diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkcua" "b/data_all_eng_slimpj/shuffled/split2/finalzzkcua" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkcua" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \n\nRecent observations with the Submillimetre Common-User Bolometer Array \n(SCUBA; Holland et al.\\ 1999) on the James Clerk Maxwell Telescope have \nhighlighted the presence of a number of submillimetre-luminous galaxies \n(Smail, Ivison \\& Blain 1997; Holland et al.\\ 1998; Barger et al.\\ 1998a; \nHughes et al.\\ 1998; Eales et al.\\ 1998). To date about forty \nsources have been found. These measurements confirm earlier suggestions \n(Blain \\& Longair 1993; Eales \\& Edmunds 1996, 1997) that submillimetre-wave \nobservations will provide an important probe of cosmology.\n\nThe optical counterparts of these SCUBA sources are faint, most with $I > 20$ \n(Smail et al. 1998). They are presumably not low-redshift sources: the best \nstudied is SMM\\,J02399$-$0136 (Ivison et al.\\ 1998), a dust-enshrouded AGN at \n$z=2.8$, perhaps similar to IRAS F10214+4724 (Rowan-Robinson et al.\\ 1991; \nLacy et al.\\ 1998). The recent detection of molecular gas in SMM\\,J02399$-$0136 \n(Frayer et al.\\ 1998) also suggests that these galaxies are similar.\n \nWhat is the nature of these SCUBA-selected sources? In particular are they \nsimilar to the ultraluminous infrared galaxies (ULIRGs) observed locally (e.g. \nSanders \\& Mirabel 1996)? They have the same submillimetre-wave flux densities \nas would have local ULIRGs seen at high redshift (Barger et al.\\ 1998a). They also have \nsimilar optical colours (Smail et al.\\ 1998) to two of the three local ULIRGs\nstudied at ultraviolet wavelengths by Trentham, Kormendy \\& Sanders\n(1998), were those galaxies to be seen at high redshift. If this interpretation is \ncorrect, then we might hope to be able to use the wealth of observational \ninformation available for local ULIRGs to help to understand the properties of the \nSCUBA sources and their relevance to cosmology. \n\nWhether the SCUBA sources are dust-enshrouded AGNs, like Markarian 231, or \ndust-enshrouded starbursts, like Arp 220, is an important question. \nThis is difficult to determine for even local ULIRGs, because there are up to \nseveral hundred magnitudes of extinction along our lines of sight to the galaxy \ncores at optical wavelengths. For local ULIRGs, simple conclusions can be \ndrawn based on the general form of the spectral energy distributions (SEDs) \n(e.g.~Sanders et al.\\ 1988a); however, this information is not available for the \nSCUBA sources. Recently, new more detailed methods based on mid-infrared \nspectroscopic diagnostics (Lutz et al.\\ 1996, Genzel et al.\\ 1998) have been used \nto resolve this question for the most local ULIRGs. \n \nWe construct redshift-dependent luminosity functions that are consistent with\nall the counts and backgrounds from a number of recent surveys at far-infrared \nand millimetre\/submillimetre wavelengths. We use non-evolving \nGaussian luminosity functions over specified redshift ranges. These are the \nsimplest luminosity functions that we could adopt, without the results \nbeing dependent on the properties of low-luminosity galaxies that are not \nprobed by the SCUBA observations. Contributions from these low-luminosity \ngalaxies are important when computing backgrounds, although they contribute \ninsignificantly to the counts. The parameterization used to compare models \nwith observations reflects this fact. Imposing all the \nconstraints simultaneously does limit the possible form of the luminosity \nfunction and we identify three plausible models.\n\nGiven the redshift-dependent luminosity functions of these models, we \ninvestigate the properties to the individual sources using observations of their \nlocal ultraluminous counterparts, and derive the cosmological implications of \nthe SCUBA galaxies under both the starburst and AGN interpretation.\n\nWe first investigate the possibility that the SCUBA sources are high-redshift \nstar-forming galaxies. The starburst models of Leitherer \\& \nHeckman (1995) are used to convert far-infrared luminosities to star-formation \nrates; a transformation that is uncertain by more than an order of magnitude.\nWe then place the SCUBA galaxies on the Madau plot, which relates the \ncosmic star formation rate to the cosmic epoch (Madau et al.\\ 1996; Madau, \nDella Valle \\& Panagia 1998), \nand predict the total density of local stars produced in such \nobjects. Most of the star formation in the SCUBA sources is observed through\nhuge amounts of internal extinction, and so will not be included in global\nstar-formation rates that are computed using the optically selected samples that \nare normally used to construct the Madau plot. Indeed Hughes et al.\\ (1998) \nshow that deriving a star-formation rate from a rest-frame ultraviolet flux \nresults in a value that is more than an order of magnitude too low.\nThe SCUBA sources are thus not accounted for in the existing samples that are \nused to construct the Madau plot.\n\nWe then investigate the possibility that the SCUBA sources are high-redshift\ndust-enshrouded AGN, as discussed by Haehnelt, Natarajan \\& Rees (1998) \nand Almaini, Lawrence \\& Boyle (1999). The three distant \nsubmillimetre-luminous galaxies that have been studied in detail -- \nAPM\\,08279+5255 (Irwin et al. 1998; Lewis et al. 1998; Downes et al. submitted), \nSMM\\,J02399$-$0136 and IRAS F10214+4724 -- all contain powerful AGN. If all \nthe sources derive their bolometric luminosity from AGN that heat their dust \nshrouds radiatively by accretion onto a massive black hole, then we can use our \nredshift-dependent luminosity functions to\ncompute the comoving integrated mass density of these black holes.\n\nWe find plausible (although not unique) scenarios that are consistent with all \nthe observations and present their cosmological implications. Finally, we \nhighlight future work which may help to distinguish between the various \nscenarios. Throughout this paper we assume an Einstein--de Sitter world model \nwith $H_0 = 50$\\,km\\,s$^{-1}$\\,Mpc$^{-1}$.\n\n\\section{Far-infrared\/submillimetre-wave counts and backgrounds\nand the ULIRG luminosity function}\n\nThere have been many recent measurements of the far-infrared and\nsubmillimetre source counts and backgrounds at a number of wavelengths:\nsee Table 1. This has been a recent field of substantial activity \nbecause of new instrumentation, both SCUBA and {\\it ISO}. If we assume that\nthe SED at far-infrared and submillimeter\nwavelengths is known for all the sources,\nwe can then use all these measurements in conjunction to constrain the\nbivariate far-infrared luminosity--redshift function of these sources.\n\n\\begin{table*}\n\\caption{Far-infrared and submillimetre-wave surveys.\nLagache et al.~(1998) and Kawara et al.~(1997, 1998) obtain a different\ncalibration for the ISO 175-$\\mu$m counts. A discussion of calibration is\ngiven in Lagache et al.~(1998). The detection limit in the 850-$\\mu$m \nsurvey by Hughes et al. (1998) is about 2\\,mJy: the count at 1\\,mJy is \nderived from a source confusion analysis. See Blain et al.~(1999b) and \nSmail et al. (1999) for a \ndirect sub-mJy 850-$\\mu$m count. For complementary \nfar-infrared\/submillimetre-wave background measurements the reader \nis referred to Puget et al.~(1996), Guiderdoni et al.~(1997), Dwek et al.~(1998) \nand Fixsen et al.~(1998).}\n{\\vskip 0.75mm}\n{$$\\vbox{\n\\halign {\\hfil #\\hfil && \\quad \\hfil #\\hfil \\cr\n\\noalign{\\hrule \\medskip} \nWavelength & Counts \/ & Flux Limit \/& Background \/ & \nTelescope\/Instrument & Reference &\\cr\n & deg$^{-2}$ & mJy & nW m$^{-2}$ sr$^{-1}$ &\n & &\\cr \n\\noalign{\\smallskip \\hrule \\smallskip} \n\\cr \n2.8\\,mm & $<162$ & 3.55 & $-$ & BIMA & Wilner \\& Wright (1997) &\\cr \n\\noalign{\\smallskip}\n850\\,$\\mu$m & $2500\\pm1400$ & 4 & $-$ & JCMT\/SCUBA & \nSmail et al.~(1997)&\\cr\n\t & $1100\\pm600$ & 8 & $-$ & & Holland et al.~(1998)&\\cr \n\t & $800^{+1100}_{-500}$ & 3 & $-$ & & Barger et al.~(1998a) & \\cr\n & $7000\\pm3000$ & 1 & $-$ & & Hughes et al.~(1998) & \\cr \n\t & $1800\\pm600$ & 2.8 & $-$ & & Eales et al.~(1998) &\\cr \n\\noalign{\\smallskip}\n850\\,$\\mu$m & $-$ & $-$ & $0.55 \\pm 0.15$ & {\\it COBE}\/FIRAS & \nFixsen et al.~(1998)\n&\\cr\n\\noalign{\\smallskip}\n450\\,$\\mu$m & $< 1000$ & 80 & $-$ & JCMT\/SCUBA & Smail et al.~(1997) &\\cr\n\t & $<360$ & 75 & $-$ & & Barger et al.~(1998a) &\\cr \n\\noalign{\\smallskip}\n240\\,$\\mu$m & $-$ & $-$ & $17 \\pm 4$ & {\\it COBE}\/DIRBE + {\\it IRAS}\/ISSA & \nSchlegel et al.~(1998) &\\cr \n & $-$ & $-$ & $14 \\pm 4$ & & Hauser et al.~(1998) &\\cr \n\\noalign{\\smallskip}\n175 $\\mu$m & $41 \\pm 6$ & 150 & $-$ & {\\it ISO}\/ISOPHOT & Kawara et\nal.~(1998) \n&\\cr\n\t & $98 \\pm 15$ & 100 & $-$ & \t & \nLagache et al.~(1998) &\\cr\n\\noalign{\\smallskip}\n140 $\\mu$m & $-$ & $-$ & $32 \\pm 13$ & {\\it COBE}\/DIRBE + {\\it IRAS}\/ISSA & \nSchlegel et al.~(1998) &\\cr\n\t & $-$ & $-$ & $24 \\pm 12$ & & Hauser et al. (1998) &\\cr\n\\noalign{\\smallskip \\hrule} \n\\noalign{\\smallskip}\\cr}}$$}\n\\end{table*} \n\nWe assume that the galaxies have the thermal SEDs of \nwarm dust that is heated by an enshrouded starburst or AGN, and adopt a \nsimple analytic form for the luminosity function between a minimum and \nmaximum redshift. All the observational constraints on the model are imposed \nthrough integral functions, and so do not require unique solutions. \nNevertheless, it is interesting to see which general classes of \nluminosity--redshift distributions are ruled out and why.\n\nTo ensure consistent definitions are used when matching models to\nobservations, first we present all the details of our computations. Much of this \nwill be familiar to many readers, but different authors define parameters in slightly \ndifferent ways. Secondly, we point out some generic features of the comparison \nbetween our models and observations, in particular the results of requiring the \nmodels to be consistent with source counts at one wavelength and with the \ninfrared background at another simultaneously. Finally, we isolate some \nplausible luminosity functions that are consistent with all the observations for \nfurther investigation.\n\nWe define a bivariate 60-$\\mu$m luminosity-redshift function \n$\\phi_z (L_{\\rm 60})$, with units Mpc$^{-3}$\\,${\\rm L}_{\\odot}^{-1}$, such that \n$\\phi_z (L_{\\rm 60}) \\, {\\rm d} L_{\\rm 60}\\, {\\rm d}z$ is the total number density of \ngalaxies with 60-$\\mu$m luminosity between $L_{\\rm 60}$ and \n$L_{\\rm 60} + {\\rm d} L_{\\rm 60}$ with redshifts between $z$ and $z + {\\rm d}z$. \nSome familiar analytic examples of this function are a \nGaussian in $\\log_{10} L_{\\rm 60}$ with no luminosity or density evolution, \n\\begin{equation} \n\\phi_z (L_{\\rm 60}) = C (1+z)^3 \\exp \\left[ - {1\\over{2 \\sigma^2}} \n\\log_{10}^2 \\left( {{L_{\\rm 60}} \\over {L_{\\rm 60}^*}}\\right) \\right] \n{1\\over{L_{\\rm 60} \\, \\rm{ln} 10}},\n\\end{equation} \nor a Saunders et al.\\ (1990) function, which is a power-law in $L_{\\rm 60}$\nwith index $\\alpha$ at the faint end and a Gaussian in $\\log_{10} L_{\\rm 60}$ at \nthe bright end, and allows for density and luminosity\nevolution,\n\\begin{eqnarray} \n\\lefteqn{\\nonumber \n\\phi_z (L_{\\rm 60}) = C(z)\\, (1+z)^{3} \n\\left( {{L_{\\rm 60}} \\over {L_{\\rm 60}^* (z)}} \\right)^{1 - \\alpha} \\times } \\\\ \n& & \\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\n \\exp \\left[ - {1\\over{2 \\sigma^2}}\n\\log_{10}^2 \\left( 1 + {{L_{\\rm 60}} \\over {L_{\\rm 60}^* (z)}}\\right) \\right]\n{1\\over{L_{\\rm 60} \\, \\rm{ln} 10}}.\n\\end{eqnarray} \nImplicit in both equations (1) and (2) above \nare a normalization constant $C$ (units Mpc$^{-3}$, which depends on\n$z$ if there is density evolution), a characteristic\nluminosity $L_{\\rm 60}^{*}$ (which depends on $z$ if there is\nluminosity evolution), and a Gaussian width $\\sigma$.\n\nOnce we have specified $\\phi_z (L_{\\rm 60})$ for some\npopulation of galaxies, we can compute their contribution to the \ncosmic infrared background at some frequency $\\nu$:\n\\begin{equation} \n\\nu {\\rm I}_{\\nu} = {{1}\\over{4 \\pi}} \\int_{0}^{\\infty} \\int_{0}^{\\infty}\n{ {l_{\\nu} (z,L_{\\rm 60})}\\over{4 \\pi d_{\\rm L} (z)^2}} \\,\\, \\phi_z (L_{\\rm 60})\n\\, \\, {\\rm d} L_{\\rm 60} \\, \\, { {{\\rm d} V}\\over{{\\rm d} z}} \\, \\, {\\rm d} z \n,\n\\end{equation}\nin units of W\\,m$^{-2}$\\,sr$^{-1}$, where, \n\\begin{equation} \nl_{\\nu} (z,L_{\\rm 60}) = \\left[ { {\\nu (1+z)} \\over {\\nu_{60}}}\\right]^4\n\\!\\! L_{\\rm 60} \n{ {k_{\\nu (1+z)}}\\over{k_{\\nu_{60}}}} \\,\n{ {\\exp \\left({{h \\nu_{60}}\\over{kT}} \\right) - 1 }\\over\n{\\exp \\left[{{h \\nu(1+z)}\\over{k T}} \\right] - 1 } }, \n\\end{equation}\nand $\\nu_{60}$ \nis the frequency corresponding to a wavelength of 60\\,$\\mu$m. \n$k_{{\\nu}}$ is the\nemissivity function of dust. We follow Hughes (1996) and \nBlain et al.\\ (1999a) in \nassuming a power law $k_{\\nu} \\sim {\\nu^{1.5}}$, and so the dust emission \nspectrum at long wavelengths is a Raleigh--Jeans power-law with spectral \nindex 3.5. We can also compute the number density of sources with flux density \nabove some threshold $S_{\\rm lim}$, which is measured in units of\nW\\,m$^{-2}$\\,Hz$^{-1}$ (or Jy), \n\\begin{equation} \nn(S_{\\rm lim}) \n= {{1}\\over{4 \\pi}} \\int_{0}^{\\infty} \\int_{L_{\\rm lim} \n[S_{\\rm lim}]}^{\\infty} \\> \n\\phi_z (L_{\\rm 60})\n\\, \\, {\\rm d} L_{\\rm 60} \\, \\, { {{\\rm d} V}\\over{{\\rm d} z}} \\, \\, {\\rm d} z\n,\n\\end{equation}\nin units of sr$^{-1}$ (or deg$^{-2}$), where, \n\\begin{eqnarray} \n\\lefteqn{\\nonumber \nL_{\\rm lim} [S_{\\rm lim}] = \n4 \\pi d_{\\rm L} (z)^2 \\, \\, S_{\\rm lim} \\, \\nu_{60} \n\\left[ { {\\nu_{60}} \\over {\\nu (1+z)} } \\right]^{3} \\times } \\\\\n& & \\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\\>\n\\>\\>\\>\\>\\>\\>\n\\, { {k_{\\nu_{60}}} \\over {k_{\\nu(1+z)}} } \\,\n{ {\\exp \\left[{{h \\nu (1+z)}\\over{k T}} \\right] - 1} \\over\n {\\exp \\left({{h \\nu_{60}}\\over{kT}} \\right) - 1 }}\n.\n\\end{eqnarray} \nThe quantities,\n\\begin{equation} \nd_{\\rm L} (z) = \n{ {2 c}\\over{H_0}} \\left( 1 - {{1}\\over{\\sqrt{1+z}}} \\right) (1+z), \n\\end{equation} \nand,\n\\begin{equation} \n{ {{\\rm d} V}\\over{{\\rm d} z}} =\n16 \\pi \\, \\left({ { c}\\over{H_0}} \\right)^3\n\\, (1+z)^{-{{3}\\over {2}}} \\, \n\\left( 1 - {{1}\\over{\\sqrt{1+z}}} \\right)^2,\n\\end{equation} \nare the luminosity distance and the\nredshift-derivative of the volume element respectively (assuming\n$\\Omega_0 = 1$).\n\nA few features of the comparison between observation and theory\nare immediately apparent. Source counts constrain the integral of \n$\\phi_z(L_{60})$ over redshift and luminosity above some redshift-dependent \nlimit. However, measurements of the background light\nconstrain the integral of the product of luminosity $L_{60}$ and\n$\\phi_z(L_{60})$ over redshift and all luminosities. For a Saunders luminosity \nfunction with pure density evolution, we compare in Fig.\\,1 the ratio\nof the predicted 850-$\\mu$m source count and 240-$\\mu$m background \nas compared with the observed values listed in Table\\,1 as a function of the \ncharacteristic luminosity $L_{*} \\equiv L_{60}^{*}$. \nThe minimum is produced by the different constraints on the integral of \n$\\phi_z(L_{60})$ imposed by the source counts and the background \nmeasurement. \n\nVery low luminosity galaxies make a negligible contribution to the source counts, \nbut a substantial one to the infrared background if $L_*$ is low in the \nSaunders function. Hence, a low normalization is required to explain the \nbackground data, but a high one is required to explain the source count data,\nand so the ratio plotted in Fig.\\,1 is very high at low $L_*$. Increasing $L_*$\nincreases the fraction of high-luminosity galaxies, and so lowers the ratio of the\nnormalizations required in the figure. At very high $L_*$, the source counts are \nproduced by a very small number of extremely luminous sources, which cause \nthe background to be extremely high based on their luminosities alone. Hence, \nthe ratio of the normalization begins to increase again, producing the minimum\nin Fig.\\,1. If the flux threshold (lower limit) in the luminosity integral in\nequation (5) is reduced, then the turn-up of the ratio at high luminosities is less \nmarked, and disappears as the lower limit tends to zero.\n\nIt is intriguing that this minimum in the ratio occurs at a very high characteristic \nluminosity $L_{*} = 10^{12}$\\,L$_\\odot$. Locally $L_{*} \\sim 10^{9}$\\,L$_\\odot$, \nand so if the luminosity function of the SCUBA sources has a Saunders form, \nhuge amounts of luminosity evolution are required to match the data. This was \nessentially one of the main conclusions of Blain et al.\\ (1999a).\n\n\\begin{figure} \n\\begin{center}\n\\vskip-2mm\n\\epsfig{file=fig1.ps, width=8.65cm}\n\\end{center}\n\\vskip-4mm\n\\caption{ \nThe normalization obtained by fitting a Saunders function \n(equation 2) with density evolution parameterized by $\\gamma$ to the \nCOBE 240-$\\mu$m background (Schlegel et al.\\ 1998), relative to the \nnormalization obtained by fitting a similar function to the SCUBA \n850-$\\mu$m counts of Smail et al.\\ (1997). A value of $\\sigma = 0.724$, \nderived by Saunders et al.\\ (1990), is assumed. A dust emissivity index \nof 1.5 is assumed when converting luminosities measured at different \nwavelengths to restframe 60-$\\mu$m luminosities. The four line styles \nrepresent the following parameter values: solid ($T=70$\\,K, $\\gamma = 0$),\nshort-dashed ($T=40$\\,K, $\\gamma = 0$), long-dashed ($T=70$\\,K, \n$\\gamma = 6$), and dot-dashed ($T=40$\\,K, $\\gamma = 6$). Locally, \n$L^{*}_{60} = 1.1 \\times 10^9$\\,L$_{\\odot}$.