Evolution of the Stellar Mass Function and Infrared Luminosity Function of Galaxies since $z = 1.2$

TL;DR

This study examines the evolution of the galaxy stellar mass and infrared luminosity functions since redshift 1.2, revealing slow growth in red galaxy mass density and luminosity, with implications for galaxy migration and merger rates.

Contribution

It provides new measurements of galaxy luminosity and stellar mass functions over a large area and redshift range, with improved error analysis and insights into galaxy evolution processes.

Findings

K-band luminosity density increased by 20% from z~1.1 to z~0.3

Massive red galaxies grew by a factor of 1.7 between z~1.1 and z~0.3

Red galaxy stellar mass density grew by a factor of 2.3 over the same period

Abstract

We measured evolution of the $K$-band luminosity function and stellar mass function for red and blue galaxies at $z<1.2$ using a sample of 353 594 $I<24$ galaxies in 8.26 square degrees of Bo\"otes. We addressed several sources of systematic and random error in measurements of total galaxy light, photometric redshift and absolute magnitude. We have found that the $K$-band luminosity density for both red and blue galaxies increased by a factor of 1.2 from $z\sim1.1$ to $z\sim0.3$, while the most luminous red (blue) galaxies decreased in luminosity by 0.19 (0.33) mag or $\times0.83 (0.74)$. These results are consistent with $z<0.2$ studies while our large sample size and area result in smaller Poisson and cosmic variance uncertainties than most $z >0.4$ luminosity and mass function measurements. Using an evolving relation for $K$-band mass to light ratios as a function of $(B-V)$ color, we found a slowly decreasing rate of growth in red galaxy stellar mass density of $\times2.3$ from $z\sim1.1$ to $z\sim0.3$, indicating a slowly decreasing rate of migration from the blue cloud to the red sequence. Unlike some studies of the stellar mass function, we find that massive red galaxies grow by a factor of $\times1.7$ from $z\sim1.1$ to $z\sim0.3$, with the rate of growth due to mergers decreasing with time. These results are comparable with measurements of merger rates and clustering, and they are also consistent with the red galaxy stellar mass growth implied by comparing $K$-band luminosity evolution with the fading of passive stellar population models.

Tables

Table 1. Table 1 Some previous measurements of the near infrared luminosity function and its evolution

reference	surveys	approx.	redshift	LF	approx	sample	approx.	subsamples
	used	redshift	typeaaSpectroscopic (s) or photometric (p) redshifts.	wavebands	faint	size	sample area
		range	(s or p)		limit (AB)bbThese faint limits are only intended to provide approximate depth comparisons between different surveys, as different authors quote survey depths to different completeness (typically $5\sigma$).		(deg2)
LOW REDSHIFT STUDIES
loved00	Cerro Tololo 1.5 m		s	$K$	$K=13.8$	$345$	$\mathrm{4\hspace{0.17em}270}$	ELG/non ELGccELG = emission line galaxy.
kocha01	2MASS		s	$K$	$K=13.1$	$\mathrm{3\hspace{0.17em}878}$	$\mathrm{6\hspace{0.17em}960}$	morphology
cole01	2MASS, 2dFGRS		s	$J,K$	$K=15$	$\mathrm{5\hspace{0.17em}683}$	$619$	-
bell03	2MASS, SDSS		s	$ugrizK$	$K=15.3$	$\mathrm{6\hspace{0.17em}282}$	$414$	morphology,
								color
huang03	Hawaii+AAO		s	$K$	$K=16.8$	$\mathrm{1\hspace{0.17em}056}$	$8.22$	morphology
	K-band GRS
eke05	2MASS, 2dFGRS		s	$J,K$	$K=15.2$	$\mathrm{15\hspace{0.17em}664}$	-	cluster or
								group size
jones06	6dFGS	$0.054$	s	$b_{J}rJHK$	$K=14.6$	$\mathrm{60\hspace{0.17em}869}$	$\mathrm{9\hspace{0.17em}075}$	-
dever09	2MASS		s	$K$	$K=11.8$	$\mathrm{1\hspace{0.17em}613}$	$\sim \mathrm{15\hspace{0.17em}000}$	morphology
smith09	UKIDSS, SDSS		s	$r,K$	$K=17.8$	$\mathrm{40\hspace{0.17em}111}$	$619$	color
hill10	MGC, UKIDSS, SDSS		s	$ugrizYJHK$	$K=17.5$	$\mathrm{1\hspace{0.17em}785}$	$28$	-
drive12	GAMA, GALEX,		s	FUV, NUV,	$K=19.9$	$\mathrm{7\hspace{0.17em}638}$	$125$	morphology
	SDSS, UKIDSS			$ugrizYJHK$
kelvi14	GAMA		s	$ugrizYJHK$	$r=19.4$	$\mathrm{3\hspace{0.17em}727}$	$144$	morphology
bonne15	2MASS		p	$K$	$K=12.6$	$\mathrm{13\hspace{0.17em}325}$	all but	morphology,
							GPddGP = Galactic plane.	color
STUDIES OF LF EVOLUTION
drory03	MUNICS		p	$K$	$K=21.3$	$\sim \mathrm{5\hspace{0.17em}000}$	$0.28$	-
pozze03	K20	$z=0.5,1.0,1.5$	s	$J,K$	$K=21.8$	$489$	$0.014$	ELG/non ELGccELG = emission line galaxy.
								color
ciras07	UKIDSS UDS		p	$K$	$K=22.5$	$\mathrm{22\hspace{0.17em}000}$	$0.6$	color
arnou07	SWIRE, VVDS,		p	$K$	$K=22.0$	$\mathrm{21\hspace{0.17em}200}$	$0.76$	SED fits
	CFHTLS, UKIDSS UDS
ciras10	UKIDSS UDS, SXDS		p	$K$	$K=23$	$\sim \mathrm{50\hspace{0.17em}000}$	$0.7$	-
mortl17	UltraVISTA,		p	$K$	$K=22.8$	$\mathrm{88\hspace{0.17em}484}$	$1.0$	-
	CANDELS, HUDF
This work	NDWFS, SDWFS,		p	$K$	$I=24,$	$\mathrm{359\hspace{0.17em}802}$	8.26	color
	NEWFIRM				$\left[3.6\mu \text{m}\right]$
					$=23.3$

Table 2. Table 2 Some previous measurements of the stellar mass function and its evolution

reference	surveys	approx.	redshift	approx	sample	approx.	subsamples	stellar
	used	redshift	typeaaSpectroscopic (s) or photometric (p) redshifts.	faint	size	sample area		mass
		range	(s or p)	limit (AB)bbThese faint limits are only intended to provide approximate depth comparisons between different surveys, as different authors quote survey depths to different completeness (typically $5\sigma$).		(deg2)		based on
LOW REDSHIFT STUDIES
cole01	2MASS, 2dFGRS		s	$K=13.1$	$\mathrm{5\hspace{0.17em}683}$	$\mathrm{6\hspace{0.17em}19}$	-	SED fit
bell03	2MASS, SDSS		s	$r=17.5$,	$\mathrm{18\hspace{0.17em}714}$	$414$	concn. index,	$M/L_g$,
				$K=13.5$			color	$M/L_K$
eke05	2MASS, 2dFGRS		s	$K=15.2$	$\mathrm{15\hspace{0.17em}664}$	-	cluster or	SED fit
							group size
li09	SDSS		s	$r=17.6$	$\mathrm{486\hspace{0.17em}840}$	$\mathrm{6\hspace{0.17em}437}$	-	kcorrect
smith09	UKIDSS, SDSS		s	$K=20.1$	$\mathrm{40\hspace{0.17em}111}$	$\mathrm{6\hspace{0.17em}19}$	color	kcorrect
baldr12	GAMA		s	$r=19.8$	$\mathrm{5\hspace{0.17em}210}$	$143$	-	$M/L_i$
STUDIES OF SMF EVOLUTION
drory05	GOODS-S, FORS		p	$I=26.8$,	$\mathrm{5\hspace{0.17em}557}$,	$0.025$	-	SED fit
				$K=25.4$	$\mathrm{3\hspace{0.17em}367}$
arnou07	SWIRE, VVDS,		p	$\left[3.6\mu \text{m}\right]$	$\sim \mathrm{21\hspace{0.17em}200}$	$0.76$	SED fits	$M/L_K$
	CFHTLS,			$=21.5$
	UKIDSS UDS
bundy06	DEEP2		s	$K=21.5$	$\sim \mathrm{8\hspace{0.17em}000}$	$1.5$	morphology,	$M/L_K$
	Palomar NIR						color, ELW
borch06	COMBO17		p	$R=24$	$\sim \mathrm{25\hspace{0.17em}000}$	$0.78$	color	$M/L$
								by SED fit
perez08	IRAC, MIPS,		p	$\left[3.6\mu \text{m}\right]$	$\mathrm{28\hspace{0.17em}000}$	$0.18$	SFR	SED fit
	Subaru, other			$=23.4$				$\text{for}\left[3.6\mu \text{m}\right]$
	optical + IR
drory09	COSMOS		p	$i^{+}=25$,	$\mathrm{138\hspace{0.17em}001}$	$1.73$	SED fits,	SED fit
				$K=24$			color
ilber10	COSMOS		p	$\left[3.6\mu \text{m}\right]$	$\mathrm{196\hspace{0.17em}000}$	$2.3$	morphology,	$M/L$
				$=23.9$			color	by SED fit
bramm11	NEWFIRM MBS		p	$K=22.8$	$\sim \mathrm{27\hspace{0.17em}000}$	$0.39$	color	SED fit
gonza11	IRAC, HST ACS,		p	$J\sim 28$	$437$	0.011	-	SED fit
	HST WFC3/IR
mortl11	HST NICMOS		p	$H=26.8$	$\mathrm{8\hspace{0.17em}298}$	$0.73$	color, SFR	SED fit
ilber13	UltraVISTA		p	$K=24$	$\mathrm{220\hspace{0.17em}000}$	$1.52$	color, SFR	SED fit
moust13	PRIMUS (+SDSS)		s	$i=23$	$\mathrm{40\hspace{0.17em}430}$	$5.5$	SFR	SED fit
muzzi13b	UltraVISTA		p	$K_s=23.4$	$\mathrm{26\hspace{0.17em}000}$	$1.62$	color	SED fit
maras13	BOSS		s	$r=17.6$	$\sim \mathrm{400\hspace{0.17em}000}$	$3275$	-	SED fit
					(massive only)
david13	VIPERS		s	$i=22.5$	$\mathrm{53\hspace{0.17em}608}$	$10.31$	color	SED fit
	(+CFHT, GALEX)
tomcz14	ZFOURGE/		p	$K\sim 25$	$\mathrm{76\hspace{0.17em}505}$	$0.09$	color	SED fit
	CANDELS
This work	NDWFS, SDWFS,		p	$I=24,$	$\mathrm{359\hspace{0.17em}802}$	8.26	color	$M/L_K$
	NEWFIRM			$\left[3.6\mu \text{m}\right]$
				$=23.3$

Table 3. Table 3 The observed colors used to determine absolute magnitudes.

restframe	effective	redshift	color
waveband	wavelength	range	$\left(m_Y-m_Z\right)$
	$\mu m$
$U$aaAs calculated in beare15	0.361	0.0 to 0.8	$B_w-R$
$U$aaAs calculated in beare15	0.361	0.8 to 1.2	$R-I$
$B$††footnotemark:	0.441	0.0 to 0.4	$B_w-I$
$B$††footnotemark:	0.441	0.4 to 1.2	$R-J$
$V$	0.551	0.2 to 0.4	$R-K_s$
$V$	0.551	0.4 to 1.2	$I-J$
$K$	2.195	0.2 to 0.6	$K_s-\left[3.6\mu \text{m}\right]$
$K$	2.195	0.6 to 1.0	$\left[3.6\mu \text{m}\right]-\left[4.5\mu \text{m}\right]$
$K$	2.195	1.0 to 1.2	$J-\left[4.5\mu \text{m}\right]$

Table 4. Table 4 K 𝐾 K -band luminosity function Schechter parameters for fixed α 𝛼 \alpha .

z	$\alpha$	${\varphi }^{\ast }$	$M_K^{\ast }-5\log h_{70}$	$M_K$ at fixed space densityaa${10}^{-4.0}{h}_{70}^3{\mathrm{Mpc}}^{-3}{\mathrm{mag}}^{-1}$  ($M_K$ at this fixed space density is measured by fitting a Schechter function with variable $\alpha$ to the bright end of the LF). (measures	$j_K$
		$\left(h_{70}^3{\mathrm{Mpc}}^{-3}{\mathrm{mag}}^{-1}\right)$		evolution of most luminous galaxies)	$L_{☉}{\mathrm{Mpc}}^{-3}$
Red galaxies - $\alpha =-0.5$
0.3	$-0.5$	$3.14\pm 0.10\times {10}^{-3}$	$-22.59\pm 0.04$	$-24.13\pm 0.03$	$3.50\pm 0.24\times {10}^8$
0.5	$-0.5$	$2.29\pm 0.10\times {10}^{-3}$	$-22.88\pm 0.08$	$-24.24\pm 0.06$	$3.33\pm 0.31\times {10}^8$
0.7	$-0.5$	$2.25\pm 0.10\times {10}^{-3}$	$-22.93\pm 0.05$	$-24.31\pm 0.03$	$3.43\pm 0.15\times {10}^8$
0.9	$-0.5$	$2.01\pm 0.20\times {10}^{-3}$	$-22.97\pm 0.05$	$-24.35\pm 0.04$	$3.19\pm 0.44\times {10}^8$
1.1	$-0.5$	$1.75\pm 0.13\times {10}^{-3}$	$-23.00\pm 0.05$	$-24.32\pm 0.03$	$2.85\pm 0.24\times {10}^8$
Blue galaxies - $\alpha =-1.2$
0.3	$-1.2$	$2.35\pm 0.06\times {10}^{-3}$	$-22.51\pm 0.05$	$-23.68\pm 0.05$	$3.19\pm 0.46\times {10}^8$
0.5	$-1.2$	$1.52\pm 0.16\times {10}^{-3}$	$-22.88\pm 0.09$	$-23.85\pm 0.08$	$2.92\pm 1.42\times {10}^8$
0.7	$-1.2$	$1.54\pm 0.02\times {10}^{-3}$	$-22.99\pm 0.05$	$-23.96\pm 0.06$	$3.28\pm 0.77\times {10}^8$
0.9	$-1.2$	$1.43\pm 0.11\times {10}^{-3}$	$-23.11\pm 0.06$	$-24.06\pm 0.04$	$3.39\pm 0.46\times {10}^8$
1.1	$-1.2$	$0.99\pm 0.14\times {10}^{-3}$	$-23.24\pm 0.09$	$-24.01\pm 0.07$	$2.62\pm 0.55\times {10}^8$
All galaxies - $\alpha =-1.0$
0.3	$-1.0$	$4.34\pm 0.25\times {10}^{-3}$	$-22.79\pm 0.06$	$-24.19\pm 0.04$	$6.58\pm 0.90\times {10}^8$
0.5	$-1.0$	$3.15\pm 0.23\times {10}^{-3}$	$-23.08\pm 0.08$	$-24.31\pm 0.07$	$6.25\pm 1.51\times {10}^8$
0.7	$-1.0$	$3.14\pm 0.09\times {10}^{-3}$	$-23.16\pm 0.04$	$-24.40\pm 0.04$	$6.64\pm 0.66\times {10}^8$
1.1	$-1.0$	$3.11\pm 0.22\times {10}^{-3}$	$-23.17\pm 0.05$	$-24.47\pm 0.04$	$6.67\pm 0.83\times {10}^8$
1.1	$-1.0$	$2.67\pm 0.19\times {10}^{-3}$	$-23.20\pm 0.06$	$-24.46\pm 0.05$	$5.89\pm 0.76\times {10}^8$

