1_{11} \n= \\frac{1}{2} h^{11} \\partial_1 h_{11}\\ . \n\\label{eq:christoffel}\n\\]\nOne can in principle add higher spatial derivative terms to the action (\\ref{eq:hlaction}). \nHowever, such terms are not necessary because the model is renormalizable in two dimensions without introducing them, \nand therefore we omit such terms. \n\nThe continuum limit of $2$d CDT can be precisely obtained if quantizing the $2$d projectable HL gravity \nwith the following identification of parameters: \n\\[\n\\Lambda = \\frac{ \\widetilde{\\Lambda} }{2(1-\\eta)}\\ , \\ \\ \\ \n\\eta < 1\\ , \\ \\ \\ \n\\kappa = 4(1-\\eta)\\ , \n\\label{eq:parameteridentification}\n\\]\nwhere $\\Lambda$ is the renormalized cosmological constant of CDT defined by eq.~(\\ref{eq:parametrization2}). \n\n\n\n\n\n\\subsection{Quantization}\n\\label{sec:quantization}\n\nLet us overview the quantization of $2$d projectable HL gravity shown in Ref.~\\cite{Ambjorn:2013joa} (see also Ref.~\\cite{Li:2014bla} for another article examining this issue). \n\nWe introduce the conjugate momentum of $\\sqrt{h}$ as $\\Pi$, which satisfy the Poisson bracket: \n\\[\n\\left\\{ \n\\sqrt{h(t,x)}, \\Pi (t,x')\n\\right\\}\n= \\delta (x-x')\\ . \n\\label{eq:poisson}\n\\] \nThrough the Legendre transformation of the Lagrangian (\\ref{eq:hlaction}), \none obtains the Hamiltonian of $2$d projectable HL gravity:\n\\[\nH = \\int dx\n\\left( \n\\Pi(t,x) \\partial_t \\sqrt{h (t,x)} \n\\right)\n- \\mathcal{L} \n= \nN(t) \\mathcal{C}(t) +\n\\int dx\\ \nN_1 (t,x) \\mathcal{C}^1 (t,x)\n\\ . \n\\label{eq:classicalhamiltonian}\n\\]\nSince $2$d projectable HL gravity is a singular system due to the invariance under FPD, \nthere exist two kinds of constraint: \n\\[\n\\mathcal{C}^1 (t,x) \n&= - \\frac{\\partial_1 \\Pi (t,x)}{\\sqrt{h (t,x)}} \\approx 0\\ , \\label{eq:momentumconst}\\\\\n\\mathcal{C} (t)\n&= \\int dx\\ \n\\left(\n\\frac{\\kappa}{4(1-\\eta)} \\Pi^2(t,x) \\sqrt{h(t,x)} + \\frac{2}{\\kappa} \\widetilde{\\Lambda} \\sqrt{h(t,x)}\n\\right)\n\\approx 0 \n\\ , \\label{eq:classicalhamiltonianconst} \n\\] \nwhere $\\mathcal{C}^1 (t,x) \\approx 0$ is the momentum constraint, \nand $\\mathcal{C} (t) \\approx 0$ is the Hamiltonian constraint which is global because of the projectable Lapse function\\footnote{\nThe Hamiltonian and the momentum constraints come from the consistency conditions that \nthe primary constraints, $\\Pi_N \\approx 0$ and $\\Pi_{N_1} \\approx 0$, should be preserved under the time flow \nwhere $\\Pi_N$ and $\\Pi_{N_1}$ are the conjugate momenta of $N$ and $N_1$, respectively. \n}. \n\nThe strategy is to solve the momentum constraint (\\ref{eq:momentumconst}) at the level of classical theory, i.e. \n\\[\n\\mathcal{C}^1 (t,x) = 0 \\ \\ \\ \n\\Rightarrow \\ \\ \\ \n\\Pi (t,x) = \\Pi (t)\\ , \n\\label{eq:solvemomentumconst} \n\\] \nmeaning that the conjugate momentum becomes a function only of time. \nApplying eq.~(\\ref{eq:solvemomentumconst}), \nthe Hamiltonian (\\ref{eq:classicalhamiltonian}) reduces to the one for the one-dimensional system: \n\\[\nH =\nN(t) \\left(\n\\frac{\\kappa}{4 (1-\\eta)} \\Pi^2(t) L(t) \n+ \\frac{2}{\\kappa} \\widetilde{\\Lambda} L(t) \n\\right)\\ , \n\\ \\ \\\n\\text{with} \\ \\ \\ \nL(t) = \\int dx \\sqrt{h(t,x)}\\ , \n\\label{eq:reducedhamiltonian}\n\\] \nwhere $L(t)$ is the invariant length of the one-dimensional universe. \nLet us discuss solutions to the Hamiltonian constraint. \nIf $(\\eta -1)\\widetilde{\\Lambda}>0$, one has a solution:\n\\[\n\\Pi^2 = \\frac{8(\\eta - 1)}{\\kappa^2}\\ \\widetilde{\\Lambda}\\ , \n\\label{eq:hamiltonianconstsol1}\n\\] \nwhich means that the extrinsic curvature is a constant. \nOn the other hand, if $(\\eta -1)\\widetilde{\\Lambda}<0$, \nthe only solution is \n\\[\nL=0\\ . \n\\label{eq:hamiltonianconstsol2}\n\\]\n\nHereafter we apply the parametrization (\\ref{eq:parameteridentification}): We choose \n$(\\eta -1)\\widetilde{\\Lambda}<0$, set the unimportant dimensionless gravitational constant as $\\kappa = 4(1-\\eta)$, and redefine the cosmological constant as $\\Lambda = \\frac{ \\widetilde{\\Lambda} }{2(1-\\eta)}$. \nSince $\\kappa > 0$, this means that $\\eta <1$ which selects the correct sign of the kinetic term, and the positive cosmological constant $\\widetilde{\\Lambda}>0$. \nThe dynamics of the classical $1$d system with the Hamiltonian (\\ref{eq:reducedhamiltonian}) can be alternatively described by the following action: \n\\[\nS= \\int^{1}_{0} dt \\left(\n\\frac{\\dot{L}^2(t)}{4N(t)L(t)} - \\Lambda N(t)L(t)\n\\right)\\ , \n\\label{eq:reducedaction}\n\\] \nwhere $\\dot{L}(t) := \\frac{d}{dt} L(t)$. \nWe then introduce the proper time:\n\\[\n\\tau (s) = \\int^{s}_{0}dt\\ N(t)\\ , \\ \\ \\ s\\in [0,1]\\ .\n\\label{eq:propertime}\n\\]\nSince the proper time (\\ref{eq:propertime}) is invariant under the reparametrization of time, $t\\to t + \\xi^0 (t)$, \nif one fixes the Lapse function as $N(\\tau)=1$, the length of the one-dimensional universe $L(\\tau)$ is also invariant under the time redefinition. \nTherefore, it makes sense to discuss the amplitude such that the one-dimensional universe with the length $L_1:=L(\\tau=0)$ propagates in the proper time $\\tau$, \nand ends up with the universe whose length is given by $L_2 := L_2(\\tau = T)$. \n\n\n\nWith this understanding, we consider such an amplitude based on the path integral. \nFor convenience, we rotate $t \\to it$, which is possible thanks to the foliation and introduce the Euclidean action:\n\\[\nS_E = \n\\int^{1}_{0} dt \\left(\n\\frac{\\dot{L}^2(t)}{4N(t)L(t)} + \\Lambda N(t)L(t)\n\\right)\\ . \n\\label{eq:euclideanaction}\n\\] \nUsing the Euclidean action (\\ref{eq:euclideanaction}), the amplitude becomes \n\\[\n\\mathcal{G}_{\\Lambda} (L_1,L_2;T)\n= \\int \\frac{\\mathcal{D}N(t)}{\\text{Diff [0,1]}} \\int^{L(1)=L_2}_{L(0)=L_1} \\mathcal{D}L(t)\\ e^{-S_E [N(t), L(t)]}\\ , \n\\label{eq:pathintegral}\n\\]\nwhere \n\\[\nT := \\int^{1}_{0} dt\\ N(t)\\ . \n\\label{eq:T}\n\\] \nWe fix the Lapse function as $N(\\tau)=1$ introducing the corresponding Faddeev-Popov (FP) determinant. \nSince the FP determinant only gives an overall constant, we will omit it in the following. \nAfter the gauge fixing, the amplitude (\\ref{eq:pathintegral}) becomes\n\\[\n\\mathcal{G}_{\\Lambda} (L_1,L_2;T) \n= \\int^{L(T)=L_2}_{L(0)=L_1} \n \\mathcal{D}L(\\tau)\\ \n \\exp \\left[ \n - \\int^T_0 d\\tau \n \\left(\n \\frac{\\dot{L}^2(\\tau)}{4L(\\tau)} + \\Lambda L(\\tau)\n \\right)\n \\right]\\ , \n \\label{eq:pathintegral2}\n\\] \nwhere $\\dot{L}(\\tau) := \\frac{d}{d \\tau} L(\\tau)$. \n\nSo far, we have not specified the integral measure $\\mathcal{D}L(\\tau)$. \nWe apply the three kinds of measure given by \n\\[\n\\mathcal{D}^{(a)}L(\\tau) \n= \\prod^{\\tau=T}_{\\tau=0} L^a(\\tau)\\ dL(\\tau)\\ , \\ \\ \\ (a=0,\\pm 1)\\ .\n\\label{eq:measure}\n\\] \nAccordingly, we consider the three kinds of amplitude, i.e. $\\mathcal{G}^{(a)}_{\\Lambda}(L_1,L_2;T)$, \nand rewrite them introducing the quantum Hamiltonian $\\hat{H}^{(a)}$:\n\\[\n\\mathcal{G}^{(a)}_{\\Lambda}(L_1,L_2;T) \n= {}_{a}\\langle L_2 | e^{-T \\hat{H}^{(a)}} |L_1 \\rangle_{a}\\ ,\n\\label{eq:amplitudematrix}\n\\] \nwhere the eigenstates of $L$ satisfy the completeness relation: \n\\[\n1 = \\int^{\\infty}_{0}L^{a}dL\\ |L\\rangle_{a} {}_{a}\\langle L | \\ \\ \\ \n\\Leftrightarrow \n\\ \\ \\ \n{}_a \\langle L' | L \\rangle_{a} = \\frac{1}{L^a} \\delta (L - L')\\ . \n\\label{eq:completenessrelations2}\n\\] \nIn order to read off the quantum Hamiltonian $\\hat{H}^{(a)}$, we discretize the proper time interval in steps of $\\varepsilon$, \nand calculate the one-step matrix element $G^{(a)}_{\\Lambda}(L,L';\\varepsilon)$. \nThe normalization can be fixed so as to satisfy the following equation:\n\\[\n\\lim_{\\varepsilon \\to 0} \\int^{\\infty}_{0} L^a dL\\ \\mathcal{G}^{(a)}_{\\Lambda}(L,L';\\varepsilon) = 1\\ , \n\\label{eq:normalization} \n\\] \nwhich comes from the completeness relation (\\ref{eq:completenessrelations2}). \nThe result is \n\\[\n\\mathcal{G}^{(a)}_{\\Lambda}(L,L';\\varepsilon) \n= \\frac{(LL')^{(1-a)\/2}}{L' \\sqrt{4\\pi \\varepsilon L'}} e^{-\\frac{(L-L')^2}{4\\varepsilon L'} - \\Lambda \\varepsilon L'}\\ . \n\\label{eq:onesteppropagator}\n\\] \nIntegrating the one-step amplitude together with a function, $\\psi_a (L) = {}_a \\langle L | \\psi \\rangle$, for $\\varepsilon \\ll 1$, \none can read off the quantum Hamiltonian: \n\\[\n\\psi_a (L'; \\varepsilon) \n&= {}_{a}\\langle L' | e^{- \\varepsilon \\hat{H}^{(a)}} | \\psi \\rangle \\notag \\\\ \n&= \\int^{\\infty}_{0} L^a dL\\ {}_{a}\\langle L' | e^{- \\varepsilon \\hat{H}^{(a)}} |L \\rangle_{a} {}_a \\langle L | \\psi \\rangle \\notag \\\\ \n& \\cong \\psi_a (L') - \\varepsilon \\hat{H}^{(a)}\\psi_a (L') + \\mathcal{O} (\\varepsilon^{3\/2})\\ . \n\\label{eq:readoffhamiltonian} \n\\] \nUsing eq.