\n}\n\\end{figure} \n\n\\begin{figure*}\n\\begin{minipage}{170mm}\n{\\vskip-3.5cm} \n\\begin{center}\n\\epsfig{file=fig2.ps, width=18.65cm}\n\\end{center}\n{\\vskip-7.2cm} \n\\caption{\nCombinations of the normalization constant $C$ and the width $\\sigma$ for \nGaussian luminosity functions with no density evolution that are consistent \nwith the 850-$\\mu$m counts. The units of $C$ are Mpc$^{-3}$. All galaxies \nare assumed to have a temperature $T = 70$\\,K. The comoving galaxy \ndensity $C$ is assumed to be constant between redshifts $z_1 = 3$ and \n$z_2 = 5$ and zero outside this range, (that is the upper and lower limits to \nthe redshift integral in equation (5) are $z_1$ and $z_2$). The solid lines show \nfits to the mean value quoted by Smail et al.\\ (1997). \nThe dotted-dashed lines are fitted to the +1$\\sigma$ values. The dashed \nlines are fitted to the $-\\sigma$ values: see Table 1. The four panels are \nplotted for different values of $L_{*}$.\n}\n\\end{minipage}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{minipage}{170mm}\n{\\vskip-3.5cm}\n\\begin{center}\n\\epsfig{file=fig3.ps, width=18.65cm}\n\\end{center}\n{\\vskip-7.2cm}\n\\caption{As Figure 2, but for $z_1=2$ and $z_2=4$.}\n\\end{minipage}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{minipage}{170mm}\n{\\vskip-3.5cm}\n\\begin{center}\n\\epsfig{file=fig4.ps, width=18.65cm}\n\\end{center}\n{\\vskip-7.2cm}\n\\caption{As Figure 2, but for $z_1=1$ and $z_2=3$.}\n\\end{minipage}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{minipage}{170mm}\n{\\vskip-3.5cm}\n\\begin{center}\n\\epsfig{file=fig5.ps, width=18.65cm}\n\\end{center}\n{\\vskip-7.2cm}\n\\caption{As Figure 2, but for $T=40$K.}\n\\end{minipage}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{minipage}{170mm}\n{\\vskip-3.5cm}\n\\begin{center}\n\\epsfig{file=fig6.ps, width=18.65cm}\n\\end{center}\n{\\vskip-7.2cm}\n\\caption{As Figure 3, but for $T=40$K.}\n\\end{minipage}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{minipage}{170mm}\n{\\vskip-3.5cm}\n\\begin{center}\n\\epsfig{file=fig7.ps, width=18.65cm}\n\\end{center}\n{\\vskip-7.2cm}\n\\caption{As Figure 4, but for $T=40$K.} \n\\end{minipage}\n\\end{figure*} \n\nHigher temperature sources systematically produce greater background fluxes\nand counts at shorter wavelengths, because their dust spectra peak at shorter \nwavelengths. Hence, lower normalizations are required in Fig.\\,1 in order to \nexplain the data for 40-K sources as compared with 70-K sources. The 40-K \ncurve is consistent with a ratio of one over a range of luminosities, suggesting\nthat if a Saunders luminosity function describes the SCUBA sources, then they \nmust have temperatures of about 40\\,K. This was another conclusion of \nBlain et al.\\ (1999a): see their Figure 4 -- in their favoured scenarios most of the \nSCUBA sources are at $z>2$ and have luminosities \n$\\sim 10^{12} {\\rm L}_{\\odot}$ and dust temperatures $\\sim 40$\\,K. \n\nFinally, note that the results in Fig.\\,1 are only weakly dependent on the\nform of density evolution parameterized by $\\gamma$.\n\n\\subsection{Comparison with the 850-$\\mu$m source counts}\n\nWe now compute the normalizations $C$ required to fit the 850-$\\mu$m source \ncount data for a number of Gaussian luminosity functions (equation 1) with no \ndensity or luminosity evolution. $C$ is constant between redshifts \n$z_1$ and $z_2$ and is zero elsewhere. This is the simplest possible form of the \nluminosity function; at present the observations do not justify a more rigorous \ntreatment. This parametrization is motivated in part by the\nfact that the local 60-$\\mu$m luminosity function (Saunders et al.\\ 1990) is \napproximately Gaussian at the bright end. The local elliptical galaxy luminosity \nfunction (Binggeli, Sandage \\& Tammann 1988, Ferguson \\& Sandage 1991) is \nalso Gaussian, which is relevant if the SCUBA sources evolve into elliptical \ngalaxies: see Section 3. One important feature of the Gaussian function is that it \nis only valid for high-luminosity galaxies. Large numbers of galaxies with low \nfar-infrared luminosities will also exist between $z_1$ and $z_2$. Although these \nwill not contribute to the source counts, they may contribute significantly to the \ninfrared backgrounds if they are very numerous. Hence, when we match our \nluminosity functions to counts (equation 5), we require the model prediction to\nequal the observed values, but when we match our luminosity functions\nto background fluxes (equation 3), we require the model prediction to be less \nthan or equal to the observed values. Our approach is fundamentally different \nfrom that of Blain et al.\\ (1999a). We are not evolving the local 60-$\\mu$m \nluminosity function to high redshift. Indeed, in Section 3, we argue that the\nsystems which have the highest far-infrared luminosities for redshifts \n$z_1 < z < z_2$ probably evolve into elliptical galaxies, \nwhich are not the most \nluminous systems at far-infrared wavelengths in the local Universe. \n\nIn Figs\\,2 to 7 we present the required normalization constants for\na Gaussian luminosity function to fit the 850-$\\mu$m data of Smail\net al.\\ (1997) for various values of $\\sigma$ and $L_{*}$ given\na constant dust temperature $T$. The uncertainties are large (see Table\\,1), \nand so the range of acceptable normalizations is large, with a peak-to-peak \nspread of about a factor of 4. Given these large uncertainties, all the \n850-$\\mu$m counts by different authors listed in\nTable 1 are fully consistent.\nSome general \nfeatures of Figs\\,2 to 7 are worth highlighting:\n\\begin{enumerate} \n\\item $\\sigma = 0$ (a $\\delta$-function) never appears to fit the data well; \n\\item lower normalizations are required for higher characteristic luminosities;\n\\item the absolute values of the normalizations depend primarily on where the \ndust spectrum peak is shifted in wavelength space. It is shifted closer to\n850\\,$\\mu$m for a source at higher redshift at either 40 or 70\\,K, and so a lower \nnormalization is required at higher $z$;\n\\item lower temperatures require lower normalizations for a given 60-$\\mu$m \nluminosity, since 850\\,$\\mu$m is closer to the blackbody peak. \n\\end{enumerate} \n\n\\subsection{Comparison with counts at other wavelengths}\n\nHaving isolated various models that are consistent with the 850-$\\mu$m \ncounts, we now consider the constraints from measurements at other\nwavelengths. We select a number of models with various values of \n$\\sigma$, $L_{*}$, $C$, and $T$, each represented by a point \non the curves in Figs\\,2 to 7, and thus consistent with the 850-$\\mu$m count. \nFor each model, we consider three separate scenarios: scenario ``0'', in which \nthe 850-$\\mu$m source counts are the mean Smail et al.\\ (1997) values;\nscenario ``+'', in which they are 1$\\sigma$ larger; and scenario ``$-$'', in which \nthey are 1$\\sigma$ smaller. The counts in the ``$-$'' scenario are very close to \nthe lowest published counts (Barger et al.\\ 1998a). In all, a total of 45 models, \neach of which generates three scenarios, are considered.\n\nIn Table 2, we show how well the models defined by these parameters account \nfor the measurements at other wavelengths. The numbers in that table \nrepresent the model prediction relative to the observed counts or background \nfluxes listed in Table 1, assuming scenario ``0''. The fractional uncertainties in \nthe numbers in Table 2 are given in Table 3. \n\nOther measurements exist in the literature that we do not use while\nmaking this comparison. For example, measurements of 15-$\\mu$m source \ncounts with {\\it ISO} have recently been made (Rowan-Robinson \net al.\\ 1997, Aussel et al.~1998). \nHowever, these measurements probe far down the Wien tail of \nthe dust spectrum, and are of very limited use in constraining the high-redshift \nluminosity function. Furthermore we would need to account for \ncontamination in the 15-$\\mu$m samples by nearby (unobscured) galaxies and \nAGN. Similarly the IRAS 60-$\\mu$m source counts (Saunders et al.\\ 1990)\nare dominated by nearby ($z < 0.1$) sources. Measurements at 1.25\\,mm offer a \npromising probe in the future, but to date measurements have only been made \nfor IRAS-selected galaxies (Franceschini, Andreani \\& Danese 1998). Unbiased \nsamples are not yet available at this wavelength.\n\nSome of the more notable general trends shown in Table 2 are:\n\\begin{enumerate}\n\\item models in which the sources are at higher redshift tend to\nproduce higher source counts or backgrounds at longer \nwavelengths, because the peak of the dust spectrum is shifted\nto longer wavelengths;\n\\item models in which the sources are at higher redshift tend to\nproduce higher backgrounds for a given source count\n{\\it at the same wavelength}. The counts probe only part of the total range of \ngalaxy luminosities, and there are a larger number\nof low-luminosity sources that contribute to the backgrounds but\nnot to the counts;\n\\item on decreasing the temperature from 70 to 40\\,K, the predicted 2.8-mm \ncounts increase, but the predicted 175-$\\mu$m counts and 140- and \n240-$\\mu$m backgrounds decrease, because the peak in the dust emission \nspectrum shifts to longer wavelengths;\n\\item Changing the temperature has a substantial effect on the relative value \nof the source counts and backgrounds at the same wavelength. Decreasing the \ntemperature has a greater relative effect on the counts, because the lower limit\nin the integral in equation 5 is decreased: see Fig.\\,8. \n\\end{enumerate} \n\nSeven of the models that we investigate are consistent with all the\nobservations: see the last column of Table 2 for details. These all have a high \ncharacteristic luminosity greater than $10^{11} {\\rm L}_{\\odot}$. \nUsing a different parametrization, Blain et al.\\ (1999a) also found that most of the\nSCUBA sources are distant galaxies with similarly high luminosities.\nThese seven models are:\n12\/40\/3\/5\/0.9``0'' (hereafter Model A);\n13\/40\/3\/5\/0.9``$-$'' (hereafter Model B);\n11\/40\/2\/4\/0.9``$-$'' (hereafter Model C);\n12\/40\/2\/4\/0.9``$-$'' (hereafter Model D);\n13\/40\/2\/4\/0.5``0'' (hereafter Model E);\n13\/40\/2\/4\/0.5``$-$'' (hereafter Model F);\n13\/40\/1\/3\/0.1``$-$'' (hereafter Model G).\nIn general, the tables suggest that models with $T = 70$\\,K overproduce the \n175-$\\mu$m counts and the 140- and 240-$\\mu$m backgrounds, given \nthe normalization from the 850-$\\mu$m source counts. Models in which the \ngalaxies are nearby, that is where $z_1 = 1$, tend to have similar problems. \nConversely, models with very low temperatures $ T < 40$\\, K would overproduce\nthe 2.8-mm counts, but such low-temperature sources are ruled out by the\nconsistency arguments of Blain et al.~(1999a --- see Section 4 of that paper).\n\n\\subsection{Comparison with the FIRAS background}\n\nFixsen et al.\\ (1998) derived the cosmic submillimetre background from \n{\\it COBE} FIRAS data. Although the background at these wavelengths is \ndominated by emission from the Galaxy and the cosmic microwave background, \nthey used three independent techniques to subtract out these signals. The \nresulting residual 850-$\\mu$m extragalactic background radiation intensity \n$\\nu I_\\nu$=$0.55 \\pm 0.15$\\,nW\\,m$^{-2}$\\,sr$^{-1}$. \n\nWe can predict 850-$\\mu$m backgrounds in our models using\nequation (3). The results are presented in Table 4. As in the previous\nsection, we require that our models do not overproduce the background. Model \nG achieves this convincingly, and models B and F are consistent within \n2$\\sigma$. \nWe select these three models for further\nstudy. In general, models with $z_1 = 3$ overpredict the 850-$\\mu$m \nbackground if the 240-$\\mu$m background is predicted correctly, because the \nobserved spectra at a temperature of 40\\,K are shifted to too long a \nwavelength. The surviving models have $z_1 = 2$ (B and F) or $z_1$ = 1 (G).\nAll these models also have $L_* = 10^{13}$\\,L$_{\\odot}$, meaning\nthat most of the far-infrared luminosity from this population comes\nfrom galaxies with 60-$\\mu$m luminosities in excess of 10$^{12}$\\,L$_{\\odot}$. \nAs discussed in Section\\,2, models with lower characteristic luminosities tend \nto overpredict the 850-$\\mu$m background if they match the 850-$\\mu$m \ncounts. \n\nNote that in the three surviving models (B, F and G) the normalization of the \n850-$\\mu$m counts is consistent with all the observations. In the best-fitting \nmodel (G), the mean redshift of the sources $z_{\\rm s} \\sim 2$, as argued by Lilly \net al.\\ (1998). One half (F,G) or more (B) of the 240-$\\mu$m background, but \nconsiderably less of the 140-$\\mu$m background, comes from the \nhigh-luminosity galaxies described by our Gaussian luminosity function. In \nmodels B and F, about half of the 175-$\\mu$m sources are the same sources \nthat contribute to the 850-$\\mu$m source counts; in model G, the two \npopulations are practically the same. In Model B, the predicted 2.8-mm \ncount is close to the observed limit (Wilner \\& Wright 1997); for the other models\nthe predicted count is much smaller. The 450-$\\mu$m source count limit\n(Barger et al.\\ 1998a) is larger than the predicted values by a factor of about 3 \nfor Model B, and by a much larger factor for the other models.\n\nThree models appear to fit all the data. Even within our simple parametrization \nwe do not find a unique best-fitting model. Furthermore, there are presumably \nmany other (non-Gaussian) models that fit all the data, for example the \nmodels of Blain et al.\\ (1999a). Nevertheless most of our models seem to be ruled \nout when we consider all the observations in conjunction. That a relatively \nnarrow region of parameter space is consistent with observation \n($L_{*} \\sim 10^{13}$\\,L$_{\\odot}$, $T \\simeq 40$\\,K, $z_{\\rm s} \\sim 2$) is \nencouraging and suggests that it will be productive to investigate these models \nin further detail. \n\n\\section{Properties of the SCUBA sources} \n\nWe have isolated three models of the redshift-dependent luminosity\nfunction of ULIRGs that are consistent with observation, and now\ninvestigate the cosmological implications. \nIn Section 3.1 we investigate the possibility that these are dusty star-forming\ngalaxies. We use the star-formation models of Leitherer \\& Heckman (1995) to\nplace the galaxies on the Madau Plot, and then examine the consequences\nof this interpretation in the context of the wider galaxy formation\npicture. In Section 3.2 we investigate the alternative possibility that the \nsources are dust-enshrouded AGNs. \n\n\\subsection{The SCUBA sources as star-forming galaxies}\n\nThe majority of local ULIRGs appear to be star-forming galaxies and not \ndust-enshrouded AGN (Sanders et al.\\ 1988a, Genzel et al.\\ 1988). Furthermore, \nthe temperatures of the SCUBA sources inferred in Section 2 ($T \\sim 40$\\,K) \nare close to, although systematically slightly cooler than, those of local \nstar-forming ULIRGs. Locally, dust-enshrouded AGN like Mrk 231 tend to have \nhigher dust temperatures than dust-enshrouded starbursts like Arp 220 \n(Sanders et at.\\ 1988a); however, note that some dusty high-redshift quasars \nappear to be fairly cold: see Benford et al.\\ (1998). Hence, it seems \nreasonable to investigate the possibility that the SCUBA sources are\nhigh-redshift star-forming galaxies.\n\n\\subsubsection{Leitherer-Heckman models}\n\nIn order to compute the cosmological star-formation rate associated with the \nSCUBA sources at high redshift, we need a recipe to convert far-infrared \nluminosities to star-formation rates for individual sources.\n\nThe transformation between these two quantities is not straightforward to \ndetermine observationally even for local ULIRGs because of the high internal \nextinction along our lines to the galaxy centers. For example, \neven the near-infrared 2.2-$\\mu$m Br$\\gamma$ line strength--far-infrared\nluminosity correlation (Goldader et al.\\ 1997a,b), which is usually very \nreliable, breaks down in ULIRGs at very high luminosities, presumably due to \ninternal extinction. We therefore need to compute this transformation using \nmodels (Leitherer \\& Heckman 1995).\n\n\\begin{table*} \n\\caption{Comparison between observations and models. All the numbers \nrepresent the predicted source counts or background flux for\nmodel ``0'' that are appropriate to the luminosity function defined by\nthe parameters $L_{*}$, $T$, $\\sigma$, $z_1$, and $z_2$, relative to the \nobserved counts and background intensities listed in Table\\,1.\nThe model parameters are listed in condensed form in the model name, which\ntakes the general form log$_{10}$($L^*_{60}$)\/$T$\/$z_1$\/$z_2$\/$\\sigma$. The \nfigure panel in which the constraints for each model are imposed are \nlisted in the second column. \nA value less than $10^{-10}$ is listed as $\\sim 0$ in the Table.\nThe errors in these quantities are listed in Table\\,3. \nAll the models have been selected {\\it a priori} to give the correct \n850-$\\mu$m counts. The comments refer to the selection or rejection of \nmodels based on the listed values. The range of acceptable values and a \ndescription of the numerical code values are given in Table\\,3. The \n140-$\\mu$m and 240-$\\mu$m background values of \nSchlegel et al.\\ (1998), the 175-$\\mu$m counts of Lagache\net al.~(1998), and the 450-$\\mu$m counts of Smail et al.