To be submitted to the Astrophysical Journal
Preprint typeset using LATEX style emulateapj v. 11/10/09

COSMOLOGICAL DEPENDENCE OF THE MEASUREMENTS OF LUMINOSITY FUNCTION, PROJECTED
CLUSTERING AND GALAXY-GALAXY LENSING SIGNAL
Surhud More 1,2
1 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan
2 surhud.more@ipmu.jp
To be submitted to the Astrophysical Journal

ABSTRACT
Observables such as the luminosity function of galaxies, Φ(M), the projected clustering of galaxies, wp (rp ),and the galaxy-galaxy lensing signal, ∆Σ(rp ), are often measured from galaxy redshift surveys assuming afiducial cosmological model for calculating distances to and between galaxies. There is a growing number ofstudies that perform joint analyses of these measurements and constrain cosmological parameters. We quantify
the amount by which such measurements systematically vary as the fiducial cosmology used for the measurements is changed, and show that these effects can be significant at high redshifts (z ∼ 0.5). Cosmologicalanalyses (or halo occupation distribution analyses) that use the luminosity function, clustering measurements
and galaxy-galaxy lensing signal but ignore such systematic effects may bias the inference of parameters. We
present a simple way to account for the difference in the cosmological model used for the measurements andthose used for the prediction of observables, thus allowing a fair comparison between models and data.
Subject headings: cosmology: observations — dark matter — large-scale structure of universe — galaxies:
distances and redshifts
1. INTRODUCTION

Large scale galaxy redshift surveys such as the Sloan Digital Sky Survey (SDSS hereafter) have revolutionized the fieldof galaxy formation and cosmology. Data from such surveys
has enabled precise measurements of the abundance of galaxies (see e.g., Blanton et al. 2003), the clustering of galaxies asa function of luminosity (see e.g., Zehavi et al. 2011) and thegalaxy-galaxy lensing signal (see e.g., Sheldon et al. 2004;
Mandelbaum et al. 2006, 2013). These measurements havebeen used to constrain an important outcome of the galaxy
formation processes, the relation between galaxy luminosity(or stellar mass) and the underlying halo mass (Cacciato et al.
2009, 2013; Leauthaud et al. 2012; Tinker et al. 2013). It has
also been argued that a joint analysis of these measurements
can be used to constrain cosmological parameters, such as thematter density parameter and the amplitude of cosmologicalfluctuations using the data on small scales (see e.g., Seljak
et al. 2005; Yoo et al. 2006; Cacciato et al. 2009; van den
Bosch et al. 2013; More et al. 2013) and from large scales(Baldauf et al. 2010; Mandelbaum et al. 2013). Cosmologicalconstraints have been obtained using the clustering of galaxies combined with other observables such as the mass-to-lightratio on cluster scales (van den Bosch et al. 2003; Tinker et al.
2005; van den Bosch et al. 2007), mass-to-number ratio (Tinker et al. 2012) or the galaxy-galaxy lensing signal (Cacciato
et al. 2013; Mandelbaum et al. 2013).
To measure observables such as the luminosity functionof galaxies, their projected clustering signal and the galaxygalaxy lensing, the galaxy angular positions and redshifts
are converted to cosmological distances between us and thegalaxies and between galaxies themselves. These conversions are dependent upon the assumed cosmological model.
It would be incorrect to assume that the measurements do
not change when the cosmological model used to analyze thedata is changed. Fitting analytical models to the same measurements with varying cosmological parameters can affectthe constraints derived on the cosmological parameters. Suchproblems can be mitigated if instead the angular clustering ofgalaxies and the angular galaxy-shear (or shear-shear) correlations are used as measurements (see e.g., Oguri & Takada
2011).
The objective of this short letter is to quantify the effect ofchanging the fiducial cosmological model used to measure theparticular set of observables, Φ(M), wp (rp ) and ∆Σ(rp ), andshow that it is straightforward to account for such biases. In
Section 2, we quantify analytically the sensitivity of each ofthese measurements to the assumed cosmological model. In
Section 3, we use the example of the projected galaxy clustering signal to demonstrate that the effect on measurementsof the changing reference model can be accounted for in a
simple manner. In Section 4, we summarize and discuss our
results. Since the particular set of observables we consider areoften measured in units which keeps their dependence on the
Hubble parameter h = H0 /100 km s−1 Mpc−1 transparent (forexample, wp (rp ) is measured in h−1 Mpc), we will not explore
differences in cosmological models with different values of h.
2. ANALYTICAL ESTIMATES
2.1. Galaxy luminosity function

