We present a general gauge invariant formalism for defining cosmological averages that are relevant for observations based on light-like signals. Such averages involve either null hypersurfaces corresponding to a family of past light-cones or compact surfaces given by their intersection with timelike hypersurfaces. Generalized Buchert-Ehlers commutation rules for derivatives of these light-cone averages are given. After introducing some adapted "geodesic light-cone" coordinates, we give explicit expressions for averaging the redshift to luminosity-distance relation and the so-called "redshift drift" in a generic inhomogeneous Universe.

Using a recently proposed gauge invariant formulation of light-cone averaging, together with adapted "geodesic light-cone" coordinates, we show how an "induced backreaction" effect emerges, in general, from correlated fluctuations in the luminosity distance and covariant integration measure. Considering a realistic stochastic spectrum of inhomogeneities of primordial (inflationary) origin we find that both the induced backreaction on the luminosity-redshift relation and the dispersion are larger than naïvely expected. On the other hand the former, at least to leading order and in the linear perturbative regime, cannot account by itself for the observed effects of dark energy at largeredshifts. A full second-order calculation, or even better a reliable estimate of contributions from the non-linear regime, appears to be necessary before firm conclusions on the correct interpretation of the data can be drawn. PACS numbers: 98.80-k, 95.36.+x, 98.80.Es I. INTRODUCTIONThe so-called concordance (or ΛCDM) model, based on a suitable combination of dark matter, dark energy and baryons for an overall critical density, has become the reference paradigm for the late -i.e. post-equality epochevolution of our Universe (see e.g. [1]). It accounts equally well for the CMB data, the Large Scale Structure and, even more significantly, for the supernovae data in terms of a cosmic acceleration [2].Strictly speaking these three tests of the concordance model are not at the same level of theoretical rigor. While the first two have to do, by definition, with the inhomogeneities present in our Universe, the third is based on an ideal homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) geometry. It is clear that a better treatment of cosmic acceleration should take inhomogeneities into account, at least in an average statistical sense. Only when this is done we can establish in a convincing way whether ΛCDM gives a simultaneous consistent description of the above-mentioned body of cosmological data.This realization has led to a vast literature about averaging cosmological observables in realistic inhomogeneous cosmologies (see e.g. [3] for recent reviews). The conclusions, however, are still rather controversial: according to some authors [4] present inhomogeneities might explain, by themselves, cosmic acceleration without any need for dark-energy contributions; according to others [5] the effect of inhomogeneities is, instead, completely negligible. The truth may lie somewhere in between, in the sense that a quantitative understanding of inhomogeneities effects could be important in order to put precise constraints on dark-energy parameters, such as the critical fraction of dark-energy density, Ω Λ , and the time evolution of its effective equation of state, w Λ (z).In the first papers studying the dynamical effects of averaging, the problem was approached mainly following Buchert's prescriptions [6], namely averaging inhomogeneities over spacelike hypersurfaces and computing the ensuing "backreaction" on the averaged ge...

After recalling a general non-perturbative expression for the luminosity-redshift relation holding in a recently proposed "geodesic light-cone" gauge, we show how it can be transformed to phenomenologically more convenient gauges in which cosmological perturbation theory is better understood. We present, in particular, the complete result on the luminosity-redshift relation in the Poisson gauge up to second order for a fairly generic perturbed cosmology, assuming that appreciable vector and tensor perturbations are only generated at second order. This relation provides a basic ingredient for the computation of the effects of stochastic inhomogeneities on precision dark-energy cosmology whose results we have anticipated in a recent letter. More generally, it can be used in connection with any physical information carried by light-like signals traveling along our past light-cone. 95.36.+x, 98.80.Es 1 Following the pioneering work of [5], d L has been already computed to first order in the longitudinal gauge (for a CDM model in [6], CDM and ΛCDM in [7] and for a generic model in [2]), and to second order in the synchronous gauge, but only for a dust-dominated Universe, in [8]. 2 Except if caustics form. It has been argued [10] that the area distance is modified when caustics are present inside the past light-cone.