~(\\ref{eq:onesteppropagator}), one obtains \n\\[\n\\hat{H}^{(-1)} (L) = - L \\frac{d^2}{dL^2} + \\Lambda L\\ , \\ \\ \\ \n\\hat{H}^{(0)} (L) = - \\frac{d}{dL} L \\frac{d}{dL} + \\Lambda L\\ , \\ \\ \\ \n\\hat{H}^{(1)} (L) = - \\frac{d^2}{dL^2} L + \\Lambda L\\ , \n\\label{eq:hamiltonianhorava}\n\\]\n\n\nThe quantum Hamiltonians (\\ref{eq:hamiltonianhorava}) obtained by quantizing $2$d projectable HL gravity are \nprecisely equivalent to those obtained by the continuum limit of $2$d CDT (see eqs.~(\\ref{hamiltonian-1}), (\\ref{hamiltonian1}) and (\\ref{hamiltonian0})). \nThe amplitudes are related as follows:\n\\[\n\\mathcal{G}^{(-1)}_{\\Lambda}(L,L';T) = L' G^{(-1)}_{\\Lambda} (L,L';T)\\ , \\ \\ \\ \n\\mathcal{G}^{(a)}_{\\Lambda}(L,L';T) = G^{(a)}_{\\Lambda} (L,L';T)\\ , \\ \\ \\ (a=0,1)\\ . \n\\label{eq:relationamplitudes}\n\\]\nThereby, we understand that the classical field theory that reproduces the continuum limit of $2$d CDT once it is quantized is indeed \n$2$d projectable HL gravity. The projectable Lapse function allows us to introduce the reparametrization-invariant proper time, \nand to reduce the $2$d field theory to the $1$d system. \n\n\n\n\n\n\\section{Sum over all wormholes and baby universes}\n\\label{sec:sumoverallgeneraingenera}\n\nIn the CDT model, the spatial topology change is not allowed to occur by definition. \nOne can generalize the $2$d CDT model in such a way that spatial topology changes do occur in keeping with the foliation structure, \nand the universality class is the same as that of $2$d CDT. \nSuch a model is called generalized CDT (GCDT). \nGCDT can be constructed as both discretized and continuum models. \nHere of course the continuum model can be obtained by the continuum limit of the discretized model, \nbut one can directly construct the continuum GCDT model promoting the one-dimensional quantum-mechanical system discussed in Sec.~\\ref{sec:2dcdt} to a $2$d field theory \nthat includes the splitting and joining interactions of the one-dimensional spatial universe. \nSuch a field theory is dubbed the string field theory for CDT, in which the string means the one-dimensional universe \\cite{Ambjorn:2008ta}. \nIn this section, we introduce the string field theory for CDT, and briefly explain the fact that one can take the sum over all wormholes (i.e. handles) and baby universes \\cite{Ambjorn:2009wi, Ambjorn:2009fm}. \nHere the baby universe is a portion of geometry that is pinched off from the ``parent universe'' and vanishes into the vacuum. \nWe also introduce an effective one-body theory that reproduces the many-body effects coming from the splitting and joining interactions. \nWe then discuss those effects in the context of HL gravity \\cite{Ambjorn:2021wou}. \nIn the end, we show that a sort of Coleman's mechanism works when taking into account all contributions of wormholes and baby universes non-perturbatively \\cite{Ambjorn:2021wdm}. \n\n\n\n\nWe introduce an operator that creates a marked closed string (i.e. a marked closed one-dimensional universe) with length $L$, $\\Psi^{\\dagger} (L)$, \nand an operator that annihilates a length-$L$ closed string without a mark, $\\Psi (L)$. \nThese operators satisfy the following commutators: \n\\[\n[\\Psi (L), \\Psi^{\\dagger} (L')] = \\delta (L-L')\\ , \\ \\ \\ \n[\\Psi (L), \\Psi (L')] \n= [\\Psi^{\\dagger} (L), \\Psi^{\\dagger} (L')] \n=0\\ . \n\\label{eq:commutators}\n\\] \nThe vacuum state $| \\text{vac} \\rangle$ is defined by the equation: \n$\\Psi (L) | \\text{vac} \\rangle = 0$. \nThe CDT amplitude (\\ref{eq:matrixelement2}) can be expressed by sandwiching the one-body Hamiltonian: \n\\[\nG^{(-1)}_{\\Lambda} (L_1,L_2;T)\n= \\langle \\text{vac} | \\Psi (L_2)\\ \ne^{-T \\mathcal{H}^{(-1)_{\\text{free}}} (L_1)}\\\n\\Psi^{\\dagger} (L_1)\n| \\text{vac} \\rangle\\ , \n\\label{eq:2ndamplitude-1}\n\\]\nwhere \n\\[\n\\mathcal{H}^{(-1)}_{\\text{free}} (L)\n= \\int^{\\infty}_{0} \\frac{dL}{L} \n\\Psi^{\\dagger} (L) \\left(\n-L \\frac{\\partial^2}{\\partial L^2} + \\Lambda L\n\\right) \n\\Psi (L)\\ . \n\\label{eq:2ndhamiltonian-1}\n\\]\nHereafter we omit the superscript $(-1)$ for avoiding notational complexity. \nAdding splitting and joining interactions into the free Hamiltonian (\\ref{eq:2ndhamiltonian-1}), \none obtains the full Hamiltonian of the string field theory for CDT:\n\\[\n\\mathcal{H}\n&=\\mathcal{H}_{\\text{free}} \n-g_s \\int^{\\infty}_{0}dL_1 \\int^{\\infty}_0 dL_2 \\Psi^{\\dagger} (L_1) \\Psi^{\\dagger} (L_2) (L_1 + L_2) \\Psi (L_1 + L_2) \\notag \\\\ \n&\\ \\ \\ - \\alpha g_s \\int^{\\infty}_{0}dL_1 \\int^{\\infty}_0 dL_2 \\Psi^{\\dagger} (L_1 + L_2) L_1 \\Psi (L_1) L_2 \\Psi (L_2) \\notag \\\\\n&\\ \\ \\ - \\int^{\\infty}_{0} dL\\delta (L)\\Psi (L)\\ ,\n\\label{eq:fullhamiltonian-1}\n\\]\nwhere the second, third and fourth terms respectively mean the splitting interaction with the string coupling constant $g_s$, \nthe joining interaction with the coupling constant $\\alpha g_s$, \nand the term associated with a string vanishing into the vacuum. \nHere the parameter $\\alpha$ is introduced for counting the number of handles (i.e. wormholes). \nOne can in principle calculate the amplitude for the process such that $m$ closed strings propagate in time and end up with \n$n$ closed strings: \n\\[\nA(L_1, \\cdots, L_m; L'_1, \\cdots, L'_n; T) \n= \\langle \\text{vac} | \n\\Psi (L'_1) \\cdots \\Psi (L'_n)\ne^{-T \\mathcal{H}} \n\\Psi^{\\dagger} (L_1) \\cdots \\Psi^{\\dagger} (L_m)\n| \\text{vac} \\rangle\\ , \n\\label{eq:fullamplitude}\n\\]\n\n\\subsection{Effective theory}\n\\label{sec:effectivetheory}\n\nLet us consider the full propagator $A(L_1, L_2;T)$ that includes the sum over all genera and baby universes. \nWe can set $\\alpha = 1$ without loss of generality since the parameter $\\alpha$ plays a supplementary role and we are interested in taking the sum over all genus contributions. \nSomewhat miraculously, the full propagator defined in the many-body system with the Hamiltonian (\\ref{eq:fullhamiltonian-1}) can be effectively described by \nthe one-body system \\cite{Ambjorn:2009wi, Ambjorn:2009fm}: \n\\[\nA(L_1,L_2;T) \n= \\langle L_2 | e^{-T \\hat{H}_{\\text{eff}} (L_1)} | L_1 \\rangle\\ , \n\\label{eq:fullpropagator}\n\\] \nwhere $\\hat{H}$ is the effective Hamiltonian given by \n\\[\n\\hat{H}_{\\text{eff}} (L) = - L \\frac{d^2}{dL^2} + \\Lambda L - g_s L^2\\ . \n\\label{eq:effectivehamiltonian}\n\\] \nThis is possible because there exists a bijection called Ambj\\o rn-Budd bijection \\cite{Ambjorn:2013csx} such that \none can map each geometry generated in GCDT to a branched polymer with loops at the discrete level. \nThe last term in the Hamiltonian (\\ref{eq:effectivehamiltonian}), $-g_s L^2$, expresses all the effects originated with the baby universes and the wormholes. \nNote that the Hamiltonian (\\ref{eq:effectivehamiltonian}) is not bounded from below because of the last term, \nbut in fact this system is known to be ``classical incomplete,'' which means that the Hamiltonian has discrete energy spectra, \nand a set of square integrable eigenfunctions (see e.g. Ref.~\\cite{Ambjorn:1992ve} for a pedagogical explanation about the classical incomplete systems). \nA similar deformation has been observed in the $c=1$ non-critical string theory \\cite{Moore:1991sf, Betzios:2020nry}. \n\nThe full propagator (\\ref{eq:effectivehamiltonian}) can be also described in terms of the path-integral:\n\\[\nA(L_1,L_2;T) \n= \\int^{L(T) = L_2}_{L(0) = L_1}\n\\mathcal{D}L(\\tau)\\ \n\\exp \\left[\n- \\int^{T}_{0}d\\tau \n\\left(\n\\frac{\\dot{L}^2 (\\tau)}{4L(\\tau)} + \\Lambda L(\\tau) - g_s L^2 (\\tau)\n\\right)\n\\right]\\ ,\n\\label{eq:pathintegralfullpropagator}\n\\]\nwhere the integral measure is given by \n\\[\n\\mathcal{D}L(\\tau) \n= \\prod^{\\tau = T}_{\\tau = 0} L^{-1} (\\tau) dL (\\tau)\\ . \n\\label{eq:measure-1full}\n\\]\nIn order for the functional integral (\\ref{eq:pathintegralfullpropagator}) to be well-defined, one need to choose the boundary conditions on $L(\\tau)$ at infinity such that \nthe kinetic term counteracts the unboundedness of the potential. \nIf one generalizes the integral measure (\\ref{eq:measure-1full}) as \n\\[\n\\mathcal{D}^{(a)}L(\\tau) \n= \\prod^{\\tau = T}_{\\tau = 0} L^{a} (\\tau) dL (\\tau)\\ , \\ \\ \\ (a=0, \\pm 1)\\ ,\n\\label{eq:measureafull}\n\\] \none can recover all possible orderings of the effective Hamiltonian (\\ref{eq:effectivehamiltonian}) following the procedure explained in Sec.