Table 5. Table 5 ORDER OF MAGNITUDE ESTIMATES OF RANDOM AND SYSTEMATIC ERRORS FOR RED GALAXIES - IN DEX

Quantity	Details of error	Random	Systematic
OBSERVATIONAL ERRORS
Photometry errors	Aperture photometry errors ($0.02$ to $0.4$ mag)	$0.01$ to $0.16$aaAperture photometry errors depend on the waveband and increase with apparent magnitude.	$0.01$
	Total flux correction from growth curves ($0.1$ to $0.2$ mag)	$0.04$ to $0.1$	$0.01$
	${\left(1+z\right)}^{-4}$ cosmological surface brightness dimming
	TOTAL	$0.04$ to $0.19$	$0.02$
Photometric redshift errors	$\lambda =\left(z_p-z_s\right)/\left(1+z_s\right)=$ 0.05 (random), 0.02 (systematic)	0.02	0.01
Space density uncertainties	Cosmic variance uncertainty - 8% by subfields	$0.03$	-
ERRORS IN DERIVED QUANTITIES FOR INDIVIDUAL GALAXIES
Luminosity errors (U, B, V, K) (in dex, not mag)	Due to photometric redshift errors	$0.04$	$0.02$
	Due to photometry errors (as above)	$0.02-0.19$aaAperture photometry errors depend on the waveband and increase with apparent magnitude.	$0.02$
	Due to K-correction errors (Equation 1)	$0.02$	$0.01$
	TOTAL	$0.05$ to $0.20$aaAperture photometry errors depend on the waveband and increase with apparent magnitude.	$0.05$
$M/L_K$ errors	Inaccuracy of $M/L_K$ formula (Equations 3 and 5)	$0.10$	$0.03$
	[Dependence of $M/L_K$ formula on assumed
	SPS model, SFH, metallicity, & dust	-	$0.30$
	(may not significantly affect relative mass growth)]	-	[$0.30$]
	[Dependence of $M/L_K$ formula on assumed IMF
	(does not affect relative mass growth)]	-	[$0.30$]
	Impact of photometric redshift errors on evolving
	$M/L_K$ formula (Equations 3 and 5)	$0.02$	$0.01$
	TOTAL (ignoring SPS and IMF error)	$0.10$	$0.04$
Stellar mass errors	Due to $M/L_K$ error (as above)	$0.10$	$0.04$
	Due to $K$-band luminosity error (as above)	$0.05$ to $0.20$aaAperture photometry errors depend on the waveband and increase with apparent magnitude.	$0.05$
	TOTAL	$0.11$ to $0.22$aaAperture photometry errors depend on the waveband and increase with apparent magnitude.	$0.09$
UNCERTAINTIES IN EVOLVING RED GALAXY LUMINOSITY DENSITIES AND STELLAR MASS DENSITIES,
Evolving $K$-band luminosity density errors for red galaxies	Due to $K$-band luminosity errors (as above)	-	$0.05$
	Due to cosmic variance	$0.03$	-
	Due to inexactness of the Schechter parameterization	$0.03$	-
	Error due to $z_{\text{phot}}$ scattering across redshift bin boundariesbbFrom Monte Carlo simulations, as shown below’	-	$0.01-0.07$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
	TOTAL	$0.04$	$0.06-0.12$
Evolving SMD errors for red galaxies	Due to stellar mass errors (as above)	-	$0.09$ddAssuming no uncertainty due to variation with redshift of SPS model predictions of stellar mass.
	Due to cosmic variance	$0.03$	-
	Due to inexactness of the Schechter parameterization	$0.03$	-
	Error due to $z_{\text{phot}}$ scattering across redshift bin boundariesbbFrom Monte Carlo simulations, as shown below’	-	$0.01-0.09$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
	TOTAL	$0.04$	$0.10-0.18$ddAssuming no uncertainty due to variation with redshift of SPS model predictions of stellar mass.
UNCERTAINTIES IN EVOLVING $K$-BAND LUMINOSITIES AND STELLAR MASSES OF LUMINOUS/MASSIVE RED GALAXIES
$K$-band luminosity errors for highly luminous red galaxies	Due to $K$-band luminosity errors		$0.05$
	Eddington bias from $\sigma =0.03$ scatter in $K$-band luminosities
	due to photometry and K-corrections (but not $z_{\text{phot}}$ errors)	-	$0.03$
	Eddington bias due to random $z_{\text{phot}}$ errors and bias due to
	$z_{\text{phot}}$ scattering across redshift bin boundariesbbFrom Monte Carlo simulations, as shown below’		$0.001-0.04$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
	TOTAL	$0.01$	$0.08-0.13$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
Stellar mass errors for massive red galaxies	Due to stellar mass errors		$0.09$ddAssuming no uncertainty due to variation with redshift of SPS model predictions of stellar mass.
	Eddington bias from $\sigma =0.1$ scatter in ${\log }_{10}\left(M/L_K\right)$ (Equation 5)	-	$0.22$
	Eddington bias due to random $z_{\text{phot}}$ errors and bias due to
	$z_{\text{phot}}$ scattering across redshift bin boundariesbbFrom Monte Carlo simulations, as shown below’		$0.001-0.04$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
	TOTAL	$0.01$	$0.11-0.15$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
NET EFFECT ON RED GALAXY RESULTS OF RANDOM FRACTIONAL PHOTOMETRIC REDSHIFT ERROR $\sigma =0.05$ (FROM SIMULATION)
Red galaxy $K$-band luminosity density		$\le 0.01$	$0.01-0.07$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
$K$-band luminosity at fixed space density for luminous red galaxies		$\le 0.01$	$0.001-0.04$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
Red galaxy stellar mass density		$\le 0.01$	$0.01-0.09$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
Stellar mass $\tilde{M}$ at fixed space density for massive red galaxies		$\le 0.01$	$0.001-0.04$ccError increases with redshift, largely due to decreasing proportion of accurate spectroscopic redshifts.
NET EFFECT ON RED GALAXY RESULTS OF SYSTEMATIC FRACTIONAL PHOTOMETRIC REDSHIFT ERROR OF +0.01 (FROM SIMULATION)
Red galaxy $K$-band luminosity density		-	$-0.02$
$K$-band luminosity at fixed space density for luminous red galaxies		-	$-0.01$
Red galaxy stellar mass density		-	$-0.03$
Stellar mass $\tilde{M}$ at fixed space density for massive red galaxies		-	$-0.01$

Table 6. Table 6 Stellar mass function Schechter parameters based on K 𝐾 K , V 𝑉 V and i 𝑖 i -band mass to light ratios.

z	$\alpha$	${\varphi }^{\ast }$	${\log }_{10}{M}^{\ast }/M_{☉}$	${\log }_{10}M/M_{☉}$	${\log }_{10}$stellar mass density
		$\left(h_{70}^3{\mathrm{Mpc}}^{-3}{\log }_{10}{M}^{-1}\right)$		at fixed space densityaa$2.5\times {10}^{-4.0}{h}_{70}^3{\mathrm{Mpc}}^{-3}{\text{dex}}^{-1}$.    (${\log }_{10}M/M_{☉}$ at this fixed space density is measured by fitting a Schechter function with variable $\alpha$ to the massive end of the SMF.) (measures	($h_{70}^3{M}_{☉}{\mathrm{Mpc}}^{-3}$)
				evolution of most massive galaxies)
Red galaxies
0.3	$-0.5$	$2.90\pm 0.11\times {10}^{-3}$	$10.78\pm 0.07$	$11.47\pm 0.07$	$8.19\pm 0.02$
0.5	$-0.5$	$2.35\pm 0.16\times {10}^{-3}$	$10.79\pm 0.04$	$11.42\pm 0.08$	$8.11\pm 0.03$
0.7	$-0.5$	$2.27\pm 0.11\times {10}^{-3}$	$10.78\pm 0.04$	$11.40\pm 0.06$	$8.08\pm 0.02$
0.9	$-0.5$	$1.88\pm 0.04\times {10}^{-3}$	$10.73\pm 0.06$	$11.31\pm 0.05$	$7.95\pm 0.01$
1.1	$-0.5$	$1.46\pm 0.13\times {10}^{-3}$	$10.71\pm 0.05$	$11.24\pm 0.10$	$7.82\pm 0.04$
Blue galaxies
0.3	$-1.2$	$1.78\pm 0.04\times {10}^{-3}$	$10.59\pm 0.06$	$11.10\pm 0.05$	$7.90\pm 0.01$
0.5	$-1.2$	$0.99\pm 0.04\times {10}^{-3}$	$10.79\pm 0.04$	$11.17\pm 0.03$	$7.85\pm 0.02$
0.7	$-1.2$	$0.97\pm 0.02\times {10}^{-3}$	$10.79\pm 0.04$	$11.16\pm 0.03$	$7.85\pm 0.01$
0.9	$-1.2$	$1.04\pm 0.04\times {10}^{-3}$	$10.67\pm 0.05$	$11.04\pm 0.04$	$7.75\pm 0.02$
1.1	$-1.2$	$0.74\pm 0.05\times {10}^{-3}$	$10.66\pm 0.08$	$10.95\pm 0.06$	$7.59\pm 0.03$
All galaxies
0.3	$-1$	$3.13\pm 0.06\times {10}^{-3}$	$10.87\pm 0.08$	$11.48\pm 0.07$	$8.36\pm 0.01$
0.5	$-1$	$2.90\pm 0.13\times {10}^{-3}$	$10.86\pm 0.03$	$11.43\pm 0.05$	$8.32\pm 0.02$
0.7	$-1$	$2.62\pm 0.06\times {10}^{-3}$	$10.88\pm 0.03$	$11.42\pm 0.04$	$8.29\pm 0.01$
0.9	$-1$	$2.36\pm 0.08\times {10}^{-3}$	$10.80\pm 0.05$	$11.33\pm 0.04$	$8.17\pm 0.01$
1.1	$-1$	$2.09\pm 0.16\times {10}^{-3}$	$10.76\pm 0.05$	$11.27\pm 0.05$	$8.08\pm 0.03$

Table 7. Table 7 Binned K 𝐾 K -band luminosity function for all galaxies.

$M_K-5\log h_{70}$		Luminosity Function (${10}^{-3}{h}_{70}^3{\mathrm{Mpc}}^{-3}{\mathrm{mag}}^{-1}$)
Min	Max
$-26.00$	$-25.75$	-	$0.001\pm 0.001$	-	-	-
$-25.75$	$-25.50$	-	-	-	-	$0.001\pm 0.001$
$-25.50$	$-25.25$	-	-	-	$0.001\pm 0.001$	$0.001\pm 0.001$
$-25.25$	$-25.00$	$0.003\pm 0.002$	$0.003\pm 0.001$	$0.004\pm 0.001$	$0.005\pm 0.001$	$0.008\pm 0.001$
$-25.00$	$-24.75$	$0.001\pm 0.001$	$0.007\pm 0.002$	$0.015\pm 0.003$	$0.022\pm 0.003$	$0.026\pm 0.003$
$-24.75$	$-24.50$	$0.018\pm 0.005$	$0.027\pm 0.004$	$0.037\pm 0.004$	$0.055\pm 0.004$	$0.052\pm 0.004$
$-24.50$	$-24.25$	$0.060\pm 0.009$	$0.072\pm 0.007$	$0.113\pm 0.007$	$0.137\pm 0.007$	$0.125\pm 0.006$
$-24.25$	$-24.00$	$0.134\pm 0.014$	$0.176\pm 0.011$	$0.241\pm 0.010$	$0.254\pm 0.009$	$0.241\pm 0.008$
$-24.00$	$-23.75$	$0.248\pm 0.019$	$0.365\pm 0.016$	$0.420\pm 0.014$	$0.436\pm 0.012$	$0.396\pm 0.011$
$-23.75$	$-23.50$	$0.453\pm 0.026$	$0.561\pm 0.020$	$0.646\pm 0.017$	$0.657\pm 0.015$	$0.573\pm 0.013$
$-23.50$	$-23.25$	$0.759\pm 0.034$	$0.860\pm 0.024$	$0.919\pm 0.020$	$0.894\pm 0.017$	$0.729\pm 0.015$
$-23.25$	$-23.00$	$1.122\pm 0.041$	$1.137\pm 0.028$	$1.183\pm 0.023$	$1.101\pm 0.019$	$0.924\pm 0.017$
$-23.00$	$-22.75$	$1.369\pm 0.045$	$1.428\pm 0.031$	$1.458\pm 0.025$	$1.293\pm 0.021$	$1.085\pm 0.018$
$-22.75$	$-22.50$	$1.617\pm 0.049$	$1.665\pm 0.034$	$1.645\pm 0.027$	$1.524\pm 0.023$	$1.149\pm 0.019$
$-22.50$	$-22.25$	$2.017\pm 0.055$	$1.737\pm 0.034$	$1.775\pm 0.028$	$1.648\pm 0.024$	-
$-22.25$	$-22.00$	$2.255\pm 0.058$	$1.969\pm 0.037$	$1.892\pm 0.029$	$1.745\pm 0.025$	-
$-22.00$	$-21.75$	$2.602\pm 0.062$	$2.012\pm 0.037$	$1.960\pm 0.030$	$1.800\pm 0.025$	-
$-21.75$	$-21.50$	$2.738\pm 0.064$	$2.074\pm 0.038$	$2.152\pm 0.031$	$1.827\pm 0.025$	-
$-21.50$	$-21.25$	$2.957\pm 0.066$	$2.121\pm 0.038$	$2.150\pm 0.031$	-	-
$-21.25$	$-21.00$	$3.195\pm 0.069$	$2.176\pm 0.039$	$2.298\pm 0.033$	-	-
$-21.00$	$-20.75$	$3.437\pm 0.072$	$2.191\pm 0.039$	$2.226\pm 0.032$	-	-
$-20.75$	$-20.50$	$3.638\pm 0.074$	$2.153\pm 0.039$	-	-	-
$-20.50$	$-20.25$	$4.152\pm 0.079$	$2.053\pm 0.038$	-	-	-
$-20.25$	$-20.00$	$4.707\pm 0.084$	$2.001\pm 0.038$	-	-	-
$-20.00$	$-19.75$	$5.062\pm 0.087$	$1.917\pm 0.037$	-	-	-
$-19.75$	$-19.50$	$5.599\pm 0.092$	-	-	-	-
$-19.50$	$-19.25$	$6.164\pm 0.097$	-	-	-	-
$-19.25$	$-19.00$	$6.661\pm 0.101$	-	-	-	-
$-19.00$	$-18.75$	$6.876\pm 0.103$	-	-	-	-
$-18.75$	$-18.50$	$7.338\pm 0.107$	-	-	-	-
$-18.50$	$-18.25$	$6.987\pm 0.105$	-	-	-	-

Table 8. Table 8 Binned K 𝐾 K -band luminosity function for red galaxies.

