3D Spherical Analysis of Baryon Acoustic Oscillations
AAstronomy & Astrophysics manuscript no. article c (cid:13)
ESO 2018October 31, 2018
3D Spherical Analysis of Baryon Acoustic Oscillations
A. Rassat , and A. Refregier Laboratoire d’Astrophysique, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Observatoire de Sauverny,CH-1290, Versoix, Switzerland. Laboratoire AIM, UMR CEA-CNRS-Paris 7, Irfu, SAp/SEDI, Service d’Astrophysique, CEA Saclay, F-91191GIF-SUR-YVETTE CEDEX, France. Institute for Astronomy, ETH Z¨urich, Wolfgang Pauli Strasse 27, CH-8093 Z¨urich, Switzerland.Preprint online version: October 31, 2018
ABSTRACT
Context.
Baryon Acoustic Oscillations (BAOs) are oscillatory features in the galaxy power spectrum which are usedas a standard rod to measure the cosmological expansion. These have been studied in Cartesian space (Fourier or realspace) or in spherical harmonic space in thin shells.
Aims.
Future wide-field surveys will cover both wide and deep regions of the sky and thus require a simultaneoustreatment of the spherical sky and of an extended radial coverage. The Spherical Fourier-Bessel (SFB) decompositionis a natural basis for the analysis of fields in this geometry and facilitates the combination of BAO surveys with othercosmological probes readily described in this basis. In this paper, we present a new way to analyse BAOs by studyingthe BAO wiggles from the SFB power spectrum.
Methods.
In SFB space, the power spectrum generally has both a radial ( k ) and tangential ( (cid:96) ) dependence and so dothe BAOs. In the deep survey limit and ignoring evolution, the SFB power spectrum is purely radial and reduces tothe Cartesian Fourier power spectrum. In the opposite limit of a thin shell, all the information is contained in thetangential modes described by the 2D spherical harmonic power spectrum. Results.
We find that the radialisation of the SFB power spectrum is still a good approximation even when consideringan evolving and biased galaxy field with a finite selection function. This effect can be observed by all-sky surveys withdepths comparable to current surveys. We also find that the BAOs radialise more rapidly than the full SFB powerspectrum.
Conclusions.
Our results suggest the first peak of the BAOs in SFB space becomes radial out to (cid:96) ∼
10 for all-skysurveys with the same depth as SDSS or 2dF, and out to (cid:96) ∼
70 for an all-sky stage IV survey. Subsequent BAO peakswill also become radial, but for shallow surveys these may be in the non-linear regime. For modes that have becomeradial, measurements at different (cid:96) ’s are useful in practice to reduce measurement errors.
Key words.
Cosmology, Baryon Acoustic Oscillations, Wide-field surveys, Statistics, Spherical Fourier-Bessel.
1. Introduction
The study of Large Scale Structure (LSS) with galaxysurveys is a promising tool to study the dark universe(Peacock et al. 2006; Albrecht et al. 2006). Baryon AcousticOscillations (BAOs) are a special feature in the galaxypower spectrum present on scales 100 h − Mpc, which aredue to oscillations in the coupled baryon-photon fluid be-fore recombination (Sunyaev & Zeldovich 1970; Peebles &Yu 1970; Eisenstein et al. 2005; Seo & Eisenstein 2003,2007). The BAOs are considered a powerful cosmologicaltool as the BAO scale acts as a standard ruler with whichto probe cosmic expansion both in the radial and tangentialdirections.BAOs were first detected by Eisenstein et al. (2005) inSDSS data (Adelman-McCarthy et al. 2008) and later with2dF galaxies (Colless et al. 2003; Cole et al. 2005) and fi-nally with both surveys (Percival et al. 2007 b), though oth-ers suggest current data cannot currently probe the BAOscale sufficiently (Cabr´e & Gazta˜naga 2011).Until now, BAOs have been studied in Cartesian space,either in Fourier space (Seo & Eisenstein 2003, 2007), or inreal space (Eisenstein et al. 