Toward a more stringent test of gravity with redshift space power spectrum: simultaneous probe of growth and amplitude of large-scale structure
YYITP-21-09
Toward a more stringent test of gravity with redshift space power spectrum:simultaneous probe of growth and amplitude of large-scale structure
Yong-Seon Song , Yi Zheng , Atsushi Taruya , ∗ Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea School of Physics and Astronomy, Sun Yat-sen University, 2 Daxue Road, Tangjia, Zhuhai, 519082, China Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan, Kavli Institute for the Physics and Mathematics of the Universe,Todai Institutes for Advanced Study, the University of Tokyo,Kashiwa, Chiba 277-8583, Japan (Kavli IPMU, WPI) (Dated: February 4, 2021)Redshift-space distortions (RSD) offers an exciting opportunity to test the gravity on cosmologicalscales. In the presence of galaxy bias, however, the RSD measurement at large scales, where thelinear theory prediction is safely applied, is known to exhibit a degeneracy between the parametersof structure growth f and fluctuation amplitude σ , and one can only constrain the parameters inthe form of fσ . In order to disentangle this degeneracy, in this paper, we go beyond the lineartheory, and consider the model of RSD applicable to a weakly nonlinear regime. Based on the Fishermatrix analysis, we show explicitly that the degeneracy of the parameter fσ can be broken, and σ is separately estimated in the presence of galaxy bias. Performing further the Markov chain MonteCarlo analysis, we verify that our model correctly reproduces the fiducial values of fσ and σ , withthe statistical errors consistent with those estimated from the Fisher matrix analysis. We showthat upcoming galaxy survey of the stage-IV class can unambiguously determine σ at the precisiondown to (cid:46)
10% at higher redshifts even if we restrict the accessible scales to k (cid:46) . h Mpc − . PACS numbers: 98.80.-k;04.50.Kd;98.65.Dx
I. INTRODUCTION
Since its discovery two decades ago [1, 2], the originof late-time cosmic acceleration has remained puzzled.While the flat Lambda cold dark matter (ΛCDM) modelis currently the best as concordant cosmological model todescribe both the cosmic expansion and structure forma-tion, consistent with observations of cosmic microwavebackground and large-scale structure, the tension withthe cosmological parameters determined at the local uni-verse has been recently highlighted, suggesting a need ofnew physics beyond ΛCDM model [3–5].Theoretically, the origin of cosmic acceleration can beexplained by either the presence of a mysterious energycomponent called dark energy or a long-distance mod-ification of gravity, referred to as modified gravity [6–16]. In order to realize the accelerated cosmic expansion,the former introduces a negative pressure support, andthe latter changes a law of gravitational physics on largescales. Observationally discriminating between two sce-narios therefore requires a simultaneous measurements ofthe cosmic expansion and growth of structure.Among various cosmological probes, the redshift-spacegalaxy clustering offers a sensible probe of both the cos-mic expansion and growth of structure. At large scales,the baryon acoustic oscillations (BAO) imprinted on theclustering pattern of galaxies appears statistically mani-fest, and it can be used for a standard ruler to determine ∗ Electronic address: [email protected] the angular diameter distance ( D A ) and Hubble param-eters H at high redshifts through the Alcock-Paczynskitest [17]. Further, the observed galaxy distribution viaspectroscopic surveys is statistically anisotropic due tothe peculiar velocity of galaxies by Doppler effect, re-ferred to as the redshift-space distortions (RSD). In lineartheory, the strength of anisotropies is solely characterizedby the growth rate f , defined by f = d ln D + /d ln a , with D + and a being linear growth factor and scale factorof the Universe, respectively. Thus, through a precisionmeasurement of redshift-space galaxy power spectrum orcorrelation function, one can in principle obtain simul-taneously the information on the three parameters (i.e., D A , H , and f ).However, in the presence of the galaxy bias, the situ-ation becomes bit complicated. To be precise, considerthe large scales where the linear theory is safely applied.Then, the galaxy power spectrum is generally modeledas P ( S ) g ( k, µ ) = (cid:2) b σ ( z ) + f σ ( z ) µ (cid:3) P m ( k ) , (1)where the variable µ is the directional cosine betweenwavevector and line-of-sight direction. The function P m is the matter power spectrum normalized at the presentday. The quantity σ ( z ) is the root-mean-square massfluctuation in spheres with radius 8 h − Mpc at redshift z , and is recast in linear theory as σ ( z ) = σ (0) D + ( z )with D + being normalized to unity at z = 0. The b is the linear bias parameter. Note