Scale-Dependent Bias of Galaxies from Baryonic Acoustic Oscillations
aa r X i v : . [ a s t r o - ph . C O ] S e p Mon. Not. R. Astron. Soc. , 1–6 (2010) Printed 7 November 2018 (MN L A TEX style file v2.2)
Scale-Dependent Bias of Galaxies from Baryonic AcousticOscillations
Rennan Barkana and Abraham Loeb ⋆ Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel Astronomy Department, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA
ABSTRACT
Baryonic acoustic oscillations (BAOs) modulate the density ratio of baryons to darkmatter across large regions of the Universe. We show that the associated variationin the mass-to-light ratio of galaxies should generate an oscillatory, scale-dependentbias of galaxies relative to the underlying distribution of dark matter. A measurementof this effect would calibrate the dependence of the characteristic mass-to-light ratioof galaxies on the baryon mass fraction in their large scale environment. This bias,though, is unlikely to significantly affect measurements of BAO peak positions.
Key words: cosmology:theory – galaxies:formation – large-scale structure of Universe
The rapid acoustic waves in the radiation-baryon fluid priorto cosmological recombination were not followed by the darkmatter at that time. Following recombination, the baryonswere freed from the strong radiation pressure and fell intothe gravitational potential fluctuations of the dark matter.As a result, the fractional difference between the densityfluctuations of baryons and dark matter decreased steadilywith cosmic time. But since the baryons amount to a sizablefraction of the total mass density of matter (Ω b / Ω m ≈ ∼
100 Mpc (corresponding to the sound horizon at recombi-nation), provides a yardstick that can be used to measure thedependence of both the angular diameter distance and Hub-ble parameter on redshift (see review by Eisenstein 2005).When analyzing galaxy surveys, it is often assumedthat galaxies are biased tracers of the underlying mat-ter distribution (Kaiser 1984), with a bias factor that ap-proaches a constant value on sufficiently large scales wheredensity fluctuations are still linear (e.g., Mo & White 1996;Tegmark & Peebles 1998; Sheth et al. 2001). However, theimprint of primordial acoustic waves on the baryon fluidat recombination introduced a scale-dependent modulationof the ratio between the density fluctuations of baryonsand dark matter that has not been completely erased bythe present time. A large-scale region with a higher baryonmass fraction than average (in the perturbations that lead ⋆ E-mail: [email protected] (RB); [email protected](AL) to galactic halos) is expected to produce more stars per unittotal mass and hence result in galaxies with a lower mass-to-light ratio.In this paper we characterize the associated scale-dependent bias in flux-limited surveys of galaxies. The ratiobetween the power spectra for fluctuations in the luminositydensity and number density of galaxies is expected to showBAO oscillations that reflect the large-scale variations in thebaryon-to-matter ratio.In §
2, we formulate the oscillatory BAO signature ongalaxy bias in terms of a simple analytical model. The quan-titative results from this model are presented in §
3. Finally,we summarize our main conclusions in § Since galaxies sample the high peaks of the underlying mat-ter density, they are biased tracers of the matter density.When the clustering of galaxies is usually analyzed, thebias is considered simply with respect to the matter density,without separating out the effects of the baryons. As longas the baryon fluctuations follow the same spatial patternas that of the dark matter, biasing with respect to each ofthem cannot be separated since this separation is degeneratewith an overall change of the bias factor, which is not knownapriori. However, since the BAOs induce a scale-dependentdifference between the baryons and dark matter, it becomesimportant to consider their influence on galaxies separately.Consider the power spectrum of fluctuations in thegalaxy number density n gal and in the luminosity density c (cid:13) Rennan Barkana and Abraham Loeb ρ L . For a given galaxy population, ρ L = n gal × h L i , (1)where h L i is the mean luminosity of the galaxies. Sincegalaxy formation is driven by halo collapse, which dependson the evolution of the overall matter perturbations, thenumber density fluctuations δ n are driven by the fluctuation δ tot in the total matter density, with a bias b n that shouldbe approximately constant on large scales (for a fixed galaxypopulation): δ n = b n δ tot . (2)The mean luminosity of galaxies may depend on their envi-ronment through their merger rate history, which is corre-lated with the local matter density. This can lead to fluctu-ations δ L in ρ L with a different bias factor that should alsoapproach a constant on large scales: δ L = ( b n + b L;t ) δ tot , (3)where the overall bias factor of the luminosity density withrespect to the total matter includes the number density bias b n as well as a possible additional bias b L;t from the depen-dence of h L i on the matter density.However, the luminosity is also affected separately bythe baryon fluctuations, since the luminosity depends onthe gas fraction in halos f b . Regions that have halos witha higher baryon fraction will proportionally have morebaryons in the galaxies within them. If, e.g., we assume thatthe star formation rate per baryon is on average constant,then h L i ∝ f b . In fact, the dependence of the luminosityon the gas fraction is likely to be non-linear. For instance,in simple models for disk formation within halos (Mo et al.1998), the disk radius is approximately independent of thegas fraction. Thus, if we assume that the disk mass is a fixedfraction of the halo gas mass, then the typical gas surfacedensity within the disk varies in proportion to the overallhalo gas fraction. According to the Schmidt-Kennicutt law(e.g., Kennicutt 1998), the star formation rate in the diskshould vary with the gas surface density to the power 1.4.Thus, in general we assume that h L i ∝ ( f b ) b L;f , (4)where these simple considerations suggest that b L;f ≈ . δ L = ( b n + b L;t ) δ tot + b L;f δ f , (5)where δ f is the perturbation in the halo gas fraction f b .Thus, b L;f is the bias factor of the luminosity density withrespect to the halo baryon fraction. Note that in our notationall the perturbations are the actual ones at the consideredredshift (i.e., we do not use the common practice of linearextrapolation to redshift zero).