2005; Xu et al. 2010; Slosar et al. 2009), and in 2D spherical harmonic space on thin sphericalshells (Dolney et al. 2006). These descriptions use differentinformation and therefore have different constraining powerfor cosmological parameters (Rassat et al. 2008).Future wide-field BAO surveys will, however, cover bothlarge and deep areas of the sky, and thus require a si-multaneous treatment of the spherical sky geometry andof extended radial coverage. The Spherical Fourier-Bessel(SFB) decomposition is a natural basis for the analysis offields in this geometry. The SFB analysis is powerful as ituses a coordinate system in which the radial selection func-tion and physical effects are naturally described. Moreover,this description facilitates the combination of BAO sur-veys with other cosmological probes which are readily de-scribed in the SFB decomposition such as the smooth powerspectrum and redshift space distortions (Heavens & Taylor1995; Fisher et al. 1995b; Percival et al. 2004; Erdo˘gdu (a)et al. 2006; Erdo˘gdu (b) et al. 2006), weak lensing (Heavens2003; Castro et al. 2005; Kitching et al. 2008), and theIntegrated Sachs-Wolfe effect (ISW, Shapiro et al. 2011).Studying BAOs from wide-field surveys with an SFB ex- a r X i v : . [ a s t r o - ph . C O ] J a n . Rassat , and A. Refregier : 3D Spherical Analysis of BAOs pansion, is therefore natural both for the geometry consid-ered, and for unifying the treatment with the other probes.In § radiali-sation when the field is statistically isotropic and homo-geneous. In § § §
2. 3D Spherical Fourier-Bessel (SFB) Expansion
Let us consider a field f ( r ) at time t , where r = ( r, θ, ϕ )in spherical polar coordinates. In practice, this field mayrepresent the galaxy or mass density (or overdensity) in theuniverse. In a flat geometry, the field can be decomposed inthe 3D SFB basis set, which is complete and orthonormal,as f ( r ) = (cid:114) π (cid:90) d k (cid:88) (cid:96)m f (cid:96)m ( k ) kj (cid:96) ( kr ) Y (cid:96)m ( θ, ϕ ) , (1)where j (cid:96) ( x ) are spherical Bessel functions of the first kind, Y (cid:96)m ( θ, ϕ ) are spherical harmonics, (cid:96) and m are multipolemoments and k is the wavenumber. The inverse relation is f (cid:96)m ( k ) = (cid:114) π (cid:90) d r f ( r ) kj (cid:96) ( kr ) Y ∗ (cid:96)m ( θ, ϕ ) , (2)where we use the same conventions as Leistedt et al. (2011)and Castro et al. (2005) (see also Heavens & Taylor 1995;Fisher et al. 1995b, who use a different convention and basisset).This decomposition can be viewed as the spherical po-lar coordinate analogue to the Fourier decomposition inCartesian coordinates given by f ( x ) = 1(2 π ) (cid:90) d k ˜ f ( k ) e i k · x , (3)˜ f ( k ) = (cid:90) d x f ( x ) e − i k · x . (4)The 3D SFB power spectrum C (cid:96) ( k ) of the field f ( r ) isgiven by the the 2-point function of the SFB coefficients f (cid:96)m ( k ) which can be written as (cid:104) f (cid:96)m ( k ) f ∗ (cid:96) (cid:48) m (cid:48) ( k (cid:48) ) (cid:105) = C (cid:96) ( k ) δ ( k − k (cid:48) ) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) , (5)when the field is Statistically Isotropic and Homogeneous(hereafter, the SIH condition). Similarly and in the samecondition, the Fourier power spectrum P ( k ) is implicitlydefined by (cid:104) ˜ f ( k ) ˜ f ∗ ( k (cid:48) ) (cid:105) = (2 π ) P ( k ) δ ( k − k (cid:48) ) . (6)These two power spectra are related by (see for e.g.,Fisher et al. 1995b; Castro et al. 2005): C (cid:96) ( k ) = P ( k ) . (7)Thus, the SFB power spectrum C (cid:96) ( k ) is independent of themultipole (cid:96) , and thus only has a radial ( k ) dependence. Thisremarkable yet recondite fact is only true if the field fulfillsthe SIH condition.In the following subsections, we discuss the impact ofthe (partial) violation of this condition in practice and theimplication for the measurements of BAO in SFB