The Flat Sky Approximation to Galaxy Number Counts
PPrepared for submission to JCAP
The Flat-Sky Approximation to GalaxyNumber Counts
William L. Matthewson and Ruth Durrer
Universit´e de Gen`eve, D´epartement de Physique Th´eorique and Centre for Astroparticle Physics,24 quai Ernest-Ansermet, CH-1211 Gen`eve 4, SwitzerlandE-mail: [email protected], [email protected]
Abstract.
In this paper we derive and test the flat sky approximation for galaxy number counts.We show that, while for the lensing term it reduces to the Limber approximation, for the stan-dard density and redshift space distortion it is different and very accurate already at low (cid:96) whilethe corresponding Limber approximation completely fails. At equal redshift the accuracy ofthe standard terms is around 0.2% at low redshifts and 0.5% for redshift z = 5, even to low (cid:96) . At unequal redshifts the precision is less impressive and can only be trusted for very smallredshift differences, ∆ z < ∆ z (cid:39) . × − (1 + z ) . , but the lensing terms dominate for∆ z > ∆ z (cid:39) . r ( z ) H ( z )) / ( z + 1). The Limber approximation achieves an accuracy of 0.5%above (cid:96) (cid:39)
40 for the pure lensing term and above (cid:96) (cid:39)
80 for the lensing-density cross-correlation.Besides being very accurate, the flat sky approximation is also very fast and can therefore be usefulfor data analysis and forecasts with MCMC methods. a r X i v : . [ a s t r o - ph . C O ] J un ontents Until today, the most successful cosmological data are the anisotropies and polarisation of thecosmic microwave background (CMB), see [1] for the latest experimental results. However, in thisdecade there are deep and large galaxy surveys planned [2–10] which may do as well as, or in someaspects better than, CMB experiments. To make optimal use of these data, a correct analysishas to be performed. On small scales and at late times, non-linearities and baryonic effects arethe most difficult challenge, while at intermediate to large scales and higher redshifts a correctrelativistic treatment is most relevant.In recent years, fully relativistic expressions for the fluctuations of the observed galaxy numbercounts and their spectra have been derived [11–14]. A detailed study of these spectra has shown [13–22] that, in nearly all situations, the large scale relativistic terms are very small and can be neglectedfor percent level accuracy. Exceptions to this are very low redshifts, z < .
1, see [22], and verylarge angular scales, (cid:96) <
10. The latter are not very relevant, at least for single tracer analyses,due to cosmic variance. Their importance is discussed in Refs. [23–26].The remaining terms which are relevant on sub-horizon scales are the density, redshift-spacedistortion (RSD) and the lensing term. These are the terms which we investigate here and for whichwe derive simple approximations that can be computed rapidly, but are nevertheless accurate atthe 0.5% level or better for equal redshift correlations. For unequal redshifts our approximationsfor density and RSD are much less precise, but the lensing terms, which can be computed with theLimber approximation, dominate for large redshift differences. In the past, the density and RSDterms have been computed mainly in Fourier space [27]. This is sufficient for small surveys in oneredshift bin. We shall see that the flat sky approximation for density and RSD, while requiringa similar numerical effort, is not equivalent and is valid also at very large angular scales down to– 1 – θ (cid:39) (cid:96) = 2. While the flat sky approximation has been derived previously [28–31], its accuracy hasnever been analysed in any detail . Doing this is the goal of the present paper.The lensing term is an integral along the line of sight and cannot, without approximations,be represented in Fourier space (see [22] for an attempt). As the truly measured quantities aredirections and redshifts, it is most consistent and model independent to represent the numbercount fluctuations as a function of direction and redshift. This is what we do in this work. Whenassuming a background cosmology, the redshift space correlation function can also be computedand may be more useful for the analysis of spectroscopic surveys [22, 32] within one redshiftbin. However, for the very promising analysis of number counts from large photometric surveys,angular–redshift power spectra will most probably become the method of choice, since they aretruly model independent. Angular and redshift fluctuations are also simple to combine with shapemeasurements in order to derive galaxy-galaxy lensing cross-correlation spectra, see e.g. [33, 34].The remainder of this paper is structured as follows: In the next section we introduce theflat sky approximation for density, RSD and lensing and we compare flat sky results with CLASS results. We first present results for equal redshift correlations which are exquisitely accurate andthen study unequal redshifts which are more problematic. In Section 3 we compare the flat skyand the Limber approximations and in Section 4 we summarize our findings and conclude.
We set the speed of light c = 1 throughout. We consider a Friedmann Universe with scalarperturbations only in longitudinal (Newtonian, Poisson) gauge, ds = a ( t ) (cid:2) − (1 + 2Ψ) dt + (1 − δ ij dx i dx j (cid:3) . (1.1)We denote conformal time by t and a derivative wrt. t by an overdot. The conformal Hubbleparameter is denoted by H = ˙ a/a while the physical Hubble parameter is H = H /a . Including RSD and lensing, but neglecting large scale relativistic effects, the linear perturbationtheory expressions for the number count fluctuations in direction n at redshift z are [13, 14, 17]∆( n , z ) = b ( z ) D ( r ( z ) n , t ( z )) + 1 H ( z ) ∂ r V r ( r ( z ) n , t ( z )) + 2(1 − γ ( z )) κ ( n , z ) . (2.1)Here b ( z ) is the linear galaxy bias which depends on the class of galaxies considered in the survey, D is the density fluctuation (in comoving gauge), V r is the radial component of the velocity field(in longitudinal gauge) and κ is the convergence,2 κ ( n , z ) = ∆ S (cid:90) r ( z )0 dr (cid:48) ( r ( z ) − r (cid:48) ) r ( z ) r (cid:48) (Ψ( r (cid:48) n , t − r (cid:48) ) + Φ( r (cid:48) n , t − r (cid:48) )) , (2.2)where ∆ S denotes the Lapace operator on the 2-sphere, i.e. wrt. n . The function γ ( z ) is theluminosity bias which is given by the logarithmic derivative of the observed galaxy population at More precisely, in Ref. [28] the authors claim that they find an accuracy of better than 1% for ∆ ν/ν = ∆ z (cid:39) − (cid:29) ∆ z ( z ) (cid:39) .