~(1997)\nare fitted. \n} \n{\\vskip -2.75mm}\n{$$\\vbox{\n\\halign {\\hfil #\\hfil && \\quad \\hfil #\\hfil \\cr\n\\noalign{\\hrule \\medskip}\nModel & Figure \n & 2.8-mm & 450-$\\mu$m & 175-$\\mu$m\n & 240-$\\mu$m & 140-$\\mu$m & comments &\\cr\n & & counts & counts & counts & background & background & (See\nTable\\,3)&\\cr\n\\noalign{\\smallskip \\hrule \\smallskip}\n\\cr\n\n10\/70\/3\/5\/0.5 & 2a & $\\sim 0$\n & $7.2 \\times 10^{-5}$ \n & $0.18$ \n & $1.5 \\times 10^{10}$ & $7.5 \\times 10^9$ & (4$^{-}$56) &\\cr \n10.70\/3\/5\/0.9 & 2a & $0.0019$\n & $0.082$\n & $2.5$ \n & $110$ & $55$ & (3$^{0+}$56) &\\cr\n11\/70\/3\/5\/0.5 & 2b & $3.2 \\times 10^{-9}$\n & $8.1 \\times 10^{-4}$\n & $0.40$ \n & $2.3 \\times 10^{5}$ & $1.1 \\times 10^{5}$ & (56) &\\cr\n11\/70\/3\/5\/0.9 & 2b & $0.011$\n & $0.17$\n & $3.9$ \n & $15$ & $7.4$ & (356) &\\cr\n12\/70\/3\/5\/0.5 & 2c & $1.3 \\times 10^{-6}$\n & 0.0090 \n & $1.0$ \n & $178$ & $87$ & (3$^{+}$56) &\\cr\n12\/70\/3\/5\/0.9 & 2c & $0.067$\n & $0.36$\n & $6.0$ \n & $6.3$ & $3.1$ & (356$^{0+}$) &\\cr\n13\/70\/3\/5\/0.1 & 2d & $\\sim 0$\n & $\\sim 0$\n & $3.2 \\times 10^{-6}$ \n & $2.9 \\times 10^{13}$ & $1.9 \\times 10^{13}$ & (456) &\\cr\n13\/70\/3\/5\/0.5 & 2d & $4.4 \\times 10^{-4}$\n & $0.094$\n & $3.0$ \n & $5.8$ & $2.8$ & (356$^{0+}$) &\\cr\n13\/70\/3\/5\/0.9 & 2d & $0.35$\n & $0.71$\n & $9.6$ \n & $7.2$ & $3.5$ & (2$^{+}$356) &\\cr\n10\/70\/2\/4\/0.5 & 3a & $\\sim 0$\n & $2.3 \\times 10^{-4}$\n & $32$ \n & $1.9 \\times 10^{10}$ & $1.6 \\times 10^{10}$ & (356) &\\cr\n10\/70\/2\/4\/0.9 & 3a & $0.0014$\n & $0.12$\n & $14$ \n & $150$ & $120$ & (356) &\\cr\n11\/70\/2\/4\/0.5 & 3b & $1.5 \\times 10^{-9}$\n & $0.0021$\n & $20$ \n & $3.0 \\times 10^{5}$ & $2.5 \\times 10^{5}$ & (356) &\\cr\n11\/70\/2\/4\/0.9 & 3b & $0.0090$\n & $0.23$\n & $14$ \n & $20$ & $17$ & (356) &\\cr\n12\/70\/2\/4\/0.5 & 3c & $7.2 \\times 10^{-7}$\n & $0.017$\n & $14$ \n & $230$ & $190$ & (356) &\\cr\n12\/70\/2\/4\/0.9 & 3c & $0.055$ \n & $0.45$\n & $16$ \n & $8.3$ & $7.0$ & (356) &\\cr\n13\/70\/2\/4\/0.1 & 3d & $\\sim 0$\n & $\\sim 0$\n & $5.6 \\times 10^{5}$ \n & $2.4 \\times 10^{13}$ & $2.1 \\times 10^{13}$ & (356) &\\cr\n13\/70\/2\/4\/0.5 & 3d & $3.0 \\times 10^{-4}$\n & $0.15$\n & $14$ \n & $7.5$ & $6.4$ & (356) &\\cr\n13\/70\/2\/4\/0.9 & 3d & $0.3$\n & $0.82$\n & $18$ \n & $9.5$ & $8.0$ & (356) &\\cr\n10\/70\/1\/3\/0.5 & 4a & $ \\sim 0$\n & $8.8 \\times 10^{-4}$\n & $3600$ \n & $1.7 \\times 10^{10}$ & $2.4 \\times 10^{10}$ & (356) &\\cr\n10\/70\/1\/3\/0.9 & 4a & $0.0011$\n & $0.17$\n & $60$ \n & $170$ & $230$ & (356) &\\cr\n11\/70\/1\/3\/0.5 & 4b & $7.6 \\times 10^{10}$\n & $0.0054$\n & $560$ \n & $3.0 \\times 10^5$ & $4.0 \\times 10^5$ & (356) &\\cr\n11\/70\/1\/3\/0.9 & 4b & $0.0073$\n & $0.31$\n & $44$ \n & $23$ & $32$ & (356) &\\cr\n12\/70\/1\/3\/0.5 & 4c & $4.5 \\times 10^{-7}$\n & $0.034$\n & $120$ \n & $250$ & $350$ & (356) &\\cr\n12\/70\/1\/3\/0.9 & 4c & $0.047$\n & $0.55$\n & $34$ \n & $10$ & $14$ & (356) &\\cr\n13\/70\/1\/3\/0.1 & 4d & $ \\sim 0 $\n & $ \\sim 0$ \n & $3.1 \\times 10^{10}$ \n & $3.0 \\times 10^{11}$ & $4.3 \\times 10^{11}$ & (356) &\\cr\n13\/70\/1\/3\/0.5 & 4d & $2.2 \\times 10^{-4}$\n & $0.21$\n & $41$ \n & $8.8$ & $12$ & (356) &\\cr\n13\/70\/1\/3\/0.9 & 4d & $0.27$\n & $0.94$\n & $29$ \n & $12$ & $16$ & (2$^{+}$356) &\\cr\n10\/40\/3\/5\/0.5 & 5a & $1.0 \\times 10^{-8}$ \n & $9.2 \\times 10^{-6}$\n & $1.1 \\times 10^{-8}$ \n & $1.1 \\times 10^{6}$ & $7.1 \\times 10^{5}$ & (456) &\\cr \n10\/40\/3\/5\/0.9 & 5a & $0.018$\n & $0.033$\n & $0.0067$ \n & $3.6$ & $0.23$ & (45) &\\cr\n11\/40\/3\/5\/0.5 & 5b & $2.1 \\times 10^{-6}$\n & $2.1 \\times 10^{-4}$\n & $1.4 \\times 10^{-6}$ \n & $200$ & $14$ & (456) &\\cr\n11\/40\/3\/5\/0.9 & 5b & $0.087$\n & $0.092$\n & $0.034$ \n & $0.99$ & $0.067$ & (45$^{+}$) &\\cr\n12\/40\/3\/5\/0.5 & 5c & $3.9 \\times 10^{-4}$\n & $0.0050$\n & $1.8 \\times 10^{-4}$ \n & $1.7$ & $0.12$ & (45$^{0+}$) &\\cr\n12\/40\/3\/5\/0.9 & 5c & $0.40$\n & $0.25$\n & $0.17$ \n & $0.84$ & $0.056$ & (4$^{-}$5$^{+}$) &\\cr\n13\/40\/3\/5\/0.1 & 5d & $\\sim 0$\n & $3.3 \\times 10^{-18}$\n & $\\sim 0$ \n & $1.14$ & $0.076$ & (45$^{+}$) &\\cr\n13\/40\/3\/5\/0.5 & 5d & $0.053$\n & $0.11$\n & $0.019$ \n & $0.50$ & $0.034$ & (4) &\\cr\n13\/40\/3\/5\/0.9 & 5d & $1.6$\n & $0.63$\n & $0.82$ \n & $1.8$ & $0.12$ & (1$^{0+}$3$^{+}$5$^{0+}$) &\\cr\n10\/40\/2\/4\/0.5 & 6a & $2.96 \\times 10^{-9}$\n & $8.6 \\times 10^{-5}$\n & $5.7 \\times 10^{-4}$ \n & $1.1 \\times 10^{6}$ & $2.1 \\times 10^{5}$ & (456) &\\cr\n10\/40\/2\/4\/0.9 & 6a & $0.012$\n & $0.068$\n & $0.22$ \n & $6.6$ & $1.2$ & (56$^{+}$) &\\cr\n11\/40\/2\/4\/0.5 & 6b & $8.7 \\times 10^{-7}$\n & $0.0012$\n & $0.0058$ \n & $280$ & $51$ & (4$^{0-}$56) &\\cr\n11\/40\/2\/4\/0.9 & 6b & $0.065$\n & $0.16$\n & $0.54$ \n & $2.0$ & $0.36$ & (5$^{0+}$) &\\cr\n12\/40\/2\/4\/0.5 & 6c & $2.2 \\times 10^{-4}$\n & $0.017$\n & $0.062$ \n & $3.0$ & $0.54$ & (4$^{0-}$5$^{0+}$) &\\cr\n12\/40\/2\/4\/0.9 & 6c & $0.33$\n & $0.38$\n & $1.4$ \n & $1.77$ & $0.32$ & (3$^{0+}$5$^{0+}$) &\\cr\n13\/40\/2\/4\/0.1 & 6d & $\\sim 0$\n & $\\sim 0$\n & $2.0 \\times 10^{-9}$ \n & $1.1$ & $0.020$ & (45$^{+}$) &\\cr\n13\/40\/2\/4\/0.5 & 6d & $0.040$\n & $0.22$\n & $0.70$ \n & $1.1$ & $0.19$ & (5$^{+}$) &\\cr\n13\/40\/2\/4\/0.9 & 6d & $1.4$\n & $0.84$\n & $3.7$ \n & $4.0$ & $0.72$ & (1$^{0+}$2$^{+}$35) &\\cr\n10\/40\/1\/3\/0.5 & 7a & $9.4 \\times 10^{-10}$\n & $8.0 \\times 10^{-4}$\n & $3.7$ \n & $9.6 \\times 10^{5}$ & $4.3 \\times 10^{5}$ & (356) &\\cr\n10\/40\/1\/3\/0.9 & 7a & $0.0086$\n & $0.13$\n & $3.6$ \n & $11$ & $4.8$ & (356) &\\cr\n11\/40\/1\/3\/0.5 & 7b & $4.0 \\times 10^{-7}$\n & $0.0062$\n & $3.0$ \n & $323$ & $147$ & (356) &\\cr\n11\/40\/1\/3\/0.5 & 7b & $0.051$\n & $0.28$\n & $4.5$ \n & $3.5$ & $1.6$ & (356$^{0+}$) &\\cr\n12\/40\/1\/3\/0.1 & 7c & $ \\sim 0$\n & $\\sim 0$\n & $140$ \n & $2.8 \\times 10^{17}$ & $1.3 \\times 10^{17}$ & (356) &\\cr\n12\/40\/1\/3\/0.5 & 7c & $1.4 \\times 10^{-4}$\n & $0.52$\n & $3.0$ \n & $4.6$ & $2.1$ & (356$^{0+}$) &\\cr\n12\/40\/1\/3\/0.9 & 7c & $0.28$\n & $0.56$\n & $6.5$ \n & $3.3$ & $1.5$ & (356$^{0+}$) &\\cr\n13\/40\/1\/3\/0.1 & 7d & $\\sim 0$\n & $6.1 \\times 10^{-6}$\n & $2.2$ \n & $1.2$ & $0.54$ & (3$^{0+}$5$^{+}$) &\\cr\n13\/40\/1\/3\/0.5 & 7d & $0.033$\n & $0.42$\n & $5.4$ \n & $2.0$ & $0.92$ & (35$^{0+}$6$^{+}$) &\\cr\n13\/40\/1\/3\/0.9 & 7d & $1.3$\n & $1.1$\n & $10$ \n & $8.0$ & $3.6$ & (356) &\\cr\n\\noalign{\\smallskip \\hrule}\n\\noalign{\\smallskip}\\cr}}$$}\n\\end{table*}\n\n\\begin{table*}\n\\caption{Fractional uncertainties in the background and source counts \nvalues used to accept or reject the models listed in Table\\,2. The acceptable \nranges of values are used to make the selection, and the relevant cases are \ndescribed by numerical codes in the last column of Table\\,2. A code is \nshown if the model is rejected on the grounds of: (1) overproduction of \nthe 2.8-mm count; (2) overprediction of the 450-$\\mu$m count; (3) \noverproduction of the 175-$\\mu$m count; (4) underprediction of the \n175-$\\mu$m count; (5) overprediction of the {\\it COBE} 240-$\\mu$m background; \nand (6) overprediction of the {\\it COBE} 140-$\\mu$m background. Although \na model that underpredicts the 175-$\\mu$m count is not formally excluded, \nsuch a model requires that the 175-$\\mu$m counts and 850-$\\mu$m counts come\nfrom entirely different populations of galaxies. This is not impossible, but we \nregard it as unlikely; we choose a factor of ten discrepancy as the cutoff\nbetween a model being acceptable and unacceptable. \nIn the last column of Table\\,2, superscripts 0, + or $-$ indicate that \nthe exclusion applies only in that scenario, corresponding to the \nvalues listed below. The absence of a superscript implies that the \nmodel is rejected in all three scenarios.} \n{\\vskip -0.75mm}\n{$$\\vbox{\n\\halign {\\hfil #\\hfil && \\quad \\hfil #\\hfil \\cr\n\\noalign{\\hrule \\medskip}\nModel & 2.8-mm\n & 450-$\\mu$m\n & 175-$\\mu$m & 240-$\\mu$m & 140-$\\mu$m &\\cr\n & counts & counts & counts & background & background & &\\cr\n\\noalign{\\smallskip \\hrule \\smallskip}\n\\cr\nFractional Error in ratio & $-$ & $-$ & 0.15 & 0.24 & 0.41 &\\cr\n & & & & & & & &\\cr\nRange of Acceptability (``0'' models) &\n$<1.0$ & $<1.0$ & [0.85,1.2] & $<1.2$ & $<1.4$ &\\cr\nRange of Acceptability (``$-$'' models) &\n$<2.4$ & $<2.4$ & [2.0,2.8] & $<3.0$ & $<3.4$ &\\cr\nRange of Acceptability (``+'' models) &\n$<0.63$ & $<0.63$ & [0.54,0.73] & $<0.78$ & $<0.89$ &\\cr\n & & & & & & & &\\cr\n\\noalign{\\smallskip \\hrule}\n\\noalign{\\smallskip}\\cr}}$$}\n\\end{table*}\n\nIn Table 5 we list a number of star-formation models that we investigate in \nfurther detail. Note that we use capital letters to represent galaxy formation \nmodels (Section 2) and roman numerals to represent the different star-formation \nmodels. There is insufficient information to determine accurately the appropriate \nset of input parameters to the star-formation models for the SCUBA sources, \nand even for local ULIRGs in many cases. The more important parameters are:\n\\begin{enumerate}\n\\item the mode of star formation. For example, the star formation may occur\ncontinuously, as an instantaneous burst, or with a more complex time\ndependence, for example an exponential decay. We investigate both the \ninstantaneous (Models I and II) and continuous (models III through X) cases.\nObservations of the core of Arp 220 (Mouri \\& Taniguchi 1992;\nPrestwich, Joseph, \\& Wright 1994; Armus et al.\\ 1995; Larkin et al.\\ 1995,\nScoville et al.\\ 1998), combined with the inferred presence of ionizing sources \nthere (Larkin et al.\\ 1995; Kim et al.\\ 1995; Goldader et al.\\ 1995), indicate that \nthe continuous models offer a more plausible description of the star formation \nin this well studied ULIRG;\n\\item the stellar initial mass function (IMF). The precise form of the IMF in \nstarburst galaxies is very much an open question: see Leitherer (1998) for a \nreview. It appears that the IMF of the most luminous star clusters in the Milky \nWay and the Magellanic Clouds follows closely the Salpeter form (Hunter et al.\\ \n1995; Hunter et al.\\ 1997; Massey et al.\\ 1995a). However, the field-star IMF may \nbe significantly steeper than the cluster IMF (Massey et al.\\ 1995b),\nand there is some evidence (Meurer et al.\\ 1995) that the most massive stars\nin starbursts form in environments that are more similar to the field than to \nthe cores of the luminous star clusters studied above. There are \nhence few direct observations to guide us to the correct IMF in starburst \ngalaxies, and we therefore consider four different scenarios in Models III to X.\nLeitherer \\& Heckman (1995) consider power-law IMFs, and \nwe investigate various possible combinations of\nthe upper and lower mass cutoffs and the slope of this power law.\nNote that if the SCUBA sources end up as elliptical galaxies (Section 3.1.3; \nBlain et al. 1999a; Lilly et al.\\ 1998), a lower limit to the IMF at \n$M_{\\rm l}$ = 3\\,M${_\\odot}$ is suggested, in agreement with the value suggested \nby Zepf \\& Silk (1996).\n\\item the inital gas metallicity. We assume a metallicity of twice solar.\nThis is close to the metallicities of the most luminous elliptical\ngalaxies (e.g.~Faber 1973, Vader 1986); \n\\item the age at which we observe the star clusters. We investigate the\ncases where we observe the galaxies at between 10$^8$ and 10$^{8.5}$\nyears after the onset of star formation. Fortunately, most of the\nproperties of the starburst, like the total bolometric luminosity, reach\na plateau soon after the onset of star formation, and so our results do not \ndepend strongly on this parameter. The median total amount\nof gas consumed by a $L_{*}$ galaxy 10$^{8.5}$ years after the onset\nof star-formation is 7 $\\times 10^{10}$\\,M$_\\odot$ in our models. This is \nconsistent with the progenitor galaxies to the SCUBA sources being gas-rich \nnormal galaxies; \n\\item the effects of extinction. Since we are only attempting to model\nthe far-infrared and submillimetre properties of the starburst, we\ncan assume that the galaxies are optically thin and do not need to\nmake any corrections for extinction. We assume that the dust absorbs\nand reradiates all the energy produced by the starburst.\n\\end{enumerate} \n\nFrom the seventh column of Table 5 it is immediately apparent that there is \nmore than an order of magnitude of uncertainty in the transformation from \nfar-infrared luminosity to star-formation rate, due to the uncertainty in the\nstar-formation parameters. The constant conversion factor of 2.2 $\\times$\n10$^9$\\,L$_{\\odot}$\\,M$_{\\odot}^{-1}$\\,yr (Rowan-Robinson et al.\\ 1997), \nused by Blain et al.\\ (1999a), corresponding to 0.22 in the seventh column of\nTable 5, is bracketed by our results.\n\n\\subsubsection{The Madau plot}\n\nWe combine our redshift-dependent luminosity functions (Models B, F and G) \nwith the star-formation parameters for the models listed in Table 5 \n(Models I to X) to compute the comoving star formation rate in the SCUBA \nsources. The results are listed in Table 6. We can then put the SCUBA \nsources on the the Madau plot: see Figs\\,9, 10 and 11. The other points on the \nMadau plot all come from optically-selected samples. As there is no luminosity \nevolution in the models between $z_1$ and $z_2$, the comoving star-formation \nrate is constant between these redshifts, and so the lines on the Madau plot are \nhorizontal. \n\nThe figures show that the uncertainty in determining where the SCUBA \nsources lie on the Madau plot is huge, more than an order of magnitude, \neven if we have specified the correct luminosity--redshift distribution model B, \nF or G. This uncertainty is solely due to our lack of knowledge regarding the \nstar-formation parameters, as discussed in Section 3.1.1. For Model B, and \npossibly Model F, if the true cosmic star-formation rate in the SCUBA sources\nis towards the higher end of our permitted range, then they dominate the \nstar-formation rate of the Universe: this is essentially the scenario proposed\nby Blain et al.\\ (1999a) and Hughes et al.\\ (1998). If the true cosmic star-formation \nrate in the SCUBA sources is towards the lower end of our range, then\nthey do not contribute significantly to the star-formation rate of the Universe \nat any redshift. In general, the instantaneous models \npredict higher star-formation rates. This is because more stars\nneed to be formed to produce a given bolometric luminosity we observe\nmore than 10$^8$ years after the starburst.\n\nIt is interesting to note that the IMF proposed by Zepf \\& Silk (1996) for \nelliptical galaxies, if valid for the SCUBA sources, results in them being at \nthe extreme low end of our proposed range. This is because these models \nunderproduce low-mass stars, and so result in a low star-formation rate for \na given amount of bolometric luminosity: the low-mass stars, when young, \ndo not contribute significantly to the bolometric luminosity. The implication \nhere is that if SCUBA sources evolve into elliptical galaxies, then they do \nnot contribute significantly to the star-formation rate of the Universe at \nany redshift given our models of the luminosity function. \n \n\\begin{figure}\n\\begin{center}\n\\epsfig{file=fig8.ps, width=8.65cm}\n\\end{center}\n\\caption{The lower limit on the integral in equation (5) as a function of the \ngalaxy dust temperature $T$, normalized to its value for $T=38$\\,K, predicted \nfrom {\\it IRAS} and {\\it ISO} counts by Blain et al.\\ (1999a).} \n\\end{figure} \n\nThe broadband optical colours of the SCUBA sources are not significantly \ndifferent from those of normal field galaxies (Smail et al.\\ 1998). Furthermore, \ntwo of the three local ULIRGs studied at ultraviolet wavelengths with {\\it HST} \nby Trentham, Kormendy \\& Sanders (1998) would also have broadband colours \nsimilar to normal field galaxes if they were placed at high redshift. Therefore,\nSCUBA sources cannot be identified as submillimetre-luminous galaxies \nbased on broadband optical colours alone. The star-formation rates given \nin Table 5 are far higher than the rates in normal galaxies. Therefore we need \nto treat the SCUBA sources as a separate population on the Madau plot. This \nis particularly important if the SCUBA sources contribute significantly to the\nstar-formation rate of the Universe at any redshift. \n\n\\begin{table}\n\\caption{Predicted 850-$\\mu$m background intensities. The observed\n850-$\\mu$m background intensity is $0.55 \\pm 0.15$\\,nW\\,m$^{-2}$\\,sr$^{-1}$ \n(Fixsen et al. 1998).} \n{$$\\vbox{\n\\halign {\\hfil #\\hfil && \\quad \\hfil #\\hfil \\cr\n\\noalign{\\hrule \\medskip}\nModel & Background \/ & Model & Background \/\\cr\n & nW\\,m$^{-2}$\\,sr$^{-1}$ & & nW\\,m$^{-2}$\\,sr$^{-1}$\\cr\n\\noalign{\\smallskip \\hrule \\smallskip}\n\\cr\nA & 3.40 & E & 1.71 \\cr\nB & 0.85 & F & 0.71 \\cr\nC & 1.32 & G & 0.34 \\cr\nD & 1.19 & & \\cr\n\\noalign{\\smallskip \\hrule}\n\\noalign{\\smallskip}\\cr}}$$}\n\\end{table}\n\n\\begin{table*}\n\\caption{Properties of star-formation models. \n$\\alpha$ is the slope of the stellar IMF and $M_{\\rm u}$ and $M_{\\rm l}$\nare the lower and upper mass cutoffs.\nThe final two columns are derived from the results of Leitherer \\& Heckman\n(1995). The star-formation\nrates (SFR) as a function of 60-$\\mu$m luminosity come from Figs\\,7 and 8,\nassuming a temperature of 70\\,K. The metal masses $M_Z$ come from \nFigs\\,53 and 54 of Leitherer \\& Heckman (1995), assuming that the mass of \nmetals produced is equal to the mass returned by winds and supernovae.\nAll models assume an initial metallicity of twice solar.\n} \n{$$\\vbox{\n\\halign {\\hfil #\\hfil && \\quad \\hfil #\\hfil \\cr\n\\noalign{\\hrule \\medskip}\nModel & SF profile & log$_{10}$(age\/yr) &\n $\\alpha$ & $M_{\\rm u}$ \/ M$_\\odot$ & $M_{\\rm l}$ \/ M$_\\odot$ & \nSFR \/ ($L_{60} \/ 10^{9} {\\rm L}_{\\odot}$) M$_\\odot$\\,yr$^{-1}$ \n& $M_Z$ \/ ($L_{60} \/ 10^{9} {\\rm L}_{\\odot}$) M$_\\odot$ &\\cr\n\\noalign{\\smallskip \\hrule \\smallskip}\n\\cr\nSF I & Instantaneous & 8.0 & 2.35 & 100 & 1 & 1.0 & $4.0 \\times 10^7$ &\\cr\nSF II & Instantaneous & 8.5 & 2.35 & 100 & 1 & 0.81 & $1.0 \\times 10^8$ &\\cr\nSF III & Continuous & 8.0 & 2.35 & 100 & 1 & 0.11 & $3.5 \\times 10^6$ &\\cr\nSF IV & Continuous & 8.5 & 2.35 & 100 & 1 & 0.072 & $7.2 \\times 10^6$ &\\cr\nSF V & Continuous & 8.0 & 2.35 & 30 & 1 & 0.15 & $4.1 \\times 10^6$ &\\cr\nSF VI & Continuous & 8.5 & 2.35 & 30 & 1 & 0.12 & $1.0 \\times 10^7$ &\\cr\nSF VII & Continuous & 8.0 & 3.3 & 100 & 1 & 0.34 & $8.0 \\times 10^5$ &\\cr\nSF VIII & Continuous & 8.5 & 3.3 & 100 & 1 & 0.25 & $1.6 \\times 10^6$ &\\cr\nSF IX & Continuous & 8.0 & 2.35 & 100 & 3 & 0.056 & $3.5 \\times 10^6$ &\\cr\nSF X & Continuous & 8.5 & 2.35 & 100 & 3 & 0.043 & $7.2 \\times 10^6$ &\\cr\n\\noalign{\\medskip \\hrule}\n\\noalign{\\smallskip}\\cr}}$$}\n\\end{table*}\n\n\\subsubsection{Why do the result differ from those of other authors} \n\nBoth Blain et al.~(1999a) and Hughes et al.~(1998) derive global star-formation rates\nat least an order of magnitude higher than we derive in the current paper.\nThere are two sources of discrepancy.