A brief review on cosmological analysis of galaxy surveys with multiple tracers
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A brief review on cosmological analysis of galaxy surveys with multiple tracers
Yuting Wang ∗ and Gong-Bo Zhao
1, 2, † National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, P.R.China University of Chinese Academy of Sciences, Beijing 100049, P.R.China
Submitted to RAAABSTRACTGalaxy redshift surveys are one of the key probes in modern cosmology. In the data analysis ofgalaxy surveys, the precision of the statistical measurement is primarily limited by the cosmic varianceon large scales. Fortunately, this limitation can in principle be evaded by observing multiple types ofbiased tracers. In this brief review, we present the idea of the multi-tracer method, outline key steps inthe data analysis, and show several worked examples based on the GAMA, BOSS and eBOSS galaxysurveys. INTRODUCTIONMapping the Universe with massive galaxy surveys provides critical cosmological information, which is key to revealthe physics governing the evolution of the Universe. In particular, galaxy surveys play a crucial role for understandingthe origin of the cosmic acceleration discovered in the late 1990s (Riess et al. 1998; Perlmutter et al. 1999), which maybe due to the existence of Dark Energy (see Huterer & Shafer 2018 for a recent review), an energy component with anegative pressure, or to the extension of General Relativity (GR), as reviewed in Koyama (2016).Cosmological information in galaxy surveys is primarily encoded in the baryon acoustic oscillations (BAO) and theredshift space distortions (RSD), which are specific clustering patterns of biased tracers (Percival et al. 2001; Peacock etal. 2001; Eisenstein et al. 2005; Cole et al. 2005; Percival et al. 2007; Beutler et al. 2011; Blake et al. 2011; Anderson etal. 2012; Padmanabhan et al. 2012; Ross et al. 2015; Alam et al. 2017). Being complementary to other probes includingthe supernovae (SNIa) (Riess et al. 1998; Perlmutter et al. 1999; Riess et al. 2007; Sullivan et al. 2011; Suzuki et al.2012; Betoule et al. 2014; Scolnic et al. 2018; D’Andrea et al. 2018) and the cosmic microwave background (CMB)(Bennett et al. 2003; Hinshaw et al. 2013; Planck Collaboration et al. 2018), the BAO, as a standard ruler, mapsthe expansion history of the Universe, thus is essential to constrain dark energy, while the RSD, caused by peculiarmotions of galaxies, is a natural probe for the nature of gravity on cosmological scales.Great progress on galaxy surveys has been made since last century. Started in the 1970’s, the Harvard-SmithsonianCenter for Astrophysics (CfA) redshift survey brought us the first large galaxy sample, which consists of several thou-sand galaxies for a clustering analysis (Huchra et al. 1983; Davis & Peebles 1983). Thanks to technical developments ofthe multi-object spectrographs and charge coupled device (CCD)-based photometry, larger galaxy surveys were builtand operated since then, including the Las Campanas Redshift Survey (Shectman et al. 1996), the Sloan Digital SkySurvey (SDSS) (York et al. 2000), the 2dF Galaxy Redshift Survey (2dFGRS) (Colless et al. 2001), the WiggleZ DarkEnergy Survey (Blake et al. 2008), the 6dF Galaxy Survey (6dFGS) (Jones et al. 2009) and the Galaxy And MassAssembly (GAMA) survey (Baldry et al. 2010), etc.
Based on the 2 . ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ a s t r o - ph . C O ] S e p Wang & Zhao
Figure 1.