The abundance of galaxies is quantified by measurementsof the luminosity function of galaxies, Φ(M)dM, which gives
the average number density of galaxies within the absolutemagnitude range M ± dM/2. The luminosity functions areoften determined from flux limited surveys and require us toconvert the apparent magnitude m, of a galaxy into an absolutemagnitude, and obtain the maximum distance out to which
a galaxy with a given luminosity could have been observed
given the flux limit of the survey. These conversions are dependent on the assumed cosmology in the following manner.
The apparent magnitude of a galaxy at redshift z is related
to the absolute magnitude via the distance modulus µ(z) =5.0 log(Dlum [z, Ω]) + 25,
M = m − µ(z) ,(1)where Dlum (z, Ω) is the luminosity distance (in units of
h−1 Mpc) in a particular cosmological model, Ω. The maxi-2More, S.

mum comoving distance, χmax to which this object can be observed given the magnitude limit mlim of the survey is givenby1χmax =
100.2(mlim −M−25)(2)
1+z
Thus differences in cosmology change the luminosity ofgalaxies and the change in χmax affects the normalization of
the luminosity function.
To estimate the amount by which the luminosity function
can change due to a change in cosmological model, let us first
assume that the luminosity function as a function of redshiftchanges extremely weakly within the survey area used to estimate it using some fiducial cosmological model 1 . The luminosity function Φ̃(M, Ω̃), measured in some fiducial cosmological model Ω̃ can be used to calculate the redshift dependence of the apparent magnitude counts N(m, z)dmdz per unit
steradian, which is the observable in a true sense,
N(m, z)dmdz = Φ̃(m − µ[z, Ω̃])dmχ̃2

dχ̃
dz .
dz

(3)

These number counts can be reinterpreted as a luminosity
function in a cosmological model, Ω other than the fiducial
model using the following equation,Z
1 zmaxΦ(M, Ω)dM =N(M + µ[z, Ω], z) dM dz(4)
V 0
Z zmax1dχ̃=Φ̃(M + µ[z, Ω] − µ[z, Ω̃]) dM χ̃2 dz.
V 0
dz
(5)
Here, zmax denotes the maximum redshift to which a galaxy
can be observed in the redshift survey, given its absolute magnitude in the cosmological model Ω, and V denotes the comoving volume encompassed by the Universe below this redshift. In Figure 1, we use the Schechter fit for the luminosity
function provided by Blanton et al. (2003), in an Ωm = 0.3
model and show the residuals of the luminosity function inΩm = 0.25 and Ωm = 0.35 models computed using the above
equation. The range of cosmological models is chosen to besuch that it is well within the ranges of cosmological constraints obtained by a number of joint analyses involving theabundance, clustering and lensing measurement. We have assumed the magnitude limit in r-band of 17.77 corresponding
to the spectroscopic sample in SDSS.
Although the measurement errors on the luminosity function are large, the difference is a systematic change in the
shape of the luminosity function, and can be as large as ∼ 20
percent at the bright end. Since the errorbars are difficult to
propagate in an integral equation such as the one above, theideal way is to change the prediction for a particular cosmological model to predict the counts in fiducial model used toobtain the luminosity function.
2.2. Projected clustering measurement

For a flat ΛCDM model, the projected and the line-of-sightcomoving separations between two galaxies separated in redshift by a small difference ∆z is given by,crp = χ(zeff )θ; π =∆z ,(6)
H0 E(zeff )1 Of course, this cannot be strictly true and is an assumption. If violated it
is better to divide up the survey in to finer redshift slices.