\n\\vskip 1pt\n\\noindent (i) differences in star-formation rates. \nFor example Blain et al.~(1999a) use a value of $M_{\\rm l}$ below 1 M$_\\odot$ and\nconsequently assume star-formation rates that are towards the upper end of our\nconsidered range.\n\\vskip 1pt\n\\noindent (ii) differences in the assumed luminosity-redshift distribution, which can \nbe broadly split further into (a) differences in the characteristic luminosity of the\nsources who dominate the contribution to the SCUBA counts , and (b) difference in the\nnormalization of the luminosity function at a given redshift. Effect (iia) is not a\nmajor source of discrepancy between our results and those of Blain et al.~(1999a) --- in\nboth cases, most of the star formation occurs in very luminous sources\nwith $L_{60} > 10^{11}$\\,L$_{\\odot}$. Effect (iib) is a much more substantial source\nof discrepancy. Blain et al.~(1999a) assume a redshift-dependent Saunders luminosity function at\nhigh redshift which is related to the local 60-$\\mu$m luminosity function by simple luminosity\nevolution. We assume a Gaussian luminosity function with no evolution over some specified\nredshift range. Our objects, when evolved to, $z=0$ are early-type stellar populations, \nquite unrelated to the objects which dominate the local 60-$\\mu$m luminosity function. \nWe consequently derive a far lower\nnormalization to our luminosity functions than do Blain et al.~(1999a).\nThe present observations do not permit us to distinguish \nbetween these models, but this should be a straightforward exercise when a \nsignificantly complete redshift distribution for the SCUBA sources is known. \n\nIn addition, Eales et al.~(1998) argue that at least 10 percent\nof the stars in the Universe\nformed in SCUBA sources since they produce about\n10 percent \nof the extragalactic background light. This number is much closer to the\nnumbers presented in this paper. Nevertheless, the assumption that the SCUBA sources\ncontribute the same fraction of the submillimeter background at all wavelengths, on which\nthe Eales et al.~calculation is based, appears to be inconsistent with the formulation that\nwe use, when the BIMA 2.8-mm counts, \nISO 175-$\\mu$m counts, and COBE 850-$\\mu$m background\nare all considered in conjuction. Furthermore, it is unclear how we should relate\nstar-formation rates derived from optical or ultraviolet luminosities of normal field\ngalaxies to star-formation rates derived from submillimetre fluxes of the SCUBA sources. \n\n\\begin{figure}\n\\begin{center}\n\\vskip -2mm\n\\epsfig{file=fig9.ps, width=8.65cm}\n\\end{center}\n\\vskip -5mm\n\\caption{The comoving star-formation density of\nthe Universe contained in the SCUBA sources when their\nluminosity-redshift distribution is as in Model B.\nThe eight dashed lines represent the SFR histories for models\n(in ascending order) X, IX, IV, III, VI, V, VIII, VII, II and I.\nThe other points come from:\nfilled triangle -- Gallego et al.~(1996);\nopen triangle -- Treyer et al.~(1998);\nopen circle -- Tresse \\& Maddox (1998);\nstars -- Lilly et al.~(1996);\nopen hexagons -- Hammer \\& Flores (1998);\nfilled squares - Connolly et al.~(1997);\nfilled circles - Madau et al.~(1996);\nopen squares - Pettini et al.~(1998; these include a global correction\nfor dust extinction). The recent work of Glazebrook et al.~(1999) \nsuggests a SFR almost coincident with the $z=0.85$ point of Hammer\n\\& Flores (1998).} \n\\end{figure} \n\n\\begin{figure}\n\\begin{center}\n\\vskip -2mm\n\\epsfig{file=fig10.ps, width=8.65cm}\n\\end{center}\n\\vskip-5mm\n\\caption{\nAs Figure 9, but for model F. The symbols\nhave the same meanings and the lines are in the same order.\n}\n\\end{figure} \n\n\\begin{figure}\n\\begin{center}\n\\vskip-2mm\n\\epsfig{file=fig11.ps, width=8.65cm}\n\\end{center}\n\\vskip-5mm\n\\caption{\nAs Figure 9, but for model G. The symbols\nhave the same meanings and the lines are in the same order.\n}\n\\end{figure} \n\n\\subsubsection{The fate of the star-forming galaxies}\n\nWe have hinted that the SCUBA sources could evolve into elliptical galaxies.\nLocal ULIRGs have gas densities that are similar to the core stellar densities \nin elliptical galaxies (Kormendy \\& Sanders 1992, Doyon et al.\\ 1994).\nIf the SCUBA sources have similar morphologies to local ULIRGs, we \nmight then expect them to evolve into elliptical galaxies. \n\nIn Table 7 we present the density parameter in stars produced by the SCUBA \nsources, in our luminosity-function and star-formation models. For most models, \nparticularly those with IMFs appropriate to elliptical galaxies (models IX and\nX), these numbers are very small as compared with the stellar density \ncontained in the local spheroid stellar population $\\Omega_{\\rm sph}$,\nthat is elliptical galaxies and bulges: $\\Omega_{\\rm sph} = 0.0036$ \n(Fukugita, Hogan \\& Peebles 1998). \nIn Models IX and X, between 1 and 4\\,per cent of $\\Omega_{\\rm sph}$\nformed in the luminous high-redshift SCUBA sources, and slightly more if a \nsolar-neighbourhood IMF (Models III and IV) is assumed. These numbers are \nsufficiently low that a scenario in which all the SCUBA sources evolve into \nelliptical galaxies is consistent with galaxy formation models that predict\nlow-redshift elliptical formation (Kauffmann 1996; Kauffmann, Charlot \\& White \n1996; Kauffmann \\& Charlot 1998). They are also low enough to be\nconsistent with the observed paucity of red galaxies in the Hubble Deep Field \nand the consequent interpretation that less than 30\\,per cent of field ellipticals\nformed at high redshift (Zepf 1997; Barger et al.\\ 1998b). \n\nVarious lines of argument suggest that the elliptical galaxies in clusters are \nvery old. Not only did the stars form a long time ago (Ellis et al.\\ 1997; \nStanford et al.\\ 1998; Kodama et al.\\ 1998), but it appears that in the richest \nclusters the most luminous elliptical galaxies themselves were assembled by \n$z \\sim 1$ (Trentham \\& Mobasher 1998). It is further possible that most 3C radio \ngalaxies are cluster ellipticals in the process of formation (Best, Longair \\& \nRottgering 1998). This leads to a natural question: are the SCUBA galaxies \nforming cluster ellipticals? Low-redshift cluster ellipticals are\nhighly clustered, with a bias of about 4, and an even greater bias at high redshifts\n(Fry 1996; Mo \\& White 1996). In comparison, the reasonably uniform detection\nrate of SCUBA galaxies in the fields studied by Smail et al.\\ (1998) suggests \nthat this is unlikely. \nThese fields contained low-redshift clusters (which magnify the background\nSCUBA sources through gravitational lensing), but these low-redshift\nclusters are unrelated to the hypothesized ellipticals in formation that\nwe are discussing here. \nInstead, we propose that the SCUBA sources are a \nforming trace population of field ellipticals. This is consistent with the \nconstraints outlined in the previous paragraph. This is not to say that if we \nhappened to point SCUBA at a cluster in formation we would not detect a large \nnumber of galaxies.\n\n\\begin{table*} \n\\caption{Comoving star-formation rates in units of \nM$_\\odot$\\,yr$^{-1}$\\,Mpc$^{-3}$ in the three well fitting models of \ndistant ULIRGs.\n}\n{$$\\vbox{\n\\halign {\\hfil #\\hfil && \\quad \\hfil #\\hfil \\cr\n\\noalign{\\hrule \\medskip}\nLF Model & SF I & SF II & SF III & SF IV & SF V & SF VI & SF VII & SF VIII & SF\nIX & SF X &\\cr\n\\noalign{\\medskip \\hrule \\smallskip}\n\\cr\nB & 0.097 & 0.078 & 0.011 & 0.0069 & 0.014 & 0.012 & 0.039 & 0.025 & 0.0064\n& 0.0041 &\\cr\nF & 0.042 & 0.034 & 0.0046 & 0.0030 & 0.0060 & 0.0050 & 0.017 & 0.010 & 0.0027\n& 0.0018 &\\cr\nG & 0.045 & 0.037 & 0.0050 & 0.0032 & 0.017 & 0.0054 & 0.018 & 0.011 & 0.0030 &\n 0.0019 &\\cr\n\\noalign{\\medskip \\hrule}\n\\noalign{\\smallskip}\\cr}}$$}\n\\end{table*}\n \n\\begin{table*} \n\\caption{The present-day values of the density parameter in stars $\\Omega_{*}$ \nproduced by each well fitting \nluminosity function (LF) model, in each star-formation model.} \n{$$\\vbox{\n\\halign {\\hfil #\\hfil && \\quad \\hfil #\\hfil \\cr\n\\noalign{\\hrule \\medskip}\nLF Model & SF I & SF II & SF III & SF IV & SF V & SF VI & SF VII & SF VIII & SF\nIX & SF X &\\cr\n\\noalign{\\medskip \\hrule \\smallskip}\n\\cr\nB & 0.0010 & 0.00082 & 0.00012 & 7.3 $\\times$ 10$^{-5}$ & 0.00015 & 0.00013 &\n0.00041 & 0.00026 & 6.7 $\\times$ 10$^{-5}$ & 4.3 $\\times$ 10$^{-5}$ &\\cr\nF & 0.00080 & 0.00064 & 8.7 $\\times$ 10$^{-5}$ & 5.7 $\\times$ 10$^{-5}$ & \n0.00011\n& 9.5 $\\times$ 10$^{-5}$ & 0.00032 & 0.00019 & 5.1 $\\times$ 10$^{-5}$ & 3.4\n$\\times$ 10$^{-5}$ &\\cr\nG & 0.0019 & 0.0016 & 0.00021 & 0.00014 & 0.00072 & 0.00023 & 0.00077 & 0.00047 \n& 0.00013 & 8.0 $\\times$ 10$^{-5}$ &\\cr\n\\noalign{\\medskip \\hrule}\n\\noalign{\\smallskip}\\cr}}$$}\n\\end{table*}\n\n\\begin{table*} \n\\caption{Cosmic enrichment from each well fitting model. The results are \nin units of the solar metallicity, assumed to be 0.0189 (Anders\n\\& Grevesse 1989). The redshift range is specified in the luminosity function \nmodel: see Table\\,2. All the enrichment due to distant ULIRGs is \ncompleted by the lower limit to this redshift range.} \n{$$\\vbox{\n\\halign {\\hfil #\\hfil && \\quad \\hfil #\\hfil \\cr\n\\noalign{\\hrule \\medskip}\nLF Model & $z$ range & SF I & SF II & SF III & SF IV & SF V & SF VI &\nSF VII & SF VIII & SF IX & SF X &\\cr\n\\noalign{\\medskip \\hrule \\smallskip}\n\\cr\nB & $30$, there exist two positive constants $\\kappa_{T,1,H}\\leq \\kappa_{T,2,H}$ depending only on $T$ and $H$, such that for any $0\\leq s0$ and the multi-index $\\mathbf{k}=(k_1,\\cdots, k_d)$ with all $k_i$ being nonnegative integers, let\n\\[\np^{(\\mathbf{k})}_{\\varepsilon}(x)=\\frac{\\partial^\\mathbf{k}}{\\partial x^{k_1}_1\\cdots \\partial x^{k_d}_d}p_{\\varepsilon}(x) =\\frac{\\iota^{|\\mathbf{k}|}}{(2\\pi)^d}\\int_{{\\mathbb R}^d} \\Big(\\prod^d_{i=1}y^{k_i}_i\\Big)\\, e^{\\iota y\\cdot x}e^{-\\frac{\\varepsilon|y|^2}{2}}\\, dy,\n\\]\nwhere $p_{\\varepsilon}(x)=\\frac{1}{(2\\pi \\varepsilon)^{\\frac{d}{2}}} e^{-\\frac{|x|^2}{2\\varepsilon}}$ and $|\\mathbf{k}|=\\sum\\limits^d_{i=1}k_i$.\n\nFor any $T>0$ and $x\\in{\\mathbb R}^d$, if\n\\begin{align}\\label{epsilon}\nL^{(\\mathbf{k})}_{\\varepsilon}(T,x):=\\int^T_0\\int^T_0 p^{(\\mathbf{k})}_{\\varepsilon}(X^{H_1}_t-\\widetilde{X}^{H_2}_s+x)\\, ds\\, dt\n\\end{align}\nconverges to some random variable in $L^2$ when $\\varepsilon\\downarrow 0$, we denote the limit by $L^{(\\mathbf{k})}(T,x)$ and call it the $\\mathbf{k}$-th derivative of local time for the $(2,d)$-Gaussian field $Z$. If it exists, $L^{(\\mathbf{k})}(T,x)$ admits the following $L^2$-representation\n\\begin{align} \\label{dlt}\nL^{(\\mathbf{k})}(T,x)=\\int^T_0\\int^T_0 \\delta^{({\\bf k})}(X^{H_1}_t-\\widetilde{X}^{H_2}_s+x)\\, ds\\, dt.\n\\end{align}\n\nThe following are main results of this paper.\n\\begin{theorem}\\label{thm1}\nAssume that $X^{H_1}=\\{X^{H_1}_t:\\, t\\geq 0\\}$ and $\\widetilde{X}^{H_2}=\\{\\widetilde{X}^{H_2}_t:\\, t\\geq 0\\}$ are two independent Gaussian processes in $G^d_{1,2}$ with parameters $H_1, H_2\\in(0,1)$, respectively. For any $x\\neq 0$, if $\\frac{H_1H_2}{H_1+H_2}(2|\\mathbf{k}|+d)\\geq 1$, then there exist positive constants $c_1$ and $c_2$ such that\n\\begin{align*}\n\\liminf_{\\varepsilon\\downarrow 0}\\frac{{{\\mathbb E}\\,}[|L^{(\\mathbf{k})}_{\\varepsilon}(T,x)|^2]}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\\geq c_1e^{-c_2|x|^2},\n\\end{align*}\nwhere\n\\begin{align} \\label{rate}\nh^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)=\n\\left\\{\\begin{array}{ll}\n\\varepsilon^{\\frac{H_1+H_2}{2H_1H_2}-\\frac{d}{2}-|\\mathbf{k}|} & \\text{if}\\; \\frac{H_1H_2}{H_1+H_2}(2|\\mathbf{k}|+d)>1\\\\ \\\\\n\\ln(1+\\varepsilon^{-\\frac{1}{2}}) & \\text{if}\\; \\frac{H_1H_2}{H_1+H_2}(2|\\mathbf{k}|+d)=1.\n\\end{array} \\right.\n\\end{align}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm2}\nAssume that $X^{H_1}=\\{X^{H_1}_t:\\, t\\geq 0\\}$ and $\\widetilde{X}^{H_2}=\\{\\widetilde{X}^{H_2}_t:\\, t\\geq 0\\}$ are two independent Gaussian processes in $G^d_{1}$ with parameters $H_1, H_2\\in(0,1)$, respectively. We further assume that (i) $|\\mathbf{k}|$ is even or (ii) ${{\\mathbb E}\\,}[X^{H_1,1}_{t}X^{H_1,1}_{s}]\\geq 0$ and ${{\\mathbb E}\\,}[\\widetilde{X}^{H_2,1}_{t}\\widetilde{X}^{H_2,1}_{s}]\\geq 0$ for any $0\\sum\\limits^N_{j=1} \\frac{1}{H_j}\\\\\n\\ln(1+\\varepsilon^{-\\frac{1}{2}}) & \\text{if}\\;\\; 2|\\mathbf{k}|+d=\\sum\\limits^N_{j=1} \\frac{1}{H_j}.\n\\end{array} \\right.$\n\n\\item[(ii)] Assume that $X^{j,H_j}_{t_j} (j=1,\\dots,N)$ are independent $d$-dimensional Gaussian processes in $G^d_{1}$, $|\\mathbf{k}|$ is even or all $X^{j,H_j}$ have nonnegative covariance functions, and $2|\\mathbf{k}|+d\\geq \\sum\\limits^N_{j=1} \\frac{1}{H_j}$. Then there exists a positive constant $c_6$ such that $\n\\liminf\\limits_{\\varepsilon\\downarrow 0}\\frac{{{\\mathbb E}\\,}[|L^{(\\mathbf{k})}_{N,\\varepsilon}(T,0)|^2]}{h^{d,|{\\bf k}|}_{H_1,H_2,\\dots, H_N}(\\varepsilon)}\\geq c_6.$\n\\end{enumerate}\nThe case $N=1$ is easy. It follows from simplified proofs of Theorems \\ref{thm1} and \\ref{thm2}. Moreover, our methodologies also work for derivatives of local times of L\\'{e}vy processes or fields.\n\\end{remark}\n\nAfter some preliminaries in Section 2, Sections 3 and 4 are devoted to the proofs of Theorems \\ref{thm1} and \\ref{thm2}, respectively. Throughout this paper, if not mentioned otherwise, the letter $c$, with or without a subscript, denotes a generic positive finite constant whose exact value may change from line to line. For any $x,y\\in{\\mathbb R}^d$, we use $x\\cdot y$ to denote the usual inner product and $|x|=(\\sum\\limits^d_{i=1}|x_i|^2)^{1\/2}$. Moreover, we use $\\iota$ to denote $\\sqrt{-1}$.\n\n\n\\section{Preliminaries}\n\nIn this section, we give three lemmas. The first two will be used in the proof of Theorem \\ref{thm1} and the last one in the proof of Theorem \\ref{thm2}.\n\\begin{lemma} \\label{lma1} Assume that $k\\in{\\mathbb N}\\cup\\{0\\}$ and $\\varepsilon>0$. Then, for any $a,b,c, x\\in{\\mathbb R}$ with $a>0$, $c>0$ and $\\Delta=c+\\varepsilon-\\frac{(b-\\varepsilon)^2}{a+2\\varepsilon}>0$, we have\n\\begin{align} \\label{integral}\n&\\frac{(-1)^k}{2\\pi}\\int_{{\\mathbb R}^2} \\exp\\Big\\{-\\frac{1}{2}(y_2^2a+2y_2 y_1b+y_1^2c)-\\frac{\\varepsilon}{2}((y_1-y_2)^2+y_2^2)+\\iota y_1x\\Big\\} y^{k}_2 (y_1-y_2)^{k} \\, dy \\nonumber \\\\\n&=\\sum^k_{\\ell=0}\\sum^{k+\\ell}_{m=0:\\text{even}}\\sum^{2k-m}_{n=0:\\text{even}} c_{k,\\ell,m,n} (a+2\\varepsilon)^{-\\frac{m+1}{2}} (\\frac{\\varepsilon-b}{a+2\\varepsilon})^{k+\\ell-m}\\Delta^{-(2k-m)-\\frac{1-n}{2}} x^{2k-m-n}e^{-\\frac{x^2}{2\\Delta}},\n\\end{align}\nwhere\n\\[\nc_{k,\\ell,m,n}=(-1)^{\\ell-\\frac{m+n}{2}} \\binom{k}{\\ell} \\binom{k+\\ell} {m}\\binom{2k-m} {n}(m-1)!! (n-1)!!\n\\]\nand we use the convention $0^0=1$ for the case $x=0\\in{\\mathbb R}$.\n\\end{lemma}\n\\begin{proof} Let $L$ be the left hand side of the equality \\eref{integral}. It is easy to show that\n\\begin{align*}\nL&=\\frac{(-1)^k}{2\\pi}\\int_{{\\mathbb R}^2} \\exp\\Big\\{-\\frac{1}{2}y_2^2(a+2\\varepsilon)-y_2 y_1(b-\\varepsilon)-\\frac{1}{2}y_1^2(c+\\varepsilon)+\\iota y_1 x\\Big\\} y^{k}_2 (y_1-y_2)^{k} \\, dy\\\\\n&=\\sum^k_{\\ell=0}\\frac{(-1)^{k+\\ell}}{2\\pi} \\binom{k}{\\ell}\\int_{{\\mathbb R}^2} \\exp\\Big\\{-\\frac{1}{2}y_2^2(a+2\\varepsilon)-y_2 y_1(b-\\varepsilon)-\\frac{1}{2}y_1^2(c+\\varepsilon)+\\iota y_1 x\\Big\\} y^{k+\\ell}_2 y_1^{k-\\ell} \\, dy\\\\\n&=\\sum^k_{\\ell=0}\\sum^{k+\\ell}_{m=0:\\, \\text{even}} \\frac{(-1)^{k+\\ell}}{\\sqrt{2\\pi}}\\binom{k}{\\ell} \\binom{k+\\ell} {m}(m-1)!! (a+2\\varepsilon)^{-k-\\ell-\\frac{1-m}{2}} (\\varepsilon-b)^{k+\\ell-m} \\int_{{\\mathbb R}} y_1^{2k-m}e^{-\\frac{1}{2} y_1^2\\Delta+\\iota y_1 x} \\, dy_1\\\\\n&=\\sum^k_{\\ell=0}\\sum^{k+\\ell}_{m=0:\\, \\text{even}}\\sum^{2k-m}_{n=0:\\, \\text{even}} c_{k,\\ell,m,n} (a+2\\varepsilon)^{-\\frac{m+1}{2}} (\\frac{\\varepsilon-b}{a+2\\varepsilon})^{k+\\ell-m}\\Delta^{-(2k-m)-\\frac{1-n}{2}} x^{2k-m-n}e^{-\\frac{x^2}{2\\Delta}}.\n\\end{align*}\n\\end{proof}\n\n\n\\begin{lemma} \\label{lma2} Assume that $k\\in{\\mathbb N}\\cup\\{0\\}$ and $\\varepsilon>0$. Then, for any $a_1,b_1,c_1, a_2,b_2,c_2, x\\in{\\mathbb R}$ with $a_1, a_2, c_1, c_2>0$ and $\\Delta'=c_1+c_2+a_2+2b_2+\\varepsilon-\\frac{(b_1-b_2-a_2-\\varepsilon)^2}{a_1+a_2+2\\varepsilon}>0$, we have\n\\begin{align*}\n&\\frac{(-1)^k}{2\\pi}\\int_{{\\mathbb R}^2} \\exp\\left\\{-\\frac{1}{2}\\big([y_2^2a_1+2y_2 y_1b_1+y_1^2c_1]+[(y_1-y_2)^2a_2+2(y_1-y_2)y_1b_2+y^2_1c_2]\\big)\\right\\}\\\\\n&\\qquad\\qquad\\times \\exp\\Big\\{-\\frac{\\varepsilon}{2}((y_1-y_2)^2+y_2^2)+\\iota y_1x\\Big\\} y^{k}_2 (y_1-y_2)^{k} \\, dy\\\\\n&=\\sum^k_{\\ell=0}\\sum^{k+\\ell}_{m=0:\\text{even}}\\sum^{2k-m}_{n=0:\\text{even}} c_{k,\\ell,m,n} (a_1+a_2+2\\varepsilon)^{-\\frac{m+1}{2}} (\\frac{\\varepsilon+b_2+a_2-b_1}{a_1+a_2+2\\varepsilon})^{k+\\ell-m}(\\Delta')^{-(2k-m)-\\frac{1-n}{2}} x^{2k-m-n}e^{-\\frac{x^2}{2\\Delta'}},\n\\end{align*}\nwhere\n\\[\nc_{k,\\ell,m,n}=(-1)^{\\ell-\\frac{m+n}{2}} \\binom{k}{\\ell} \\binom{k+\\ell} {m}\\binom{2k-m} {n}(m-1)!! (n-1)!!\n\\]\nand we use the convention $0^0=1$ for the case $x=0\\in{\\mathbb R}$.\n\\end{lemma}\n\\begin{proof} Note that the integral in the above statement can be written as\n\\begin{align*}\n\\int_{{\\mathbb R}^2} \\exp\\Big\\{-\\frac{1}{2}y_2^2(a_1+a_2+2\\varepsilon)-y_2 y_1(b_1-a_2-b_2-\\varepsilon)-\\frac{1}{2}y_1^2(c_1+c_2+a_2+2b_2+\\varepsilon)+\\iota y_1 x\\Big\\} y^{k}_2 (y_1-y_2)^{k} \\, dy.\n\\end{align*}\nThen the desired result follows from Lemma \\ref{lma1}.\n\\end{proof}\n\n\\begin{lemma} \\label{lma3} Assume that $k\\in{\\mathbb N}\\cup\\{0\\}$ and $\\varepsilon>0$. Then, for any $a,b,c\\in{\\mathbb R}$ with $a,c>0$ and $(a+\\varepsilon)(c+\\varepsilon)-b^2>0$, we have\n\\begin{align*}\n&\\frac{(-1)^k}{2\\pi}\\int_{{\\mathbb R}^2} \\exp\\Big\\{-\\frac{1}{2}(y_2^2a+2y_2 y_1b+y_1^2c)-\\frac{\\varepsilon}{2}(y_2^2+y_1^2)\\Big\\} y^{k}_2 y_1^{k} \\, dy\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad=\\sum^k_{\\ell=0, \\text{even}} \\frac{c_{k,\\ell} \\,b^{k-\\ell}}{((a+\\varepsilon)(c+\\varepsilon)-b^2)^{\\frac{2k-\\ell+1}{2}}},\n\\end{align*}\nwhere $c_{k,\\ell}=(\\ell-1)!! \\binom{k}{\\ell} (2k-\\ell-1)!!