$M_K-5\log h_{70}$		Luminosity Function (${10}^{-3}{h}_{70}^3{\mathrm{Mpc}}^{-3}{\mathrm{mag}}^{-1}$)
Min	Max
$-26.00$	$-25.75$	-	$0.001\pm 0.001$	-	-	-
$-25.75$	$-25.50$	-	-	-	-	-
$-25.50$	$-25.25$	-	-	-	$0.001\pm 0.001$	$0.001\pm 0.001$
$-25.25$	$-25.00$	$0.003\pm 0.002$	$0.003\pm 0.001$	$0.004\pm 0.001$	$0.003\pm 0.001$	$0.004\pm 0.001$
$-25.00$	$-24.75$	$0.001\pm 0.001$	$0.007\pm 0.002$	$0.011\pm 0.002$	$0.016\pm 0.002$	$0.017\pm 0.002$
$-24.75$	$-24.50$	$0.015\pm 0.005$	$0.020\pm 0.004$	$0.028\pm 0.004$	$0.035\pm 0.003$	$0.036\pm 0.003$
$-24.50$	$-24.25$	$0.052\pm 0.009$	$0.063\pm 0.007$	$0.085\pm 0.006$	$0.102\pm 0.006$	$0.084\pm 0.005$
$-24.25$	$-24.00$	$0.111\pm 0.013$	$0.142\pm 0.010$	$0.178\pm 0.009$	$0.176\pm 0.008$	$0.158\pm 0.007$
$-24.00$	$-23.75$	$0.205\pm 0.017$	$0.276\pm 0.014$	$0.286\pm 0.011$	$0.276\pm 0.010$	$0.259\pm 0.009$
$-23.75$	$-23.50$	$0.324\pm 0.022$	$0.382\pm 0.016$	$0.432\pm 0.014$	$0.403\pm 0.012$	$0.377\pm 0.010$
$-23.50$	$-23.25$	$0.541\pm 0.028$	$0.558\pm 0.019$	$0.594\pm 0.016$	$0.540\pm 0.014$	$0.451\pm 0.012$
$-23.25$	$-23.00$	$0.764\pm 0.034$	$0.732\pm 0.022$	$0.717\pm 0.018$	$0.617\pm 0.015$	$0.543\pm 0.013$
$-23.00$	$-22.75$	$0.888\pm 0.036$	$0.859\pm 0.024$	$0.846\pm 0.019$	$0.694\pm 0.015$	$0.616\pm 0.014$
$-22.75$	$-22.50$	$1.013\pm 0.039$	$0.955\pm 0.025$	$0.888\pm 0.020$	$0.757\pm 0.016$	$0.522\pm 0.013$
$-22.50$	$-22.25$	$1.119\pm 0.041$	$0.948\pm 0.025$	$0.854\pm 0.019$	$0.742\pm 0.016$	-
$-22.25$	$-22.00$	$1.157\pm 0.042$	$0.979\pm 0.026$	$0.844\pm 0.019$	$0.704\pm 0.016$	-
$-22.00$	$-21.75$	$1.212\pm 0.043$	$0.898\pm 0.025$	$0.785\pm 0.019$	-	-
$-21.75$	$-21.50$	$1.152\pm 0.041$	$0.810\pm 0.024$	$0.786\pm 0.019$	-	-
$-21.50$	$-21.25$	$1.097\pm 0.040$	$0.728\pm 0.022$	$0.672\pm 0.018$	-	-
$-21.25$	$-21.00$	$1.087\pm 0.040$	$0.707\pm 0.022$	$0.626\pm 0.017$	-	-
$-21.00$	$-20.75$	$1.055\pm 0.040$	$0.598\pm 0.021$	-	-	-
$-20.75$	$-20.50$	$0.963\pm 0.038$	$0.530\pm 0.019$	-	-	-
$-20.50$	$-20.25$	$1.018\pm 0.039$	$0.393\pm 0.017$	-	-	-
$-20.25$	$-20.00$	$1.076\pm 0.040$	$0.329\pm 0.016$	-	-	-
$-20.00$	$-19.75$	$0.919\pm 0.037$	-	-	-	-
$-19.75$	$-19.50$	$0.901\pm 0.037$	-	-	-	-
$-19.50$	$-19.25$	$0.896\pm 0.037$	-	-	-	-
$-19.25$	$-19.00$	$0.901\pm 0.037$	-	-	-	-
$-19.00$	$-18.75$	$0.867\pm 0.037$	-	-	-	-
$-18.75$	$-18.50$	$0.808\pm 0.036$	-	-	-	-
$-18.50$	$-18.25$	$0.720\pm 0.034$	-	-	-	-
$-18.25$	$-18.00$	$0.630\pm 0.032$	-	-	-	-

Table 9. Table 9 Binned K 𝐾 K -band luminosity function for blue galaxies.

$M_K-5\log h_{70}$		Luminosity Function (${10}^{-3}{h}_{70}^3{\mathrm{Mpc}}^{-3}{\mathrm{mag}}^{-1}$)
Min	Max
$-25.75$	$-25.50$	-	-	-	-	$0.001\pm 0.001$
$-25.50$	$-25.25$	-	-	-	$-$	-
$-25.25$	$-25.00$	-	-	$0.001\pm 0.001$	$0.001\pm 0.001$	$0.004\pm 0.001$
$-25.00$	$-24.75$	-	$0.001\pm 0.001$	$0.004\pm 0.001$	$0.007\pm 0.001$	$0.008\pm 0.001$
$-24.75$	$-24.50$	$0.003\pm 0.002$	$0.007\pm 0.002$	$0.008\pm 0.002$	$0.020\pm 0.003$	$0.016\pm 0.002$
$-24.50$	$-24.25$	$0.007\pm 0.003$	$0.010\pm 0.003$	$0.028\pm 0.003$	$0.035\pm 0.003$	$0.042\pm 0.003$
$-24.25$	$-24.00$	$0.024\pm 0.006$	$0.035\pm 0.005$	$0.063\pm 0.005$	$0.079\pm 0.005$	$0.083\pm 0.005$
$-24.00$	$-23.75$	$0.043\pm 0.008$	$0.088\pm 0.008$	$0.134\pm 0.008$	$0.161\pm 0.007$	$0.137\pm 0.006$
$-23.75$	$-23.50$	$0.129\pm 0.014$	$0.180\pm 0.011$	$0.215\pm 0.010$	$0.254\pm 0.009$	$0.197\pm 0.007$
$-23.50$	$-23.25$	$0.218\pm 0.018$	$0.302\pm 0.014$	$0.324\pm 0.012$	$0.354\pm 0.011$	$0.278\pm 0.009$
$-23.25$	$-23.00$	$0.359\pm 0.023$	$0.405\pm 0.017$	$0.466\pm 0.014$	$0.484\pm 0.013$	$0.382\pm 0.010$
$-23.00$	$-22.75$	$0.481\pm 0.027$	$0.570\pm 0.020$	$0.612\pm 0.016$	$0.599\pm 0.014$	$0.468\pm 0.012$
$-22.75$	$-22.50$	$0.604\pm 0.030$	$0.710\pm 0.022$	$0.757\pm 0.018$	$0.767\pm 0.016$	$0.628\pm 0.014$
$-22.50$	$-22.25$	$0.898\pm 0.037$	$0.789\pm 0.023$	$0.921\pm 0.020$	$0.906\pm 0.018$	$0.701\pm 0.014$
$-22.25$	$-22.00$	$1.098\pm 0.040$	$0.990\pm 0.026$	$1.048\pm 0.022$	$1.040\pm 0.019$	$0.760\pm 0.015$
$-22.00$	$-21.75$	$1.390\pm 0.046$	$1.114\pm 0.028$	$1.175\pm 0.023$	$1.185\pm 0.020$	-
$-21.75$	$-21.50$	$1.585\pm 0.049$	$1.264\pm 0.029$	$1.365\pm 0.025$	$1.343\pm 0.022$	-
$-21.50$	$-21.25$	$1.860\pm 0.053$	$1.393\pm 0.031$	$1.478\pm 0.026$	$1.447\pm 0.023$	-
$-21.25$	$-21.00$	$2.109\pm 0.056$	$1.469\pm 0.032$	$1.672\pm 0.028$	-	-
$-21.00$	$-20.75$	$2.383\pm 0.060$	$1.592\pm 0.033$	$1.695\pm 0.028$	-	-
$-20.75$	$-20.50$	$2.675\pm 0.063$	$1.623\pm 0.034$	$1.734\pm 0.028$	-	-
$-20.50$	$-20.25$	$3.134\pm 0.068$	$1.659\pm 0.034$	$1.825\pm 0.029$	-	-
$-20.25$	$-20.00$	$3.632\pm 0.074$	$1.672\pm 0.034$	-	-	-
$-20.00$	$-19.75$	$4.143\pm 0.079$	$1.677\pm 0.035$	-	-	-
$-19.75$	$-19.50$	$4.698\pm 0.084$	$1.717\pm 0.035$	-	-	-
$-19.50$	$-19.25$	$5.268\pm 0.090$	-	-	-	-
$-19.25$	$-19.00$	$5.759\pm 0.094$	-	-	-	-
$-19.00$	$-18.75$	$6.009\pm 0.097$	-	-	-	-
$-18.75$	$-18.50$	$6.531\pm 0.101$	-	-	-	-
$-18.50$	$-18.25$	$6.268\pm 0.100$	-	-	-	-
$-18.25$	$-18.00$	$5.133\pm 0.090$	-	-	-	-

Table 10. Table 10 Binned SMF for all galaxies.

$\log M$		SMF ($h_{70}^3{\mathrm{Mpc}}^{-3}{\log }_{10}{M}^{-1}$)
Min	Max
$8.80$	$8.90$	$14.734\pm 0.235$	-	-	-	-
$8.90$	$9.00$	$14.424\pm 0.232$	-	-	-	-
$9.00$	$9.10$	$13.725\pm 0.226$	-	-	-	-
$9.10$	$9.20$	$12.765\pm 0.218$	-	-	-	-
$9.20$	$9.30$	$11.559\pm 0.208$	-	-	-	-
$9.30$	$9.40$	$11.077\pm 0.203$	-	-	-	-
$9.40$	$9.50$	$9.709\pm 0.190$	$4.011\pm 0.083$	-	-	-
$9.50$	$9.60$	$8.966\pm 0.183$	$3.981\pm 0.082$	-	-	-
$9.60$	$9.70$	$8.491\pm 0.178$	$4.134\pm 0.084$	-	-	-
$9.70$	$9.80$	$7.856\pm 0.171$	$4.212\pm 0.085$	-	-	-
$9.80$	$9.90$	$7.016\pm 0.162$	$4.102\pm 0.084$	-	-	-
$9.90$	$10.00$	$6.698\pm 0.158$	$4.083\pm 0.083$	$3.919\pm 0.065$	-	-
$10.00$	$10.10$	$6.489\pm 0.156$	$4.187\pm 0.084$	$3.725\pm 0.064$	-	-
$10.10$	$10.20$	$5.906\pm 0.149$	$4.187\pm 0.084$	$3.849\pm 0.065$	-	-
$10.20$	$10.30$	$5.264\pm 0.140$	$4.107\pm 0.084$	$3.761\pm 0.064$	-	-
$10.30$	$10.40$	$5.391\pm 0.142$	$4.057\pm 0.083$	$3.812\pm 0.064$	-	-
$10.40$	$10.50$	$4.946\pm 0.136$	$3.954\pm 0.082$	$3.738\pm 0.064$	-	-
$10.50$	$10.60$	$4.214\pm 0.125$	$3.802\pm 0.080$	$3.512\pm 0.062$	$2.968\pm 0.050$	-
$10.60$	$10.70$	$3.982\pm 0.122$	$3.583\pm 0.078$	$3.324\pm 0.060$	$2.633\pm 0.047$	-
$10.70$	$10.80$	$3.250\pm 0.110$	$3.321\pm 0.075$	$2.878\pm 0.056$	$2.276\pm 0.043$	-
$10.80$	$10.90$	$2.817\pm 0.103$	$2.708\pm 0.068$	$2.524\pm 0.052$	$1.827\pm 0.039$	$1.360\pm 0.031$
$10.90$	$11.00$	$2.223\pm 0.091$	$2.120\pm 0.060$	$2.022\pm 0.047$	$1.404\pm 0.034$	$1.029\pm 0.027$
$11.00$	$11.10$	$1.558\pm 0.076$	$1.443\pm 0.050$	$1.473\pm 0.040$	$0.939\pm 0.028$	$0.690\pm 0.022$
$11.10$	$11.20$	$0.997\pm 0.061$	$1.007\pm 0.041$	$0.945\pm 0.032$	$0.584\pm 0.022$	$0.402\pm 0.017$
$11.20$	$11.30$	$0.650\pm 0.049$	$0.531\pm 0.030$	$0.530\pm 0.024$	$0.315\pm 0.016$	$0.203\pm 0.012$
$11.30$	$11.40$	$0.280\pm 0.032$	$0.255\pm 0.021$	$0.276\pm 0.017$	$0.164\pm 0.012$	$0.103\pm 0.008$
$11.40$	$11.50$	$0.202\pm 0.027$	$0.133\pm 0.015$	$0.105\pm 0.011$	$0.052\pm 0.007$	$0.037\pm 0.005$
$11.50$	$11.60$	$0.075\pm 0.017$	$0.037\pm 0.008$	$0.044\pm 0.007$	$0.014\pm 0.003$	$0.013\pm 0.003$
$11.60$	$11.70$	$0.011\pm 0.006$	$0.015\pm 0.005$	$0.009\pm 0.003$	$0.006\pm 0.002$	$0.004\pm 0.002$
$11.70$	$11.80$	-	$0.005\pm 0.003$	$0.004\pm 0.002$	-	-
$11.80$	$11.90$	$0.004\pm 0.004$	-	$0.002\pm 0.002$	-	-
$11.90$	$12.00$	$0.004\pm 0.004$	-	-	-