respectively, where χ(zeff ) is the comoving distance to the effective redshift zeff , θ denotes the angular separation of thetwo galaxies, c denotes the speed of light, H0 is the Hubble
constant and E(z) is the expansion function. Here we haveignored the finite redshift width of the galaxy sample and assumed a particular effective redshift to convert the angular positions and redshift into distances. These equations can thenbe used to count the number of pairs of galaxies at a given
separation vector (rp , π), and compare it to the number of pairs
expected if the galaxies were distributed in a random manner.
The excess number of pairs over those expected in a random
distribution gives the clustering signal at the effective redshift,ξ(rp , π, z). The projected clustering is then obtained by integrating ξ(rp , π, z) along the line-of-sight,
Z πmaxwp (rp , zeff ) = 2
ξ(rp , π, zeff )dπ ,(7)0Z ∆zmaxc=2
ξ(rp , π, zeff )d(∆z) . (8)H0 E(zeff )0
The changes to the luminosity of galaxies due to a change incosmology can slightly alter which galaxies enter a volumelimited sample. In addition, there are three different ways
the wp measurements get affected. If a cosmology other than
the fiducial is used to analyze the data, then a given angularscale corresponds to a different comoving projected separation scale. This difference can be small at low redshift (seee.g. Zehavi et al. 2011, ∼ 1 percent due to a change in Ωmfrom 0.25 to 0.3 at z = 0.15). As wp ∝ rp−1 , the change inprojected separation scale roughly corresponds to a similarchange in the value of wp (rp ) when the cosmology is changed.
The second effect is due to change in the factor E(z) as cosmology is changed, and this changes the value of wp (rp ) by amultiplicative constant at all scales. This effect is important,especially at higher redshift (at z ∼ 0.5, the difference in the
expansion functions is 3.5 per cent between Ωm of 0.25 and0.3). Such effects due to the change of transverse (and lineof-sight) scales are usually taken into account when analysing
baryon acoustic oscillations (see e.g., Blake & Glazebrook
2003; Eisenstein et al. 2005; Percival et al. 2007; Anderson
et al. 2012) and redshift space distortions (see e.g., Ballinger
et al. 1996; Tegmark et al. 2006; Blake et al. 2011).
The third effect on wp (rp ) is subtle and is related to thechange in the integration limit in the above equation, as agiven value of πmax corresponds to a different value of ∆zmax .
For sufficiently large values of πmax as are employed in observations (typically 60 ∼ 100 h−1 Mpc) this effect is quite small
as the value of ξ(rp , π) at large values of π does not dominate
the wp (rp ) integral. Note however, that even this small difference can be easily accounted for by adopting a different πmax
value when computing the analytical prediction.
In Table 1, we present the ratios of the comoving distanceand the expansion function for three different cosmologicalΩm =0.30
models, at different redshifts. The fractions fE(z)andΩm =0.30fχ(z)are defined in the caption, and are chosen such thatthey correspond to the deviations in the clustering signal expected when the data is analysed in two different cosmologies(this is approximate since wp is not exactly proportional torp−1 ). The two effects change the clustering signal in the same
direction.
It can be seen that the combination of the first two effects
are small ∼ 1 per cent differences between Ωm = 0.25 andCosmological dependence of observables from redshift surveys
m =0.251.15

m =0.35

Blanton et al. 2003

m =0.30

1.10

3

/

1.05
1.00
0.95
0.90
0.85
0.80

17

18

19

20

0.1 Mr -5 log h

21

22

23

Figure 1. The ratio of the SDSS luminosity function from Blanton et al. (2003) derived in a Ωm = 0.3 model with respect to the luminosity function expected if
the data was analyzed using Ωm = 0.25 or Ωm = 0.35 model.

m =0.301.15

m =0.35m =0.25

1.10

/g

1.05

1.00

0.95

Future uncertainties

0.90
0.85
0.80

Current uncertainties
100

10-1

101

rp h-1 Mpc

Figure 2. The ratio of the galaxy-galaxy lensing signal around BOSS galaxies expected assuming an Ωm = 0.3 model with respect to that expected if the data
was analyzed using Ωm = 0.25 or Ωm = 0.35 model. The light and dark grey band corresponds to current and future uncertainties.

