Decomposition of Spectra from Redshift Distortion Maps
aa r X i v : . [ a s t r o - ph . C O ] M a r Mon. Not. R. Astron. Soc. , ?? – ?? (0000) Printed 7 November 2018 (MN L A TEX style file v2.2)
Decomposition of Spectra from Redshift Distortion Maps
Yong-Seon Song and Issha Kayo Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwanoha, Chiba 277-8582, Japan
ABSTRACT
We develop an optimized technique to extract density–density and velocity–velocity spectraout of observed spectra in redshift space. The measured spectra of the distribution of ha-los from redshift distorted mock map are binned into 2–dimensional coordinates in Fourierspace so as to be decomposed into both spectra using angular projection dependence. Withthe threshold limit introduced to minimize nonlinear suppression, the decomposed velocity–velocity spectra are reasonably well measured up to scale k = . h Mpc − , and the measuredvariances using our method are consistent with errors predicted from a Fisher matrix analysis.The detectability is extendable to k ∼ . h Mpc − with more conservative bounds at the costof weakened constraint. Key words: cosmology: large-scale structure
The evolution of large scale structure, as revealed in the cluster-ing of galaxies observed in wide–deep redshift surveys has beenone of key cosmological probes. Structure formation is driven by acompetition between gravitational attraction and the expansion ofspace-time, which enables us to test our model of gravity at cosmo-logical scales and the expansion of history of the Universe (Wang2008; Linder 2008; Guzzo et al. 2008; Song & Percival 2009;Simpson & Peacock 2009; Guzik et al. 2010; McDonald & Seljak2009; Stril et al. 2009; Bean & Tangmatitham 2010).Maps of galaxies where distances have been measured fromredshifts show anisotropic deviations from the true galaxy dis-tribution (York 2000; Peacock 2001; Colless 2003; Hawkins2003; Percival 2004; Zehavi 2005; Le F`evre 2005; Tegmark2006; Okumura 2008; Gaztanaga & Cabre 2008; Garilli 2008;Guzzo et al. 2008), because galaxy recession velocities includecomponents from both the Hubble flow and peculiar velocities.In linear theory, a distant observer should expect a multiplicativeenhancement of the overdensity field of tracers due to the pecu-liar motion along the line of sight (Davis & Peebles 1982; Kaiser1987; Lilje & Efstathiou 1989; McGill 1990; Lahav et al. 1991;Hamilton 1992; Fisher et al. 1994; Fisher 1995). In principle, theobserved spectra in redshift space can be decomposed into bothdensity–density and velocity–velocity spectra using angular projec-tion dependence (Song & Percival 2009; Percival & White 2008;White et al. 2009; Song et al. 2010). With a local linear bias, thereal-space galaxy density field is affected, while the peculiar veloc-ity term is not. In this paper, we attempt to extract velocity–velocityspectra as an unbiased tool to trace the history of structure forma-tion. A theoretical formalism (White et al. 2009) was derived forforecasting errors when extracting velocity–velocity spectra out of the observed redshift space distortion maps. However, it is not yetfully understood what the optimal technique is to practically de-compose the spectra as theory predicts. We propose a statisticaltechnique to extract it up to the limit of theoretical estimation.Our method utilizes the distinct angular dependence of density–density and velocity–veclocity spectra to decompose them fromtwo–dimensional redshift power spectra, and is consistent with thetheoretical estimate from Fisher matrix analysis.We present the detailed formalism in the next section. TheFisher matrix analysis to decompose spectra is briefly reviewed,then we present the method to decompose spectra in an optimalway with mock data. We discuss statistical method to minimize theeffect by nonlinear suppression.
The observed power spectrum in redshift space is decomposed intospectra of density fluctuations and peculiar velocity fields in realspace. The observed power spectra in redshift space, ˜ P , is given by,˜ P ( k , µ, z ) = n P gg ( k , z ) + µ r ( k ) h P gg ( k , z ) P ΘΘ ( k , z ) i / + µ P ΘΘ ( k , z ) o G ( k , µ, σ v ) , (1)where P gg is the galaxy–galaxy density spectrum, P ΘΘ is thevelocity–velocity spectrum ( Θ is the divergence of velocity mapin unit of aH ), and µ denotes the cosine of the angle between ori-entation of the wave vector and the line of sight. Because this de-composition is valid only at large scale and when the rotation of thevelocity field is negligible, we focus on modes of k < . h Mpc − (Pueblas & Scoccimarro 2009). The cross-correlation coefficient c (cid:13) Song and Kayo
Figure 1.
