An Introduction to General Relativity and Cosmology

Physical Review D, 2011

In dynamic spacetimes in which asymmetric gravitational collapse/expansion is taking place, the timelike geodesic equation appears to exhibit an interesting property: Relative to the collapsing configuration, free test particles undergo gravitational "acceleration" and form a double-jet configuration parallel to the axis of collapse.

Physical Review D, 2014

We investigate the asymptotic behavior of peculiar velocities in certain physically significant time-dependent gravitational fields. Previous studies of the motion of free test particles have focused on the collapse scenario, according to which a double-jet pattern with Lorentz factor γ → ∞ develops asymptotically along the direction of complete gravitational collapse. In the present work, we identify a second wave scenario, in which a single-jet pattern with Lorentz factor γ → ∞ develops asymptotically along the direction of wave propagation. The possibility of a connection between the two scenarios for the formation of cosmic jets is critically examined.

Journal of Cosmology and Astroparticle Physics, 2011

The Szekeres inhomogeneous models can be used to model the true lumpy universe that we observe. This family of exact solutions to Einstein's equations was originally derived with a general metric that has no symmetries. In this work, we develop and use a framework to integrate the angular diameter and luminosity distances in the general Szekeres models. We use the affine null geodesic equations in order to derive a set of first-order ordinary differential equations that can be integrated numerically to calculate the partial derivatives of the null vector components. These equations allow the integration in all generality of the distances in the Szekeres models and some examples are given. The redshift is determined from simultaneous integration of the null geodesic equations. This work does not assume spherical or axial symmetry, and the results will be useful for comparisons of the general Szekeres inhomogeneous models to current and future cosmological data. PACS numbers: 98.80.Es,95.36.+x,98.80.-k * Electronic address: mishak@utdallas.edu

Physical Review D, 2008

We use the Szekeres inhomogeneous relativistic models in order to fit supernova combined data sets. We show that with a choice of the spatial curvature function that is guided by current observations, the models fit the supernova data almost as well as the LCDM model without requiring a dark energy component. The Szekeres models were originally derived as an exact solution to Einstein's equations with a general metric that has no symmetries and are regarded as good candidates to model the true lumpy universe that we observe. The null geodesics in these models are not radial. The best fit model found is also consistent with the requirement of spatial flatness at CMB scales. The first results presented here seem to encourage further investigations of apparent acceleration using various inhomogeneous models and other constraints from CMB and large structure need to be explored next. PACS numbers: 98.80.Es,95.36.+x,

Physical Review D, 2012

This study belongs to a series devoted to using the Szekeres inhomogeneous models in order to develop a theoretical framework where cosmological observations can be investigated with a wider range of possible interpretations. While our previous work addressed the question of cosmological distances versus redshift in these models, the current study is a start at looking into the growth rate of large-scale structure. The Szekeres models are exact solutions to Einstein's equations that were originally derived with no symmetries. We use here a formulation of the Szekeres models that is due to Goode and Wainwright, who considered the models as exact perturbations of a Friedmann-Lemaître-Robertson-Walker (FLRW) background. Using the Raychaudhuri equation we write, for the two classes of the models, exact growth equations in terms of the under/overdensity and measurable cosmological parameters. The new equations in the overdensity split into two informative parts. The first part, while exact, is identical to the growth equation in the usual linearly perturbed FLRW models, while the second part constitutes exact non-linear perturbations. We integrate numerically the full exact growth rate equations for the flat and curved cases. We find that for the matter-dominated cosmic era, the Szekeres growth rate is up to a factor of three to five stronger than the usual linearly perturbed FLRW cases, reflecting the effect of exact Szekeres non-linear perturbations. We also find that the Szekeres growth rate with an Einstein-de Sitter background is stronger than that of the well-known non-linear spherical collapse model, and the difference between the two increases with time. This highlights the distinction when we use general inhomogeneous models where shear and a tidal gravitational field are present and contribute to the gravitational clustering. Additionally, it is worth observing that the enhancement of the growth found in the Szekeres models during the matter-dominated era could suggest a substitute to the argument that dark matter is needed when using FLRW models to explain the enhanced growth and resulting large-scale structures that we observe today. PACS numbers: 98.80.Es,98.80.-k,95.30.Sf * Electronic address: mishak@utdallas.edu