that taking furtherthe Alcock-Paczynski effect into account, the projectedwavenumbers perpendicular and parallel to the line-of-sight direction, k ⊥ and k (cid:107) = µ k , are respectively re-placed with D A /D A, fid k ⊥ and ( H/H fid ) − k (cid:107) , and the a r X i v : . [ a s t r o - ph . C O ] F e b power spectrum given above is further multiplied by( H/H fid )( D A /D A, fid ) − , where the quantities with sub-script indicate those estimated in a fiducial cosmologicalmodel.The structure of Eq. (1) indicates that the parameter σ is degenerated with growth rate f and bias b , and onecan only determine the combinations of the parameters,i.e., b σ and f σ , through the observed power spectrum.In other words, unless the bias parameter b is known apriori, one cannot break the degeneracy between growthrate f and σ . Note that the Alcock-Paczynski effectinduces distinctive anisotropies in the measured powerspectrum, and making use of BAO features, one can sep-arately determine D A /D A, fid and H/H fid without anydegeneracy with σ .Toward a solid test of gravity and cosmic acceleration,the degeneracy between f and σ has to be broken. Todo this, one simple approach is to combine the powerspectrum with other cosmological probe. In this respect,the use of galaxy bispectrum would be obviously impor-tant, and this can also provide an additional cosmologi-cal information, further tightening the cosmological con-straints [18, 19]. Another approach is to stick to thepower spectrum, and to use the small-scale informationbeyond the linear regime. Recalling that Eq. (1) is validonly at large scales, if we go to nonlinear regime, thereappear corrections involving the parameters σ and f butwith a different combination. In fact, using the perturba-tion theory calculation, we can identify the directional-dependent terms proportional to f σ or f σ in thematter power spectrum [20, 21]. Thus, provided an ac-curate theoretical model, accessing (weakly) nonlinearscales would give a way to break the degeneracy between f and σ . A price to pay is, however, the new degreesof freedom to characterize the galaxy bias at nonlinearscales. That is, beyond linear scales, we need to introduceseveral bias parameters describing the nonlinear modifi-cation to power spectrum, and one has to marginalizethem to determine parameters sensitive to the cosmol-ogy. It is thus not trivial at all that the growth rate f isuniquely determined without any degeneracy.In this paper, based on a nonlinear theoretical modelof the redshift-space galaxy power spectrum, we explic-itly demonstrate that the degeneracy between f and σ isbroken at weakly nonlinear scales. The model specificallyconsidered here is called the hybrid RSD model that hasbeen developed by Ref. [22–24]. This is a perturbationtheory based model [20, 21], but partly including the cor-rection terms calibrated by N -body simulations, the ac-curacy is improved and the applicable range is extended[23]. In this paper, incorporating further the nonlinearbias prescription into the hybrid RSD model, we will in-vestigate how the degeneracy between f and σ is broken,and quantify the expected constraints on parameters σ , f σ as well as D A and H for a representative galaxysurvey (Dark Energy Spectroscopy Instrument, DESI).Related to the present study, one may comment on thefull-shape analysis that recently performed using SDSS BOSS galaxies [25, 26]. In this study, based on the ef-fective field theory of large-scale structure, the input lin-ear power spectrum is allowed to vary, enabling us todirectly constrain each cosmological parameters. Goingbeyond linear regime, not only tightening constraints butalso breaking parameter degeneracy is shown to be man-ifest [27], assuming the underlying theory of gravity. Thespirit of this approach is close to the present paper, butwe here stick to the consistent test of gravity, and take f or f σ to be a free parameter, independent of othercosmological parameters.This paper is organized as follows. Sec. II presentsa model of the redshift-space galaxy power spectrumapplicable to the weakly nonlinear regime. Based onthis model as a theoretical template, in Sec. III, we usethe Fisher matrix formalism to quantitatively investigatehow well one can break the degeneracy of the parameters f σ . Sec. IV examines the Markov chain Monte Carloanalysis to estimate the cosmological parameters in thehal catalog. We verify that our model of RSD faithfullyreproduces the fiducial parameters in the N -body simula-tions, consistent with the Fisher forecast results. Finally,Sec. V is devoted to conclusion and discussion. II. HYBRID RSD MODEL FOR GALAXYCLUSTERING
In this section, we present the model of redshift-spacegalaxy power spectrum beyond the linear regime. Af-ter briefly reviewing the hybrid RSD model for matterfluctuations in Sec. II A, we extend it to incorporate thenonlinear galaxy bias in Sec. II B.