06 at redshift z = 10. We roughly agree with this as we shall see later in the present paper. – 2 –he flux limit of the given survey, γ ( z, F lim ) ≡ − ∂ log N ( z, F > F lim ) ∂ log F (cid:12)(cid:12)(cid:12)(cid:12) F lim , (2.3)where N denotes the mean density of galaxies which are seen with a flux F > F lim from redshift z , i.e., the density of galaxies with luminosity L > L lim . The luminosity is related to the flux asusual via F = L/ (4 πD ( z ) ). Clearly, this function is survey-dependent, but γ ( z ) is also directlyobservable and does not depend on the background cosmology (which determines e.g. D ( z )).The number count fluctuation can be expanded in spherical harmonics,∆( n , z ) = (cid:88) (cid:96)m a (cid:96)m ( z ) Y (cid:96)m ( n ) (2.4) a (cid:96)m ( z ) = (cid:90) S ∆( n , z ) Y ∗ (cid:96)m ( n ) d Ω n , (2.5)and the angular redshift power spectrum is given by (cid:104) a (cid:96)m ( z ) a ∗ (cid:96) (cid:48) m (cid:48) ( z (cid:48) ) (cid:105) = C (cid:96) ( z, z (cid:48) ) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) . (2.6)Like for the CMB, the Kronecker-deltas are a consequence of statistical isotropy.In the flat sky approximation we replace the direction n by n = e z + α where α lives in theplane normal to e z , the flat sky. The direction e z is a reference direction out to the center of oursurvey. In the flat sky, (cid:96) is the dimensionless variable conjugate to α and the spherical harmonictransform (2.5) of an arbitrary variable X becomes a 2d Fourier transform, a X ( (cid:96) , z ) (cid:39) π (cid:90) d αe i (cid:96) · α X ( α , z ) , X ( α , z ) (cid:39) π (cid:90) d (cid:96)e − i (cid:96) · α a X ( (cid:96) , z ) . (2.7)Let us first consider a variable X ( x , t ) defined in all of space at any time with transfer function T X ( k, z ) so that X is given by X ( k , z ) = T X ( k, z ) R ( k ) , (2.8) X ( x , z ) = 1(2 π ) (cid:90) d ke − i x · k T X ( k, z ) R ( k ) . (2.9)Here R ( k ) is the initial curvature fluctuation after inflation. Its power spectrum is defined by (cid:104)R ( k ) R ∗ ( k (cid:48) ) (cid:105) = (2 π ) δ ( k − k (cid:48) ) P R ( k ) (2.10) k π P R ( k ) = P R ( k ) . (2.11)The normalization of the dimensionless power spectrum P R is such that the correlation functionof R in real space is simply (cid:104)R ( x ) R ( y ) (cid:105) = (cid:90) ∞ dkk j ( kr ) P R ( k ) , r = | x − y | , (2.12)without any pre-factor. Here j is the spherical Bessel function of order 0, see [35]. The scalarperturbation amplitude A s and the scalar spectral index n s are defined by P R ( k ) = A s ( k/k ∗ ) n s − , (2.13)– 3 –here k ∗ is called the pivot scale. For numerical examples in this paper we shall use the Planckvalues for k ∗ = 0 . / Mpc, log(10 A s ) = 3 . , n s = 0 . . (2.14)Inserting X ( α , z ) = X ( r ( z )( e z + α ) , z ) in (2.9) we find X ( α , z ) = 1(2 π ) (cid:90) d ke − ir ( z )( e z + α ) · k T X ( k, z ) R ( k ) . (2.15)Comparing this with (2.7) and identifying k = k (cid:107) e z + (cid:96) /r ( z ) = k (cid:107) e z + k ⊥ , we find a X ( (cid:96) , z ) = 1(2 π ) (cid:90) d ke − ir ( z )( e z + α ) · k T X ( k, z ) R ( k ) δ ( (cid:96) − r ( z ) k ⊥ ) (2.16)= 1(2 π ) r ( z ) (cid:90) ∞−∞ dk (cid:107) e − ir ( z ) k (cid:107) T X ( k, z ) R ( k ) (2.17)where k = k (cid:107) e z + (cid:96) /r ( z ) and k = (cid:113) k (cid:107) + (cid:96) /r ( z ) . Let us first correlate two variables X and Y at the same redshift, a X ( (cid:96) , z ) and a Y ∗ ( (cid:96) (cid:48) , z ). Using(2.16) we obtain (cid:104) a X ( (cid:96) , z ) a Y ∗ ( (cid:96) (cid:48) , z ) (cid:105) = 12 π (cid:90) d kP R ( k ) δ ( (cid:96) − r ( z ) k ⊥ ) δ ( (cid:96) (cid:48) − r ( z ) k ⊥ ) T X ( k, z ) T ∗ Y ( k, z ) (2.18)= δ ( (cid:96) − (cid:96) (cid:48) ) 1 πr ( z ) (cid:90) ∞ dk (cid:107) P R ( k ) T X ( k, z ) T ∗ Y ( k, z ) , (2.19) C XY(cid:96) ( z, z ) = 1 πr ( z ) (cid:90) ∞ dk (cid:107) P R ( k ) T X ( k, z ) T ∗ Y ( k, z ) . (2.20)The situation is more complicated when we consider different redshifts z (cid:54) = z (cid:48) , and we defer adiscussion of this case to Section 2.3.To determine the number counts we have to apply our formalism to the density fluctuationin comoving gauge, D , the the radial component of the velocity, V r and to the Weyl potential,Ψ W = (Ψ + Φ) /
2. The latter then has to be integrated over the lightcone in order to obtain κ : a D ( (cid:96) , z ) = (cid:90) d k (2 π ) δ ( (cid:96) − r ( z ) k ⊥ ) e − ik (cid:107) r ( z ) T D ( k, z ) R ( k ) (2.21) a rsd ( (cid:96) , z ) = H − (cid:90) d k (2 π ) δ ( (cid:96) − r ( z ) k ⊥ ) e − ik (cid:107) r ( z ) k (cid:107) k T V ( k, z ) R ( k ) , (2.22) a κ ( (cid:96) , z ) = 2(1 − γ ( z )) (cid:96) (cid:90) d k (2 π ) (cid:90) r ( z )0 dr r ( z ) − rr ( z ) r δ ( (cid:96) − r ( z ) k ⊥ ) e − ik (cid:107) r T Ψ W ( k, z ( r )) R ( k ) . (2.23)For Eq. (2.22) we have used that V ( k , z ) = i ˆ k T V ( k, z ) R ( k ), hence ( ∂ r V r )( k ) = ( k (cid:107) /k ) T V ( k, z ) R ( k ).In (2.23), z ( r ) is the redshift of the comoving distance r , i.e. r ( z ( r )) ≡ r .We now consider each term, first by itself and then its correlation with the other contributions.To compute a spectrum numerically we use (2.14) for P R ( k ) and employ the numerical transferfunction from class for the variable X . The remaining integral (2.20) is then computed with asimple Python code. For the lensing term this would lead to triple or double integrals and wetherefore perform additional simplifications, as set out in Section 2.2.3 below.– 4 – .2.1 Density Fluctuations Figure 1 . Left: the density in the flat sky approximation (solid) compared to the class result (dashed)at redshifts z = 1 , , , In Fig. 1 we plot the density power spectrum at redshifts z = 1 , , ,
5. The solid linespresent our approximation (2.20) with X = Y = D while the dashed line are the numerical resultfrom class . The agreement is clearly excellent. To be more quantitative we also show the relativedifferences in the right panel. The difference is about 0.4% for z = 5, about 0.1% for z = 1 andtypically 0.2% for 1 < z ≤
3. Note that 0.1% is roughly the accuracy of the class code itself,hence our agreement is as good as we can expect.
Here we repeat the same analysis for the redshift space distortion, where we have X ( z, k ) = k (cid:107) V r / H = k H ( z ) (cid:18) k (cid:107) k (cid:19) T V ( k, z ) R ( k ) . (2.24)In Fig. 2 we plot the RSD power spectrum at redshifts z = 1 , , ,
5. The solid lines presentour approximation (2.20) with X = Y = rsd while the dashed line are the numerical result from class . Again we see very good agreement with the numerical result from class . At low (cid:96) theapproximation degrades somewhat, but the error is never larger than about 1%. In general thedifference is 0.5% or less.Again at high redshift our result is slightly below the class value while at low redshift it isslightly above. – 5 – igure 2 . Left: the RSD term in the flat sky approximation (solid) compared to the class result (dashed)at redshifts z = 1 , , , To obtain the lensing term we write the flat sky approximation as (cid:104) a κ ( (cid:96) , z ) a κ ∗ ( (cid:96) (cid:48) , z (cid:48) ) (cid:105) = δ ( (cid:96) − (cid:96) (cid:48) ) 2 (cid:96) π (cid:90) ∞ dk (cid:107) (cid:90) r ( z )0 dr (cid:90) r ( z (cid:48) )0 dr (cid:48) ( r ( z ) − r )( r ( z (cid:48) ) − r (cid:48) ) r ( z ) r ( z (cid:48) )( rr (cid:48) ) e ik (cid:107) ( r − r (cid:48) ) P R ( k ) × T Ψ W ( k, z ( r )) T ∗ Ψ W ( k, z ( r (cid:48) )) . (2.25)Eq. (2.25) is a triple integral of a rapidly oscillating function and hence very time-consuming numer-ically. To simplify it, we integrate (2.25) over k (cid:107) neglecting the dependence of the transfer functionsand the power spectrum on k (cid:107) , i.e., simply setting k (cid:107) = 0 in the expression k = (cid:113) k (cid:107) + ( (cid:96) /r ) .The integral of the exponential over k (cid:107) then yields a δ -function in the resulting expression. Thiscorresponds to setting (cid:90) dk (cid:107) f ( r (cid:48) , r, k ) exp( ik (cid:107) ( r − r (cid:48) )) (cid:39) πf ( r, r, | k ⊥ | ) δ ( r − r (cid:48) )which is a good approximation for a slowly varying function f ( r, r (cid:48) , k )and for (cid:96)/r (cid:29) k (cid:107) . Theintegral over r (cid:48) then simply eliminates the δ -function and we obtain C κ(cid:96) ( z, z (cid:48) ) = 4 (cid:96) (cid:90) r min dr ( r ( z ) − r )( r ( z (cid:48) ) − r ) r ( z ) r ( z (cid:48) ) r P R ( k ) | T Ψ W ( k, z ( r )) | , (2.26)where now k = (cid:96)/r and r min = min { r ( z ) , r ( z (cid:48) ) } . As we shall see in Section 3, (2.26) simplycorresponds to the Limber approximation [36, 37] which is often used for lensing. Eq. (2.26) is asingle integral of a positive definite slowly-varying function which does not pose any problem andcan be calculated with a simple Python code. – 6 – igure 3 . Left: the lensing term in the flat sky approximation (solid) compared to the class result(dashed) at redshifts z = 1 , , , In Fig. 3 we show C κ(cid:96) ( z, z ) for the redshifts z = 1 , , ,
