aa r X i v : . [ a s t r o - ph ] S e p Constraining the growth fa tor with baryon os illationsDomeni o Sapone ∗ and Lu a Amendola † Départment de Physique Théorique, Université de Genève,24 quai Ernest Ansermet, CH-1211, Genève 4, Switzerland and INAF/Osservatorio Astronomi o di Roma Via Fras ati 33 - 00040 Monteporzio Catone (Roma) - ItalyThe growth fa tor of linear (cid:29)u tuations is probably one of the least known quantity in observa-tional osmology. Here we dis uss the ontraints that baryon os illations in galaxy power spe trafrom future surveys an put on a onveniently parametrized growth fa tor. We (cid:28)nd that spe tro-s opi surveys of 5000 deg extending to z ≈ ould estimate the growth index γ within . ; asimilar photometri survey would give ∆ γ ≈ . . This test provides an important onsisten y he k for the standard osmologi al model and ould onstrain modi(cid:28)ed gravity models. We dis ussthe errors and the (cid:28)gure of merit for various ombinations of redshift errors and survey sizes.I. INTRODUCTIONThe hara terization of dark energy (DE) has been sofar based almost uniquely on ba kground tests at ratherlow redshifts ( z ≤ . : Riess et al. 1998, Perlmutter etal. 1999, Tonry et al. 2003, Riess et al. 2004, Astier etal. 2006, Eisenstein et al. 2005) or very large redshifts( z ≈ : e.g. Netter(cid:28)eld et al. 2002, Halverson et al.2002, Lee et al. 2002, Bennett et al. 2003, Spergel et al.2006). These tests are based essentially on estimationsof luminosity D L ( z ) or angular-diameter distan es D ( z ) ,i.e. on integrals of the Hubble fun tion H ( z ) whi h, inturn, ontain integrals of the equation of state. Only veryre ently tests involving the linear perturbations have be-gun to be dis ussed, using methods based on the inte-grated Sa hs-Wolfe e(cid:27)e t, weak lensing and high-redshiftpower spe tra (e.g. Boughn & Crittenden 2004, Refregieret al. 2006, Crotts et al. 2005 ). However, it is fair to saythat the growth fun tion is still one of the least knownquantity in osmology. So far, it is possible to quote onlytwo published results that put limits on it: the value at z ≈ . obtained in 2dF (Hawkins et al. 2003; Verdeet al. 2002) and the z ≈ result from Lyman − α louds(M Donald et al. 2005). De(cid:28)ning G ( z ) = δ ( z ) /δ (0) ( δ being the matter density ontrast) we have for f ≡ d log Gd log a (1)the value f = 0 . ± . for 2dF at z ≈ . and f = 1 . ± . for the Lyman- α at z ≈ . These re-sults show learly how large is the degree of un ertainty.A tually the un ertainty is mu h larger than it appearsfrom the quoted statisti al errors. In the ase of the low- z estimate, the result is obtained by estimating the biasfrom higher-order statisti s, whi h is known to be parti -ularly sensitive to the sele tion e(cid:27)e ts, to in ompletenesset .; di(cid:27)erent methods give in fa t quite di(cid:27)erent results(see dis ussion in Hawkins et al. 2003). In the ase of ∗ Ele troni address: domeni o.saponephysi s.unige. h † Ele troni address: amendolamporzio.astro.it the high- z estimation, the main problem is the re on-stru tion of the bias fa tor from numeri al simulationswhi h, by their nature, are performed only in a limitedrange of (cid:28)du ial models. It is therefore important to testthe growth fa tor with other methods and with improveddatasets.A test of the growth fa tor would be important both asa onsisten y he k for the standard