Equations

$$(M_W+D_M)-m_Z=a(m_Y-m_Z{)}^2+b(m_Y-m_Z)+c.$$

$${\log }_{10}(M/L_K)=-0.854+0.728(M_B-M_V).$$

$${\log }_{10}(M/L_K)=-[0.05+0.30(M_B-M_V)]z+(M_B-M_V)-\mathrm{1.00.}$$

$${\log }_{10}(M/L_K)=-0.287+0.135(M_B-M_V).$$

$${\log }_{10}(M/L_K)=-[0.11+0.3(M_B-M_V)]z+(M_B-M_V)-\mathrm{0.9.}$$

$$(M_{\mathrm{U}}-M_{\mathrm{B}})>1.074-0.18z-0.03(M_{\mathrm{B}}+19.4).$$

$${\varphi }_K\left(L_K\right)d{L}_K=\left(\frac{{\varphi }_K^{\ast }}{L_K^{\ast }}\right){\left(\frac{L_K}{L_K^{\ast }}\right)}^{\alpha }\exp \left(\frac{-L_K}{L_K^{\ast }}\right)d{L}_K.$$

$$\begin{gathered} {\varphi }_K\left(M_K\right)d{M}_K \\ =-0.4\ln 10{\varphi }_K^{\ast }10^{-0.4(\alpha +1)(M_K-M_K^{\ast })}\exp (-10^{-0.4(M_K-M_K^{\ast })})d{M}_K. \end{gathered}$$

$$j_K={\varphi }_K^{\ast }{L}_K^{\ast }\Gamma (\alpha +2).$$

$${\varphi }_M\left(M\right)dM=\left(\frac{{\varphi }^{\ast }}{M^{\ast }}\right){\left(\frac{M}{M^{\ast }}\right)}^{\alpha }\exp \left(\frac{-M}{M^{\ast }}\right)dM.$$

$${\varphi }_{\mu }(\mu )d\mu =({\varphi }_{\mu }^{\ast }\ln 10)10^{(\alpha +1)(\mu -{\mu }^{\ast })}\exp (-10^{(\mu -{\mu }^{\ast })})d\mu .$$

$${\rho }_M={\varphi }^{\ast }{M}^{\ast }\Gamma (\alpha +2).$$

Full text

††slugcomment: Accepted for publication in the Astrophysical Journal44affiliationtext: Email: richard@beares.net

1. INTRODUCTION

Measurements of the optical and infrared luminosity functions (LFs) and the stellar mass function (SMF) at different redshifts provide important observational tests of large scale simulations of galaxy formation and evolution. Such simulations can be either cosmological hydrodynamical models, which attempt to model the detailed physical and chemical processes involved in galaxy formation, such as ILLUSTRIS (vogel14) and EAGLE (schay15), or semi-analytic models (SAMs, e.g. croto06; guo08; lacey16). SAMs include simple empirical representations of physical processes and these are ‘added onto’ the dark matter merger trees resulting from hierarchical N-body simulations (e.g. the Millennium Simulation, sprin05). Both types of simulation make predictions of LF and SMF evolution that can be tested against observational measurements. Discrepancies between observations and simulations then provide the motivation for refining the models incorporated in the simulations. In this way, measurements of optical and infrared LF evolution have in the past motivated significant improvements in our understanding of the physical processes occurring in galaxy formation and evolution (e.g. croto06; lacey16).

As benchmarks for testing and calibrating simulations, LFs have the advantage over SMFs that they have only limited model dependencies. SMFs require the determination of stellar masses from photometry and this involves use of a number of physical models, notably stellar population synthesis (SPS) models (e.g. fioc97; bruzu03; maras05), stellar initial mass functions (IMFs; e.g. salpe55; kenni83; chabr03), and dust attenuation laws (e.g. calze00). SMFs are therefore subject to significant model uncertainties in addition to the observational uncertainties inherent in LFs.

Table 1 and Table 2 (respectively) summarise several recent measurements of the near-infrared LF and the SMF and their evolution. Note that different studies have differentiated quiescent and star forming galaxies in different ways, e.g. by color, morphology or emission line strengths. Also included in these tables are low redshift ($z\leq 0.2$) studies that provide an accurate low redshift “anchor” for evolutionary studies.

A number of the studies in Table 2 derived SMFs by fitting theoretical stellar population synthesis (SPS) models to available photometry, e.g. using the kcorrect software of blant07 which fits observed photometry with combinations of five template SEDs that are derived from several hundred SPS models. Others used stellar mass to light $M/L$ ratios given as a function of observed color (e.g. bell03; taylo11) or redshift for red/quiescent and blue/active galaxies (e.g. arnou07). It should be noted that empirical $M/L$ ratios are also derived with the aid of SPS models.

Near infrared (especially $K$-band) $M/L$ ratios have often been preferred to optical ones for determining stellar masses because they are a much weaker function of stellar population color than optical ones. For example, bell01 found that $K$-band $M/L$ ratios amongst spiral galaxies varied by a factor of $\sim 2$ while those in the $B$-band varied by a factor of $\sim 7$. In this work we base our measurements of SMF evolution on stellar masses calculated using $M/L_{K}$ given as a function of $(B-V)$ color. Both the LF and the SMF can be approximated by schec76 functions, although the faint end power-law index is poorly constrained when survey magnitude limits are close to the characteristic magnitude or log stellar mass ($M_{K}^{*}$ or $\log_{10}M^{*}$ respectively). Given the steep faint-end slope for star forming galaxies, uncertainty in the power-law index has a significant impact on estimates of the luminosity and mass densities of these galaxies.

There is general agreement in the literature that the evolution of quiescent stellar mass density (SMD) has been more rapid than that of star-forming SMD, (e.g. bramm11; muzzi13b; tomcz14; moust13). There is agreement that the rate of increase of the space density of quiescent galaxies has depended strongly on galaxy mass, with smaller galaxies increasing more rapidly in numbers than larger ones (e.g. muzzi13b). The most massive quiescent galaxies have increased relatively slowly in stellar mass, implying that most of their stellar mass was already in place at higher redshift with very little stellar mass being added since (e.g. brown07; moust13; muzzi13b).

In this study we have provided precise measurements of the evolution of the $K$-band LF and the galaxy SMF between $z=0.2$ and $z=1.2$. Our very large sample of 353 594 galaxies spanning an area of 8.26 deg2 in Boötes enabled us to significantly reduce Poisson errors and the effects of cosmic variance. The significant depth $I=24.0$ of our survey allowed us to measure precisely the evolution of red and blue galaxies, both combined and separately, from $z=1.2$. For this work we used the low redshift kocha01 LF and the cole01 SMF as $z\sim 0$ reference points.

We have employed the same data and similar methods to those that beare15 used to measure optical $B$-band LF evolution over the redshift range $0.2<z<1.2$. We used the same optical and near-infrared photometry, our apparent magnitudes and colors were based on the same measurements of total flux, and we used the same method of calculating absolute magnitudes (beare14). For this paper we have used improved photometric redshifts, employing the Bayesian EAZY code to match our photometry to the 129 empirical galaxy templates of brown14.

The structure of this paper is as follows. Section 2 describes our imaging and catalogs, Section 3 our photometric redshifts, Sections 4 and 5 how we measured galaxy luminosities and stellar masses, and Section 6 our sample selection. Sections 7 and 8 describe how we measured evolution of the $K$-band LF and the SMF. Sections 9 and 10 present our results for evolution of $K$-band luminosities and stellar masses and discuss their significance. Finally we summarise our work and conclusions in Section 11.

Our results were determined assuming a cosmology111Conversions to other cosmologies can be made as described in croto13. with $\Omega_{0}=0.3$, $\Omega_{\textrm{k}}=0$, $H_{0}=70\,\,\textrm{km s}^{-1}\,\textrm{Mpc}^{-1}$ which is similar to that measured by WMAP (benne13), and presented using AB-based magnitudes and units in which $h_{70}=H_{0}/70$.

2. Imaging and catalogs

The images, catalogs and photometry were identical to those used in beare15. They provided a very large sample of 353 594 galaxies covering a substantial area of 8.26 deg2 in Boötes, surveyed to a depth of $I=24.0$. Our sample was an excellent one for measuring LF and SMF evolution because it was deep enough to provide precise photometry out to $z=1.2$. and because its large size minimised random (Poisson) error and the effects of cosmic variance. Furthermore, we were able to utilise photometry in 13 optical and near infrared wavebands, and this enabled us to obtain precise photometric redshifts, to calculate precise K-corrections (beare14) for determining restframe magnitudes and colors, and to apply cuts to exclude stars and AGN. Below we provide only a brief summary of our data. We refer the reader to beare15 for a more thorough description.

We used an update of the galaxy catalog produced by brown07 which includes additional images, photometry and minor refinements to the photometry code. brown07 detected sources in the Boötes field using SExtractor 2.3.2 (berti96) run on $I$-band images from the NWDFS third data release. They removed regions surrounding very extended galaxies and saturated stars in order to minimize contamination and the final sample covered an area of 8.26 deg2 over a $2.9^{\circ}\times 3.6^{\circ}$ field of view.

Optical photometry was based on $B_{W}RI$-band imaging from the NOAO Deep Wide Field Survey (NDWFS, jannu99). Near-infrared photometry was derived from $J$, $H$ and $K_{s}$-band imaging from the NEWFIRM Boötes Imaging Survey (Gonzalez et al. 2011, in prep), $u$ and $y$-band images were from the $2\times 8.4$ m Large Binocular Telescope (LBT; bian13), $z$-band data were from the 8.2 m Subaru Telescope (miyaz12), and 3.6, 4.5, 5.8 and 8.0 $\mu$m imaging was from the IRAC camera of the Spitzer Deep Wide Field Survey (SDWFS; ashby09; eisen08).

We used the method described in beare15 to measure the total flux from each galaxy. This employs magnitude dependent aperture sizes and applies corrections based on growth curves of apparent magnitude with aperture size to precisely account for flux falling outside the photometric aperture.

3. Determining photometric redshifts

Measurements of galaxy distances, and hence restframe magnitudes, rely on accurate redshifts. For the Boötes field we had to rely on photometric redshifts for the vast majority of our galaxies, as spectroscopic redshifts were only available for 3.4% of galaxies in the redshift range $0.2\leq z\leq 1.2$ (12 191 in all). Our photometric redshifts were determined using the Bayesian EAZY code (bramm08; taylo09) to model photometry in 13 optical to near-infrared wavebands using the 129 empirical template SEDs from brown14. Figure 1 shows that our photometric redshifts have $(z_{\textrm{phot}}-z_{\textrm{spec}})\,/\,(1+z_{\textrm{specz}})$ systematic errors of less than 0.01 at $z_{\rm{phot}}<1.0$ while at $1.0<z_{\rm{phot}}<1.2$ the systematic errors are less than 0.02. Our 1-$\sigma$ random errors were less than $\sim 0.05$ over the whole of our redshift range. The percentage of catastrophic errors, defined using the $|{z_{\textrm{phot}}-z_{\textrm{spec}}|\,/\,(1+z_{\textrm{spec}}})>0.15$ criterion of ilber13 was 1.50 (0.75, 2.17)% for all (red, blue) galaxies.

We chose to use Bayesian photometric redshifts because these do not exhibit the significant aliasing or bunching at specific redshifts seen in photometric redshifts based on frequentist least-squares fitting (as in beare15). Both sets of photometric redshifts exhibit similar systematic and random errors but the even distribution of the Bayesian redshifts ensures that bunching around preferred values does not affect the numbers of galaxies allocated to different redshift bins.

When available we used spectroscopic redshifts in preference to photometric redshifts. These were mainly from the AGN and Galaxy Evolution Survey (AGES, Kochanek et al. 2012), with several hundred additional redshifts from SDSS and a variety of programmes with the Gemini, Keck and Kitt Peak National Observatory telescopes.

4. Measuring absolute magnitudes

In order to measure evolution of the $K$-band LF we first used the method of beare14 to determine the absolute magnitudes of our galaxies. This method enables the absolute magnitude $M_{W}$ in a waveband $W$ to be precisely measured using a single, carefully chosen, observed color $(m_{Y}-m_{Z})$ and apparent magnitude $m_{Z}$. As Figure 2 shows, $M_{W}$ is determined from the second degree polynomial best fit to a plot of $(M_{W}+D_{M})-m_{Z}$ against $(m_{Y}-m_{Z})$ for the 129 template galaxies of brown14, $D_{M}$ being the distance modulus:

[TABLE]

Table 3 lists the observed colors we used at different redshifts to determine absolute magnitudes in different restframe wavebands. The method allows precise determination of the uncertainties due to photometric error, redshift error and intrinsic galaxy variability. (The $Y$ and $Z$-band filter transmission functions we used took account of atmospheric absorption but the $W$-band restframe filter did not.)

We used the same method to determine absolute magnitudes in the $B$ and $V$-bands as these were needed for calculating galaxy masses based on stellar mass to light ratios given as a function of restframe $(B-V)$ color.

The template galaxy substantially redder in $([3.6\mu m]-[4.5\mu m])$ color than the others in Figure 2 is the ultra-luminous infrared galaxy UGC 5101. This exhibits substantial emission from hot ($\sim$a few 100 K) dust in the near-infrared, indicating the presence of a powerful AGN. This outlier breaks the assumption that infrared light from galaxies is always dominated by stellar emission. However, it does in fact play a useful role in “anchoring” the polynomial used to determine absolute $K$-band magnitudes, preventing it being poorly constrained.

5. Measuring galaxy stellar masses

taylo11 showed that optical and near infrared stellar mass to light ratios ($M/L$) can be determined to within 0.2 dex using a linear function of a single restframe color. For example, Figure 3 for GAMA galaxies shows how $M/L_{K}$ can be determined from restframe $(B-V)$ color using the following relation (in Solar units):

[TABLE]

Determining stellar masses from color dependent mass to light ratios is simpler than determining them from detailed SPS modelling of individual galaxies. However, it must be remembered that relationships such as Equation (2) are derived by averaging the results of detailed SPS modelling and consequently suffer from similar problems while at the same time increasing uncertainty as a result of the loss of detailed color information.

The stellar masses in Figure 3 were derived using bruzu03 SPS models fitted to $ugrizYJHK$ optical to near infrared photometry. The advantage of a relation such as Equation (2) over full SED fitting is that photometry is required in many fewer wavebands - in this case just those required to compute the absolute magnitudes $M_{B}$ and $M_{V}$ using K-corrections (Equation 1).