analysis. In practice, a cosmological field, such as the galaxy densityfield, will only be partially observed due the finite surveyvolume. In this case the observed field f obs ( r ) can be de-scribed by f obs ( r ) = φ ( r ) f ( r ) , (8)where φ ( r ) is the radial selection function of the survey,and f ( r ) is assumed to fulfill the SIH condition for now(we will address the effect of bias and evolution in § f obs ( r ) is thus no longer homogeneousbecause of the radial the selection function. There may alsobe a tangential selection function to account for regions ofmissing data, but we assume here that the data is availableon the full sky. For convenience, the selection function isnormalised as (cid:90) d r φ ( r ) = V, (9)where V is a characteristic volume of the survey chosensuch that φ → V → ∞ , such thatlim V →∞ f obs ( r ) = f ( r ) . (10)In this case, the homogeneity condition (SIH) is nolonger valid in the radial direction and the observed 2-pointfunction can be written as (cid:104) f obs (cid:96)m ( k ) f obs ∗ (cid:96) (cid:48) m (cid:48) ( k (cid:48) ) (cid:105) = C obs (cid:96) ( k, k (cid:48) ) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) (11)where the observed SFB power spectrum C obs (cid:96) ( k, k (cid:48) ) de-pends this time on (cid:96) , k and k (cid:48) , as opposed to only on (cid:96) and k as in Equation 5. It can be shown that it can be expressedas C obs (cid:96) ( k, k (cid:48) ) = (cid:18) π (cid:19) (cid:90) d k (cid:48)(cid:48) k (cid:48)(cid:48) P ( k (cid:48)(cid:48) ) W (cid:96) ( k, k (cid:48)(cid:48) ) W (cid:96) ( k (cid:48) , k (cid:48)(cid:48) ) , (12)where P ( k ) is defined in Equation 6 and the window func-tion W (cid:96) ( k, k (cid:48) ) is defined as W (cid:96) ( k, k (cid:48) ) = (cid:90) d r r φ ( r ) kj (cid:96) ( kr ) j (cid:96) ( k (cid:48) r ) . (13)In practice, C obs (cid:96) ( k, k (cid:48) ) tends to fall off rapidly away fromthe diagonal k = k (cid:48) and we will often only compute ¯ C obs (cid:96) ( k )defined by:¯ C obs (cid:96) ( k ) ≡ C obs (cid:96) ( k, k ) . (14)We next evaluate the window function and the observedpower spectrum for three special cases for the selectionfunction. As a first example, let us consider a Gaussian selection func-tion defined as φ ( r ) = e − ( r/r ) , (15)where r is a radius parameter and the normalisation obeysEquation 9 (with V = π r ). In this case, the window
2. Rassat , and A. Refregier : 3D Spherical Analysis of BAOs function can be integrated analytically, and Equation 13becomes: W (cid:96) ( k, k (cid:48) ) = πr (cid:114) kk (cid:48) exp (cid:20) − r k + k (cid:48) (cid:21) I (cid:96) + (cid:18) r kk (cid:48) (cid:19) , (16)where I ν ( x ) is the modified Bessel function of the first kind.This analytical form considerably facilitates the evaluationof Equation 12 (see § A for a way to numerically calculatethe above Equation for large arguments of I ν ( x )). In the case where the radial selection function correspondsto full radial coverage (i.e. φ ( r ) = 1 , ∀ r ), the window func-tion becomes W (cid:96) ( k, k (cid:48) ) = π k (cid:48) δ ( k − k (cid:48) ) . (17)Inserting this expression into Equation 12, we recoverEquation 7, namely C obs (cid:96) ( k, k (cid:48) ) = C (cid:96) ( k ) δ ( k − k (cid:48) ) = P ( k ) δ ( k − k (cid:48) ) , (18)meaning that the 3D spherical spectrum is only dependenton radial coordinate k , as discussed above.As we show in the appendix § A, this limit can also beobtained by taking the limit r → ∞ for the Gaussianweight function of Equation 15 and is achieved in practicewhen the condition (see Equation A.4) r k (cid:29) (cid:112) (cid:96) ( (cid:96) + 1) (19)is satisfied. We refer to this as the radialisation of the fieldand discuss this more in § In the other extreme case, the radial selection function cov-ers only a thin shell of the field at a distance r ∗ φ ( r ) = r ∗ δ ( r − r ∗ ) , (20)where the normalisation of Equation 9 was chosen with V = 4 πr ∗ . Equation 12 then becomes C obs (cid:96) ( k, k (cid:48) ) = (cid:18) π (cid:19) kk (cid:48) j (cid:96) ( kr ∗ ) j (cid:96) ( k (cid:48) r ∗ ) × (cid:90) d k (cid:48)(cid:48) k (cid:48)(cid:48) P ( k (cid:48)(cid:48) ) (cid:2) r ∗ j (cid:96) ( k (cid:48)(cid:48) r ∗ ) (cid:3) . (21)This can be related to the statistics of the 2D projectedfield f ( θ, ϕ ) = (cid:90) d rr φ ( r ) f ( r, θ, ϕ ) , (22)whose spherical harmonic decomposition is given by f ( θ, ϕ ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) f (cid:96)m Y (cid:96)m ( θ, ϕ ) , (23) f (cid:96)m = (cid:90) dΩ f ( θ, ϕ ) Y ∗ (cid:96)m ( θ, ϕ ) , (24) and whose 2D spherical harmonic power spectrum C (cid:96) isdefined by (cid:104) f (cid:96)m f ∗ (cid:96) (cid:48) m (cid:48) (cid:105) = C (cid:96) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) . (25)This power spectrum is related to the 3D Fourier spacepower spectrum P ( k ) by: C (cid:96) = 2 π (cid:90) d kk P ( k ) (cid:2) W (cid:96) ( k ) (cid:3) , (26)where the 2D window function W (cid:96) ( k ) is given by W (cid:96) ( k ) = (cid:90) d rr φ ( r ) j (cid:96) ( kr ) . (27)Note that due to our choice of normalisation for φ ( r ) inEquation 9, W (cid:96) ( k ) has units of volume.In the case of a thin-shell selection function (Equation20), this reduces to C (cid:96) = 2 π (cid:90) d kk P ( k ) (cid:2) r ∗ j (cid:96) ( kr ∗ ) (cid:3) . (28)Thus, the SFB power spectrum (Equation 21) is simplyrelated to the 2D spherical harmonic power spectrum by C obs (cid:96) ( k, k (cid:48) ) = 2 π kk (cid:48) j (cid:96) ( kr ∗ ) j (cid:96) ( k (cid:48) r ∗ ) C (cid:96) . (29)The term kk (cid:48) j (cid:96) ( kr ∗ ) j (cid:96) ( k (cid:48) r ∗ ) is a geometric factor whichdoes not depend on the field f . Thus, in this case, all theinformation in the 3D SFB spectrum C (cid:96) ( k, k (cid:48) ) is containedin the 2D power spectrum C (cid:96) , i.e. depends solely on (cid:96) . For a realistic galaxy field, the SIH condition will not bemet due to the radial evolution of the field from the growthof structure and redshift dependent bias.For a biased and evolving field in the linear regime, thegalaxy power spectrum P evol ( k, r ) will have an explicit dis-tance dependence: P evol ( k, r ) = b ( r ) D ( r ) P ( k ) , (30)where the term b ( r ) is the linear bias, assumed to bescale-independent, D ( r ) is the growth factor and P ( k ) = P ( k, r = 0) is the matter power spectrum at r = z = 0. Thelinear approximation will hold up to a redshift-dependentscale k max ( z ). In the standard cosmological model, we find k max ( z = 0) (cid:39) . h Mpc − (i.e., before the second BAOpeak) and k max ( z = 2) (cid:39) . h Mpc − .For the realistic galaxy field, an approximation for thespherical 3D power spectrum for the galaxy field is stillgiven by Equation 12, but the window function will incor-porate the linear growth and bias functions as W evol (cid:96) ( k (cid:48) , k ) = (cid:90) d r r φ ( r ) kD ( r ) b ( r ) j (cid:96) ( kr ) j (cid:96) ( k (cid:48) r ) . (31) Equation 29 is conceptually different from Equation B3 inKitching et al. (2011): the former is an exact solution for a thinshell, while the latter uses the Limber approximation. 3. Rassat , and A. Refregier : 3D Spherical Analysis of BAOs Fig. 1.
Ratio R C(cid:96) ( k ) of SFB spectrum with and without the physical effects of baryons in ( (cid:96), k ) space for a wide andshallow survey (left, r = 100 h − Mpc) and a wide and deep survey (right, r = 1400 h − Mpc) using a Gaussian selectionfunction in r . The baryonic wiggles are seen both in the radial ( k ) and the tangential ( (cid:96) ) directions, but as r is increased,only radial wiggles - with increased amplitude - persist. This radialisation of the information is due to mode-cancellingalong the line of sight for deep surveys.