5. Our approximation (2.26) isexcellent for (cid:96) >
20 where the relative difference are typically about 0.5% and only for z = 1 largerthan 1%. The same is true for unequal redshifts which are shown in Fig. 6. The agreement is badat low (cid:96) ≤
10, but this is not so surprising as for small (cid:96) , neglecting k (cid:107) wrt. k ⊥ = (cid:96)/r is certainlynot a good approximation. For z = 1 the error also raises above 1% for 20 < (cid:96) <
50. But lensingfrom z = 1 is very subdominant (compare the amplitudes in Figs. 1 and3) so that this error doesnot contribute significantly to the total error budget.Even though this approximation is much better than the flat sky approximation for unequalredshifts which we discuss below, it cannot capture the behavior of the lensing term at low (cid:96) .Nevertheless, we shall see that the low (cid:96) contribution from the lensing terms is subdominant sothat we can still achieve a good approximation for the power spectrum of the full number count C ∆ (cid:96) ( z, z ). Note that while density and RSD decrease with increasing redshift, the situation isreversed for the integrated lensing term. While at z = 1, the lensing is about 100 times smallerthan the density term, at redshift z = 5 it is only about twice smaller at low (cid:96) . The number count expression (2.1) implies that the full number count power spectrum is given by C ∆ (cid:96) ( z, z (cid:48) ) = b ( z ) b ( z (cid:48) ) C D(cid:96) ( z, z (cid:48) ) + b ( z ) C D, rsd (cid:96) ( z, z (cid:48) ) + b ( z (cid:48) ) C rsd ,D(cid:96) ( z, z (cid:48) ) + C rsd (cid:96) ( z, z (cid:48) )+ b ( z )(1 − γ ( z (cid:48) )) C D,κ(cid:96) ( z, z (cid:48) ) + (1 − γ ( z )) b ( z (cid:48) ) C κ,D(cid:96) ( z, z (cid:48) ) + (1 − γ ( z (cid:48) )) C rsd ,κ(cid:96) ( z, z (cid:48) )+(1 − γ ( z )) C κ, rsd (cid:96) ( z, z (cid:48) ) + (1 − γ ( z ))(1 − γ ( z (cid:48) )) C κ(cid:96) ( z, z (cid:48) ) (2.27)For simplicity and in order to be as model independent as possible, we set b ( z ) = 1 and γ ( z ) = 0 inthe following, but they are easily re-introduced for any specific example. In this section we shownumerical examples of the correlation spectra, C X,Y(cid:96) ( z, z ) for X (cid:54) = Y . As a consequence of the– 7 –ontinuity equation, the velocity transfer function is very simply related to the density transferfunction via T V ( k, z ) = f ( z ) T D ( k, z ) , f ( z ) = − d log D ( z ) d log(1 + z ) , (2.28)where D ( z ) is the growth function of density perturbations. In a matter-dominated universe D ∝ / (1 + z ), while during dark energy domination D grows slower and tends to a constant asΩ m →
0. With this the correlation C D, rsd (cid:96) ( z, z ) simply becomes C D, rsd (cid:96) ( z, z ) (cid:39) f ( z ) πr ( z ) H ( z (cid:48) ) (cid:90) ∞ dk (cid:107) k (cid:107) k P R ( k ) T D ( k, z ) T ∗ D ( k, z ) . (2.29) Figure 4 . Left: the density-RSD cross spectrum in the flat sky approximation (solid) compared to the class result (dashed) at redshifts z = 1 , , , In Figure 4 we show some examples of the density-RSD cross-correlation spectrum for equalredshifts. Not surprisingly, the errors are like the ones for density or RSD terms, i.e. for (cid:96) ≥ (cid:96) . More precisely, the largest error at (cid:96) = 2 and z = 5 is 0.7%.Let us consider the lensing-density cross-correlation next. Inserting (2.21) and (2.23) in (2.20)we obtain C D,κ(cid:96) ( z, z (cid:48) ) (cid:39) − (cid:96) π (cid:90) ∞−∞ dk (cid:107) (cid:90) r ( z (cid:48) )0 dr r ( z (cid:48) ) − rr ( z (cid:48) ) rR P R ( k ) T D ( k, z ) T ∗ Ψ W ( k, z ( r )) exp( ik (cid:107) ( r ( z ) − r )) . (2.30)Here R = (cid:112) r ( z ) r , hence we cannot take P R ( k ) in front of the r -integration, as k = (cid:113) k (cid:107) + (cid:96) /R .We perform the same simplification as in the lensing integral. We neglect the dependence on k (cid:107) in k and integrate the exponential over k (cid:107) which then yields 2 πδ ( r ( z ) − r ) so that we end up with C D,κ(cid:96) ( z, z (cid:48) ) (cid:39) (cid:40) − (cid:96) r ( z (cid:48) ) − r ( z ) r ( z (cid:48) ) r ( z ) P R ( (cid:96)/r ( z )) T D ( (cid:96)/r ( z ) , z ) T ∗ Ψ W ( (cid:96)/r ( z ) , z ) if z < z (cid:48) z ≥ z (cid:48) . (2.31)– 8 –hus, for equal redshifts in this approximation, the contribution from the lensing-densitycross-correlation vanishes. This is well-justified as the spectrum for this term is between two andthree orders of magnitude smaller than the density or RSD autocorrelations individually, and cansafely be neglected for equal redshifts. Figure 5 . Left: the full power spectrum, C ∆ (cid:96) ( z, z ) in the flat sky approximation (solid) compared to the class result (dashed) at redshifts z = 1 , , , This same approximation, due to the factor k (cid:107) in the transfer function, yields C rsd ,κ(cid:96) ( z, z (cid:48) ) (cid:39) . (2.32)In Figure 5 we show the total power spectrum result. The precision is always better than about0.5%. Hence, neglecting the lensing-density and lensing-RSD terms does not degrade the accuracy.At the highest redshift, z = 5, lensing contributes about 10% to the total result, while at z = 1 itdrops to about 0.1% and thus below the accuracy of our approximation. Let us also consider unequal redshifts. For lensing and lensing-density cross-correlations we simplyuse approximations (2.26) and (2.31). The results are shown in Figs. 6 and 7. While the lensing-lensing term is growing with redshift, the amplitude of the density-lensing term is more complex:the density decreases with increasing redshift while lensing increases. These two competing effectslead to a maximal signal (in amplitude, the sign of this term is always negative) at ( z, z (cid:48) ) = (1 , . z, z (cid:48) ) = (3 ,
4) the signal is smallest, while ( z, z (cid:48) ) = (1 , .
1) yields the second smallest signal(for (cid:96) >
40) and ( z, z (cid:48) ) = (2 ,
3) is the second largest signal. More precisely, the signals for( z, z (cid:48) ) = (1 , .
5) and ( z, z (cid:48) ) = (2 ,
3) cross at (cid:96) (cid:39)
30. Note also that for z ≥
2, the positive lensing-lensing term always dominates over the negative density-lensing term, while for ( z, z (cid:48) ) = (1 , . (cid:96) (cid:38)
100 and for ( z, z (cid:48) ) = (1 , .
1) it dominates already for (cid:96) (cid:38)
60. This can be seen in Figure 8, which shows the sum of these two terms.– 9 –he precision of the lensing-lensing approximation at different redshifts is as good as the oneat equal redshifts, namely on the order of 1% for 10 ≤ (cid:96) ≤
100 and around 0 .
5% for (cid:96) > z ≥ (cid:96) . Figure 6 . Left: the lensing term in the flat sky approximation (solid) compared to the class result(dashed) at redshifts ( z, z (cid:48) ) = (1 , . z, z (cid:48) ) = (1 , . z, z (cid:48) ) = (1 , z, z (cid:48) ) = (3 ,
4) from bottom to top.Right: the relative differences.