$.\n\\end{lemma}\n\\begin{proof} This follows from similar arguments as in the proof of Lemma \\ref{lma1}.\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm1}}\n\nIn this section, we give the proof of Theorem \\ref{thm1}.\n\n\\begin{proof} We divide the proof into several steps.\n\n\n\\noindent\n{\\bf Step 1.}\nRecall the definition of $L^{(\\mathbf{k})}_{\\varepsilon}(T,x)$ in (\\ref{epsilon}). Using Fourier transform,\n\\begin{align*}\nL^{(\\mathbf{k})}_{\\varepsilon}(T,x)\n&=\\frac{\\iota^{|\\mathbf{k}|}}{(2\\pi)^d} \\int^T_0\\int^T_0\\int_{{\\mathbb R}^d} e^{\\iota z\\cdot (X^{H_1}_u-\\widetilde{X}^{H_2}_v+x)}e^{-\\frac{\\varepsilon|z|^2}{2}} \\prod^d_{i=1}z^{k_i}_i\\, dz\\, du\\, dv.\n\\end{align*}\nHence\n\\begin{align*}\n{{\\mathbb E}\\,}[|L^{(\\mathbf{k})}_{\\varepsilon}(T,x)|^2]\n&=\\frac{(-1)^{|\\mathbf{k}|}}{(2\\pi)^{2d}}\\int_{[0,T]^4}\\int_{{\\mathbb R}^{2d}} e^{-\\frac{1}{2}\\big[{{\\mathbb E}\\,}(z_2\\cdot X^{H_1}_{t_2}+z_1\\cdot X^{H_1}_{t_1})^2+{{\\mathbb E}\\,}(z_2\\cdot \\widetilde{X}^{H_2}_{s_2}+z_1\\cdot \\widetilde{X}^{H_2}_{s_1})^2\\big]}\\\\\n&\\qquad\\qquad \\times e^{-\\frac{\\varepsilon}{2}(|z_2|^2+|z_1|^2)+\\iota(z_1+z_2)\\cdot x} \\prod\\limits^d_{i=1}z^{k_i}_{2,i} \\prod\\limits^d_{i=1}z^{k_i}_{1,i}\\, dz_2\\, dz_1\\, dt\\, ds,\n\\end{align*}\nwhere $z_1=(z_{1,1},\\cdots,z_{1,d})$ and $z_2=(z_{2,1},\\cdots,z_{2,d})$.\n\nFor $i=1,\\cdots, d$, we first introduce the following notations\n\\begin{align*}\nI_i(H,t_2,t_1,z_2,z_1)&=e^{-\\frac{1}{2} {{\\mathbb E}\\,}[z_{2,i}\\cdot (X^{H,i}_{t_2}-X^{H,i}_{t_1})+z_{1,i}\\cdot X^{H,i}_{t_1}]^2}\\\\\n\\widetilde{I}_i(H,t_2,t_1,z_2,z_1)&=e^{-\\frac{1}{2} {{\\mathbb E}\\,}[z_{2,i}\\cdot (\\widetilde{X}^{H,i}_{t_2}-\\widetilde{X}^{H,i}_{t_1})+z_{1,i}\\cdot \\widetilde{X}^{H,i}_{t_1}]^2}\\\\\nK_i(\\varepsilon,z_2, z_1)&=e^{-\\frac{\\varepsilon}{2}(z^2_{2,i}+z^2_{1,i})+\\iota(z_{1,i}+z_{2,i})x_i}z^{k_i}_{2,i}z^{k_i}_{1,i}.\n\\end{align*}\nThen we define\n\\begin{align*}\nF_1(t_2,t_1, s_2,s_1, x_i)&=\\frac{(-1)^{k_i}}{2\\pi}\\int_{{\\mathbb R}^2}I_i(H_1,t_2,t_1,z_2,z_1+z_2)\\widetilde{I}_i(H_2,s_2,s_1,z_2,z_1+z_2)K_i(\\varepsilon,z_2, z_1) \\, dz_{2,i}\\, dz_{1,i}\\\\\nF_2(t_2,t_1, s_2,s_1, x_i)&=\\frac{(-1)^{k_i}}{2\\pi}\\int_{{\\mathbb R}^2}I_i(H_1,t_2,t_1,z_2,z_1+z_2)\\widetilde{I}_i(H_2,s_2,s_1,z_1,z_1+z_2)K_i(\\varepsilon,z_2, z_1)\\, dz_{2,i}\\, dz_{1,i}.\n\\end{align*}\nNow we can obtain that\n\\begin{align} \\label{e1}\n{{\\mathbb E}\\,}\\Big[ |L^{(\\mathbf{k})}_{\\varepsilon}(T,x)|^2\\Big]&=\\frac{2}{(2\\pi)^d}\\left[\\int_{D}\\prod^d_{i=1}F_1(t_2,t_1, s_2,s_1, x_i)\\, dt\\, ds+\\int_{D}\\prod^d_{i=1}F_2(t_2,t_1, s_2,s_1, x_i)\\, dt\\, ds\\right] \\nonumber \\\\\n&=:\\frac{2}{(2\\pi)^d}(I_1(\\varepsilon)+I_2(\\varepsilon)),\n\\end{align}\nwhere $D=\\{00$ and\n\\begin{align*}\n\\Delta=\\frac{ac+a\\varepsilon+2c\\varepsilon+\\varepsilon^2-b^2+2b\\varepsilon}{a+2\\varepsilon}\\geq \\frac{c\\varepsilon+\\varepsilon^2}{a+2\\varepsilon}>0,\n\\end{align*}\nwhere we use $|b|\\leq \\sqrt{ac}\\leq \\frac{a+c}{2}$ in the first inequality.\n\nBy Lemma \\ref{lma1}, $F_1(t_2,t_1, s_2,s_1, x_i)$ equals\n\\begin{align*}\n\\sum^{k_i}_{\\ell=0}\\sum^{k_i+\\ell}_{m=0:\\text{even}}\\sum^{2k_i-m}_{n=0:\\text{even}} c_{k_i,\\ell,m,n} (a+2\\varepsilon)^{-\\frac{m+1}{2}} (\\frac{\\varepsilon-b}{a+2\\varepsilon})^{k_i+\\ell-m}\\Delta^{-(2k_i-m)-\\frac{1-n}{2}} x^{2k_i-m-n}_ie^{-\\frac{x^2_i}{2\\Delta}},\n\\end{align*}\nwhere $c_{k_i,\\ell,m,n}=(-1)^{\\ell-\\frac{m+n}{2}} \\binom{k_i}{\\ell} \\binom{k_i+\\ell} {m}\\binom{2k_i-m} {n}(m-1)!! (n-1)!!$.\n\n\nFor any $\\gamma>1$ and $(t_2,t_1,s_2,s_1)\\in D$, using the Cauchy-Schwartz inequality and properties {\\bf (P1)} and {\\bf (P2)}, we can show that\n\\begin{align*}\n\\big|{{\\mathbb E}\\,}[(X^{H_1,1}_{t_2}-X^{H_1,1}_{t_1})X^{H_1,1}_{t_1}]\\big|\\leq c_1\\big[(\\gamma^{H_1}+\\gamma^{-H_1})a+\\beta(\\gamma)a^{\\frac{1}{2}}\\big],\n\\end{align*}\nwhere $\\gamma^{H_1}a$ comes from the case $\\frac{1}{\\gamma}<\\frac{t_2-t_1}{t_1}<\\gamma$, $\\gamma^{-H_1}a$ from the case $\\frac{t_2-t_1}{t_1}\\geq \\gamma$, and $\\beta(\\gamma)a^{\\frac{1}{2}}$ from the case $\\frac{t_2-t_1}{t_1}\\leq \\frac{1}{\\gamma}$. Similarly,\n\\begin{align*}\n\\big|{{\\mathbb E}\\,}[(\\widetilde{X}^{H_2,1}_{s_2}-\\widetilde{X}^{H_2,1}_{s_1})\\widetilde{X}^{H_2,1}_{s_1}]\\big|\\leq c_2\\big[(\\gamma^{H_1}+\\gamma^{-H_1})a+\\beta(\\gamma)a^{\\frac{1}{2}}\\big].\n\\end{align*}\nHence\n\\begin{align*}\n\\Big|\\frac{\\varepsilon-b}{a+2\\varepsilon}\\Big|\n&\\leq \\frac{1}{2}+c_3\\frac{(\\gamma^{H_1}+\\gamma^{-H_1}+\\gamma^{H_2}+\\gamma^{-H_2})a+\\beta(\\gamma)a^{\\frac{1}{2}}}{a+2\\varepsilon}\\leq c_4\\Big(\\gamma^{H_1}+\\gamma^{H_2}+\\frac{\\beta(\\gamma)}{(a+2\\varepsilon)^{\\frac{1}{2}}}\\Big).\n\\end{align*}\n\nLet \\begin{align} \\label{dg}\nD_{\\gamma}=D\\cap\\Big\\{0<\\frac{t_2-t_1}{t_1}<\\frac{T\\wedge 1}{2\\gamma}, \\frac{T}{4}0$, the function $h(w)=\\frac{1}{w^{\\alpha}}e^{-\\frac{1}{w}}\\in(0,\\alpha^{\\alpha}e^{-\\alpha}]$ when $w\\in(0,+\\infty)$. Choosing $\\gamma$ large enough gives\n\\begin{align*}\n\\liminf_{\\varepsilon\\downarrow 0}\\frac{I_1(\\varepsilon)}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\n&\\geq \\liminf_{\\varepsilon\\downarrow 0} \\frac{(1-c_4\\beta(\\gamma))^d}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\\int_{D} (a+2\\varepsilon)^{-|{\\bf k}|-\\frac{d}{2}}\\, \\Delta^{-\\frac{d}{2}}e^{-\\frac{|x|^2}{2\\Delta}}dt\\, ds,\n\\end{align*}\nwhere $00$ and\n\\begin{align*}\n\\Delta'&\\geq \\frac{a_1e_2+a_1a_2+a_2e_1+2a_1b_2+2a_2b_1+2b_1b_2+\\varepsilon(e_1+e_2)+\\varepsilon^2}{a_1+a_2+2\\varepsilon}\\geq \\frac{\\varepsilon(e_1+e_2)+\\varepsilon^2}{a_1+a_2+2\\varepsilon}>0,\n\\end{align*}\nwhere we use $|b_1|\\leq \\sqrt{a_1e_1}\\leq \\frac{a_1+e_1}{2}$ and $|b_2|\\leq \\sqrt{a_2e_2}\\leq \\frac{a_2+e_2}{2}$ in the first two inequalities.\n\n\n\n\nBy Lemma \\ref{lma1}, $F_2(t_2,t_1, s_2,s_1, x_i)$ equals\n\\begin{align*}\n\\sum^{k_i}_{\\ell=0}\\sum^{k_i+\\ell}_{m=0:\\text{even}}\\sum^{2k_i-m}_{n=0:\\text{even}} c_{k_i,\\ell,m,n} (a_1+a_2+2\\varepsilon)^{-\\frac{m+1}{2}} (\\frac{\\varepsilon+b_2+a_2-b_1}{a_1+a_2+2\\varepsilon})^{k_i+\\ell-m}(\\Delta')^{-(2k_i-m)-\\frac{1-n}{2}} x^{2k_i-m-n}e^{-\\frac{x^2}{2\\Delta'}},\n\\end{align*}\nwhere $c_{k_i,\\ell,m,n}=(-1)^{\\ell-\\frac{m+n}{2}} \\binom{k_i}{\\ell} \\binom{k_i+\\ell} {m}\\binom{2k_i-m} {n}(m-1)!! (n-1)!!$.\n\nFor any $(t_2,t_1,s_2,s_1)\\in D$, using the Cauchy-Schwartz inequality and properties {\\bf (P1)} and {\\bf (P2)}, we can show that\n\\begin{align*}\n\\Big|\\frac{\\varepsilon+b_2+a_2-b_1}{a_1+a_2+2\\varepsilon}\\Big|\n&\\leq 1+\\frac{|b_2-b_1|}{a_1+a_2+2\\varepsilon}\\leq c_{10}\\Big(\\gamma^{H_1}+\\gamma^{H_2}+\\frac{\\beta(\\gamma)}{(a_1+a_2+2\\varepsilon)^{\\frac{1}{2}}}\\Big).\n\\end{align*}\nTherefore,\n\\begin{align*}\n\\liminf_{\\varepsilon\\downarrow 0}\\frac{I_2(\\varepsilon)}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\n&\\geq \\liminf_{\\varepsilon\\downarrow 0} \\frac{1}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\\int_{D} (a_1+a_2+2\\varepsilon)^{-|{\\bf k}|-\\frac{d}{2}}\\, \\Delta^{-\\frac{d}{2}}e^{-\\frac{|x|^2}{2\\Delta}}dt\\, ds\\\\\n&\\qquad\\qquad-c_{11}\\beta(\\gamma)\\limsup_{\\varepsilon\\downarrow 0} \\frac{1}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\\int_{D} (a_1+a_2+2\\varepsilon)^{-|{\\bf k}|-\\frac{d}{2}}\\, dt\\, ds\\\\\n&\\geq -c_{12}\\beta(\\gamma)\\limsup_{\\varepsilon\\downarrow 0} \\frac{1}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\\int_{D} ((t_2-t_1)^{2H_1}+(s_2-s_1)^{2H_2}+2\\varepsilon)^{-|{\\bf k}|-\\frac{d}{2}}\\,dt\\, ds\\\\\n&\\geq -c_{13}\\beta(\\gamma).\n\\end{align*}\nLetting $\\gamma\\uparrow+\\infty$ gives\n\\begin{align} \\label{e3}\n\\liminf_{\\varepsilon\\downarrow 0}\\frac{I_2(\\varepsilon)}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\\geq 0.\n\\end{align}\n\n\\noindent\n{\\bf Step 3.} Combining \\eref{e1}, \\eref{e2} and \\eref{e3} gives\n\\begin{align*}\n\\liminf_{\\varepsilon\\downarrow 0}\\frac{{{\\mathbb E}\\,}[|L^{(\\mathbf{k})}_{\\varepsilon}(T,x)|^2]}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\n&\\geq \\liminf_{\\varepsilon\\downarrow 0}\\frac{I_1(\\varepsilon)}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}+\\liminf_{\\varepsilon\\downarrow 0}\\frac{I_2(\\varepsilon)}{h^{d,|{\\bf k}|}_{H_1,H_2}(\\varepsilon)}\\geq c_{14}e^{-\\frac{|x|^2}{2c_{5}(T^{2H_1}+T^{2H_2})}}.\n\\end{align*}\nThis completes the proof.\n\\end{proof}\n\n\n\\section{Proof of Theorem \\ref{thm2}}\n\nIn this section, we give the proof of Theorem \\ref{thm2}.\n\n\\begin{proof}\nBy Lemma \\ref{lma3},\n\\begin{align*}\n{{\\mathbb E}\\,}[|L^{(\\mathbf{k})}_{\\varepsilon}(T,0)|^2]\n&=\\frac{(-1)^{|\\mathbf{k}|}}{(2\\pi)^{2d}}\\int_{[0,T]^4}\\int_{{\\mathbb R}^{2d}} e^{-\\frac{1}{2}\\big[{{\\mathbb E}\\,}(z_2\\cdot X^{H_1}_{t_2}+z_1\\cdot X^{H_1}_{t_1})^2+{{\\mathbb E}\\,}(z_2\\cdot \\widetilde{X}^{H_2}_{s_2}+z_1\\cdot \\widetilde{X}^{H_2}_{s_1})^2\\big]}\\\\\n&\\qquad\\qquad \\times e^{-\\frac{\\varepsilon}{2}(|z_2|^2+|z_1|^2)} \\prod\\limits^d_{i=1}z^{k_i}_{2,i} \\prod\\limits^d_{i=1}z^{k_i}_{1,i}\\, dz_2\\, dz_1\\, dt\\, ds\\\\\n&=\\frac{1}{(2\\pi)^d}\\int_{[0,T]^4} \\prod^d_{i=1}\\Big(\\sum^{k_i}_{\\ell=0, \\text{even}} \\frac{c_{k_i,\\ell} \\,b^{k_i-\\ell}}{((a+\\varepsilon)(c+\\varepsilon)-b^2)^{\\frac{2k_i-\\ell+1}{2}}}\n\\Big)dt\\, ds,\n\\end{align*}\nwhere $a={{\\mathbb E}\\,}[(X^{H_1,1}_{t_2})^2]+{{\\mathbb E}\\,}[(\\widetilde{X}^{H_2,1}_{s_2})^2]$, $b={{\\mathbb E}\\,}[X^{H_1,1}_{t_2}X^{H_1,1}_{t_1}]+{{\\mathbb E}\\,}[\\widetilde{X}^{H_2,1}_{s_2}\\widetilde{X}^{H_2,1}_{s_1}]$ and $c={{\\mathbb E}\\,}[(\\widetilde{X}^{H_2,1}_{s_1})^2]+{{\\mathbb E}\\,}[(X^{H_1,1}_{t_1})^2]$. According to the assumption (i) $|\\mathbf{k}|$ is even or (ii) ${{\\mathbb E}\\,}[X^{H_1,1}_{t}X^{H_1,1}_{s}]\\geq 0$ and ${{\\mathbb E}\\,}[\\widetilde{X}^{H_2,1}_{t}\\widetilde{X}^{H_2,1}_{s}]\\geq 0$ for any $05\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$. The novelty of our study over previous ones is two-fold. First, we include information about the structural distortion of each galaxy to trace merger remnants (besides considering morphology and star formation properties). And secondly, as most objects in their evolution towards the Red Sequence must have gone over nearby Green Valley locations transitorily (F07), we have analysed the red galaxies both lying on the Red Sequence and at close positions on the Green Valley. Therefore, red galaxies in the context of this paper include both the galaxies on the Red Sequence and at its neighbourhood. The galaxy classes resulting from the combination of morphological, structural, and star-formation activity properties allow us to trace the evolution of intermediate stages of major mergers and of their final remnants since $z\\sim 1.5$. Finally, the observed number density evolution experienced by each galaxy type is used to carry out a set of novel observational tests defined on the basis of the expectations of hierarchical models, which provide observationally and for the first time main evolutionary paths among the different red galaxy types that have occurred in the last $\\sim 9$\\,Gyr. \n\nThe paper is organized as follows. In \\S\\ref{Sec:sample}, we provide a brief description of the survey. Section \\S\\ref{Sec:RGSelection} is devoted to the definition of the mass-limited red galaxy sample. In \\S\\ref{Sec:classification}, we define the galaxy classes according to the global morphology, structural distortion level, and star formation enhancement of the red galaxies. In \\S\\ref{Sec:Error}, we comment on the sources of errors and uncertainties. Section \\S\\ref{Sec:Tests} presents three novel tests to check the existence of any evolutionary links between the different red galaxy types, based on the expectations of hierarchical models of galaxy formation. The results of the study are presented in \\S\\ref{Sec:Results}. In particular, the results of the three tests proposed for the hierarchical scenario of E-S0 formation can be found in \\S\\ref{Sec:TestsResults}. The discussion and the main conclusions of the study are finally exposed in \\S\\S\\ref{Sec:Discussion} and \\ref{Sec:Conclusions}, respectively. Magnitudes are provided in the Vega system throughout the paper. We assume the concordance cosmology \\citep[$\\Omega_\\mathrm{m} = 0.3$, $\\Omega_\\Lambda = 0.7$, and $H_0 = 70$\\,km s$^{-1}$ Mpc$^{-1}$, see][]{2007ApJS..170..377S}. \n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth,angle=0]{star_galaxy_separation.eps}\n\\caption{Color-color diagram for distinguishing stars from galaxies ($I-K_{s}$ vs.~$V-I$). \\emph{Dots}: Data from the original $K$-band selected catalogue. \\emph{Solid line}: Color cut defined to isolate stars (i.e., the sequence of data located below the line) from galaxies (data lying above it). \\emph{Circles}: Objects classified as \"stars\" or \"compact\" visually (see \\S\\ref{Sec:Morphology}). All objects identified as \"stars\" visually are located in the stellar region of this diagram. }\\label{Fig:star-galaxy}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth,angle=0]{zspecRainbow_vs_zpRainbow_GRs_mag.eps}\n\\caption{Spectroscopic redshifts vs. photometric redshifts for the red galaxies in the sample having spectroscopic confirmation in the DEEP2 catalog \\citep{2003SPIE.4834..161D,2007ApJ...660L...1D}. Bright and faint galaxies in each one of the three wide redshift bins under consideration in the study are plotted with different symbols. The typical photometric redshift uncertainty for the red galaxy sample is below $\\Delta(z) \/ (1+z) <0.03$ at all redshifts for both bright and faint sources (see the estimates in the figure).}\\label{Fig:zspec-zphot}\n\\end{figure}\n\n\\section{The sample}\n\\label{Sec:sample}\n\nWe have combined multiwavelength data from the Rainbow Extragalactic Database\\footnote{Rainbow Extragalactic Database:\\\\https:\/\/rainbowx.fis.ucm.es\/\\-Rainbow\\_Database\/\\-Home.html} \\citep{2011ApJS..193...13B,2011ApJS..193...30B} and the GOYA photometric survey\\footnote{GOYA project (Galaxy Origins and Young Assembly):\\\\ http:\/\/www.astro.ufl.edu\/GOYA\/home.html} \\citep[see][]{2002INGN....6...11B} over an area of $\\sim 155$\\,arcmin$^{2}$ in the Groth Strip \\citep[$\\alpha=14^h 16^m 38.8^s$ and $\\delta=52^o 16' 52''$, see][]{1994AAS...185.5309G,1999AJ....118...86R,2002ApJS..142....1S}. The Rainbow Extragalactic Database compiles multi-wavelength photometric data from the UV to the FIR (and, in particular, in Spitzer\/MIPS 24$\\mu m$ band) over this sky area, providing analysis of spectral energy distributions of nearly 80,000 IRAC 3.6$+$4.5 $\\mu m$ selected galaxies. This study considers the photometric redshifts available in the Rainbow Database, which have a typical photometric redshift accuracy of $<\\Delta z\/(1+z)> = 0.03$ \\citep{2011ApJS..193...30B}, as derived for the sources with spectroscopic redshifts available in the DEEP2 Galaxy Redshift Survey \\citep{2003SPIE.4834..161D,2007ApJ...660L...1D}. The GOYA Survey is a survey covering the Groth Strip compiling photometry in four optical bands ($U$, $B$, $F606W$, and $F814W$) and in two near infrared ones ($J$ and $K_{s}$) with visual classifications available, reaching similar depths to the Rainbow data in similar bands \\citep[$U\\sim 25$, $B\\sim 25.5$, $K\\sim 21$ mag, see][]{2003ApJ...595...71C,2006ApJ...639..644E,2007RMxAC..29..165A,2008A&A...488.1167D}.\n\nWe have performed the sample selection starting from a $K$-band selected catalog in the field, reaching a limiting magnitude for 50\\% detection efficiency at $K\\sim 21$\\,mag. Several cuts have been performed to the original catalogue to obtain a mass-limited red galaxy sample. Firstly, red galaxies are selected as detailed in \\S\\ref{Sec:RGSelection}. This selection determines the redshift interval of the study, as it is restricted to the redshifts where the obtained number of red galaxies is statistically significant (i.e., to $0.3\\lesssim z\\lesssim 1.5$, see \\S\\ref{Sec:ClassificationSED}). \n\nAccording to the $M_{K}$-$z$ distribution of the red galaxies sample, the faintest absolute magnitude for which the catalogue is complete in luminosity up to $z\\sim 1.5$ corresponds to $M_{K,\\mathrm{lim}}\\sim -24$\\,mag. According to the redshift evolution of the mass-to-light relation (assuming a Salpeter IMF) derived by \\citet{2007A&A...476..137A} for a sample of quiescent bright galaxies, a red passive galaxy with this $K$-band absolute magnitude at $z\\sim 1.5$ has a stellar mass of $M_{*,\\mathrm{lim}}\\sim 5\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$. Therefore, we have selected red galaxies with masses higher than $M_{*}\\sim 5\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at each $z$ just considering all galaxies with $M_{K,\\mathrm{cut}} (z) = -23.3 - 0.45 z$ to account for their luminosity evolution. This luminosity cut is very similar to the one obtained by \\citet{2007MNRAS.380..585C}. Analogously, the equivalent luminosity cut to obtain a complete red galaxy sample for $M_{*,\\mathrm{lim}}=10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ would be: $M_K = -24-0.45 z$ (we will use this selection for comparing our results with those reported in other studies, see \\S\\ref{Sec:RGSelection}). The detection efficiency in the $K$-band drops below 0.9 for $m_K >20.5$\\,mag \\citep{2003ApJ...595...71C}, so we have checked that all galaxies in our mass-limited red sample exhibit apparent magnitudes brighter than this limit. \n\nWe have used the color-color diagram shown in Fig.\\,\\ref{Fig:star-galaxy} to remove stars from the sample. It represents the $I-K_{s}$ vs.~$V-I$ distribution for all the sources in the mass-limited red galaxy sample. Stars typically exhibit NIR colors bluer than galaxies, so they populate the lower region in the diagram. Attending to this bimodality of star-galaxy colors, we have defined a color-color cut to isolate galaxies from stars (see the solid line in the figure). The marked points include the stars and compact objects in the sample. We have checked that the stars identified according to it include all the objects at lower redshifts that have been classified as \"stars\" in the morphological classification performed in \\S\\ref{Sec:Morphology} (they are marked in Fig.