Mapping the Universe in three dimensions with SDSS (1998-2020) using different types of tracers including galaxies,quasars, and Lyman- α As key quantities for cosmological implications, BAO and RSD are measured from almost all available wide-fieldgalaxy surveys. The statistical uncertainties of BAO and RSD parameters are inherited from the uncertainty in thetwo-point statistics of galaxy clustering, which consists of two components: the shot noise and the cosmic variance.While the former can be suppressed by increasing the number density of tracers in the observation, the latter cannotbe reduced statistically, as long as only one tracer is observed, because of the limited number of pairs on large scales.However, if at least two tracers with an overlapping cosmic volume are observed, the cosmic variance can be reducedto some extent, depending on the level of the shot noise (Seljak 2009; McDonald & Seljak 2009). This ‘multi-tracermethod’ works if at least two differently biased tracers of the same underlying dark matter field are available. Bycomparing the galaxy clustering of these tracers, one is able to determine a ratio between the ‘effective biases’, whichmay include the RSD parameter β , and the primordial non-Gaussianity parameter f NL , to an infinitive accuracy inthe zero noise limit, as detailed in Sec. 2. The validity of this method with quantitative forecasts has been extensivelystudied in theory using the Fisher information matrix approach (White et al. 2009; Gil-Mar´ın et al. 2010; Bernstein &Cai 2011; Hamaus et al. 2011; Abramo 2012; Cai & Bernstein 2012; Abramo & Leonard 2013; Ferramacho et al. 2014;Yamauchi et al. 2014; Abramo et al. 2016; Zhao et al. 2016; Alarcon et al. 2018; Abramo & Amendola 2019; Boschettiet al. 2020; Viljoen et al. 2020).Applying this method to actual galaxy surveys requires finding at least two differently biased tracers covering thesame patch of the sky and redshift range. This is challenging for most existing surveys, which are designed andoptimized for a single tracer within a given footprint and redshift range. One possibility to create such ‘multi-tracer’samples from a single-tracer survey is to split the sample into subsamples by luminosity or color (Blake et al. 2013;Ross et al. 2014), but this may be subject to a limited relative galaxy bias, as samples in a single-tracer survey usuallydo not differ much in the galaxy bias. Alternatively, one can combine tracers observed by different surveys (Mar´ın etal. 2016; Beutler et al. 2016). However, this approach could be limited by the small overlapping area, as most galaxysurveys are designed to be complementary to each other, in terms of the sky coverage and/or redshift range.Fortunately, the SDSS-IV/eBOSS survey has created a great opportunity for a proper multi-tracer analysis, as it isthe first galaxy survey to observe multiple tracers with a large overlap in the cosmic volume. Targeting at both theluminous red galaxies (LRGs, denoted as ‘Old Red Galaxies’ in Fig. 1 at 0 . < z < . . < z < . ) in the same redshift range (Dawson et al. 2016). A multi-tracer brief multi-tracer review THE MULTI-TRACER METHODWe start with a matter over-density field, namely, δ m ≡ ( ρ − ¯ ρ ) / ¯ ρ with ¯ ρ being the mean density. The observedgalaxy traces the matter field up to an effective bias factor b g , relating the over-density of the observed galaxy δ g to matter δ m on linear scales, i . e . , δ g = b g δ m . The measured 2-point statistics of δ g , e . g . , the power spectrum, P g ≡ (cid:104) δ g δ g (cid:105) = b P m , is subject to the cosmic variance and the shot noise, which dominates the error budget on thelarge and small scales, respectively.If two tracers for the same underlying matter density field are available, then the ratio between δ g2 and δ g1 is, δ g2 δ g1 = b g2 δ m + (cid:15) g2 b g1 δ m + (cid:15) g1 , (1)where (cid:15) denotes the Poisson noise. We can immediately see that in the low-noise limit, i . e . , (cid:15) →
0, the ratio becomes δ g2 /δ g1 = b g2 /b g1 , thus is free from the cosmic variance, as δ m is cancelled out. Because the RSD parameter β ≡ f /b and the primordial non-Gaussianity parameter f NL are part of the effective bias here, the multi-tracer method can inprinciple improve the constraint on β (McDonald & Seljak 2009) and f NL (Seljak 2009). In practice, the dependenceon δ m can not be completely eliminated due to the shot noise, thus the gain from multiple tracers can be downgradedby various factors including the signal-to-noise ratio of each tracer, the overlapping volume and the relative bias, etc . (Gil-Mar´ın et al. 2010).The general data covariance matrix for a 2-tracer system is, C ≡ (cid:34) (cid:10) δ (cid:11) (cid:104) δ g1 δ g2 (cid:105)(cid:104) δ g2 δ g1 (cid:105) (cid:10) δ (cid:11) (cid:35) , (2)and the Fisher matrix can be evaluated as (Tegmark et al. 1997), F λλ (cid:48) = 12 Tr (cid:2) C ,λ C − C ,λ (cid:48) C − (cid:3) (3)where C ,λ ≡ dC/dλ .In what follows, we shall apply this result to cases of RSD and the primordial non-Gaussianity, respectively.2.1. Determining the RSD parameter with multi-tracer surveys