Classical and Quantum Gravity, 2010

We analyze the big bounce transition of the quantum FRW model in the setting of the nonstandard loop quantum cosmology (LQC). Elementary observables are used to quantize composite observables. The spectrum of the energy density operator is bounded and continuous. The spectrum of the volume operator is bounded from below and discrete. It has equally distant levels defining a quantum of the volume. The discreteness may imply a foamy structure of spacetime at semiclassical level which may be detected in astro-cosmo observations. The nonstandard LQC method has a free parameter that should be fixed in some way to specify the big bounce transition.

Classical and Quantum Gravity, 2015

We recast the system of Einstein field equations for Locally Rotationally Symmetric spacetimes into an autonomous system of covariantly defined geometrical variables. The analysis of this autonomous system gives all the important global features of the maximal extension of these spacetimes. We conclude that the dynamical system analysis can be a powerful mathematical tool for qualitative understanding of the global structure of spacetimes covariantly, without actually solving the field equations.

Physical Review D

We examine the gravitational collapse and black hole formation of multiple non-spherical configurations constructed from Szekeres dust models with positive spatial curvature that smoothly match to a Schwarzschild exterior. These configurations are made of an almost spherical central core region surrounded by a network of "pancake-like" overdensities and voids with spatial positions prescribed through standard initial conditions. We show that a full collapse into a focusing singularity, without shell crossings appearing before the formation of an apparent horizon, is not possible unless the full configuration becomes exactly or almost spherical. Seeking for black hole formation, we demand that shell crossings are covered by the apparent horizon. This requires very special fine-tuned initial conditions that impose very strong and unrealistic constraints on the total black hole mass and full collapse time. As a consequence, non-spherical non-rotating dust sources cannot furnish even minimally realistic toy models of black hole formation at astrophysical scales: demanding realistic collapse time scales yields huge unrealistic black hole masses, while simulations of typical astrophysical black hole masses collapse in unrealistically small times. We note, however, that the resulting time-mass constraint is compatible with early Universe models of primordial black hole formation, suitable in early dust-like environments. Finally, we argue that the shell crossings appearing when non-spherical dust structures collapse are an indicator that such structures do not form galactic mass black holes but virialise into stable stationary objects.

General Relativity and Gravitation, 2015

It is common to think of our universe according to the "block universe" concept, which says that spacetime consists of many "stacked" 3-surfaces, labelled by some kind of proper time, τ. Standard ideas do not distinguish past and future, but Ellis' "evolving block universe" tries to make a fundamental distinction. One proposal for this proper time is the proper time measured along the timelike Ricci eigenlines, starting from the big bang. This work investigates the shape of the "Ricci time" surfaces relative to the the null surfaces. We use the Lemaître-Tolman metric as our inhomogeneous spacetime model, and we find the necessary and sufficient conditions for these {τ = constant} surfaces, S(τ), to be spacelike or timelike. Furthermore, we look at the effect of strong gravity domains by determining the location of timelike S regions relative to apparent horizons. We find that constant Ricci time surfaces are always spacelike near the big bang, while at late times (near the crunch or the extreme far future), they are only timelike under special circumstances. At intermediate times, timelike S regions are common unless the variation of the bang time is restricted. The regions where these surfaces become timelike are often adjacent to apparent horizons, but always outside them, and in particular timelike S regions do not occur inside black holes.

International Journal of Modern Physics E, 2012

The pseudo-complex General Relativity (pc-GR) is further considered. A new projection method is proposed. It is shown that the pc-GR introduces automatically terms into the system which can be interpreted as dark energy. The modified pseudo-complex Schwarzschild solution is investigated. The dark energy part is treated as a liquid and possible solutions are discussed. As a consequence, the collapse of a large stellar mass into a singularity at r = 0 is avoided and no event-horizon is formed. Thus, black holes do not exist. The resulting object can be viewed as a gray star. It contains no singularity which emphasizes, again, that it is not a black hole. The corrections implied by a charged large mass object (Reissner–Nordström) and a rotating gray star (Kerr) are presented. For the latter, a special solution is presented. Finally, we will consider the orbital speed of a mass in a circular orbit and suggest a possible experimental verification.