~\\ref{sec:quantization}. \n\n\nInterestingly, one can reproduce the full propagator (\\ref{eq:pathintegralfullpropagator}) \nif one considers that the cosmological constant $\\Lambda$ in eq.~(\\ref{eq:pathintegral2}) is not a constant but fluctuates independently in time around $\\Lambda$, \nfollowing the Gaussian distributions with a standard deviation $\\sigma = 2\\sqrt{g_s}$: \n\\[\nA(L_1,L_2;T) \n= \\int \\mathcal{D}\\nu (\\tau)\\ e^{- \\frac{1}{4g_s} \\int^{T}_{0} d\\tau\\ \\nu^2(\\tau)} \\mathcal{G}_{\\Lambda+\\nu} (L_1,L_2;T)\\ ,\n\\label{eq:fluctuations}\n\\]\nwhere \n\\[\n\\mathcal{G}_{\\Lambda+\\nu} (L_1,L_2;T) \n:= \\int^{L(T)=L_2}_{L(0)=L_1} \n \\mathcal{D}L(\\tau)\\ \n \\exp \\left[ \n - \\int^T_0 d\\tau \n \\left(\n \\frac{\\dot{L}^2(\\tau)}{4L(\\tau)} + (\\Lambda + \\nu (\\tau)) L(\\tau)\n \\right)\n \\right]\\ .\n \\label{eq:pathintegralnu}\n\\]\nTherefore, all the contributions coming from the sum over all wormholes and baby universes \ncan be fully taken in if the cosmological ``constant'' in (the continuum limit of) $2$d CDT or projectable HL quantum gravity where no wormholes and baby universes exist \nis not really a constant but fluctuates in time. This would lead to a realization of Coleman's mechanism that will be discussed in Sec.~\\ref{sec:coleman}. \n\nIn the next section, we will show that the full propagator can be also obtained if quantizing $2$d projectable HL gravity with an effective wormhole interaction term. \n\n\n\n\n\n\\subsection{Wormhole interaction in $2$d projectable HL gravity}\n\\label{sec:wormholeinteraction}\n\nLet us consider the $2$d projectable HL gravity with a space-like wormhole interaction given by \nthe following action: \n\\[\nI_{\\text{w}} &=\n\\frac{1}{\\kappa} \n\\int dtdx\\ \nN(t) \\sqrt{h(t,x)} \n\\left(\n(1-\\eta) K^2(t,x) - 2 \\widetilde{\\Lambda}\n\\right)\\notag \\\\\n&\\ \\ \\ + \\beta \\int dt N(t) \\int dx_1 dx_2 \\sqrt{h(t,x_1)} \\sqrt{h(t,x_2)}\n\\ , \n\\label{eq:wormholeaction}\n\\]\nwhere $\\beta$ is a dimension-full coupling constant. \nThe last bi-local term can be interpreted as an effective interaction term for a space-like wormhole connecting two distant regions at a given $t$. \nThis bi-local term is allowed to be included since it is invariant under FPD. \n\n\n\nFollowing essentially the same procedure explained in Sec.~\\ref{sec:quantization}, \nlet us quantize the system defined by the action (\\ref{eq:wormholeaction}). \nIntroducing the conjugate momentum of the density $\\sqrt{h}$ as $\\Pi$, we introduce the Poisson bracket (\\ref{eq:poisson}). \nImplementing the Legendre transform, one obtains the corresponding Hamiltonian: \n\\[\nH_{\\text{w}} \n= N(t)\\ \\mathcal{C}_{\\text{w}} (t) \n+ \\int dx N_1 (t,x)\\ \\mathcal{C}^1_{\\text{w}} (t,x)\\ , \n\\label{eq:hw}\n\\]\nwhere \n\\[\n\\mathcal{C}^1_{\\text{w}} (t,x) \n&= - \\frac{\\partial_1 \\Pi (t,x)}{\\sqrt{h (t,x)}} \\approx 0\\ , \\label{eq:momentumconst2}\\\\\n\\mathcal{C}_{\\text{w}} (t)\n&= \\int dx\\ \n\\biggl(\n\\frac{\\kappa}{4(1-\\eta)} \\Pi^2(t,x) \\sqrt{h(t,x)} + \\frac{2}{\\kappa} \\widetilde{\\Lambda} \\sqrt{h(t,x)} \\notag \\\\\n& \\ \\ \\ \n- \\beta \\sqrt{h(t,x)} \n\\int dx_2 \\sqrt{h(t,x_2)}\n\\biggl)\n\\approx 0 \n\\ . \\label{eq:classicalhamiltonianconst2} \n\\] \nThe constraints (\\ref{eq:momentumconst2}) and (\\ref{eq:classicalhamiltonianconst2}) are the momentum constraint and the Hamiltonian constraint, respectively. \nSolving the momentum constraint (\\ref{eq:momentumconst2}) at the classical level as before, \nthe Hamiltonian (\\ref{eq:hw}) reduces to the following one-dimensional one:\n\\[\nH_{\\text{w}}\n= N(t) \\left(\n\\frac{\\kappa}{4(1-\\eta)} \\Pi^2 (t) L(t) \n+ \\frac{2}{\\kappa} \\widetilde{\\Lambda} L(t) \n- \\beta L^2 (t)\n\\right)\\ , \n\\label{eq:hw2}\n\\] \nwhere $L(t) := \\int dx \\sqrt{h(t,x)}$. The Hamiltonian (\\ref{eq:hw2}) is subject to the Hamiltonian constraint: \n\\[\nL(t)\n\\left(\n\\frac{\\kappa}{4(1-\\eta)} \\Pi^2 (t) \n+ \\frac{2}{\\kappa} \\widetilde{\\Lambda} \n- \\beta L (t) \n\\right)\n\\approx 0 \n\\ . \n\\label{eq:classicalhamiltonianconst3}\n\\] \nHere we choose the CDT parametrization (\\ref{eq:parameteridentification}). \nA solution to the Hamiltonian constraint (\\ref{eq:classicalhamiltonianconst3}) is \n\\[\n\\Pi^2 = - \\Lambda + \\beta L \\ge 0\\ , \\ \\ \\ \\text{for}\\ \\ \\ \\sqrt{\\Lambda} L \\ge 1\/\\xi\\ , \n\\label{eq:solutiontohamiltonianconstw1}\n\\] \nwhere $\\xi$ is a dimensionless parameter given by $\\xi = \\beta\/ \\Lambda^{3\/2}$. \nFor $\\sqrt{\\Lambda} L < 1\/\\xi$, the only allowed solution is $L=0$. \n\n\nWhen quantizing the system, if we follow the same procedure described in Sec.~\\ref{sec:quantization}, and set \n$\\beta = g_s$, one can reproduce the path-integral of the full propagator (\\ref{eq:pathintegralfullpropagator}). \nRemember the boundary condition for the path-integral (\\ref{eq:pathintegralfullpropagator}), i.e. \nthe kinetic term should counteract the unboundedness of the potential term at $L=\\infty$. \nThis balance between the kinetic and potential terms is precisely what is reflected in the classical Hamiltonian constraint (\\ref{eq:solutiontohamiltonianconstw1}). \n \n\n\n\n\n\\subsection{Coleman's mechanism} \n\\label{sec:coleman}\n\nIn this section, we discuss a sort of Coleman's mechanism in the context of two-dimensional gravity based on CDT briefly. \n\nLet us define the two kinds of Wheeler-deWitt equation: \n\\[\n\\hat{H} W_0 (L) = 0\\ , \\ \\ \\ \\hat{H}_{\\text{eff}} W(L) = 0\\ , \n\\label{eq:wdw}\n\\]\nwhere $\\hat{H} := \\hat{H}^{(-1)}$ introduced in eq.~(\\ref{eq:hamiltonians}). \nThe solutions to the Wheeler-deWitt equations are the Hartle-Hawking wave functions given by \n\\[\nW_0 (L) = e^{-\\sqrt{\\Lambda} L}\\ , \\ \\ \\ \nW(L) = \\frac{\\text{Bi} (\\xi^{-2\/3} - \\xi^{1\/3} \\sqrt{\\Lambda} L)}{ \\text{Bi} (\\xi^{-2\/3}) } \n+ c\\ \\text{Ai} ( \\xi^{-2\/3} - \\xi^{1\/3} \\sqrt{\\Lambda} L)\\ , \n\\label{eq:hartlehawking}\n\\] \nwhere $\\text{Ai}$ and $\\text{Bi}$ are the standard Airy functions, \n$\\xi$ is a dimensionless string coupling constant measured by the cosmological constant, i.e. \n$\\xi := g_s\/ \\Lambda^{3\/2}$, \nand $c$ is an undetermined dimensionless constant. \nThe Hartle-Hawking wave function $W_0 (L)$ is the one for (the continuum theory of) $2$d CDT, \ni.e. neither baby universe nor wormhole contributions are included. \nOn the other hand, $W(L)$ is the Hartle-Hawking wave function including all possible contributions of baby universes and wormholes non-perturbatively. \n\nWe wish to explore the behavior of the non-perturbative Hartle-Hawking wave function $W(L)$ (see Fig.~\\ref{fig:HH}).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=.4]{HH.pdf}\n\\caption{A plot of the Hartle-Hawking wave functions, $W_0$ (dashed line) and $W$ (solid line), for $\\xi=1\/3$ and $c=0$: \nThe horizontal axis is $\\sqrt{\\Lambda} L$ and the vertical axis is either $W_0$ or $W$. \n}\n\\label{fig:HH} \n\\end{figure} \nFor $\\sqrt{\\Lambda}L \\ll 1\/\\xi$, one obtains the asymptotic expansion: \n\\[\nW(L) \\sim e^{-\\sqrt{\\Lambda}L} = W_0(L)\\ . \n\\label{eq:asymptoticexpansion}\n\\] \nTherefore, when the size of the one-dimensional universe is small enough, \nthe physics is very closed to the one without baby universes and wormholes, and it is essentially governed by the cosmological constant. \nThe wave function in this region decreases exponentially, and this behavior does not change at any finite order of perturbation. \n \nHowever, once the size of the universe is large enough, i.e. $\\sqrt{\\Lambda} > 1\/\\xi$, the wave function starts oscillating, and the behavior is governed by \nthe string coupling constant instead of the cosmological constant:\n\\[\nW(L) \\sim 1\/(g^{1\/3}_s L)^{1\/4}\\ . \n\\label{eq:largebehavior}\n\\] \nThis drastic change happens due to the infinitely many wormholes and baby universes. \nThe similar behavior has been observed in the context of non-critical string theory \\cite{Moore:1991sf, Betzios:2020nry}. \n\nFrom the discussion above, we observe that a sort of Coleman's mechanism works: \nFor a large universe, the cosmological constant is not important enough to govern the physics\\footnote{\nThe Hartle-Hawking wave function $W(L)$ is not normalizable, but similar arguments are valid on the normalizable energy eigenstates \\cite{Ambjorn:2021wdm}. \n}. \n\n\n\\section{Summary}\n\\label{sec:summary} \n\nWe have reviewed the relation between the two-dimensional causal dynamical triangulations ($2$d CDT) and the two-dimensional projectable Ho\\v rava-Lifshitz quantum gravity ($2$d projectable HL QG). \n\nIn the first part, it has been shown that the physics described by the continuum limit of $2$d CDT coincides with the one obtained quantizing $2$d projectable HL gravity. \nThis is confirmed because the quantum Hamiltonians of both models are exactly the same. \nThe system is expressed in terms of quantum mechanics of a $1$d extended object, i.e. a $1$d universe. \nIt would be too hasty to consider that this scenario also holds for the higher dimensional cases. \nIn fact, it has been shown that in $2+1$ dimensions numerical studies of the so-called locally causal dynamical triangulations (LCDT) that relaxes the proper-time foliation of CDT and requires the local causality reproduce an intriguing specialty of CDT, \nan emergence of the de-Sitter-like geometry \\cite{Jordan:2013awa} (see e.g. Ref.