Theoretical and Observational Aspects of Dark Energy

Thesis

Reference

Theoretical and Observational Aspects of Dark Energy

BONVIN, Camille

Abstract

Durant ma thèse, j'ai travaillé sur le problème de l'accélération de l'Univers, également appelé problème de l'énergie sombre. Au travers de trois projets, j'ai étudié certains aspects théoriques et observationnels de ce problème. D'un point de vue plutôt observationnel, j'ai calculé les fluctuations de la distance lumineuse des supernovae, générées par les perturbations de densité de matière dans l'Univers. Le but de ce projet était d'utiliser ces fluctuations dans la distance lumineuse pour extraire de l'information sur le contenu de l'Univers. Dans mes deux autres projets, plus théoriques, j'ai exploré certaines propriétés de deux modèles spécifiques. Le premier est un modèle d'énergie sombre, appelé k-essence, pour lequel j'ai démontré l'existence de signaux superluminaux, générant des problèmes de causalité. Le deuxième modèle étudié est un modèle de modification de la gravitation, la théorie généralisée de l'aether, dont j'ai montré la compatibilité avec les observations du système solaire.

BONVIN, Camille. Theoretical and Observational Aspects of Dark Energy. Thèse de doctorat : Univ. Genève, 2008, no. Sc. 3992

URN : urn:nbn:ch:unige-6091

DOI : 10.13097/archive-ouverte/unige:609

Available at:

http://archive-ouverte.unige.ch/unige:609

Disclaimer: layout of this document may differ from the published version.

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Theoretical and Observational Aspects of Dark Energy

TH`ESE

pr´esent´ee `a la Facult´e des sciences de l’Universit´e de Gen`eve pour obtenir le grade de

Docteur `es sciences, mention physique

par

Camille BONVIN

de

Chermignon (VS)

Th`ese N^◦ 3992

GEN`EVE

Atelier de reproduction de la Section de physique 2008

Remerciements

Je tiens en premier lieu `a remercier de tout coeur ma directrice de th`ese Ruth Durrer, pour sa comp´etence, sa disponibilit´e et sa gentillesse, ainsi que pour sa fabuleuse aptitude

`

a transmettre sa passion pour la recherche. J’ai eu un plaisir immense `a travailler sous son œil attentif durant ces quatre ann´ees de th`ese.

Un grand merci ´egalement `a Pedro Ferreira, qui m’a chaleureusement accueillie `a l’Universit´e d’Oxford durant trois mois et grˆace `a qui j’ai pu tirer le meilleur parti de ce s´ejour. Ce fˆut un r´eel bonheur de profiter de ses conseils et de sa connaissance.

C’est avec grand plaisir que je remercie Chiara Caprini, Jean-Pierre Eckmann, Alice Gasparini, Martin Kunz, Danail Obreschkow, Glenn Starkman et Tom Zlosnik avec qui j’ai eu la chance de travailler et d’´echanger de fructueuses discussions: j’ai ´enorm´ement b´en´efici´e de ces interactions.

Je tiens ´egalement `a remercier les membres pr´esents et pass´es du groupe de cosmologie, grˆace auxquels l’ambiance au d´epartement est si agr´eable: Thierry Baertschiger, Umberto Cannella, Cyril Cartier, Florian Dubath, Elisa Fenu, Stefano Foffa, Michele Maggiore, Syksy R¨as¨anen, Hillary Sanctuary, Domenico Sapone, Natalia Shuhmaher, Riccardo Stu- rani, Sebastian Szybka, Roberto Trotta, Antti Va¨ıhk¨onen, ainsi que Marcus Ruser, mon sympathique compagnon de bureau avec qui j’ai partag´e de tr`es bons moments au cours de ces quatre ann´ees. Merci aussi `a Andreas Malaspinas, Nathalie Chaduiron, Dani`ele Cheva- lier, Francine Gennai-Nicole, C´ecile Jaggi et Christine Schaffter, les perles du d´epartement, pour leur aide ind´efectible et efficace.

Je remercie enfin du fond du coeur mes parents Marianne et Raymond et mon petit fr`ere Vivien, qui ont toujours cru en moi et m’ont soutenue avec tendresse et fiert´e. Je leur dois la chance de pr´esenter cette th`ese.

Et puis, un merci tout particulier `a Marco, mon ami, mon ˆame soeur, qui me comprend et m’accompagne si bien et qui m’a apport´e une aide inestimable tout au long de cette th`ese.

Examinateurs

Le jury de cette th`ese se compose de

• Prof. Ruth Durrer, Universit´e de Gen`eve, Suisse.

• Dr. Pedro Ferreira, Oxford University, UK.

• Dr. Martin Kunz, Sussex University, UK.

• Prof. Michele Maggiore, Universit´e de Gen`eve, Suisse.

Je les remercie d’avoir accept´e de faire partie du jury et d’avoir consacr´e du temps `a la lecture de cette th`ese.

R´ esum´ e

La cosmologie est l’´etude de l’Univers consid´er´e dans son ensemble, qui combine recher- che th´eorique et observationnelle. Durant ces derni`eres d´ecennies, la confrontation entre th´eorie et exp´erience a mis `a jour l’existence d’importantes interrogations concernant notre compr´ehension de l’Univers. Au cours de ma th`ese, j’ai ´etudi´e diff´erents aspects de l’une de ces questions fondamentales qu’est l’expansion acc´el´er´ee de notre Univers. Ce com- portement, observ´e pour la premi`ere fois en 1998, est en effet `a l’oppos´e des pr´edictions th´eoriques, puisque l’attraction gravifique agissant sur le contenu (mati`ere et radiation) de l’Univers est suppos´ee ralentir l’expansion. Trois pistes sont actuellement explor´ees dans le but d’expliquer ce comportement inattendu. La premi`ere solution consiste `a introduire une nouvelle forme d’´energie dans l’Univers, appel´ee ´energie sombre. La seconde solu- tion implique la modification de la th´eorie de la Relativit´e G´en´erale `a grandes ´echelles.

Quant `a la derni`ere approche, elle explore l’id´ee que les inhomog´en´eit´es dans la densit´e de mati`ere pr´esente dans l’Univers pourrait reproduire, par r´etroaction (backreaction), un effet d’acc´el´eration. Durant ma th`ese, j’ai ´etudi´e le probl`eme de l’acc´el´eration de l’Univers de deux points de vue, l’un plutˆot observationnel, dans le but d’identifier quelle informa- tion l’on peut obtenir sur l’acc´el´eration de l’Univers `a partir de la distance lumineuse des supernovae; et l’autre plus th´eorique, o`u j’ai explor´e certaines propri´et´es de deux mod`eles sp´ecifiques: l’un d’´energie sombre et l’autre de modification de la gravitation. Ces trois projets m’ont permis de construire une image exhaustive des probl`emes et d´efis de la cos- mologie actuelle.