Figure 7 . Left: the density-lensing cross spectrum in our approximation (solid) compared to the class result (dashed) at redshifts ( z, z (cid:48) ) = (1 , . , ( z, z (cid:48) ) = (1 , . , ( z, z (cid:48) ) = (1 , , ( z, z (cid:48) ) = (3 ,
4) from top tobottom. Right: the relative differences.
The density-lensing cross-correlations are less accurate than the pure lensing term, at low (cid:96) .The reason here is that Limber approximation is less accurate for this case. This is especially true– 10 – igure 8 . The sum of the lensing-lensing and density-lensing cross spectra as produced by class (solid)at redshifts ( z, z (cid:48) ) = (1 , . , ( z, z (cid:48) ) = (1 , . , ( z, z (cid:48) ) = (1 , , ( z, z (cid:48) ) = (3 , at low redshift, z = 1 where for (cid:96) <
60 the error is larger than 1% and it exceeds 4% for (cid:96) < (cid:96) >
50. For (cid:96) >
50, the density-lensing cross-correlation is as accurate as the pure lensing term. ComparingFigs 6 and 7 , we see that the pure lensing term actually dominates in all the cases presented inthese figures. Only at very low redshift, z <
1, does the density-lensing term become larger. Wehave checked that the RSD-lensing term always remains very subdominant and we neglect it inour approximation.Note also that these unequal-redshift correlations are always at least one or two orders ofmagnitude smaller than the full equal redshift result, which is dominated by the density and RSDterms. Therefore, the signal to noise of individual unequal-redshift terms is always very small.On the other hand, their number scales quadratic with the number of redshift bins and they canbecome relevant if we consider more than 10 bins. In Fig. 8 we plot the sum of the two lensingterms. Since the lensing-lensing term is positive while the lensing density term is negative, therecan be significant cancellation. Especially, for ( z, z (cid:48) ) = (1 , . (cid:96) > z, z (cid:48) ) = (2 ,
3) the same cancellation is effective for (cid:96) > r α − r (cid:48) α (cid:48) where r = r ( z ) and r (cid:48) = r ( z (cid:48) ). For z (cid:54) = z (cid:48) thisis not proportional to α − α (cid:48) . Therefore, upon Fourier transforming, we will not obtain a deltafunction δ ( (cid:96) − (cid:96) (cid:48) ), since we break the flat sky analog of statistical isotropy. The reason for this isthat in principle we now consider correlations of functions that live on two different skies: one atcomoving distance r ( z ) and the other at r ( z (cid:48) ). In order to restore statistical isotropy we have to– 11 –roject them onto one sky at some fiducial common distance R .To do this we introduce the angles ˜ α = α r/R and ˜ α (cid:48) = α (cid:48) r (cid:48) /R , so that r α − r (cid:48) α (cid:48) = R ( ˜ α − ˜ α (cid:48) ).Isotropy is now equivalent to translation invariance in the ˜ α plane. Furthermore, we can write a X ( (cid:96) , z ) (cid:39) π (cid:90) d ˜ αe − i (cid:96) · ˜ α X ( ˜ α , z ) (2.33)= (cid:90) d ˜ α (cid:90) d k (2 π ) e i ( (cid:96) − R k ⊥ ) · ˜ α e − ik (cid:107) r ( z ) T X ( k, z ) R ( k ) . (2.34)The integration of d ˜ α just generates a Dirac- δ function (times (2 π ) ) so that a X ( (cid:96) , z ) = (cid:90) d k (2 π ) δ ( (cid:96) − R k ⊥ ) e − ik (cid:107) r ( z ) T X ( k, z ) R ( k ) . (2.35)This corresponds to Eq. (2.16) except that now, in the Dirac- δ function, r ( z ) is replaced by R .Correlating two variables X and Y at redshifts z and z (cid:48) now yields (cid:104) a X ( (cid:96) , z ) a Y ∗ ( (cid:96) (cid:48) , z (cid:48) ) (cid:105) = 12 π (cid:90) d kP R ( k ) δ ( (cid:96) − R k ⊥ ) δ ( (cid:96) (cid:48) − R k ⊥ ) e − ik (cid:107) ( r − r (cid:48) ) × T X ( k, z ) T ∗ Y ( k, z (cid:48) ) (2.36)= δ ( (cid:96) − (cid:96) (cid:48) ) 1 πR (cid:90) ∞ dk (cid:107) P R ( k ) T X ( k, z ) T ∗ Y ( k, z (cid:48) ) cos( k (cid:107) ( r − r (cid:48) )) , (2.37)where k = (cid:113) (cid:96) /R + k (cid:107) . The power spectrum at unequal redshift is therefore given by C XY(cid:96) ( z, z (cid:48) ) (cid:39) πR (cid:90) ∞ dk (cid:107) P R ( k ) T X ( k, z ) T ∗ X ( k, z (cid:48) ) cos( k (cid:107) ( r − r (cid:48) )) (2.38)= 2 πR (cid:90) ∞ dk (cid:107) k P R ( k ) T X ( k, z ) T ∗ X ( k, z (cid:48) ) cos( k (cid:107) ( r − r (cid:48) )) . (2.39)This expression has two problems. First of all, the result depends on the choice of R via k andvia the pre-factor 1 /R . If z = z (cid:48) , we can simply choose R = r ( z ) which is the true physicaldistance of the flat sky. However, for z (cid:54) = z (cid:48) , there are different possibilities. The simplest choiceis R = √ rr (cid:48) . We shall see below that this is also the choice motivated by the exact expression.The second problem is that the integrand is now rapidly oscillating and, in what concerns thenumerical computation, nothing is really gained wrt. the exact calculation performed by class .Therefore, let us go back and consider a term on the surface at redshift z , for example thedensity perturbation. Writing the exponential as a sum of Legendre polynomials and sphericalBessel functions it is easy to obtain the exact standard result [13] for two local (not integrated)variables X and Y :Replacing in (2.15) e z + α = n by n and expanding the exponential in Legendre polynomialsand spherical Bessel functions, we obtain the exact expression X ( n , z ) = (cid:90) d k (2 π ) e − ir ( z ) n · k T X ( k, z ) R ( k ) (2.40)= (cid:88) (cid:96) i (cid:96) (2 (cid:96) + 1) (cid:90) d k (2 π ) P (cid:96) ( n · ˆ k ) j (cid:96) ( kr ( z )) T X ( k, z ) R ( k ) , (2.41)– 12 –here P (cid:96) is the Legendre polynomial of degree (cid:96) . Applying the addition theorem for sphericalharmonics, we find for the correlation function (cid:104) X ( n , z ) Y ( n (cid:48) , z (cid:48) ) (cid:105) = 12 π (cid:88) (cid:96) (2 (cid:96) + 1) P (cid:96) ( n · n (cid:48) ) (cid:90) ∞ dkk j (cid:96) ( kr ( z )) j (cid:96) ( kr ( z (cid:48) )) T X ( k, z ) T ∗ Y ( k, z (cid:48) ) P R ( k ) , so that C X,Y(cid:96) ( z, z (cid:48) ) = 4 π (cid:90) ∞ dkk j (cid:96) ( kr ( z )) j (cid:96) ( kr ( z (cid:48) )) T X ( k, z ) T ∗ Y ( k, z (cid:48) ) P R ( k ) . (2.42)For the last equation we used (2.11) and the fact that the correlation function is related to thepower spectrum by (cid:104) X ( n , z ) Y ( n (cid:48) , z (cid:48) ) (cid:105) = 14 π (cid:88) (2 (cid:96) + 1) P (cid:96) ( n · n (cid:48) ) C X,Y(cid:96) ( z, z (cid:48) ) . (2.43)So far, no approximation has been made and class actually calculates the power spectra usingexpression (2.42).We now use the following approximation for the Bessel functions, see [38]: j (cid:96) ( x ) (cid:39) (cid:40) , x < L , L = (cid:96) + 1 / cos [ √ x − L − L arccos ( Lx ) − π/ ] √ x ( x − L ) / , x > L (2.44)This approximation has a singularity at x → L , but is an excellent approximation for x (cid:38) L + 1.Inserting it into (2.42) we obtain C XY(cid:96) ( z, z (cid:48) ) (cid:39) πrr (cid:48) (cid:90) ∞ (cid:96) +1 / r min dkk (cid:113) k (cid:107) k (cid:48)(cid:107) P R ( k ) T X ( k, z ) T ∗ Y ( k, z (cid:48) ) cos (cid:18) rk (cid:107) − rk ⊥ arccos (cid:18) k ⊥ k (cid:19) − π/ (cid:19) × cos (cid:18) r (cid:48) k (cid:48)(cid:107) − r (cid:48) k (cid:48)⊥ arccos (cid:18) k (cid:48)⊥ k (cid:19) − π/ (cid:19) (2.45)= 2 πrr (cid:48) (cid:90) ∞ (cid:96) +1 / r min dkk (cid:113) k (cid:107) k (cid:48)(cid:107) P R ( k ) T X ( k, z ) T ∗ Y ( k, z (cid:48) ) (cid:40) cos (cid:34) rk (cid:107) − r (cid:48) k (cid:48)(cid:107) − rk ⊥ (cid:18) arccos (cid:18) k ⊥ k (cid:19) − arccos (cid:18) k (cid:48)⊥ k (cid:19)(cid:19) (cid:35) + sin (cid:34) rk (cid:107) + r (cid:48) k (cid:48)(cid:107) − rk ⊥ (cid:18) arccos (cid:18) k ⊥ k (cid:19) + arccos (cid:18) k (cid:48)⊥ k (cid:19)(cid:19) (cid:35)(cid:41) . (2.46)Here r min = min { r, r (cid:48) } and we have defined k ⊥ = (cid:96) + 1 / r , k (cid:48)⊥ = (cid:96) + 1 / r (cid:48) , k (cid:107) = (cid:113) k − k ⊥ , k (cid:48)(cid:107) = (cid:113) k − k (cid:48) ⊥ . (2.47)Note rk ⊥ = r (cid:48) k (cid:48)⊥ = (cid:96) + 1 /