Galaxies in the $z<0.4$ GAMA survey have redshifts with a median value of 0.2 (taylo11) and evolution of the relation Equation (2) is not apparent in the data. However our data extend to $z=1.2$ and it cannot be assumed that evolution of $\log_{10}(M/L_{K})$ as a function of restframe $(B-V)$ is not significant over this larger redshift range. Indeed the SFHs of the stellar populations that lead to Equation (2) at $z\sim 0.2$ may not be the same as those of a comparable sample at $z\sim 1.2$ so some evolution in Equation (2) is to be expected.

We therefore endeavoured to measure evolution of the dependence of $\log_{10}(M/L_{K})$ on $(B-V)$ using the G10 catalog, which contains consistent total flux measurements across 38 far-UV to far-IR for sources in a  1 deg2 subset of the COSMOS region (davie15a; andre17). The G10 catalog employed the LAMBDAR code (wrigh16) to largely eliminate systematic error arising from the different flux measurements and reduction methods used by constituent COSMOS surveys. It aimed to produce a catalog extending to $z=1$ which was consistent with the low redshift ($z<0.4$) spectroscopic GAMA survey (drive11; liske15).

Stellar masses for galaxies in the G10 catalog were calculated using MAGPHYS (dacun08). As shown by the example in Figure 4, for each redshift bin, we plotted G10 values of $\log_{10}(M/L_{K})$ and 1-$\sigma$ deviations against restframe $(B-V)$ color and determined the median y-axis values of $\log_{10}(M/L_{K})$ for different color bins on the x-axis (solid points and vertical bars).

We performed a linear best fit to the median $\log_{10}(M/L_{K})$ values for colors in the range 0.3 to 0.7. Figure 5 shows how these median $\log_{10}(M/L_{K})$ values and best fit relationships have evolved with redshift.

To measure evolution of $\log_{10}(M/L_{K})$ as a function of restframe color, the best fit median $\log_{10}(M/L_{K})$ values corresponding to $(B-V)$ colors of 0.3, 0.4, 0.5, 0.6 and 0.7 were plotted against redshift, as in Figure 6. (Each curve in this plot corresponds to the intersections of a vertical line of constant color with the best fit lines in the preceding plot.)

Equations for the linear best fits to the five plots in Figure 6 are shown in the top right hand corner. It turns out that they can all be described to within 0.01 dex by just one function of $(B-V)$ and redshift, namely:

[TABLE]

We derived Equation (3) using $(B-V)$ colors between 0.3 and 0.7 because a primary focus of this paper is to study evolution of the stellar mass in red galaxies and this restframe color range is sufficient to encompass all red sequence galaxies at $z<1.2$. Also, a close linear fit was apparent for colors between 0.3 and 0.7, random error being $\sim 0.2$ dex and systematic error being up to $\sim 0.03$ dex. For galaxies bluer than $(B-V)=0.3$, the random error is similar but the systematic error greater - up to $\sim 0.06$ dex.

As the example plot in Figure 4 illustrates, the 1-$\sigma$ scatter in $\log_{10}(M/L_{K})$ values for given restframe color is generally $\sim$0.2 dex. However, some of this variation is due to random error in the G10 photometry rather than just intrinsic variability amongst galaxies. taylo11 found a smaller variation of only $\sim$0.1 dex for GAMA galaxies at $z<0.4$, as can be seen in Figure 3. We take their value $\sigma=0.1$ dex as intrinsic scatter of $\log_{10}M_{K}/L$ at fixed $(B-V)$.

As Figure 5 shows, our evolving relationship, Equation (3), exhibits a broadly similar dependence on restframe color to the GAMA one, but it evolves at a rate that depends on color, this rate being slightly faster for red galaxies than for blue. This is more clearly seen in Figure 6.

As is clear from Figure 5, the 2MASS based $\log_{10}(M/L_{K})$ values from bell03 vary little with $(M_{B}-M_{V})$ color, whereas our values and those based on GAMA are a strong function of color. bell03 used the observed SDSS+2MASS colors of galaxies and masses determined by SED fitting, and we suspect that the difference is due to 2MASS underestimating the fluxes of star forming galaxies causing their $M/L_{K}$ values to be greater than ours by $\sim$0.4 dex for the bluest galaxies but less than $\sim$0.1 dex for the reddest galaxies. When converted for a Chabrier IMF, the bell03 relation is:

[TABLE]

Equation (3) is based on the mean properties of an ever-changing population of red G10 galaxies, i.e. a population which is continually being augmented by blue galaxies that have ceased star formation, and one in which major and minor mergers and possibly bursts of star formation are changing the demographics of the population.

By contrast, in the case of massive red galaxies, we are predominantly measuring how individual massive galaxies fade and grow via mergers. As the evolutionary history of these massive red galaxies is different to that of the majority of red galaxies we expect their dependence of $\log_{10}(M/L_{K})$ on restframe $(B-V)$ color to evolve differently with time. We therefore investigated this separately using a subsample with $\log_{10}M>10.75$. This mass cut-off was chosen to ensure that a range of masses was included on both sides of the mass corresponding to the fixed space density of $\widetilde{\phi}=2.5\times 10^{-4.0}{h_{70}}^{3}{\rm{Mpc}}^{-3}\textrm{dex}^{-1}$ that we used to measure evolution of massive galaxies (see Section 8 below).

At $z>0.8$, only the reddest ($B-V>0.6$) massive galaxies exist in significant numbers in the G10 data to permit a reliable analysis, so we confined ourselves to studying massive red galaxies with a single restframe color of $(B-V)=0.7$. For massive red galaxies we found:

[TABLE]

We used Equation (3) for our measurement of red galaxy SMD evolution but Equation (5) for our measurement of the evolution of the massive red galaxies ($\log_{10}M>10.75$). As explained later in Section 8, these two measurements are based on entirely separate calculations involving Schechter function fitting with different mass ranges and different $\alpha$ constraints (fixed and variable respectively). We used only Equation (3) when measuring evolution of red galaxy SMD, and did not treat massive red galaxies separately. To do so would have introduced a discontinuity into our stellar mass measurements, and would not in any case have affected our red galaxy SMD measurements, because the presence of a small number of massive galaxies contributes a negligible fraction to overall SMD.

We note that massive red galaxies of restframe color $B-V=0.7$ evolve more rapidly in $(M/L_{K})$ (0.32 dex per unit redshift) than red galaxies of the same restframe color as a whole (0.26 dex per unit redshift). This is to be expected because, to some extent, new arrivals from the blue cloud start off their red sequence life with similar properties, whether they arrive at z = 1.1 or z = 0.1. We therefore expect the properties of red galaxies as a whole to evolve less rapidly than those of individual massive red galaxies that are not being “diluted” by new arrivals and affected by major mergers, and this is what we observe.

It is instructive to compare the rate of $(M/L_{K})$ evolution for individual massive red galaxies to those of bruzu03 SPS models with different metallicities and formation redshifts. For solar metallicity simple stellar populations (SSPs) the rates of evolution in $M/L_{K}$ from $z=1.2$ to $z=0$ are -0.20, -0.24 and -0.51 dex per unit redshift for formation redshifts $z_{f}=10.0$, 4.1 and 1.4 respectively.

From Equation (5), calculated rates of evolution of $\log_{10}(M/L_{K})$ for individual galaxies depend on redshift and the evolving restframe $(B-V)$ color. If we assume reddening of 0.1 mag per unit redshift for the most luminous red galaxies, as observed in our Boötes data and as predicted by SSPs, the net rate of change in $\log_{10}(M/L_{K})$ is $\sim-0.4$ dex per unit redshift. This lies between the rates of change for solar metallicity SSPs with $z_{f}=4.1$ and $z_{f}=1.4$. We conclude therefore that the empirical Equation (5) is broadly consistent with the predictions of bruzu03 SSP models.

Our models of an evolving $M/L_{K}$ relation that is a function of rest-frame $(B-V)$ extend the work of taylo11, showing $M/L$ relations can be determined with 0.2 dex accuracy using a single restframe color out to $z\sim 1$. It should be mentioned that lopez18 reached a conflicting conclusion, i.e. that the dependence of $M/L_{i}$ on restframe $(g-i)$ color has not evolved since $z=1.5$.

We find in this study that it is essential to include evolution of $M/L_{K}$ as a function of $(B-V)$ if our measurements of SMF evolution since $z=1.2$ are to agree with previous studies that derive stellar masses directly from SPS modelling rather than via mass to light ratios. An evolving $M/L_{K}$ versus $(B-V)$ relationship is also essential if our measurements of rates of red galaxy stellar mass evolution are to agree with those inferred from comparisons with passively evolving stellar populations.

6. Sample selection

The same cuts as in beare15 were applied to limit the sample to objects brighter than $I=24.0$ and $[3.6\mu\textrm{m}]=23.3$, and to exclude stars. We used the modified stern05 mid-infrared selection criteria that we used previously in beare15 to exclude AGN, with additional AGN identifications being made with SDSS and AGES spectroscopy. The AGN cuts removed less that 1% of our galaxy sample. Quiescent and star-forming galaxies were separated using the same evolving red-blue color cut in restframe color-magnitude space:

[TABLE]

The final sample size for galaxies at redshifts $0.2\leq z<1.2$ with absolute magnitudes $-24\leq M_{B}<14$ was 353 594. The sample was 85% complete at our faint limit of $I=24.0$ and $\simeq 100\%$ complete at $I=21.5$. We corrected for the small degree of incompleteness over the range $21.5<I<24.0$.

7. Determining the $K$-band luminosity function

$K$-band LFs were determined for red and blue galaxy sub-samples separately, as well as for the total sample. In each case, galaxies in the redshift range $0.2\leq z<1.2$ were allocated to five redshift bins of equal width $\Delta z=0.2$. For each redshift bin, empirical binned LFs, $\Phi(M_{K})$, were obtained by dividing the (completeness corrected) numbers of galaxies $N$ in $K$-band absolute magnitude bins of width $\Delta M_{K}$ by the comoving volume $\Delta V$ corresponding to the given redshift range $\Delta z$, i.e. $\Phi(M_{K})=N/\Delta V$.

For each redshift bin, we used the maximum likelihood method (marsh83) to fit schec76 functions to the (unbinned) absolute magnitude data over a magnitude range no wider than that over which the sample was complete to $I=24.0$. This range was determined from a plot of $(M_{K}+D_{\rm{M}}-I)$ against redshift. In terms of luminosities, the Schechter function is:

[TABLE]

Here $\phi_{K}$ is the comoving space density per unit increment in luminosity $L_{K}$, $\phi_{K}^{*}$ is a normalising factor (the characteristic space density), $L_{K}^{*}$ is the characteristic luminosity corresponding roughly to the transition from a power law LF to an exponential one, and $\alpha$ determines the slope of the power law variation at the faint end. The characteristic space density $\phi_{K}^{*}$ provides an approximate measure of the space density close to the characteristic magnitude (more specifically $\phi_{K}=1.086\phi_{K}^{*}$ at $M_{K}=M_{K}^{*}$). The maximum likelihood method provides an estimate of Schechter fit parameter uncertainties.

It is often more useful to write the Schechter function in terms of absolute magnitudes $M_{K}$:

[TABLE]

Integrating over all luminosities gives the total luminosity density $j_{K}$ in waveband $K$:

[TABLE]

Because $\alpha$ becomes increasingly poorly constrained as redshift increases, we kept $\alpha$ fixed at values corresponding to those for the lowest two redshift bins ($0.2\leq z<0.4$ and $0.4\leq z<0.6$) where $\alpha$ was a free parameter, namely -0.5,  -1.2 and -1.0 for red, blue and all galaxies respectively.

To measure luminosity evolution of the most luminous galaxies we determined how the restframe magnitude $\widetilde{M}_{K}$ corresponding to a fixed space density of $\widetilde{\phi}=10^{-4.0}{h_{70}}^{3}{\rm{Mpc}}^{-3}{\rm{mag}}^{-1}$ had evolved. In order to do this as precisely as possible, we fitted a Schechter function with variable $\alpha$ parameter just to the bright end of the LF (i.e. using only data points brighter than $K$-band magnitude -23.0). This procedure effectively uses the Schechter function as a convenient measuring tool, the precise values of the Schechter parameters not having any other relevance).

8. Determining the SMF

We measured binned SMFs of red, blue and all galaxies in the same manner as the binned LFs, including determining mass limits using the apparent magnitude limits and the mass-to-light ratios of passive galaxies. To parameterise our SMFs and measure their evolution we used the maximum likelihood method to fit Schechter functions to the (unbinned) galaxy masses over a mass range no wider than that over which the sample was complete to $I=24.0$:

[TABLE]

Here $\phi_{M}$ is the comoving space density per unit increment in stellar mass $M$, $\phi_{M}^{*}$ is a normalising factor, $M^{*}$ is a characteristic mass, and $\alpha$ determines the slope of the power law variation at the low mass end. We used fixed $\alpha$ values of of -0.5, -1.2 and -1.0 for red, blue and all galaxies respectively.

Given the large range of galaxy masses and to avoid confusion between mass and absolute magnitude, we rewrite Equation (10) in terms of logarithms of the mass $\mu=\log_{10}(M/M_{\sun})$, so the comoving density per unit $\mu$ is:

[TABLE]

Here $\phi_{\mu}^{*}$ is the normalising factor and $\mu^{*}=\log_{10}M^{*}$. At the value $\mu=\mu^{*}$, $\phi=\phi^{*}\ln{10}=2.30\phi^{*}$, so $\phi_{\mu}^{*}$ effectively measures the space density of galaxies per unit $\mu=\log_{10}M$ at the characteristic mass $M^{*}$.

Integrating Equation 10 over all masses gives the total stellar mass density in galaxies:

[TABLE]

To measure evolution of the mass of the most massive galaxies we determined how the value $\log_{10}\widetilde{M}$ corresponding to a fixed space density of $\widetilde{\phi}=2.5\times 10^{-4.0}{h_{70}}^{3}{\rm{Mpc}}^{-3}\textrm{dex}^{-1}$ had evolved. In order to do this as precisely as possible, we fitted a Schechter function with variable $\alpha$ parameter just to the massive end of the SMF, i.e. using only data points for red (blue, all) galaxies for which $\log_{10}(M/M_{\sun})>$ 11.0 (10.5, 10.5). (Because a luminosity ratio of 1.0 dex is equivalent to 2.5 mag, an LF space density of $10^{-4.0}{h_{70}}^{3}{\rm{Mpc}}^{-3}{\rm{mag}}^{-1}$ is equivalent to an SMF space density of $\widetilde{\phi}=2.5\times 10^{-4.0}{h_{70}}^{3}{\rm{Mpc}}^{-3}\textrm{dex}^{-1}$, assuming a fixed stellar $M/L$ ratio. The fixed space densities used for the LF and SMF were therefore equivalent.)