osmologi al model(sin e f is determined by H ( z ) in a standard osmol-ogy) and as a onstraint on non-standard models likee.g. modi(cid:28)ed gravity. In fa t, models that modify thePoisson equation will also generi ally modify the pertur-bation equation for the matter density ontrast δ . As anexample, models in whi h dark energy is oupled to mat-ter display a growth index whi h deviates from the stan-dard ase at all epo hs (see e.g. Amendola & To hini-Valentini 2003; Demianski et al. 2004; Nunes & Mota2004). Several other papers dis ussed the parametriza-tion of the perturbation equations in modi(cid:28)ed gravitymodels, see e g. Ishak et al. (2005), Heavens, Kit h-ing and Taylor (2006), Taylor et al. (2007); Heavens,Kit hing, Verde (2007) , Caldwell, Cooray and Mel hiorri(2007), Amendola, Kunz and Sapone (2007), Zhang et al.(2007).In this paper we investigate the extent to whi h baryonos illations an set limits to G ( z ) in future large-s ale ob-servations at z up to 3. The method we use is based onre ent proposals (Linder 2003, Blake & Glazebrook 2003,Seo & Eisenstein 2003) to exploit the baryon a ousti os- illations (BAOs) in the power spe trum as a standardruler alibrated through CMB a ousti peaks. In parti -ular, Seo & Eisenstein (2003; SE) have shown the feasi-bility of large (100 to 1000 square degrees) spe tros opi surveys at z ≈ and z ≈ to put stringent limits tothe equation of state w ( z ) and its derivative. As it iswell-known, BAOs have been dete ted at low z in SDSS(Eisenstein et al. 2005); the dete tion at large z , wheremore peaks at smaller s ales an be obtained, is likely tobe ome one of the most interesting astrophysi al endeav-ours of the next years.II. BACKGROUND EQUATIONHere we review the basi equations and notation forthe ba kground evolution and for the linear (cid:29)u tuations.The evolution of the dark energy an be expressed by thepresent dark energy density Ω DE and by a time-varyingequation of state (see Copeland, Sami & Tsujikawa 2006for a re ent review): w ( z ) = pρ (2)Given w ( z ) , the dark energy density equation is ρ ( z ) = ρ (0) a − w ) where ˆ w ( z ) = 1log(1 + z ) Z z w ( z ′ )1 + z ′ dz ′ (3)The Hubble parameter and the angular diameter dis-tan e, H ( z ) and D A ( z ) , assuming a (cid:29)at universe Ω m +Ω DE = 1 , be ome respe tively: H ( z ) = H [Ω m (1 + z ) + (1 − Ω m )(1 + z ) w ) ] (4)and D A ( z ) = c z Z z dz ′ H ( z ′ ) (5)where the total matter density is Ω m ( z ) = Ω m Ω m + (1 − Ω m )(1 + z ) w (6)It is well known that a good approximation to thegrowth index for sub-horizon s ales in (cid:29)at models is givenby (Lahav et al. 1991, Wang and Steinhardt 1998 ) f ≡ ∂ log G∂ log a = Ω m ( a ) γ (7)This introdu es a new parameter γ , beside those that hara terize the ba kground model (see also Linder 2005,Per ival 2005) .We remark that a re ent analysis of most of the ex-tant data produ ed the result γ = 0 . +0 . − . (Di Porto &Amendola 2007) .III. FISHER MATRIX FORMALISMFollowing Seo & Eisenstein (2003; hereinafter SE) wewrite s hemati ally the observed galaxy power spe trumas: P obs ( z, k r ) = D Ar ( z ) H ( z ) D A ( z ) H r ( z ) G ( z ) b ( z ) (cid:0) βµ (cid:1) P r ( k )+ P shot ( z ) (8)where the subs ript r refers to the values assumed for thereferen e osmologi al model, i.e. the model at whi h we evaluate the Fisher matrix. Here P shot is the shot noisedue to dis reteness in