A. Hybrid RSD model for matter power spectrum
In order to model the observed galaxy power spec-trum, one critical ingredient is the redshift-space distor-tions (RSD). In principle, all the effects of RSD is ac-counted for by the simple relation between real-space ( r )and redshift-space ( s ) positions: s = r + v · ˆ zaH ˆ z, (2)where the quantities v , a and H respectively denote thephysical peculiar velocity, the scale factor of the Universe,and the Hubble parameter. In this paper, we will takethe distant-observer limit, and choose the z -direction asthe line-of-sight direction. With the mapping relationgiven above, the power spectrum in redshift space is gen-erally expressed in terms of the real-space quantities (e.g.,Ref. [20]): P (S) ( k, µ ) = (cid:90) d x e i k · x (cid:10) e j A A A (cid:11) , (3)where the variable j and functions A i are defined asfollows: j = − i kµ,A = u z ( r ) − u z ( r (cid:48) ) ,A = δ ( r ) + ∇ z u z ( r ) ,A = δ ( r (cid:48) ) + ∇ z u z ( r (cid:48) ) . The quantities x and u are given respectively by x = r − r (cid:48) and u ≡ − v / ( aH ). The function u z is the line-of-sight component of u .Based on Eq. (3), a rigorous calculation of theredshift-space power spectrum generally requires a non-perturbative treatment. This is true even if the den-sity and velocity follow the Gaussian statistics. We thusemploy the perturbative treatment and derive the ex-pression relevant to the weakly nonlinear regime. To dothis, one proposition made in Refs. [20, 28] is that a partof the zero-lag correlation in the exponent is kept as anon-perturbative contribution, while rest of the terms isTaylor-expanded. The resultant expression for the powerspectrum, relevant at the next-to-next-to-leading orderbecomes [20, 21, 28] P (S) ( k, µ ) = D FoG ( kµσ z ) (cid:104) P δδ ( k ) + 2 µ P δ Θ ( k ) + µ P ΘΘ ( k )+ A ( k, µ ) + B ( k, µ ) + T ( k, µ ) + F ( k, µ ) (cid:105) , where the quantity Θ is the velocity-divergence field,Θ ≡ −∇ · v / ( aH ) = ∇ · u . The spectra, P δδ , P δ Θ ,and P ΘΘ are respectively the auto-power spectrum ofdensity, velocity-divergence fields, and their cross-powerspectrum. The first line in the bracket is originated fromthe term (cid:104) A A (cid:105) c , and is obtained assuming the irro-tational flow. At the linear order, it is reduced to thesquashing Kaiser term, i.e., Eq. (1). The rest of the termsin the bracket are the higher-order corrections character-izing the nonlinear correlations between the density andvelocity fields, defined by A ( k, µ ) = j (cid:90) d x e i k · x (cid:104) A A A (cid:105) c ,B ( k, µ ) = j (cid:90) d x e i k · x (cid:104) A A (cid:105) c (cid:104) A A (cid:105) c ,T ( k, µ ) = 12 j (cid:90) d x e i k · x (cid:104) A A A (cid:105) c ,F ( k, µ ) = − j (cid:90) d x e i k · x (cid:104) u z u (cid:48) z (cid:105) c (cid:104) A A (cid:105) c . Note that the factorized term D FoG in Eq. (5) representsa non-perturbative contribution coming from the zero-lagcorrelation of velocity fields, and it plays a role to sup-press the overall amplitude at small scales. The explicitfunctional form will be later specified (see Sec. III).To explicitly compute Eq. (5) in the case of the matterfluctuations, in Ref. [23], we have used the re-summedperturbation theory treatment by the multi-point prop-agator expansion [21, 29, 30]. To account further for a small but non-negligible flaw in the perturbative cal-culations, we have added the corrections calibrated bythe N -body simulations, which enables us to predict theredshift-space matter power spectrum at the precision of (cid:46)
1% down to k (cid:39) . h Mpc − . In Appendix A, thedescription of our hybrid model is presented for the mat-ter power spectrum, and the explicit dependence on thequantities σ ( z ) and f ( z ) are shown.Although the expressions of the higher-order contribu-tions are rather intricate, these terms involve the contri-butions expressed in the polynomial form of f m σ n , withthe power-law indices m and n running respectively from1 to 4 and 4 to 6 for the perturbative calculations atnext-to-next-to-leading order. Besides, beyond the lin-ear order, even the auto-power spectra P δδ and P ΘΘ donot simply scale as σ and ( f σ ) , respectively. Thus,if one gets access to the weakly nonlinear regime wherethe higher-order terms play a role, we expect that thedegeneracy of the parameter f σ at Eq. (1) is broken,and f and σ can be separately determined. B. Modeling galaxy bias
