Observable Correlation Function

2

3(β1+β2) + 4

7β1β2)P2(ˆx₁⋅x)ˆ ξ₂²(x) +

8

35β1β2P4(ˆx₁⋅x)ˆ ξ₄²(x), (3.41) which, neglecting galaxy bias and setting x1 ≈x2, agrees with [136, 270]. Indeed, it is straightforward to check numerically that the functions ξ₂²(x)and ξ₄²(x) coincide with the respective ones employed in [136], although formally different.

We finally add that additional terms to the plane-parallel approximation obtained when neglecting the α-dependent terms in eq. (3.23), are usually calledwide angle ef-fects since they represent the deviation from only line-of-sight direction in the distant-observer approximation. The α-dependent terms themselves represent instead addi-tional velocity contributions, as discussed in section 2.4.

3.2.4 Observable Correlation Function

In a galaxy survey, the fundamental observable is the number of galaxies within a certain redshift bin and within a solid angle. Hence, to compare theoretical predic-tions to observapredic-tions, one should integrate the three-dimensional correlation function ξ∆∆(z1, φ1, z2, φ2, x) given in equation (3.32) over the redshift bins around the fiducial redshifts ¯z1, ¯z2. Then, the galaxy separation should be written as x=x(z1, z2, φ1, φ2). This can be done using the cosine rule (see also Figure 3.1):

x(z1, z2, ϑ) =

√

x1(z1)²+x2(z2)²−2x1(z1)x2(z2)cos(ϑ), (3.42) where we defined ϑ≡φ2−φ1, andx1(z1),x2(z2)can be obtained in function of z1 and z2, respectively, using equation (B.24).

We can now write the observable angular correlation function as

ξ∆∆(ϑ,z¯1,z¯2) = ∫ dz1W(z1,z¯1,∆¯z1)b(z¯1)G(z1) ∫ dz2W(z2,z¯2,∆¯z2)b(z¯2)G(z2)

×Ξ(φ1(z1, z2, ϑ), φ2(z1, z2, ϑ), x(z1, z2, ϑ)), (3.43)

where W(z,z,¯ ∆¯z) is the radial window function, e.g., a Tophat of width ∆¯z around the mean redshift ¯z, and normalized to 1 within the bin of width ∆¯z. We also defined

Ξ(φ1, φ2, x) ≡ ∑

n1,n2=0,1,2

[an1n2cos(n1φ1)cos(n2φ2)

+bn1n2sin(n1φ1)sin(n2φ2)] . (3.44) Note that introducingϑ, we are mixing variables from different coordinate systems.

In fact, in the coordinate system defined in Figure 5.1, Ξ depends on φ1 and φ2 sep-arately, and not only on the combination ϑ=φ2−φ1. Nevertheless, to compare with observations it is useful to study the angular dependence of the correlation function on ϑ, then using eq. (3.25) we write the angles φ1, φ2 in function of x1(z1), x2(z2), x(z1, z2, ϑ) as, e.g.,³

φ2=sin⁻¹[x1

x sinϑ] , φ1=φ2−ϑ=sin⁻¹[x1

x sinϑ] −ϑ=sin⁻¹[x2

x sinϑ] , (3.45) for φ1 ≤ φ2, and use property iii. discussed below equation (3.26) if φ1 > φ2. This justify the choice of writing ξ∆∆(ϑ,z¯1,z¯2) as function of the only angular variable ϑ instead of φ1, φ2 separately.

One could have considered from the beginning a system such that the ˆz-axis points between the first two unit vectors, ˆz∝xˆ₁+xˆ₂, and that all the vectors ˆx₁, ˆx₂, ˆx lay in the φ=0 plane. With this choice we would have

Sl1l2l(ˆx₁,xˆ₂,x) =ˆ Sl1l2l({θ,0},{θ, π},{γ,0}), (3.46) where φ2=π ensures that the ˆz-axis is between the two first unit vectors. The tripolar expansion would be a generalization of the double Legendre expansion obtained in [269], and it would result in a less compact form than equation (3.32) [270]. This justify the choice of coordinates in Figure 5.1 to calculate the tripolar expansion.