General Relativity and Gravitation, 2009

General Relativity and Gravitation, 2010

The 1+3 covariant approach and the covariant gauge-invariant approach to perturbations are used to analyze in depth conformal transformations in cosmology. Such techniques allow us to obtain insights on the physical meaning of these transformations when applied to non-standard gravity. The results obtained lead to a number of general conclusions on the change of some key quantities describing any two conformally related cosmological models. For example, even if some of the geometrical properties of the cosmology are preserved (homogeneous and isotropic Universes are mapped into homogeneous and isotropic universes), it can happen that decelerating cosmologies can be mapped into accelerated ones. From the point of view of the cosmological perturbations it is shown how these fluctuation transform. We find that first-order vector and tensor perturbations equations are left unchanged in their structure by the conformal transformation, but this cannot be said of the scalar perturbations, which present differences in their evolutionary features. The results obtained are then explicitly interpreted and verified with the help of some clarifying examples based on f (R)-gravity cosmologies. Keywords conformal transformations • modified gravity • cosmology • 1+3 covariant approach • covariant gauge invariant theory of perturbations

Physical Review D, 2013

The Oppenheimer-Snyder solution models a homogeneous round dust cloud collapsing to a black hole. Inside its event horizon there is a region through which trapped surfaces pass. We try to determine exactly where the boundary of this region meets the centre of the cloud. We present explicit examples of the relevant trapped (topological) spheres; they extend into the exterior vacuum region, and are carefully matched at the junction between the cloud and the vacuum.

Journal of High Energy Physics

We study the effect of the simplest geometry which is imposed via the topology of the universe by gauging non-relativistic particle model on torus and 3-torus with the help of symplectic formalism of constrained systems. Also, we obtain generators of gauge transformations for gauged models. Extracting corresponding Poisson structure of existed constraints, we show the effect of the shape of the universe on canonical structure of phase-spaces of models and suggest some phenomenology to prove the topology of the universe and probable non-commutative structure of the space. In addition, we show that the number of extra dimensions in the phase-spaces of gauged embedded models are exactly two. Moreover, in classical form, we talk over modification of Newton's second law in order to study the origin of the terms appeared in the gauged theory.

Classical and Quantum Gravity, 2011

We consider spherically symmetric inhomogeneous dust models with a positive cosmological constant, Λ, given by the Lemaître-Tolman-Bondi metric. These configurations provide a simple but useful generalization of the Λ-CDM model describing cold dark matter (CDM) and a Λ term, which seems to fit current cosmological observations. The dynamics of these models can be fully described by scalar evolution equations that can be given in the form of a proper dynamical system associated with a 4-dimensional phase space whose critical points and invariant subspaces are examined and classified. The phase space evolution of various configurations is studied in detail by means of two 2-dimensional subspaces: a projection into the invariant homogeneous subspace associated with Λ-CDM solutions with FLRW metric, and a projection into a subspace generated by suitably defined fluctuations that convey the effects of inhomogeneity. We look at cases with perpetual expansion, bouncing and loitering behavior, as well as configurations with "mixed" kinematic patters, such as a collapsing region in an expanding background. In all cases, phase space trajectories emerge from and converge to stable past and future attractors in a qualitatively analogous way as in the case of the FLRW limit. However, we can identify in both projections of the phase space various qualitative features absent in the FLRW limit that can be useful in the construction of toy models of astrophysical and cosmological inhomogeneities.

