Buchert Equations Research Articles

Mori-Zwanzig Formalism for General Relativity: A New Approach to the Averaging Problem.

Cosmology relies on a coarse-grained description of the universe, assumed to be valid on large length scales. However, the nonlinearity of general relativity makes coarse graining extremely difficult. We here address this problem by extending the Mori-Zwanzig projection operator formalism, a highly successful coarse-graining method from statistical mechanics, towards general relativity. Using the Buchert equations, we derive a new dynamic equation for the Hubble parameter which captures the effects of averaging through a memory function. This gives an empirical prediction for the cosmic jerk.

Timescape cosmology with radiation fluid

The timescape cosmology represents a potentially viable alternative to the standard homogeneous cosmology, without the need for dark energy. Although average cosmic evolution in the timescape scenario only differs substantially from that of Friedmann–Lemaître model at relatively late epochs when the contribution from the energy density of radiation is negligible, a full solution of the Buchert equations to incorporate radiation is necessary to smoothly match parameters to the epoch of photon decoupling and to obtain constraints from cosmic microwave background data. Here we extend the matter-dominated solution found in earlier work to include radiation, providing series solutions at early times and an efficient numerical integration strategy for generating the complete solution. The numerical solution is used to directly calculate the scale of the sound horizon at decoupling, and at the baryon drag epoch. The constraints on these scales from the Planck satellite data yield bounds on the timescape cosmological parameters, which are found to also agree with the best-fit values from a recent analysis of SDSS-II supernova data, while avoiding the problem of a primordial lithium-7 abundance anomaly.

The fractal bubble model with a cosmological constant

We generalize the fractal bubble (FB) model, recently proposed in the literature as an alternative to the standard ΛCDM cosmology, to include a non-zero cosmological constant. We retain the same volume partition of voids and walls as the original FB model, and the same matching conditions for null geodesics, but do not include effects associated with a nonuniform time flow arising from differences of quasilocal gravitational energy that may arise in the coarse-graining process. The Buchert equations are written and partially integrated and the asymptotic behavior of the solutions is given. For a universe with Λ = 0, as is the case in the FB model, an initial void fraction with hyperbolic curvature evolves in such a way that it asymptotically fills completely our particle horizon. Conversely, in the presence of a nonvanishing Λ, we show that this does not happen and the voids fill a finite fraction , where the value of is expected to depend on Λ and the initial fraction fvi and also to be small. For its determination, a numerical integration of the equations is necessary. Finally, an interesting prediction of our model is a formula giving a minimum allowed value of present day dark energy as a function of the age of the universe and of the matter and curvature density parameters at our time.To my mother Cristina Pergolini

Bayesian analysis of the backreaction models

We present the Bayesian analysis of four different types of backreation models, which are based on the Buchert equations. In this approach, one considers a solution to the Einstein equations for a general matter distribution and then an average of various observable quantities is taken. Such an approach became of considerable interest when it was shown that it could lead to agreement with observations without resorting to dark energy. In this paper we compare the LambdaCDM model and the backreation models with SNIa, BAO, and CMB data, and find that the former is favoured. However, the tested models were based on some particular assumptions about the relation between the average spatial curvature and the backreaction, as well as the relation between the curvature and curvature index. In this paper we modified the latter assumption, leaving the former unchanged. We find that, by varying the relation between the curvature and curvature index, we can obtain a better fit. Therefore, some further work is still needed -- in particular the relation between the backreaction and the curvature should be revisited in order to fully determine the feasibility of the backreaction models to mimic dark energy.

The effect of inhomogeneous expansion on the supernova observations

We consider an inhomogeneous but spherically symmetric Lemaitre–Tolman–Bondi modelto demonstrate that spatial variations of the expansion rate can have a significant effect onthe cosmological supernova observations. A model with no dark energy but a local Hubbleparameter about 15% larger than its global value fits the supernova data better than thehomogeneous model with the cosmological constant. The goodness of the fit is not sensitiveto inhomogeneities in the present-day matter density, and our best fit model hasΩM(r) ∼ 0.3, in agreement with galaxy surveys. We also compute the averaged expansion rate, definedby the Buchert equations, of the best fit model and show explicitly that there is no averageacceleration.

Cosmic clocks, cosmic variance and cosmic averages

Cosmic acceleration is explained quantitatively, purely in general relativity with matter obeying the strong energy condition, as an apparent effect due to quasilocal gravitational energy differences that arise in the decoupling of bound systems from the global expansion of the universe. ‘Dark energy’ is recognized as a misidentification of those aspects of gravitational energy which by virtue of the equivalence principle cannot be localized. Matter is modelled as an inhomogeneous distribution of clusters of galaxies in bubble walls surrounding voids, as we observe. Gravitational energy differences between observers in bound systems, such as galaxies, and volume-averaged comoving locations in freely expanding space can be so large that the time dilation between the two significantly affects the parameters of any effective homogeneous isotropic model one fits to the universe. A new approach to cosmological averaging is presented, which implicitly solves the Sandage–de Vaucouleurs paradox. Comoving test particles in freely expanding space, which observe an isotropic cosmic microwave background (CMB), possess a quasilocal ‘rest’ energy E=⟨γ(τ,x)⟩mc2 on the spatial hypersurfaces of homogeneity. Here : the lower bound refers to fiducial reference observers at ‘finite infinity’, which is defined technically in relation to the demarcation scale between bound systems and expanding space. Within voids γ>1, representing the quasilocal gravitational energy of expansion and spatial curvature variations. Since all our cosmological measurements apart from the CMB involve photons exchanged between objects in bound systems, and since clocks in bound systems are largely unaffected, this is entirely consistent with observation. When combined with a non-linear scheme for cosmological evolution with back-reaction via the Buchert equations, a new observationally viable model of the universe is obtained, without ‘dark energy’. A quantitative scheme is presented for the recalibration of average cosmological parameters. It uses boundary conditions at the time of last scattering consistent with primordial inflation. The expansion age is increased, allowing more time for structure formation. The baryon density fraction obtained from primordial nucleosynthesis bounds can be significantly larger, yet consistent with primordial lithium abundance measurements. The angular scale of the first Doppler peak in the CMB anisotropy spectrum fits the new model despite an average negative spatial curvature at late epochs, resolving the anomaly associated with ellipticity in the CMB anisotropies. Non-baryonic dark matter to baryonic matter ratios of about 3:1 are typically favoured by observational tests. A number of other testable consequences are discussed, with the potential to profoundly change the whole of theoretical and observational cosmology.

COSMOLOGICAL ACCELERATION FROM STRUCTURE FORMATION

We discuss the Buchert equations, which describe the average expansion of an inhomogeneous dust universe. In the limit of small perturbations, they reduce to the Friedmann–Robertson–Walker equations. However, when the universe is very inhomogeneous, the behavior can be qualitatively different from the FRW case. In particular, the average expansion rate can accelerate even though the local expansion rate decelerates everywhere. We clarify the physical meaning of this paradoxical feature with a simple toy model, and demonstrate how acceleration is intimately connected with gravitational collapse. This provides a link to structure formation, which in turn has a preferred time around the era when acceleration has been observed to start.