In redshift surveys, the radial distance of a galaxy is inferred from the observed redshift, which is determined bythe underlying Hubble flow and the peculiar velocity of a galaxy along the line of sight. Hence, the observed galaxyposition in redshift space, x s is x s = x + v · ˆ rH , (4)where x is the galaxy position in real-space. v the peculiar velocity of a galaxy, and ˆ r the unit vector along the lineof sight. Statistically there is an enhancement of galaxy clustering along the line of sight on large scales due to thispeculiar motion, dubbed the RSD. According to the linear perturbation theory, relation between the velocity andmatter over-density reads, − ik v = f Hδ m k , (5)where f = d ln δ m /d ln a is the growth rate of structure, and k is the wave-number in Fourier space. Then theredshift-space galaxy density fluctuation on linear scales is given by (Kaiser 1987), δ g = (cid:0) b g + f µ (cid:1) δ m + (cid:15) g , (6)where µ is the cosine of the angle between k and the line of sight. Now the data covariance matrix becomes, C = P θθ (cid:34) (cid:0) β − + µ (cid:1) (cid:0) β − + µ (cid:1) (cid:0) αβ − + µ (cid:1)(cid:0) β − + µ (cid:1) (cid:0) αβ − + µ (cid:1) (cid:0) αβ − + µ (cid:1) (cid:35) + N , (7) Wang & Zhao -4 -3 -2 -1 k [h/Mpc] P g ( k ) [ M p c / h ] f NL = 2.0f NL = 1.0f NL = 0.5f NL = 0.0 Figure 2.
The galaxy power spectrum for different values of f NL , as shown in the legend. where β ≡ f /b g1 , α ≡ b g2 /b g1 , P θθ ≡ f (cid:10) δ (cid:11) , and N ij ≡ (cid:104) (cid:15) i (cid:15) j (cid:105) . Denoting X ij = N ij /b i b j P m , McDonald &Seljak (2009) shows that using only the transverse and the radial modes ( i . e . the modes with µ = 0 and µ = 1), theuncertainty of β is (with α and P θθ marginalized over), σ β β = (cid:2) α (1 + β ) + ( α + β ) (cid:3) X − (cid:2) α (1 + β ) + α (1 + β )( α + β ) (cid:3) X + 2 α (1 + β ) X β ( α − , (8)which goes to zero if X →
0, meaning that β can be measured without the cosmic variance from a multi-tracer survey.2.2. Determining f NL with multi-tracer surveys The local-type of the primordial non-Gaussianity (see Wands 2010 for a review) can be described by a quadraticcorrection to the Gaussian field φ , i . e . , Φ = φ + f NL (cid:0) φ − (cid:10) φ (cid:11)(cid:1) , in which f NL describes the amplitude of the non-Gaussian correction. This leads to a scale-dependent galaxy bias, i . e . , b g → b g + ∆ b ( k ) with (Dalal et al. 2008; Slosaret al. 2008), ∆ b ( k ) = 3 f NL ( b g − p ) δ c Ω m k T ( k ) D ( z ) (cid:18) H c (cid:19) , (9)where p depends on the type of tracer, δ c is the critical linear over-density for the spherical collapse, T ( k ) is thematter transfer function (normalized to unity on large scales), and D ( z ) is the growth function (normalized to a in thematter-dominated era). The galaxy over-density thus receives a k -dependent correction, say, δ g = [ b g + ∆ b ( k, f NL )] δ m ,which alters the shape of the power spectrum on large scales, as shown in Fig. 2. This makes it possible to constrain f NL using galaxy surveys (Nikoloudakis et al. 2013; Ross et al. 2013; Karagiannis et al. 2014; Mueller et al. 2019).As f NL primarily affects the large-scale modes, its constraint can be significantly tightened if the cosmic variance isreduced. For example, the ratio of σ ( f NL ) (with other relavant parameters marginalized over) derived from a two-tracersurvey to that from a single-tracer survey using only one mode is (Seljak 2009), σ ( f NL ) σ ( f NL ) = (cid:114) (cid:104) ( n g2 P g2 ) − + ( n g1 P g1 ) − (cid:105) , (10)which clearly shows that the gain can be significant in the low-noise limit. THE PROCEDURE FOR A MULTI-TRACER ANALYSISIn this section, we show the key steps in the multi-tracer analysis, including the measurement of the 2-point statistics,modeling and parameter estimation. brief multi-tracer review E L G S G C C R O S S S G C L R G p C M A S S S G C - 7 0- 5 0- 3 0- 1 01 03 05 07 0 - 1 5 0 - 1 0 0 - 5 0- 1 5 0- 1 0 0- 5 0 s|| [ h -1 Mpc] - 8 0- 6 0- 4 0- 2 002 04 06 08 0 s ^ [ h - 1 M p c ] - 1 5 0 - 1 0 0 - 5 0 - 9 0- 7 0- 5 0- 3 0- 1 01 03 05 07 09 0 - 1 5 0 - 1 0 0 - 5 0 s x ( s ^ , s || ) Figure 3.