Classical and Quantum Gravity, 2011

We provide a thorough examination of the conditions for the existence of back-reaction and an "effective" acceleration (in the context of Buchert's averaging formalism) in regular generic spherically symmetric Lemaître-Tolman-Bondi (LTB) dust models. By considering arbitrary spherical comoving domains, we verify rigorously the fulfillment of these conditions expressed in terms of suitable scalar variables that are evaluated at the domains' boundaries. Effective deceleration necessarily occurs in all domains in: (a) the asymptotic radial range of models converging to a FLRW background, (b) the asymptotic time range of non-vacuum hyperbolic models, (c) LTB self-similar solutions and (d) near a simultaneous big bang. Accelerating domains are proven to exist in the following scenarios: (i) central vacuum regions, (ii) central (non-vacuum) density voids, (iii) the intermediate radial range of models converging to a FLRW background, (iv) the asymptotic radial range of models converging to a Minkowski vacuum and (v) domains near and/or intersecting a non-simultaneous big bang. All these scenarios occur in hyperbolic models with negative averaged and local spatial curvature, though scenarios (iv) and (v) are also possible in low density regions of a class of elliptic models in which local spatial curvature is negative but its average is positive. Rough numerical estimates between −0.003 and −0.5 were found for the effective deceleration parameter. While the existence of accelerating domains cannot be ruled out in models converging to an Einstein de Sitter background and in domains undergoing gravitational collapse, the conditions for this are very restrictive. The results obtained may provide important theoretical clues on the effects of back-reaction and averaging in more general non-spherical models.

Classical and Quantum Gravity, 2010

We undertake a comprehensive and rigorous analytic study of the evolution of radial profiles of covariant scalars in regular Lemaître-Tolman-Bondi dust models. We consider specifically the phenomenon of "profile inversions" in which an initial clump profile of density, spatial curvature or the expansion scalar, might evolve into a void profile (and vice versa). Previous work in the literature on models with density void profiles and/or allowing for density profile inversions is given full generalization, with some erroneous results corrected. We prove rigorously that if an evolution without shell crossings is assumed, then only the 'clump to void' inversion can occur in density profiles, and only in hyperbolic models or regions with negative spatial curvature. The profiles of spatial curvature follow similar patterns as those of the density, with 'clump to void' inversions only possible for hyperbolic models or regions. However, profiles of the expansion scalar are less restrictive, with profile inversions necessarily taking place in elliptic models. We also examine radial profiles in special LTB configurations: closed elliptic models, models with a simultaneous big bang singularity, as well as a locally collapsing elliptic region surrounded by an expanding hyperbolic background. The general analytic statements that we obtain allow for setting up the right initial conditions to construct fully regular LTB models with any specific qualitative requirements for the profiles of all scalars and their time evolution. The results presented can be very useful in guiding future numerical work on these models and in revising previous analytic work on all their applications.

Classical and Quantum Gravity, 2014

We consider generic Lemaître-Tolman-Bondi (LTB) dust models to probe the gravitational entropy proposals of Clifton, Ellis and Tavakol (CET) and of Hosoya and Buchert (HB). We also consider a variant of the HB proposal based on a suitable quasi-local scalar weighted average. We show that the conditions for entropy growth for all proposals are directly related to a negative correlation of similar fluctuations of the energy density and Hubble scalar. While this correlation is evaluated locally for the CET proposal, it must be evaluated in a non-local domain dependent manner for the two HB proposals. By looking at the fulfilment of these conditions at the relevant asymptotic limits we are able to provide a well grounded qualitative description of the full time evolution and radial asymptotic scaling of the three entropies in generic models. The following rigorous analytic results are obtained for the three proposals: (i) entropy grows when the density growing mode is dominant, (ii) all ever-expanding hyperbolic models reach a stable terminal equilibrium characterized by an inhomogeneous entropy maximum in their late time evolution; (iii) regions with decaying modes and collapsing elliptic models exhibit unstable equilibria associated with an entropy minimum (iv) near singularities the CET entropy diverges while the HB entropies converge; (v) the CET entropy converges for all models in the radial asymptotic range, whereas the HB entropies only converge for models asymptotic to a FLRW background. The fact that different independent proposals yield fairly similar conditions for entropy production, time evolution and radial scaling in generic LTB models seems to suggest that their common notion of a "gravitational entropy" may be a theoretically robust concept applicable to more general spacetimes.

