Computing the distance redshift relation

In this appendix we explain in more detail how we calculate the luminosity distance in theN-body code. Details for the exact relativistic fluid solution can be found in Ref. [5].

The relativistic N-body scheme we use has the advantage that the metric is explicitly computed, and one can therefore directly integrate the null geodesic equa-tion numerically to obtain the path of a photon in the perturbed geometry. The difficulty, when constructing observables like the distance–redshift relation, is that they are defined on the past light cone of an observer, whereas the simulation, on the other hand, evolves forward in time. In the perturbed Universe, one does not know exactly whether a point lies on the past light cone of an event until one has actually found a photon path which connects the two. Naturally, one would do this construction backwards in time, starting form the observer, but for this one needs to save a part of the four-dimensional geometry with high resolution in space and time. Although this is a possible way to go, we chose a different approach which one may call a “shooting method”.

Somewhere close to the highest redshift which we want to plot in the distance–

redshift diagram, we choose an initial point located at spacetime coordinates which would be connected to the observer event by a null ray in theunperturbed geometry.

This location can simply be read off from the distance–redshift relation obtained in an exact FL universe. We then shoot a light ray directed at the observer by integrating the null geodesic equation in the perturbed geometry, along with the N-body simulation. When the simulation reaches the observer event, we usually find that we have missed the observer by some small spacelike distance due to the perturbations of the photon path. We then restart the simulation, correcting the coordinates of the initial point by the amount by which we have missed the observer. Rerunning the simulation will now bring the perturbed light ray almost to the observer event, up to second-order perturbations. This procedure can be iterated to close in on the observer event to arbitrary precision. For our purposes, a single iteration was enough.

At the last iteration, we also save the change of the photon energy and some infor-mation about the geometry, in particular the terms which enter the Sachs equation, along the light ray with high resolution. Additionally, we also save the peculiar

3.5. Appendix 65

velocities of sources which lie on the path. Since baryonic physics are completely neglected in our simple setup, we assume that observable sources have the same distribution as CDM particles. With this set of data, once we have reached the end of the simulation, we can integrate the Sachs equations backwards in time along the line of sight given by the light ray. Eliminating the affine parameter s in the equations in favor of the coordinate time τ, the angular distance evolves according to

d¨A+n˙⁰

n⁰d˙A+ |σ|²+R dA

(n⁰)² = 0, (3.21)

where n⁰ = dτ /ds is the τ-component of the photon null vector which has to be determined from the null geodesic equation,

n˙⁰+

Φ˙ −Ψ + 2˙ ∇ⁿΨ + 2H

n⁰ = 0, (3.22)

where n denotes the spatial direction of the photon vector. This equation takes care of the path-dependent contribution to the redshift. The total redshift (the ratio between photon energies measured in the rest frames of source and observer) is given by Eq. (3.14), which yields

1 +z =

hn⁰a

1 + Ψ−n·v+^v₂²i _∗ n⁰a 1 + Ψ−n·v+^v₂²

0

. (3.23)

Here, v is the peculiar velocity vector in the longitudinal gauge, n·v denotes its projection on the photon direction, and the subscripts ∗ and 0 indicate that the entire expression has to be evaluated at the source and observer event, respectively.

The last two equations have been truncated at our approximation order.

We finally need an evolution equation for the complex shear, see [49, 50]. For the purpose of solving Eq. (3.21), it is useful to write it as

d dτ

σ n⁰

+ n˙⁰ n⁰ + 2

d˙A

dA

! σ

n⁰ + F

(n⁰)² = 0, (3.24) where F = ¹₂Cκλµν^κn^λµn^ν is a contraction of the Weyl tensor with the complex screen vector and the photon four velocity.

In order to solve this coupled system of differential equations, it is sufficient to know four real-valued quantities along the line of sight (as a function of τ) which can all be obtained from the knowledge of the metric and the photon direction:

n˙⁰/n⁰,R/(n⁰)², and F/(n⁰)². The last quantity is complex in general and therefore corresponds to two real-valued quantities. However, for the particular lines of sight we chose to study in this work, owing to the symmetry of our setup, we can choose the screen vectors such thatF/(n⁰)² remains real-valued. In particular, for the light

ray perpendicular to the plane of symmetry, we find R

(n⁰)² = 4πGa²

−(1 + 2Ψ)T₀⁰+ 2T₀¹ +T₁¹ , F

(n⁰)² = 0, (3.25)

up to terms which are neglected in our approximation scheme. In this case, the shear remains zero and the angular distance is entirely governed by convergence. For the light ray parallel to the plane of symmetry, we can choose an orthogonal basis of the screen where one basis vector remains orthogonal to the plane of symmetry, while the other basis vector remains parallel to it. This is possible because the center of the underdense region has an additionalZ2 symmetry which guarantees that the ray remains parallel to the plane of symmetry (even though this path is unstable under small perturbations). Using such a basis, we find