Power spectra from mock map in 2D cartesian coordinate ( k ⊥ , k k ). r ( k ) is defined as r ( k ) ≡ P g Θ / p P gg P ΘΘ . The density and velocitydivergence are highly correlated for k < . h Mpc − so we assumethat both are perfectly correlated, r ( k ) ∼ P gg and P ΘΘ . As Scoccimarro (2004) clearly pointed out, the redshiftspace power spectrum is suppressed along line-of-sight due to thevelocity dispersion of large-scale flow, and we follow his model byintroducing a function G = exp( − k µ σ v ) where σ v will be calcu-lated from linear theory. Considering the possibility that nonlineardynamics, like Finger-of-Gods effect, might contaminate the powerspectrum, we use this term to find a cut-off scale of µ to excludedata which could be affected strongly by nonlinear dynamics. In-deed, Taruya et al. (2009) pointed out that σ v calculated by lineartheory does not match with result from N-body simulations if onetries to model the power spectrum at & . h Mpc − . This cut-offedge µ cut is defined by µ cut ≡ σ th / k σ v , where the value of σ th willbe discussed later.We estimate the accuracy of decomposition of P gg and P ΘΘ from ˜ P using Fisher matrix analysis determining the sensitivity of aparticular measurement. Fisher matrix for this decomposition, F dec αβ ,is written as, F dec αβ = Z µ cut − µ cut d µ Z ∂ ˜ P ( k , µ ) ∂ p α ∂ ˜ P ( k , µ ) ∂ p β V e ff ( ˜ P )˜ P ( k , µ ) k dk π ) , (2)where p α = ( P gg , P ΘΘ ). The effective volume V e ff ( ˜ P ) is given by, V e ff ( ˜ P ) = " n ˜ Pn ˜ P + V survey , (3)where n denotes galaxy number density.Derivative terms in Eq. (2) are given by, ∂ ln ˜ P ( k i , µ, z j ) ∂ P gg ( k i , z j ) = P ( k i , µ, z j ) + µ s P ΘΘ ( k i , z j ) P gg ( k i , z j ) ∂ ln ˜ P ( k i , z j ) ∂ P ΘΘ ( k i , z j ) = µ ˜ P ( k i , µ, z j ) s P gg ( k i , z j ) P ΘΘ ( k i , z j ) + µ . (4) Figure 2.
The observed power spectra at scales, ¯ k =0.03, 0.05, 0.07 and0.09 h Mpc − (from top to bottom) are plotted with error bars at various µ .Solid curves are ˜ P th ( k , µ ) (Kaiser effect alone) and dash curves are ˜ P fit ( k , µ )(including dispersion effect) from best fitting bias b ( k ). The diagonal elements of the inverse Fisher matrix indicate the esti-mated errors of decomposion accuracy. The variances of P gg ( k i , z j )and P ΘΘ ( k i , z j ) is given by, σ [ P gg ( k i , z j )] = q F dec − gg ( k i , z j ) σ [ P ΘΘ ( k i , z j )] = q F dec − ΘΘ ( k i , z j ) . (5) We use the halo catalogue from the time-streaming mock map ofthe Horizon simulation (Teyssier et al. 2008), and cut 1 (Gpc/ h ) cubic box at the median redshift ¯ z = .
83, which contains 2.2 mil-lion halos. The fiducial cosmological parameters of the simulationare given by ( Ω m = . , Ω k = , h = . , σ = . , n S = . k ⊥ , k k ) space. The density fluctuation field is constructedby assigning the halos to 512 grids for the fast Fourier transforma-tion (FFT) using the nearest grid point (NGP) method. Fig. 1 showsthe resulting power spectrum. While linearly spaced bins in ( k ⊥ , k k )are used in this plot for presentation purpose, we use bins in k and µ for the following analysis. k is divided in ∆ k = . h Mpc − lin-early equally spaced bins from k = . h Mpc − to 0 . h Mpc − and µ is in 5 linear-bins from 0 to 1 with equal spacing. The measured2D power spectra in ( k , µ ) coordinate are shown in Fig. 2.The Gaussian variance is used to derive errors for each binshown as error bars in Fig. 2, σ [ ˜ P ob ( k , µ )] = ˜ P ( k , µ ) p / N ( k , µ )where N ( k , µ ) is number of modes in Fourier space. We test thisusing an alternative method, jack–knife errors (we do not attemp togenerate more samples as we are interested in mocking real observ-ables in a single patch). A total 64 jack–knife samples are preparedout of a single mock map by dividing each coordinate into 4 pieces. c (cid:13) , ?? – ?? ecomposition of Spectra from Redshift Distortion Maps k ( h Mpc − ) 0 .