General Relativity and Gravitation, 2010

We examine the radial asymptotic behavior of spherically symmetric Lemaître-Tolman-Bondi dust models by looking at their covariant scalars along radial rays, which are spacelike geodesics parametrized by proper length , orthogonal to the 4-velocity and to the orbits of SO(3). By introducing quasilocal scalars defined as integral functions along the rays, we obtain a complete and covariant representation of the models, leading to an initial value parametrization in which all scalars can be given by scaling laws depending on two metric scale factors and two basic initial value functions. Considering regular "open" LTB models whose space slices allow for a diverging , we provide the conditions on the radial coordinate so that its asymptotic limit corresponds to the limit as → ∞. The "asymptotic state" is then defined as this limit, together with asymptotic series expansion around it, evaluated for all metric functions, covariant scalars (local and quasi-local) and their fluctuations. By looking at different sets of initial conditions, we examine and classify the asymptotic states of parabolic, hyperbolic and open elliptic models admitting a symmetry center. We show that in the radial direction the models can be asymptotic to any one of the following spacetimes: FLRW dust cosmologies with zero or negative spatial curvature, sections of Minkowski flat space (including Milne's space), sections of the Schwarzschild-Kruskal manifold or self-similar dust solutions.

Physical Review D, 2015

We examine the relation between the dynamics of Lemaître-Tolman-Bondi (LTB) dust models (with and without Λ) and the dynamics of dust perturbations in two of the more familiar formalisms used in cosmology: the metric based Cosmological Perturbation Theory (CPT) and the Covariant Gauge Invariant (GIC) perturbations. For this purpose we recast the evolution of LTB models in terms of a covariant and gauge invariant formalism of local and non-local "exact fluctuations" on a Friedmann-Lemaître-Robertson-Walker (FLRW) background defined by suitable averages of covariant scalars. We examine the properties of these fluctuations, which can be defined for a confined comoving domain or for an asymptotic domain extending to whole time slices. In particular, the non-local density fluctuation provides a covariant and precise definition for the notion of the "density contrast". We show that in their linear regime these LTB exact fluctuations (local and non-local) are fully equivalent to the conventional cosmological perturbations in the synchronouscomoving gauge of CPT and to GIC perturbations. As an immediate consequence, we show the time-invariance of the spatial curvature perturbation in a simple form. The present work may provide important theoretical connections between the exact and perturbative (linear or no-linear) approach to the dynamics of dust sources in General Relativity.

Astronomische Nachrichten

We apply a recent proposal to define “gravitational entropy” to the expansion of cosmic voids within the framework of nonperturbative General Relativity. By considering CDM void configurations compatible with basic observational constraints, we show that this entropy grows from post-inflationary conditions towards a final asymptotic value in a late time fully nonlinear regime described by the Lemaître-Tolman-Bondi (LTB) dust models. A qualitatively analogous behavior occurs if we assume a positive cosmological constant consistent with a Λ-CDM background model. However, the Λ term introduces a significant suppression of entropy growth with the terminal equilibrium value reached at a much faster rate. (© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Physical Review D, 2015

We examine the spatial extrema (local maxima, minima and saddle points) of the covariant scalars (density, Hubble expansion, spatial curvature and eigenvalues of the shear and electric Weyl tensors) of the quasi-spherical Szekeres dust models. Sufficient conditions are obtained for the existence of distributions of multiple extrema in spatial comoving locations that can be prescribed through initial conditions. These distributions evolve without shell crossing singularities at least for ever expanding models (with or without cosmological constant) in the full evolution range where the models are valid. By considering the local maxima and minima of the density, our results allow for setting up elaborated networks of "pancake" shaped evolving cold dark matter over-densities and density voids whose spatial distribution and amplitudes can be controlled from initial data compatible with standard early Universe initial conditions. We believe that these results have an enormous range of potential application by providing a fully relativistic non-perturbative coarse grained modelling of cosmic structure at all scales.