We would expect the baryon fraction within halos to reflectthat of their surroundings, but the precise relation is com-plex due to the non-linear process of halo collapse. Here weemploy reasonable simplifications to derive an approximateanalytical result, which is partly verified and quantified bysimulation results shown in §
3. We find it useful to analyze the baryon fraction in sev-eral steps, where the first step is to avoid halo collapse andsimply consider γ b ≡ ρ b ρ tot , (6)where we use γ b for the general baryon fraction and reserve f b for the baryon fraction inside halos. The mean of thisquantity is the cosmic mean baryon fraction:¯ γ b = Ω b Ω m , (7)and its fluctuation is simply δ γ = δ b − δ tot = rδ tot . (8)Here we have measured the fractional difference between thebaryonic and total matter fluctuations with r ≡ ( δ b /δ tot ) − k and redshift.In reality, halos form out of perturbations that eventu-ally grow to an overdensity of hundreds, making the contri-bution of the mean density negligible, and thus we expectthe baryon fraction to reflect the relative mass of the baryonperturbation that formed the halo: f b = Ω b δ b Ω m δ tot = ¯ γ b δ b δ tot . (9)Before discussing non-linear collapse, we wish to apply thisequation to the linear perturbations that will form a halo,but even in the linear case we cannot easily apply this equa-tion in Fourier space, since halos form out of a sum of per-turbations on all scales, and taking a ratio as in equation (9)is a non-linear operation.To make further progress, we make a separation of scales(also called a peak-background split; Cole & Kaiser 1989),where we assume that the fluctuations that we wish to ob-serve (in the galaxy luminosity, etc.) are on much largerscales than the (initial comoving) scales that formed the ha-los. Typically, we are interested in measuring fluctuationson BAO scales, which are ∼ δ tot = δ l tot + δ s tot , (10) δ b = δ l b + δ s b = (1 + r l ) δ l tot + (1 + r s ) δ s tot , (11)where the relative difference between the baryonic and totalmatter perturbations is r l and r s on large and small scales,respectively.We now use the standard result of spherical collapse,that a forming halo has a linear δ tot = δ c , where the criticaldensity of collapse δ c is independent of mass (and equals 1.69in the Einstein de-Sitter limit, valid over a wide range of red-shifts ). We also assume that we are considering sufficientlylarge scales so that δ l tot can be treated as a perturbation of The value of δ c decreases at low redshift due to the cosmolog-ical constant, and at very high redshift due to the effects of thebaryons and radiation. However, at all z <
20 the change is below1% (Naoz & Barkana 2007). c (cid:13) , 1–6 cale-Dependent Bias from BAOs δ tot (or δ s tot ), and that r l and r s are also small quantities.Then the mean baryon fraction in halos is¯ f b = ¯ γ b (1 + r s ) , (12)and the lowest order perturbation is derived to be δ f = r l − r s δ c δ l tot . (13)We now use the actual value of r ( k ) (see § k ) that we denote r LSS (for Large Scale Structure) follow-ing Naoz & Barkana (2007). Thus, in the just-derived equa-tions we can treat r s = r LSS as a constant (at a given red-shift), since most of the density δ c needed to form a halocomes from scales well below the BAO scale. Thus, the meanbaryon fraction in halos is¯ f b = ¯ γ b (1 + r LSS ) , (14)while on large scales (i.e., small k ) the fluctuation is δ f = r ( k ) − r LSS δ c δ tot . (15)The remaining issue is the effect of non-linear collapse,and the relation between the baryon fraction in the linearly-extrapolated halo perturbation and the baryon fraction inthe actual virialized halo. We show simulation results in § r LSS that is amplified by a factor of several.One way to understand this enhancement is to consider thevariation of r LSS with time. It declines (in absolute value)approximately as r ∝ /a (where a = 1 / (1 + z ) is the scalefactor), since ( δ tot − δ b ) ≈ const while δ tot ∝ a (until the cos-mological constant becomes significant at low redshift). Thedecline of r LSS with time is of critical importance, since weare computing it according to