On top of the hybrid RSD model in previous subsec-tion, there is one more step toward a practical applicationto the observed galaxy power spectrum. Since the galaxydistribution is a biased tracer of matter distribution, ac-counting for the galaxy bias is another crucial task. Inorder to do this, one may recall that the expression givenat Eq. (5) is fairly general. Replacing the matter densityfield δ with the galaxy/halo density field δ h , Eq. (5) canbe applied to the observed power spectrum, and henceour hybrid model of matter power spectrum is used as abuilding block to compute accurately the redshift-spacegalaxy power spectrum.For our interest at the weakly nonlinear regime, a per-turbative description of the tracer field is valid, and δ h is expanded in powers of the matter density field δ , in-cluding the non-local contributions. Here, we adopt theprescription proposed by Ref. [31, 32], which has been ap-plied to the SDSS BOSS (e.g., [19]) and eBOSS galaxies[33]. Apart from the stochastic terms, this is a generalperturbative expansion valid at the next-to-leading or-der. While the hybrid model for the matter power spec-trum in Sec. II A and Appendix A includes the correc-tions valid at next-to-next-to-leading order, we shall be-low consider the scales where the next-to-leading order isimportant, but the next-to-next-to-leading order is stillsubdominant. A fully consistent treatment including thebias at next-to-next-to-leading order will be discussed inour future work.Based on Refs. [31, 32], the auto-power spectrum ofthe tracer density field, P ˜ δ h ˜ δ h , is expressed as follows: P ˜ δ h ˜ δ h ( k ) = P δ h δ h ( k ) + P (cid:15)(cid:15) , (4)where the term P (cid:15)(cid:15) is the stochastic contribution mainlycharacterizing the shot noise, which is usually constant, FIG. 1:
Left : Logarithmic derivative of the real-space halo power spectrum with respect to the linear growth factor G δ (black)and linear bias parameter b (blue), plotted as function of the wavenumber k . In each panel, the solid lines represent the resultsfor hybrid model, P hybrid δδ , while dashed and dotted lines are for the linear theory prediction, P lin δδ . Right : logarithmic derivativeof the redshift-space halo power spectra with respect to G δ , b and G Θ , respectively shown from top to bottom panels. Bothof the Kaiser (linear theory) and hybrid models are shown as blue and red curves, specifically fixing the directional cosine to µ = 0 . . . . and is estimated, assuming the Poisson noise, from thenumber density of tracer field. The first term, P δ h δ h , isthe deterministic part, and is given in the following form: P δ h δ h ( k ) = b P δδ ( k ) + 2 b b P b ,δ ( k ) + 2 b b s P bs ,δ ( k )+ 2 b b σ ( k ) P Lm ( k ) + b P b ( k )+ 2 b b s P b s ( k ) + b s P b ( k ) . (5)On the other hand, the cross-power spectrum, P δ h θ h , con-tains only the deterministic contribution, whose expres-sion is given by P δ h Θ h ( k ) = b P δ Θ ( k ) + b P b , Θ ( k ) + b s P bs , Θ ( k )+ b σ ( k ) P Lm ( k ) . (6)Note that in the absence of velocity bias, the auto-powerspectrum of galaxy/halo velocity field, P Θ h Θ h , is identi-cal to P ΘΘ . In Eqs. (4) and (6), while the first terms, b P δδ and b P δ Θ , as well as P Θ h Θ h , are the leading-orderbias contribution, and can be computed with the hybridRSD model for the matter power spectrum, the termsinvolving parameters b , b s , and b are the higher-order bias terms. The explicit expressions for their scale-dependent functions, P b ,δ , P bs ,δ , σ , P b , ..., can befound in Ref. [31, 32].Regarding the A , B , T and F terms in Eq. (5), theyare all regarded as the higher-order corrections. We only apply the linear bias b to these terms [24], A h ( k, µ ) = b A ( k, µ, f /b ) , (7) B h ( k, µ ) = b B ( k, µ, f /b ) , (8) F h ( k, µ ) = b F ( k, µ, f /b ) , (9) T h ( k, µ ) = b T ( k, µ, f /b ) , (10)where all the subscripts h at left-hand-side represent thehigher-order terms of the tracer fields. The accuracy ofthis treatment and its impacts on the RSD model wasdiscussed in Ref. [24].Incorporating the bias prescription given above intoEq. (5), our hybrid RSD model in Sec. II A and AppendixA enables us to predict the redshift-space galaxy/halopower spectrum beyond the linear regime. To see the be-havior of the model beyond the linear bias prescription,left panel of Fig. 1 shows the logarithmic response of thereal-space power spectrum to the quantities b and G δ ,depicted as respectively blue and black lines. In lineartheory, there is no difference between these responses,which exactly give 2 (dashed and dotted). On the otherhand, the hybrid model involving the galaxy bias expan-sion exhibits a wiggle feature for the response to G δ ,showing a small but visible deviation from 2. This iscaused by non–linear smoothing around BAO peaks. Onthe other hand, the response to b still lies at 2 even inthe hybrid model, indicating that the dependence of G δ or σ and b becomes distinguishable beyond the linearregime. The selected fiducial values of galaxy biases are FIG. 2:
Left : the expected errors on fσ for DESI, with σ marginalized over. The result derived from the Fisher matrix,depicted as the error bar around the fiducial value, are plotted against redshift. The upper and lower panels respectively showthe estimated constraints on fσ derived from the LRG and ELG type galaxies. Right : Same as in the left panel, but theparameter σ is fixed, and is not treated as a free parameter. presented in Table I. The fiducial values for coherent bias b are taken from DESI prediction [34], and the redshiftdependence of b biases are fitting results from halo cat-alogue.To feel some flavors on how the degeneracy is broken,left panel of Fig. 1 presents the response of the redshift-space power spectrum to the variation of G δ and G Θ .The quantity G δ is the linear growth factor normalized atprimordial epoch as G δ ( z i )(1 + z i ) = 1, and is related to σ ( z ) through σ ( z ) = [ G δ ( z ) /G δ (0)] σ (0). In left panelof Fig. 1, the response of the redshift-space galaxy/halopower spectrum to the variation of G δ is shown for bothlinear theory prediction and hybrid model, depicted asblack dotted and solid curves respectively. The func-tion G Θ is related to the linear growth rate f through G Θ = f G δ . On the other hand, right panel of Fig. 1plots the response of the 2D redshift-space power spec-trum to the variations of G δ and G Θ at top and bottompanels respectively, specifically fixing the directional co-sine µ to 0 .
0, 0 .
3, 0 . .
9. The distinct behavior ofthe power spectrum is observed by varying µ , and theresults of hybrid model start to deviate from linear the-ory predictions, indicating that the parameter degener-acy between G δ and G Θ is broken, and so is the case for σ and f σ . III. FORECAST CONSTRAINTS ON σ AND fσ In this section, based on the model described in Sec. IIas a theoretical template, we demonstrate explicitly howwell the degeneracy of the parameter f σ can be broken,adopting specifically the Dark Energy Spectroscopy In-strument (DESI) [34], which is a representative galaxyredshift survey of the so-called stage-IV class [36], dedi-cated for a precision measurement of BAO and RSD uti- z n LRG g n ELG g V b
LRG1 b ELG1 b LRG2 b ELG2 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × −
10. — 1.70 — 5.68TABLE I: The expected number density of LRG and ELGgalaxies n LRG g [ h Mpc − ] and n ELG g [ h Mpc − ], and survey vol-ume V [ h − Gpc ] at each redshift bin used in the Fisher ma-trix analysis. These specific values are taken from those as-sumed in the DESI experiment [34]. The fiducial values ofbias for LRG and ELG are presented as well, calculated bythe formulas b LRG1 ( z ) D ( z ) = 1 . b ELG1 ( z ) D ( z ) = 0 .
84 [34],with D being the linear growth factor normalized to unity atpresent time, i.e., D ( z ) = G δ ( z ) /G δ (0). Concerned with b LRG2 and b ELG2 , we exploits the halo fit results. To compute theseparameters, Planck ΛCDM model [35] is adopted as the fidu-cial cosmology.