Dans mon premier projet, j’ai investigu´e l’id´ee d’utiliser les fluctuations de la dis- tance lumineuse des supernovae pour extraire de l’information sur la vitesse d’expansion de l’Univers et sur son contenu. Ce travail repose sur la constatation que les perturbations de densit´e dans l’Univers modifient le flux d’´energie que l’on re¸coit des supernovae, et donc elles affectent leur distance lumineuse. En cons´equence, on peut consid´erer les fluctuations dans la distance lumineuse, non plus comme un bruit qui perturbe le signal (ce qui est le cas actuellement), mais comme un signal suppl´ementaire qui contient des informations sur l’´evolution temporelle des perturbations de densit´e. Comme cette ´evolution d´epend

´etroitement du m´echanisme responsable de l’acc´el´eration de l’Univers, les fluctuations de la distance lumineuse constituent un nouvel outil permettant d’une part de diff´erencier entre ´energie sombre et modification de la gravitation, et d’autre part de mesurer certains param`etres cosmologiques avec pr´ecision. Dans mon travail, j’ai ainsi calcul´e la distance lumineuse dans un Univers perturb´e, dont j’ai identifi´e les diff´erentes contributions [1].

Un calcul semi-analytique dans un mod`ele simplifi´e d’Univers (constitu´e exclusivement de mati`ere sombre), a mis en ´evidence que la contribution dominante vient du lensing, c’est-`a-dire de la d´eviation des rayons lumineux par les inhomog´en´eit´es. J’ai ´egalement d´emontr´e que la partie dipolaire des fluctuations, qui d´ecoule de la vitesse de l’observateur par rapport au r´ef´erentiel cosmologique, est extrˆement int´eressante puisqu’elle permet de mesurer de mani`ere directe le param`etre de HubbleH(z) comme fonction du redshiftz[2].

Ceci constitue une am´elioration d´ecisive par rapport au monopole de la distance lumineuse qui ne fournit qu’une mesure int´egr´ee de H(z), et qui par cons´equent ne permet pas une d´etermination pr´ecise de ses variations temporelles.

Dans mon deuxi`eme projet, j’ai ´etudi´e la propagation d’information dans un mod`ele d’´energie sombre appel´e k-essence. Dans ce mod`ele, un champ scalaire auquel sont as- soci´es des termes d’´energie cin´etique non-quadratiques est responsable de l’acc´el´eration de l’Univers. L’int´erˆet du mod`ele de k-essence r´eside dans son aptitude `a r´esoudre le probl`eme de co¨ıncidence; il explique en effet naturellement pourquoi l’acc´el´eration de l’Univers a commenc´e seulement tr`es r´ecemment. N´eanmoins, j’ai d´emontr´e dans mon travail que la r´esolution du probl`eme de co¨ıncidence s’accompagne in´evitablement de la pr´esence de sig- naux se propageant dans l’Univers `a une vitesse superluminale [3]. Comme un tel comporte- ment est en contradiction avec les lois de la relativit´e restreinte, j’ai explor´e ses cons´equences sur la causalit´e [4]. Dans ce travail, j’argumente que deux positions peuvent ˆetre adopt´ees face `a l’existence de propagation superluminale. La premi`ere approche respecte l’un des principes fondamentaux de la relativit´e qui affirme que tous les observateurs (autrement dit tous les syst`emes de r´ef´erence) sont ´equivalents et par cons´equent admissibles pour d´ecrire les lois de la physique. Le respect de ce principe m`ene alors directement `a l’existence de courbes ferm´ees le long desquelles de l’information peut se propager. Ainsi, dans le contexte de la relativit´e restreinte, l’existence de propagation superluminale viole la causalit´e, ce qui rend le mod`ele de k-essence non viable. Je pr´esente ´egalement une deuxi`eme approche qui permet d’exclure la pr´esence de courbes ferm´ees et donc de sauver le mod`ele de k-essence, mais dont les cons´equences sont importantes puisqu’elles impliquent l’abandon du principe de relativit´e au travers du choix d’un r´ef´erentiel privil´egi´e.

Enfin, dans mon dernier projet, j’ai ´etudi´e une th´eorie de la gravitation modifi´ee, ap- pel´ee th´eorie g´en´eralis´ee de l’´ether¹. Dans ce mod`ele le Lagrangian poss`ede, en suppl´ement de la m´etrique, un champ vectoriel de type temps. Ce champ permet de modifier les lois de la gravitation aux ´echelles galactiques et cosmologiques, et par cons´equent, de r´esoudre le probl`eme de l’acc´el´eration de l’Univers autant que celui de la mati`ere noire. Dans mon travail, j’ai ´etudi´e le comportement de l’´ether dans le syst`eme solaire afin de d´eterminer si ce mod`ele est en accord avec les contraintes obtenues `a partir des observations de l’avance du p´erih´elie de Mercure ainsi que du temps de retard des signaux radars [5]. J’ai d´emontr´e qu’il existe une classe de solutions compatibles avec ces contraintes. La particularit´e de ces solutions r´eside dans la pr´esence de corrections au potentiel Newtonien qui croissent avec la distance. Ce comportement, bien que non-standard `a l’´echelle du syst`eme solaire, est n´eanmoins en parfait accord avec une amplification de l’interaction gravifique aux ´echelles galactiques, n´ecessaire pour r´esoudre le probl`eme de la mati`ere noire.

Ces trois projets m’ont permis, chacun `a leur mani`ere, d’appr´ehender certains des d´efis de la cosmologie actuelle, comme par exemple, les difficult´es li´ees `a l’´elaboration de mod`eles cosmologiques en accord, autant avec les principes fondamentaux de la physique qu’avec de nombreuses contraintes observationnelles extrˆemement pr´ecises. Ils ont ´egalement mis en

´evidence l’importance des deux aspects de la recherche cosmologique que sont la recherche th´eorique et la recherche observationnelle, ainsi que la n´ecessit´e de combiner ces deux approches pour pouvoir construire un mod`ele consistent de notre Univers.

1Generalized Einstein-Aether theories

1 Introduction 11

1.1 Luminosity distance . . . 13

1.2 K-essence dark energy . . . 17

1.3 Modified gravity theories . . . 21

2 Fluctuations of the luminosity distance 25 2.1 Introduction . . . 27

2.2 The luminosity distance in inhomogeneous geometries . . . 28

2.3 The luminosity distance in a perturbed Friedmann universe . . . 32

2.3.1 Conformally related luminosity distances . . . 32

2.3.2 The Jacobi map in a perturbed Friedmann universe . . . 34

2.4 The luminosity distance power spectrum . . . 40

2.4.1 The dipole . . . 40

2.4.2 The higher multipoles . . . 42

2.5 Results for a pure CDM universe . . . 44

2.6 Conclusions and outlook . . . 50

2.7 Appendix . . . 52

2.7.1 Christoffel symbols and the Riemann tensor . . . 52

2.7.2 The derivation of Eq. (2.59) . . . 53

2.7.3 The power spectrum . . . 54

2.7.4 Details for the power spectrum . . . 56

2.7.5 Integrals and approximations . . . 58

3 Dipole of the luminosity distance: a direct measure of H(z) 69 4 No-go theorem for k-essence dark energy 79 5 Superluminal motion and closed signal curves 89 5.1 Introduction . . . 91

5.2 Closed signal curves from superluminal velocities . . . 92

5.2.1 Lagrangians which allow for superluminal motion . . . 94

5.3 Closed signal curves on a background . . . 95

5.3.1 k-essence . . . 96

5.4 Conclusions . . . 100

6 Generalized Einstein-Aether theories and the solar system 103 6.1 Introduction . . . 105

6.2 Field equations . . . 106

6.3 Spherically symmetric static metric . . . 107

6.3.1 Weak field approximation . . . 109

6.3.2 Additional terms . . . 110

6.4 Conclusion . . . 115

6.5 Appendix . . . 117

6.5.1 On Asymptotic Flatness . . . 117

6.5.2 Specific expression of the equations of motion . . . 118

6.5.3 Order of magnitude of the vector field perturbations . . . 119

6.5.4 Resolution . . . 120

7 Conclusions and outlook 135

Introduction

Cosmology is the study of the Universe as a whole. It combines theoretical calculations with precise observations. On the theoretical side, cosmology requires a theory of matter in order to describe the content of the Universe, as well as a theory of gravity in order to govern its evolution. The simplest complete theory of matter is nowadays the Standard Model of particle physics. And the most well known and accepted theory of gravity is General Relativity, which links in a precise way the content of the Universe to its geometry.