linear theory, and it may notbe appropriate to extrapolate r LSS all the way to the haloformation time when we evaluate it in equation (14). Thebaryon fluctuations, which were erased on small scales be-fore cosmic recombination, later continuously catch up withthe dark matter (and thus with the total matter as well)in linear perturbation theory. However, once a perturbationbegins to form a halo and enters the non-linear stage of col-lapse, we expect that the rapid collapse will bring with itonly the baryons already present within the perturbation,and the continued decline of the linear-theory r LSS will be-come irrelevant for the halo gas content. The upshot is thatthe simulations suggest that if we use the linear-theory r LSS (and, we assume, more generally for r ( k )) then we mustmultiply it by an effective amplification factor A r :¯ f b = ¯ γ b (1 + A r r LSS ) , (16) δ f = A r δ c [ r ( k ) − r LSS ] δ tot . (17)The resulting fluctuations in the luminosity density (equa-tion 5) are δ L = ( b n + b L;t ) δ tot + b L;∆ [ r ( k ) − r LSS ] δ tot , (18)where b L;∆ ≡ b L;f A r δ c (19) is an effective bias factor that measures the overall depen-dence of galaxy luminosity on the underlying difference ∆between the baryon and total density fluctuations. We have assumed thus far that we observe a fixed galaxypopulation, regardless of the varying luminosity of its mem-bers. In reality, observed samples are limited by flux, orequivalently by luminosity if for simplicity we considergalaxies at a single redshift. Suppose the fraction of galaxiesabove luminosity L is F ( L ) = Z ∞ L ′ = L φ ( L ′ ) dL ′ , (20)where φ is the luminosity function. Then the observed num-ber density of galaxies is n obs = n gal F ( L ) , (21)and the luminosity density of these galaxies is ρ L = n gal h L i F ( L ) , (22)where h L i = 1 F ( L ) Z ∞ L ′ = L L ′ φ ( L ′ ) dL ′ . (23)We assume for simplicity that the same luminosity dis-tribution holds in different regions, except that the lumi-nosity of all galaxies is enhanced or diminished uniformlyin response to changes in the total density and the halobaryon fraction, as discussed in § L min , then wecan analyze the variations of F ( L ) by keeping φ fixed andvarying the effective threshold L min , while in ρ L we alsoinclude the perturbation in the luminosity of each galaxy.From equation (20) we obtain a relative fluctuation δ F = C min [ b L;t δ tot + b L;f δ f ] , (24)where the dimensionless coefficient C min = L min φ ( L min ) F ( L min ) . (25)The dependence of luminosity on the halo baryon frac-tion introduces a dependence of the galaxy number densityon the baryon fluctuations (i.e., on r ( k )). Putting our resultstogether, for a flux-limited survey we find δ n = ( b n + C min b L;t ) δ tot + C min b L;∆ [ r ( k ) − r LSS ] δ tot , (26)and δ L = [ b n +(1+ D min ) b L;t ] δ tot +(1+ D min ) b L;∆ [ r ( k ) − r LSS ] δ tot , (27)where D min = L min h L i C min , (28)with h L i evaluated for L = L min .In the limit where L min is well below the peak of theluminosity function, C min and D min both approach zero,and these expressions simplify to the previous ones (equa-tions 2 and 18). In the opposite limit, e.g., in the exponen-tial tail of a Schechter function, we can approximately set φ ( L ) ∝ e − L/L ∗ , and then C min = L min /L ∗ and D min = C min L min / ( L min + L ∗ ) are both ≫ L min ≫ L ∗ . c (cid:13) , 1–6 Rennan Barkana and Abraham Loeb
As we have shown, both the galaxy luminosity density and(for a flux-limited sample) number density depend on thehalo gas fraction. The scale-dependence of the relation be-tween the baryon and dark matter fluctuations implies thatthe BAOs can be observed in ratios that previously wouldhave been expected to be scale-independent.One proposal is to compare the power spectrum of fluc-tuations in the galaxy number density ( P n ) with that of theluminosity density ( P L ), with both measured for the samegalaxy sample. Taking the ratio may help to clear away somesystematic effects that affect both power spectra. Their ratio(square-rooted) should have the form (assuming r ( k ) ≪ (cid:18) P L P n (cid:19) / = B { B [ r ( k ) − r LSS ] } , (29)where the various bias factors enter into the coefficients B and B . If we denote the bias ratio b r ≡ b L;t /b n , then B = 1 + (1 + D min ) b r C min b r , (30)and B = b L;∆ b n D min − C min (1 + C min b r ) · [1 + (1 + D min ) b r ] . (31)Note that in the limit where most of the galaxy popula-tion is observed (i.e., the flux limits are unimportant), theseexpressions simplify to B = 1 + b r and B = b L;∆ / ( b n B ).In practice, using these expressions is not as dauntingas it may appear. For a given galaxy sample, C min and D min can be calculated from the measured luminosity function.This leaves two unknowns, b r and the ratio b L;∆ /b n . Withinthe ratio, we have a well-motivated expectation for b L;∆ = b L;f A r /δ c , given that δ c ≈ . b L;f ≈ . § A r ≈ § r were independent ofscale, then we could only measure a degenerate combinationof the unknown quantities. However, a precise measurementof the power spectrum ratio can separate out the constantand BAO terms, thus yielding B and B separately, whichin turn yields b r and the ratio b L;∆ /b n .Although it is implicit in the equations, r ( k ) and r LSS are also declining functions of time. However, even at lowredshift r ( k ) contains a signature of the BAOs, since theBAOs are still imprinted more strongly in the baryon fluc-tuations than in those of the dark matter or the total matter.This clear signature offers a chance to detect this effect, evenif the various bias factors that we have assumed to be con-stant actually vary slowly with k . A detection of the effectcan be combined with an estimate of b n from comparing P n with the underlying matter power spectrum (e.g., as mea-sured with weak lensing on large scales). Extraction of thevalue of b L;∆ would yield a new quantity in galaxy forma-tion, a combination of the way in which the luminosity ofa galaxy depends on the baryonic content of its host halo,and of how this baryonic content depends on the underlyingdifference between the baryon and total density fluctuations.Another possibility is to compare the power spectraof luminosity density (or flux-limited number density) be-tween two different samples. Their ratio should again have aform similar to equation (29), from which the constant andBAO term can be separately measured. It is well known thatgalaxy bias depends on galaxy luminosity (Lahav 1996), but
Figure 1.
The fractional baryon deviation r ( k ) = ( δ b /δ tot ) − k , at various redshifts ( z = 0, 0.5, 1, 3, and 6,from top to bottom). here the bias would be scale dependent in a way that de-pends on L min . For our quantitative results, we use the CAMB linear per-turbation code (Lewis et al. 2000), with the WMAP 5-yearcosmological parameters (Komatsu 2009), matching the sim-ulation that we compare with below.We show the dependence of r on both wavenumber andredshift in Figure 1. At a given redshift, r ( k ) approaches aconstant value ( r LSS ) at k > ∼ . r LSS (itselfa function only of redshift) we can separate out the twovariables k and z in their effect on r , as shown in Figure 2.The function [ r ( k ) /r LSS ] − k dependence of r is determined by a single, fixedfunction of k . Thus, the redshift dependence of r is the sameat all k , and it suffices to show the dependence of r LSS .Figure 2 shows that, as noted in the previous section, r LSS indeed varies approximately in proportion to 1 /a , but indetail the variation with redshift is slightly slower than that.As noted in the previous section, we expect the non-linear evolution that takes place during halo formation tomagnify the gas depletion effect compared to the linear the-ory calculation. We can test this effect using the hydrody-namical simulation of Naoz, Yoshida, & Barkana (2010). Al-though superficially it appears that they studied a quite dif-ferent regime (low-mass halos forming at high redshift), theirresults should be applicable here. In the linear theory, thegas depletion factor r LSS is constant all the way from nearlythe BAO scale ( k ∼ . k > ∼
100 h/Mpc). Naoz, Yoshida, & Barkana(2010) investigated the gas depletion in virialized halos from c (cid:13) , 1–6 cale-Dependent Bias from BAOs Figure 2.