\\,\\ref{Fig:star-galaxy}). \nAs the number of compact objects found in the sample is not statistically significant (see \\S\\ref{Sec:Morphology}), they have been excluded from this study. From an initial sample of 1809 sources from the original $K$-band selected catalogue, we finally have a mass-limited red galaxy sample of 257 systems at $0.32.0$, unless there is a large population of dust-reddened star-forming objects at all redshifts at $0.410^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$, but for different mass-to-light transformation relations. Panel c presents the results obtained with the same selection criteria as panel b, but for red galaxies with $\\mathrm{M}_*>5\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$. All data plotted in the same panel use equivalent color and mass selection criteria. \\emph{Filled black circles}: Results for our $K$-band selected data for each selection. \\emph{Rest of symbols}: Results obtained by \\citet{2007A&A...476..137A}, \\citet{2007MNRAS.380..585C}, F07, \\citet{2009ApJ...694.1171T}, \\citet{2010ApJ...709..644I}, \\citet{2010MNRAS.405..100M}, and \\citet{2011ApJ...727...51N}. Plotted data assume $h\\equiv H_0\/100 = 0.7$ and a Salpeter initial mass function. Cosmic variance and sample incompleteness contribute to the large dispersion found among different studies at $z<0.4$.\n\\label{Fig:comparacion}}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{Comparison with other studies}\n\\label{Sec:Comparison}\n\nThe red galaxy selection made in this study cannot be directly compared to the red galaxy samples obtained by most studies in the literature because, first, we have included red galaxies adjacent to the Red Sequence to study objects at transitory stages of their evolution towards it (which is not usual, see \\S\\ref{Sec:introduction}), and secondly, we have estimated masses using the $\\mathrm{M}_*\/L_K$-z relation derived by \\citet{2007A&A...476..137A} for different redshifts, whereas most authors use the $\\mathrm{M}_*\/L_V$-color relation derived by \\citet{2001ApJ...550..212B} or an equivalent relation. Moreover, most studies report the number evolution of red galaxies for masses $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$, instead of for masses $\\mathrm{M}_*>5\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ (as our case). In order to check out our results, we have made alternative red galaxy selections for $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$, adopting the color cuts and\/or the mass-to-light relation used by other authors.\n\nThe three panels of Fig.\\,\\ref{Fig:comparacion} compare the redshift evolution of the number density of red galaxies derived from our data with the results of different authors, for analogous mass and color selections in each case. In panel a, we have assumed the $U-B$ color evolution derived by \\citet{2001ApJ...553...90V} to select red galaxies (following F07), and the masses are estimated using the $\\mathrm{M}_*\/L_V$-color relation by \\citeauthor[][]{2001ApJ...550..212B}. Only red galaxies with $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at each redshift are considered in this panel. Panel b of the figure also assumes the color cut by \\citeauthor{2001ApJ...553...90V} for selecting red galaxies, but the mass estimates assume the $\\mathrm{M}_*\/L_K$ relation by \\citeauthor{2007A&A...476..137A}, which includes evolutive corrections \\citep[it is equivalent to the one derived by][]{2007MNRAS.380..585C}. The number densities of red galaxies shown in panel b are also for galaxies with $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at each redshift. Finally, panel c of the figure use the same selection criteria as panel b, but the number densities of red galaxies have been computed for $\\mathrm{M}_*>5\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$. Note that the results of our color selection (including galaxies on the Red Sequence and at nearby locations) are not plotted in this figure (see \\S\\ref{Sec:Results} and Table\\,\\ref{Tab:densities}).\n\nThe data from \\citet{2007A&A...476..137A}, \\citet{2007MNRAS.380..585C}, and F07 have been obtained by integrating their red galaxy luminosity functions at each redshift for $\\log(\\mathcal{M}_*\/\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi)>11$. In this case, the absolute magnitudes have been transformed into stellar masses using the expression derived for local E-S0's by \\citet{2006A&A...453L..29C}, considering the L-evolution of red galaxies derived by F07 and the redshift evolution of the $(B-K)$ color expected for E-S0's \\citep[]{1998ApJ...501..578S,2003A&A...404..831D,2007A&A...476..137A}. AB magnitudes have been transformed to the Vega system in the $B$ and $K$ bands according to \\citet{2007AJ....133..734B} transformations and considering galaxy colors derived for E-S0's by \\citet{1995PASP..107..945F}.\n\nThe good agreement between our results and those from independent studies for similar selection criteria supports the reliability of our methodology and completeness of our nominal red sample (compare the black filled circles with the rest of studies in all panels of Fig.\\,\\ref{Fig:comparacion}). However, although our first data point in panels b and c in the figure is inside the cloud of points within errors, it does not follow the trend of the other authors. So, our data at this redshift are probably affected by volume and cosmic variance effects.\n\n\\section{Classification of red galaxy types}\n\\label{Sec:classification}\n\nThe hierarchical picture of galaxy formation predicts that massive E-S0's are the result of the most violent and massive merging histories in the Universe (see references in \\S1 in EM10). To test this scenario, we need to distinguish between galaxies undergoing a major merger and normal E-S0's (see \\S\\ref{Sec:Tests}). Normal relaxed galaxies and major mergers differ basically in their structural distortion level. Major mergers also exhibit different global morphology and star formation enhancement depending on the gas content of the progenitors and the evolutionary stage of the encounter.\n\nA gas-rich major merger is expected to turn into a dust-reddened star-forming disk with noticeable structural distortions at intermediate and advanced stages of the encounter, basically since the coalesce of the two galaxies into an unique galaxy body (this stage is known as the merging-nuclei phase). In earlier phases of the merger, the two galaxies can develop noticeable tidal tails and asymmetric structures, but the two bodies can still be distinguished and are not expected to suffer from enough dust reddening to lie nearby or on top of the Red Sequence. During the latest phase of the encounter (post-merger), the star formation is quenched and the remnant gets a more relaxed spheroidal structure until it transforms into a typical E-S0 \\citep[see][]{2008MNRAS.384..386C,2008MNRAS.391.1137L,2010MNRAS.404..590L,2010MNRAS.404..575L}. Intermediate-to-late stages of gas-poor major mergers present a distorted spheroidal morphology and negligible levels of star formation, thus being quite red too \\citep{2005AJ....130.2647V}. By the contrary, typical E-S0's present a spheroidal-dominated relaxed morphology, although they are also expected to be quite red due to their negligible star formation. Therefore, a gas-rich major merger is expected to be quite red from its merging-nuclei phase until its transformation into a typical E-S0, so we have traced major mergers once the two merging galaxies have merged into a unique remnant, because they are expected to be quite red in any case. Accounting for this, we have classified the galaxies in our red sample attending to their global morphology, structural distortion level, and star formation enhancement to distinguish among normal galaxies and intermediate-to-advanced phases of major mergers.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth,angle=0]{stamps2.eps}\n\\caption{False-color postage stamps of some red galaxies in our sample, obtained using the $V$ and $I$ bands. One example representative of each type is shown for each wide redshift bin used in the present study ($0.3100$, see][]{2000ApJ...529..886C,2003ApJS..147....1C,2009MNRAS.394.1956C}. But the galaxies in our red sample have $S\/N \\sim 40-50$ at most (as $I$-band magnitude errors are $\\sim 0.02-0.03$\\,mag typically), making our asymmetry estimates quite uncertain. Moreover, asymmetry indices are sensitive to environmental influences in the galaxy outskirts. This means that some galaxies identified as regulars according to our criteria (because they do not exhibit noticeable distortions in its whole body) may have a high asymmetry index because of tidal features in the outer parts. This obviously smudges the correlation between visual irregularity and computed asymmetries. \n\nWe have adapted the method by \\citet{1993ApJ...418...72Z} to quantify the irregularity level of galaxy morphology. These authors developed a procedure to classify an elliptical galaxy as regular or irregular, attending to the distortion level of their isophotes with respect to perfect ellipses. They fitted ellipses to the isophotes of each elliptical to obtain the radial profiles of the coefficients $a_3$, $b_3$, $a_4$, and $b_4$ of their Fourier expansion series. The peak value of each Fourier coefficient was identified along the radial profile. These authors considered the following criteria to distinguish among regular and irregular galaxies:\n\n\n\\begin{enumerate}\n \\item If all the peaks of the coefficients were small, this meant that the isophotes exhibited small deviations \n from perfect ellipses. Therefore, these galaxies could be considered as regulars.\n \\item If one of them was not small, then they differentiated between two possible cases:\n \\begin{itemize}\n \\item If the maximum value of the peaks of all coefficients did not correspond to $b_4$, then the isophotes deviated noticeably from ellipses, meaning that the galaxy was irregular.\n \\item If the peak of $b_4$ was the maximum among all the peaks, its classification depended on its trend with the radial position in the galaxy. If the profile of this coefficient changed from one type to another within 1.5 effective radius of the galaxy, then the galaxy was irregular. If not, it just implied that the galaxy was boxy or disky (depending on the sign of $b_4$), but the galaxy morphology could be considered as regular.\n \\end{itemize}\n\\end{enumerate}\n\nWe can adopt this method for our galaxies, as we are considering irregularities that must affect to the galaxy as a whole. We have limited the analysis in each red galaxy to the isophotes with a mean signal higher than 1.5 times the standard deviation of the sky signal per pixel. We have used the IRAF task \\texttt{ellipse} for fitting ellipses to the isophotes and for getting the third and fourth order coefficients of their Fourier expansion series. We have identified the peaks of each coefficient in the galaxy radial profile. \n\nThe task \\texttt{ellipse} uses a normalization to the surface brightness of the isophote that directly measure the deviations of the isophote from perfect ellipticity. According to de \\citet{1990AJ....100.1091P}, these deviations can be considered negligible if they are $< 5$\\%, a value that can be translated directly to a value of 0.05 in these coefficients. Therefore, we have adopted this limit as the critical value to distinguish between small and high values. We have defined the irregularity index $C_\\mathrm{irr,isoph}$ as the peak value of maximum absolute value among the peaks of the four Fourier coefficients. Therefore, according to \\citet{1993ApJ...418...72Z}, any galaxy that has $|C_\\mathrm{irr,isoph}|<0.05$ is regular. If not, it is irregular, except if $C_\\mathrm{irr,isoph}$ corresponds to the $b_4$ coefficient and this coefficient does not change between $|b_4|>0.5$ and $|b_4|<0.5$ values or viceversa along its radial profile. \n\nWe compare the results of the visual and quantitative classifications concerning the irregularity level of our red galaxies in Fig.\\,\\ref{Fig:IrregularClassification} for each wide redshift bin. The percentage of agreement between the visual and quantitative classifications into the regular type is $\\sim 77$\\% (decreasing from 83\\% to 70\\% from low to high redshifts). This percentage is slightly lower in the irregular type: $\\sim $66\\% (rising from 58\\% to 74\\% from low to high redshifts). The miss-classifications between both methods are $\\sim $23\\% for visually-identified regular galaxies (rising the confussion percentage from 17\\% to 30\\% from low to high z), and $\\sim $34\\% for visual irregulars (dropping from 42\\% at low z to 26\\% at high z). In general, both procedures coincide in $\\sim 78$\\% of the galaxy classifications at $0.3 1.5$ at all redshifts. This implies that the quantitative method is not biased towards more irregular types at high redshifts, as it is limited to the isophotes with enough $S\/N$ at all redshifts. Obviously, $C_\\mathrm{irr,isoph}$ is derived from an intrinsic physical region in the galaxies smaller at high redshift than at low redshift, just because of cosmological effects. But, as commented above, we consider as irregular galaxies only the stages of advanced major mergers, which imply a noticeable distortion level in its whole body. The effects of cosmological dimming and resolution loss on the classification are analysed in \\S\\ref{Sec:TestVisual2}.\n\nIn conclusion, this test proves the robustness of the visual classifications into regular and irregular types at all redshifts.\n\n\\subsubsection{Robustness of the observed morphology against cosmological effects}\n\\label{Sec:TestVisual2}\n\nIn order to find out how the loss of spatial resolution, the cosmological dimming, and the change of rest-frame band with redshift are affecting to our classification, we have simulated images of galaxies at different redshifts in the observed $I$-band. We have used \\texttt{COSMOPACK} \\citep{2003RMxAC..16..259B}, an IRAF package that transforms images of real galaxies to depict their appearance at a given redshift as observed with a given telescope, camera, and filter. The transformation includes K-corrections, change of observing band, repixelation to the scale of the observing system, convolution by the seeing, and noise from sky, detector, and dark current. \n\nStarting from the $I$-band image of a galaxy representative of one type at the lowest wide redshift bin ($0.311.1$ derived by \\citet{2011ApJ...730...61K}. \\emph{Yellow circles}: Redshift evolution of the SFR of galaxies with $10.8 < \\log (\\mathcal{M}_*\/\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi) < 11.1$ derived by the same authors. \\emph{Grey stars}: Galaxy data from \\citet{2005ApJ...630...82P} with $F(24\\mu m)> 85$\\,$\\mu$J. \\emph{Black stars}: Galaxy data from the same authors with $F(24\\mu m)< 85$\\,$\\mu$J. This limiting flux in 24\\,$\\mu$m naturally isolates galaxies with high (enhanced) SFR compared to the average value of the whole galaxy population at each redshift from those with low SFR, for the mass range considered here.} \n\\end{center}\n\\label{Fig:ksfr}\n\\end{figure}\n\n\\subsection{Classification according to star formation activity}\n\\label{Sec:ClassificationSED}\n\nA typical problem of studies based on red galaxy samples is to disentangle dust-reddened star-forming galaxies from quiescent ones \\citep[][]{2000MNRAS.317L..17P,2004ApJ...617..746D,2007ApJ...665..265F,2011MNRAS.412..591P}, because both galaxy populations are indistinguishable using only broad-band photometry at wavelengths $\\lesssim 10$\\,$\\mu$m \\citep{2006AJ....132.1405S}. In particular, the rest-frame $U-B$ color cannot differentiate between both red galaxy types adequately at $z>0.8$ (F07). Therefore, different selection techniques based on color indices, mid-IR data, or SED fitting have been developed to isolate both populations in red galaxy samples \\citep{1999ApJ...518..533L,2003A&A...401...73W,2006A&A...455..879Z,2009ApJ...691.1879W}.\n\nThe mid-IR emission and the SFR of a galaxy are known to be tightly correlated \\citep[][]{1998ARA&A..36..189K,2005ApJ...630...82P}. The higher sensitivities and spatial resolutions achieved by IR instruments in the last years have allowed the development of SFR indicators based on the emission of a galaxy in a single mid-IR band \\citep[]{2005ApJ...633..871C,2007ApJ...666..870C,2006ApJ...648..987P}. In particular, the 24\\,$\\mu$m band of the Multiband Imaging Photometer in the \\emph{Spitzer} Space Telescope (MIPS) is found to be a good tracer of the infrared emission coming from the dust heated by star-forming stellar populations \\citep{2006ApJ...650..835A}. \n\nIn this study, we have identified galaxies with noticeable star formation compared to the average SFR exhibited by the galaxies with similar masses at the same redshift, because this is an evidence of that mechanisms different to passive evolution are triggering it (such as tidal interactions, mergers, gas infall, or stripping). The SFR of a galaxy changes noticeably with redshift due to the natural evolution of their stellar populations. The specific SFR has decayed with cosmic time as $\\sim (1+z)^{n}$ since $z=3$, being $n=4.3$ for all galaxies and $n=3.5$ for star-forming sources \\citep{2011ApJ...730...61K}. This means that we must have into account the intrinsic rise of SFR with redshift to define when a galaxy is forming stars more efficiently than the average of the whole galaxy population at each redshift. Therefore, we have used {\\it Spitzer}\/MIPS 24~$\\mu$m data to discriminate between red galaxies with enhanced SFRs from those that have lower SFRs compared to the average SFR of the whole galaxy population at each redshift \\citep[see also][]{2009ApJ...691.1879W}. A limiting flux in 24\\,$\\mu$m of $\\sim 60$\\,$\\mu$Jy corresponds to the 5-$\\sigma$ detection level on our MIPS 24~$\\mu$m data \\citep[][]{2011ApJS..193...13B}.