Both of these theories have successfully passed numerous observational tests, which have legitimated them. However, confrontation with cosmological observations has revealed fundamental problems. One of the most famous evidence that our Universe behaves in a very unexpected way comes probably from supernovae observations in 1998. At this time, two groups of astronomers [6, 7] concluded from their observations of luminosity distance of supernovae that the Universe is presently undergoing a phase of accelerated expansion.

We know since Hubble’s observations in 1929 [8] that the Universe is in expansion. This fact is well explained by General Relativity, since an expanding Universe is a solution of Einstein’s equation [9]. However, the discovery that the expansion was accelerating came as a complete surprise. Indeed, gravitational interactions acting on the content (matter and radiation) of the Universe was expected to slow down the expansion. Since the first observations in 1998, the acceleration of the Universe has been corroborated by other experiments. Among them, the Cosmic Microwave Background (CMB), coupled with other measurements, provided a strong evidence towards acceleration [10].

The other big puzzle in our Universe is related to observations at galactic scales. In 1933, Zwicky observed for the first time that the amount of visible matter inside the Coma galaxy cluster was not sufficient to explain the velocity of galaxies [11, 12]. Forty years later, observations of the motion of stars within galaxies revealed similar discrepancies between the amount of visible matter and the strength of gravitational interaction [13]. Calculations relating the rotational velocity of stars around the center of a galaxy to the amount of matter inside the galaxy using Newtonian’s limit of General Relativity, showed indeed that the velocity of stars in the outer part of the galaxy were much larger than expected.

Recently, it was also observed that the visible matter present in clusters of galaxies is

not sufficient to explain the growth of structures or the light deflection of distant objects.

Hence observations at galactic and cosmological scales reveal a number of imperfections in our understanding of the Universe if we adopt General Relativity as the theory of gravity and the Standard Model of particle physics as the theory of matter.

Three directions are currently explored to explain the acceleration of the expansion of the Universe. The first approach consists in invoking the presence of a new exotic form of energy, called dark energy, able to drive the acceleration [14]. The second widely studied proposition is modification of General Relativity at cosmological scales. An alternative theory of gravity could indeed be compatible with acceleration [15]-[18]. Finally, the last approach studies the possibility that both the Standard Model and General Relativity are correct, but that discrepancies with observations follow from erroneous assumptions in the calculation of the Universe expansion rate [19]. The motivation for these three different approaches follows from Einstein’s equations

Gµν = 8πGTµν . (1.1)

Here Gis Newton’s constant,G_µν represents Einstein’s tensor, which defines the geometry of the Universe andT_µν denotes the energy momentum tensor, which describes the matter and energy content of the Universe. Hence Einstein’s equations relate the geometry of the Universe to its content. For a homogeneous and isotropic Universe, these equations give rise to the Friedmann equations

a˙ a

2

+K

a² = 8πG

3 ρ (1.2)

¨ a

a = −4πG

3 (ρ+ 3P) , (1.3)

where ais the scale factor of the Universe, ρ its energy density, P its pressure and K is the curvature. A dot denotes derivative with respect to cosmic time. The left hand side of Friedmann equations follows directly from the left hand side of Einstein’s equations.

Hence the geometry of a homogeneous and isotropic Universe is completely described by the evolution of its scale factor a. Equivalently, the right hand side of Friedmann equa- tions, which follows from the right hand side of Einstein’s equations, describes the content of the Universe. An accelerating Universe is characterized by ¨a >0. Equation (1.3) tells us that this cannot be achieved by usual matter and radiation, since it requires a fluid with negative pressureP <−^ρ₃. The first solution to the acceleration problem consists then in modifying the right hand side of Friedmann equations by adding a new component with negative pressure (dark energy) in the Universe, leading to ¨a > 0. The second approach tends to modify the left hand side of Friedmann equations, by changing the laws of gravity.

Concretely it implies to modify either Einstein’s tensor G_µν or the relation between G_µν and Tµν. Finally the third approach suggests that Einstein’s equations are correct but that Friedmann equations do not describe properly the evolution of our Universe. Indeed, those are valid under the assumption that the Universe is approximately homogeneous and isotropic and that density perturbations have no impact on its global evolution. However,

nonlinearities of Einstein’s equations could result in a backreaction effect of these per- turbations on the expansion rate of the Universe and consequently invalidate Friedmann equations. Hence discrepancies between observations and calculations could be solved by using Einstein’s equations to compute the expansion rate without assuming a homogeneous and isotropic Universe.

There are two different and complementary ways of testing the pertinence and consis- tency of the three approaches. The first relies on current and future observations. Indeed, the first step towards a solution to the acceleration problem consists in a better knowledge of the expansion rate of the Universe, its evolution and its variations as a function of red- shift. This is crucial in order to determine whether the acceleration is due to dark energy, modification of gravity or backreaction. Since experiments provide more and more precise data, new methods have to be constructed in order to exploit intelligently and completely the available information. The other direction of making progress is to construct new cos- mological models able to reproduce acceleration and to confront them with fundamental principles, as well as observational constraints. This highlights the main difficulties and challenges of the three approaches, and consequently it provides a first insight in their feasibility and internal consistency.

In this thesis, we present some work on these two aspects of cosmological research. First we propose a new method to get information on the evolution of the Universe, by using fluctuations of the luminosity distance of supernovae due to density perturbations in the Universe. Then we study some properties of two different cosmological models. We explore causality in a dark energy model called k-essence and we test compatibility of a modified theory of gravity (the generalized Einstein-Aether theory) with solar system constraints.

These three projects provide a general view on the different features of the problem of acceleration.

1.1 Luminosity distance

Supernovae type Ia are exploding stars which can be used as indicators of distances in the Universe. They can indeed be regarded as modified standard candles. More precisely we can deduce their intrinsic luminosity (the total energy emitted per unit of time), by observing their light curves. Consequently, it is sufficient to measure their flux to have information on their distance from us. In an expanding Universe, the notion of distance has to be carefully defined. One meaningful definition is the luminosity distance

d_L= r L

4πF , (1.4)

whereLis the intrinsic luminosity andF is the flux measured at the observer position. In a static Euclidian Universe, luminosity distance reduces to ordinary distance. The energy emitted by the supernova is indeed distributed on a sphere centered on the source and therefore the flux received by an observer sitting at distance dfrom the source is simply the intrinsic luminosity L divided by the surface of the sphere 4πd². The luminosity

distance dL generalizes this way of computing distances to an expanding Universe. One can easily understand that d_L is affected by the expansion. Indeed, the photon number emitted by a source per unit of surface is diluted as spacetime expands. As a result, the flux measured by the observer is reduced by the expansion, leading to an increase of the luminosity distance. Hence luminosity distance is a measure of distance sensitive to the expansion of the Universe.