Top panel: [ r ( k ) /r LSS ] − k , at thesame redshifts as in Figure 1 (the curves all lie on top of eachother). Bottom panel: The quantity 100 r LSS versus 1 + z (solidcurve), or equivalently, the value of r LSS in units of percent. Alsoshown is the function − . /a (dotted curve). For both panels,in practice we set r LSS ≡ r ( k = 1h / Mpc). below the Filtering mass (which is a time-averaged Jeansmass) up to a thousand times higher mass scale. Thus, themost massive halos in their simulation were well into thelarge-scale structure regime, where pressure is negligible,and the effect we are interested in (i.e., non-linear gas de-pletion on large scales) should operate.Figure 3 shows that the fractional gas deple-tion measured in virialized halos in the simulation ofNaoz, Yoshida, & Barkana (2010) was much larger than thedepletion r LSS predicted for linear perturbations at thehalo virialization redshift z vir . The simulated results canbe reasonably fit either by multiplying r LSS by a factorof 3.2, or by adopting r LSS from a higher redshift z [where(1 + z ) = 3 . z vir )]. Additional simulations are requiredto test whether these results can indeed be extrapolated toour regime of much more massive halos at low redshift, butthese results suggest that the gas depletion in halos is am-plified by a factor > ∼ C min = D min = 0, b n = 2, and b L;t = 1. As notedin § b L;∆ ∼ .
6, and also b r = 0 . B = 1 . B = 0 .
9. Thus, the oscillations in the square-rooted ratioof the luminosity and number density power spectra are atthe level of 0 .
4% at z = 1 (measured from the first peak, i.e.at the lowest k , to the following trough; the variation from k = 0 to the first peak is roughly twice as large). This is a There is a hint of a different slope with redshift in the simula-tion results compared to the fits. However, this mainly dependson a single point (at z vir = 12) and needs to be checked withfurther simulations. Figure 3.
The fractional gas depletion in halos ver-sus redshift. We show the results from the simulation ofNaoz, Yoshida, & Barkana (2010) (data points), where the cor-responding redshift (at which the virialized halos are identifiedin the simulation) is denoted z vir . We compare the depletion asmeasured in the simulation to r LSS at z vir (dotted curve), 3 . r LSS at z vir (long-dashed curve), and r LSS at a value of (1 + z ) equalto 3 . z vir ) (short-dashed curve). weaker effect by about a factor of five compared to the nor-mal BAOs in the total matter power spectrum. Thus, if highprecision is achieved in the regular BAO measurement, thenthe scale-dependent bias that we have highlighted shouldalso be measurable.This scale-dependent bias is unlikely to significantly af-fect the standard BAO measurements. Such measurementsare usually carried out on the power spectrum of the galaxynumber density. Scale-dependent bias enters this quantityonly in proportion to C min (see equation 26), so it wouldbe present only in a sample for which the flux limit playsa significant role. Even then, the effect on the BAO peakpositions would be quite weak, since the BAOs in δ tot arephysically a result of the influence of the baryons on thedark matter. Thus, the peak positions in δ tot and in δ b arenearly identical. For instance, even in the case that in equa-tion (26) the coefficients ( b n + C min b L;t ) and C min b L;∆ areequal, the BAO peak positions are shifted by only ∼ . We have shown that the variation in the baryon to mat-ter ratio imprinted by acoustic waves prior to cosmologicalrecombination should result today in an oscillatory, scale-dependent bias of galaxies relative to the underlying matterdistribution (see Figs. 1 & 2). The percent-level amplitudeof this signature depends on how the typical luminosity ofgalaxies scales with the baryon mass fraction in the large-scale region in which they reside. Simulations suggest thatthis signature is significantly amplified by non-linear effects c (cid:13) , 1–6 Rennan Barkana and Abraham Loeb during halo collapse (Fig. 3). The resulting amplitude canbe measured from the ratio between the power spectra offluctuations in the luminosity density and number densityof galaxies (equation 29). An observational calibration ofthis amplitude would offer a new cosmological probe of thephysics of galaxy formation.This effect may be marginally observable with currentdata, but it should certainly be feasible using future galaxysurveys (such as BOSS or BigBOSS ). However, since thebaryonic and the matter fluctuations have nearly identicalBAO peak positions, the scale-dependent bias is unlikely tosignificantly affect the standard BAO measurements, evenat percent-level precision. ACKNOWLEDGMENTS
We thank the US-Israel Binational Science Foundation forgrant support that enabled this collaboration (BSF grant2004386). This work was also supported in part by Is-rael Science Foundation grant 823/09 (for R.B.), and NSFgrant AST-0907890 and NASA grants NNX08AL43G andNNA09DB30A (for A.L.).
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