\n\nIn Fig.\\,\\ref{Fig:ksfr}, we plot the $z$-evolution of the average SFR of galaxy populations with different mass ranges overlapping with ours (which is $\\log (\\mathcal{M}_*\/\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi)>10.7$), as derived by \\citet{2011ApJ...730...61K} from a deep 3.6\\,$\\mu$m-selected sample in the COSMOS field. We have overplotted the location of the galaxies from \\citet{2005ApJ...630...82P} in the diagram, differentiating those with emission fluxes in MIPS 24~$\\mu$m above 85\\,$\\mu$J from those below it. Note that this limiting flux follows pretty well the redshift evolution of the average SFR of the whole galaxy population for our mass range, being a straightforward way for distinguishing galaxies with enhanced SFRs from those with low SFRs, compared to the average value of the galaxy population at each redshift. Therefore, we have considered red galaxies with a 24~$\\mu$m MIPS flux above $85$\\,$\\mu$Jy as systems with enhanced star formation compared to the global galaxy population at the same redshift (High Star-Forming galaxies or HSF, hereafter), whereas the galaxies with a 24~$\\mu$m flux below this limit are not substantially star-forming compared to it (so, they are Low Star-Forming ones or LSF, henceforth). \n\n\\begin{table}\n\\begin{minipage}{0.5\\textwidth}\n\\caption{Percentages of AGN contamination in the subsample of HSF red galaxies by morphological types.\\label{Tab:agns}}\n\\begin{center}\n\\begin{tabular}{lcccccccc}\n\\hline\n \\multirow{2}{*}{HSF} & \\multicolumn{2}{c}{$0.3 \\cdot (1+z_\\mathrm{phot}$), where $<\\Delta(z)\/(1+z)>$ is the average value obtained for this normalized dispersion at the redshift bin of the galaxy (see Fig.\\,\\ref{Fig:zspec-zphot}). Then, we have obtained the number densities corresponding to each simulated catalogue for each galaxy type and at each redshift bin, accounting for the different redshifts of each catalogue. The dispersion of the 100 values obtained for the number density at each redshift bin and galaxy type represents an estimate of the error associated to the photometric redshift uncertainties. \n\nStatistics of massive red galaxies can be dramatically affected by cosmic variance due to their high clustering \\citep{2004ApJ...600L.171S}. We have estimated cosmic variance using the model by \\citet{2011ApJ...731..113M}, which provides estimates of cosmic variance for a given galaxy population using predictions from cold dark matter theory and the galaxy bias. They have developed a simple recipe to compute cosmic variance for a survey as a function of the angular dimensions of the field and its geometry, the mean redshift and the width of the considered redshift interval, and the stellar mass of the galaxy population. We have considered the geometry and angular dimensions of our field, as well as the different redshift bins analysed in each case to estimate the cosmic variance. \\citeauthor{2011ApJ...731..113M} software provides these estimates in two mass ranges overlapping with ours: $10.5<\\log (\\mathcal{M}_*\/\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi)<11$ and $11<\\log (\\mathcal{M}_*\/\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi)<11.5$. Therefore, we have considered the mean cosmic variance of both mass ranges as a representative value of the cosmic variance of our mass-limited sample at each redshift bin. Cosmic variance depends on the redshift. At $0.310^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at $z=0$, just accounting for the effects of the major mergers strictly reported by observations since $z\\sim 1.2$ \\citep{2009ApJ...694..643L}. This model reproduces the observed evolution of the massive end of the galaxy luminosity function by color and morphological types. The evolutionary track described by F07 appears naturally in the model, as it considers the relative contribution of gas-poor and gas-rich mergers at each redshift reported by \\citet{2008ApJ...681..232L} and their different effects on galaxy evolution. \n\nThe advantage of this model is that its predictions are in excellent agreement with cosmological hierarchical models (despite being based on observational major merger fractions), reproducing observational data at the same time \\citep[see EM10;][]{2010arXiv1003.0686E}. Based on these predictions, we have defined some tests that observational data must fulfill if most massive E-S0's have really derived from major mergers occurred at relatively late epochs in the cosmic history. These predictions are the following ones:\n\n\\begin{enumerate}\n \\item Most present-day E-S0's with $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ are the result of at least one gas-rich major merger that place them on the Red Sequence since $z\\sim 1.2$.\n \\item In addition, $\\sim 75$\\% of the remnants resulting from these gas-rich events have been involved in a subsequent gas-poor major merger, occurred quite immediately. The remaining $\\sim 25$\\% have thus continued their evolution towards an E-S0 passing through a quiet post-merger phase. \n \\item The bulk of these major mergers are at intermediate-to-late stages during the $\\sim 2$\\,Gyr period elapsed at $0.710^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at $z=0$ must have been definitely built up during the $\\sim 2.2$\\,Gyr time period elapsed at $0.75\\times10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at each redshift. This means that we can ensure that we are in a position to trace back in time the potential progenitors of the present-day E-S0's with $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at $z\\sim 0$ that could have merged to create them during the last $\\sim 9$\\,Gyr (see \\S\\ref{Sec:sample}).\n\nTherefore, we have estimated the cumulative distribution of ID's and of IS's, and we have compared them with the redshift evolution of the number density of RS's since $z\\sim 1.5$. In the case that major mergers have not driven the assembly of the massive E-S0's as proposed by the EM10 model, the previous three tests must fail. On the contrary, if these three distributions agree pretty well, these tests will support strongly the existence of an evolutionary link between major mergers and the appearance of massive E-S0's, as expected by hierarchical scenarios of galaxy formation. The results of these tests are presented in \\S\\ref{Sec:TestsResults}. \n\nWe must remark that the EM10 model exclusively quantifies the effects of major mergers on galaxy evolution at $z<1.2$. Hence, it does not discard the contribution of different evolutionary processes to the definitive assembly of massive E-S0's, although it predicts that it must have been low, as most of their number density evolution can be explained just accounting for the effects of the major mergers. This seems to be confirmed by observations, as other evolutionary mechanisms (such as minor mergers, ram pressure stripping, or bars) seem to have been significant for the formation of the Red Sequence only for low and intermediate masses, but not for high masses \\citep[][]{2004ARA&A..42..603K,2007ApJ...660.1151D,2008A&A...489.1003D,2009ApJ...697.1369B,2009A&A...508.1141S,2010MNRAS.409..346C,2011MNRAS.411.2148K}. Moreover, the model does not exclude disk rebuilding after the major merger either. On the contrary, it is probably required for giving rise to a S0 instead of an elliptical, as indicated by observations \\citep{2005A&A...430..115H,2009A&A...507.1313H,2009A&A...496..381H,2009A&A...501..437Y}.\n\nMoreover, the EM10 model assumes that intermediate-to-late stages of major mergers are red and will produce an E-S0, on the basis of many observational and computational studies \\citep[see references in EM10 and][]{2002A&A...381L..68C,2007MNRAS.382.1415S,2010ApJ...714L.108S,2010A&A...518A..61C}. These assumptions are crucial for the model, as they are necessary to reproduce the redshift evolution of the luminosity functions selected by color and morphological type. Therefore, testing if our data is coherent with the existence of an evolutionary link between the advanced stages of major mergers in our red sample and the definitive buildup of massive E-S0's, we are also indirectly testing these assumptions of the EM10 model.\n\n\n\\subsection{Observational considerations for the tests}\n\\label{Sec:RSEvolution}\n\nAccording to \\citet{2006ApJ...652..270B}, the average number density of major mergers at a given redshift centered at $z$, $n _\\mathrm{m}(z)$, is related to the number density of the major mergers detected at certain intermediate phase of the encounter in that redshift bin, $$, as follows:\n\n\\begin{equation}\\label{eq:density}\nn_\\mathrm{m}(z) = \\frac{t([z_1,z_2])}{\\tau _\\mathrm{det}},\n\\end{equation}\n\n\\noindent with $t([z_1,z_2])$ being the time elapsed in the redshift bin, and $\\tau _\\mathrm{det}$ representing the detectability time of the intermediate merger stage under consideration. We have used this equation in our tests.\n\nWe find that the number densities of ID's and IS's remain quite constant with redshift (see \\S\\ref{Sec:Results}). As ID's and IS's correspond to intermediate merger stages, this means that the major merger rate must evolve smoothly with redshift, in good agreement with observational estimates of merger rates \\citep{2008ApJ...681..232L,2011ApJ...739...24B,2011ApJ...742..103L}. This also indicates that the net flux of irregular galaxies appearing on the Red Sequence or at nearby locations on the Green Valley (i.e., the number of red irregulars created and destroyed per unit time in the red sample) must have been nearly constant at $0.35\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$, according to their morphology and structural distortion level, for the three wide redshift bins considered in the study ($0.35\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ per redshift bins at $0.35\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at $0.3 10^ {-4}$\\,Mpc$^{-3}$. Although the fractions of red irregular spheroids and disks decrease with cosmic time, their densities remain quite constant at $0.35\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at $0.35\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$. \n\nAs commented in \\S\\ref{Sec:introduction}, the rise of the number density of red regular types with cosmic time and the constancy of that of irregular ones has been interpreted as a sign pointing to the conversion of irregulars into regulars with time. We provide observational evidence supporting the existence of this evolutionary link in \\S\\ref{Sec:TestsResults}. \n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics*[width=0.24\\textwidth,angle=0]{mip_RS_v2.eps}\n\\includegraphics*[width=0.24\\textwidth,angle=0]{mip_RD_v2.eps}\n\\includegraphics*[width=0.24\\textwidth,angle=0]{mip_IS_v2.eps}\n\\includegraphics*[width=0.24\\textwidth,angle=0]{mip_ID_v2.eps}\n\\caption{Redshift evolution of the comoving number density of red galaxies with $\\mathrm{M}_*>5\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$, according to their morphological, structural, and star formation activity properties. \\emph{Solid lines}: Number densities for LSF galaxies of each type. \\emph{Dashed lines}: Number densities of HSF galaxies of each morphological type. \\emph{Panel a)}: Regular spheroids (RS's). \\emph{Panel b)}: Regular disks (RD's). \\emph{Panel c)}: Irregular spheroids (IS's). \\emph{Panel d)}: Irregular disks (ID's). }\\label{Fig:mip_p_classification}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Star formation activity according to red galaxy types since $z\\sim 1.5$}\n\\label{Sec:SFandMorphology}\n\nIn Fig.\\,\\ref{Fig:mip_p_classification} we show the redshift evolution of the number density of massive red galaxies for each morphological type defined in this study, attending to its star formation activity. We remark that HSF's are galaxies that show enhanced SFRs compared to the average SFR of the galaxy population at each redshift (\\S\\ref{Sec:ClassificationSED}). All types, except RS's, host a significant number of HSF galaxies, with percentages varying depending on type and redshift. Red regular spheroids are the galaxy types hosting the lowest fractions of HSF systems at all redshifts since $z\\sim 1.5$, as expected. Curiously, red RD's exhibit a noticeable increase of the HSF systems fraction at $z<0.7$ (panel b). Irregular types harbour enhanced SFRs typically (both spheroidal and disk systems), coherently with their merger-related nature (panels c and d in the figure). The percentage of HSF objects in these types has not changed at $0.35\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$), and for an alternative one which is identical to our nominal one in all aspects, except in the mass range ($\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$).\n\nThe EM10 model traces back in time the evolution of the E-S0's that have $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at $z=0$ (\\S\\ref{Sec:F07}). According to the model, these galaxies have been mostly assembled through major mergers at $0.70.7$, which have masses lower by a factor of $\\sim 2$ compared to the E-S0's resulting from the merger. Consequently, the model predictions on the number density of E-S0's at $z>0.9$ can be compared with our results for $\\mathrm{M}_*\\gtrsim 5\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ (as both studies trace similar mass ranges globally). But at lower redshifts, the EM10 model traces E-S0's that already have $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$, so it is comparable to the results with a mass selection $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$. \n\nAs Fig.\\,\\ref{Fig:acumIDSF} c shows, at $z>0.7$, the model reproduces much better the settlement of the RS's with $\\mathrm{M}_*\\gtrsim 5\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$, whereas at $z<0.7$ it clearly follows the trend of RS's with $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$, as expected from the arguments given above. However, our data seem to have completeness problems at $z<0.5$, so we cannot assure that the number density of E-S0's with $\\mathrm{M}_*> 10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ has remained constant since $z\\sim 0.7$. \n\nNevertheless, the number density of E-S0's with $\\mathrm{M}_*>10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ at $z\\sim 0.6$ estimated with our data is\nquite similar to the one estimated for these galaxies at $z=0$ \\citep{2003ApJ...599..997M}, as shown in panel c of Fig.\\,\\ref{Fig:acumIDSF}. This means that the number density of these objects has remained nearly constant since $z\\sim 0.6$, in good agreement with the predictions of the EM10 model.\n\nAlthough better data at $z<0.6$ are required to directly confirm this result, our data are coherent with the fact that E-S0's with $\\mathrm{M}_*> 10^{11}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$ have been definitively assembled since $z\\sim 0.6$, supporting that the bulk of the assembly of red massive E-S0's has occurred during the $\\sim 2.2$\\,Gyr period elapsed at $0.7 11$ seem to have been in place since $z\\sim 0.6$ (see \\S\\ref{Sec:TestsResults}). This is supported by the fact that many of these processes are observed to be relevant only for the evolution of galaxies with lower masses \\citep[in particular, the fading of stellar populations, see]{2002ApJ...577..651B,1998ApJ...496L..93D,2006A&A...458..101A}. \n\nWe must also remark that our red galaxy sample does not trace the evolution of S0's observed in the clusters, which seems to have been relevant since $z\\sim 0.4$-0.5 due to the environmental star-formation quenching of the spirals that fall into the clusters \\citep{2007ApJ...660.1151D,2009ApJ...697L.137P,2009A&A...508.1141S,2011MNRAS.412..246V}. Cluster S0's have typical masses lower than those selected here \\citep[$\\mathrm{M}_*\\lesssim 5\\times 10^{10}\\ifmmode{\\mathrm M_\\odot}\\else{M$_\\odot$}\\fi$, see][]{1999ApJS..122...51D,2006MNRAS.373.1125B}, and hence, our results do not apply to them in general. Moreover, some studies indicate that the fraction of S0's in groups is similar to that of clusters at $z < 0.5$ \\citep{2009ApJ...692..298W}. Considering that most galaxies reside in groups \\citep[$\\sim 70$\\%, see][]{2006ApJS..167....1B,2007ApJ...655..790C}, this means that the majority of S0's in the Universe are located in groups (not in clusters). Nevertheless, note that this evolution in clusters does not contradict the EM10 model at all, because this model exclusively analyses the effects of the major mergers on galaxy evolution since $z\\sim 1.2$, independently on the relevance of other evolutionary processes. \n\nTo summarize, our study supports that major mergers have been the main drivers of the evolution of the massive end of the Red Sequence since $z\\sim 1.5$, although other processes can also have contributed to it significantly at intermediate-to-low masses (especially, since $z\\sim 0.6$). Our tests support observationally a late definitive buildup of the massive E-S0's through major mergers (mostly at $0.6$ 1 MeV.}}\n\\label{fig:Nhits}\n\\end{figure}\nFigure~\\ref{fig:Nhits} shows the distribution of the number of hits in the drift chamber ( $N_{\\rm Hits}$ ) obtained for events with detected energy in the LYSO crystals $E_{\\rm LYSO} > $ 1 MeV. Events with $N_{\\rm Hits} >$ 9 are selected for the analysis of the charged secondary particles.\n\n\\noindent\nIn order to evaluate the setup acceptance and efficiency, and to optimize the particle identification analysis a detailed simulation has been developed using the FLUKA software release 2011.2~\\cite{Fasso'2003,Ferrari2005}. The detailed geometry description with the setup materials (air included) together with the trigger logic, the time resolution of the scintillator as well as the experimental space resolution of the drift chamber have been considered. The quenching effect in the scintillator has also been introduced in the Monte Carlo according to \\cite{Koba2011}.