In 1998, the luminosity distance of about 50 supernovae has been measured by two groups of astrophysicists [6, 7]. Comparison of the data with theoretical predictions has lead to the claim that the Universe is currently in acceleration. A crucial assumption has however been used for this calculation: the Universe has been taken as homogeneous and isotropic. In this case, the relation between the luminosity distance and the Hubble parameter H, which describes the expansion of the Universe, is given by [20]

dL(z) = 1 +z

p|Ω_K|H₀ χ p

|ΩK|H0

Z z 0

dz^′ (1 +z^′)H(z^′)

!

, where (1.5)

χ(x) =







x in the Euclidean case, K= 0 sin(x) in the spherical case, K >0 sinh(x) in the hyperbolic case, K <0.

(1.6)

Here Ω_K =−_H^K2

0a²₀,H₀ is the value of the Hubble parameter today andzis the redshift of the supernova. Hence measurements of luminosity distance of supernovae at different redshift z allow to compute the evolution of the Hubble parameter as a function of z and therefore to infer whether the Universe is accelerating or decelerating. However, the as- sumption that the Universe is homogeneous and isotropic is not completely correct, even on large scales. The matter energy density in the Universe exhibits indeed small fluctuations, which have an effect on the luminosity distance. They modify the photon trajectory from the supernova to the observer, and consequently they affect the flux measured by the ob- server. Hence density perturbations in the Universe create luminosity distance fluctuations and modify therefore equation (1.5). Our idea is to use luminosity distance fluctuations, which will be observable in future supernovae experiments [21, 22], to get information on density perturbations in the Universe.

The technical details of the computation of luminosity distance in an inhomogeneous and anisotropic Universe are given in paper [1] which is reproduced in Chapter 2. We perform a calculation at first order in the metric perturbations, on a Friedmann-Lemaˆıtre- Robertson-Walker (FLRW) background. This reveals different contributions to the lumi- nosity distance fluctuations. First, peculiar motions of the supernova and the observer with respect to the Hubble flow affect the luminosity distance through a Doppler effect. We will discuss in more detail hereafter the Doppler contribution of the observer velocity, which initiates a new tool to measure the Hubble parameter directly. Secondly, we find a gravita- tional redshift contribution, simply due to the difference of gravitational potential between the supernova and the observer. We also find effects involving the integrated gravitational potential along the photon path, as well as its integrated time derivatives. Those are similar

to the integrated Sachs-Wolf effect in the Cosmic Microwave Background (CMB). Finally we identify the gravitational lensing contribution, which ensues from deviation of light by density perturbations.

Contrary to the background contribution in equation (1.5) which depends only on the redshift z of the supernova, all the perturbed effects depend also on the direction of observation n. This simply follows from anisotropies in density perturbations. It is therefore meaningful to perform a multipole expansion of luminosity distance fluctuations, similar to what is done with the CMB [20]. We write

d_L(z,n) =X

ℓm

a_ℓm(z)Y_ℓm(n) , (1.7)

where Y_ℓm(n) are the spherical harmonics. We can then construct the angular power spectrum at redshift z

C_ℓ(z) =ha_ℓm(z)a^∗_ℓm(z)i . (1.8) Here the h·i denotes a statistical average. Like for the CMB, statistical isotropy implies that the C_ℓ’s are independent ofm. The main difference with respect to the CMB is that the angular power spectrum is now a function of redshift. Technically this means that we collect supernovae situated at the same redshift (or in the same bin of redshifts) but in different directions, and we expand the resulting signal in multipoles on the sphere.

We evaluate then the angular power spectrum of all contributions to the luminosity distance fluctuations in a very simple model of Universe: a cold dark matter (CDM) Uni- verse. In this specific case, the Universe is completely filled with cold dark matter; and other contributions like radiation are neglected. Moreover there is no dark energy compo- nent. This model is therefore not realistic, but its interest lies in the fact that the different effects can be calculated semi-analytically. We determine that in a CDM Universe, the gravitational lensing term constitutes, at large redshifts z > 0.4, the largest contribution, and that it affects mainly small scales ℓ > 100 . This effect is up to five hundred times larger than the variance of density perturbations at linear level, hence it is potentially observable. The reason for this enhancement is the large number of inhomogeneities that a single ray encounters during his travel from the supernova to the observer. At small redshiftsz <0.4, the dominant effect comes from the peculiar velocity of the observer and the supernova, and mainly large scales ℓ <80 are affected.

The principal interest of luminosity distance fluctuations lies in the information they provide on matter perturbations in the Universe, and more precisely on their time evolu- tion. Measurements of the angular power spectrum at different redshifts translate indeed into measurements of the evolution of density perturbations. This latter is crucial to under- stand properly the mechanism leading to the Universe acceleration. In order to distinguish between dark energy and modified gravity theories, we need indeed to know the effect of acceleration on both the expansion history of the Universe and the evolution of density per- turbations. General Relativity predicts a definite relation between these two terms, which is generically violated by theories of modified gravity. Therefore, observations sensible to the expansion of the Universe as well as to the growth rate of perturbations are necessary to constrain dark energy and modified gravity theories [23]. The strength of luminosity

distance observations is that they give access to both of these terms: the averaged expan- sion of the Universe is probed by the background value of the luminosity distance, and also, as explained below, by the dipole term of the fluctuations; whereas the evolution of perturbations is tested through higher multipoles. Hence luminosity distance fluctuations supply strong basis to discriminate between dark energy and modified gravity.

As mentioned above, the dipole part of the fluctuations allows to measure the expansion rate of the Universe. Contrary to higher multipoles, the dipole is not caused by the effect of density perturbations on the trajectory of light, but rather by their impact on the observer velocity. In a homogeneous and isotropic Universe, the observer and the supernova move in the same way as the Universe expands: they follow geodesics of the FLRW Universe. This collective motion is called the Hubble flow. In a perturbed Universe, density fluctuations generate a peculiar motion of both the observer and of the supernova with respect to the Hubble flow. This peculiar velocity affects the luminosity distance through a Doppler effect. The relative motion of the observer and the supernova modifies indeed the energy density of photons received by the observer. The peculiar velocity of the observer creates simply a dipole term, whereas the one of the supernova generates larger multipoles. The amplitude of the dipole is given by

d⁽¹⁾_L (z) = |v₀|(1 +z)²

H(z) , (1.9)

and its direction is e=v₀/|v₀|, where v₀ is the observer peculiar velocity. The derivation of equation (1.9) is given in paper [2] which is reproduced in Chapter 3. The velocity v₀ can be measured from the CMB dipole which is due to the same motion. It follows that the dipole in the supernovae data provides a direct measure of H(z). The dependence onH(z) can be understood as follow: when the observer selects supernovae at the same redshift but in different direction, he observes supernovae which are not situated at the same conformal time. The sphere of constant redshift is indeed slightly shifted with respect to the sphere of constant conformal time, due to the observer velocity (see fig.1.1). Supernovae sitting at the same redshift (measured by the observer) do therefore not move with the same Hubble velocity: they do not experience exactly the same expansion rate. This difference generates the H(z) term in the dipole of the luminosity distance.

