Appendix: Wide-angle RSD using the tripolar expansion

Dans le document Relativistic effects in galaxy clustering (Page 94-102)

We used the redshift dependent angular correlation function to determine the trans-verse and the longitudinal acoustic peak positions as function of the redshift. We have proposed to use these functions to perform an Alcock-Paczy´nski test. We have fully taken into account redshift space distortions. Even though these modify significantly the shape of the correlation function, they are not very important for the position of the acoustic peak.

We have found that photometric redshift determinations withσz/(z+1) ≃0.03 are sufficient to determine the BAO peak in the transverse correlation function, but in order to cleanly determine the position of the longitudinal peak, spectroscopic precision with σz/(z+1) ≃0.001 are needed. Furthermore, we have seen that the peak position of the transverse correlation function slightly depends on the window function for the redshift determination. To achieve an accuracy better than about 6% for the transversal BAO scale, one has the employ the correction suggested in this work. This, together with the correctly symmetrized longitudinal correlation function allows to obtain an estimation of F(z) = (1+z)H(z)DA(z) which is accurate to about 2%, when performing the Alcock-Paczy´nski test.

Acknowledgments. We thank Chris Clarkson, Enea Di Dio, Roy Maartens and Rom´an Scoccimarro for discussions. We also aknowledge Camille Bonvin for sharing with us her codes, and the anonymous Referee for pointing out the lack of discussion about the covariance matrix in a previous draft of the paper. The GNU Scientific Library (GSL) has been used for the numerical implementation. This work is supported by the Swiss National Science Foundation.

5.5 Appendix: Wide-angle RSD using the tripolar expansion

In this appendix we derive expressions for the parameters an1n2 and bn1n2 used in the calculation of the correlation function via Eq. (5.17).

We use the coordinates introduced in section 5.2 and shown in Fig. 5.1. We can characterize the correlation function, equation (5.11), by the two directions n1, n2 from the observer and the comoving distance between the two galaxies r=r1−r2 =rn.

As pointed out in [137, 269], the correlation function in redshift space depends on the triangle formed by the observer and the two galaxies. This triangle is invariant under rotation about the observer and hence can be described by one size and two shape parameters. For the former we use the separation of the galaxies r, while for the shapes we choose the angles φ1, φ2, e.g., between n1, n2 and n, respectively, see Fig. 5.1. We suppose that φ1 ≤φ2 and the distances are r1 =r(z1) and r2=r(z2). For

fixed redshifts z1 and z2, the distance r is constrained by∣r1−r2∣ ≤r≤r1+r2. We can evaluate the galaxy correlation function as given in Eq. (5.11),

ξgal(n1,n2, r) ≡ b1G1b2G2∫ d3k

(2π)3P(k)eik⋅r

× [1+β1

3 +2β1

3 P2(n1⋅k) −ˆ iα1H1β1

r1k P1(n1⋅k)]ˆ

× [1+β2

3 +2β2

3 P2(n2⋅k) +ˆ iα2H2β2

r2k P1(n2⋅k)]ˆ .

(5.27) The most direct way would be to replace the occurrences of ˆk in the integrand of equation (5.27) in terms of the gradient ∇r. This is indeed straightforward in the plane-parallel limit, because Fourier modes are eigenfunctions of the plane-parallel distortion operator, that corresponds to the case in which the terms of equation (5.11) multiplied by Legendre polynomials are negligible, and can be written in terms of the inverse Laplacian ∇−2r [137]. However, it is non-trivial to obtain an expression for ξgal

beyond the plane-parallel limit proceeding in this way. In fact, the computation would lead to a large number of terms that, although straightforward, are cumbersome to evaluate numerically. Despite the non-locality of the power spectrum including the wide-angle RSD, the corresponding correlation function has still been calculated in spherical coordinates and expressed in a closed form, e.g., in [269] by the introduction of a convenient spherical tensor and in the approximation of small redshifts z ≪ 1.