Ωm = 0.30 models (although are systematically in the same
direction on all scales) for low redshift (z ∼ 0.1) analyses. Atz ∼ 0.5, the effects cause ∼ 5 percent differences between thetwo cosmological models above and can be very important
given the statistical errors in the measurements of wp (rp ) withcurrent and upcoming large surveys.
2.3. Galaxy-galaxy lensing measurement

The primary observable for the galaxy-galaxy lensing signal is the tangential ellipticity of background galaxies in thevicinity of foreground galaxies. The galaxy-galaxy lensing
signal is often reported as the excess surface density, ∆Σ byaveraging the tangential component of ellipticity around an
ensemble of galaxies∆Σ(rp ) = Σ̃crit (zl , zs )hi(rp ) ,(9)where rp denotes the projected comoving separation rp between the two galaxies at the redshift of the foreground lens,and Σcrit (zl , zs ) is a cosmology dependent factor called the critical surface density which is defined asΣcrit (zl , zs ) =c2 DA (zs )(1 + zl )−2.
4πG DA (zl , zs )DA (zl )(10)Here DA (zl ), DA (zs ) and DA (zl , zs ) are the angular diameter
distances to the lens, the source, and between the lens andsource, respectively, and the (1 + zl )−2 factor arises from theuse of comoving units.
The cosmology dependence enters the measurement of ∆Σ
in two ways. The first one is similar to that discussed in theprevious subsection. A given angular scale on the sky corresponds to different comoving projected scales in different
cosmological models. Since the excess surface density is alsoroughly proportional to rp−1 , the percentage change in ∆Σ issimilar to the percentage in the comoving distances. The second effect leads to a change in the normalization of ∆Σ due
to the dependence of Σcrit on the cosmological parameters and4More, S.

Table 1
Cosmological dependence of comoving distance and the expansion function

RedshiftΩmΩm =0.3fχ(z)Ωm =0.3
fE(z)

0.1
0.1
0.1
0.3
0.3
0.3
0.5
0.5
0.5

0.25
0.30
0.35
0.25
0.30
0.35
0.25
0.30
0.35

0.996
1.000
1.004
0.989
1.000
1.011
0.982
1.000
1.017

0.992
1.000
1.007
0.978
1.000
1.022
0.965
1.000
1.034

Ωm =0.30
The fraction fE(z)is defined as the ratio of the expansion function in agiven cosmology to that in Ωm = 0.30 cosmology, i.e., E(z, Ωm )/E(z, Ωm =
Ωm =0.300.30). The fraction fχ(z)is defined as the ratio of the comoving distancein Ωm = 0.30 cosmology, to that in another cosmology i.e., χ(z, Ωm =0.30)/χ(z, Ωm ).

Table 2
Cosmological dependence of comoving distance and the critical density

can be also seen in real data (Miyatake et al., in preparation).
Ideally to account for the cosmological dependence of thissignal, one needs to also consider both the source and lens redshift distributions. However if the lens redshift range is narrow enough, one can assume the lenses to be located at a similar effective redshift, zeff . In addition, note that fΣΩcritm =0.30(zl ,zs ) forsame lens redshift and the same cosmology is a very weakly
varying function of source redshift (see Table 2). This implies
that the amplitude correction can be safely assumed to be aconstant normalization shift fairly independent of the source
redshift distribution. This correction can then be calculated at
the median redshift of the source galaxy population and applied to the model before comparing to data. In Fig. 2, wecompare the differences in ∆Σ at z ∼ 0.5 expected based on
the above discussion, when the fiducial cosmological model
used for the measurements is changed. We find that the deviations can be of the order of ∼ 2 − 2.5 percent betweenΩm = 0.25 and Ωm = 0.30 models. Although small compared to errors possible with existing data, it is important tonote, that they systematically go in the opposite direction as
the clustering signal. The errors are expected to go down to
5 percent or better with upcoming surveys such as the Hyper
Suprime-Cam survey 2 (Miyazaki et al. 2012).
3. TESTS ON REAL DATA FOR THE PROJECTED CLUSTERING

zl

zs

Ωm

Ωm =0.3fχ(z)
lfΣΩm =0.3(z ,zs )