the survey, µ is the dire tion osinewithin the survey, P is the present spe trum for the(cid:28)du ial osmology. For the linear matter power spe trumwe adopt the (cid:28)t by Eisenstein & Hu (1999) (with nomassive neutrinos and also negle ting any hange of theshape of the spe trum from small deviation around w = − ).The wavenumber k is also to be transformed betweenthe (cid:28)du ial osmology and the general one (SE; see alsoAmendola, Quer ellini, Giallongo 2004, hereinafter AQG,for more details). The bias fa tor is de(cid:28)ned as: b ( z ) = Ω m ( z ) γ β ( z ) (9)and for the (cid:28)du ial model is estimated by omparing the Mp /h ell varian e σ ,g of the galaxies orre ted forthe linear redshift distortion with the same quantity forthe total matter. Clearly, the growth fun tion is degener-ate with the bias ex ept for the redshift orre tion fa tor (1+ βµ ) . Sin e we marginalize over β , it is lear that theredshift orre tion plays a ru ial role for as on ern theestimation of the growth fa tor. The linear orre tion weuse should therefore be onsidered only a (cid:28)rst approxi-mation and more work to go beyond Kaiser's small-angleand Gaussian approximation is needed, as dis ussed inHamilton & Culhane (1996), Zaroubi & Ho(cid:27)man (1996),Tegmark et al. (2004) and S o imarro (2004).The total galaxy power spe trum in luding the errorson redshift an be written as (SE) P ( z, k ) = P obs ( z, k ) e k µ σ r (10)where σ r = δzH ( z ) is the absolute error on the measure-ment of the distan e and δz is the absolute error on red-shift. Given the un ertainties of our observations, wenow want to propagate these errors to ompute the on-straints on osmologi al parameters. The Fisher matrixprovides a useful method for doing this. Assuming thelikelihood fun tion to be Gaussian, the Fisher matrix is(Eisenstein, Hu & Tegmark 1998; Tegmark 1997) F ij = 2 π Z k max k min ∂ log P ( k n ) ∂θ i ∂ log P ( k n ) ∂θ j · V eff · k π · dk (11)where the derivatives are evaluated at the parameter val-ues of the (cid:28)du ial model and V eff is the e(cid:27)e tive volumeof the survey, given by: V eff = Z (cid:20) n ( ~r ) P ( k, µ ) n ( ~r ) P ( k, µ ) + 1 (cid:21) d~r == (cid:20) n ( ~r ) P ( k, µ ) n ( ~r ) P ( k, µ ) + 1 (cid:21) V survey (12)where the last equality holds only if the omoving numberdensity is onstant in position and where µ = ~k · b r/k , b r be-ing the unit ve tor along the line of sight and k the waveParameters1 total matter density ω m = Ω m h ω b = Ω b h τ n s Ω m For ea h redshift bin6 shot noise P s log D A log H log D
10 bias log β Table I: Cosmologi al parametersParameters1 total matter density ω m = Ω m h ω b = Ω b h τ n s Ω m T /S log D A log A s Table II: CMB parametersve tor. The highest frequen y k max ( z ) is hosen to benear the s ale of non-linearity at z : we hoose values from . h /Mp for small z bins to . h /Mp for the highestredshift bins. Any submatrix of F − ij gives the orrela-tion matrix for the parameters orresponding to rows and olumns on that submatrix. The eigenve tors and eigen-values of this orrelation matrix give the orientation andthe size of the semiaxes of the on(cid:28)den e region ellipsoid.This automati ally marginalizes over the remaining pa-rameters. The parameters that we use for evaluating theFisher matrix are shown in Tab. (I). Our (cid:28)du ial model orresponds to the Λ CDM WMAP3y best-(cid:28)t parameters(Spergel