FIG. 3:
Left : The fractional errors on D A from DESI LRG (upper) and ELG (lower) samples. The results in units of percentageare plotted as function of redshift. Solid and dashed curves represent the cases with σ marginalized over and fixed, respectively. Right : Same as in left panel, but the fractional errors on H − is shown. lizing the emission-line galaxy (ELG) and luminous redgalaxy (LRG) samples (see Table I for parameter speci-fication, which is discussed below). A. Fisher matrix formalism
To estimate quantitatively the size of the expected er-rors, we use the Fisher matrix formalism. Regardingthe model in Sec. II as an observed power spectrum, theFisher matrix is evaluated with F αβ = (cid:88) k,µ ∂P obs ( k, µ ) ∂x α C − P P ∂P obs ( k, µ ) ∂x β (11)where the derivative of the power spectrum is taken withrespect to the parameter to estimate, x α . The observedpower spectrum P obs is related to P (S) given at Sec. IIthrough the Alcock-Paczynski effect (Ref. [17], see alsoSec. I): P obs ( k, µ ) = (cid:16) HH fid (cid:17)(cid:16) D A D A, fid (cid:17) − P (S) ( q, ν ) ; (12) q = (cid:110)(cid:16) D A D A, fid (cid:17) (1 − µ ) + (cid:16) HH fid (cid:17) − µ (cid:111) / k,ν = kq (cid:16) HH fid (cid:17) − µ. In Eq. (11), the quantity C P P describes the error co-variance of the measured power spectra, whose domi-nant contributions are the shot noise arising from the discreteness of galaxy distribution, and the cosmic vari-ance due to the limited number of Fourier modes for afinite-volume survey. While the non-Gaussian contribu-tion, leading to the non-zero off-diagonal components,is known to play an important role beyond the linearregime, we will work with the linear Gaussian covariancegiven below, C PP = 1 N P (cid:20) P (S) ( k, µ ) + 1 n Xg (cid:21) , (13)where the quantity n Xg denotes the number density ofgalaxies for a specific galaxy type X , i.e., X =LRG orELG for DESI, whose values are summarized in TableI. The quantity N P represents the number of availableFourier modes, which is estimated to be N P = V survey π ) k ∆ k ∆ µ (14)with ∆ k and ∆ µ being the bin width of the powerspectrum data given in the ( k, µ ) plane, with ∆ k =0 . h Mpc − and ∆ µ = 0 .
1. Here, the quantity V survey isthe survey volume, whose values is also listed in Table I.Adapting the Gaussian covariance at Eq. (13), the fore-cast results of our Fisher matrix analysis would be op-timistic. Nevertheless, at the scales where the pertur-bative corrections of the next-to-next-to-leading order isstill sub-dominant, the impact of non-Gaussian covari-ance would be small. The forecast results presented be-low can thus give an important guideline toward a morequantitative analysis. Indeed, to compute the Fisher ma- FIG. 4: The expected error on σ for the DESI LRG (upper)and ELG (lower) type samples. trix at Eq. (11) below, we will conservatively fix the max-imum wavenumber to k max = 0 . h Mpc − , at whichthe non-linearity is still mild at the range of redshifts,0 . ≤ z ≤ . P (S) ( k, µ ) in Eq. 5. As the free parametersto estimate, we consider, at each redshift slice, x α = { G δ , G Θ , D A , H, b , b , σ z } , where the former two aresimply proportional to σ ( z ) and f σ ( z ). The parame-ter σ z quantifies the non-perturbative suppression of thepower spectrum by the so-called Fingers-of-God dampingthat appears in the function D FoG at Eq. (5). We shallbelow adopt the Gaussian form of the damping function: D FoG ( x ) = exp( − x ) . (15)where the fiducial σ p = 3 . h Mpc − computed using lin-ear theory only. Then, the derivative of the power spec-trum P obs , given at Eq. (12), is taken with respect to theparameters, x α . Here, the higher-order bias parameters, b s and b , are fixed, adopting the following relations: b s = −
47 ( b − , b = 32315 ( b − , which are derived assuming the Lagrangian local bias,and are shown to be a good approximation for halos andgalaxies residing at the halo center (e.g., [32, 37, 38]).The fiducial values of the linear bias parameters for ELGand LRG are presented in Table I. B. Results
With the setup described in Sec. III A, the one-dimensional marginalized errors on the parameters f σ and σ , as well as the geometric distances, are obtainedfrom the inverse of Fisher matrix, ∆ x α = (cid:112) F − αα .Let us first see how the expected error on the parame-ter f σ is altered when the parameter σ or G δ is takento be free, and is marginalized. Fig. 2 plots the resultswith σ marginalized (left) and fixed (right), the latterof which is obtained from the inverse of the sub-matrix F αβ subtracting the component of G δ . In each panel,the one-dimensional error on f σ is shown at each red-shift slice for the LRG (top) and ELG (bottom) samples.Remarkably, we found that the size of the error on f σ almost remains unchanged irrespective of the treatmentof the parameter σ , leading to the constraint at the levelof a few percent in both ELG and LRG samples, whichis expected from the stage-IV class surveys.To elucidate further the impact of marginalizing the σ parameter, we also plot in Fig. 3 the constraints onthe geometric distances. Here, the fractional errors onthe angular diameter distances and Hubble parameters ,( D A − D A, fid ) /D A, fid and ( H − H fid ) /H fid , are respectivelyshown in left and right panels as function of the redshift.In each panel, the results with σ marginalized and fixedare depicted as solid and dashed lines. Again, no notabledifference is found between the two cases, and even theresults marginalizing σ reach at 1 % precision for theconstraints on both D A and H . In particular, with theELG samples, the constraint is improved as increasingthe redshift out to z ∼ .