The 2D auto- and cross-correlation functions ξ ( s, µ ) assembled using the monopole, quadrupole and hexadecapolemeasured using the estimator in Eq. (11), where s = s (cid:107) + s ⊥ , from the samples of eBOSS DR16 ELG (left), LRG (right), andtheir cross-correlation (middle) in South Galactic Gap. This figure is adopted from Wang et al. (2020). Measuring the galaxy clustering
Most of the information in the clustering of galaxies is carried by its two-point correlation function, ξ ( s ), or equiva-lently, the power spectrum in Fourier space, P ( k ).3.1.1. The correlation function
The 2-point correlation function is measured by counting pairs of galaxies at a given comoving separation, s , andan unbiased estimator for two tracers, named as A and B , is (Landy & Szalay 1993), ξ AB ( s, µ ) = D A D B − D A R B − D B R A + R A R B R A R B , (11)where DD , DR , and RR are the normalized number pairs of galaxy-galaxy, galaxy-random and random-random,respectively, within a separation whose central value is s , and the cosine between the concerning pair and the line ofsight vector is µ . The commonly-used estimator for a single tracer system is a special case of Eq. (11) with A = B ,namely, ξ ( s, µ ) = DD − DR + RRRR . (12)Fig. 3 shows the 2D auto- and cross-correlation functions measured from the eBOSS DR16 LRG and ELG samplesusing the above estimator Eq. (11). 3.1.2.
The power spectrum
In Fourier space, the estimated power spectrum with shot noise corrected is given in (FKP estimator; Feldman etal. 1994) (cid:98) P ( k ) = (cid:10) | F ( k ) | (cid:11) − P shot , (13)where F ( k ) is the Fourier transformation of the weighted galaxy fluctuation field, F ( r ) F ( r ) = w ( r ) I / [ n ( r ) − αn s ( r )] , (14)with n ( r ) and n s ( r ) being the observed number density field for the galaxy catalog and synthetic random catalog,respectively, and I is a normalisation factor. The estimator for the auto- and cross-power spectrum for tracers A and B is (Zhao et al. 2020), (cid:98) P AB (cid:96) ( k ) = 2 (cid:96) + 12 I AB (cid:90) dΩ k π [ F , A ( k ) F (cid:96), B ( − k ) + F , B ( k ) F (cid:96), A ( − k )] , (15)with F (cid:96) ( k ) ≡ (cid:90) dr F ( r ) e i k · r L (cid:96) (ˆ k · ˆ r ) = 4 π (cid:96) + 1 (cid:96) (cid:88) m = − (cid:96) Y (cid:96)m (ˆ k ) (cid:90) dr F ( r ) Y ∗ (cid:96)m (ˆ r ) e i k · r . (16) Wang & Zhao - 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2- 0 . 2- 0 . 10 . 00 . 10 . 2 k || [ h Mpc-1] k ^ [ h M p c - 1 ] D R 1 6 L R G x E L G N G C , 0 . 6 < z < 1 . 0 D R 1 6 L R G x E L G S G C , 0 . 6 < z < 1 . 0- 0 . 2 - 0 . 1 0 . 0 0 . 1 0 . 2 k ^ [ h M p c - 1 ] Figure 4.
The cross power spectrum between the eBOSS DR16 LRGs and ELGs in the Northern (left) and Southern GalacticCap (right) measured using the estimator Eq. (15). This figure is adopted from Zhao et al. (2020).