R

(n⁰)² = −4πGa²(1 + 2Ψ)T₀⁰, F

(n⁰)² = 1

2 + 2Φ + Ψ

∆Φ + 1

2+ Φ

∆Ψ. (3.26)

In all these explicit expressions we have used the symmetries of our setup to simplify them. Note also that we assume T₀¹ is of order^1/2 and T₁¹ is of order ; components with spatial indices 2, 3 vanish by symmetry. Using this information, the distance–

redshift relation is constructed by integrating Eqs. (3.21), (3.22), (3.24) backwards in time. To this end, the final conditions are fixed at the observer as dA(τ0) = 0, d˙A(τ0) = −a(1 + Ψ−n·v+v²/2)|⁰, σ(τ0) = 0, and n⁰(τ0) > 0 (arbitrary). The solution for the ellipticityein terms of the real-valued shear follows from Eq. (3.20):

e= 2 sinh

2 Z τ∗

τ0

σ n⁰dτ

. (3.27)

4

Vector and Tensor Contributions to the Luminosity

Distance

PHYSICAL REVIEW D86, 023510 (2012)

Vector and Tensor Contributions to the Luminosity Distance

Enea Di Dio and Ruth Durrer

We compute the vector and tensor contributions to the luminosity distance fluctua-tions in first order perturbation theory, and we expand them in spherical harmonics.

This work presents the formalism with a first application to a stochastic background of primordial gravitational waves.

4.1 Introduction

The distance–redshift relation for far away objects plays an important role in cos-mology. It has led Hubble, or rather Lemaˆıtre [32], to discover the expansion of the Universe; and the distance–redshift relation to far away Supernovae type Ia is at the origin of last year’s Nobel Prize in physics for the discovery of the accelerated expansion of the Universe [2, 3, 4, 79, 81, 82].

A next step that has been initiated recently considers the angular and redshift fluctuations of the luminosity distance, which may also contain important informa-tion about our Universe [8, 9, 10]. One important unsolved problem is the quesinforma-tion how strongly the distance–redshift relation may be affected by the fact that the actual Universe is not homogeneous and isotropic, but the matter distribution and also the geometry have fluctuations. To first order in perturbation theory these fluctuations can average out in the mean and are therefore expected to be small.

However, it has been found that they are significantly larger than the naively expected value that would be of the order of the gravitational potential, namely,

∼10⁻⁵. An analysis in first order gave fluctuations of the order of 10⁻³, hence 100 times larger than the naive estimate [9]. Recently, Ben-Dayan et al. [11] have cal-culated a second order contribution to the distance–redshift relation of the order of

∼10⁻³. Evidently, if the second order term is as large as the first order, this means that perturbation theory cannot be trusted. On the other hand, fully nonlinear toy models, which have been studied in the past, always gave relatively small modifica-tions of the luminosity distance if the size of the fluctuamodifica-tions, spherical voids [112] or parallel walls [5], is small compared to the Hubble scale. Hence the problem remains open.

So far, the perturbative analyses of the distance–redshift relation have concen-trated on scalar perturbations. In this work, we want to study the contributions from vector and tensor perturbations on a Friedmann–Lemaˆıtre (FL) universe. This is interesting for several reasons. First of all, tensor perturbations are generically

produced during inflation, and hence their contribution has to be added for com-pleteness. Second, a passing gravitational wave from some arbitrary source does generate a tensor perturbation in the distance–redshift relation to any far away ob-ject and could, at least in principle, be detected in this way. For single binary sources we have found that this effect is very small [113]; however, a stochastic background might lead to a detectable effect. Even though vector perturbations are usually not generated during inflation (and if they are they decay during the subsequent radi-ation dominated phase), they are relevant in many models with sources like, e.g., cosmic strings or primordial magnetic fields. A third important motivation to study vector and tensor contributions comes from the fact that at second order in pertur-bation theory, scalars also generate vector and tensor perturpertur-bations [114, 115]. In a complete second order treatment these have to be included. With the formalism developed in this work, such an inclusion is straight forward. We plan to report on the result of these second order contributions in a forthcoming paper [116]. A similar program is carried out in Refs. [117, 118]. There the authors discuss scalar, vector, and tensor perturbations and split them into E and B modes. The treatment of these papers is, however, more adapted to describe distortions of surveys and weak lensing, but the convergence calculated there is related to our distance fluctuations.

The paper is organized as follows. In the next section we discuss the luminosity-redshift relation perturbatively at first order. In Sec. 4.3 we apply these results to tensor perturbations. We first derive the general first order expressions, which we then expand in spherical harmonics. We also give a numerical example for the grav-itational wave background from inflation. In Sec. 4.4 we treat vector perturbations and in Sec. 4.5 we conclude. Some lengthy calculations and some details are deferred to four Appendices.

Notation: We use the metric signature (−,+,+,+). We denote the derivative w.r.t. the conformal time η with a dot.