2, but rk (cid:107) (cid:54) = r (cid:48) k (cid:48)(cid:107) . For z = z (cid:48) , after replacing (cid:96) → (cid:96) + 1 /
2, neglecting therapidly oscillating sin-term and making the variable transform kdk = k (cid:107) dk (cid:107) , this becomes exactlyour flat sky approximation (2.20). – 13 –owever, when z (cid:54) = z (cid:48) , the case of interest here, the original flat sky approximation (2.38)for z (cid:54) = z (cid:48) is obtained for R = rr (cid:48) only if we neglect the sin-term, set k (cid:48)(cid:107) = k (cid:107) and drop fullythe contributions in the argument of the cos which are proportional to rk ⊥ which corresponds tosetting k ⊥ = k (cid:48)⊥ . We have found that while keeping the sin term is not crucial, the differencesbetween k (cid:48)(cid:107) and k (cid:107) , as well as between k ⊥ and k (cid:48)⊥ , are.For redshifts that are sufficiently well-separated, terms of the above form become very small.In this case the spectrum is dominated by the integrated lensing and lensing × density terms whichwe can compute with Eqs. (2.31) and (2.26), while neglecting the contributions from the localterms. However, if the redshifts are fairly close, the local terms like D × D cannot be neglected,and we must use Eq. (2.46) to calculate them. Henceforth we refer to the collection of the terms( C D(cid:96) , C rsd (cid:96) , C D, rsd (cid:96) ) as the standard terms. We want to estimate their contribution for small redshiftdifferences and also the redshift difference above which they can be neglected with respect to thelensing and lensing × density contributions.For this we examine (2.46) in the particular case when r and r (cid:48) are close, r (cid:48) = r (1 + (cid:15) ) 0 < (cid:15) (cid:28) . (2.48)Without loss of generality we assume r < r (cid:48) and hence (cid:15) >
0. Expanding the argument of thecos-term in (2.46), let us call it a − , to second order in (cid:15) we find a − = rk (cid:107) − r (cid:48) k (cid:48)(cid:107) − rk ⊥ (cid:18) arccos (cid:18) k ⊥ k (cid:19) − arccos (cid:18) k (cid:48)⊥ k (cid:19)(cid:19) (cid:39) − (cid:15)rk (cid:107) − (cid:15) k ⊥ rk (cid:107) + O ( (cid:15) ) . (2.49)Considering only the term ∝ (cid:15) results exactly in approximation (2.38) . Expanding the argumentof the sin-term, let us call it a + , to second order in (cid:15) we find a + = rk (cid:107) + r (cid:48) k (cid:48)(cid:107) − rk ⊥ (cid:18) arccos (cid:18) k ⊥ k (cid:19) + arccos (cid:18) k (cid:48)⊥ k (cid:19)(cid:19) (cid:39) rk (cid:107) − (2 (cid:96) + 1) arccos (cid:18) k ⊥ k (cid:19) + (cid:15)rk (cid:107) + (cid:15) k ⊥ rk (cid:107) + O ( (cid:15) ) . (2.50)The sin-term oscillates with a frequency of the order of 2 (cid:96) which is very rapid and we can neglect itwhen k (cid:107) is not very small so that k > k ⊥ . The cos term oscillates slowly for small (cid:15) and we shouldtake it into account. Nevertheless, our expansion in (cid:15) cannot be trusted in the regime of verysmall k (cid:107) since the (cid:15) terms diverge when k (cid:107) →
0. Furthermore, in this limit the term multipliedby (2 (cid:96) + 1) in the sin tends to arccos(1) = 0 and is not rapidly oscillating anymore. Before wecan believe the approximation we therefore must request that the higher order terms in (cid:15) becomesmaller as the expansion proceeds. The series expansion is of the general form a − = x (cid:107) L ∞ (cid:88) n =1 α n (cid:34)(cid:32) L x (cid:107) (cid:33) (1 + O ( x (cid:107) /L )) (cid:15) (cid:35) n (2.51) a + = 2 x (cid:107) + x (cid:107) L ∞ (cid:88) n =1 β n (cid:34)(cid:32) L x (cid:107) (cid:33) (1 + O ( x (cid:107) /L )) (cid:15) (cid:35) n (2.52)with coefficients | α n | ≤ | β n | ≤
1. Here we have introduced x = kr , x (cid:48) = kr (cid:48) = x (1 + (cid:15) ), x ⊥ = k ⊥ r = k (cid:48)⊥ r (cid:48) = x (cid:48)⊥ = (cid:96) + 1 / ≡ L . Note that x (cid:48)(cid:107) = (cid:113) x (cid:107) (1 + (cid:15) ) + (2 (cid:15) + (cid:15) ) L . (2.53)– 14 –he terms in square brackets in the series are small when L x (cid:107) (cid:15) (cid:28) x (cid:107) →
0. Wetherefore must request that x (cid:107) > L √ (cid:15) at least, for these series to converge. On the other hand,even when x (cid:107) = L √ (cid:15) , for small L the arguments become a − ( x (cid:107) = L √ (cid:15) ) ∼ L √ (cid:15) (1 + O ( (cid:15) )) (2.54) a + ( x (cid:107) = L √ (cid:15) ) ∼ L √ (cid:15) (1 + O ( (cid:15) )) , (2.55)which may still be smaller than 1, especially for a − . We can neglect the integrals only when theystart oscillating (rapidly). For x (cid:107) > /(cid:15) we find a − ( x (cid:107) = 1 /(cid:15) ) ∼ − − L (cid:15) O ( (cid:15) )) while a + ( x (cid:107) = 1 /(cid:15) ) ∼ (cid:15) (1 + O ( L(cid:15) )) . (2.56)Hence, we choose an x max (cid:107) ≡ × max { L √ (cid:15), /(cid:15) } above which both arguments of the cosine canbecome large and the contributions can be neglected.Numerical testing shows that while the second (the sin) term does improve the overall am-plitude of the approximate spectrum somewhat (largest effect for large epsilon) the integrationintroduces additional oscillations in (cid:96) , and increases the calculation time. Therefore we neglect thesecond term and examine the resulting approximation. Our the final expression for the approxi-mation is: C XY(cid:96) ( z, z (cid:48) ) (cid:39) πr r (cid:48) (cid:40) (cid:90) x max (cid:107) dx (cid:107) T ( k, z, z (cid:48) ) √ x (cid:107) x (cid:113) x (cid:48)(cid:107) × cos (cid:20) x (cid:107) − x (cid:48)(cid:107) − ( (cid:96) + 1 / (cid:18) arccos (cid:18) (cid:96) + 1 / x (cid:19) − arccos (cid:18) (cid:96) + 1 / x (cid:48) (cid:19)(cid:19)(cid:21) (cid:41) , (2.57)where T ( k, z, z (cid:48) ) = P R ( k ) T X ( k, z ) T ∗ Y ( k, z (cid:48) ).Finally, one can convert (cid:15) into the redshift difference ∆ z = z (cid:48) − z . At first order (cid:15) and ∆ z are related by r (cid:48) = r (1 + (cid:15) ) = r + drdz ∆ z = r + 1 H ( z ) ∆ z or (cid:15) = ∆ zr ( z ) H ( z ) . (2.58)For C XY(cid:96) ( z, z (cid:48) ) we have also converted the integral over x = rk into an integral over x (cid:107) using x (cid:107) dx (cid:107) = xdx .In Fig. 9 we plot the power spectrum of the standard terms for five different redshift pairs( z, z (cid:48) ). The solid lines present our approximation (2.57) with X = Y = D + ∂ r V r / H . Clearly theaccuracy is much worse than for equal redshifts, when the redshift difference is not very small, butnote also that the amplitude is small, especially for the problematic terms with the two largestredshift separations. As we argue below, for significant redshift differences the unequal redshiftterms are largely dominated by the lensing terms. We use adjusted relative differences to determinethe error. These are given by ( C (cid:96), app − C (cid:96), ex ) / ¯ C (cid:96), ex , where the (cid:96) -band average ¯ C (cid:96), ex is defined by¯ C (cid:96), ex = (cid:113) (101) − (cid:80) (cid:96) +50 (cid:96) − ( C (cid:96), ex ) for (cid:96) >