This is based on the Yamamono estimator (Yamamoto et al. 2006), and makes use of the Addition Theorem to reducethe number of Fast Fourier Transform (FFTs) required in the calculation (Hand et al. 2017) . The cross powerspectrum, measured from the eBOSS DR16 samples using Eq. (15), is shown in Fig. 4.For the power spectrum analysis, one has to model the effect from the survey geometry carefully, as this windowfunction effect can alter the power spectrum multipoles , as formulated in Wilson et al. (2017). The survey windowfunction can be measured from the randoms by a pair-counting approach (Wilson et al. 2017; Zhao et al. 2020;Gil-Mar´ın et al. 2020), namely, W AB (cid:96) ( s ) = (2 (cid:96) + 1) I AB η − N ran (cid:88) i,j w Atot ( x i ) w Btot ( x j + s )4 πs ∆(log s ) L (cid:96) (ˆ x los · ˆ s ) , (17)where η is the ratio of the weighted numbers of the data and random. Note that the same normalization factor I AB appears in both Eqs. (15) and (17) , to guarantee that the measured and the theoretical power spectrum arenormalized in the same way (de Mattia & Ruhlmann-Kleider 2019).3.2. Modeling the galaxy clustering
This section describes the models commonly used for a multi-tracer survey in both configuration space and Fourierspace. 3.2.1.
Modeling the 2-point correlation function
In configuration space, the Gaussian Streaming Model (GSM) (Reid & White 2011; Wang et al. 2014) is widely usedfor modeling the full-shape of the 2-point correlation function,1 + ξ ( s ⊥ , s (cid:107) ) = (cid:90) dy (cid:112) π [ σ ( r, µ ) + σ ] [1 + ξ ( r )] × exp (cid:40) − (cid:2) s (cid:107) − y − µv ( r ) (cid:3) σ ( r, µ ) + σ ] (cid:41) , (18)where s || ≡ sµ and s ⊥ ≡ s (cid:112) (1 − µ ) denotes the separation of pairs along and across the LOS, respectively; ξ ( r ) isthe real-space correlation function as a function of the real-space separation r ; v ( r ) is the mean infall velocity ofgalaxies separated by r ; and σ ( r, µ ) is the pairwise velocity dispersion of galaxies. The parameter σ FOG is to account The FFT-based estimator for the power spectrum multipoles was first developed in Bianchi et al. (2015); Scoccimarro (2015) using adecomposition of L (cid:96) (ˆ k · ˆ r ) in Cartesian coordinates, thus it requires a larger number of FFTs than the estimator shown in Eqs. (15) and(16). Note that the window function has no effect on the correlation function, because the effect changes the DD ( DR ) and the RR pairs inthe same, multiplicative way, thus it is perfectly cancelled out in the Landy & Szalay estimator shown in Eq. (11). brief multi-tracer review ξ ( r ) , v ( r ) , σ ( r, µ ) are evaluated using theConvolution Lagrangian Perturbation Theory (CLPT) (Carlson et al. 2013; Wang et al. 2014), which requires a linearpower spectrum, and two bias parameters (cid:104) F (cid:48) (cid:105) and (cid:104) F (cid:48)(cid:48) (cid:105) . The bias parameters can either be treated as independentparameters in the fitting, or be related using the peak-background split argument (Matsubara 2008).To generalize Eq. (18) for the multi-tracer case, one only needs to perform the following transformation on the biasparameters, as appeared in the CLPT calculation (Carlson et al. 2013; Wang et al. 2014), (cid:104) F (cid:48) (cid:105) →
12 ( (cid:104) F (cid:48) A (cid:105) + (cid:104) F (cid:48) B (cid:105) ) (cid:104) F (cid:48)(cid:48) (cid:105) →
12 ( (cid:104) F (cid:48)(cid:48) A (cid:105) + (cid:104) F (cid:48)(cid:48) B (cid:105) ) (cid:104) F (cid:48) (cid:105) → (cid:104) F (cid:48) A (cid:105) (cid:104) F (cid:48) B (cid:105)(cid:104) F (cid:48)(cid:48) (cid:105) → (cid:104) F (cid:48)(cid:48) A (cid:105) (cid:104) F (cid:48)(cid:48) B (cid:105)(cid:104) F (cid:48) (cid:105) (cid:104) F (cid:48)(cid:48) (cid:105) →