O

z = cst v0

η = cst η = cst

O

Figure 1.1: Shift of the sphere of constant redshiftzwith respect to the sphere of constant conformal time η due to the observer velocity v₀.

This direct dependence of the dipole in the Hubble parameter is extremely interesting.

The equation of state of dark energy depends directly on the Hubble parameter and on its first derivative with respect toz[2]. Hence it is crucial to have good measurements ofH(z) in order to compute the evolution of the equation of state and consequently to constrain

dark energy models. The monopole of the luminosity distance has the disadvantage to depend on the Hubble parameter through an integration over redshifts (see equation 1.5).

This renders it rather insensitive to small localized variations ofH(z). The direct relation between the dipole and the Hubble parameter offers therefore a great improvement in measurements of the Universe expansion rate.

Hence our work on luminosity distance shows that not only the monopole part but also the directional dependence of luminosity distance provide information on the expansion of the Universe and on its content. Luminosity distance fluctuations can therefore be considered as a new signal rich in information, rather than as noise on the data. They supply consequently a novel observational tool to determine cosmological parameters. The disadvantage of this method lies in the large number of supernovae needed to measure fluctuations with a good accuracy. At present, far not enough supernovae have been observed to achieve precise measurements. However, future supernovae experiments are expected to deliver enough data to render this method interesting [21, 22].

1.2 K-essence dark energy

Supernovae observations indicate that acceleration of the Universe started only recently around redshift 0.5, and that before this period the Universe was in deceleration [24]. Dark energy models have then to reproduce correctly this transition from a decelerating to an accelerating Universe. Most dark energy candidates require extraordinary fine-tuning of some of their parameters or of their initial conditions, to lead to a phase of acceleration precisely today. This problematic is called the coincidence problem. For example, the value of the cosmological constant, the simplest model of dark energy, has to be accurately tuned in order to become dominant exactly today. Equivalently, in quintessence dark energy [25]-[28], some parameter of the field potential has to be chosen very carefully to get acceleration at the required time.

K-essence provides an elegant solution to the coincidence problem. In this model pro- posed by Mukhanov, Steinhard and Picon [29, 30], the Universe contains in addition to usual matter and radiation, a scalar field with non-canonical kinetic terms. Such terms, which are motivated by string and supergravity theories, result in very interesting equations of motion. Indeed, under some conditions, field equations possess fixed points, i.e. solutions which remain constant as the Universe evolves. Some of these fixed points are attractors, and therefore every solution that enters their basin of attraction quickly approaches them and get trapped. They can therefore be used to construct a dynamical explanation to the coincidence problem. Each fixed point is characterized by two quantities, which are fixed by the background content of the Universe. One is the equation of state of k-essence w_k= ^P_ρ^k

k, and the other its energy density ratio Ω_k= _ρ^ρk

tot. HereP_k denotes the pressure of k-essence, ρ_k its energy density andρtot the total energy density in the Universe. The values of w_k and Ω_k differ when the Universe is dominated by radiation, matter or k-essence itself. In the early Universe, when radiation is dominant, the fixed point is characterized by w_k= ¹₃. Hence, k-essence mimics radiation. This point is therefore called the radiation fixed point.

The value of Ω_k depends on the particular form of the non-canonical kinetic terms. When the Universe is dominated by matter, we find two other fixed points. The first one, called de Sitter fixed point, is only an approximative solution of the equation of motion. It is given by w_k≃ −1 and Ω_k≃0. The second one, called the matter fixed point has w_k= 0, meaning that k-essence mimics matter. The value of Ω_kat the matter fixed point is again dependant on the particular form of the non-canonical kinetic terms. Finally, when the Universe is dominated by k-essence, there exists another fixed point called the k-attractor characterized by w_k <0 and Ω_k ≃1. At the k-attractor, k-essence dominates completely the Universe.

Using these fixed points, it is then possible to construct an evolution of the k-essence field compatible with observations. In the early Universe, initial conditions are chosen in the basin of attraction of the radiation fixed point. Hence k-essence quickly reaches this attractor and remains there as long as the Universe is dominated by radiation. It is possible to choose kinetic terms such that at the radiation fixed point Ω_k . 0.1, in order not to violate nucleosynthesis bounds. Then the transition from radiation to matter domination forces the field to leave the radiation fixed point, which is no more a solution of the equations of motion, and to evolve to the de Sitter fixed point. After some time, since the de Sitter fixed point is not an exact solution, the field naturally evolves to another stage. This can be either the matter fixed point or the k-attractor, depending on the particular form of the non-canonical kinetic terms. Those can also be adjusted such that today the field is on its way from the de Sitter fixed point to the matter or k-attractor, and has w_k<−¹₃ and Ω_k≃0.7, in agreement with the observed values. Hence, in this scenario k-essence drives the acceleration of the Universe today: it plays the role of dark energy.

Moreover, the fact that acceleration takes place precisely today is not a coincidence. It is closely related to the transition from radiation to matter domination, which forces the field to leave the radiation fixed point, and consequently to reach acceleration only sometimes after the transition, i.e. today. Furthermore, the evolution of the field is insensitive to the particular choice of initial conditions, as long as they lay in the basin of attraction of the radiation fixed point. K-essence seems therefore a good candidate to explain the current acceleration of the Universe without fine-tuning.

However, in our paper [3] which is reproduced in Chapter 4, we demonstrate that perturbations of the k-essence field propagate superluminally during some stage of the Universe evolution, leading to causality problems. We determine the linearized evolution equations of k-essence perturbations which (in the limit of high wave-number) propagate with the k-essence sound speed. We then prove that this sound speed becomes larger than the speed of light, for all kind of models described above. We establish furthermore that superluminal propagation is not directly due the presence of non-canonical kinetic terms in the action. It follows rather from the transition from a decelerating to an accelerating Universe. Indeed, to evolve from the radiation fixed point, with w_k = ¹₃, to a period of acceleration with w_k < −¹₃, the field has to pass by the matter fixed point w_k = 0.

However, if k-essence falls on the matter fixed point directly after leaving the radiation fixed point, it will stay there for ever and mimic matter. Consequently it can not drive the acceleration of the Universe. Hence, the action for k-essence has to be chosen such that the field avoids the matter attractor after leaving the radiation fixed point. We have demonstrated that this choice results in an increase of the sound speed above the value of

the speed of light shortly after the radiation-matter transition. Hence, the very property which makes k-essence valuable, i.e. its ability to evolve naturally from a decelerating to an accelerating Universe, is responsible for superluminal motion of the field perturbations.

Special Relativity states that nothing can propagate faster than the speed of light [31].