The remarkable properties of the recently proposed geodesic light-cone (GLC) gauge allow to explicitly solve the geodesic-deviation equation, and thus to derive an exact expression for the Jacobi map J A B (s, o) connecting a generic source s to a geodesic observer o in a generic space time. In this gauge J A B factorizes into the product of a local quantity at s times one at o, implying similarly factorized expressions for the area and luminosity distance. In any other coordinate system J A B is simply given by expressing the GLC quantities in terms of the corresponding ones in the new coordinates. This is explicitly done, at first and second order, respectively, for the synchronous and Poisson gauge-fixing of a perturbed, spatially-flat cosmological background, and the consistency of the two outcomes is checked. Our results slightly amend previous calculations of the luminosity-redshift relation and suggest a possible non-perturbative way for computing the effects of inhomogeneities on observations based on light-like signals.

Abstract. We determine the number counts to second order in cosmological perturbation theory in the Poisson gauge and allowing for anisotropic stress. The calculation is performed using an innovative approach based on the recently proposed "geodesic light-cone" gauge. This allows us to determine the number counts in a purely geometric way, without using Einstein's equation. The result is valid for general dark energy models and (most) modified gravity models. We then evaluate numerically some relevant contributions to the number counts bispectrum. In particular we consider the terms involving the density, redshift space distortion and lensing.

Starting from the luminosity-redshift relation recently given up to second order in the Poisson gauge, we calculate the effects of the realistic stochastic background of perturbations of the socalled concordance model on the combined light-cone and ensemble average of various functions of the luminosity distance, and on their variance, as functions of redshift. We apply a gauge-invariant light-cone averaging prescription which is free from infrared and ultraviolet divergences, making our results robust with respect to changes of the corresponding cutoffs. Our main conclusions, in part already anticipated in a recent letter for the case of a perturbation spectrum computed in the linear regime, are that such inhomogeneities not only cannot avoid the need for dark energy, but also cannot prevent, in principle, the determination of its parameters down to an accuracy of order 10 −3 − 10 −5 , depending on the averaged observable and on the regime considered for the power spectrum. However, taking into account the appropriate corrections arising in the non-linear regime, we predict an irreducible scatter of the data approaching the 10% level which, for limited statistics, will necessarily limit the attainable precision. The predicted dispersion appears to be in good agreement with current observational estimates of the distance-modulus variance due to Doppler and lensing effects (at low and high redshifts, respectively), and represents a challenge for future precision measurements. 95.36.+x, 98.80.Es

The effect of a stochastic background of cosmological perturbations on the luminosity-redshift relation is computed to second order through a recently proposed covariant and gauge-invariant light-cone averaging procedure. The resulting expressions are free from both ultraviolet and infrared divergences, implying that such perturbations cannot mimic a sizable fraction of dark energy. Different averages are estimated and depend on the particular function of the luminosity distance being averaged. The energy flux being minimally affected by perturbations at large z is proposed as the best choice for precision estimates of dark-energy parameters. Nonetheless, its irreducible (stochastic) variance induces statistical errors on Ω(Λ)(z) typically lying in the few-percent range.

We prove that the stochastic and standard field-theoretical approaches produce exactly the same results for the amount of light massive scalar field fluctuations generated during inflation in the leading order of the slow-roll approximation. This is true both in the case for which this field is a test one and inflation is driven by another field, and the case for which the field plays the role of inflaton itself. In the latter case, in order to calculate the mean square of the gauge-invariant inflaton fluctuations, the logarithm of the scale factor a has to be used as the time variable in the Fokker-Planck equation in the stochastic approach. The implications of particle production during inflation for the second stage of inflation and for the moduli problem are also discussed. The case of a massless self-interacting test scalar field in de Sitter background with a zero initial renormalized mean square is also considered in order to show how the stochastic approach can easily produce results corresponding to diagrams with an arbitrary number of scalar field loops in the field-theoretical approach (explicit results up to four loops included are presented)