12 ( (cid:104) F (cid:48) A (cid:105) (cid:104) F (cid:48)(cid:48) B (cid:105) + (cid:104) F (cid:48)(cid:48) A (cid:105) (cid:104) F (cid:48) B (cid:105) ) (19)Finally, ξ ( s ⊥ , s (cid:107) ) needs to be replaced with ξ ( s (cid:48)⊥ , s (cid:48)(cid:107) ) to account for the Alcock-Paczynski (AP) effect (Alcock &Paczynski 1979), which is an anisotropy in the clustering if a wrong cosmology is used to convert the redshifts intodistances. The AP effect transforms the coordinates in the following way, s (cid:48)⊥ = α ⊥ s ⊥ , s (cid:48)(cid:107) = α (cid:107) s (cid:107) , (20)with α ⊥ = D M ( z ) r fidd D fid M ( z ) r d , α (cid:107) = D H ( z ) r fidd D fid H ( z ) r d . (21)where r d denotes the sound horizon at recombination, D M ( z ) ≡ (1 + z ) D A ( z ), and D A ( z ) is the angular diameterdistance. D H ( z ) = c/H ( z ), H ( z ) is the Hubble expansion parameter. The superscript fid denotes the correspondingvalues in a fiducial cosmology. 3.2.2. Modeling the power spectrum
In Fourier space, the TNS model (Taruya et al. 2010) is proven to be robust up to quasi-nonlinear scales, thus hasbeen widely used by the BOSS and eBOSS collaboration for data analysis (Beutler et al. 2017; Gil-Mar´ın et al. 2018;Zhao et al. 2019; Gil-Mar´ın et al. 2020; de Mattia et al. 2020; Zhao et al. 2020). An extension of the TNS model for amulti-tracer survey is recently developed by Zhao et al. (2020), P ABg ( k, µ ) = D FoG ( k, µ ) (cid:2) P ABg ,δδ ( k ) + 2 f µ P ABg ,δθ ( k ) + f µ P AB θθ ( k ) + A AB ( k, µ ) + B AB ( k, µ ) (cid:3) , (22)where P ABg ,δδ ( k ) = b A1 b B1 P δδ ( k ) + (cid:0) b A1 b B2 + b B1 b A2 (cid:1) P b2 ,δ ( k ) + (cid:0) b As2 b B1 + b Bs2 b A1 (cid:1) P bs2 ,δ ( k )+ (cid:0) b As2 b B2 + b Bs2 b A2 (cid:1) P b2s2 ( k ) + (cid:0) b A3nl b B1 + b B3nl b A1 (cid:1) σ ( k ) P Lm ( k )+ b A2 b B2 P b22 ( k ) + b As2 b Bs2 P bs22 ( k ) + N AB P ABg ,δθ ( k ) = 12 (cid:2)(cid:0) b A1 + b B1 (cid:1) , P δθ ( k ) + (cid:0) b A2 + b B2 (cid:1) P b2 ,θ ( k ) + (cid:0) b As2 + b Bs2 (cid:1) P bs2 ,θ ( k ) , + (cid:0) b A3nl + b B3nl (cid:1) σ ( k ) P Lm ( k ) (cid:3) P g ,θθ ( k ) = P θθ ( k ) ,D FoG ( k, µ ) = (cid:110) kµσ v ] / (cid:111) − . (23)The subscripts δ and θ are the over-density and velocity divergence fields, respectively, and P δδ , P δθ and P θθ denotethe quasi-nonlinear auto- or cross-power spectrum, evaluated using tools such as the regularized perturbation theory Publicly available at https://github.com/wll745881210/CLPT GSRSD.
Wang & Zhao (RegPT) (Taruya et al. 2012). P Lm ( k ) denotes the linear power spectrum, and terms b and b stand for the linear biasand the second-order local bias, respectively. The second-order non-local bias b s2 and the third-order non-local bias b can be related to the linear bias via (Chan et al. 2012), b s2 = −
47 ( b − , b = 32315 ( b − . (24)The correction A, B terms for a multi-tracer survey requires a non-trivial extension, and we refer the readers toAppendix A of Zhao et al. (2020) for a full derivation and result. As for the correlation functions, the AP effect distorts the power spectrum by changing ( k, µ ) to ( k (cid:48) , µ (cid:48) ) in thefollowing way (Ballinger et al. 1996), k (cid:48) = kα ⊥ (cid:20) µ (cid:18) F − (cid:19)(cid:21) / ; µ (cid:48) = µF (cid:20) µ (cid:18) F − (cid:19)(cid:21) − / , (25)where F = α (cid:107) /α ⊥ . The theoretical power spectrum multipoles are, P AB (cid:96) ( k ) = (2 (cid:96) + 1)2 α ⊥ α (cid:107) (cid:90) − d µP ABg [ k (cid:48) ( k, µ ) , µ (cid:48) ( µ )] L (cid:96) ( µ ) , (26)where L (cid:96) is the Legendre polynomial of order (cid:96) .As mentioned in Sec. 3.1.2, the measured power spectrum multipoles are the true ones convolved with the surveywindow function shown in Eq. (17), therefore the same convolution needs to be applied to the theoretical prediction,for a fair comparison. An efficient way, which is based on the FFTLog algorithm Hamilton (2000) , for evaluating thisconvolution is developed in Wilson et al. (2017).3.3. Parameter estimation
Given measurements of the 2-point correlation function or power spectrum multipoles from a galaxy sample, thecorresponding data covariance matrix measured from the mock catalogs (see Zhao et al. 2020 for an example forthis procedure) and a theoretical template, one can preform a likelihood analysis to estimate parameters including α || , α ⊥ , f σ , as well as f NL using efficient algorithms including the Monte Carlo Markov Chain (MCMC), as imple-mented in the CosmoMC (Lewis & Bridle 2002) and Getdist (Lewis 2019) packages. WORKED EXAMPLESThis section shows a few worked examples of recent multi-tracer analyses, based on the GAMA, BOSS and eBOSSsurveys, respectively. 4.1.