50, and for smaller (cid:96) ’s as many neighbouring points as areavailable are used. This effectively removes large spikes in the errors caused by zero-crossings of thespectra. The redshift pairs are chosen as follows: The (blue) pair ( z, z (cid:48) ) = (1 , . igure 9 . Left: the standard terms (density and RSD) in the flat sky approximation (solid) comparedto the class result (dashed) at redshifts ( z, z (cid:48) ) = (1 , . z, z (cid:48) ) = (2 , . z, z (cid:48) ) = (2 , . z, z (cid:48) ) =(2 , z, z (cid:48) ) = (5 , . z, z (cid:48) ) = (2 , . z, z (cid:48) ) = (2 ,
3) have been enhanced by a factor of 50 to facilitate interpretation of the figure. Right: theadjusted relative differences in %. For the redshifts ( z, z (cid:48) ) = (2 , .
2) and ( z, z (cid:48) ) = (2 ,
3) the error has beenreduced by a factor 10 − to make it into the plot. The true maximum error is therefore more than a factorof 1000. ( z, z (cid:48) ) = (2 , . z, z (cid:48) ) = (5 , . z, z (cid:48) ) = (2 , .
2) and ( z, z (cid:48) ) = (2 ,
3) show theinvalidity of the approximation when the separation in redshift increases. However, the signal atthese redshift differences is also very small, which makes this problem less relevant. In the figurethe signal is enhanced by a factor of 50 for better visibility, while the error is reduced by a factor10 , so that the maxima at 40% indicate an error of a factor of 4000. Our approximation has highand low frequency oscillations in (cid:96) which are not present in the standard result. If one averagesthe approximation over a rather large band of ∆ (cid:96) ∼ | z − z (cid:48) | = ∆ z > ∆ z (cid:39) . r ( z ) H ( z )1 + z and (cid:96) ∼ (cid:96) (cid:38)
20, but there are exceptions as we shall discuss.The approximation for the standard terms at unequal redshifts is only valid (to within 10%)for small redshift separations. We find that this second critical value, below which the redshift will– 16 –ave at most 10% error, follows a power law: ∆ z (cid:39) . × − (1 + z ) . .In Fig. 10 we show these critical separations ∆ z and ∆ z as functions of redshift. Figure 10 . The critical redshift differences, ∆ z (cid:39) . r ( z ) H ( z ) / (1 + z ) (orange) and ∆ z = 3 . × − (1 + z ) . (blue) respectively showing the values: above which the standard terms can be neglected(for mean values between (cid:96) = 90 and (cid:96) = 110, ¯ C ST D(cid:96) = 1% ¯ C T OT(cid:96) ) and below which the approximation isaccurate to 10% or less, in the unequal time correlators.
For redshift differences larger than ∆ z we neglect the standard contribution, as we can againobtain a very good accuracy (1% or better) when including only the lensing terms. However, ourapproximation for the standard terms can only be trusted for redshift differences smaller than∆ z , where it is accurate to 10% at least for z (cid:54) = z (cid:48) and better than 0.5% for z = z (cid:48) . Therefore,for redshift differences | z − z (cid:48) | ∈ [∆ z , ∆ z ] we have no satisfactory approximation and these casesneed to be computed with the class (or CAMB) code.In Fig. 11 we show the total power spectrum for unequal redshifts. Again, the adjustedrelative differences are used, and the redshift pairs are chosen to illustrate: the accuracy of theapproximation in the limit approaching equal redshifts with separations close to ∆ z (( z, z (cid:48) ) =(1 , . z, z (cid:48) ) = (2 , . z, z (cid:48) ) = (5 , . z > ∆ z (( z, z (cid:48) ) = (2 ,
3) and ( z, z (cid:48) ) = (3 , − − z (2) and ∆ z (3) correspondingly, we neglect the contribution from the standardterms which is inaccurate in this regime. At sufficiently small separations (the blue, orange andpurple curves), the additional contribution of the lensing terms is small but the error, mainly dueto the standard term, remains below 10%. For large separations, on the other hand, we neglectthe standard terms and include only the approximation of the remaining lensing terms. Theapproximation for ( z, z (cid:48) ) = (3 ,
4) is very good, especially above (cid:96) ∼
30. However for ( z, z (cid:48) ) = (2 , (cid:96) > igure 11 . Left: the total power spectrum result in the flat sky approximation (solid) compared to the class result (dashed) at redshifts ( z, z (cid:48) ) = (1 , . z, z (cid:48) ) = (2 , . z, z (cid:48) ) = (2 , . z, z (cid:48) ) = (2 , z, z (cid:48) ) = (5 , . z, z (cid:48) ) = (2 , .
2) and( z, z (cid:48) ) = (2 ,
3) have been enhanced by a factor of 300 to facilitate interpretation of the figure. Right: theadjusted relative differences. nearly cancel for (cid:96) >
200 (this can be seen in Fig. 8) so that the standard terms which we neglecthere contribute up to 15%. We have checked this fact with class which produces the samedifference when neglecting the standard terms for this redshift pair. For higher redshifts thepositive lensing-lensing term dominates, while for lower redshifts the negative density-lensing termdominates. In the pair (3 , z (3) (cid:39) .
5, the lensing-lensing andlensing-density contributions for this redshift pair no longer cancel. The lensing-lensing term istruly the dominant contribution. As a result we see that, while the error increases for low (cid:96) , itremains on the order of a few percent for all (cid:96) >
50, and does not exhibit the particular structurethat we see for the redshift pair ( z, z (cid:48) ) = (2 , , . z (2) (cid:39) . z (2) (cid:39) .
03. For this case the error is very large at low (cid:96) butreaches a level below 5% for (cid:96) > (cid:96) the negative contribution of the standard termsto the total spectrum cannot be neglected, but is also not well modelled by our approximation. Ifsuch cases are relevant, they have to be computed with class . As the separation increases evenfurther (above the critical separation), the lensing terms, which are well approximated, make up asufficiently significant portion of the total spectrum such that the standard terms can be neglectedand the results has an error below 5%. This is true also for ( z, z (cid:48) ) = (2 , .
2) and (cid:96) > z < ∆ z < ∆ z (e.g. the example not plotted, with ( z, z (cid:48) ) = (2 , .
2) while ∆ z (2) (cid:39) . z (2) (cid:39) . (cid:96) > z, z (cid:48) ) = (2 ,
3) where the lensing terms nearly cancel for (cid:96) >
200 which degradesthe approximation and induces an error of up to 15% (green curve in Fig. 11). There are alsoother redshift pairs at lower z , e.g. ( z, z (cid:48) ) = (1 , .