5, with the statistical errordown to a sub-percent level.In Fig. 4, we plot the statistical errors on σ that isseparately determined as a free parameter together with f σ . As opposed to the tight constraint on f σ , the σ appears poorly constrained. Considering the fact thatthe nonlinear corrections in the power spectrum templatethat can break the parameter degeneracy are small, thisresult is reasonable. Rather, accessing the scales to theweakly nonlinear regime, the number of available Fouriermodes gets increasing, and the constraints on f σ , aswell as geometric distances are improved, as we have seenin Figs. 2 and 3. Nevertheless, using the ELG, the ex-pected error on σ becomes also improved with redshifts,achieving 10% precision at z = 1 .
5. Since constrainingthe growth history of structure at higher redshifts partic-ularly helps testing and constraining the gravity as wellas the cosmic acceleration, the simultaneous determina-tion of f σ and σ is beneficial. Combining the powerspectrum with bispectrum, the constraining power willbe further improved. . . . . . . . . f D A (1/ Mpc ) H ( / M p c ) FIG. 5:
Left:
The measured and estimated cosmological constraints on σ and fσ at 68% and 95 % confidence levels arepresented with the blue filled and the black unfilled contours respectively. The true values are presented with a black dot in themiddle of contours. Those constraints are computed after fully marginalizing other parameters such as ( D A , H − , σ p , b , b Right:
The same presentation as for the left panel, but for D A and H − . Those constraints are computed after fully marginalizingother parameters such as ( σ , fσ , σ p , b , b ). IV. BREAKING THE f σ DEGENERACY IN N -BODY SIMULATIONS Given the survey setup and the theoretical templateof power spectrum, the Fisher matrix analysis in previ-ous section predicts the statistical errors on each free pa-rameter and their parameter degeneracy, but cannot tellhow the best-fit parameters are accurately determined.In this section, to check the validity of the forecast re-sults in Sec. III as well as to test the hybrid RSD modelof power spectrum in Sec. II, we here examine the pa-rameter estimation analysis, based on the Markov chainMonte Carlo (MCMC) technique.For this purpose, the cosmological N -body simulationis carried out by the publicly available code, GADGET-2 [39], and we ran in total 100 simulations, with 1024 particles in comoving periodic cubes of the side length1 , h − Mpc. Using the output results at z = 0 .