A multi-tracer analysis for the GAMA survey
The first multi-tracer analysis was performed on the Galaxy and Mass Assembly (GAMA) survey (Blake et al. 2013).GAMA provided a sample of 178 ,
579 galaxies within the redshift range of 0 < z < . . To meet the requirement for a multi-tracer analysis, the entire sample was split into two subsamples by colourand luminosity. A joint constraint on f (the logarithmic growth rate) and σ v (the FoG damping parameter) is shownin Fig. 5, where two subsamples, split by colour (left panels) and luminosity (right panels), respectively, are usedfor a multi-tracer analysis in two redshift slices 0 < z < .
25 and 0 . < z < .
5, respectively. It is found that theprecision of f from the joint fits (black solid lines) can be improved by 10 − A multi-tracer analysis for the BOSS survey
Also splitting samples by the colour, Ross et al. (2014) performed a multi-tracer analysis on the ‘CMASS’ samplereleased in SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) DR10. This original CMASS DR10 sample ∼ atsushi.taruya/cpt pack.html Publicly available at https://jila.colorado.edu/ ∼ ajsh/FFTLog/ Publicly available at https://cosmologist.info/cosmomc/ Publicly available at https://github.com/cmbant/getdist brief multi-tracer review Figure 5.
The joint constraint on f and σ v presented in Blake et al. (2013). The GAMA sample is split into two using colour(left panels) and luminosity (right), respectively. The blue dashed, red dotted and black solid contours show constraints usingtracer 1 (tr-1), tracer 2 (tr-2) and the combined (Joint), respectively. The analysis is performed using galaxies in two redshiftslices: 0 < z < .
25 (upper) and 0 . < z < . consists of 540 ,
505 galaxies within 0 . < z < .
7. Based on a color-selection, ‘Red’ and ‘Blue’ subsamples are createdwith a relative bias being b Red /b Blue = 1 . ± .
04. Note that the color-selected samples only have 254 ,
936 galaxies intotal, which halves the number of the original, unsplit sample.The monopole and quadrupole of the color-selected samples and their cross-correlation are shown in Fig. 6, whichyields a constraint on the RSD parameter (with relevant parameters marginalized over) (Ross et al. 2014), f σ , Red = 0 . ± . f σ , Blue = 0 . ± . f σ , Cross = 0 . ± . . (27)This well demonstrates the importance of the cross correlation function for constraining the RSD parameter: it providesa better constraint on f σ on its own, namely, the uncertainty on f σ is reduced by ∼
25% using the cross-correlationfunction, compared to that using the auto-correlation function. Combining the auto- and cross-correlation furtherimproves the constraint to f σ , comb = 0 . ± . f σ , full = 0 . ± . A multi-tracer analysis for the eBOSS survey
As mentioned in Sec. 1, the SDSS-IV/eBOSS delivered observations of multiple tracers within one galaxy survey.eBOSS observes three types of discrete tracers, the LRGs, ELGs, and quasars (Dawson et al. 2016). The LRGs andELGs significantly overlap, namely, the overlapping sky coverage is 730 deg within the redshift range of 0 . < z < . . Wang & Zhao
Figure 6.
The correlation function monopole (left) and quadrupole (right), measured from the BOSS DR10 color-split samples.The top and bottom panels are for the ‘Red’, ‘Blue’ samples, respectively, while the middle panels show the measurement fromthe cross-correlation. Figure adopted from Ross et al. (2014).