0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

0.5
0.5
0.5
0.7
0.7
0.7
0.9
0.9
0.9
0.8
0.8
0.8
1.0
1.0
1.0
2.0
2.0
2.0

0.25
0.30
0.35
0.25
0.30
0.35
0.25
0.30
0.35
0.25
0.30
0.35
0.25
0.30
0.35
0.25
0.30
0.35

0.996
1.000
1.004
0.996
1.000
1.004
0.996
1.000
1.004
0.982
1.000
1.017
0.982
1.000
1.017
0.982
1.000
1.017

1.00771.00000.9925
1.00771.00000.9926
1.00771.00000.99261.03621.00000.96761.03581.00000.96801.03411.00000.9698

crit lΩm =0.30
The fraction fχ(z)is defined the same way as in Table 1. The frac-tion fΣΩm =0.30(z ,zs ) is defined as the ratio of the critical surface density in
crit lthe Ωm = 0.30 model to that in another cosmology, i.e., fΣΩm =0.30=
crit (zl ,zs )Σcrit (zl , zs , Ωm = 0.30)/Σcrit (zl , zs , Ωm ).

depends upon both the source and lens redshift distribution.
In Table 2, we calculate the ratios of the comoving distanceand the critical surface density for three different cosmological models and for different combinations of source and lens
redshifts, to quantify the effect it can have on the galaxygalaxy lensing signal. The fraction fΣΩcritm =0.30(zl ,zs ) is defined as theratio of the critical surface density in a cosmological modelto that in the Ωm = 0.30 model. The two effects go in opposite direction making the lensing signal less sensitive to thecosmological parameters than the clustering signal, and eventhough the current statistical errors are large, these systematic
dependence of the lensing signal on the fiducial cosmology

FUNCTION

We now use galaxies from the SDSS-III Baryon Oscillation Spectroscopic Survey project Data Release 9 (hereafter
BOSS; Dawson et al. 2013; Ahn et al. 2012), and demonstrate
for the case of the projected clustering measurement how wellthe effects mentioned in the Section 2.2 capture the relevantchanges to the measurement. We analyze the projected clustering using three different cosmological models and showthat they differ by the amount expected from the discussion
in the previous section. We choose all galaxies in the northern region of BOSS with redshifts between z ∈ [0.47, 0.59]
and M∗ > 1010.2 M where M∗ denotes the stellar mass calculated using the stellar population synthesis models by thePortsmouth group (Maraston et al. 2012). The effective redshift of the sample is zeff = 0.53.
In the top panel of Figure 3, we show the clustering ofgalaxies measured by assuming three different flat ΛCDM
cosmological models with varying Ωm while converting theangular positions and redshifts to distances. The solid circles
in the lower panel denote the differences between the measured clustering signal in a given cosmological model to thatin the Ωm = 0.30 model. The measurement errorbars are
small enough that the differences between the cosmologicalmodels are larger than the statistical error and are systematicin nature. Fitting a constant to the (inverse variance-weighted)
residuals results in 0.062 ± 0.005 (−0.042 ± 0.003) for theΩm = 0.25 (0.35) model. 3
Next we take the measurements in the Ωm = 0.25 model,
and predict the clustering expected in the Ωm = 0.30 model
as follows. To account for the first effect discussed in Section 2.2, we change the projected comoving scale of the measurement from the Ωm = 0.25 analysis torpcorr = rp (Ωm = 0.25)

χ(zeff , Ωm = 0.30).
χ(zeff , Ωm = 0.25)(11)

2 http://www.naoj.org/Projects/HSC/index.html
3 With the catalogs that use DR11, which is an internal data release in the
BOSS collaboration, the differences are even more statistically significant.