It is therefore important to investigate the consequences of superluminal propagation in order to decide if k-essence is ruled out or not as a good dark energy model. In our preprint paper [4] which is reproduced in Chapter 5, we study in details this problematic. We show that two different positions can be adopted regarding superluminal propagation. The first approach, which respects the principle of Einstein’s Relativity, leads to the existence of closed curves along which a signal can propagate, i.e. closed signal curves. This entails a clear violation of causality, therefore, superluminal motion rules out the theory. The other proposal enables one to save a theory with superluminal propagation by preventing the existence of closed signal curves. However, the consequence of this approach is violation of Lorentz invariance.

The origin of these two approaches resides in the ambiguity to define a time direction of propagation for superluminal signals. The equations of motion define the sound speed of perturbationsc_s, but not the time direction of propagation. The sound speed is indeed the same for a signal propagating into the future (dt >0) and to the right (dx >0) or into the past (dt <0) and to the left (dx <0). If the sound speed is smaller than the speed of light, we can simply impose that all signals have to propagate into the future. This choice can be made for all observers at once, independently of their motion. Past and future are indeed well defined notions and therefore, if dt >0 in one reference frame, after a Lorentz transformation dt^′ > 0. On the contrary, if the sound speed is larger than the speed of light, it is no more possible to define past and future without ambiguity for all observers.

One can for example choose dt > 0 for an observer, and after a Lorentz transformation obtain dt^′ < 0. Moving observers therefore disagree on the notion of past and future for a signal propagating at a speed c_s > c. Hence, we need to define a new rule in order to describe the propagation of superluminal signals. The existence or not of closed signal curves depends then on the rule chosen.

One of the basic principle of Einstein’s relativity states that every reference frame is suitable to describe physics. All observers are indeed on equal footing and therefore they follow the same laws of physics, independently of their motion. The only way to respect this principle is then to construct a rule describing superluminal motion which is the same for all observers. Each observer can define his proper time, as the time given by a clock at rest with respect to himself. This notion of proper time allows him to determine his proper past and his proper future. We define then propagation of superluminal signals such that an observer always sends signal into his proper future. This rule removes the ambiguity about time direction of propagation while respecting Einstein’s principle: all observers behave in the same way. Of course an observer moving with respect to the sender can see the signal propagating into his past. But this does not contradict the rule, since only the observer who sends the signal has to see it propagating into his future. However, the direct consequence of this rule is the existence of closed signal curves. In figure 5.1 (see chapter 5), the construction of such a curve is presented. It involves two observers, one moving with

respect to the other, sending signals with two different superluminal velocities. Hence we conclude that, in the context of Einstein’s relativity, the existence of superluminal motion leads to violation of causality and therefore rules out the theory.

However, if we are ready to give up Einstein’s Relativity principle, it is possible to con- struct a rule governing superluminal propagation, which excludes the existence of closed signal curves. We can indeed choose a preferred reference frame (for example the cosmo- logical frame in which the k-essence background is homogeneous and isotropic) and define past and future with respect to this frame. All observers have then to send signals into the future as defined by this frame. Consequently no closed signal curve can be constructed, since no signal can propagate into the past in this preferred frame. This rule is in contra- diction with Einstein’s Relativity principle since it does not regard observers as equivalent.

Some moving observers can for example not use their proper time as a suitable time coor- dinate to describe propagation of signals. According to their proper time they indeed have to send superluminal signals into their proper past. They have then to use the proper time of the preferred observer to describe physics. It is therefore not possible to have a theory with superluminal motion which satisfies both causality and Lorentz invariance. Even if the Lagrangian is invariant under Lorentz transformation, the choice of a preferred frame breaks the symmetry. Hence it is no longer equivalent to solve the equations of motion in the preferred frame or in a moving frame. In the preferred frame, we can solve the equa- tions of motion and choose the retarded Green function as the physical solution, whereas in some moving frames this procedure does not work any more. The rule states indeed that the retarded Green function is not always the one which describes properly signal propagation. In some moving frame, we have to take a mixture of retarded and advanced Green function [32]. Hence there is only a class of reference frames in which the Green function is determined by initial conditions in the past.

The other strange consequence of this approach is the important role played by the cosmological (preferred) frame on small scales. Up to now we never observed any effect of this frame on our scale. It is in principle meaningful only on cosmological scales, where it is related to the homogeneous and isotropic background. In the context of superluminal propagation, the cosmological frame becomes however also crucial on small scales, since it defines in which time direction signals have to propagate. Hence, even on our scale Lorentz symmetry is broken. However boost symmetry is tested with very high accuracy in particles colliders, which implies that interactions of k-essence with Standard Model particles must be extremely small in order not to generate an observable breaking of this symmetry at the currently available energies.

Hence even if the second approach does not contain any logical contradiction and is con- sequently appealing to save a theory with superluminal propagation, it leads to non trivial consequences on the fundamental Relativity principle of Einstein and therefore seems to us very unphysical. In the end we will have to rely on observations to know if superlumi- nal motion exists or not. If Einstein’s principles are correct, we know that we will never observe superluminal motion, since it would mean that causality is violated. On the other hand, if we do observe superluminal motion, then we will know that different observers are not equivalent and that Lorentz invariance has to be given up.

1.3 Modified gravity theories

The second way of generating acceleration of the expansion of the Universe is by modifying the laws of gravity at cosmological scales. Such a modification is not only motivated by accelerated expansion, but also by observations at galactic scales. The amount of visible matter (stars, gas, ...) that we detect directly seems indeed not sufficient to explain gravitational interactions in structures like galaxies and clusters of galaxies [33]. As for the problem of acceleration of the Universe, disagreement between observations and theoretical calculations can be explained either by invoking the presence of a new form of matter, called dark matter, or by modifying the laws of gravity at galactic scales. It is then appealing to look for a modified theory of gravity which could explain discrepancies at both galactic and cosmological scales. Modifications of gravity can be achieved in a covariant way either by modifying the form of the Einstein-Hilbert action, like in f(R) theories [34], or by introducing extra fields in the action which alter (directly or indirectly) the relation between the metric and the matter content. An example of modified theory of gravity containing an scalar field is Brans-Dicke theory [35]. Einstein-Aether theories [36] require the presence of a vector field, whereas TeVeS Bekenstein theory [37] contains a scalar and a vector field in addition to the metric field. Hence a plethora of modified theories of gravity have been constructed, which all have to satisfy observational constraints. In our paper [5] which is reproduced in Chapter 6, we study the compatibility of one class of theories, the generalized Einstein-Aether theories, with solar system experiments.

Generalized Einstein-Aether theories are characterized by the presence in the action of a timelike vector field coupled to gravity. The primordial motivation of this vector field was to introduce spontaneous breaking of Lorentz invariance, leading to variations in the speed of light. It has then been shown that such a dynamical vector field could generate modifications of gravity. Contrary to Einstein-Aether theory, in which the vector field possesses only quadratic kinetic terms, the generalized version allow for non-canonical kinetic terms. The interest of those lies in the possibility to modify the laws of gravity at both galactic and cosmological scales. Generalized Einstein-Aether theories provide consequently a promising alternative theory of gravity. As with General Relativity, they must therefore satisfy the constraints one infers from precise observations in the solar system. In our work we study solutions of generalized Einstein-Aether theories in the solar system. We use then solar system tests to place constraints on parameters of the theories.

