a non-perturbative level and define the background space-time (which can be related, e.g., to the FLRW coordinates through perturbations).6 This framework needs further investigation when caustics appear on the past light cones, as wbecomes multi-valued.
As an example, in the GLC gauge the redshift and volume discussed in sections2.1.1 and 2.1.2 takes a very simple form
1+z= Υo
Υs , v=
√
∣γ∣ Υ2s
Υo∂τΥs , (4.12)
where the subscript indicates if the quantity is evaluated at the source or at the ob-server, and γ denotes the determinant of the 2-dimensional matrixγab. Our expression for the number counts derived in this way are in principle fully non-perturbative. The drawback is that Einstein equations are not know in the GLC gauge yet. Then we have to perform a second order coordinate transformation to translate the result to the fa-miliar Bardeen potentials, i.e., the metric perturbations in the Poisson (or generalized Newtonian longitudinal) gauge. We also allow the possibility of having two different Bardeen potentials, which can be used to model modifications of gravity. We write the final result fully in terms of the Bardeen potentials, the peculiar velocities and the density fluctuations. The final expressions remain complicated. There may also be a different gauge from the Poisson one more suited for the computation, where the ex-pressions may be more simple. One of the advantages of the Poisson gauge is that the results can be readily used in numerical codes and compared to the literature. We also provide a simplified expression, obtained by using the fact that Bardeen potential are small and nearly constant in time, while their spatial derivatives can be large, together with the fact that lensing-like term can be relevant on large radial extensions. This corresponds to a large-redshift and small-angle limit.
We describe the number counts as a function of direction and observed redshift.
We use these expressions to compute some of the most important terms for the bis-pectrum in harmonic space, which is a function of three redshifts and multipoles B`1`2`3(z1, z2, z3). We study the contribution from redshift space distortion and the one from lensing, and we evaluate them numerically. They are compared to the usual second order density term. We find that for narrow redshift bins, the intrinsic den-sity and redshift-space distortions are the dominant contribution. However, when the bispectrum is evaluated within a large redshift bin 0.2<z <3, the relativistic lensing signal contributes significantly to the total bispectrum. This analysis further moti-vates the effort to include other poorly studied velocity and density terms at second order that are not taken into account in our numerical analysis, and also novel second order lensing-like contributions that were not considered before in the literature and
6This is the inverse perspective usually considered in perturbation theory, where the fictitious FLRW coordinates define the background space-time, related to the true one via perturbations.
that can be substantial on large radial separations. For more details about this study, see [104].
Chapter 5
A new Method for the Alcock-Paczynski Test
Based on:
[200] F. Montanari and R. Durrer, “A new method for the Alcock-Paczynski test,”
Phys. Rev. D 86 (2012) 063503 [arXiv:1206.3545[astro-ph.CO]].
Abstract. We argue that from observations alone, only the transverse power spec-trumC`(z1, z2)and the corresponding correlation functionξ(θ, z1, z2)can be measured and that these contain the full three dimensional information. We determine the two point galaxy correlation function at linear order in perturbation theory. Redshift space distortions are taken into account for arbitrary angular and redshift separations. We discuss the shape of the longitudinal and the transversal correlation functions which are very different from each other and from their real space counterpart. We then go on and suggest how to measure both, the Hubble parameter, H(z), and the angular diameter distance, DA(z), separately from these correlation functions and perform an Alcock-Paczy´nski test.
5.1 Introduction
Cosmology has become a data driven science. After the amazing success story of the cosmic microwave background (CMB), see [158, 161, 110], which is still ongoing [8], we now also want to profit in an optimal way from actual and future galaxy catalogs.
Contrary to the CMB which comes from the two dimensional 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
55
distribution and about gravitational clustering can be gained by observations of 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, a hypothesis which might turn out to be too simple [35].
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 to a distance. For small redshift, the simple relation D=z/H0 can be used. Redshift space distortions (RSD) can be taken into account with a convenient expansion in tripolar spherical harmonics [270,211]. This gives an accurate description of the correlation function at small scales. Apart from RSD, a wrong measurements of H0 will just rescale the entire catalog but not distort its clustering properties.
However, if we go out to high redshifts,z ≳1, non-linear terms inzbecome relevant, and wrong assumptions about the distance redshift relation can bias the entire catalog.
We therefore believe that it is important to analyze the truly observed catalog, either using theC`(z1, z2)spectra introduced in Ref. [64], or the angular correlations functions ξ(θ, z1, z2) to describe the observations, and to compare them with their theoretically obtained counterparts. In this way we truly compare observations with their theoretical modeling. If, on the other hand, we determine a power spectrum in Fourier space for the observed catalog,P(k), we have already assumed a cosmology to convert observable redshifts into length scales. Therefore, e.g., cosmological parameter estimations using P(k)can at best be viewed as a consistency check. If the cosmological parameters used to obtain P(k) agree with those revealed in a Markov Chain–Monte Carlo parameter estimation, the model is consistent. A thorough error estimation seems however, quite tricky.
We therefore advocate to abandon this ‘mixed’ method in future, high redshift catalogs in favor of the more direct procedure which compares theoretical models with the directly observed two-point statistics, C`(z1, z2) and/or ξ(θ, z1, z2).
This paper is structured as follows: in the next section we explain how to com-pute ξ(θ, z1, z2) and its covariance matrix from the theoretical power spectrum. In Section 5.3 we discuss the longitudinal and transverse correlation functions and we show how the baryon acoustic oscillations in these correlation functions can be used as an Alcock-Paczy´nski test [15]. In Section 5.4 we conclude. In Appendix 5.5 we generalize the expansion in tripolar spherical harmonics of [211] to model the RSD of the correlation function at arbitrary redshifts.