et al. 2006): Ω m = 0 . , h = 0 . , Ω DE = 0 . , Ω K = 0 , Ω b h = 0 . , τ = 0 . , n s = 0 . and T /S = 0 and as anti ipated γ = 0 . . Beside theBAO from large s ale stru ture, we also employ the CMBFisher matrix, following the method in Eisenstein, Hu &Tegmark (1999) and assuming a Plan k-like experiment.The osmologi al parameters we use for CMB are listedin Tab. (II). The total Fisher matrix is given simply bythe addition of the two matri es.The derivatives of the spe trum with respe t to the os-mologi al parameters p i (i.e. ω m = Ω m h , ω b = Ω b h , τ , n s , Ω m plus P s , β, G, D, H for ea h redshift bin) areevaluated using the (cid:28)t of Eisenstein & Hu (1999). a f(a) f(a) Ω m0.545 Ω m γ (a) Figure 1: Growth index vs the s ale fa tor for a DE modelwith a varying equation of state, w ( z ) = w + w z with w = − . and w = 1 . The bla k solid line refers to the solutionobtained by the di(cid:27)erential equation for γ (Per ival 2005).The red dashed line refers to the growth index given by eq.(7) and the green dotted line is the growth index with a γ fa tor given by eq. (17). The matter density is given by eq.(6).Sin e we want to propagate the errors to the osmo-logi ally relevant set of parameters q i = { w , w , γ } (13)we need to hange parameter spa e. This will be donetaking the inverse of the Fisher Matrix F − ij and thenextra ting a submatrix, alled F − mn ontaining only therows and olumns with the parameters that depend on q i , namely D A , H and G . The root mean square of thediagonal elements of the inverse of the submatrix give theerrors on D A , H , and G . Then we ontra t the inverseof the submatrix with the new set of parameters q i ; thenew Fisher matrix will be given by F DE ; ij = ∂p m ∂q i F mn ∂p n ∂q j (14)This automati ally marginalizes over all the remainingparameters.The derivatives of the Hubble parameter and for theangular diameter distan e an be written as ∂ log H∂q i = 1 H ∂H∂q i (15) ∂ log D A ∂q i = − z ) D A Z ∂ log H∂q i H dz (16)IV. GROWTH FACTORWe onsider now separately two ases: in Case 1 thegrowth rate depends on w (assumed onstant); in Case2 the growth rate is free and we fore ast the onstraintsthat future experiments an put on it.Surveys z V s (Gp /h) n − . · − . − . . · − . − . . · − . − . . · − . − . . · − . − . · − Table III: Details of the surveys. - - - - - p - - - w z = +
3; A = deg - - - - - p - - - w z = +
3; A = deg - - - - - p - - - w z = +
3; A = deg - - - - - p - - - w z =
3; A = deg - - - - - p - - - w z =
3; A = deg - - - - - p - - - w z =
3; A = deg Figure 2: Con(cid:28)den e level for w p and w for surveys of 1000,5000 and 10000 deg and for di(cid:27)erent ombinations of red-shift bins ( ase 1). The solid red urve refers to spe tros opi surveys and the dashed blue urve to photometri surveys, δz = 0 and δz/z = 0 . respe tively.A. Case 1In general, the exponent γ depends on the osmologi alparameters. To see this, we just need to onsider theequation of perturbations and insert the growth indexde(cid:28)ned by eq. (1). Then we obtain the approximateanalyti solution (Wang & Steinhardt 1998) : F O M W m h W b h n s W m Figure 3: Fom for w p − w vs marginalized parameters. deg deg deg δz w p w w p w w p w z = 1 + 30 % .
036 0 .
313 0 .
018 0 .
208 0 .
012 0 . % .
054 1 .
523 0 .
028 0 .
726 0 .
018 0 . z = 30 % .
044 0 .
579 0 .
033 0 .
257 0 .
030 0 . % .
061 2 .
614 0 .
051 1 .
463 0 .