This work has been generalized by [191, 192] to the case of arbitrary galaxy redshifts z1,z2also for non-flat cosmologies (theα-term there is a selection function). Even with these results, however, the symmetry of the problem, namely the dependence of ξgal

on the rotationally invariant triangle determined by {n1,n2,n}, suggests that a more natural way to proceed is to expand the expression in tripolar spherical harmonics. In particular, we use the functions defined in [270,211]

S`1`2`(n1,n2,n) = ∑

m1,m2,m

(`1 `2 ` m1 m2 m)

× C`1m1(n1)C`2m2(n2)C`m(n),

(5.28) whereC`m(n) =

2`+1Y`mare conveniently normalized spherical harmonics and(m`11 m`22 m` ) is the Wigner 3-j symbol. These function form an complete orthogonal basis for ex-panding spherically symmetric functions depending on three vectors, which are invari-ant under global rotations. In [270,211] it has been shown that, in the approximation z ≪ 1, this expansion leads to a more compact form for ξgal than those obtained in

5.5. APPENDIX: WIDE-ANGLE RSD USING THE TRIPOLAR EXPANSION 75 previous works. We generalize the work of [270, 211] by allowing arbitrary galaxy redshifts z1,z2 and a generic function α(z).

After some algebra one finds that the only non-vanishing coefficients of the expan-sion

ξgal(n1,n2,n, r) =b1G1b2G2

× ∑

`1`2`

B`1`2`(r, z1, z2)S`1`2`(n1,n2,n), (5.29)

are

B000(r, z1, z2) = (1+ 1

1) (1+ 1

2) ζ02(r), B220(r, z1, z2) = 4

9√

1β2ζ02(r), B222(r, z1, z2) =

4√ 10 9√

7 β1β2ζ22(r), B224(r, z1, z2) = 4√

2

35β1β2ζ42(r), B202(r, z1, z2) = − (

2 3β1+

2 9β1β2)

22(r), B022(r, z1, z2) = − (

2 3β2+

2 9β1β2)

22(r), (5.30)

which are independent of the angles (as they do not involve the α-terms), and 2/H, this result agrees with [211]. We also verified the consistency with [50].

To proceed further we must choose a system of coordinates. As shown in [270], however, once the coefficients of the tripolar expansion have been calculated, it is trivial to write the correlation function for a given coordinate system. In fact, this step involves only simple algebra that, although tedious, can easily be carried out using a computer algebra package.

We use the coordinate system a) discussed in [270] since it gives the most compact form for ξgal. As shown in Figure 5.1, the direction ˆz is orthogonal to the plane that contain the triangle formed by the observer O and the two galaxies. Hence, all the vectors r1, r2, r have latitude ϑ = π/2. The galaxy separation vector r has zero longitude φ=0, i.e., it is parallel to the abscissas.

With this choice of coordinates we can evaluate the elements of the tripolar basis, equation (5.28), which so far have been written in terms of the re-normalized spherical harmonics C`m(n). Using the notationn= {ϑ, φ}, we have

S`1`2`(n1,n2,n) =S`1`2`({π/2, φ1},{π/2, φ2},{π/2,0}).

Evaluating these expressions, the summation over`1, `2 and`in Eq. (5.29), using (5.30)

5.5. APPENDIX: WIDE-ANGLE RSD USING THE TRIPOLAR EXPANSION 77 and (5.31) for the non-vanishing functions B`1`2` leads to

ξgal(z1, φ1, z2, φ2, r) =b(z1)G(z1)b(z2)G(z2)

× ∑

n1,n2=0,1,2

[an1n2cos(n1φ1)cos(n2φ2) +bn1n2sin(n1φ1)sin(n2φ2)] .

(5.32)

The coefficients an1n2 and bn1n2 can be easily calculated with a numerical algebra package. We first write all the non-vanishing coefficients which do not involveα-terms:

a00 = (1+1

3(β12) + 2

15β1β202(r)

− ( 1

6(β12) + 2

21β1β222(r) +

3

140β1β2 ζ42(r), a20 = − (

1 2β1+

3

14β1β222(r) + 1

28β1β2ζ42(r), a02 = − (1

2+ 3

14β1β222(r) + 1

28β1β2ζ42(r), a22 =

1

15β1β2 ζ02(r) − 1

21β1β2 ζ22(r) + 19

140β1β2 ζ42(r), b22 =

1

15β1β2 ζ02(r) − 1

21β1β2 ζ22(r) − 4

35β1β2 ζ42(r),

(5.33)

The coefficients that involve α-terms are: are assuming φ1≤φ2 (Figure 5.1), this constraint can be inverted using the symmetry ξgal(n2,n1,n, r) =ξgal(−n1,−n2,n, r)evident from equation (5.11).