For further discussion and applications we refer to section 5, in which the main steps of the derivation showed above are also outlined. In particular, the transverse correlation ξ∆∆(ϑ,z,¯ z¯) and the longitudinal one ξ∆∆(0,z¯1,z¯2) will be introduced and thoroughly examined.

3As a technical note, we stress that attention should be paid when computing numerically eq. (3.45), as the usual range of principle values of sin⁻¹xin numerical libraries is−π/2<x<π/2.

Part II

Applications

41

Chapter 4

Overview

Here we summarize our main lines of research. The common point of interest is eq. (2.55) describing the full relativistic number counts. In the following chapters we reproduce the publications describing our work. We neglect curvature K =0, whose effects are nevertheless discussed in [221]. The notation may vary compared to the review given in part I, but each chapter is self-contained.

We organize the work into three main research lines. In section4.1we discuss which observable is better suited to constrain our cosmological model through fully relativistic number counts. This quest led in particular to development of a code, Classgal, for fast and accurate computations of galaxy angular power spectra. In section 4.2, we forecast the detectability of those effect poorly exploited in standard galaxy clustering analyses. Assuming realistic surveys specifications, we show in particular how lensing convergence can be constrained using future galaxy surveys like Euclid and SKA. In section 4.3, as the physics of LSS is inherently non-linear, we evaluate at what level relativistic terms can contribute to the study of non-Gaussianities generated by gravity through the galaxy bispectrum.

4.1 Model independent observations

As motivated in chapter 3, when observing galaxies we measure their redshift and angular position. To convert this into a three-dimensional galaxy catalog we must make an assumption to relate the observed redshift z to a distance. For small redshift, the simple relation d = z/H0, where H0 is the present Hubble parameter, can be used. However, if we go out to high redshifts, z > 1, non-linear terms in z become relevant, and wrong assumptions about the distance-redshift relation can bias the entire catalog. This is usually taken into account by recursive procedures: a set of cosmological parameters (usually the best fit parameters from CMB observations) is

43

z

=z

θ

z z

= z - Δz/2 z

= z + Δz/2

Figure 4.1: Left: the transverse correlation function investigates the correlations be-tween galaxies in the same redshift bin as a function of the angular separation θ.

Right: the longitudinal correlation function studies the cross-correlations between dif-ferent redshift bins within a small fixed angular bin ∆θ∼0, as a function of the redshift bins separation ∆z. The observer is placed at the bottom of the figures.

chosen, the power spectrum is determined under the assumption that this set correctly describes the background cosmology, and then a new set of cosmological parameters is estimated together with their errors. The process is repeated by choosing few different starting cosmologies, and by verifying that the final estimates are in agreement. This allows fine parameter constraints, but it is also important to consider complementary analysis taking into account the truly observed catalog, either using the angular power spectra or the respective angular correlation function.

In reference [200], reproduced in chapter 5, we analyze the two point correlation functionξ(θ, z1, z2), whereθis the galaxy angular separation andz1,z2are the redshifts of the two galaxies. We include wide-angle velocity terms additional to redshift-space distortions in the Kaiser approximation through an expansion in tripolar spherical harmonics, showing that they may be substantial at very large angular separations θ ≳ 10^○. A more detailed analysis of their signal-to-noise, discussed in the following works, is needed to conclude about their detectability, as on these scales cosmic variance is important. We further study the transverse correlation, i.e., the auto-correlation of the same bin around redshift z as a function of the angular separation ξ(θ, z, z), and the longitudinal correlation ξ(0, z−∆z/2, z+∆z/2), i.e., fixing the angular scale and varying the redshift difference between galaxies ∆z = ∣z2−z1∣. This is depicted

4.1. MODEL INDEPENDENT OBSERVATIONS 45

L=ΔzL(z)/ H(z)

L

=

θL(z)×(1+z)DA(z)

Figure 4.2: The Alcock-Paczy´nski test. Even without the knowledge of the comoving scale Lcharacterizing a spherical distribution, the product (1+z)H(z)DA(z) can be constrained in terms of the observable ∆zL(z)/θL(z). This provides a consistency check of our cosmological model.