20 40 60 80 100 120 140 160 s [Mpc/h]15010050050100 ( s ) s [ M p c / h ] Wang et al. (2020)(plot by Jiamin Hou, MPE)
LRGpCMASSxELG model k [ h Mpc ]500050010001500 k P ( k )[ M p c / h ] Zhao et al. (2020)(plot by Jiamin Hou, MPE)
LRGpCMASSxELG model P P P Figure 7.
A cross-correlation signal is well detected between the eBOSS LRGs and ELGs in both configuration (Wang et al.2020) and Fourier space (Zhao et al. 2020), as shown in Fig. 7 (data points with error bars) together with theorycurves of the best-fit model (solid lines). The cross-correlation can not only reduce the statistical error of the RSDparameters, it also helps with mitigating the systematic errors, because the contamination in the photometry of eachtracer, which is one possible source of the observational systematics, does not correlate with each other. Take theeBOSS ELG sample for example, it is known to be affected by photometric systematics, thus the correlation function brief multi-tracer review D R 1 4 Q S O 4 - z D R 1 2 c o n s e n s u s D R 1 2 9 - z P L C 1 8 D R 1 6 L R G x E L G ( Z h a o + , 2 0 2 0 ) D R 1 6 L R G x E L G ( W a n g + , 2 0 2 0 ) D R 1 6 Q S O f s r e d s h i f t z Figure 8.
A compilation of fσ measurements from the SDSS BOSS and eBOSS surveys, including those from the latestmulti-tracer analysis of the eBOSS DR16 sample in configuration space (DR16 LRG × ELG) (Wang et al. 2020) and Fourier-space (DR16 LRG × ELG) (Zhao et al. 2020), the consensus result from eBOSS DR16 QSOs (DR16 QSO) (Hou et al. 2020),the tomographic RSD measurement from the eBOSS DR14 QSO sample using the optimal redshift weighting method (DR14QSO 4- z ) (Zhao et al. 2019), and the tomographic RSD measurement from the BOSS DR12 sample in Fourier space (DR129- z ) (Zheng et al. 2019). For a reference, the blue band is the 68% CL constraint derived from Planck 2018 observations in aΛCDM cosmology (Planck Collaboration et al. 2018). and power spectrum of the ELG sample has to be specifically processed for a cosmological analysis (Tamone et al.2020; de Mattia et al. 2020). However, the cross correlation function and cross power spectrum are largely immune tothis type of systematics, as demonstrated using the mock catalogs (Wang et al. 2020; Zhao et al. 2020).Fig. 8 shows the measurement of f σ from the eBOSS multi-tracer analyses, with other recent measurements fromBOSS and eBOSS programs. Note that the effective redshifts of the LRG and ELG samples are different, and Zhao etal. (2020) and Wang et al. (2020) take different approaches to account for this effect (both approaches are validatedby the mock tests), thus the resultant measurements are at different effective redshifts, e . g . , Zhao et al. (2020) reportstwo measurements at z = 0 .
70 and 0 . z = 0 .
77. Whilea direct comparison is not straightforward at this level, Zhao et al. (2020) performed an additional measurement at z = 0 .
77, and found a consistency with Wang et al. (2020). In this analysis, it is found that the constraint on f σ getsimproved by approximately 12%, compared to that using the LRGs alone (Wang et al. 2020). CONCLUSION AND DISCUSSIONSStepping into the stage-IV era of galaxy surveys, we are seeking statistical and theoretical methods to improve theprecision on the measured cosmological parameters, which is limited by the cosmic variance and the shot noise. Onepromising way to beat the cosmic variance is to contrast the 2-point statistics of different tracers, yielding a quantityof cosmological importance, which is free from the cosmic variance.In this brief review, we present the basic idea of the multi-tracer method, outline a procedure for performing themulti-tracer analysis, followed by a few worked examples based on large galaxy surveys, which has well demonstratedthe efficacy of this method.The newly-started Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration et al. 2016 a,b) is a typicalmulti-tracer survey, whose LRGs and ELGs targets do overlap significantly across a wide redshift range with a muchhigher number density than that of eBOSS. This makes it ideal for a multi-tracer analysis for improving the statisticalprecision of the RSD and primordial non-Gaussianity. The multi-tracer analysis also helps with the observational sys-tematics through the cross-correlation between tracers, in addition to mitigating the systematics by a better modelingand pipeline for processing the raw observations.2
Wang & Zhao