1) where the lensing terms nearly cancel, butthese redshift separations are smaller and the standard terms are expected to be the most relevantcontribution there. For z > . Comparing Figs. 5 and 11 we see that there is a large difference in the accuracy of the approxima-tions for equal and unequal redshifts. While for equal redshifts in the range of redshifts considered, z ∈ [1 , class result is always better than ∼ . z ≤ ∆ z or ∆ z ≥ ∆ z . For intermediate redshift differences, ∆ z < ∆ z < ∆ z and (cid:96) < z ≤ ∆ z or in the otherextreme where the separation is sufficiently large, ∆ z ≥ ∆ z , such that the standard terms may besafely neglected. We also show the exception to this rule, namely the redshift pair ( z, z (cid:48) ) = (2 , z = 2 and z (cid:48) > .
5, the positive lensing-lensing term and the negative density-lensing termnearly cancel for (cid:96) >
200 so that the approximation neglecting the standard terms becomes worseagain and the error increases to nearly 15% for some values of (cid:96) .In a real observation we cannot measure the C (cid:96) ( z, z (cid:48) ) at exact redshifts z, z (cid:48) without error.First of all, even for spectroscopic surveys the redshift accuracy is finite, of order ∆ z ∼ − (1+ z ).For photometric surveys redshift determination is much less accurate, of order ∆ z ∼ . z )in the most optimistic case. But even if redshift accuracy is very high, in a too slim redshift binthere are few galaxies, and shot noise will prevent the determination of the C (cid:96) ’s especially at high (cid:96) . We therefore have also investigated windowed C (cid:96) ’s defined by C (cid:96) ( z, ∆ z ) = (cid:90) dz dz C (cid:96) ( z , z ) W ∆ z ( z, z ) W ∆ z ( z, z ) (2.59)where W ∆ z ( z, z (cid:48) ) denotes a (normalized) window function of full width ∆ z centred at z . Typicallyone chooses a Gaussian or a tophat window. In the windowed C (cid:96) ’s unequal redshift correlatorsalways enter and our reduced accuracy for them therefore affects the windowed C (cid:96) ’s. If we choosea slim window, ∆ z (cid:46) z , as shown in Fig. 12, the approximation is good with an error of 4.5%or less for z ≤
3. For z = 5, the error decreases to less than 4%.However, if we choose a photometric window width, ∆ z (cid:38) . z ), the accuracy degradessignificantly, up to 17%, see Fig. 13.The reason for this is clear, we enter the regime ∆ z < ∆ z < ∆ z for which we have nogood approximation. Neglecting the standard terms already for ∆ z > ∆ z is also not a goodapproximation. The result then underestimated the true windowed C (cid:96) ’s be nearly a factor of 10.This somewhat surprising finding shows that the standard terms do contribute significantly (about90% in total) also for z (cid:54) = z (cid:48) in the interval ∆ z < ∆ z < ∆ z . Also multiplying the result from the∆ z -window by a factor ∆ z/ ∆ z does not give an accurate approximation. Therefore, a windowed– 19 – igure 12 . Left: the windowed full power spectrum, C ∆ (cid:96) ( z, ∆ z ) in the flat sky approximation (solid)compared to the class result (dashed) at redshifts z = 1 , , , z = 2∆ z has been applied. Right: the relative differences. Figure 13 . Left: the windowed full power spectrum, C ∆ (cid:96) ( z, z ) in the flat sky approximation (solid)compared to the class result (dashed) at redshifts z = 1 , , , z = 0 . z ) has been applied. Right: the relative differences. Clearly theapproximation is not satisfactory. – 20 –orrelation function cannot be determined with an accuracy better than 17% with the flat skyapproximation except for very narrow redshift bins. An approximation which is well-known, especially for lensing, is the so-called Limber approxi-mation [36, 37]. We shall see that while this approximation is equivalent to the flat sky one forlensing and density-lensing correlations, it is very different and actually a bad approximation forthe density and RSD terms. The fact that the Limber approximation does not work for densityand RSD has already been noted in Refs. [39, 40].We start with the exact expression (2.42) which is also used in class to calculates the powerspectra. The Limber approximation now makes use of the fact that, for a sufficiently slowly varyingfunction f ( x ) one can approximate (cid:90) x f ( x ) j (cid:96) ( yx ) j (cid:96) ( y (cid:48) x ) dx (cid:39) π y δ ( y − y (cid:48) ) f (cid:18) (cid:96) + 1 / y (cid:19) (3.1)This equation is exact if f is a constant. Using it in (2.42) for X = Y = D yields C D(cid:96) ( z, z (cid:48) ) = δ ( r − r (cid:48) ) r ( z ) P R (cid:18) (cid:96) + 1 / r ( z ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) T D (cid:18) (cid:96) + 1 / r ( z ) , z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (3.2)= δ ( z − z (cid:48) ) H ( z ) r ( z ) P R (cid:18) (cid:96) + 1 / r ( z ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) T D (cid:18) (cid:96) + 1 / r ( z ) , z (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (3.3)Up to (cid:96) → (cid:96) + 1 /
2, this is obtained from (2.38) if we neglect k (cid:107) in k , i.e. we set k (cid:39) (cid:96)/R in P R ( k )and T D ( k, z ) and integrate the cosine over k (cid:107) . Hence, the Limber approximation is analogous tothe flat sky approximation which we used for the lensing terms. The δ -function pre-factor meansthat for a physically sensible result we have to introduce a window function such that C D(cid:96) ( z, z (cid:48) , ∆ z ) = (cid:90) dz dz (cid:48) W ∆ z ( z, z ) W ∆ z ( z (cid:48) , z (cid:48) ) C D(cid:96) ( z , z (cid:48) )= (cid:90) dz W ∆ z ( z, z ) W ∆ z ( z (cid:48) , z ) H ( z ) r ( z ) P R (cid:18) (cid:96) + 1 / r ( z ) (cid:19) T D (cid:18) (cid:96) + 1 / r ( z ) , z (cid:19) . (3.4)In other words, density fluctuations are only considered correlated only if they are at equal redshiftand for unequal redshifts correlations are due entirely to overlapping window functions.For the lensing spectra we find C κ(cid:96) ( z, z (cid:48) ) = 4[( (cid:96) + 1) (cid:96) ] (cid:90) r min drr ( r ( z ) − r )( r ( z (cid:48) ) − r ) r ( z ) r ( z (cid:48) ) r P R (cid:18) (cid:96) + 1 / r (cid:19) × (cid:12)(cid:12)(cid:12)(cid:12) T Ψ W (cid:18) (cid:96) + 1 / r , z ( r ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (3.5) C D,κ(cid:96) ( z, z (cid:48) ) = − (cid:96) ( (cid:96) + 1) r ( z (cid:48) ) − r ( z ) r ( z (cid:48) ) r ( z ) P R (cid:16) (cid:96) +1 / r ( z ) (cid:17) T D (cid:16) (cid:96) +1 / r ( z ) , z (cid:17) T ∗ Ψ W (cid:16) (cid:96) +1 / r ( z ) , z (cid:17) if z < z (cid:48) z ≥ z (cid:48) , (3.6)(3.7)– 21 – C C , a pp C , e x C , e x × ( % ) ×0.01×0.1 z = 0.001 z = 0.01 z = 0.1 z = 0.3CLASS Figure 14 . Left: Density term at redshifts z = 1 in a window of width ∆ z = 10 − , 10 − , 10 − , and 0 . class result (black, dashed). Right: therelative differences between class and the Limber approximation (solid). Note: The relative differenceshave been adjusted to facilitate interpretation, in particular the % errors of the two smallest window widths(blue and orange) have been reduced by multiplication by factors 0.01 and 0.1 respectively in order to fitinto the plot window. Contrary to the Limber approximation of the density, for z = z (cid:48) these are identical to the simplifiedflat sky approximations given in the previous section (up to (cid:96) → (cid:96) + 1 / z = 1 and different window widths ∆ z . Clearly, the Limber approximation cannotbe trusted even at quite high (cid:96) if ∆ z is not very large.In Fig. 14 we see that the Limber approximation of the density term is extremely bad forslim windows, but gets increasingly more accurate as the windows widen. The accuracy is within ∼
6% above (cid:96) = 100 for the widest window with ∆ z = 0 .