in figure 4.1. The shapes of the longitudinal and transverse correlation function in redshift space are not only very different from each other, but also very different from the real space correlation function.¹ The integral of the latter has to vanish, so that the correlation changes sign ξ<0 for large separations. However, in redshift space we have that ξ(θ, z, z) > 0 ∀ θ and ξ(θ = 0, z−∆z/2, z+∆z/2) < 0 ∀ ∆z >0.01, around redshifts z∼1. Interestingly, this implies that the BAO peak already corresponds to a negative longitudinal correlation. It is worth recalling that in practice a binning must be performed in order to have enough galaxies to beat shot-noise. Angular correlations can be investigate also with relatively deep redshift bins of width larger than typical photometric redshift errorsσz/ (1+z) ≃0.05. Instead, longitudinal correlations require a more fine redshift binning, ideally given by spectroscopic determinationsσz/ (1+z) ≃ 0.001.

As an application that can optimally profit from these directly observable corre-lations, we consider an Alcock-Paczy´nski test. As illustrated in figure 4.2, given a spherical object of comoving size L in the sky at redshift z, the redshift difference of its front and back is then given by

∆zL(z) =LH(z), (4.1)

and we see it under an angle

θL(z) = L

(1+z)DA(z) , (4.2)

1As in section2.2, here real spacerefers to the correlation determined by intrinsic clustering only, whileredshift space refers to the case when also peculiar velocity contributions are included.

whereHandDAare the Hubble parameter and angular diameter distance, respectively.

Even without any knowledge of L, we can infer the product (1+z)H(z)DA(z) = ∆zL(z)

θL(z)

, (4.3)

where the right hand side is directly observable. The left hand side can be estimated theoretically by assuming a cosmological model, whose consistency can be tested by eq. (4.3). Furthermore, combining eq. (4.3) with a measurement of, e.g., the luminosity distanceDL(z) = (1+z)²DA(z)from supernova data, we can break the degeneracy be-tweenH(z)andDA(z). The test was first applied to galaxy clusters [15]. Nevertheless, the assumption that L describes a spherical distribution can also apply statistically.

Hence, a standard ruler such as Baryon Acoustic Oscillations (BAO) also provides the scale L. From the transverse correlation function we can measure the BAO angular extensionθBAO at a given redshift, and from the longitudinal correlation we can obtain the BAO redshift extension ∆zBAO. By carefully taking into account that the peak position slightly depends on the width of redshift bins, this information can be used as an independent consistency check of our cosmological model via eq. (4.3).

In reference [106], reproduced in section 6, we present a new version of the Cos-mic Linear Anisotropy Solving System (Class) code [165, 60], incorporating the full relativistic expression given in eq. (2.55). This code is called Classgal and it is pub-licly available on a dedicated website.² The principal features have been merged with the main Class branch.³ It computes the full observable angular power spectrum C`(z1, z2) at first order in perturbation theory, including redshift-space distortions, lensing and other relativistic and observational effects. For realistic redshift bins of finite thickness described by a set of window functions Wi(z)—e.g., a Tophat or a Gaussian centered at some redshift zi—and a given number density of galaxies per redshift interval dN/dz (the integral of the productWi(z)dN/dz being normalized to unity), the power spectrum can be written as

C_`^ij =4π∫ dk

k P (k)∆ⁱ_{`</sub>(k)∆<sup>j</sup><sub>`}(k). (4.4) HereP (k) =As(k/k∗)^ns−1is the primordial power spectrum of curvature perturbations, written in terms of the primordial amplitudeAsand spectral indexnsat the pivot scale k∗

4, and

∆ⁱ_`(k) = ∫ dzdN

dz Wi(z)∆_`(z, k), (4.5)

2http://cosmology.unige.ch/content/classgal

3http://class-code.net

4We recall that the pivot scalek∗ is arbitrary, and usually chosen to be the scale at which a given survey is most sensitive.

4.2. FORECASTS WITH RELATIVISTIC CORRELATIONS 47