Chapter 6

The CLASS gal code for Relativistic Cosmological Large Scale Structure

Based on:

[106] E. Di Dio, F. Montanari, J. Lesgourgues and R. Durrer, “The CLASSgal code for Relativistic Cosmological Large Scale Structure,” JCAP 1311 (2013) 044 [arXiv:1307.1459 [astro-ph.CO]].

Here we include corrections to the published version and arXiv:1307.1459v3 [astro-ph.CO]. In particular, note that the term −3j`(kr(z))Hk has been replaced by (fevoN − 3)j`(kr(z))H

k in eq. (6.17), (6.33), (6.44), (6.49), (6.50); a sign error has been corrected in eq. (6.40); an erroneus rS factor has been removed from eq. (6.54).

Abstract. We present accurate and efficient computations of large scale structure observables, obtained with a modified version of theclasscode which is made publicly available. This code includes all relativistic corrections and computes both the power spectrumC`(z1, z2)and the corresponding correlation functionξ(θ, z1, z2)of the matter density and the galaxy number fluctuations in linear perturbation theory. For Gaussian initial perturbations, these quantities contain the full information encoded in the large scale matter distribution at the level of linear perturbation theory. We illustrate the usefulness of our code for cosmological parameter estimation through a few simple examples.

6.1 Introduction

Since many years now, cosmology is a data driven science. This became especially evident with the discovery of the apparent acceleration of the expansion of the universe,

79

which was found by observations and still remains very puzzling on the theoretical side.

The most blatant success story of cosmology, however, remains the agreement between predictions and observations of the cosmic microwave background (CMB) anisotropies, see [158, 161, 110], which is confirmed with the new Planck results [6, 8].

We now want to profit also in an optimal way from actual and future galaxy cat-alogs which contain information on the large scale matter distribution, termed large scale structure (LSS). Contrary to the CMB which is two dimensional, coming mainly from the surface of last scattering, galaxy catalogs are three dimensional and therefore contain potentially more, richer information. On the other hand, galaxy formation is a complicated non-linear process, and it is not clear how much cosmological information about the underlying matter distribution and about gravitational clustering can be inferred from the galaxy distribution. This is the problem of biasing which we do not address in this paper. Here we simply assume that on large enough scales, biasing is linear and local, an hypothesis which might turn out to be too simple [35].

When observing galaxies, we measure their redshift z and their angular position

−n = (sinθcosφ,sinθsinφ,cosθ). Note that n is the photon direction, so from the source to the observer. Hence we see a galaxy in direction −n. This observed three-dimensional data does not only contain information on the galaxy position, but also on the cosmic velocity field (redshift space distortions) and on perturbations of the geometry, e.g., lensing effects. Therefore, by making optimal use of galaxy catalogs, we can learn not only about the large scale matter distribution but also about the velocity field and the geometry. Since Einstein’s equations relate these quantities, this allows us to test general relativity or more generally the ΛCDM hypothesis, and to estimate cosmological parameters.

In this paper we present a new version of the Cosmic Linear Anisotropy Solving Sys-tem (class) code1 [165, 60], incorporating several correction terms already presented in the theoretical works of Ref. [293, 290, 64, 80, 50]. This code is called classgal and is made publicly available on a dedicated website2. The parts that concern the CMB have not been changed with respect to the mainclassdistribution. Several new features of classgal will be merged with the main code in future class versions.

In the next section, we describe the equations solved by classgal and the initial input and final output. In section 6.3, we discuss the relevance of the different con-tributions to the observed power spectrum. In section 6.4, we present some forecasts for parameter estimation with future catalogs, in order to illustrate the usefulness of our code. The topic of this section is worked out in more detail in an accompanying publication [105]. In Section 6.5, we conclude with an outlook to future possibilities using our code. The detailed description of the modifications of theclasscode as well as some derivations are deferred to two appendices.

1http://class-code.net

2http://cosmology.unige.ch/tools/

Dans le document Relativistic effects in galaxy clustering (Page 94-102)