03 0 .
05 0 .
07 0 . b ( k ) 1 . ± . . ± . . ± . . ± . Table 1.
Best fitting biases b ( k ) at given scales k from k = .
03 to0.09 h Mpc − . Both errors agrees well, and different bins weakly correlate witheach other.Halo distribution is a biased tracer of the dark matter distribu-tion. Theoretical ˜ P th ( k , µ ) from Kaiser effect only is given by,˜ P th ( k , µ ) = b P mm + b µ r h p P mm P ΘΘ + µ P ΘΘ , (6)where P mm ( k ) is the dark matter density–density spectra and b = b ( k ) is the halo bias for each given scale. Spectra P mm ( k ) and P ΘΘ ( k )are given from the cosmological parameters used for the simula-tion, and the halo cross–correlation parameter r h is set to be unity.It has been tested that r for dark matter– Θ is perfectly correlated atlinear scales k < . h Mpc − from simulation. Unfortunately, thesame sanity check is not applicable for halo maps due to the insuffi-cient number of halo in each grid for direct velocity power spectra.Instead, the theoretical ˜ P th ( k , µ ) is derived based upon r h ( k ) = P ΘΘ ( k ) at linear scales.The tracer bias is assumed not to be determined by theoret-ical formalism or by other experiment. Instead of applying scaleindependent bias, b ( k ) is varied independently for each k –bin. Wefit b ( k ) for each mode to get ˜ P th ( k , µ ) (solid curves in Fig. 2).In Table 1, the best fit b ( k ) is given with 1– σ confidence level.Theoretical ˜ P th ( k , µ ) with fitted b ( k ) is over–plotted with the mea-sured ˜ P ob ( k , µ ) from the simulation in Fig. 2. We cut out scales k < . h Mpc − due to our limited box size and k > . h Mpc − due to non-linear effects.Using ˜ P th ( k , µ ), theoretical errors are estimated from Fishermatrix analysis. Un-filled black contours in Fig. 3 represent thetheoretical expectation around b ( k ) P mm ( k ) and P ΘΘ ( k ). As it isprediced from halo bias model, measured bias is nearly scale in-dependent. Spectra P gg ( k i ) and P ΘΘ ( k i ) are fitted simultaneously to ˜ P ob ( k i , µ p )where i and p denote k and µ bins respectively. Bias is not pa-rameterized to fit ˜ P ob ( k i , µ p ), instead, we use P gg ( k i ). The fitting˜ P fit ( k i , µ p ) is given by˜ P fit ( k i , µ p ) = h P gg ( k i ) + µ p p P gg P ΘΘ + µ p P ΘΘ ( k i ) i × G ( k , µ j , σ v ) . (7)We consider the velocity dispersion effect from one–dimensionalvelocity dispersion σ v which is given by, (cid:18) σ v aH (cid:19) = · π Z P ΘΘ ( k , z ) dk . (8)This formula needs P ΘΘ which is what we want to measure. Wewill discuss how we calculate this term in the next paragraph. Eq. 7is expected to be invalidated beyond some threshold. The observedmodes are cut out when it goes beyond given the threshold limit σ th as k i µ cut σ v > σ th . The fiducial value is σ th = .
24 which representsconfidence of theoretical prediction up to 6% drop of G ( k i , µ cut , σ v )from unity.The most important factor in the integration Eq. 8 is the ampli-tude of P ΘΘ , as scale–dependent factor of P ΘΘ is tightly constrained Figure 3.
Contour plots are shown for decomposed P gg ( k i ) and P ΘΘ ( k i ) at k = .