~\\cite{Loll:2019rdj} for the higher-dimensional CDT). \n\nIn the second part, we have introduced the generalized CDT (GCDT) that permits baby universes and wormholes to form in keeping with the foliation, \nand in particular the construction based on the string field theory for CDT has been explained. \nHere the string means the $1$d universe, and the string field theory is constructed in such a way that the free part reproduces the CDT amplitudes, \nand the splitting and joining interactions of string are introduced to create baby universes and wormholes. \n\n\nFocusing on the loop-to-loop amplitude, we have introduced an effective $1$d theory that includes all the contributions coming from the sum over all possible baby universes and wormholes. \nFrom the point of view of HL gravity, the effective theory can be precisely reproduced if introducing a bi-local interaction term into the action of $2$d projectable HL gravity and if quantizing the system. \nIn addition, the effective theory can be also obtained considering that the cosmological constant of $2$d CDT is not a constant but it fluctuates in time. \nThis leads to Coleman's mechanism in $2$d CDT such that for a large universe the cosmological constant is not important enough to govern the physics. \n \n \nAlthough we have not discussed issues of the coupling to matter, $2$d CDT coupled to Yang-Mills theory has been solved analytically in Ref.~\\cite{Ambjorn:2013rma}, \nand it has been shown that the quantum Hamiltonian obtained in Ref.~\\cite{Ambjorn:2013rma} can be reproduced quantizing $2$d projectable HL gravity coupled to Yang-Mills theory \\cite{Ipsen:2015ckl}. \nIn fact, we know very little about the analytical treatment of the coupling to matter compared to the situation of the $2$d dynamical triangulations and the Liouville quantum gravity. \nThis direction need to be explored in the future. \n \n\nWhat is remarkable is that following the standard Wilsonian renormalization group, one can take the continuum limit of the lattice model, $2$d CDT, \nand find the continuum quantum field theory, $2$d projectable HL QG, which is in the same universality class of $2$d CDT. \nA missing piece is the continuum quantum field theory of GCDT that is described by metric components and allows us to compute all the amplitudes defined by the string field theory for CDT, \nalthough we have the effective field theory, the $2$d projectable HL gravity with a bi-local interaction, that reproduces the restricted class of GCDT amplitudes once it is quantized. \nWe wish to unveil the underlying continuum quantum field theory of GCDT, through which we can understand something inherently interesting about quantum geometries for sure. \n\n\n \n\n\n\\begin{acknowledgement}\nYS would like to thank \nJan Ambj\\o{}rn, \nLisa Glaser, \nYuki Hiraga, \nYoshiyasu Ito \nand \nYoshiyuki Watabiki, for wonderful collaborations on the topics related to this chapter. \nGreat ideas appeared in this chapter attribute to them. However, if conceptual mistakes exist, YS is responsible for them. \n\\end{acknowledgement}\n\n\n\n\\input{references}\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{}\n\\subsection{}\n\n\\begin{theorem}[Optional addition to theorem head]\n\\end{theorem}\n\n\\begin{proof}[Optional replacement proof heading]\n\\end{proof}\n\n\\begin{figure}\n\\includegraphics{filename}\n\\caption{text of caption}\n\\label{}\n\\end{figure}\n\n\n\\begin{equation}\n\\end{equation}\n\n\\begin{equation*}\n\\end{equation*}\n\n\\begin{align}\n & \\\\\n &\n\\end{align}\n\n\\section{Introduction}\nThe motivation behind this note is the following question: what are the measures $\\mu$ on the Heisenberg group ${\\mathbb{H}}^n$ which guarantee that the (correct notion of) Riesz transform is bounded from $L^2(\\mu)$ to itself? This question (or some variant of it) with $\\mathbb{R}^n$ instead of ${\\mathbb{H}}^n$, was one of the major starting point of the theory that came to be known as \\textit{quantitative rectifiability}. This area of geometric measure theory has seen an impressive development in the past thirty years, starting with the landmark works of Peter Jones \\cite{jones} and David and Semmes \\cite{david-semmes91}, \\cite{david-semmes93}, through the solution of fundamental questions in complex analysis, such as the Painlev\\'{e} problem (see \\cite{mmv}, \\cite{david98}, \\cite{tolsa03}), to more recent applications to harmonic analysis, see for example \\cite{ntv} and \\cite{ahmmt}.\n\nIn the last years, there has been an increasing interest in developing such a quantitative theory in different contexts than that of Euclidean spaces; examples of these are parabolic spaces and Heisenberg groups, or, more generally, Carnot groups. The former appear in the study of caloric measure. The latter arise naturally in the study of certain hypoelliptic operators, in the sense that the natural translations and dilations for these operators are those characterising the spaces; the Heisenberg group is the most important prototypical example, and the related operator is the so-called Kohn Laplacian; see \\cite{blu} for a comprehensive study of stratified Lie groups and the corresponding operators. \n\nWe should mention that the study of Heisenberg geometry can be approached from different perspectives and with different applications in mind; for example, see \\cite{ny} for a connection with theoretical computer science. \n\nTo be a little more specific: the starting motivation to develop a theory of quantitative rectifiability connected to our initial question, is to understand basic issues such as the removable sets for harmonic function (with respect to the relevant sub-Laplacian), or to give a characterisation of those domains where the Dirichlet problem (again, for the relevant sub-Laplacian) is well-posed. We want to underline, however, that a theory of quantitative rectifiability in the Heisenberg setting has its own, purely geometric, intrinsic appeal.\n\nIn the last couple of years, there has been some progress towards an answer to our initial question; see for example \\cite{cfo}, \\cite{fo18} and \\cite{o18}. In this note we give a necessary condition to be imposed on a Radon measure $\\mu$ on ${\\mathbb{H}}^n$ for the Riesz transform to be $L^2(\\mu)$ bounded. Here $R_\\mu$ is the singular integral operator whose kernel is the horizontal gradient of the fundamental solution of the Heisenberg sub-Laplacian, as defined in \\cite{chousionis2012singular}. See Section \\ref{preliminaries} for precise definitions. \n\n\\begin{theorem} \\label{t:main}\nLet $\\mu$ be a Radon measure on ${\\mathbb{H}}^n$ such that $R_\\mu$ is bounded on $L^2(\\mu)$ with norm $C_1$, and such that $\\mu(F) = 0$ whenever $\\dim_H(F) \\leq 2$. Then there exists a constant $C_2$ such that for all balls $B(x,r) \\subset {\\mathbb{H}}^n$, we have\n\\begin{align}\\label{e:main}\n \\mu(B(x,r)) \\leq C_2 r^{2n +1}.\n\\end{align}\nHere $C_2$ depends only on $n$ and $C_1$, and the ball $B(x,r)$ is defined with respect to the Kor\\'{a}nyi metric, see Section \\ref{preliminaries}.\n\\end{theorem}\n\nA corresponding statement holds in the Euclidean setting, and is a result of David, \\cite{david-wavelets}, Part III, Proposition 1.4. See \\cite{orponen-notes}, Proposition 6.9 for a more detailed proof. Let $\\mathcal{R}^{d}_\\mu$ denote the standard $d$-dimensional Riesz transform in $\\mathbb{R}^n$.