9. $K$-band luminosity evolution - results and discussion

9.1. $K$-band LF evolution - space density and characteristic magnitude

Figures 7 to 9 show our binned $K$-band LFs for all, red and blue galaxies, together with results from a variety of previous studies. Maximum likelihood fits to our (unbinned) data are over-plotted as continuous red lines. To provide a fixed reference we show the local LFs of kocha01 for all, red and blue galaxies in each bin. We only plot bins for which 97.7% (the $2-\sigma$ limit) of the measured absolute magnitudes have observed magnitudes brighter than our faint limit of $I=24.0$.

Our LFs are often brighter than the prior literature, with differences of up to $\sim$0.5 mag evident for $z\sim 0.3$, where bright galaxies are relatively well-resolved. We attribute this to our careful accounting for light falling outside the photometric aperture by the use of magnitude dependent aperture sizes and corrections based on growth curves of measured magnitude with aperture diameter (beare15). Figure 10 shows more clearly the difference between our LF and those of other authors for all galaxies at $0.2<z\leq 0.4$. (Our binned data points lie above unity for the faintest galaxies, i.e. $M_{K}-5\log h_{70}>-21$, because our maximum likelihood Schechter fit in Figure 7 underestimates the space density at fainter magnitudes.) We measure significantly greater space densities for the brightest galaxies than other authors.

Figures 11 to 13 show our maximum likelihood Schechter fits (continuous lines) as well as our binned space densities (data points), with all redshift bins on a single plot to make the evolution of the LFs more apparent.

We see from Figures 14 and 15 and Table 4 that the characteristic space density $\phi_{K}^{*}$ approximately doubled for both red and blue galaxies from $z\sim 1.1$ to $z\sim 0.3$, while the characteristic magnitude $M_{K}^{*}$ of red galaxies faded by $\sim$0.4 mag and that of blue by $\sim$0.7 mag. The more rapid fading of $M_{K}^{*}$ for blue galaxies is consistent with downsizing (cowie96), i.e. the greater proportion of high mass galaxies seen in the star-forming population at $z\sim 2$ than at $z\sim 0$.

Luminosity density, as calculated by Equation 9, is a more physically meaningful quantity than the three individual Schechter parameters, because these relate to a specific functional representation of the LF rather than measurable physical quantities. Furthermore, luminosity density is relatively insensitive to degeneracy amongst the three Schechter parameters, because it is effectively the luminosity weighted area under the LF. This is illustrated by the fact that jones06 obtained a value for $\phi_{K}^{*}$ for all galaxies which is half that of bell03, and an $\alpha$ value of -1.16 as compared with -0.77, but the two studies obtained luminosity densities within 4% of each other.

9.2. $K$-band luminosity density evolution

Figure 16 plots evolution of the total luminosity density $j_{K}$, showing that it increased by 20% $\pm$10% (0.08 dex) from $z\sim 1.1$ to $z\sim 0.3$ for all, red and blue galaxies. Also plotted are the results of ciras10, arnou07 and drory03, and luminosity densities for the same six low redshift studies as in Figures 14 and 15.

For red galaxies, the measured luminosity density increase by a factor of $\sim 1.2$ implies a build up of stellar mass, because a passively evolving stellar population fades as it evolves. As in beare15, we estimated the increase in SMD, $\rho$, by comparison with a passively fading quiescent population. Taking a single burst bruzu03 SSP with formation redshift $z_{f}=3.0$, Solar metallicity $Z=0.02$ and Chabrier IMF to be representative of quiescent stellar populations, evolution of the stellar mass to light ratio $\Upsilon=\rho/j_{K}$ is given by $d\Upsilon/dz=-0.24$, resulting in passive fading of $\sim$0.56 mag from $z=1.1$ to $z=0.3$. From this we deduce that red galaxy SMD increased by a factor of $\sim$2.1 from $z=1.1$ to $z=0.3$. Varying $z_{f}$ between 4.0 and 1.5 or adopting a very low metallicity of $Z=0.0001$ alters the measured SMD increase by less than 20%.

9.3. $K$-band luminosity evolution of highly luminous galaxies

As argued by bell04, luminous red galaxies have evolved by a combination of passive stellar fading and the addition of stellar mass through mergers, because highly luminous blue galaxies are too rare to account for the growth in luminous red galaxy numbers via cessation of star formation. The evolution of the bright end of the red galaxy LF thus approximates the luminosity evolution of the most luminous red galaxies. Figure 17 shows how we used a Schechter function with variable $\alpha$ fitted to the bright end of the LF to measure evolution of the bright end of the red galaxy LF at a space density of $\widetilde{\phi}=10^{-4.0}\,h_{70}^{3}\,\rm{Mpc}^{-3}\,{\rm{mag}}^{-1}$.

Figure 18 plots the evolving absolute magnitude of galaxies having a space density of $\widetilde{\phi}=10^{-4.0}\,h_{70}^{3}\,\rm{Mpc}^{-3}\,{\rm{mag}}^{-1}$, which measures the luminosity evolution of the most luminous galaxies. For red galaxies and blue galaxies there are decreases in luminosity of 0.19 mag and 0.33 mag (0.08 and 0.13 dex) respectively from $z\sim 1.1$ to $z\sim 0.3$.

Figure 18 indicates that highly luminous red galaxies fade at an ever-increasing rate from $z\sim 1.1$ to $z\sim 0.3$. For a passively evolving model with a formation redshift of $z_{f}=3.0$, we expect $\sim$0.56 mag of fading in the $K$-band between $z=1.1$ and $z=0.3$. Given the $\sim$0.1 mag fading we see in the absolute magnitudes of the brightest red galaxies, this implies that the most massive red galaxies increased in mass by a factor of $\sim$1.4 between $z=1.1$ and $z=0.3$.

9.4. $K$-band luminosity evolution - errors

Random and/or systematic uncertainties arise from the aperture photometry, photometric redshifts, cosmic variance, absolute magnitude determinations (K-corrections), Schechter parameterization of the LF, choice of red/blue cut, and Eddington bias at the bright end. These uncertainties impact our measurements of evolution of the LF, the red galaxy luminosity density and the luminosity of highly luminous red galaxies.

As an important focus in this paper is to measure evolution of luminosity and stellar mass in red galaxies, we give order of magnitude estimates for uncertainties for red galaxies in Table 5. The table indicates how uncertainties propagate through our calculations of LF and red galaxy stellar mass evolution, and shows which uncertainties are of greatest significance. Note that all uncertainties in the table have been given in dex for consistency (e.g. magnitudes have been multiplied by 0.4). Most of the uncertainties are discussed in detail in beare14 and beare15 but we now discuss each of them in the context of the present paper.

** Photometry* We discuss photometry first as it forms the observational basis of all our work, enabling us to determine both galaxy luminosities and the photometric redshifts we used for the majority of galaxies. As described in beare15, our apparent magnitudes were measured using variable size photometric apertures and corrections for total flux. These aperture sizes and corrections were based on analysis of growth curves of measured magnitude with aperture diameter for isolated galaxy images (i.e. those that did not overlap other objects). This method has the advantage that it is empirically based, and does not assume any specific light profile (e.g. Sersic, de Vaucoleurs, Petrosian, SDSS modelmag). Further, as described in brown07, the use of SExtractor segmentation maps largely eliminates flux from neighbouring objects and corrects for the excluded flux.

We do not consider $(1+z)^{-4}$ cosmological surface brightness dimming (e.g. Calvi et al. 2014) to be a significant issue as the angular size at $z\sim 1$ of a large galaxy with half-light radius of 10 kpc is only 0.6 arcsec, and this is less than the FWHM of our 1.35 arcsec point spread function. Such a galaxy would only be partially resolved and be approximately a point source, which largely mitigates the effect of cosmological surface brightness dimming.

Random uncertainties in measured apparent magnitudes arise from the aperture photometry and from the total flux corrections we make. Aperture photometry uncertainties are greater in some wavebands than others. Overall, averaged over all the galaxies with different apparent magnitudes in our sample, 1-$\sigma$ uncertainties are $\sim$0.1 mag for $B_{W}$, $R$, $I$ and $([3.6\mu m]$, $\sim$0.2 mag for $J$ and $[4.5\mu m])$, $\sim$0.4 mag for $K$. Uncertainties are significantly less for galaxies with brighter apparent magnitudes, as used in measuring the bright end of the $K$-band LF (and the massive end of the SMF). For example, 1-$\sigma$ $K$-band uncertainties for galaxies with reliable photometry in different redshift bins (the filled circles in Figures 7 to 9 are $\sim$0.3 mag, but for galaxies with $M_{K}<-23$, as used in measuring evolution of the bright end of the LF, $K$-band uncertainties increase from 0.04 mag at $z\sim 0.3$ to 0.15 mag at $z\sim 1.1$. From our growth curves of magnitude with aperture diameter we estimate that random uncertainties in the total flux corrections range from $\sim$0.04 mag for the brightest objects to $\sim$0.1 mag for the faintest. Adding these in quadrature to the uncertainties in aperture photometry, we obtain total random photometry errors ranging from $\sim$0.1 mag to $\sim$0.54 mag ($\sim$0.04 dex to $\sim$0.22 dex in Table 5).

Systematic uncertainties in aperture photometry will vary between different observing campaigns and we do not attempt to estimate them here. However, we note that their major effect will be to shift computed luminosities by a constant factor without affecting evolutionary rates of change. We take systematic magnitude uncertainty due to our total flux corrections to be zero as these corrections are as likely to be overestimates as underestimates.

** Poisson uncertainties and cosmic variance* Poisson uncertainties in both the binned and the maximum likelihood space densities are greater at the bright end of the LF where galaxy numbers are fewer, as can be seen in Figures 7 to 13. Cosmic variance produces additional random uncertainty in space densities, which beare15 found using subfields to be $\sim$3% for all galaxies (0.01 dex), and somewhat more for red galaxies that are more strongly clustered, i.e. $\sim$8% (0.03 dex). In combination, these two random uncertainties dominate in all redshift bins over systematic uncertainties arising from our very small systematic fractional photometric redshift errors which are 0.02 at most. This size of systematic error changes the number of galaxies in a redshift bin of width 0.2 by less than 0.01 dex, (e.g. at $z\sim 1$, a shift of 0.02 in the upper boundary of the bin, changes the number in the bin by a factor of $\sim 0.202/0.200$ and this equates to 0.0043 dex).

** Luminosity errors* Errors in $U,B,V$ and $K$-band luminosity measurements arise from photometric redshift uncertainties, photometry uncertainties, and uncertainties in K-correction calculations (Equation 1). All these uncertainties are discussed in detail in beare14. Table 5 indicates how the three sources of uncertainty combine in the case of red galaxy luminosity calculations. K-correction uncertainties are due to the scatter of individual galaxies about the polynomials used to measure absolute magnitudes (Equation 1). In Figure 2 the 1-$\sigma$ K-correction uncertainty is given in the top left hand corner of the plot.

** Photometric redshifts* Photometric redshift uncertainties impact our calculations in several ways: calculation of galaxy luminosities (K-corrections), random and systematic uncertainties in binned and maximum likelihood LF numbers, Eddington bias at the bright end of the LF and scattering of galaxies from one redshift bin to another.

As Figure 1 shows, our 1-$\sigma$ random fractional photometric redshift errors ($\lambda=[z_{\textrm{phot}}-z_{\textrm{spec}}]\,/\,[1+z_{\textrm{phot}}]$) are $\sim 0.05$ (0.02 dex) or less. Our systematic fractional $z_{\textrm{phot}}$ errors are very small, being $<0.01$ (0.004 dex) for $0.2<z_{\rm{phot}}<1.0$ and $<0.02$ (0.008 dex) for $1.0<z_{\rm{phot}}<1.2$.

The impact of photometric redshift errors is mitigated, especially at the bright end of the LF, by the use of spectroscopic redshifts which are available for $\sim$75% of red and blue $I<19.5$ galaxies and $\sim$45% of $I<20.5$ galaxies.

Photometric redshift uncertainties perturb measured $K$-band restrame magnitudes by altering distance moduli $D_{M}$ in Equation (1). Assuming an approximate inverse square law, the fractional uncertainties in measured luminosities will be twice the fractional uncertainties in $z_{\textrm{phot}}$ values, i.e. random fractional uncertainties of $\sim$0.02 dex, and systematic fractional uncertainties of 0.01 dex for $0.2<z_{\rm{phot}}<1.0$ and $\sim$0.02 dex for $1.0<z_{\rm{phot}}<1.2$.

It is important to realise that these estimated errors arising from $z_{\textrm{phot}}$ errors are for individual galaxies. In the case of systematic redshift errors, the impact on evolutionary measurements depends on the distribution of $z_{\textrm{phot}}$ errors with redshift. Figure 1 shows that photometric redshifts are very slightly overestimated at $z\sim 0.25$ and $z\sim 0.7$, and underestimated at $z\sim 1.0$. We do not pursue this further here, beyond noting that the effect of $z_{\textrm{phot}}$ errors on evolutionary measurements will potentially be comparable with that on individual values.

To gauge the impact of systematic $z_{\textrm{phot}}$ errors, we repeated all our calculations twice using $z_{\textrm{phot}}$ values increased and decreased by the fractional systematic error over most of the redshift range (i.e. $0.2<z\leq 1.0$) as shown in Figure 1. This was $\lambda=[z_{\textrm{phot}}-z_{\textrm{spec}}]\,/\,[1+z_{\textrm{phot}}]=0.01$. For the increased (decreased) $z_{\textrm{phot}}$ values, red galaxy luminosity density values decreased (increased) by up to $\sim$0.02 dex while massive red galaxies showed a luminosity increase (decrease) of up to $\sim$0.01 dex.

To measure the effect of the random $z_{\textrm{phot}}$ uncertainties $\lambda=[z_{\textrm{phot}}-z_{\textrm{spec}}]\,/\,[1+z_{\textrm{phot}}]=0.05$ we repeated our calculations ten times, each time applying normally distributed random fractional errors ($\sigma=0.05$) to individual $z_{\textrm{phot}}$ values. Our results are shown in the penultimate section of Table 5. We found that individual measured values of both red galaxy luminosity density and highly luminous red galaxy luminosity differed between simulations by less than 0.01 dex, indicating that random photometric redshift errors did not produce significant scatter in these two measurements.

However, due to random $z_{\textrm{phot}}$ uncertainties, the measured luminosities of luminous red galaxies (at fixed space density) showed a small systematic shift which increased from $\sim$0.01 dex for $0.2<z\leq 0.4$ to $\sim$0.04 for $1.0<z\leq 1.2$. This shift is partly an Eddington shift (eddin13) caused by greater numbers of galaxies being randomly scattered to higher luminosities than to lower luminosities, on account of the steeply declining exponential shape of the LF at high luminosity. The remainder of the observed systematic shift is due to galaxies being scattered between redshift bins: if the LF evolves with redshift this scattering across redshift bin boundaries will modify the measured LFs in individual redshift bins. As the fraction of red galaxies with spectroscopic redshifts decreases with increasing redshift, the impact of redshift errors on the bright end of the luminosity function increases with redshift. All these different effects of random redshift errors are highly correlated and very difficult to analyse analytically. This is the reason why we used Monte Carlo simulations to measure the overall effect of random $z_{\textrm{phot}}$ uncertainties.