034 1 . Table IV: Values of σ w p and σ w for spe tros opi surveys δz = 0 and photometri surveys δz = 4 % on the redshiftestimate and for several survey areas ( ase 1). We onsidertwo di(cid:27)erent ombinations of redshift ( z = 1 + 3 and z = 3 only). γ ( z ) = 35 − w ( z )1 − w ( z ) (17)and for a Λ CDM model γ = 0 . . The behavior of thegrowth index for a w ( z ) model is shown in Fig. (1). We an see that there is almost no di(cid:27)eren e in behavior be-tween the urves obtained with the approximation (17).Be ause of the dependen e of γ on the dark energy pa-rameters, the derivatives of the growth fa tor are givenby: ∂ log G∂q i = − Z (cid:20) ∂γ∂q i log Ω m ( z )+ γ ∂ log Ω m ( z ) ∂q i (cid:21) Ω m ( z ) γ dz (1 + z ) (18)In this ase the new set of parameters is q i = { w , w } and we assume as (cid:28)du ial model w = − , w = 0 . Thefa tor γ , in this ase, depends only on the dark energyparameters w and w ; this means the only non-vanishingderivatives are ∂γ∂w and ∂γ∂w . In Fig. (2) the on(cid:28)den eregions are shown for di(cid:27)erent ombination of redshiftand area. Instead of ( w , w ) we use the pivot parameters w p − w (proje tion of w − w on the pivot point, de(cid:28)nedas the value of z for whi h the un ertainty in w ( z ) issmallest). - - - Γ z = +
3; A = - - - Γ z = +
3; A = - - - Γ z = +
3; A = - - - Γ z =
3; A = - - - Γ z =
3; A = - - - Γ z =
3; A = Figure 4: Con(cid:28)den e level for w and γ for surveys of 1000,5000 and 10000 deg and for di(cid:27)erent ombinations of red-shift bins ( ase 2). The solid red urve refers to spe tros opi surveys and the dashed blue urve to photometri surveys, δz = 0 and δz/z = 0 . respe tively.B. Case 2We want now to put onstraints on γ as a free param-eter. We assume here w = constant and again w = − as (cid:28)du ial value. The new set of parameters is there-fore q i = { w , γ } . The derivatives with respe t to the(cid:28)rst three parameters are given by the eq. (18). Thederivative for the growth fa tor with respe t to γ is: ∂ log G∂γ = − Z ∂∂γ exp [ γ log Ω m ( z )] dz (1 + z ) == − Z log Ω m ( z ) Ω m ( z ) γ dz (1 + z ) (19)V. RESULTS AND CONCLUSIONSThe main aim of this work is to give marginalized on-straints on the dark energy parameters ( w p − w ) and deg deg deg δz w γ w γ w γz = 1 + 30 % .
045 0 .
099 0 .
016 0 .
059 0 .
004 0 . % .
128 0 .
301 0 .
062 0 .
153 0 .
044 0 . z = 30 % .
089 0 .
188 0 .
039 0 .
092 0 .
026 0 . % .
152 0 .
344 0 .
076 0 .
18 0 .
081 0 . Table V: Values of σ w and σ γ for spe tros opi surveys δz =0 and photometri surveys δz/z = 4 % on the measure ofthe redshift and for several areas ( ase 2). We onsider twodi(cid:27)erent ombinations of redshift bins ( z = 1 + 3 and z = 3 only). - - - - - - - p - - w z = + Figure 5: Con(cid:28)den e level for w p and w for surveys 20000 deg (DETF ase). The solid red urve refers to spe tros opi surveys.most importantly on the growth fa tor itself, for several ombinations of surveys, redshift errors and area. Fol-lowing SE and Amendola, Quer ellini, Giallongo (2004)we onsider several binned surveys with average redshiftdepth around z = 1 and z = 3 plus a SDSS-like surveyat z < . , as detailed in Table III. More details an befound in AQG. We onsider both spe tros opi surveys( δz = 0 ) and photometri surveys ( δz/z = 0 . ) andthree areas ( , , deg ). These features arewell within the range of proposed experiments like JDEMand DUNE (Crotts at al. 2005; Réfrégier et al. 2006; seealso DETF Report Albre ht et al. 2006)We (cid:28)rst onsider Case I, in whi h the growth fa tor isnot an independent quantity but is a fun tion of w ( z ) .The two-dimensional regions of on(cid:28)den e are shown inFig (2) and the (cid:28)nal marginalized errors are summarizedin