3. Application to Baryon Acoustic Oscillations
The matter power spectrum in Equation 12 includes thephysical effects of baryons, which create oscillations inFourier space (Sunyaev & Zeldovich 1970; Peebles & Yu1970; Seo & Eisenstein 2003; Eisenstein et al. 2005; Seo& Eisenstein 2007). These Baryon Acoustic Oscillations(BAOs) can be isolated by considering the ratio R P ( k ) inFourier space: R P ( k ) = P b ( k ) P nob ( k ) , (32)where P b ( k ) is the galaxy (or matter) power spectrum in-cluding the physical effect of BAOs, and P nob ( k ) is broadband or ‘smooth’ part of the galaxy (or matter) power spec-trum. In linear theory the growth and bias terms cancel outso that Equation 32 is independent of z . These oscillationscan also be probed in 2D spherical harmonic space (Dolneyet al. 2006). We are interested here to see if they can simi-larly be probed in SFB space as well.In SFB space, we consider the ratio R C(cid:96) ( k ) given by: R C(cid:96) ( k ) = ¯ C obs , b (cid:96) ( k )¯ C obs , nob (cid:96) ( k ) , (33) where similarly to Fourier space, ¯ C obs , b (cid:96) ( k ) is the diagonalSFB power spectrum (Equation 14) including the physicaleffects of baryons and ¯ C obs , nob (cid:96) ( k ) is the ‘smooth’ part ofthe SFB power spectrum. These are calculated by using P b ( k ) or P nob ( k ) instead of P ( k ) in Equation 12 and thenconsidering Equation 14. The SFB decomposition suggeststhat the BAOs can manifest themselves in both k and (cid:96) space.In Figure 1, we plot R C(cid:96) ( k ) for two wide-field galaxy sur-veys with a shallow (left) and deep (right) galaxy Gaussianselection functions with r = 100 h − Mpc, and 1400 h − Mpcrespectively (see § m = 0 .
25, Ω DE = 0 . b = 0 . w = − . w a = 0 . h = 0 . σ = 0 . n s = 1 andno running spectral index. We consider both to be ‘realis-tic’ surveys, i.e. with evolution due to growth taken intoaccount and a linear galaxy bias (taken as b ( z ) = √ z ),using Equation 31 for the window function.In the LHS of Figure 1 (narrow survey), the BAOs de-pend on both k and (cid:96) modes, illustrating that the wigglescan be measured simultaneously in the (cid:96) and k directions,which is the first result of this paper.However, in the RHS of Figure 1 (wide survey), theBAOs appear to have only a radial ( k ) dependence, exceptat very large physical scales ( k < . h Mpc − ), where thedependence is both radial and tangential. This is a prac-
4. Rassat , and A. Refregier : 3D Spherical Analysis of BAOs Fig. 2.
Slices in R C(cid:96) ( k ) (solid, red) compared to R P ( k ) (dashed, black) for (cid:96) = 5 ,
50 ( top, bottom ), for a wide and shallowgalaxy survey (left, r = 100 h − Mpc, or z m ∼ .
05) and a wide and deep galaxy survey (right, r = 1400 h − Mpc or z m ∼ . radialisation of the power spectrum, where we have usedthe baryonic wiggles to probe the radialisation. We discussthe radialisation limit in more detail in § § b ( z ) = (cid:0) √ z (cid:1) n , where n = − ,
2, and findthis still has little effect on the radialisation of R C(cid:96) ( k ). Thegrowth values for standard concordance cosmology do notseem to have an effect on reaching the radialisation limit. We are interested in quantifying the effect of the radialisa-tion on the BAOs themselves. In Figure 2, we consider slicesin (cid:96) and k through R C(cid:96) ( k ) in order to investigate how rapidlythe limit in Equation 18 is reached. The black (dashed) linesin Figure 2 correspond to R P ( k ) (which are independent of (cid:96) and of the survey selection function) and the red (solid)lines correspond to R C(cid:96) ( k ) for different slices in (cid:96) space andfor narrow (LHS) and wide (RHS) selection functions. Asthe survey selection function is widened, we find that:lim r →∞ R C(cid:96) ( k ) = R P ( k ) , (34)i.e. R C(cid:96) ( k ) tends towards the oscillations in R P ( k ), not onlyin phase, but also in amplitude. This is noticeable in Figure1: the amplitude of the first peak for example is higher forthe deep survey ( > < k, (cid:96) ) space for different surveys
5. Rassat , and A. Refregier : 3D Spherical Analysis of BAOs Fig. 3.
Radialisation limit as defined by Equation 19 (di-agonal lines) for Gaussian surveys (Equation 15) with r = 40 , , , , h − Mpc, i.e. roughly correspond-ing to the following surveys: IRAS ( z mean ∼ . z mean ∼ . z mean ∼ . z mean ∼ . z mean ∼ . r = 40 , , , , (cid:2) h − Mpc (cid:3) . We have chosen thesevalues as they correspond roughly to the following sur-veys (for the chosen fiducial survey): IRAS ( z mean ∼ . z mean ∼ .