Monte Carlo simulations also show that random $z_{\textrm{phot}}$ errors give rise to a systematic decrease in measured luminosity density. This systematic error ranges from $\sim$0.01 dex at $0.2<z\leq 0.4$, where a significant proportion of galaxies have accurate spectroscopic redshifts, to 0.07 dex at $1.0<z\leq 1.2$, where few galaxies have spectroscopic redshifts. The error in measured luminosity densities can be attributed to galaxies being scattered across redshift bin boundaries with changed measured luminosities. For example, a galaxy with $\Delta z_{\textrm{phot}}=0.05$ and $z=0.98$ will be scattered upwards across the $z=1$ boundary into the next redshift bin and its measured luminosity will be increased by $\sim$10%, assuming an approximate inverse square law. The space density at the faint end of the LF increases with increasing luminosity, rather than decreases, so at this end there is a net flow of galaxies to lower redshift bins and lower luminosities. The observed systematic change in luminosity density in each redshift bin is the net result of perturbed luminosity values and of galaxies being scattered in and out of the redshift bin at the upper and lower bin boundaries.

In addition to random photometric redshift errors, random photometry and K-correction errors also contribute to the random errors in our $K$-band luminosity measurements. This increases the Eddington shift in our measurements of the bright end of the LF for red galaxies. We modelled the photometric and K-correction errors by convolving the Schechter function for red galaxies with a Gaussian with $\sigma=0.03$ dex (Table 5) and found that the resultant shift in the luminosity corresponding to our fixed space density of $\widetilde{\phi}=10^{-4.0}{h_{70}}^{3}{\rm{Mpc}}^{-3}{\rm{mag}}^{-1}$ was increased by 0.03 dex.

** Schechter parameterization* In measuring evolution of the $K$-band LF and the $K$-band luminosity density, uncertainty arises from parameterizing the LF using a single Schechter function and using this parameterization to compute these quantities. beare15 found that the $B$-band luminosity density varied by less than 0.03 dex when $\alpha$ was varied by $\pm 0.1$, pointing out that for red galaxies it is insensitive to the exact value of $\alpha$ chosen, even though $M^{*}$ and $\phi^{*}$ vary considerably. We assume that similar conclusions will apply in the case of the $K$-band LF and take 0.03 dex as the random uncertainty in $K$-band luminosity density due to the inexactness of the Schechter parameterization.

** Red-blue cut* Our evolving red-blue cut is intended to separate quiescent and star forming galaxies, but the correspondence is not exact due to factors such as residual star formation in some quiescent galaxies, and dust obscuration which causes some star forming galaxies to appear redder than they really are. The best position to adopt for the red-blue cut involves some uncertainty. We investigated this in beare15 and found that moving the position of the cut up or down by 0.05 mag made 0.05 dex difference to the measured optical luminosity density of red galaxies and 0.02 dex difference to the measured luminosity of luminous red galaxies. We assume here that similar differences will apply to the $K$-band LF and the SMF.

** Conclusion* Table 5 shows that systematic errors in red galaxy luminosity density range from $\sim$0.06 dex at $z=0.3$ to $\sim$0.12 dex at $z=1.1$ and are larger than the random uncertainties of $\sim$0.04 dex. Potentially, correcting for the change in systematic error with redshift could decrease the measured increase of luminosity density from $z=1.1$ to $z=0.3$ by $\sim$0.06 dex, altering the luminosity density growth from $0.08\pm 0.04$ dex ($\times 1.20\pm 0.11$) to $0.02\pm 0.04$ dex ($\times 1.05\pm 0.10$).

Systematic errors in the measured luminosity of highly luminous red galaxies range from $\sim$0.08 dex at $z=0.3$ to $\sim$0.13 dex at $z=1.1$ and dominate the random uncertainties which Monte Carlo simulations show to be less than 0.01 dex. Potentially, correcting for the change in systematic error with redshift could decrease the measured luminosity fading of highly luminous red galaxies from $z=1.1$ to $z=0.3$ by $\sim$0.05 dex, altering the measured luminosity decrease from $0.08\pm 0.01$ dex ($\times 0.83\pm 0.03$) to $0.03\pm 0.01$ dex ($\times 0.93\pm 0.02$).

To summarise, systematic errors could significantly reduce the 20% growth in red galaxy luminosity density to 5% and significantly reduce the 17% fading of luminous red galaxies to 7%.

9.5. $K$-band luminosity evolution - comparison with the literature

As Figures 7 and 10 show, the bright end of our LF for all galaxies at $0.2\leq z<0.4$ is $\sim$0.3 mag brighter than the LFs reported by several authors for the low redshift ($z<0.2$) Universe.

A small part of this difference will be due to evolution of the LF between $z\sim 0.3$ and $z\sim 0.1$, but we expect our measured magnitudes to be brighter than those from studies such as kocha01, cole01, bell03 and bonne15 which used the Two Micron All Sky Survey extended source catalog data (2MASS, jarre00). Because 2MASS is relatively shallow, the faint outer parts of extended sources become lost in the 2MASS sky background. andre02 demonstrated that the isophotal magnitude models used by authors such as kocha01 and cole01 underestimate the flux from extended sources by even more than the $\sim$0.2 mag that these authors predicted, even in the case of bright galaxies. andre02 also showed that 2MASS failed to detect low surface brightness galaxies.

By contrast, we accounted for light falling outside the photometric aperture by using growth curves of measured apparent magnitude with aperture size (beare15). E.g. for a galaxy with an apparent $I$-band magnitude of 20.5 mag we used an aperture of diameter 8 arcsec and made a correction for missing flux of -0.070 mag, while at magnitude -23.5 we used a 3 arcsec aperture and made a correction of -0.302. In beare15 we compared our photometry with MAG_AUTO and found it to be brighter by $\sim$0.06 mag for $I<20$, rising to $\sim$0.13 at $I=24$. These factors adequately account for the difference between our LFs and those based on 2MASS, as seen in Figures 7 to 10.

Local surveys do have a strong dependence on the type of photometry used, but an advantage of a deep survey like ours with partially resolved galaxies is a reduced dependence on the intrinsic light profiles of the galaxies. For example, when the galaxy is far smaller than the PSF, aperture photometry with PSF corrections is adequate. A deep survey, with high signal-to-noise, can also afford to use oversized apertures for photometry of large galaxies whereas shallower surveys using comparable apertures could be swamped by noise.

Figure 7 shows that at higher redshifts agreement with the evolving LFs for all galaxies of drory03 and ciras10 is somewhat uneven: it is closer at some redshifts than others. Both these studies used photometric redshifts as we did, but their sample sizes were much smaller than ours ($\sim$70 and $\sim$10 times smaller respectively), so larger Poisson errors and cosmic variance can explain why there were greater differences from our work at some redshifts than others.

Figures 14 and 15 compare our measurements of the evolution of $\phi_{K}^{*}$ and $M_{K}^{*}$ with those of ciras10, arnou07 and drory03 and show values for the low redshift Universe from several authors. ciras10 and arnou07 fitted an evolving functional form to their data so that their parameters were forced to vary smoothly with redshift.

It is difficult to compare $\phi_{K}^{*}$ and $M_{K}^{*}$ evolution measurements from different studies because the well-known degeneracy between the Schechter parameters means that different $\alpha$ value choices give rise to different measured values for $\phi^{*}$ and $M^{*}$, e.g. for the $B$-band LF of all galaxies at $0.2\leq z<0.4$, beare15 found that increasing $\alpha$ from -1.1 to -1.0 decreased $\phi_{B}^{*}$ by $\sim$20% while making $M_{B}^{*}$ $\sim$0.1 mag brighter. The value $\alpha=-0.9$ adopted by ciras10 and drory03 for all galaxies is 0.1 greater than our value of -1.0 and this explains their fainter $M_{K}^{*}$ and smaller $\phi_{K}^{*}$ values. Similarly, at low redshift, discrepancies between different authors are accounted for by differences in the $\alpha$ values adopted, with jones06 and bonne15 using the largest (most negative) values (see values in Table 6 of bonne15).

Taking into account the degeneracy between $\phi_{K}^{*}$, $M_{K}^{*}$ and $\alpha$, the lower faint end space densities and fainter magnitudes expected from studies based on 2MASS, and the different subsample criteria used by different authors, we conclude that our $\phi_{K}^{*}$ and $M_{K}^{*}$ measurements are not inconsistent with those from previous studies.

Figure 16 shows that we measured a higher luminosity density at all redshifts for all galaxies than the studies of ciras10 and drory03, while obtaining comparable values to arnou07, but much smoother evolution. Our measured luminosity density at low redshift is higher than the literature, much of which utilises the relatively shallow 2MASS data.

As Figure 18 shows, bright end galaxies were 0.2 to 0.3 mag brighter than those in ciras10 and drory03 and varied more smoothly in luminosity than those of arnou07. Recent studies (e.g. dsouz15; loved15; berna16) have shown that measurements of the bright end of the LF are highly sensitive to the photometric model used (e.g. S rsic, Petrosian, SDSS cmodel) as this affects how much light from the faint outer edges of extended galaxies is measured. Our photometric corrections based on growth curves were model independent and should have provided good estimates of total galaxy light, even though we did not derive different corrections for quiescent and star-forming galaxies which generally exhibit de Vaucouleurs and S rsic light profiles respectively. We therefore expect the bright end of our LFs to be brighter than much of the literature and Figures 7 to 9 show that this is the case by $\sim$0.3 mag.

10. Stellar mass evolution - results and discussion

10.1. Evolution of the stellar mass function

In order to compare our SMF results with the literature we plot our binned SMFs alongside those from a variety of previous studies in Figures 19 to 22. Maximum likelihood fits to our (unbinned) data are over-plotted as continuous red lines. To provide a fixed reference in the plots we show the local SMF for all galaxies (i.e. red and blue combined) from cole01 in each bin. We only plot bins for which 97.7% (the $2\sigma$ limit) or more of the measured masses correspond to magnitudes brighter than our faint limit of $I=24.0$.

In order to make evolution of our SMFs more apparent, we show our maximum likelihood Schechter fits (continuous lines) and binned space densities (data points) in Figures 23 to 25 with all redshift bins on a single plot.

Table 6 shows that from $z\sim 1.1$ to $z\sim 0.3$ the characteristic space density $\phi_{M}^{*}$ approximately doubled for red galaxies and increased by somewhat more ($\sim\times 2.4$) for blue galaxies. At the same time the characteristic mass $M^{*}$ of both red and blue galaxies changed by no more than 0.07 dex or $\times$ 1.17.

10.2. Evolution of stellar mass density

We quantified the growth of stellar mass within the red and blue galaxy populations with the SMD, which has a clear physical meaning and is effectively the area under the SMF curve. Figure 26 and Table 6 show our results based on $K$-band $M/L$ ratios. SMD is not as prone to degeneracies as the Schechter parameters are. For example, we showed in beare15 how varying the adopted fixed value for $\alpha$ significantly affected the measured values of $M^{*}$ and $\phi^{*}$ in the case of the $B$-band LF, but hardly affected the measured luminosity density at all. Similar behaviour is to be expected in the equivalent case of SMFs and SMD.

We found an increase of $\sim$0.37 ($\sim$0.31) dex in SMD for red (blue) galaxies from $z=1.1$ to $z=0.3$, i.e. a factor of $\sim$2.3 ($\sim$2.1). We note that the red galaxy SMD growth of $\times 2.1$ implied by comparison of $K$-band luminosity evolution with a passively evolving stellar population (Section 9.2) is very close to the $\times 2.3$ growth deduced here from evolution of the SMF. Figure 26 shows that for red, blue and all galaxies the rate at which SMD is growing decreases slowly with time. For red galaxies, this indicates that the rate at which blue galaxies move to the red sequence as they cease star formation decreases slowly with time.

10.3. Evolution of massive galaxies

In order to look quantitatively at the mass growth of the most massive galaxies, in Figure 27 we show the redshift evolution of stellar mass at a fixed comoving space density of $\widetilde{\phi}=2.5\times 10^{-4}{h_{70}}^{3}{\rm{Mpc}}^{-3}\textrm{dex}^{-1}$. This figure is directly comparable to the evolving luminosity at fixed density shown in Figure 18. These results indicate that these most massive red galaxies grew in stellar mass from $z=1.1$ to $z=0.3$ by 0.23 dex, i.e. a factor of $\sim$1.7. Our results also indicate stellar mass growth of 0.15 dex ($\times 1.4$) for massive blue galaxies and 0.21 dex ($\times 1.6$) for all massive galaxies, but it must be remembered that only for red galaxies is the measured stellar mass growth that for individual massive galaxies.

At a fixed space density threshold of density of $\widetilde{\phi}=2.5\times 10^{-4}{h_{70}}^{3}{\rm{Mpc}}^{-3}\textrm{dex}^{-1}$, the most massive blue galaxies are $\sim$0.3 dex lower in stellar mass than the most massive red galaxies, i.e. half the mass. While we expect the most massive red galaxies to remain on the red sequence and increase in mass via mergers, it is likely that the most massive blue galaxies have star formation quenched and then move onto the red sequence.

The rate at which massive red galaxies increase in stellar mass through mergers with smaller galaxies appears to slow somewhat from $\sim$0.4 dex per unit redshift for $z=1.1$ to $z=0.7$ to $\sim$0.2 dex per unit redshift for $z=0.7$ to $z=0.3$.

We know that massive early-type galaxies must have grown in stellar mass because observations clearly show them experiencing mergers sufficient in frequency and mass ratio to give rise to a significant increase in stellar mass. In fact merger studies have produced a range of estimates of stellar mass growth. For example, dokku05 found a stellar mass increase due to mergers in massive red galaxies of $\Delta M/M=0.09\pm 0.04$ per Gyr, which implies a stellar mass increase over the 4.7 Gyr from $z=1.1$ to $z=0.3$ of $\times\sim 1.4$. Similarly, lopez12 found stellar mass growth in massive early type galaxies due to mergers of $\times\sim 1.3$ from $z=1$ to the present. On the other hand, masje08 found that luminous red galaxies were growing due to merger activity at a much slower rate: at least $1.7\pm 0.1h$ per cent per Gyr on average at redshift $\sim$0.25, implying at least 8% growth from $z=1.3$ to $z=0.3$.

The lack of evolution (in comoving coordinates) of the spatial correlation function of massive red galaxies also clearly indicates that the most massive red galaxies must be undergoing mergers. white07 used observations of the clustering of luminous red galaxies to show that about one third of the most luminous satellite galaxies appear to have undergone merging or disruption with massive halos between $z=0.9$ and $z=0.5$, while brown08 found that massive red galaxies grew by $\times 1.3$ from $z=1.0$ to $z=0$.