03, 0.05, 0.07 and 0.09 h Mpc − . Unfilled black contours representtheoretical prediction from Fisher matrix analysis, and filled blue contoursrepresent measured P gg ( k i ) and P ΘΘ ( k i ) from mock map in redshift space. by CMB physics. The shape of the power spectra is determined be-fore the epoch of matter–radiation equality. When the initial fluc-tuations reach the coherent evolution epoch after matter-radiationequality, they experience a scale-dependent shift from the momentthey re-enter the horizon to the equality epoch. Gravitational in-stability is governed by the interplay between radiative pressureresistance and gravitational infall. The different duration of modesduring this period results in a shape dependence on the power spec-trum. This shape dependence is determined by the ratio betweenmatter and radiation energy densities and sets the location of thematter-radiation equality in the time coordinate (Song et al. 2010).One way to estimate σ v will be to use fitted P ΘΘ for each fit-ting step. Our measurement is, however, limited at scale of k . . h Mpc − and the contribution to σ v from P ΘΘ at k & . h Mpc − is small but not negligible ( ∼ σ v us-ing the linear shape of P ΘΘ with an amplitude which is estimated ateach fitting step as follows.For each P ΘΘ we want to test, we calculate the amplitude fac-tor g Θ ( k i , z ) defined by P ΘΘ ( k i , z ) = g Θ ( k i , z ) P ΘΘ ( k i , z lss ) , (9)and constrain the amplitude by calculating a weighted average of¯ g Θ ( z ) = P i max i = i min (cid:16) g Θ ( k i , z ) /σ g Θ ( k i , z ) (cid:17)P i max i = i min /σ g Θ ( k i , z ) . (10)Here σ g Θ ( k i , z ) is given by σ g Θ ( k i , z ) = g fid Θ ( k i , z ) σ [ P fid ΘΘ ( k i , z )] P fid ΘΘ ( k i , z ) , (11)and σ [ P fid ΘΘ ( k i , z j )] is given by theoretical estimation in Eq. 5 andsuperscript ‘fid’ denotes the fiducial model for Fisher matrix analy-sis. We would not expect that fractional error of P ΘΘ ( k i , z ) is muchdependent on different fiducial models. The value of σ v at the bestfitted power spectra is 2 . h − Mpc (the linear theory prediction is3 . h − Mpc). c (cid:13) , ?? ––
03, 0.05, 0.07 and 0.09 h Mpc − . Unfilled black contours representtheoretical prediction from Fisher matrix analysis, and filled blue contoursrepresent measured P gg ( k i ) and P ΘΘ ( k i ) from mock map in redshift space. by CMB physics. The shape of the power spectra is determined be-fore the epoch of matter–radiation equality. When the initial fluc-tuations reach the coherent evolution epoch after matter-radiationequality, they experience a scale-dependent shift from the momentthey re-enter the horizon to the equality epoch. Gravitational in-stability is governed by the interplay between radiative pressureresistance and gravitational infall. The different duration of modesduring this period results in a shape dependence on the power spec-trum. This shape dependence is determined by the ratio betweenmatter and radiation energy densities and sets the location of thematter-radiation equality in the time coordinate (Song et al. 2010).One way to estimate σ v will be to use fitted P ΘΘ for each fit-ting step. Our measurement is, however, limited at scale of k . . h Mpc − and the contribution to σ v from P ΘΘ at k & . h Mpc − is small but not negligible ( ∼ σ v us-ing the linear shape of P ΘΘ with an amplitude which is estimated ateach fitting step as follows.For each P ΘΘ we want to test, we calculate the amplitude fac-tor g Θ ( k i , z ) defined by P ΘΘ ( k i , z ) = g Θ ( k i , z ) P ΘΘ ( k i , z lss ) , (9)and constrain the amplitude by calculating a weighted average of¯ g Θ ( z ) = P i max i = i min (cid:16) g Θ ( k i , z ) /σ g Θ ( k i , z ) (cid:17)P i max i = i min /σ g Θ ( k i , z ) . (10)Here σ g Θ ( k i , z ) is given by σ g Θ ( k i , z ) = g fid Θ ( k i , z ) σ [ P fid ΘΘ ( k i , z )] P fid ΘΘ ( k i , z ) , (11)and σ [ P fid ΘΘ ( k i , z j )] is given by theoretical estimation in Eq. 5 andsuperscript ‘fid’ denotes the fiducial model for Fisher matrix analy-sis. We would not expect that fractional error of P ΘΘ ( k i , z ) is muchdependent on different fiducial models. The value of σ v at the bestfitted power spectra is 2 . h − Mpc (the linear theory prediction is3 . h − Mpc). c (cid:13) , ?? –– ?? Song and Kayo P gg ( k i ) determines the overall amplitude of ˜ P fit ( k i , µ j ), and P ΘΘ ( k i ) determines the running of ˜ P fit ( k i , µ j ) in the µ direction.These distinct contribution allows us to separate information of P gg ( k i ) and P ΘΘ ( k i ) from 5 different µ bins at each k i bin. We findthese P gg ( k i ) and P ΘΘ ( k i ) by minimizing χ = i max X i = i min X p = X q = [ ˜ P ob ( k i , µ p ) − ˜ P fit ( k i , µ p )] × Cov − pq ( k i )[ ˜ P ob ( k i , µ q ) − ˜ P fit ( k i , µ q )] , (12)where k i min = . h Mpc − and k i max = . h Mpc − . Off diagonalelements of the covariance matrix are nearly negligible and thosediagonal elements are written asCov − pp ( k i ) = σ [ ˜ P ob ( k i , µ p )] . (13)We present the difference between ˜ P th ( k i , µ p ) (Kaiser effect) and˜ P fit ( k i , µ p ) (including dispersion effects) in Fig. 2. With the fiducial σ th = .