03, Huchra et al. 2011),SDSS (Sloan Digital Sky Survey, z mean ∼ .
10, Adelman-McCarthy et al. 2008), 2dF (2 degree Field, z mean ∼ . z mean ∼ .
80, Albrecht et al. 2006). Note that, in principle, thesesurveys would have to be all-sky to obey the radialisationlimits plotted in Figure 3.The area above each diagonal line corresponds the ra-dial limit, i.e. where the radialisation has taken place andEquation 18 holds. We overplot the radial scales corre-sponding to the first three peaks of the BAOs to show howthese will be probed for different survey depths.By comparing the diagonal lines in Figure 3 with theBAO ‘turnover’ (i.e. the (cid:96) -dependent k -scale at which theBAOs switch from being mostly tangential to mostly ra-dial) in Figure 1, we notice however that the BAOs seem to become radial before the full SFB power spectrum does, es-pecially for large (cid:96) . This surprising effect, further motivatesthe use of the SFB analysis for BAOs.
4. Conclusion
In this paper, we have presented a new way to study BaryonAcoustic Oscillations (BAOs) using a Spherical Fourier-Bessel (SFB) decomposition of a wide-field deep galaxysurvey. The BAO signal in SFB space can be studied byconsidering the ratio R C(cid:96) ( k ) of the SFB power spectrumwith wiggles (cid:0) C b (cid:96) ( k ) (cid:1) to the smooth SFB power spectrum (cid:0) C nob (cid:96) ( k ) (cid:1) , similarly to what is done in Fourier space. Inthis decomposition, BAOs can be observed simultaneouslyin both the radial ( k ) and tangential ( (cid:96) ) modes.For a field which is statistically isotropic and homo-geneous (SIH), the SFB power spectrum is purely radial,i.e. independent of tangential modes, and reduces to theCartesian Fourier power spectrum. In the other extremelimit where the field is in a thin shell, we showed that allthe SFB information is contained in the tangential modesand is simply related to the 2D spherical harmonic powerspectrum.Considering the practical case where the field is ob-served with a radial selection function (thus partially vi-olating the SIH condition), we find that the radial limitcan still be reached for selection functions covering a largeradial range (see Appendix A), which we refer to as the ra-dialisation of the power spectrum. The radialisation willbe limited to a region in ( (cid:96), k ) space corresponding to k (cid:29) (cid:112) (cid:96) ( (cid:96) + 1) /r , where r parameterises the radial se-lection function. We find that radialisation is a good ap-proximation for these modes even when evolution due togrowth and bias are considered, both of which can be con-sidered as a further violation of the SIH condition. To studythis limit, we have derived an analytic solution to the win-dow function of a non-evolving, unbiased galaxy field whenthe radial selection function is a Gaussian in r centered onthe observer (Equation 16).We also find the BAOs considered in SFB space radialiseas the survey depth is increased, meaning both the phaseand amplitude of the BAOs tend towards the Fourier spaceratio R P ( k ). This means that BAOs for a wide-field shallowsurvey have smaller amplitude and are spread across the( (cid:96), k ) space, while BAOs for a wide-field deep survey havea larger amplitude and are confined to the radial modes(Figures 1 and 2).We study the radial limit analytically (Figure 3) andfind that it can in principle be observed (for a limited (cid:96) -range and for small physical scales) with all-sky surveyswith current surveys depths or for future stage-IV surveys.This suggest that the first BAO peak becomes radial up to (cid:96) ∼
10 for an all-sky survey with similar depth to SDSSor 2dF, and up to (cid:96) ∼
70 for an all-sky stage IV survey.In practice though, the radialisation might be observableto even higher (cid:96) since we observe that the BAOs radialisemore rapidly than the full SFB power spectrum, especiallyat large (cid:96) . Subsequent BAO peaks also become radial forlarge values of (cid:96) and for shallower surveys, though thesemay be already be in the non-linear regime. For modes thathave become radial, measurements at different (cid:96) ’s are usefulin practice, to reduce measurement errors due to cosmicvariance and shot noise.