\nThe interaction of a sample of $10^9$ carbon ions with 80 MeV\/u, equivalent to $10^3$ s of data taking at the typical 1 MHz rate of beam, has been simulated. To identify the charged particles reconstructed in the drift chamber, we exploit the distribution of the detected energy in the LYSO detector $E_{\\rm LYSO}$ as a function of Time of Flight (ToF), Figure~\\ref{fig:Etof}. \nIn the data sample (left panel) a fast low-energy component due to electrons\nis clearly visible for ToF values around zero, in the area delimited by the first dashed line. These electrons are produced by Compton scattering of the de-excitation photon induced by beam interactions in the PMMA material. \nThe central most populated band, delimited by the two dashed lines, is made by protons with detected energy within a very wide range, originating also the clearly visible saturation of the LYSO crystals QDC for $E_{\\rm LYSO} >$ 24 MeV. The FLUKA simulation (right panel) shows similar populations in the (ToF , $E_{\\rm LYSO}$) plane with an additional component of deuterons, above the second dashed line, which is not present in data.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics [width = 1\\textwidth] {Fig4ab.pdf}\n\\caption{\\small{ Distribution of the detected energy in the LYSO crystals as a function of the Time of Flight: Data (Left) and FLUKA Simulation (Right).\n}}\n\\label{fig:Etof}\n\\end{center}\n\\end{figure}\n\\noindent\nWe have then identified as proton a charged secondary particle with ToF and $E_{\\rm LYSO}$ values inside the area delimited by the two dashed lines in Figure~\\ref{fig:Etof}. \nThe systematic uncertainty on the proton\/deuteron identification has been estimated using the data events in the deuterons area of the (ToF , $E_{\\rm LYSO}$) plane.\nFigure~\\ref{fig:beta} shows the distributions of $\\beta = \\frac{v}{c}$ and the corresponding detected kinetic energy $E_{\\rm kin}$ for the identified protons, obtained using the ToF measurement together with the distance between LYSO crystals and PMMA.\nThis detected kinetic energy can be related to the proton kinetic energy at emission time, $E^{\\rm Prod}_{\\rm kin}$, considering the energy loss in the PMMA and the quenching effect of the scintillating light for low energy protons. The minimum required energy to detect a proton in the LYSO crystals is $E^{\\rm Prod}_{\\rm kin} = 7.0\\pm 0.5$ MeV, evaluated using the FLUKA simulation, and a proton with an average detected kinetic energy $E_{\\rm kin}$ = 60 MeV has been emitted with $E_{\\rm kin}^{\\rm Prod} 83\\pm 5$ MeV. The uncertainty is mainly due to the finite size of both the beam spot $\\mathcal{O}$(1 cm) and profile.\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics [width = \\textwidth] {Fig5ab.pdf}\n\\caption{\\small{Distribution of $\\beta = \\frac{v}{c}$ (left) and kinetic energy (right) of charged secondary particles identified as protons.}}\n\\label{fig:beta}\n\\end{center}\n\\end{figure}\n\n\\noindent\nIn order to use the secondary protons for monitoring purposes, the crossing of some centimeters of patient's tissue has to be considered and therefore the range $E_{\\rm kin} > $ 60 MeV of the detected kinetic energy distribution is the most interesting for the above-mentioned application.\nIn the following the proton kinetic energy detected in the LYSO crystals will be referred to as the kinetic energy.\n\\section{Production region of charged secondary particles}\n\\label{region}\nTracks reconstructed in the drift chamber are backward extrapolated to the PMMA position, to find the production region of charged secondary particles along the path of the carbon ion beam. The PMMA is mounted on a single axis movement stage allowing position scans along the x-axis to be performed with a 0.2 mm accuracy (Figure~\\ref{fig:Schema}). In the configuration with the centers of PMMA, drift chamber and LYSO crystals aligned along the z-axis, the PMMA position in the stage reference frame is taken as 0 and will be referred to as the reference configuration. \n\n\\noindent From each track reconstructed in the drift chamber and backward extrapolated to the beam axis we can measure the x and y coordinates of the estimated emission point of the charged secondary particle, named $x_{\\rm PMMA}$ and $y_{\\rm PMMA}$. The expected position of the Bragg peak obtained with the FLUKA simulation \\cite{Fasso'2003} is located at $(11.0 \\pm 0.5)$ mm from the beam entrance face of the PMMA. With the setup in the reference configuration, the expected position of the Bragg peak in our coordinate system is $x_{\\rm Bragg}|^{\\rm Ref} = (9.0 \\pm 0.5)$ mm. \nFigure~\\ref{fig:peak} shows the distribution of the reconstructed $x_{\\rm PMMA}$, compared to the expected distribution of the dose deposition in the PMMA, both obtained with the setup in the reference configuration. \n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics [width = 0.8\\textwidth] {Fig6.pdf}\n\\caption{\\small{Expected dose deposition in the PMMA evaluated with FLUKA (hatched) compared to the distribution of $x_{\\rm PMMA}$ (solid), the emission point of charged secondary particles along the x-axis. The beam entrance and exit faces of the PMMA are at $x_{\\rm PMMA}$ = 2 cm and $x_{\\rm PMMA}$ = -2 cm, respectively.}}\n\\label{fig:peak}\n\\end{center}\n\\end{figure}\nThe mean of the gaussian fit to the distribution is $ \\bar{x}_{\\rm PMMA} = 17.1\\pm0.2$ mm, and consequently the separation between the BP and the peak from secondary proton emission is $\\Delta_{\\rm ProtonBragg} = 8.1 \\pm 0.5$ mm.\n Figure~\\ref{fig:reso_ekin} shows the distribution of the reconstructed $x_{\\rm PMMA}$ and $y_{\\rm PMMA}$ for all identified protons (solid line), for protons with $E_{\\rm kin} > $ 60 MeV (hatched) and for protons with $E_{\\rm kin} >$ 100 MeV (grey). The beam entrance and exit faces of the PMMA are at $x_{\\rm PMMA}$ = 2 cm and $x_{\\rm PMMA}$ = -2 cm, and $y_{\\rm PMMA}$ = 1.6 cm and $y_{\\rm PMMA}$ = -2.4 cm. The $x_{\\rm PMMA}$ distribution is related to the range of the beam while the $y_{\\rm PMMA}$ to its transversal profile. \nQuite remarkably the shape of the distribution of the emission point is approximately the same for protons emitted with different kinetic energies, e.g. the resolution on $x_{\\rm PMMA}$ does not depend critically on the $E_{\\rm kin}$ variable.%\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics [width = 1 \\textwidth] {Fig7ab.pdf}\n\\caption{\\small{Distribution of $x_{\\rm PMMA}$ (Left) and $y_{\\rm PMMA}$ (Right) obtained for all charged particles identified as protons (black solid line), for protons with $E_{\\rm kin} > $ 60 MeV (dashed line) and with $E_{\\rm kin} > $ 100 MeV (grey). The beam entrance and exit faces of the PMMA are at $x_{\\rm PMMA}$ = 2 cm and $x_{\\rm PMMA}$ = -2 cm, and $y_{\\rm PMMA}$ = 1.6 cm and $y_{\\rm PMMA}$ = -2.4 cm.}}\n\\label{fig:reso_ekin}\n\\end{center}\n\\end{figure}\n\n\\noindent\nThe existence of a relationship between the expected BP position and the peak of the $x_{\\rm PMMA}$ distribution, as a function of the PMMA position, in principle could allow us to follow the BP position using the $x_{\\rm PMMA}$ measurements. To estimate the accuracy of this method\n, a position scan has been performed acquiring several data runs moving the PMMA by means of the translation stage.\n\n\\noindent\nFor each run with different PMMA position, the production region of the protons have been monitored using the mean values of the gaussian fit to $x_{\\rm PMMA}$ and $y_{\\rm PMMA}$ distributions, $\\bar{x}_{\\rm PMMA}$ and $\\bar{y}_{\\rm PMMA}$. \nSince $\\bar{y}_{\\rm PMMA}$ is the coordinate of the proton emission point along the vertical axis, and is related to the fixed beam profile in the transverse plane, its behaviour as a function of the PMMA position provides an estimate of the method's systematic uncertainty.\n\n\\noindent\nEach PMMA position in the stage reference frame can be translated into the expected Bragg peak position $x_{\\rm Bragg}$ for that given PMMA position.\nFigure~\\ref{fig:ADD} shows the results obtained for $\\bar{x}_{\\rm PMMA}$ and $\\bar{y}_{\\rm PMMA}$ as a function of $x_{\\rm Bragg}$, with $E_{\\rm kin} >$ 60 MeV protons. A clear linear relationship is observed between $\\bar{x}_{\\rm PMMA}$ and $x_{\\rm Bragg}$, indicating that the charged secondary particles emission reconstructed with the drift chamber follows accurately the BP movement.\nNo dependence of the $\\bar{y}_{\\rm PMMA}$ values on the Bragg peak position is observed, as expected from a translation of the PMMA along the x-axis only. \nSimilar results can be obtained using protons with different $E_{\\rm kin}$ selection, as it can be inferred from Figure~\\ref{fig:reso_ekin}.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics [width = 1 \\textwidth] {Fig8.pdf}\n\\caption{\\small{Reconstructed peak position of the secondary proton emission distribution $\\bar{x}_{\\rm PMMA}$,$\\bar{y}_{\\rm PMMA}$ as a function of the expected Bragg Peak position $x_{\\rm Bragg}$, with $E_{\\rm kin} >$ 60 MeV.}}\n\\label{fig:ADD}\n\\end{center}\n\\end{figure} \n\n\\noindent\nTo estimate the achievable accuracy on the BP determination several contributions need to be considered.\nWe evaluated the difference $\\Delta_{\\rm ProtonBragg} = \\bar{x}_{\\rm PMMA} - x_{\\rm Bragg}$ for all identified protons and for the proton sample with $E_{\\rm kin} >$ 60 MeV. The $\\Delta_{\\rm ProtonBragg}$ root mean square is $\\sigma_{\\rm \\Delta_{\\rm ProtonBragg}}\\simeq$ 0.9 mm for both samples.\nThis can be explained as follows: in the sample with all identified protons the contribution to the total uncertainty due to the scattering is partially compensated by the larger statistics with respect to the sample with $E_{\\rm kin} >$ 60 MeV. Table \\ref{TAB:NPStat} reports the number of identified protons with $E_{\\rm kin} >$ 60 MeV obtained with the position scan data.\n\\begin{table}[!hbt]\n\\caption{Statistics of identified protons with $E_{\\rm kin} >$ 60 MeV, obtained with the position scan data.}\n\\begin{center}\n\\begin{tabular}{cccccccccccc}\n\\hline\n\\bs\n$x_{\\rm Bragg}$ (mm) & -19 & -15 & -11 & -9 & -5 & -3 & 1 & 5& 9& 11 & 13 \\\\\n\\bs\n\\hline\n\\bs\n$N^{\\rm 60~MeV}_{\\rm Protons}$ & 67& 77& 88& 61& 92& 75& 113& 154& 1223& 130& 83 \\\\\n\\bs\n\\hline\n\\end{tabular}\n\\label{TAB:NPStat}\n\\end{center}\n\\end{table}\n\n\\noindent\nThe uncertainty $\\sigma_{\\rm Extrapol}$ due to the backward extrapolation of the track from the drift chamber to the beam line can be estimated from the root mean square of the $\\bar{y}_{\\rm PMMA}$ values, $\\sigma_{\\bar{y}_{\\rm PMMA}} = \\sigma_{\\rm Extrapol}$ = 0.5 mm. The latter contributes to the $\\Delta_{\\rm ProtonBragg}$ distribution, together with $\\sigma_{\\rm Stage}$ = 0.2 mm from the uncertainty on the PMMA positioning. \nWe can then estimate the contribution to the total uncertainty coming from the shape of the distribution of the emission point of charged secondary particles as: \n\\begin{equation}\n\\sigma_{\\rm Emission} = \\sqrt{\\sigma_{\\rm \\Delta_{\\rm ProtonBragg}}^2 - \\sigma_{\\rm Extrapol}^2 - \\sigma_{\\rm Stage}^2} \\sim 0.7 \\text{mm}\n\\end{equation}\n\n\\noindent It must be stressed that this value represents only an indication of the precision achievable in the BP determination using secondary protons, due to the target thickness and homogeneity in the present setup, with respect to a possible clinical application.\n\\section{Flux of charged secondary particles}\n\\label{flux}\nThe flux of the secondary protons emitted from the beam interaction with the PMMA has been measured at 90$\\degree$ with respect to the beam direction and in the geometrical acceptance of the triggering LYSO crystals, configuration maximizing the sensitivity to the Bragg peak position. The surface of the LYSO is 3x3 cm$^2$, corresponding to a solid angle $\\Omega_{\\rm LYSO} = 1.3\\times 10^{-4}$ sr at a distance of 74 cm. The proton's kinetic energy spectrum measured with data has been inserted in the FLUKA simulation to evaluate the detection efficiency in the LYSO crystals for protons with $E_{\\rm LYSO} >$ 1 MeV: $\\epsilon_{\\rm LYSO}= (98.5\\pm 1.5)\\%$, with the uncertainty mainly due to the Monte Carlo statistics. To properly evaluate the rate of charged secondary particles reaching the LYSO crystals, the number of carbon ions reaching the PMMA target ($N_{\\rm C}$) has been computed according to \\cite{Agodi2012a}: counting the number of signals in the Start Counter ($N_{\\rm SC}$) within randomly-triggered time-windows of $T_w=2\\ \\micro\\second$, corrected for the Start Counter efficiency $\\epsilon_{\\rm SC} = (96 \\pm 1)\\%$, and the acquisition dead time. \nThe number of emitted secondary protons $N_{\\rm P}$ has been measured with the $x_{\\rm PMMA}$ distribution counts, corrected for $\\epsilon_{\\rm SC}$, $\\epsilon_{\\rm LYSO}$, the tracking efficiency $\\epsilon_{\\rm Track} = (98 \\pm 1)\\%$~\\cite{Abou-Haidar2012} and the acquisition dead time.\n\n\\noindent\nThe double differential production rate of secondary protons emitted at 90$\\degree$ with respect to the beam line is estimated as:\n\\begin{equation}\n\\frac{d^2N_{\\rm P}}{dN_{\\rm C}d\\Omega}(\\theta=90\\degree)=\\frac{N_{\\rm P}}{N_{\\rm C}~~\\Omega_{\\rm LYSO}}.\n\\end{equation}\nFigure~\\ref{fig:flusso} shows the double differential production rate of secondary protons, emitted at 90$\\degree$ with respect to the beam line, as a function of the rate of the carbon ions $R_{\\rm C}$ reaching the PMMA: all identified protons and protons with $E_{kin} >$ 60 MeV.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics [width = 1\\textwidth] {Fig9.pdf}\n\\caption{\\small{Double differential production rate for secondary particles emitted at 90$\\degree$ with respect to the beam line, as a function of the rate of the carbon ions $R_{\\rm C}$ reaching the PMMA target: all identified protons (triangles) and protons with $E_{\\rm kin} >$ 60 MeV (circles).\n}}\n\\label{fig:flusso}\n\\end{center}\n\\end{figure}\n\n\\noindent Expressing these results in terms of the secondary proton's kinetic energy at emission $E^{\\rm Prod}_{\\rm kin}$, we obtain:\n\\begin{eqnarray}\n\\fl\n\\frac{dN_{\\rm P}}{dN_{\\rm C}d\\Omega}(E^{\\rm Prod}_{\\rm kin} > 7 {\\rm ~MeV}, \\theta=90\\degree) = (9.56\\pm 0.18_{\\rm stat} \\pm 0.40_{\\rm sys})\\times 10^{-4} sr^{-1} \\\\\n\\fl\n \\frac{dN_{\\rm P}}{dN_{\\rm C}d\\Omega}(E^{\\rm Prod}_{\\rm kin} > 83 {\\rm ~MeV}, \\theta=90\\degree) = (2.69\\pm 0.08_{\\rm stat} \\pm 0.12_{\\rm sys})\\times 10^{-4} sr^{-1}\n\\end{eqnarray}\nwith the systematic contribution mainly due to proton identification and to the uncertainty on the production kinetic energy related to the beam's transversal profile uncertainty. \n\n\\noindent\nThe same experimental setup described in Section \\ref{setup} has been used to measure the differential production rate for prompt photons, with energy $E_{\\rm LYSO} >$ 2 MeV and emitted at 90$\\degree$ with respect to the beam line: $dN_{\\rm \\gamma}\/(dN_{\\rm C}d\\Omega)(E_{\\rm LYSO} > 2 {\\rm ~MeV} , \\theta=90\\degree) = (2.92 \\pm 0.19)\\times 10^{-4} sr^{-1}$~\\cite{Agodi2012a}.\n\n\\section{Discussion and conclusions}\nWe reported the study of secondary charged particles produced by the interaction of 80 MeV\/u fully stripped carbon ion beam of INFN-LNS laboratory in Catania with a PMMA target. \nProtons have been identified exploiting the energy and time of flight measured with a plastic scintillator together with LYSO crystals, and their direction has been reconstructed with a drift chamber. A detailed simulation of the setup based on the FLUKA package has been done to evaluate its acceptance and efficiency, and to optimize secondary particle's identification.\n\\\\\n\\noindent\nIt has been shown that the backtracking of secondary protons allows their emission region in the target to be reconstructed. Moreover the existence of a correlation between the reconstructed production region of secondary protons and the Bragg peak position has been observed, performing a position scan of the target.\nThe achievable accuracy on the Bragg peak determination exploting this procedure has been estimated to be in the submillimeter range, using the described setup and selecting secondary protons with kinetic energy at emission $E^{\\rm Prod}_{\\rm kin} >$ 83 MeV. \n\\\\\n\\noindent\nThe obtained accuracy on the position of the released dose should be regarded as an indication of the achievable accuracy for possible applications of this technique to monitor the BP position in hadrontherapy treatment. In fact in clinical application the secondary particles should cross a larger amount of material (patient tissue) resulting in an increased multiple scattering contribution worsening the BP resolution by, at most, a factor 2-3. On the other hand an optimized device allowing a closer positioning to the patient could greatly improve the collected statistics of protons produced with $E^{\\rm Prod}_{\\rm kin} >$ 80 MeV, reducing multiple scattering effects. Furthermore the intrinsic good tracking resolution and high detection efficiency easily achievable in charged particles detectors, make this monitoring option worthwhile of further investigations.\n\n\\noindent\nThe measured differential production rate for protons with $E^{\\rm Prod}_{\\rm kin} >$ 83 MeV and emitted at 90$\\degree$ with respect to the beam line is: $dN_{\\rm P}\/(dN_{\\rm C}d\\Omega)(E^{\\rm Prod}_{\\rm kin} > 83 MeV , \\theta=90\\degree) = (2.69\\pm 0.08_{\\rm stat} \\pm 0.12_{\\rm sys})\\times 10^{-4} sr^{-1}$.\n\n\\ack\nWe would like to thank the precious cooperation of the staff of the INFN-LNS (Catania, Italy) accelerator group. The authors would like to thank Dr. M.~Pillon and Dr. M.~Angelone (ENEA-Fra\\-scati, Italy) for allowing us to validate the response of our detector to neutrons on the Frascati Neutron Generator; C.~Piscitelli (INFN-Roma, Italy) for the realization of the mechanical support; M.~Anelli (INFN-LNF, Frascati) for the drift chamber construction. \n\n\\noindent\nThis work has been supported by the ``Museo storico della fisica e Centro di studi e ricerche Enrico Fermi''. \n\\section*{References}\n\\bibliographystyle{jphysicsBforPMB}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}