\n\\begin{theorem}\nAssume that $\\mu$ is a non-atomic Radon measure on $\\mathbb{R}^n$ such that $\\mathcal{R}^{d}_\\mu$ is bounded on $L^2(\\mu)$ with norm $C_1$. Then, for all Euclidean balls $B_{\\mathbb{R}^n}(x,r)\\subset\\mathbb{R}^n$ we have\n\\begin{align}\\label{e:growth}\n\\mu (B_{\\mathbb{R}^n}(x,r)) \\leq C_2 r^d \n\\end{align}\nHere $C_2$ depends only on $C_1$, $n$, and $d$. \n\\end{theorem}\nA measure satisfying \\eqref{e:growth} (or \\eqref{e:main}) is said to have \\textit{polynomial growth}. \nLet us give a couple of remarks. \n\\begin{remark}\nAlthough the result itself (both in the Euclidean and Heisenberg case) is neither hard nor deep, it is nevertheless very useful. For example, most tools developed in the last two decades that take quantitative rectifiability beyond Ahlfors regular measures still need polynomial growth\\footnote{With some exceptions, see for example \\cite{azzam-schul18}, or \\cite{badger-schul15}.} (see for example the book by Tolsa \\cite{tolsa-book}). Thus, we expect that our result will be quite useful, too. \n\\end{remark}\n\\begin{remark}\nWhile the two results above look similar, there is actually a difference, in the sense that, in the Heisenberg case, there actually exist lower dimensional measures which give a bounded Riesz transform, but are not atomic.\n\nThis is \\textit{not} a byproduct of the proof, but rather a fact of the Heisenberg geometry. Indeed, the $2$-dimensional $t$-axis (or any Heisenberg translate of it) gives a bounded $(2n+1)$-dimensional Riesz transform; this is simply because on these sets the kernel vanishes identically, see \\eqref{e:K-norm}.\n\nOne can construct a more interesting example in the vertical plane of the one dimensional Heisenberg group ${\\mathbb{H}}$, say. Consider a tube of height $1$ and radius $\\varepsilon_1^2$ around the $t$-axis, and take the intersection with the vertical plane. Call the resulting rectangle $R_{1,1}$. Cut out from $R_1$ two smaller rectangles $R_{2,1}$ and $R_{2,2}$, one in the top right corner and one in the bottom left corner, both of height $\\varepsilon_2$ and width $\\varepsilon_2^2$, for some $\\varepsilon_2\\le \\varepsilon_1\/4$. We proceed in this way, so that after $k$ steps we have $2^{k-1}$ disjoint rectangles $\\{R_{k,i}\\}_i$ of height $\\varepsilon_k$ and width $\\varepsilon_k^2$. Consider the natural probability measure $\\mu$ on the Cantor-like set $C=\\bigcap_k\\bigcup_i R_{k,i}$. It is not difficult to show that, if $\\varepsilon_k\\to 0$ are small enough, the Heisenberg Riesz transform is bounded on $L^2(\\mu)$; the idea is that the set is concentrated along the $t$-axis, and thus the kernel is very small (see \\eqref{e:K-norm} below). Depending on the choice of $(\\varepsilon_k)$ we have $\\dim_H(C)\\in [0,2].$\n\\end{remark}\n\n\\subsection*{Organisation of the article} In Section \\ref{preliminaries} we briefly recall basic facts about Heisenberg groups and the Riesz transform. We also introduce a family of ``dyadic cubes'' suitable to our setting. \n\nSection \\ref{s:main lemma} is dedicated to Lemma \\ref{l:main lemma}, our main technical lemma. Roughly speaking, we show that if a measure $\\mu$ is such that $R_{\\mu}$ is bounded on $L^2(\\mu)$, and there is some cube $Q_0$ with a very high concentration of $\\mu$ (i.e. $\\mu(Q_0)\\gg \\ell(Q_0)^{2n+1}$), then we can find a family ${\\mathsf{HD}}(Q_0)$ of much smaller cubes, contained in $Q_0$, such that\n\\begin{enumerate}\n \\item[a)] a very large portion of measure $\\mu$ on $Q_0$ is concentrated on the cubes from ${\\mathsf{HD}}(Q_0)$,\n \\item[b)] the family ${\\mathsf{HD}}(Q_0)$ is relatively small, in the sense that it consists of few cubes.\n\\end{enumerate}\n\nIn Section \\ref{s:iteration} we show that if the polynomial growth condition \\eqref{e:main} is not satisfied, then we can find a cube satisfying the assumptions of our main lemma. This in turn allows us to start an iteration algorithm, consisting of using the main lemma countably many times, that results in constructing a set $Z$ with $\\mu(Z)>0$ and $\\dim_H(Z)\\le 2.$ This finishes the proof of Theorem \\ref{t:main}.\n\n\n\\subsection*{Acknowledgements}\nWe thank Katrin F\\\"{a}ssler and Tuomas Orponen for introducing us to the Heisenberg group and for suggesting the problem. \n\nThe bulk of this work was done while the two authors were attending the Simons Semester in Geometric Analysis at IMPAN in the Autumn 2019. We thank Tomasz Adamowicz and the staff at the institute for their hospitality.\n\n\\section{Preliminaries}\\label{preliminaries}\nIn our estimates we will often use the notation $f\\lesssim g$ which means that there exists some absolute constant $C$ for which $f\\le Cg.$ If the constant $C$ depends on some parameter $t$, we will write $f\\lesssim_t g.$ Notation $f\\approx g$ will stand for $f\\lesssim g\\lesssim f,$ and $f\\approx_t g$ is defined analogously. For simplicity, in our estimates we will suppress the dependence on dimension $n$ and on absolute constants $\\lambda,\\ \\Lambda$ (see \\eqref{e:balls in cubes}).\n\n\\subsection{Heisenberg group}\nIn this paper we consider the $n$-th Heisenberg group with exponential coordinates (see \\cite{cdpt} or \\cite{faessler-notes} for a swift introduction to the Heisenberg group in a context close to ours). In practice, we will denote a point $p \\in {\\mathbb{H}}^n$ as $(z,t) \\in \\mathbb{R}^{2n} \\times \\mathbb{R}$, and $z=(x_1,...,x_n,y_1,...,y_n)$. In these coordinates the group law in ${\\mathbb{H}}^n$ takes the form \n\\begin{align*}\n p \\cdot q = \\ps{z + z', t+t' + \\frac{1}{2} \\sum_{i=1}^n (x_i y_i' + y_i x_i')},\n\\end{align*}\nwhere $p=(z,t)$ and $q=(z',t')$. The identity element is the origin $(0,0)$ and the inverse is given by $p^{-1}=(-z,-t)$. We make ${\\mathbb{H}}^n$ into a metric space by setting $ d(p,q) := \\|q^{-1}\\cdot p\\|_{\\mathbb{H}}$, where \n\\begin{align}\\label{e:korany}\n \\|p\\|_{\\mathbb{H}}^4 := |z|^4 + 16t^2,\n\\end{align}\nand $|z|$ denotes the Euclidean norm of $z\\in\\mathbb{R}^{2n}$.\n\nNote that $\\|\\cdot\\|_{\\mathbb{H}}$ is $1$-homogeneous with respect to the anisotropic dilation $p \\mapsto \\lambda p = (\\lambda z, \\lambda^2 t)$, $\\lambda >0$. The metric $d$ is sometimes called the Kor\\'{a}nyi metric.\n\nGiven $p\\in{\\mathbb{H}}^n$ and $r>0$ we set\n\\begin{equation*}\n B(p,r)=\\ck{q\\ |\\ d(p,q)\\le r},\\quad U(p,r) = \\ck{q\\ |\\ d(p,q)< r}.\n\\end{equation*}\nFor $\\alpha>0$ we will write $\\mathcal{H}^{\\alpha}$ to denote the usual $\\alpha$-dimensional Hausdorff measure with respect to metric $d$. For $A\\subset{\\mathbb{H}}^n$ we set $\\dim_H(A)$ to be the Hausdorff dimension of $A$. \n\nIt follows easily from the definition of the Kor\\'{a}nyi metric that for all $p\\in{\\mathbb{H}}^n$ and $r>0$ we have\n\\begin{equation}\\label{e:measure of balls}\n \\mathcal{H}^{2n+2}(B(p,r)) = \\mathcal{H}^{2n+2}(B(0,1))\\, r^{2n+2}.\n\\end{equation}\nThus, even though the topological dimension of ${\\mathbb{H}}^n$ is $2n+1$, the Hausdorff dimension of ${\\mathbb{H}}^n$ is equal to $2n+2$. For the sake of brevity we set $D:=2n+2.$ Usually one denotes the Hausdorff dimension of ${\\mathbb{H}}^n$ by $Q$, but we have decided to save that letter for cubes; hence the non-standard notation.\n\nIt is also easy to check that if $\\mathcal{L}^{2n+1}$ denotes the usual Lebesgue measure on $\\mathbb{R}^{2n+1}\\simeq {\\mathbb{H}}^n,$ then we have a constant $C>0$ such that\n\\begin{equation}\\label{e:lebesgue and hausdorff}\n \\mathcal{L}^{2n+1}=C\\mathcal{H}^{D}.\n\\end{equation}\n\n\\subsection{Heisenberg Riesz transform}\nRecall that, for a function $u : {\\mathbb{H}}^n \\to \\mathbb{R}$, the horizontal gradient of $u$ is given by\n\\begin{align*}\n \\nabla_{\\mathbb{H}} u := \\ps{ X_1 u,...,X_n u, Y_1 u, ..., Y_n u},\n\\end{align*}\nwhere the vector fields $X_1,\\dots,X_n,Y_1,\\dots, Y_n$ and $\\frac{\\partial}{\\partial t}$ represent the left invariant translates of the canonical basis at the identity. In particular, $X_1,\\dots,X_n,Y_1,\\dots, Y_n$ span the horizontal distribution in ${\\mathbb{H}}^n$. \n\nThe Heisenberg sublaplacian $\\Delta_{\\mathbb{H}}$ is given by $\\sum_{i=1}^n X_i^2 + Y_i^2$, and its fundamental solution is \n\\begin{align*}\n G(p) := c_n \\|p\\|_{\\mathbb{H}}^{2-D}.\n\\end{align*}\n\nThe $(D-1)$-dimensional Riesz kernel in ${\\mathbb{H}}^n$, first considered in \\cite{chousionis2012singular}, is given by\n $K(p)=\\nabla_{{\\mathbb{H}}} G(p).