6. Rassat , and A. Refregier : 3D Spherical Analysis of BAOs We note that we have ignored redshift-space distortionsin our analysis though the prescription for these in SFBspace is well known, (see for e.g., Heavens & Taylor 1995).These distortions may affect the radialisation as they willintroduce mode-mixing, albeit with a distinct signature andare readily described in the SFB basis. Further (cid:96) -mode mix-ing will occur for incomplete sky coverage, which can becorrected for using the mask geometry.
Acknowledgements.
The authors are grateful to Pirin Erdo˘gdu, AlanHeavens, Ofer Lahav, Fran¸cois Lanusse, Boris Leistedt and AdamAmara for useful discussions about SFB decompositions. The SFBcalculations use the discrete spherical Bessel transform (DSBT) pre-sented in Lanusse et al. (2011) and the authors are grateful to Fran¸coisLanusse for help implementing this. We extended iCosmo softwarefor our calculations (Refregier et al. 2011). This research is in partsupported by the Swiss National Science Foundation (SNSF). Appendix A: Radial Limit for a Gaussian SelectionFunction
As shown in § φ ( r ) = e − ( r/r ) , it is the galaxy selection function(Equation 13) reduces to W (cid:96) ( k, k (cid:48) ) = πr (cid:114) kk (cid:48) exp (cid:20) − r k + k (cid:48) (cid:21) I (cid:96) + (cid:18) r kk (cid:48) (cid:19) , (A.1)where I ν ( x ) is the modified Bessel function of the first kind.For numerical reasons, it is often useful to evaluate theexponentially scaled modified Bessel function of the firstkind ˜ I ν ( x ) = exp( − x ) I ν ( x ) instead of I ν ( x ), in which caseit is useful to re-write the above Equation as: W (cid:96) ( k, k (cid:48) ) = πr (cid:114) kk (cid:48) exp (cid:34) − r (cid:18) k − k (cid:48) (cid:19) (cid:35) ˜ I (cid:96) + (cid:18) r kk (cid:48) (cid:19) . (A.2)Figure A.1 shows the window function for two values of r , (cid:96) = 3 as a function of k, k (cid:48) . As r becomes large, it canbe seen that the window function tends towards a deltafunction k (cid:48) δ ( k − k (cid:48) ).To study this limit more precisely, we use the asymp-totic form for the modified Bessel function I α ( x ) (cid:39) e x / √ πx for x (cid:29) | α − / | , which gives W (cid:96) ( k, k (cid:48) ) (cid:39) √ π r k (cid:48) exp (cid:34) − r (cid:18) k − k (cid:48) (cid:19) (cid:35) , (A.3)in the limit where r kk (cid:48) (cid:29) (cid:96) ( (cid:96) + 1), or approximately, r k (cid:29) (cid:112) (cid:96) ( (cid:96) + 1) . (A.4)Using the definition of a Dirac delta function, namely (cid:82) dx h ( x ) δ ( x − x ) = h ( x ) for an arbitrary function h ( x ),the window function becomes, in the limit r → ∞ , W (cid:96) ( k, k (cid:48) ) (cid:39) π k (cid:48) δ ( k − k (cid:48) ) , (A.5)in agreement with Equation 17 for the radial case discussedin § C obs (cid:96) ( k, k (cid:48) ) becomes radial (i.e independentof (cid:96) ) and equal to C (cid:96) ( k ) δ ( k − k (cid:48) ) = P ( k ) δ ( k − k (cid:48) ) as inEquation 18. References
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7. Rassat , and A. Refregier : 3D Spherical Analysis of BAOs Fig. A.1.
Window function W (cid:96) ( k, k (cid:48) ) for the Gaussian selection function with for (cid:96) = 3 and r = 100 h − Mpc (left) and r = 1400 h − Mpc (right). As the selection function parameter r becomes larger (i.e., increasing the redshift coveragefrom z med ∼ .
05 to z med ∼ . π k (cid:48) δ ( k (cid:48) − k ) for scales where k (cid:29) (cid:112) (cid:96) ( (cid:96) + 1) /r ..