Cosmological dependence of observables from redshift surveys
In addition we also change the amplitude of wp such thatwcorr
= wp (Ωm = 0.25)pE(zeff , Ωm = 0.25).
E(zeff , Ωm = 0.30)(12)

We also use similar corrections to the clustering measurements in the Ωm = 0.35 model to deduce the clustering
measurements in the Ωm = 0.30 model. The filled triangles in the bottom panel of Figure 3 show the difference between these corrected clustering measurement and the projected clustering measurement in the Ωm = 0.30 model using filled triangles (the green [red] triangles correspond to theΩm = 0.25 [0.35] model corrected to that in Ωm = 0.30).
We see that this recovers the clustering measurement very
accurately. A constant model fit to the residuals now gives
0.004 ± 0.005 (0.005 ± 0.004) for the Ωm = 0.25 (0.35) modeland the residuals no longer have either just positive or negative sign.
Although in our case, we have corrected the data for thecosmological dependence, in modelling applications, it is better to account for the differences in the model itself. Errors,
typically done using a jack-knife estimator (with regions ofequal areal coverage), will not depend upon the cosmologicalmodel. But errors obtained using covariances with mock simulations populated with galaxies, may require a revision too.
Exploring these details is beyond the scope of this short letter.
4. SUMMARY

We have presented analytical estimates for the cosmological dependence of the galaxy luminosity function, Φ(M),the projected clustering measurement, wp (rp ), and the galaxygalaxy lensing signal ∆Σ obtained from a galaxy redshift survey. We showed that these measurements change in different
cosmological models due to the difference in the comoving
distances, χ(z), the expansion functions, E(z), and the change
to the critical surface density between lens and source galaxies used to measure the lensing signal.
These changes are small for low redshift surveys such as
SDSS-I, but given the systematic nature of the shifts can bias
the cosmological constraints obtained from a joint analysisof Φ(M), wp (rp ) and ∆Σ. These systematic effects can bevery important at higher redshifts which use these observables and for ongoing and future surveys which can measurethese observables with ever-increasing precision. Performing a cosmological analysis with these observables requires
one to account for the cosmological parameter dependence ofthe observables themselves. We have presented analytical expressions which can be used to change the predictions for aparticular cosmological model to the ones in the fiducial cosmological model used for the analysis, thus allowing a faircomparison between model and data.
We tested these expressions for the specific case of the projected clustering measurement wp (rp ) using existing data fromthe SDSS-BOSS survey. We analyzed this data in the context
of three different flat ΛCDM cosmological models. We foundthat the differences in the measurements are systematic in nature and significant given the errorbars. We also found that
our expressions can predict these differences correctly.

5

In cosmological analyses it is best to correct the predictions
to account for the fiducial cosmological model used to makethe measurements of Φ, wp and ∆Σ. If such analyses are run
without accounting for the systematic issues we highlight, one
runs the risk of biasing the cosmological parameters to values
close to the ones assumed in the fiducial cosmology. In future
work, we plan to quantify how the recent cosmological constraints obtained using joint fits to abundance, clustering andlensing of galaxies by Cacciato et al. (2013) may be affected
due to these systematic issues. We also plan to investigate theeffects of these systematics on the joint analysis of clustering and lensing on large scales in Mandelbaum et al. (2013),
especially for the high redshift sample.
ACKNOWLEDGMENTS

SM would like to acknowledge very useful and prompt inputs and comments on an early draft version of the paper by
Chris Blake, Frank van den Bosch, Marcello Cacciato, Alexie
Leauthaud and Hironao Miyatake.
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6More, S.

300

wp rp (h-2 Mpc2 )250

m =0.25m =0.30m =0.35

200
150
100
50

wp /wp (m =0.30)

0.2

Before correction
After correction

0.1
0.0
0.1
0.2
100

rp (h-1 Mpc)

101

102

Figure 3. The projected clustering (shown as wp rp to reduce the dynamical scale) of an approximately volume limited subsample of CMASS galaxies, analyzed
using different flat ΛCDM cosmological models are shown in the top panel with symbols of different colors. The clustering when the data is analyzed withΩm = 0.25 (0.35) is the largest (smallest). The solid circles in the bottom panel show the difference between the measurements with respect to the measurementsin the Ωm = 0.30 model. Accounting for the effects discussed in Section 2, we can use the measurements performed using Ωm = 0.25 or Ωm = 0.35 model andaccurately recover the measurements in the Ωm = 0.30 model. The differences between these corrected measurements and the clustering in the Ωm = 0.30 model
are shown using filled triangles. There is a small offset added in the x-direction in the lower panel for clarity.