$\nThe Riesz transform is formally defined as\n\\begin{align*}\n R_\\mu f (p) = \\int_{\\mathbb{H}^n} K(q^{-1}\\cdot p)f(q) \\, d\\mu(q).\n\\end{align*}\nSince it is not clear whether the integral above converges, one considers the truncated Riesz transform given by the formula\n\\begin{align*}\n R_{\\mu, \\delta}f(p)= \\int_{{\\mathbb{H}}^n \\setminus B(p,\\delta)} K(q^{-1}\\cdot p)f(q)\\ d\\mu(q),\n\\end{align*}\nfor $\\delta>0$. We say that $R_{\\mu}$ is bounded on $L^2(\\mu)$ if the truncated operators $R_{\\mu, \\delta}$ are bounded on $L^2(\\mu)$ uniformly in $\\delta>0.$\n\nOne can easily check that the Riesz kernel is actually equal to\n\\begin{align*}\n & K(z, t) \\\\\n & = n\\, \\ps{ \\frac{ -2x_1 |z|^2 + 8y_1 t }{\\|(z,t)\\|_{\\mathbb{H}}^{2n+4\n }}, \\,\\cdots , \\frac{ -2x_n |z|^2 + 8y_n t }{\\|(z,t)\\|_{\\mathbb{H}}^{2n+4\n }}, \\, \\frac{ -2y_1 |z|^2 - 8x_1 t }{\\|(z,t)\\|_{\\mathbb{H}}^{2n+4}}, \\cdots , \\frac{ -2y_n |z|^2 - 8x_n t }{\\|(z,t)\\|_{\\mathbb{H}}^{2n+4}}}.\n\\end{align*}\nHence,\n\\begin{align} \\label{e:K-norm}\n |K(z,t)|^2 = n^2 \\, \\frac{ 4 |z|^2 }{(|z|^4 + 16t^2)^{n+1}}.\n\\end{align}\nThis implies the curious fact that \n $|K(z,t)| \\leq C$\nwhenever\n\\begin{align} \\label{e:paraboloid}\n |z| \\leq \\, 16 |t|^{n+1},\n\\end{align}\nwhich is a `paraboloidal' double cone around $t$-axis with vertex at the origin. This fact will play a key role in the subsequent analysis. \n\nChousionis and Mattila showed in \\cite[Proposition 3.11]{chousionis2012singular} that the Riesz kernel is a standard kernel. In particular, it satisfies the following continuity property: whenever $q_1, q_2\\neq p\\in {\\mathbb{H}}^n$, we have\n\\begin{equation*}\\label{e:kernel continuous}\n |K(p^{-1}\\cdot q_1) - K(p^{-1}\\cdot q_2)| \\lesssim \\max\\bigg\\{\\frac{d(q_1,q_2)}{d(p,q_1)^{D}},\\frac{d(q_1,q_2)}{d(p,q_2)^{D}} \\bigg\\}.\n\\end{equation*}\nTaking $p=0$ and $q_1 = \\tilde{q_1}^{-1}\\cdot \\tilde{p},\\ q_2 = \\tilde{q_2}^{-1}\\cdot \\tilde{p},$ one gets immediately that for all $\\tilde{q_1}, \\tilde{q_2}\\neq \\tilde{p}\\in {\\mathbb{H}}^n$\n\\begin{equation}\\label{e:kernel continuous 2}\n |K(\\tilde{q_1}^{-1}\\cdot \\tilde{p}) - K(\\tilde{q_2}^{-1}\\cdot \\tilde{p})| \\lesssim \\max\\bigg\\{\\frac{d(\\tilde{q_1},\\tilde{q_2})}{d(\\tilde{p},\\tilde{q_1})^{D}},\\frac{d(\\tilde{q_1},\\tilde{q_2})}{d(\\tilde{p},\\tilde{q_2})^{D}} \\bigg\\}.\n\\end{equation}\n\n\\subsection{Dyadic cubes}\nWe are going to use a family of decompositions of ${\\mathbb{H}}^n$ into subsets that share many properties with the standard dyadic cubes from $\\mathbb{R}^n$. The most classical constructions of this kind are due to Chirst \\cite{christ1990b} and David \\cite{David1988}, but for us it will be more convenient to use the ``cubes'' constructed in \\cite{kaenmaki2012}.\n\nFirst, note that given any ball $B(p,2r)$, one may use the $5r$-covering lemma and the property \\eqref{e:measure of balls} to conclude that there exists some absolute constant $m$ such that $B(p,2r)$ may be covered by $m$ balls $B(p_i,r)$, where $\\ck{p_i}_{i=1}^m$ are points in $B(p,2r)$. That is, ${\\mathbb{H}}^n$ is geometrically doubling. In particular, we can use {\\cite[Theorem 2.1, Remark 2.2]{kaenmaki2012}}.\n\n\\begin{lemma}[\\cite{{kaenmaki2012}}] \\label{l:cubes}\n\tFor all $k\\in\\mathbb{Z}$ there exists a family of subsets of ${\\mathbb{H}}^n$, denoted by $\\mathcal{D}_k$, such that\n\t\\begin{enumerate}\n\t\t\\item ${\\mathbb{H}}^n = \\bigcup_{Q\\in\\mathcal{D}_k}Q$,\n\t\t\\item if $k\\ge l$, and $Q\\in \\mathcal{D}_k,\\ P\\in\\mathcal{D}_l$, then either $Q\\cap P=\\varnothing$ or $Q\\subset P$,\n\t\t\\item for every $Q\\in \\mathcal{D}_k$ there exists $p_Q\\in Q$ such that\n\t\t\\begin{equation}\\label{e:balls in cubes}\n\t\tU(p_Q,\\lambda2^{-k})\\subset Q\\subset B(p_Q,\\Lambda2^{-k})\n\t\t\\end{equation}\n\t\tfor some absolute constants $\\lambda,\\Lambda>0$.\n\t\\end{enumerate}\n\\end{lemma}\n\nLet us stress once more that we will not keep track of how various parameters appearing in the proof depend on $\\lambda$ and $\\Lambda$.\n\nWe set $\\mathcal{D} = \\bigcup_k \\mathcal{D}_k$. For $Q\\in\\mathcal{D}_k$ we define the sidelength of $Q$ as $\\ell(Q)=2^{-k}$. Clearly, by \\eqref{e:measure of balls} and \\eqref{e:balls in cubes}, for $Q\\in\\mathcal{D}$ we have\n\\begin{equation*}\n \\mathcal{H}^{{D}}(Q) \\approx \\ell(Q)^{{D}}.\n\\end{equation*}\nIt follows that if $Q\\in\\mathcal{D}$, then for $k\\ge 0$\n\\begin{equation}\\label{e:dyadic property}\n\\#\\ck{P\\in\\mathcal{D}\\ |\\ P\\subset Q,\\ \\ell(P)=2^{-k}\\ell(Q) }\\approx 2^{kD}.\n\\end{equation}\n\nGiven a Radon measure $\\mu$ and $Q\\in\\mathcal{D}$ we will denote the $(D-1)$-dimensional density of $\\mu$ in $Q$ by\n\\begin{equation*}\n \\Theta_{\\mu}(Q) = \\frac{\\mu(Q)}{\\ell(Q)^{D-1}}.\n\\end{equation*}\nFor simplicity, we will suppress the dependence on $\\mu$ and simply write $\\Theta(Q)$.\n\n\\section{Main lemma}\\label{s:main lemma}\n Our main tool in the proof of Theorem \\ref{t:main} is the following lemma.\n\t\\begin{lemma}\\label{l:main lemma}\n\t\tLet $\\mu$ be a Radon measure on ${\\mathbb{H}}^n$ such that $R_{\\mu}$ is bounded on $L^2(\\mu)$ with norm $C_1$. There exist constants $A=A(n)>1,\\ s=s(A,n)\\in (0,1\/2)$ and $M=M(C_1,n)>100$ such that the following holds.\n\t\t\n\t\tSuppose that $Q_0\\in\\mathcal{D}$ satisfies $\\Theta(Q_0)\\ge M$. Set $N = \\floor{A^{-2} \\log(\\Theta(Q_0))}$. Then, the family of high density cubes\n\t\t\\begin{equation*}\n\t\t{\\mathsf{HD}}(Q_0) = \\ck{Q\\in \\mathcal{D}\\ |\\ Q\\subset Q_0,\\ \\ell(Q)=2^{-N}\\ell(Q_0),\\ \\Theta(Q)>2\\,\\Theta(Q_0)}\n\t\t\\end{equation*}\n\t\tsatisfies\n\t\t\\begin{equation}\\label{e:HD are thicc}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\mu(Q) \\ge (1-\\Theta(Q_0)^{-s})\\mu(Q_0).\n\t\t\\end{equation}\n\t\tMoreover, we have\n\t\t\\begin{equation}\\label{e:HD packing}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\ell(Q)^2\\le C_p\\, \\ell(Q_0)^2\n\t\t\\end{equation}\n\t\tfor some dimensional constant $C_p$ (``$p$'' stands for ``packing'').\n\t\\end{lemma}\n\t\n\tThe rest of this section is dedicated to proving the lemma above. For brevity of notation, we set $\\Theta_0=\\Theta(Q_0)$. Observe that the integer $N$ was chosen in such a way that\n\t\\begin{equation}\\label{e:choice of N}\n\t2^{A^2N}\\approx \\Theta_0\\ge M.\n\t\\end{equation}\n\tIn particular, we have $N\\ge N_0$ for some very big $N_0$ depending on $M$ and $A$.\n\t%\n\n\n\n\n\t\n\tWe split the proof of Lemma \\ref{l:main lemma} into several steps.\n\t\n\tFirst, note that by the pigeonhole principle and \\eqref{e:dyadic property}, we can find a cube $Q_1 \\in \\mathcal{D}$ with sidelength $\\ell(Q_1) = 2^{-AN} \\ell(Q_0)$ such that \n\t\\begin{align}\\label{e:assumption Q1}\n\t\\mu(Q_1) \\gtrsim \\frac{\\mu(Q_0)}{2^{AND}}. \n\t\\end{align}\n\tWithout loss of generality, by applying the appropriate translation, we can assume that $Q_1$ is centred at the origin, i.e. $p_{Q_1}=0$.\n\tSet\n\t\\begin{align*}\n\tT:=\\ck{ (z,t) \\in Q_0 \\, |\\, |z| \\leq 2^{-N} \\ell(Q_0) }\n\t\\end{align*}\n\tand for any $\\kappa>0$ set \n\t\\begin{equation*}\n\t{ T_{\\kappa}}:=\\ck{ (z,t) \\in Q_0 \\, |\\, |z| \\leq \\kappa\\, 2^{-N} \\ell(Q_0) }.\n\t\\end{equation*}\n\tObserve that $Q_1\\subset T.$ In a sense, $T$ can be seen as a tube with vertical axis passing through $p_{Q_1}=0$. Note also that for any cube $Q\\subset Q_0 \\setminus T$ we have $\\dist(Q,Q_1)\\gtrsim 2^{-N}\\ell(Q_0).$\n\t\n\tWe start by proving a few preliminary results.\n\t\\begin{lemma}\\label{c:number-cubes}\n\tThere are at most $C(\\kappa)\\, 2^{2N}$ cubes of sidelength $2^{-N}\\ell(Q_0)$ contained in $T_\\kappa$.\n\t\\end{lemma}\n\t\\begin{proof}\n\tObserve that since $0\\in Q_0$, and by \\eqref{e:balls in cubes} $Q_0\\subset B(p_{Q_0},\\Lambda\\ell(Q_0))$, we have $Q_0\\subset B(0,2\\Lambda\\ell(Q_0)).$ Hence,\n\t\\begin{align*}\n\t T_{\\kappa} &\\subset \\ck{ (z,t)\\in B(0,2\\Lambda\\ell(Q_0))\\ |\\ |z|\\le \\kappa\\, 2^{-N} \\ell(Q_0) }\\\\\n\t &\\subset \\ck{ (z,t)\\in {\\mathbb{H}}^n\\ |\\ |z|\\le \\kappa\\, 2^{-N} \\ell(Q_0),\\ 16|t|^2\\le (2\\Lambda \\ell(Q_0))^4 }=: \\widetilde{T}_{\\kappa}.\n\t\\end{align*}\n\tBy \\eqref{e:lebesgue and hausdorff},\n\t\\begin{equation*}\n\t \\mathcal{H}^{D}(\\widetilde{T}_{\\kappa}) = C\\mathcal{L}^{2n+1}(\\widetilde{T}_{\\kappa}) \\approx (\\kappa 2^{-N} \\ell(Q_0))^{2n}(2\\Lambda \\ell(Q_0))^2 \\approx_{\\kappa} 2^{-2nN}\\ell(Q_0)^{D}.