9, thehalo catalog was created using the halo finder, ROCK-STAR [40], with a halo mass range of 10 h − M (cid:12) 0. The number density of our halo sam-ple is smaller than those expected from DESI LRG andELG samples (see Table I), but this does not affect theestimation of statistical errors, as we consider the scales where the shot noise is subdominant.The created halo catalog at z = 0 . G δ , G Θ , D A , H, b , b , σ z ), similarly to whathas been done in Sec. III, assuming the Lagrangian localbias. Adopting the same maximum wavenumber as usedin Sec. III, i.e., k max = 0 . h Mpc − , the MCMC re-sults marginalized over other parameters are shown inFig. 5, where we plot the two-dimensional error con-tours on ( σ , f σ ) (left) and ( D A , H − ) (right). Theblue shaded regions represent the 68% (dark) and 95%(light) credible regions obtained from the MCMC analy-sis, which exhibit roughly theIn Fig. 5, we also plot the forecast results for the Fishermatrix analysis. The solid and dashed contours centeredat the fiducial parameters, indicated by the black filledcircles, are respectively the the 68% and 95% credible re-gions for the forecast errors. The MCMC results repro-duce well the forecast results of the Fisher matrix analy-sis, and the 68% credible regions consistently include thefiducial parameters. Thus, with the hybrid RSD modelpresented in Sec. II, unbiased parameter estimation isshown to be possible, with the degeneracy between theparameters f and σ broken. Although the statistical er-ror of σ is large, this is the first demonstration that thesimultaneous determination of f and σ is possible onlywith the power spectrum at weakly nonlinear scales. V. CONCLUSION Redshift-space distortions (RSD) that appear in theobserved galaxy distributions via spectroscopic surveysoffer an important clue to test the gravity on cosmologicalscales. Combining the measurement of baryon acousticosculations (BAO), RSD can be also used to clarify thenature of cosmic acceleration. Toward an unambiguousestimation of cosmological parameters, a crucial issue isnot only to improve the precision of RSD measurement,but also to exploit the method to disentangle the param-eter degeneracy inherent in the observables.In this paper, on the basis of an accurate template forthe galaxy/halo redshift-space power spectrum, we getaccess to the weakly nonlinear regime, and showed thatthe parameter degeneracy inherent in the linear-theorypower spectrum can be broken. To be precise, in lineartheory, the linear growth rate f and the fluctuation am-plitude σ appear in the form of f σ [Eq. (1)], and thisdegeneracy cannot be broken unless the galaxy bias pa-rameter is a priori known or is accurately determined. Inorder to break the degeneracy, one way is to go beyondthe linear theory. Here, we use the hybrid RSD model ofpower spectrum that has been developed in our previouspapers. Based on the perturbation theory calculations,the model incorporates the higher-order corrections cali-brated with N -body simulations into the power spectrumexpressions, and the accuracy of predictions is improved.Taking further the galaxy bias into account, the hybridmodel enables us to get access to the observed galaxypower spectrum at the weakly nonlinear regime.Employing the Fisher matrix analysis, we show explic-itly that the degeneracy of the parameter f σ can bebroken, and σ is separately estimated in the presenceof galaxy bias. The statistical errors on f σ as well asthe geometric distances D A and H , determined from theBAO via the Alcock-Paczynski effect, are found to re-main unchanged, irrespective of whether we treat σ orthe growth factor D δ as a free parameter to marginal-ize or not. As a result, we have shown that the DarkEnergy Survey Instrument, as a representative stage-IVclass galaxy survey, can unambiguously determine σ atthe precision of ∼ 10% at higher redshifts even if we re-strict the accessible scales to k (cid:46) . h Mpc − . Further,performing the Markov chain Monte Carlo analysis, weexplicitly demonstrate that with the hybrid RSD model,the parameters f σ and σ are simultaneously estimated,and their fiducial values can be properly recovered, withthe statistical errors fully consistent with the forecast re-sults of the Fisher matrix analysis.While the analysis in the present paper gives a first ex-plicit demonstration on how well the parameter degener-acy in the measurement of RSD can be broken, the abilityto achieve this heavily relies on the theoretical templateof the observed power spectrum. Toward a further im-provement of the cosmological constraints in an unbiasedway, one needs to develop a model that can get access tosmaller scales. Though the present paper considered the perturbation theory based model aided by the N -bodysimulations, a simulation based model such as the so-called emulator would be certainly powerful (e.g. [41]).With such a model, the accessible range of wavenumberbecomes broader, and the simultaneous constraints on f σ and σ , as well as the geometric distances, will beimproved in a greater precision. The discussion along thedirection of this should be done in the near future. Acknowledgments We would like to thank Takahiro Nishimichi for use-ful discussions and comments. Numerical calculationswere performed by using a high performance comput-ing cluster in the Korea Astronomy and Space Sci-ence Institute. YZ acknowledges the support from theGuangdong Basic and Applied Basic Research Founda-tion No.2019A1515111098. AT acknowledges the supportfrom MEXT/JSPS KAKENHI Grant No. JP16H03977,JP17H06359 and JP20H05861. AT was also supported byJST AIP Acceleration Research Grant NO. JP20317829,Japan. Appendix A: Description of hybrid RSD model formatter power spectrum