As with SMD evolution, our stellar mass growth measurements for massive red galaxies based on SMF evolution ($\times 1.7$) are comparable with the those implied by comparison of the $K$-band LF with passive evolution ($\times 1.4$).

10.4. Stellar mass evolution - errors

In addition to the sources of error inherent in measuring $K$-band luminosity evolution (Section 9.4), we have one very significant additional source of error in measuring stellar mass evolution, namely uncertainty in stellar mass to light ratios $M/L_{K}$, and we include this in Table 5 for red galaxies. $M/L_{K}$ uncertainties arise from the evolving $M/L_{K}$ - restframe $(B-V)$ relationships in Equations 3 and 5.

We take the intrinsic random variation in $\log_{10}M_{K}/L$ between galaxies to be 0.1 dex (Section 5). Systematic uncertainties arise from the fact that Equations 3 and 5 are derived from SPS models, and differences occur between different SPS models and the parameters used in them, notably SFH, metallicity and dust obscuration. A number of authors have investigated the relative impact of these different factors and arrived at various conclusions. conro09 found differences in stellar mass estimates of up to 0.3 dex between different models and different parameter inputs. In contrast, moust13 found that varying the SPS model, the SFH and the metallicity had little effect, except that the inclusion of bursts of star formation in the SFH did have a significant impact on the derived SMF. muzzi13b, surveying redshifts up to $z=4$, found that the precise SPS model used was significant, with maras05 models producing stellar masses that are 0.2 dex lower ($\times 0.65$) than those of bruzu03 models. They also found that metallicity and delayed bursts of star formation in the SFH made little difference.

Differences in stellar mass estimates at the 0.3 dex level arising from the use of different SPS models and different SFH, metallicity and dust model inputs constitute a very significant source of uncertainty in our measurements of stellar mass for individual galaxies. However, unless the stellar mass differences between models vary with redshift, they will not impact measurements of stellar mass evolution. We do not attempt to estimate how stellar mass differences between models might vary with redshift (if indeed they do) and therefore do not include them in total error budget for red galaxy stellar mass evolution in Table 5. Also implicit in the use of SPS models is the adoption of a specific stellar IMF, (e.g. salpe55; kenni83; chabr03). However, different choices of IMF effectively only produce offsets in calculated values of $\log_{10}M_{K}/L$ (i.e. stellar masses differ by constant multiplying factors). The choice of IMF does not therefore impact conclusions regarding the percentage stellar mass growth in galaxies (bell01; bell03) and we do not take them into account in the total error budget for red galaxy stellar mass evolution in Table 5.

We note that, although the relationship between $\log_{10}M_{K}/L$ and restframe $(B-V)$ color is redshift dependent (Equations 3 and 5), the effect of $z_{\textrm{phot}}$ errors on $M/L_{K}$ is small (random error of $<0.02$ dex and systematic error of $<0.01$ dex).

It is important to realise that the estimated errors arising from $z_{\textrm{phot}}$ errors are for individual galaxies. In the case of systematic redshift errors, the impact on evolutionary measurements depends on the distribution of $z_{\textrm{phot}}$ errors with redshift. Figure 1 shows that photometric redshifts are very slightly overestimated at $z\sim 0.25$ and $z\sim 0.7$, and underestimated at $z\sim 1.0$. We do not pursue this further here, beyond noting that the effect of systematic $z_{\textrm{phot}}$ errors on evolutionary measurements will potentially be comparable with that on individual values.

As with our measurements of $K$-band luminosity evolution, in order to gauge the overall potential impact of systematic photometric redshift errors, we repeated all our calculations twice using $z_{\textrm{phot}}$ values increased and decreased by the fractional systematic error over most of the redshift range (i.e. $0.2<z\leq 1.0$) as shown in Figure 1. This was $\lambda=[z_{\textrm{phot}}-z_{\textrm{spec}}]\,/\,[1+z_{\textrm{phot}}]=0.01$. The last section of Table 5 shows that for the increased (decreased) $z_{\textrm{phot}}$ values, red galaxy stellar mass density values decreased (increased) by up to $\sim$0.03 dex while massive red galaxies showed a stellar mass decrease (increase) of up to $\sim$0.01 dex.

Again, as with $K$-band luminosities, to measure the effect on stellar masses of random $z_{\textrm{phot}}$ errors of $\lambda=[z_{\textrm{phot}}-z_{\textrm{spec}}]\,/\,[1+z_{\textrm{phot}}]=0.05$ we repeated our calculations ten times, each time applying normally distributed random fractional errors ($\sigma=0.05$) to individual $z_{\textrm{phot}}$ values. We found that individual measured values of both red galaxy SMD and the stellar mass of massive red galaxies differed between simulations by less than 0.01 dex, indicating that random photometric redshift errors did not produce significant scatter in these two measurements.

As with the LF, random photometric errors shift the massive end of the SMF due to Eddington bias and scattering of galaxies across redshift bin boundaries. As the penultimate section of Table 5 indicates, this shift was found to increase from $\sim$0.001 at $0.2<z\leq 0.4$ to 0.04 dex at $1.0<z\leq 1.2$. The increase with redshift is due to the fact that the proportion of massive red galaxies with accurate spectroscopic redshifts decreases from $z\sim 0.2$ to $z\sim 1.2$.

We assume that the difference between stellar masses derived from SED fitting and those derived from the $M_{K}/L$-color relation (Equation 5) has scatter $\sigma=0.1$ dex. We can measure the effect of this scatter on our results by convolving a $\sigma=0.1$ dex Gaussian with our measured Schechter functions. We find an additional contribution of $\sim$0.09 dex to the Eddington shift in the stellar mass of massive red galaxies in all redshift bins.

Monte Carlo simulations also show that random $z_{\textrm{phot}}$ errors give rise to a systematic decrease in measured SMD. This systematic error ranges from $\sim$0.01 dex at $0.2<z\leq 0.4$, where a significant proportion of galaxies have accurate spectroscopic redshifts, to 0.09 dex at $1.0<z\leq 1.2$, where few galaxies have spectroscopic redshifts. As with luminosity density (Section 9.4), the observed systematic change in SMD in each redshift bin is the net result of galaxies being scattered in and out of the bin at the upper and lower bin boundaries with perturbed stellar mass values.

** Conclusion* Table 5 shows systematic errors in red galaxy SMD range from $\sim$0.10 dex at $z=0.3$ to $\sim$0.18 dex at $z=1.1$ and dominate random uncertainties of $\sim$0.04 dex. Potentially, correcting for the change in systematic error with redshift could decrease the measured SMD growth from $z=1.1$ to $z=0.3$ by $\sim$0.08 dex, altering the SMD growth from $0.37\pm 0.04$ dex ($\times 2.34\pm 0.22$) to $0.29\pm 0.04$ dex ($\times 1.95\pm 0.18$).

Systematic errors in the measured stellar mass of massive red galaxies range from $\sim$0.11 dex at $z=0.3$ to $\sim$0.15 dex at $z=1.1$ and dominate over the random errors which Monte Carlo simulations show to be less than 0.01 dex. Potentially, correcting for the change in systematic error with redshift could increase the measured stellar mass growth of massive red galaxies from $z=1.1$ to $z=0.3$ by $\sim$0.04 dex, altering the measured growth from $0.23\pm 0.01$ dex ($\times 1.70\pm 0.04$) to $0.27\pm 0.01$ dex ($\times 1.86\pm 0.04$).

To summarise, systematic errors could have only a small effect on our conclusions for red galaxies, slightly reducing the $\times 2.3$ growth in red galaxy SMD, and slightly increasing the 70% mass growth in massive red galaxies.

10.5. Stellar mass evolution - comparison with the literature

Figures 19 and 22 show that over the mass range $9<\log M<11$, our SMF for all galaxies at $0.2\leq z<0.4$ differs by less than $\sim$0.2 dex ($\sim 50\%$) in space density from other evolutionary SMF studies and from the $z<0.2$ Universe studies of cole01, bell03 and baldr12. As can be seen from Figure 19, the agreement with other studies is also very good at $z>0.4$, apart from the much lower space densities seen in the $0.3\leq z<1.5$ VIPERS study of david13. It is noticeable that our SMFs vary more smoothly with stellar mass than most other studies due to our very large sample size and area.

Figures 20 and 21 show offsets of up to 0.3 dex with the quiescent and star-forming subsamples in other studies, which may well be due to the different subsample criteria they use. We used a restframe color criterion, as did bell03, bundy06, david13 and tomcz14 and this caused our red subsample to include dust reddened star forming galaxies. Such galaxies were explicitly rejected by the multi-wavelength color criterion of ilber13 and the SED modelling approach of moust13.

We note that there is no large difference between our SMFs and the low redshift SMFs of cole01 and bell03 despite these authors’ use of 2MASS $K$-band luminosity values which, even with isophotal magnitude model corrections (Section 9.5), miss much of the light from fainter outer regions of galaxies (andre02). We attribute this agreement in measured SMFs between their work and ours to the fact that their $M/L_{K}$ relationship was also calibrated using their 2MASS $K$-band luminosity values which are lower than those that we would measure.

In Figure 27 we compare our measurements for the mass evolution of very massive galaxies with measurements calculated using the Schechter parameters given by bundy06, perez08, muzzi13b and tomcz14. For massive red galaxies we see stellar mass growth of 0.23 dex ($\times 1.7$), a little more than tomcz14 and considerably more than muzzi13b.

For blue galaxies, and red and blue galaxies combined, we see similar stellar mass growth (0.22 and 0.21 dex respectively) whereas many other studies see little change, except for the growth of $\sim$0.16 dex seen by muzzi13b for red and blue galaxies combined.

Similarly, using the Stripe 82 Massive Galaxy Catalog of 41 770 massive galaxies with SDSS and UKIDSS data, bundy17 recently concluded that the stellar mass in all ($\log M>11.3$) massive galaxies changed by less than 9% from $z=0.65$ to $z=0.3$. They investigated in detail several potential sources of random and systematic error in the determination of stellar masses from photometry, including photometric redshift errors, differences due to the use of different SPS models, and differences due to differing assumed star formation histories in SPS models. They concluded that the latter source of error was the most significant. However they also speculated missing light in their photometry could strongly impact their conclusions.

It must be remembered that the results of other studies of massive galaxies are based on Schechter fits to the whole of the measured SMF, whereas we have based our mass growth measurements on variable $\alpha$ Schechter fits to just the massive end of the SMF. This will be considerably more accurate as we have only been using the Schechter parameterization as a tool to produce a very close fit in order to measure a small section of the SMF at the high mass end (Section 8).

11. Summary

We measured evolution of the $K$-band LF and the galaxy SMF from $z=1.2$ to $z=0.2$ using a very large sample of 353 594 galaxies covering a large area of 8.26 deg2 in Boötes, surveyed to a depth of $I=24$. The imaging, catalogs and photometry were identical to those in beare15 and derived from various optical and infrared surveys. Our very large sample size and area minimised both Poisson errors and the effects of cosmic variance ($\sim$3% for all galaxies and $\sim$8% for red galaxies, which are more strongly clustered).

We used a magnitude dependent aperture diameter and curve of growth analysis to measure precise photometry in 13 optical and infrared bands. Using this photometry and the 129 SED templates from brown14 we were able to precisely determine photometric redshifts (and luminosity distances) using the Bayesian EAZY code.

Galaxy luminosities were derived from apparent magnitudes and redshifts using the method of beare14 and the 129 SED templates of brown14. We used GAMA/G10 COSMOS data to derive the evolving dependence of $K$-band mass to light ratios on restframe $(B-V)$ color and used this to measure stellar masses.

Binned $K$-band LFs and SMFs were produced for five redshift bins between $z=0.2$ and $z=1.2$ and found to be consistent with LFs and SMFs from the literature. LF and SMF evolution were measured using maximum likelihood Schechter function fits within each redshift bin. Red and blue galaxies were differentiated using an evolving restframe $(U-B)$ color-magnitude cut, as in beare15.

Luminosity densities and stellar mass densities were calculated from the Schechter parameters and their evolution measured. Evolution of the bright end of the LF and the massive end of the SMF were measured by finding the luminosity and stellar mass (respectively) corresponding to a fixed space density. This was done by fitting a maximum likelihood Schechter function with variable $\alpha$ parameter to the luminous and massive ends of the LF and SMF (respectively).

As a main focus of our work has been to measure evolution of the $K$-band luminosity and the stellar mass of red galaxies, we made detailed estimates of the various random and systematic errors for red galaxies, tabulating these in Table 5.

The total luminosity density of both red and blue galaxies increased by a modest 0.08 dex, i.e. a factor of 1.2, from $z=1.1$ to $z=0.3$. Over the same redshift range the luminosity of highly luminous red (blue) galaxies decreased by 0.19 (0.33) mag, which equates to 0.08 (0.13) dex or a factor of $\times 0.83$ ($\times 0.74$). Highly luminous red galaxies fade at an ever-increasing rate from $z\sim 1.1$ to $z\sim 0.3$. Comparison with a passively evolving population implied a factor of $\sim$2.1 growth in red galaxy stellar mass density, and a factor of $\sim$1.4 growth in the stellar mass of highly luminous massive red galaxies.

Using our evolving SMFs, we found an increase of 0.37 (0.31) dex in SMD for red (blue) galaxies, i.e. a factor 2.3 (2.1), from $z=1.1$ to $z=0.3$. The rate at which SMD is growing decreases slowly with time. For red galaxies, this indicates that the rate at which blue galaxies move to the red sequence as they cease star formation decreases slowly with time.

The mass of the most massive red (blue) galaxies increased by 0.23 (0.15) dex, i.e. a factor of $\times 1.7$ ($\times 1.4$), from $z=1.1$ to $z=0.3$. The rate at which massive red galaxies increase in stellar mass through mergers with smaller galaxies slows from $\sim$0.4 dex per unit redshift for $z=1.1$ to $z=0.7$ to $\sim$0.2 dex per unit redshift for $z=0.7$ to $z=0.3$.

12. ACKNOWLEDGEMENTS

We thank the anonymous referee for suggestions which have greatly improved the quality of our results and enhanced the clarity of this paper. Richard Beare wishes to thank Monash University for financial support from MGS and MIPRS postgraduate research scholarships. Michael Brown acknowledges financial support from The Australian Research Council (FT100100280) and the Monash Research Accelerator Program (MRA). Kevin Pimbblet acknowledges the support of STFC, through the University of Hull’s Consolidated Grant ST/R000840/1. We thank colleagues on the NDWFS, SDWFS, NEWFIRM Boötes, and AGES teams, in particular M. L. N. Ashby, R. J. Cool, A. Dey, P. R. Eisenhardt, D. J. Eisenstein, A. H. Gonzalez, B. T. Jannuzi, C. S. Kochanek and D. Stern. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This research was supported by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under a cooperative agreement with the National Science Foundation.