At present, mainly three accurate observations allow to test the laws of gravity in the solar system [38]. The first one concerns the perihelion shift of Mercury. The orbit of Mercury is indeed not exactly an ellipse, since it does not close completely. Consequently, the perihelion of the orbit rotates slowly around the Sun. Newton’s law of gravity does not explain this shift, which is however predicted by General Relativity. The extremely good agreement between the observed and the predicted shift value has been a very strong evidence in favor of General Relativity. The second observation is related to deflection of light. One can measure, during solar eclipses, deflection of rays emitted by stars located behind the Sun. Calculations in Newtonian’s gravity predict only half of the observed deflection, whereas General Relativity produces a completely compatible value. Finally, the third observation uses the time delay of radio signals passing through a gravitational

potential to test gravity in the solar system. One sends a signal from the earth toward a planet or a satellite situated at the other side of the Sun, and then observes the reflected signal. General Relativity predicts a time delay of the signal with respect to the Newtonian time, due to its trajectory through the gravitational potential of the Sun. The observed time delay is in excellent agreement with predictions of General Relativity. Hence when modifying General Relativity and constructing a new theory of gravity, one needs to satisfy these three observational constraints provided by the solar system. Observations require therefore a theory which reproduces General Relativity at solar system scales but differs substantially at galactic and cosmological scales.

In our work, we investigate the behavior of generalized Einstein-Aether theories in the Solar System. We study spherically symmetric and static solutions of the equations of motion. We work, as a first step, within the framework of the Post-Newtonian Parame- terization (PPN). This method proposed by Nordtvedt [39] and generalized by Will [38]

uses spherically symmetric and static solutions of General Relativity as a guide for solu- tions in modified theories of gravity. Einstein’s equations can indeed be solved exactly in the solar system, providing the so-called Schwarzschild solution. This is a decreasing function of the distance from the Sun r. Hence one can calculate its weak field limit, by expanding it in inverse powers of r. Solar system tests provide constraints on the first coefficients of this expansion. The idea of PPN is then to replace numerical coefficients of Schwarzschild expansion by parameters and to use it as an ansatz solution for modified theories of gravity. One can then solve the modified equations of motion which determine the parameters. Those are called the Post-Newtonian parameters. A modified theory of gravity is then compatible with solar system observations only if its Post-Newtonian pa- rameters are in agreement with the values inferred from observations. However, in the case of generalized Einstein-Aether theories, we show that the Post-Newtonian ansatz is not a physically acceptable solution of the equations of motion. This ansatz forces indeed one parameter (c1) of the generalized Einstein-Aether Lagrangian to vanish. However it has been shown that c₁ must be strictly negative in order for the theory to possess a positive definite Hamiltonian [40]. We therefore conclude that spherically symmetric and static solutions of generalized Einstein-Aether equations can not be expanded in negative powers of distance from the Sun, except in the special case where the non-canonical kinetic terms reduce to a quadratic one, i.e. in usual Einstein-Aether theory.

We then study a generalization of the Post-Newtonian expansion by adding positive powers of the distance from the Sun. Mathematically, such an expansion makes sense since positive and negative powers define a complete basis. Physically, positive powers are motivated by the dark matter problem. They are indeed required in order to generate an increase of the gravitational potential at large distances from the gravitational source.

However, those terms are generally neglected in the solar system. Coefficients in front of positive powers of r must indeed be very small in order not to interfere with solar system tests. One consequently assumes that they do not play any role in the equations of motion at solar system scale [41, 42, 43]. This is however not correct for generalized Einstein-Aether theories, where positive powers modify the equations in a crucial way, even if their associated coefficients are small. This follows from the presence of a mass parameter in the Lagrangian, whose value is fixed in order to modify gravity at galactic scales. This mass parameter is then small enough to mix the large coefficients in front of

the negative powers with the small coefficients in front of the positive powers, and as a result to modify the equations of motion in a measurable way. In other words, in order to modify gravity at galactic scales generalized Einstein-Aether theories require the presence of a small mass parameter in the Lagrangian. This affects the equations of motion at solar system scales in such a way that positive powers of r are no more negligible. An interesting consequence of positive powers in the expansion is the fact that the parameter c1 of the Lagrangian does not vanish any more and consequently that the Hamiltonian can be positive definite. Furthermore, using positive and negative powers of r, we find a set of solutions compatible with solar system observations. Hence generalized Einstein- Aether theories are in agreement with solar system constraints and are consequently viable alternative theories to General Relativity.

Another important aspect of our work on generalized Einstein-Aether theories is the method we developed in order to prove that an ansatz solution satisfies the equations of motion at each order. It is indeed not sufficient to calculate the Post-Newtonian parameters and to compare them with measured values to show that a solution is compatible with solar system constraints. We need also to be sure that the following coefficients in the expansion do not lead to any inconsistency. In our work we present a method to test whether an expanding solution satisfies the equations of motion at each order. The interest of this method is that it requires only to calculate some of the first coefficients of the expansion, as well as their corresponding equations, to prove that the complete expansion satisfies the equations of motion. It is consequently not needed to solve each order explicitly. Our method is valid for expansion in positive and negative powers of r and is therefore very general.

Fluctuations of the luminosity

distance

PHYSICAL REVIEW D73, 023523 (2006)

Fluctuations of the luminosity distance

Camille Bonvin, Ruth Durrer and M. Alice Gasparini

We derive an expression for the luminosity distance in a perturbed Friedmann universe. We define the correlation function and the power spectrum of the luminosity distance fluctuations and express them in terms of the initial spectrum of the Bardeen potential. We present semi- analytical results for the case of a pure CDM (cold dark matter) universe. We argue that the luminosity distance power spectrum represents a new observational tool which can be used to determine cosmological parameters. In addition, our results shed some light into the debate whether second order small scale fluctuations can mimic an accelerating universe.

DOI: 10.1103/PhysRevD.73.023523 PACS numbers: 98.80.-k, 98.62.En, 98.80.Es, 98.62.Py

2.1 Introduction

Some years ago, to the biggest surprise for the physics community, measurements of lumi- nosity distances to far away type Ia supernovae have indicated that the Universe presently undergoes a phase of accelerated expansion [6, 7, 44]. If the Universe is homogeneous and isotropic, i.e. a Friedmann-Lemaˆıtre universe, this means that the energy density is domi- nated by some exotic ’dark energy’ which obeys an equation of state of the formP <−ρ/3.

The best known dark energy candidate is vacuum energy or, equivalently, a cosmological constant. This discovery has lately been supported by several other combined data sets, like the cosmic microwave background (CMB) anisotropies combined with either large scale structure or measurements of the Hubble parameter [45].

On the other hand, since quite some time, it is known that locally measured cosmological parameters likeH0or the deceleration parameterq0might not be the ones of the underlying Friedmann universe, but they might be dressed by local fluctuations [46, 47, 48]. Therefore, it is of great importance to derive a general formula of the luminosity distance in a universe with perturbations. To some extent, this has been done in several papers before [49]-[57]

But the formula which we derive here is new. We shall comment on the relations later on.

Lately, it has even been argued that second order perturbations might be responsible for the observed acceleration and that no cosmological constant or dark energy is needed [58]- [62]. This claim is very surprising, as it seems to require that back reaction leads to big perturbations out to very large scales, contrary to what is observed in the CMB. This proposal has thus promptly initiated a heated debate [63]-[66].