\n\t\\end{equation*}\n\tIt follows that $\\mathcal{H}^{D}(T_{\\kappa})\\lesssim_{\\kappa}2^{-2nN}\\ell(Q_0)^{D}.$ On the other hand, recall that for any cube $Q$ with sidelength $\\ell(Q)=2^{-N}\\ell(Q_0)$ we have $\\mathcal{H}^{D}(Q)\\approx 2^{-ND}\\ell(Q_0)^{D}$. Since all such cubes are pairwise disjoint, we get\n\t\\begin{equation*}\n\t \\#\\ck{Q\\in\\mathcal{D}\\ |\\ \\ell(Q)=2^{-N}\\ell(Q_0),\\ Q\\subset T_{\\kappa} } \\lesssim\t\\fr{\\mathcal{H}^{D}(T_{\\kappa})}{2^{-ND}\\ell(Q_0)^{D}}\\lesssim_{\\kappa} \\fr{2^{-2nN}\\ell(Q_0)^{D}}{2^{-N(2n+2)}\\ell(Q_0)^{D}}=2^{2N}.\n\t \\end{equation*}\n\t\\end{proof}\n\t\\begin{lemma}\\label{c:dense-tube}\n\t\tLet $Q\\in\\mathcal{D}$ satisfy $Q \\subset Q_0 \\setminus T$ and $\\ell(Q) = \\ell(Q_1)=2^{-AN}\\ell(Q_0)$. Then\n\t\t\\begin{equation*}\n\t\t \\mu(Q) \\le \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{AND}}.\n\t\t\\end{equation*}\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tSuppose the claim above is false. Then we can find a cube $Q_2 \\subset Q_0 \\setminus T$ with $\\ell(Q_2)=2^{-AN}\\ell(Q_0)$ such that\n\t\t\\begin{equation}\\label{e:assumption Q2}\n\t\t\\mu(Q_2) \\geq \\frac{\\mu(Q_0)}{\\Theta_0 \\,2^{AND}}.\n\t\t\\end{equation}\n\t\tLet $0< \\delta < \\dist(Q_1, Q_2)$, let $p\\in Q_2$ be arbitrary, and consider \n\t\t\\begin{equation*}\n\t\tR_{\\mu,\\delta}(\\mathbbm{1}_{Q_1})(p) = \\int_{Q_1} K(q^{-1} \\cdot p) \\, d\\mu(q). \n\t\t\\end{equation*}\n\t\tBy triangle inequality,\n\t\t\\begin{equation}\\label{e:split}\n\t\t|R_{\\mu,\\delta}(\\mathbbm{1}_{Q_1})(p)| \\geq \\av{ \\int_{Q_1} K(p) \\, d\\mu(q) } - \\av{ \\int_{Q_1} K(q^{-1}\\cdot p) - K(p) \\, d\\mu(q) }.\n\t\t\\end{equation}\n\t\tWe estimate the first term as follows. Note that, since $p \\in Q_2$ and $Q_2$ lies outside $T$, then, writing $p=(z, t)$ and using \\eqref{e:K-norm}, we have\n\t\t\\begin{align*}\n\t\t|K(p)|^2 \\approx \\frac{|z|^2}{(|z|^4 + 16t^2)^{n+1}} \\gtrsim \\frac{|z|^2}{\\ell(Q_0)^{4(n+1)}} \\geq 2^{-2N}\\ell(Q_0)^{-4n-2} = 2^{-2N}\\ell(Q_0)^{-2D+2}.\n\t\t\\end{align*}\n\t\tAnd thus we also have \n\t\t\\begin{equation}\\label{e:first term estimate}\n\t\t\\av{ \\int_{Q_1} K(p)\\, d\\mu(q) } = \\av{K(p)}\\mu(Q_1) \\gtrsim 2^{-N} \\, \\frac{\\mu(Q_1)}{\\ell(Q_0)^{D-1}}.\n\t\t\\end{equation}\n\t\t\n\t\tFor the second term in \\eqref{e:split} we use the continuity of the kernel $K$ \\eqref{e:kernel continuous 2} and the fact that $d(p,q) \\approx \\|p\\|_{{\\mathbb{H}}}\\ge 2^{-N}\\ell(Q_0)$ (because $p\\in Q_2\\subset Q_0\\setminus T$):\n\t\t\\begin{align}\n\t\t|K(q^{-1} \\cdot p) - K(p)| \\lesssim \\fr{\\|q\\|_{{\\mathbb{H}}}}{\\min(\\|p\\|_{{\\mathbb{H}}}, d(p,q))^{D}}\\lesssim \\fr{2^{-AN}\\ell(Q_0)}{(2^{-N}\\ell(Q_0))^{D}} = \\fr{2^{-AN+DN}}{\\ell(Q_0)^{D-1}}.\n\t\t\\end{align}\n\t\tTaking $A\\ge 2D$ we get\n\t\t\\begin{equation*}\n\t\t\\av{ \\int_{Q_1} K(q^{-1} \\cdot p) - K(p) \\, d\\mu(q) }\\lesssim 2^{-AN\/2}\\fr{\\mu(Q_1)}{\\ell(Q_0)^{D-1}}.\n\t\t\\end{equation*}\n\t\tTogether with \\eqref{e:first term estimate} and \\eqref{e:split}, assuming $N_0$ bigger than some absolute constant (recall that $N\\ge N_0$), this gives\n\t\t\\begin{equation*}\n\t\t|R_{\\mu,\\delta}(\\mathbbm{1}_{Q_1})(p)|\\gtrsim 2^{-N} \\, \\frac{\\mu(Q_1)}{\\ell(Q_0)^{D-1}}\n\t\t\\end{equation*}\n\t\tfor all $p\\in Q_2$.\n\t\t\n\t\tNow, we use the estimate above and the $L^2(\\mu)$ boundedness of $R_{\\mu}$ to get\n\t\t\\begin{equation*}\n\t\t2^{-N}\\frac{\\mu(Q_1)}{\\ell(Q_0)^{D-1}}\\mu(Q_2)^{\\frac{1}{2}} \\lesssim \\ps{\\int |R_{\\mu,\\delta}(\\mathbbm{1}_{Q_1})(p)|^2 \\, d\\mu(p)}^{\\frac{1}{2}} \\leq C_1 \\mu(Q_1)^{\\frac{1}{2}}.\n\t\t\\end{equation*}\n\t\tOur assumptions on $Q_1$ \\eqref{e:assumption Q1} and $Q_2$ \\eqref{e:assumption Q2} yield\n\t\t\\begin{align*}\\label{e:est-cauchy}\n\t\tC_1 & \\gtrsim 2^{-N}\\frac{\\mu(Q_1)^{\\frac{1}{2}}\\mu(Q_2)^{\\frac{1}{2}}}{\\ell(Q_0)^{D-1}}\\gtrsim 2^{-N}\\fr{\\mu(Q_0)}{2^{AND}\\ell(Q_0)^{D-1}}\\Theta_0^{-1\/2} = 2^{-AND-N}\\Theta_0^{1\/2} \\notag\\\\\n\t& \\overset{\\eqref{e:choice of N}}{\\approx} 2^{-AND-N}\\, 2^{A^2 N\/2}.\n\t\t\\end{align*}\n\t\tTaking $A\\ge 5D$ we can bound the last term from below in the following way:\n\t\t\\begin{equation*}\n\t\t 2^{-AND-N+A^2 N\/2}\\ge 2^{A^2N\/4}\\overset{\\eqref{e:choice of N}}{\\gtrsim} M^{1\/4}.\n\t\t\\end{equation*}\n\t\tPutting together the estimates above gives $C_1\\gtrsim M^{1\/4}$, which is a contradiction for $M=M(C_1,n)$ big enough.\n\t\\end{proof}\n\t\n\tWe immediately get the following corollary.\n\t\\begin{corollary}\n\tWe have\n\t\\begin{equation}\\label{e:T2 thicc}\n\t\\mu({T_2})\\ge (1-\\Theta_0^{-1})\\mu(Q_0).\n\t\\end{equation}\n\t\\end{corollary}\n\t\\begin{proof}\n\tObserve that if $Q\\in\\mathcal{D}$ satisfies $\\ell(Q) = \\ell(Q_1) = 2^{-AN}\\ell(Q_0)$ and $Q\\not\\subset T_2$, then we have $Q\\cap T=\\varnothing$ (assuming $A$ large enough with respect to $\\Lambda$). It follows that $Q$ satisfies the assumptions of Lemma \\ref{c:dense-tube}, and so\n\t\\begin{equation*}\n\t \\mu(Q)\\le2^{-AND}\\Theta_0^{-1}\\mu(Q_0).\n\t\\end{equation*}\n\tSumming over all such $Q$ and using \\eqref{e:dyadic property} yields\n\t\\begin{equation*}\n\t \\mu(Q_0\\setminus T_2) \\le \\Theta_0^{-1}\\mu(Q_0).\n\t\\end{equation*}\n\t\\end{proof}\n\tRecall that\n\t\\begin{equation*}\n\t{\\mathsf{HD}}(Q_0) = \\ck{Q\\in \\mathcal{D}\\ |\\ Q\\subset Q_0,\\ \\ell(Q)=2^{-N}\\ell(Q_0),\\ \\Theta(Q)>2\\Theta_0 },\n\t\\end{equation*}\n\tand that $\\Lambda$ is the absolute constant such that $Q\\subset B(p_Q,\\Lambda\\ell(Q)).$ Without loss of generality, we may assume $\\Lambda>2$.\n\t\n\n\tWe are ready to prove the first part of Lemma \\ref{l:main lemma}, the estimate \\eqref{e:HD are thicc}.\n\t\\begin{lemma}\\label{l:measureHD}\n\t\tThere exists $s=s(A,n)\\in (0,1\/2)$ such that\n\t\t\\begin{equation}\\label{e:HD are thicc 2}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\mu(Q) \\ge (1-\\Theta_0^{-s})\\mu(Q_0).\n\t\t\\end{equation}\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tWe will prove \\eqref{e:HD are thicc 2} by contradiction. Suppose that\n\t\t\\begin{equation}\\label{e:HD not thicc :(}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\mu(Q) < (1-\\Theta_0^{-s})\\mu(Q_0).\n\t\t\\end{equation}\n\t\tSet\n\t\t\\begin{equation*}\n\t\t{\\mathsf{LD}}(Q_0) = \\ck{Q\\in\\mathcal{D}\\ |\\ Q\\subset T_{2\\Lambda},\\ \\ell(Q)=2^{-N}\\ell(Q_0),\\ \\Theta(Q)\\le 2\\Theta_0 }.\n\t\t\\end{equation*}\n\t\tIt is easy to see that the cubes from ${\\mathsf{HD}}(Q_0)\\cup {\\mathsf{LD}}(Q_0)$ cover $T_{2}$. If we assume $\\Theta_0\\ge M>100$, and $s<1\/2$, then $\\Theta_0^{-s}\/2\\ge \\Theta_0^{-1}$, and so by \\eqref{e:T2 thicc} and \\eqref{e:HD not thicc :(} we get\n\t\t\\begin{equation}\\label{e:LD has plenty mass}\n\t\t\\sum_{Q\\in{\\mathsf{LD}}(Q_0)}\\mu(Q) \\ge \\fr{\\Theta_0^{-s}}{2}\\mu(Q_0).\n\t\t\\end{equation}\n\t\t\n\t\tOn the other hand, recall from Lemma \\ref{c:number-cubes} that there are at most $C2^{2N}$ cubes of sidelength $2^{-N}\\ell(Q_0)$ contained in $T_{2\\Lambda}$, where $C=C(\\Lambda,n)$. Moreover, for any $Q\\in {\\mathsf{LD}}(Q_0)$ we have\n\t\t\\begin{equation*}\n\t\t\\mu(Q)\\le 2 \\Theta_0 \\ell(Q)^{D-1}= 2\\, \\mu(Q_0)\\fr{\\ell(Q)^{D-1}}{\\ell(Q_0)^{D-1}} =2^{-N(D-1)+1}\\mu(Q_0).\n\t\t\\end{equation*}\n\t\tIn consequence,\n\t\t\\begin{equation*}\n\t\t\\sum_{Q\\in{\\mathsf{LD}}(Q_0)}\\mu(Q) \\le C 2^{2N} 2^{-N(D-1)+1}\\mu(Q_0).\n\t\t\\end{equation*}\n\t\tThis contradicts \\eqref{e:LD has plenty mass} because\n\t\t\\begin{equation*}\n\t\tC\\, 2^{-ND+3N+1} = 2\\, C\\, (2^{-A^2N})^{(-D+3)A^{-2}} \\overset{\\eqref{e:choice of N}}{\\le}\\widetilde{C}(n)\\Theta_0^{(-D+3)A^{-2}}\\le \\fr{\\Theta_0^{-s}}{2},\n\t\t\\end{equation*}\n\t\tchoosing $s=s(A,n)$ small enough.\n\t\\end{proof}\n\tWe move on to the second part of Lemma \\ref{l:main lemma}, i.e. the packing estimate \\eqref{e:HD packing}.\n\t\\begin{lemma}\\label{l:HD-in-tube}\n\tWe have\n\t\\begin{equation}\\label{e:HD-in-tube}\n\t \\bigcup_{Q\\in{\\mathsf{HD}}(Q_0)}Q\\subset T_{2\\Lambda}.\n\t\\end{equation}\n\tIn consequence, \n\t\\begin{equation}\\label{e:HD packing 2}\n\t\t\\sum_{Q\\in{\\mathsf{HD}}(Q_0)}\\ell(Q)^2\\lesssim \\ell(Q_0)^2.\n\t\t\\end{equation}\n\t\\end{lemma}\n\t\\begin{proof}\n\tWe will prove that for $Q\\in{\\mathsf{HD}}(Q_0)$ we have $Q\\cap T_2\\neq\\varnothing.$ Then, since $\\ell(Q)=2^{-N}\\ell(Q_0)$, it follows easily from \\eqref{e:balls in cubes} that indeed $Q\\subset T_{\\Lambda+2}(Q_0)\\subset T_{2\\Lambda}(Q_0)$.