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
dL= 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 dL 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|ΩK|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 ΩK =−HK2
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 gravitagravita-tional 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
dL(z,n) =X
ℓm
aℓm(z)Yℓm(n) , (1.7)
where Yℓm(n) are the spherical harmonics. We can then construct the angular power spectrum at redshift z
Cℓ(z) =haℓm(z)a∗ℓm(z)i . (1.8) Here the h·i denotes a statistical average. Like for the CMB, statistical isotropy implies that the Cℓ’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].