We make a detailed study of matter density perturbations in both metric and Palatini formalisms. Considering general theories whose Lagrangian density is a general function, fR, of the Ricci scalar R, we derive the equation of matter density perturbations in each case, in a number of gauges, including comoving, longitudinal and uniform density gauges. We show that for viable fR models that satisfy cosmological and local gravity constraints (LGC), matter perturbation equations derived under a subhorizon approximation are valid even for super-Hubble scales provided the oscillating mode (scalaron) does not dominate over the matter-induced mode. Such approximate equations are especially reliable in the Palatini formalism because of the absence of scalarons. Using these equations we make a comparative study of the behavior of matter density perturbations as well as gravitational potentials for a number of classes of fR theories. In the metric formalism the quantity m Rf ;RR =f ;R that characterizes the deviation from the CDM model is constrained to be very small during a matter era in order to ensure compatibility with LGC, but the models in which m grows to the order of 10 ÿ1 around the present epoch can be allowed. These models also suffer from an additional fine-tuning due to the presence of scalaron oscillating modes which are absent in the Palatini case. In Palatini formalism LGC and background cosmological constraints provide only weak bounds on jmj by constraining it to be smaller than 0:1. This is in contrast to matter density perturbations which, on galactic scales, place far more stringent constraints on the present deviation parameter m of the order of jmj & 10 ÿ5 -10 ÿ4 . This is due to the peculiar evolution of matter perturbations in the Palatini case, which exhibits a rapid growth or a damped oscillation depending on the sign of m.

Closed, singularity-free, inflationary cosmological models have recently been studied in the context of general relativity. Despite their appeal, these so called emergent models suffer from a number of limitations. These include the fact that they rely on an initial Einstein static state to describe the past eternal phase of the universe. Given the instability of such a state within the context of general relativity, this amounts to a very severe fine tuning. Also in order to be able to study the dynamics of the universe within the context of general relativity, they set the initial conditions for the universe in the classical phase. Here we study the existence and stability of such models in the context of Loop Quantum Cosmology and show that both these limitations can be partially remedied, once semi-classical effects are taken into account. An important consequence of these effects is to give rise to a static solution (not present in GR), which dynamically is a centre equilibrium point and located in the more natural semi-classical regime. This allows the construction of emergent models in which the universe oscillates indefinitely about such an initial static state. We construct an explicit emergent model of this type, in which a non-singular past eternal oscillating universe enters a phase where the symmetry of the oscillations is broken, leading to an emergent inflationary epoch, while satisfying all observational and semi-classical constraints. We also discuss emergent models in which the universe possesses both early- and late-time accelerating phases.Comment: 11 pages, 8 figure

An important open question in cosmology is the degree to which the Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions of Einstein's equations are able to model the large-scale behaviour of the locally inhomogeneous observable universe. We investigate this problem by considering a range of exact n-body solutions of Einstein's constraint equations. These solutions contain discrete masses, and so allow arbitrarily large density contrasts to be modelled. We restrict our study to regularly arranged distributions of masses in topological 3-spheres. This has the benefit of allowing straightforward comparisons to be made with FLRW solutions, as both spacetimes admit a discrete group of symmetries. It also provides a time-symmetric hypersurface at the moment of maximum expansion that allows the constraint equations to be solved exactly. We find that when all the mass in the universe is condensed into a small number of objects (<10) then the amount of backreaction in dust models can be large, with O(1) deviations from the predictions of the corresponding FLRW solutions. When the number of masses is large (>100), however, then our measures of backreaction become small (<1%). This result does not rely on any averaging procedures, which are notoriously hard to define uniquely in general relativity, and so provides (to the best of our knowledge) the first exact and unambiguous demonstration of backreaction in general relativistic cosmological modelling. Discrete models such as these can therefore be used as laboratories to test ideas about backreaction that could be applied in more complicated and realistic settings.Comment: 13 pages, 9 figures. Corrections made to Tables IV and

We construct a family of viable scalar-tensor models of dark energy (DE) which possess a phase of late-time acceleration preceded by a standard matter era, while at the same time satisfying the local gravity constraints (LGC). The coupling Q between the scalar field and the non-relativistic matter in the Einstein frame is assumed to be constant in our scenario, which is a generalization of f (R) gravity theories corresponding to the coupling Q = −1/ √ 6. We find that these models can be made compatible with local gravity constraints even when |Q| is of the order of unity through a chameleon mechanism, if the scalar-field potential is chosen to have a sufficiently large mass in the high-curvature regions. We show that these models generally lead to the divergence of the equation of state of DE, which occur at smaller redshifts as the deviation from the ΛCDM model become more significant. We also study the evolution of matter density perturbations and employ them to place bounds on the coupling |Q| as well as model parameters of the field potential from observations of the matter power spectrum and the CMB anisotropies. We find that, as long as |Q| is smaller than the order of unity, there exist allowed parameter regions that are consistent with both observational and local gravity constraints.

We make a systematic study of the cosmological dynamics for a number of f (R) gravity theories in Palatini formalism, using phase space analysis as well as numerical simulations. Considering homogeneous and isotropic models, we find a number of interesting results: (i) models based on theories of the type (a) f (R) = R − β/R n and (b) f (R) = R + αln R − β, unlike the metric formalism, are capable of producing the sequence of radiation-dominated, matter-dominated and de-Sitter periods, and (ii) models based on theories of the type (c) f (R) = R + αR m − β/R n can produce early as well as late accelerating phases but an early inflationary epoch does not seem to be compatible with the presence of a subsequent radiation-dominated era. Thus for the classes of models considered here, we have been unable to find the sequence of all four dynamical epochs required to account for the complete cosmological dynamics, even though three out of four phases are possible.We also place observational constraints on these models using the recently released supernovae data by the Supernova Legacy Survey as well as the baryon acoustic oscillation peak in the SDSS luminous red galaxy sample and the CMB shift parameter. The best-fit values are found to be n = 0.027, α = 4.63 for the models based on (a) and α = 0.11, β = 4.62 for the models based on (b), neither of which are significantly preferred over the ΛCDM model. Moreover, the logarithmic term alone is not capable of explaining the late acceleration. The models based on (c) are also consistent with the data with suitable choices of their parameters.We also find that some of the models for which the radiation-dominated epoch is absent prior to the matter-dominated era also fit the data. The reason for this apparent contradiction is that the combination of the data considered here does not place stringent enough constraints on the cosmological evolution prior to the decoupling epoch, which highlights the importance of our combined theoretical-observational approach to constrain models.