3. As expected the Limber approximationalso improves with increasing (cid:96) . For ∆ z = 0 .
1, an accuracy better than ∼
6% is achieved onlyabove (cid:96) = 200.
In this paper we have investigated the ’flat space’ approximation for the galaxy number counts,which allows the calculation of the C ∆ (cid:96) ( z, z (cid:48) ) with a simple, not heavily oscillating 1d numericalintegral once the transfer function is known. For the lensing terms, the flat sky approximationis equivalent to the Limber approximation, but for density and RSD it is very different. For the– 22 –ensity term, the Limber approximation becomes good to about 5% above (cid:96) ∼
100 only for verywide redshift bins of ∆ z (cid:38) . z = 1.For equal redshifts our approximation for the standard terms, i.e., density and redshift spacedistortions, up to z = 3 is accurate within about 0.3% which is close to the accuracy of class itself,for all values of (cid:96) ≥
2. Including the lensing terms and going to z = 5, for (cid:96) >
10, the approximationis still excellent, better than 0.5%. This is our first main result.However, for unequal redshifts, while the approximation for the lensing terms remains verygood, the approximation of density and RSD degrades rapidly with growing redshift difference. Forsufficiently large ∆ z , i.e. ∆ z > ∆ z (cid:39) . r ( z ) H ( z ) / ( z + 1), the lensing terms in general dominateand the unequal redshift C (cid:96) ’s can be approximated well by neglecting the standard terms. Hencealso for large redshift differences we have a good approximation with errors of about 0.5% or lessfor (cid:96) (cid:38)
60. For redshift differences beyond ∆ z (cid:39) . × − (1 + z ) . , the error of the standardterm approximation becomes larger than 10% and the flat sky approximation deviates by severalorders of magnitude for (cid:96) >
100 once we reach ∆ z . This is our second main result: for thestandard terms, the flat sky approximation is very bad for unequal redshifts.These results are valid for Dirac- δ windows in redshift space, which is equivalent to redshifterrors of less than about 10 − , i.e. spectroscopic redshifts. The approximation remains good,within a few percent, for windows slimmer than ∆ z = 2∆ z . If we want to study photometricredshift bins, we have to include a window function of width ∆ z = 0 . z ) or more. Doingthis will always include redshift differences for which the flat sky approximation is not valid. Wehave found that the approximation underestimates the true result by up to 17%. This somewhatdisappointing result is shown in Fig. 13. It is our third main result: for photometric windows, theunequal redshift standard terms are sufficiently important to degrade the the flat sky approximationconsiderably.Clearly, the flat sky approximation cannot be used to estimate the C (cid:96) ’s at equal redshiftswith photometric bin width. As a next step we plan to find an approximation that works alsofor unequal redshifts where we clearly have to go beyond both, the flat-space and the Limberapproximations. This is needed to obtain a useful approximation also for photometric redshifts. Itis also surprising that including only correlations of standard terms with redshift differences up to∆ z underestimates the windowed C (cid:96) ’s by as much as a factor 10 for bin widths of ∆ z = 0 . z ).For spectroscopic redshifts it is often more useful to employ the correlation function sincevery narrow redshifts bins are plagued by shot noise on the one hand and by non-linearities in theradial direction on the other hand [31]. It will therefore be useful to investigate how the flat-skyapproximation can be used for the correlation function as calculated e.g. in Ref. [32]. Acknowledgement
It is a pleasure to thank Francesca Lepori for enlightening discussions. The authors acknowledgefinancial support from the Swiss National Science Foundation grant n o A The Limber approximation for the redshift space distortion
The exact expression for the RSD is not of the form (3.1) but (cid:90) x F ( x ) j (cid:48)(cid:48) (cid:96) ( yx ) j (cid:48)(cid:48) (cid:96) ( y (cid:48) x ) dx . (A.1)– 23 –ore precisely, from the expressions in [13] one finds C rsd (cid:96) ( z , z ) = 4 πf ( z ) f ( z ) (cid:90) dkk j (cid:48)(cid:48) (cid:96) ( kr ) j (cid:48)(cid:48) (cid:96) ( kr ) P R ( k ) T D ( k, z ) T D ( k, z ) and (A.2) C D, rsd (cid:96) ( z , z ) = 4 πf ( z ) (cid:90) dkk j (cid:96) ( kr ) j (cid:48)(cid:48) (cid:96) ( kr ) P R ( k ) T D ( k, z ) T D ( k, z ) . (A.3)Here f ( z ) is the growth function defined in (2.28). In a ΛCDM universe one has [13] f ( z ) = 1 + H − ˙ T Ψ T Ψ . (A.4)In order to find an approximation for integrals of the form (A.1), we use the identity j (cid:48)(cid:48) (cid:96) ( x ) = (cid:96) − (cid:96) − x x j (cid:96) ( x ) + 2 x j (cid:96) +1 ( x ) . (A.5)To perform the required integrals we not only need a formula for integrals with equal (cid:96) ’s but alsowith (cid:96) and (cid:96) + 1. One might be tempted to neglect the latter, but a numerical study actuallyshows that both contribution to the above integral are of the same order. We therefore follow [40]and use the following crude approximation for the spherical Bessel functions to obtain integrals ofunequal (cid:96) j (cid:96) ( x ) ∼ (cid:114) π (cid:96) + 1 δ ( (cid:96) + 1 / − x ) which yields (A.6)2 π (cid:90) ∞ dkk F ( k ) j (cid:96) ( kr ) j (cid:96) +1 ( kr ) ∼ (cid:96) + 12 (cid:96) + 3 F (cid:18) (cid:96) + 1 / r (cid:19) δ (cid:16) r (cid:96) +32 (cid:96) +1 − r (cid:17) r . (A.7)Inserting (A.7) and (A.5) in (A.2) we obtain ( r = r ( z ) and r = r ( z )) C rsd (cid:96) ( z , z ) ∼ f ( z ) (cid:96) (cid:34) C D(cid:96) ( z , z ) + C D(cid:96) +1 ( z , z ) − (cid:40) δ (cid:16) r (cid:96) +32 (cid:96) +1 − r (cid:17) r ( z ) P R (cid:18) (cid:96) + 1 / r (cid:19) × T D (cid:18) (cid:96) + 1 / r , z (cid:19) T D (cid:18) (cid:96) + 1 / r , z (cid:19) + ( r ↔ r ) (cid:41)(cid:35) . (A.8)For the second term we have used that both f ( z ) and T D ( k, z ) are slowly varying functions andwe have neglected the difference between z and z in them. We have also neglected higher orderterms in 1 /(cid:96) . Note that if we neglect the difference between (cid:96) + 3 / (cid:96) + 1 /
2, the secondterm just cancels the first term. Numerical evaluation also has shown, that for F (cid:39) constant, theintegral (A.7) as a function of r peaks even closer to r = r than to r = r (cid:96) +32 (cid:96) +1 . Therefore, RSDare strongly suppressed in the Limber approximation. Numerically one finds that, like for densityperturbations, the Limber approximation is valid only in very wide windows where redshift spacedistortions are indeed strongly suppressed.Inserting (A.5) in (A.3), we obtain for the density-RSD correlations in the Limber approxi-mation C rsd D(cid:96) ( z , z ) ∼ − f ( z ) (cid:96) (cid:34) C D(cid:96) ( z , z ) − δ (cid:16) r (cid:96) +32 (cid:96) +1 − r (cid:17) r ( z ) P R (cid:18) (cid:96) + 1 / r (cid:19) × T D (cid:18) (cid:96) + 1 / r , z (cid:19) T D (cid:18) (cid:96) + 1 / r , z (cid:19) (cid:35) . (A.9)– 24 –s for the pure RSD term, when neglecting the difference between r and r , this term vanishesand therefore, for sufficiently high (cid:96) , where the Limber approximation is applicable, it is negligible.Like for the density term, the RSD Limber approximation has to be integrated over a windowwith some finite width ∆ z to become physically meaningful. But for large window sizes, where theLimber approximation for the density becomes reasonably accurate, the contribution from RSDcan actually be neglected. Also the Limber approximation of lensing–RSD is always very small.Therefore, the Limber approximation for RSD is either very bad (for slim redshift bins) or toosmall to be relevant. For the wide redshift bins where the Limber approximation for the dominantdensity term can be sufficiently accurate, the RSD contribution is never relevant, and neglectingit is a good approximation. References [1]
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