\n\t\n\tWe argue by contradiction. Suppose that $Q\\in {\\mathsf{HD}}(Q_0)$ and $Q\\cap T_2=\\varnothing$. Consider the cubes $\\{P_i\\}_{i\\in I}$ with $\\ell(P_i)=2^{-AN}\\ell(Q_0) = 2^{-(A-1)N}\\ell(Q)$ and $P_i\\subset Q$. Then, $Q=\\bigcup_i P_i,$ for all $i\\in I$ we have $P_i\\cap T_2=\\varnothing,$ and $\\# I\\approx 2^{(A-1)ND}$ by \\eqref{e:dyadic property}. \n\t\n\tWe use Lemma \\ref{c:dense-tube} to conclude that for all $i\\in I$\n\t\\begin{equation*}\n\t \\mu(P_i) \\le \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{AND}}.\n\t\\end{equation*}\n\tSumming over $i\\in I$ yields\n\t\\begin{equation*}\n\t \\mu(Q) = \\sum_{i\\in I}\\mu(P_i)\\le \\# I\\cdot \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{AND}} \\approx 2^{(A-1)ND} \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{AND}} = \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{ND}},\n\t\\end{equation*}\n\tso that\n\t\\begin{equation*}\n\t \\Theta(Q) = \\fr{\\mu(Q)}{(2^{-N}\\ell(Q_0))^{D-1}} \\lesssim \\frac{\\mu(Q_0)}{\\Theta_0 \\, 2^{ND}}\\cdot \\frac{1}{2^{-N(D-1)}\\ell(Q_0)^{D-1}} = \\frac{\\Theta_0}{\\Theta_0 \\, 2^{N}}=2^{-N}\\le 1.\n\t\\end{equation*}\n\tBut this contradicts the assumption $Q\\in {\\mathsf{HD}}(Q_0)$:\n\t\\begin{equation*}\n\t \\Theta(Q)\\ge 2\\Theta_0\\ge 2M>1,\n\t\\end{equation*}\n\tand so the proof of \\eqref{e:HD-in-tube} is finished.\n\t\n\tConcerning \\eqref{e:HD packing 2}, note that by \\eqref{e:HD-in-tube} and Lemma \\ref{c:number-cubes} we have\n\t\\begin{equation} \\label{e:HD-card}\n\t \\# {\\mathsf{HD}}(Q_0) \\lesssim 2^{2N}.\n\t\\end{equation}\n\tHence,\n\t\\begin{align*}\n\t \\sum_{Q \\in {\\mathsf{HD}}(Q_0)} \\ell(Q)^2 = \\ell(Q_0)^2\\, 2^{-2N} \\sum_{Q \\in {\\mathsf{HD}}(Q_0)} 1 \\lesssim \\ell(Q_0)^2. \n\t\\end{align*}\n\t\\end{proof}\n\\section{Iteration argument}\\label{s:iteration}\nTo complete the proof of Theorem \\ref{t:main}, we assume that the measure $\\mu$ does not satisfy the polynomial growth condition \\eqref{e:main}. Then we will use Lemma \\ref{l:main lemma} countably many times to construct a set $Z$ with positive $\\mu$-measure and with Hausdorff dimension at most $2$. \n\nSuppose that there exists a ball $B(x,r)$ with $\\mu(B(x,r)) \\geq C_2 r^{2n+1}$; if $C_2$ is big enough, we can find a cube $Q_0 \\in \\mathcal{D},\\ Q\\subset B(x,r)$ such that \n\t\\begin{align*}\n\t\\Theta(Q_0) \\ge M, \n\t\\end{align*}\n\twhere $M$ is the constant from Lemma \\ref{l:main lemma}.\n\nLet $A>1$ be as in Lemma \\ref{l:main lemma}. Following the notation of Lemma \\ref{l:main lemma}, for an arbitrary cube $Q\\in\\mathcal{D}$ with $\\Theta(Q)\\ge M$, set\n\\begin{equation*}\n N(Q) := \\floor{A^{-2} \\log(\\Theta(Q))}\n\\end{equation*}\nand\n\\begin{equation*}\n {\\mathsf{HD}}(Q):=\\left\\{ P \\in \\mathcal{D} \\, |\\, P\\subset Q,\\, \\ell(P) =2^{-N(Q)}\\ell(Q), \\, \\Theta(P) > 2 \\Theta(Q) \\right\\}.\n\\end{equation*}\nPut $Z_{0}:=Q_0,\\ {\\mathsf{HD}}_0:=\\{Q_0\\},\\ {\\mathsf{HD}}_1:={\\mathsf{HD}}(Q_0),$ and $Z_1:=\\bigcup_{Q\\in{\\mathsf{HD}}_1} Q$. Proceeding inductively, for all $j\\ge 2$ we define\n\\begin{gather*}\n {\\mathsf{HD}}_{j} := \\bigcup_{Q \\in {\\mathsf{HD}}_{j-1}} {\\mathsf{HD}}(Q),\\\\\n Z_{j} := \\bigcup_{Q\\in{\\mathsf{HD}}_j} Q.\n\\end{gather*}\nNote that for each $j$ the cubes in ${\\mathsf{HD}}_j$ form a disjoint family. Moreover, $\\{Z_j\\}_{j\\ge 0}$ form a decreasing sequence of sets, that is $Z_{j+1}\\subset Z_j.$ Define\n\\begin{equation*}\n Z := \\bigcap_{j \\geq 0} Z_j.\n\\end{equation*}\n\\begin{claim}\\label{c:positive}\nWe have\n\\begin{equation*}\n \\mu(Z)\\gtrsim_{M,s}\\mu(Q_0).\n\\end{equation*}\n\\end{claim}\n\\begin{proof}\nObserve that for $Q\\in{\\mathsf{HD}}_j$ we have\n\\begin{equation}\\label{e:estimate densities}\n \\Theta(Q)\\ge 2^{j}\\Theta(Q_0)\\ge 2^{j}M.\n\\end{equation}\nIn particular, $\\Theta(Q)\\ge M$ and so we may apply Lemma \\ref{l:main lemma} to $Q$. It follows that for any $j \\geq 0$ we have\n\\begin{multline*}\n \\mu(Z_{j+1})=\\sum_{Q \\in {\\mathsf{HD}}_{j+1}} \\mu(Q) = \\sum_{Q \\in {\\mathsf{HD}}_{j}} \\sum_{P \\in {\\mathsf{HD}}(Q)} \\mu(P) \\stackrel{\\eqref{e:HD are thicc}}{\\geq} \\sum_{Q \\in {\\mathsf{HD}}_{j}} (1-\\Theta(Q)^{-s})\\mu(Q)\\\\\n \\overset{\\eqref{e:estimate densities}}{\\ge} \\sum_{Q \\in {\\mathsf{HD}}_{j}} (1-2^{-js}M^{-s})\\mu(Q) = (1-2^{-js}M^{-s}) \\mu(Z_{j}).\n\\end{multline*}\nUsing this estimate $(j+1)$ times we arrive at\n\\begin{equation}\\label{e:HD_jbig}\n \\mu(Z_{j+1}) \\geq \\prod_{i=0}^{j}(1-2^{-is}M^{-s}) \\mu(Q_0).\n\\end{equation}\nSince $Z_j$ form a sequence of decreasing sets, we get by the continuity of measure \n\\begin{equation*}\n \\mu(Z) = \\lim_{j\\to\\infty} \\mu(Z_j)\\ge \\prod_{i=0}^{\\infty}(1-2^{-is}M^{-s}) \\mu(Q_0) = C(s,M)\\mu(Q_0),\n\\end{equation*}\nwhere $C(s,M)$ is positive and finite because $\\sum_{i=0}^{\\infty} 2^{-is}<\\infty$.\n\\end{proof}\n\\begin{claim}\\label{c:dim} We have\n\\begin{equation*}\n \\textstyle{\\dim_H(Z)} \\leq 2.\n\\end{equation*}\n\\end{claim}\n\\begin{proof}\nRecall that $N(Q) = \\floor{A^{-2} \\log(\\Theta(Q))}$. It follows from \\eqref{e:estimate densities} that for $Q\\in{\\mathsf{HD}}_j$ we have $N(Q)\\ge C_3jA^{-2}$ for some absolute constant $C_3>0$. Thus, for $Q\\in{\\mathsf{HD}}_j$ and $P\\in{\\mathsf{HD}}(Q)$\n\\begin{equation*}\n \\ell(P)=2^{-N(Q)}\\ell(Q)\\le 2^{-C_3jA^{-2}}\\ell(Q).\n\\end{equation*}\nUsing this observation $j$ times we get that for $P\\in{\\mathsf{HD}}_{j+1}$\n\\begin{equation*}\n \\ell(P)\\le 2^{-C_4 j(j+1)A^{-2}}\\ell(Q_0),\n\\end{equation*}\nwhere $C_4=C_3\/2.$\nHence, the cubes from ${\\mathsf{HD}}_j$ form coverings of $Z$ with decreasing diameters, well suited for estimating the Hausdorff measure of $Z$. \n\nLet $0<\\varepsilon<1,\\ 0<\\delta<1$ be small. Let $j\\ge 0$ be so big that for $Q\\in{\\mathsf{HD}}_{j}$ we have $\\mathrm{\\, diam}(Q)\\le\\Lambda\\ell(Q)\\le\\delta$. Then,\n\\begin{equation}\\label{e:Hausdorff estimate}\n \\mathcal{H}_{\\delta}^{2+\\ve}(Z) \\le \\Lambda^{2+\\varepsilon}\\sum_{Q\\in{\\mathsf{HD}}_j}\\ell(Q)^{2+\\varepsilon}\\le \\Lambda^{2+\\varepsilon}(2^{-C_4j(j-1)A^{-2}}\\ell(Q_0))^{\\varepsilon}\\sum_{Q\\in{\\mathsf{HD}}_j}\\ell(Q)^{2}.\n\\end{equation}\nIt follows by \\eqref{e:HD packing} that\n\\begin{equation*}\n \\sum_{Q\\in{\\mathsf{HD}}_j}\\ell(Q)^{2} = \\sum_{P\\in{\\mathsf{HD}}_{j-1}}\\sum_{Q\\in{\\mathsf{HD}}(P)}\\ell(Q)^{2}\\le C_p\\sum_{P\\in{\\mathsf{HD}}_{j-1}}\\ell(P)^{2}.\n\\end{equation*}\nUsing the estimate above $j$ times, and putting it together with \\eqref{e:Hausdorff estimate} we arrive at\n\\begin{equation*}\n \\mathcal{H}_{\\delta}^{2+\\ve}(Z) \\le \\Lambda^{2+\\varepsilon} (C_p)^j\\, \\,2^{-\\varepsilon C_4 j(j-1)A^{-2}}\\, \\ell(Q_0)^{2+\\varepsilon}.\n\\end{equation*}\nThe right hand side above converges to 0 as $j\\to\\infty$ (just note that the exponent at $C_p$ is linear in $j$ while the exponent at $2$ is quadratic in $j$). Hence, $\\mathcal{H}_{\\delta}^{2+\\ve}(Z)=0.$ Letting $\\delta\\to 0$ we get $\\mathcal{H}^{2+\\ve}(Z)=0.$ Since this is true for arbitrarily small $\\ve>0$, it follows that \n\\begin{equation*}\n \\textstyle{\\dim_H(Z)} = \\inf\\{t\\ge 0\\ :\\ \\mathcal{H}^t(Z)=0\\} \\le 2.\n\\end{equation*}\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{t:main}]\nWe have found a set $Z \\subset \\mathbb{H}^{n}$ of dimension smaller than or equal to $2$ (Claim \\ref{c:dim}) but which nevertheless has positive $\\mu$-measure (Claim \\ref{c:positive}). This contradicts the assumptions of Theorem \\ref{t:main}. Thus, there exists $C_2=C_2(n,C_1)$ such that $\\mu(B(x,r)) \\leq C_2 r^{2n+1}$ for all $x\\in{\\mathbb{H}}^n$ and $r>0$.\n\\end{proof}\n\n\\subsection*{Funding}\nD. D\\k{a}browski was supported by Spanish Ministry of Economy and Competitiveness, through the Mar\\'ia de Maeztu Programme for Units of Excellence in R\\&D (grant MDM-2014-0445), and also partially supported by the Catalan Agency for Management of University and Research Grants (grant 2017-SGR-0395), and by the Spanish Ministry of Science, Innovation and Universities (grant MTM-2016-77635-P).\n\nM. Villa was supported by The Maxwell Institute Graduate School in Analysis and its\nApplications, a Centre for Doctoral Training funded by the UK Engineering and Physical\nSciences Research Council (grant EP\/L016508\/01), the Scottish Funding Council, Heriot-Watt\nUniversity and the University of Edinburgh.\n\nBoth authors were partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}