• perturbation theory
• luminosity distance
• local
• lensing
• Hubble
• geodesic light cone
• cosmology
• cosmic variance
• redshift
• supernova

The Universe, seen on a large enough scale, seems to be homogeneous and isotropic. This together with the hypothesis that General Relativity is valid also at cosmological scale, are the foundation of the Standard Model of Cosmology. Such a model is compatible with a non-stationary universe whose spatial sections could be imagined as 3-dimensional surfaces of constant curvature. Topology tells us that only three kinds of such surfaces exist: hyperboloid, hyperplane, hypersphere. The straight way to determine which one is the “true'' shape, is to analyze how the expansion rate of the universe (often addressed as Hubble constant or Hubble rate) changed over time. In the light of the observations of Ia-type Supernovae (SNe) at high redshift, the universe seems to be spatially flat and in accelerating expansion. This result is in agreement with different kind of observations, the Cosmic Microwave Background (CMB) being just an example of that.

The present Hubble rate H_0 is expected to be a universal value by the Standard Model of Cosmology. However, we find an important discrepancy when comparing between its global-determined value, obtained through the CMB spectra analysis, and the local-determined one, obtained measuring the luminosity-redshift relation of Ia-type Supernovae [Riess et al. 2018]. One hypothesis is that this discrepancy could be interpreted as an effect of the universe inhomogeneities, as these inhomogeneities have an impact on the luminosity-redshift relation. In particular, through the so-called cosmic variance which gives us the theoretical error to be associated to our measurement as a consequence of the fact that we observe the Universe from a single point. An important effort in this direction has been conducted in [Ben-Dayan et al. 2014] through the evaluation of the Hubble rate cosmic variance induced by Doppler effect, which is the dominant effect at low redshifts. However, the recent improvements in the field of high redshift observation, make necessary to consider the variance induced by the other contributions, especially weak lensing.

In a contest of cosmological perturbations theory, we use the innovative Geodesic Light-Cone (GLC) gauge to obtain an exact value for the so-called Jacobi map, the function that holds the information about how the geodesics link the observer to the sources. This exact value, once expressed up to first order in the Newtonian gauge, allows us to evaluate perturbatively how much the observed luminosity-distance redshift relation should be different from its fiducial model value.

We then show that the local estimation of the Hubble rate cosmic variance depends on the averaged correlation matrix of these corrections. We also give the first explicit expression of all the terms involved in the variance determination. The most interesting one is the lensing term, as it is expected to carry the greatest contribution at high redshifts. This lensing contribution can be expressed in its simplest form as a 3-dimensional integral involving spherical Bessel functions of non rational argument. Hence, it cannot be evaluated analytically in general. By the way we find two interesting limiting cases. In the first we consider an isotropic distribution of SNe, while in the second we solve the integral for SNe along the same line-of-sight. This provides interesting constraints to the lensing contribution.

Our results are crucial to estimate the error of the local measured Hubble rate and, as a consequence, to understand if the CMB-determined value could be explained in a context of cosmological perturbations theory. Otherwise it will underline the need for new physics or the presence of systematics in the local measurement of H_0. Since, in the near future, several high redshifts Supernovae surveys will be available, our aim is also to provide a redshift-distribution based prediction for the cosmic variance of those surveys.