24, only one bin of mode k i = . h Mpc − at µ p = . k bins through σ v , those are minimallycorrelated and the results shown Fig. 3 are consistent with theoret-ical predictions. Velocity–velocity spectra are remarkably well extracted out ofmeasured spectra in redshift space at scales k = . , . . h Mpc − , and relatively well extracted at scale k = . h Mpc − with more conservative confidence on the thresholdlimit. Filled blue contours in Fig. 3 represent fitted value of P gg ( k i )and P ΘΘ ( k i ), and unfilled black contours represent estimation fromtheory with central values given by simulation. For scales from k = .
03 to 0 . h Mpc − , the decomposed P ΘΘ ( k i ) though our fit-ting strategy is trustable, which suggests that the few assumptionsmade in this paper are valid for those scales: • The assumption of perfect correlation between halo distribu-tion and velocity field is correct. The agreement of P ΘΘ ( k i ) betweenfitted and true values supports our assumption of r h ∼ • Dispersion effect is reasonably modelled at scales within ourconfidence limits, which enables us to extract P ΘΘ ( k i ) in model in-dependent way using estimated σ v .For k = . h Mpc − , more conservative threshold limitsshould be applied to remove non-linear supression. In Fig. 4, wepresent best fit P ΘΘ ( k i ) with different threshold limits of σ th = . σ th = .
18 (right panel). With σ th = .
24, only onebin at µ j = . µ j = . k = . h Mpc − which can be re-moved by more conservative bound σ th = .
18. Shown in the rightpanel of Fig. 4, true P ΘΘ ( k i ) is restored at the cost of weakenedconstraint.Theoretical estimation from Fisher matrix analysis is an op-timistic bound on errors. It is noticeable that measured varinaces(filled blue contours in Fig. 3) are consistent with estimated vari-ances (unfillled black contours in Fig. 3), which assures us that ourmethod is optimized extraction of P ΘΘ ( k i ) for the given simulationspecification. Figure 4.
Contour plots are shown for decomposed P gg ( k i ) and P ΘΘ ( k i )at k = . h Mpc − with σ th = .
24 (left panel) and 0.18 (right panel).Unfilled black and filled blue contours represent the same in Fig. 3.
We propose a statistical tool to decompose P gg ( k ) and P ΘΘ ( k ) prac-tically out of redshift distortion maps, with a few assumptions:1) perfect correlation between density and velocity fluctuations,2) confidence on theoretical prediction of velocity dispersion ef-fect within threshold limit. The results show that the true value ofvelocity–velocity spectra up to k = . h Mpc − are successfullyrecovered using theoretical dispersion effect. The detectability isextendable up to k ∼ . h Mpc − with more conservative thresholdlimit at the cost of weakened constraint. We find that the theoreti-cal dispersion effect can be estimated from P ΘΘ ( k ) parameters us-ing weighted average at k < . h Mpc − . In linear regime, P ΘΘ ( k )is well–measured with this estimated σ v as much as with the truefixed σ v of the simulation.We find that the biased measurement of P ΘΘ ( k ) is mainlycaused by the unpredictable non–linear supression effect at k > . h Mpc − . The detectability limit in scale can be extended by pa-rameterizing this effect (Tang et al. 2010), but we scope our rangeof interest in linear regime in this paper. ACKNOWLEDGMENTS
We would like to thank Romain Teyssier for offering simulationmap of Horizon ( ), and to thank PrinaPatel for comment on the manuscript. Y-S.S. is supported by STFCand I.K. acknowledges support by JSPS Research Fellowship andWPI Initiative, MEXT, Japan.
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