On the one hand, the present work is a contribution in this context. We calculate the measurable luminosity distance in a perturbed Friedmann universe and determine

its fluctuations (within linear perturbation theory). We show that these remain smaller than one and therefore higher order perturbations are probably not relevant. The main point of our procedure is that we use only measurable quantities and not some abstract averaged expansion rate to determine the deceleration parameter. We actually calculate the luminosity distance dL(n, z) where n defines the direction of the observed supernova and z its redshift. We then determine the power spectrumC_ℓ(z, z^′) defined by

d_L(n, z) = X

ℓm

a_ℓm(z)Y_ℓm(n) (2.1)

C_ℓ(z, z^′) = ha_ℓm(z)a^∗_ℓm(z^′)i . (2.2) Here the h·i denotes a statistical average. Like for the cosmic microwave background, statistical isotropy implies that the C_ℓ’s are independent ofm.

We then analyze whether the deviations of the angular diameter distance from its background value can be sufficient to fake an accelerating universe.

Aside from this problem, the new variable which is defined and calculated in this paper, might in principle present an interesting and novel observational tool to determine cosmological parameters. And this is actually the main point of our work. We hope to initiate a new observational effort, the measurement of the luminosity distance power spectrum, with this paper. A detailed numerical calculation of thed_L power spectrum and the implementation of a parameter search algorithm are postponed to future work. Here we simply show that for large redshifts, z ≥ 0.4 and sufficiently high multipoles, ℓ > 10 the lensing effect dominates. However, at smaller redshift and especially at low ℓ’s other terms can become important, most notably the Doppler term due to the peculiar motion of the supernova.

The paper is organized as follows. In Section 2.2 we derive a general formula for the luminosity distance valid in (nearly) arbitrary geometries. In the next section we apply the formula to a perturbed Friedmann universe. In Section 2.4 we derive general expressions for the dL power spectrum in terms of the Bardeen potentials. We then evaluate our expressions in terms of relatively crude approximations and some numerical calculations for a simple ΩM = 1 CDM model in Section 2.5. In Section 2.6 we discuss our results and conclude.

Notation: We denote 4-vectors by arbitrary letters, sometimes with and sometimes with- out Greek indices,k= (k^µ). Three-dimensional vectors are denoted bold face or with Latin indices, y = (yⁱ). We use the metric signature (−,+,+,+). The covariant derivative of the 4-vector kin direction of the 4-vector nis often denoted by ∇nk≡(n^µk^α_;µ).

2.2 The luminosity distance in inhomogeneous geometries

We consider an inhomogeneous and anisotropic universe with geometry ds² =gµνdx^µdx^ν. We place a standard candle emitting with total luminosityL( energy per unit proper time)

O

S dΩS

dA_O

Figure 2.1: A light beam emitted at the source event S ending on the observer O. At the source position, the plane normal to the source four-velocity is indicated.

at spacetime positionS. Its four-velocity is u_S. An observer at spacetime positionO with four velocity u_O(see Fig. 2.1) receives the energy flux F (energy per unit proper time and per surface). The luminosity distance between the source at S and the observer at O is defined by

d_L(S, O) = r L

4πF . (2.3)

The observer measures the flux F and ’knows’ the intrinsic luminosity L of the standard candle. Furthermore, she determines the source redshift z and direction n and thereby obtains the function d_L(n, z), which we now want to express in terms of the spacetime geometry.

Be dΩS the infinitesimal solid angle around the source and dA(x) the infinitesimal surface element on the surface normal to the photon beam at the position x along the photon trajectory from S to O, then

d²_L(S, O) = dA_O dΩS

(1 +z)²=|detJ(O, S)|(1 +z)² . (2.4) Here J is the so called Jacobi map mapping initial directions δθ_S^α around the source into vectors δx^µ_O transversal to the photon beam at the observer position [67],

δx^µ_O=J^µα(O, S)δθ_S^α . (2.5)

The factor 1 + z = ω_S/ω_O is the redshift of the source. There is a factor 1 +z due to the redshift of the emitted energy and a second factor due to the time dilatation in F ∝ dE_O/dτ_O with respect to L = dE_S/dτ_S. If k denotes the 4-vector of the photon momentum and u_S and u_O are the source and observer 4-velocities respectively, we have

−ωS ≡(k·uS) =gµν(S)k^µ(S)u^ν_S(S) and (2.6)

−ω_O≡(k·u_O) =g_µν(O)k^µ(O)u^ν_O(O) (2.7) If we have a standard candle source of which we know L and we measure F, we can therefore determine |detJ(O, S)|^1/2ωS/ωO, which contains information about the space- time geometry. Of course it also depends on the source and observer velocities. The Jacobi

map J^µα(O, S) maps direction vectors normal to the photons direction and normal touS

at S into vectors normal to the photon direction and u_O at O. It depends on the source velocity uS and on the curvature tensor along the photon geodesic from S to O. As we shall see, it does not depend on the observer velocity u_O.

Even though in the form (2.5), J is given by the 4×4 matrix J^µα(O, S), we have to take into account that the vectors δx^µ_O as well asδθ^α_S live in the two dimensional subspace normal to u_O respectively u_S and normal to the photon direction atO and S. The latter are given by

n_O = 1

ω_O(k(O) + (k(O)·u_O)u_O) and (2.8) n_S = 1

ω_S (k(S) + (k(S)·u_S)u_S) . (2.9) The photon direction vectors nS and nO are normalized spacelike vectors pointing into the photon direction in the reference frame of the source at S and of the observer at O respectively. Denoting the projectors onto the subspaces normal to uS, nS and uO, nO by P_S andP_O we have

(PS)^µν = δ^µν+u^µ_SuSν−n^µ_SnSν and (2.10) (PO)^µν = δ^µν+u^µ_OuOν−n^µ_OnOν . (2.11) The true Jacobi map isJ(O, S) =P_OJP_S understood as two dimensional linear map. For convenience we shall write it as four-dimensional application and determine its determinant as the product of the two non-vanishing eigen-values.

To determine the Jacobi map we now derive a differential equation for the evolution of the difference vector δx^µ(λ) in a given direction δθ_S^α along the photon trajectory. The final valueδx^µ(λ_O) then depends linearly on the initial conditionsδθ_S^α. For this we denote the photon trajectory by f^α(λ,0) and parameterize neighboring light-like geodesics by f^α(λ, δy). The 4-vector

k^α(δy) = ∂f^α(λ, δy)

∂λ is the tangent of neighboring photons at δy and

δx^α = ∂f^α

∂yⁱδyⁱ

connects the geodesicsf^α(λ,0) andf^α(λ, δy). Since the ’beam’f^α(λ,y) describes photons which are all emitted at the same event S they have the same phase (eikonal) S. With k_α=−∇αS we therefore have

0 =∇δxS ≡δx^α∇αS=−δx^αk_α . (2.12) In order for the 4-vectors δx^α(y) to sweep a surface normal to u_O at the observer event O at λ=λ_O, we also need (δx(λ_O)·u_O) = 0. This is a priori not true. However, we can re-parameterize f by

λ→¯λ=λ+h(y) and y→y¯ =g(y) . (2.13)