Tab. (IV). The errors on w p redu e from 0.036 to0.012 for the spe tros opi ase for surveys that extendfrom 1000 to 10000 deg and from 0.054 to 0.018 in thephotometri ase.Then we onsider Case II, in whi h γ is a free onstantas in eq. (1). In Fig. (4) we show the on(cid:28)den e regions - - - - Γ z = + Figure 6: Con(cid:28)den e level for w and γ for surveys 20000 deg (DETF ase). The solid red urve refers to spe tros opi surveys.for w − γ . The errors are given in Tab. (V) . We seethat the errors on γ redu e from 0.099 to 0.05 for spe -tros opi surveys and from 0.301 to 0.114 in the photo- z ase. These errors are way too large to produ e an inde-pendent onstraint on w (in fa t one has approximately ∆ w ≈ γ near w = − ) but, beside being a gen-eral test of onsisten y for the osmologi al model, theywould ertainly give interesting onstraints on modelsthat predi t growths di(cid:27)erent from standard, like modi-(cid:28)ed gravity models (Koyama & Maartens 2005; Maartens2006; Amendola, Charmousis & Davis 2005; Amendola,Polarski, Tsujikawa 2006). In Fig. (3) we show theFOM for w p − w , for only one survey ( deg ) andonly one ombination of redshift ( z = 1 − ), (cid:28)rst whenall the other parameters are (cid:28)xed and then su essivelymarginalizing over the parameter indi ated and over allthose on the left (eg the third olumn represents themarginalization over ω m , ω b ).We an ompare our results to those obtained re entlyby Huterer and Linder (2006). Using a ombinationof weak lensing, SNIa and CMB methods, they predi t σ ( γ ) = 0 . for future experiments. With large-s aletomographi weak lensing alone, Amendola, Kunz, andSapone (2007) predi t σ ( γ ) = 0 . at on(cid:28)den elevel. These values are omparable to those obtainedhere with the BAO method and onsidering a spe tro-s opi survey of 5000 deg , σ ( γ ) = 0 . .We noti e that the di(cid:27)eren e on the growth index γ be-tween General Relativity and an extradimensional grav-ity model (as DGP, where γ = 0 . , see Linder & Cahn2007) is ∆ γ = 0 . ; if we ompare our results shownin Tab. (V) we see that the errors on γ for a photo-metri survey are within this range, meaning that DGPmodel annot be es luded. Things get slightly better ifwe onsider spe tros opi surveys, where errors de reasewith about ; however in this ase we require a largesurvey extended from z = 1 to z = 3 with an area of deg to distinguish with su(cid:30) ient on(cid:28)den e DGP Area F O M δ z = 0; z = 1 - 3 δ z = 0.04; z = 1 - 3 δ z = 0; z = 3 δ z = 0.04; z = 3 Figure 7: E(cid:27)e t of Survey geometry on the dark energy FOM.We plotter the FOM ( w p − w ) for spe tros opi surveys ( δz =0 ) and photometri surveys ( δz = 0 . ) as a fun tion of thearea. Area F O M δ z = 0; z = 1 - 3 δ z = 0.04; z = 1 - 3 δ z = 0; z = 3 δ z = 0.04; z = 3 Figure 8: E(cid:27)e t of Survey geometry on the dark energy FOM.We plotter the FOM ( w − γ ) for spe tros opi surveys ( δz =0 ) and photometri surveys ( δz = 0 . ) as a fun tion of thearea.from Λ CDM. In Fig. (6) is shown the on(cid:28)den e regionfor w − γ for a survey extended from z = 0 to z = 1 . and an area of deg (DETF ase): the error on γ redu es to σ ( γ ) = 0 . .In Fig. (7) we also show the (cid:28)gure-of-merit (FOM) sug-gested by the Dark Energy Task For e report ( Albre htet al. 2006) as a simple measure of the onstraining powerof an experiment. The FOM is de(cid:28)ned as the inverse ofthe area that en loses the 95% on(cid:28)den e region and anbe found simply as (6 . π √ det F ) − . In Fig. (8) weplot the FOM for w and γ . The general trend is thatthe FOM for spe tros opi surveys are roughly 4-6 timeshigher than for similar 4% error photo- z surveys. It willbe interesting to ompare our FOM on the plane w , γ, γ