Journal of Cosmology and Astroparticle Physics, 2016

We show that the full dynamical freedom of the well known Szekeres models allows for the description of elaborated 3-dimensional networks of cold dark matter structures (over-densities and/or density voids) undergoing "pancake" collapse. By reducing Einstein's field equations to a set of evolution equations, which themselves reduce in the linear limit to evolution equations for linear perturbations, we determine the dynamics of such structures, with the spatial comoving location of each structure uniquely specified by standard early Universe initial conditions. By means of a representative example we examine in detail the density contrast, the Hubble flow and peculiar velocities of structures that evolved, from linear initial data at the last scattering surface, to fully non-linear 10-20 Mpc scale configurations today. To motivate further research, we provide a qualitative discussion on the connection of Szekeres models with linear perturbations and the pancake collapse of the Zeldovich approximation. This type of structure modelling provides a coarse grained-but fully relativistic non-linear and non-perturbative-description of evolving large scale cosmic structures before their virialisation, and as such it has an enormous potential for applications in cosmological research.

Physical Review D

We study the differences and equivalences between the non-perturbative description of the evolution of cosmic structure furnished by the Szekeres dust models (a non-spherical exact solution of Einstein's equations) and the dynamics of Cosmological Perturbation Theory (CPT) for dust sources in a ΛCDM background. We show how the dynamics of Szekeres models can be described by evolution equations given in terms of "exact fluctuations" that identically reduce (at all orders) to evolution equations of CPT in the comoving isochronous gauge. We explicitly show how Szekeres linearised exact fluctuations are specific (deterministic) realisations of standard linear perturbations of CPT given as random fields but, as opposed to the latter perturbations, they can be evolved exactly into the full non-linear regime. We prove two important results: (i) the conservation of the curvature perturbation (at all scales) also holds for the appropriate linear approximation of the exact Szekeres fluctuations in a ΛCDM background, and (ii) the different collapse morphologies of Szekeres models yields, at nonlinear order, different functional forms for the growth factor that follows from the study of redshift space distortions. The metric based potentials used in linear CPT are computed in terms of the parameters of the linearised Szekeres models, thus allowing us to relate our results to linear CPT results in other gauges. We believe that these results provide a solid starting stage to examine the role of non-perturbative General Relativity in current cosmological research.

The European Physical Journal C

We examine the evolution of peculiar velocities of cold dark matter (CDM) in localized arrays of inhomogeneous cosmic structures in a $$\varLambda $$ Λ CDM background that can be identified as a frame comoving with the Cosmic Microwave (CMB). These arrays are constructed by smoothly matching to this cosmological background regions of Szekeres-II models whose source is an imperfect fluid reinterpreted as non-comoving dust, keeping only first order terms in v/c. Considering a single Szekeres-II region matched along two comoving interfaces to a $$\varLambda $$ Λ CDM background, the magnitudes of peculiar velocities within this region are compatible with values reported in the literature, while the present day Hubble expansion scalar differs from that of the $$\varLambda $$ Λ CDM background value by a 10% factor, a result that might provide useful information to the ongoing debate on the $$H_0$$ H 0 tension. While the models cannot describe the virialization process, we show through a r...

Physical review. E, 2021

We study the classical analog of the quantum metric tensor and its scalar curvature for two well-known quantum physics models. First, we analyze the geometry of the parameter space for the Dicke model with the aid of the classical and quantum metrics and find that, in the thermodynamic limit, they have the same divergent behavior near the quantum phase transition, as opposed to their corresponding scalar curvatures which are not divergent there. On the contrary, under resonance conditions, both scalar curvatures exhibit a divergence at the critical point. Second, we present the classical and quantum metrics for the Lipkin-Meshkov-Glick model in the thermodynamic limit and find a perfect agreement between them. We also show that the scalar curvature is only defined on one of the system's phases and that it approaches a negative constant value. Finally, we carry out a numerical analysis for the system's finite sizes, which clearly shows the precursors of the quantum phase tran...

International Journal of Modern Physics A, 2020

Explicit expressions for the bending angle of light deflection arising from phenomenologically deformed black hole metrics, subject to possible weak and strong quantum gravity effects, respectively, are obtained, by a highly effective method. The accuracy and effectiveness of these expressions are then illustrated by numerically solving the differential equation governing the deflection angle directly in the weak quantum-gravity effect situation.

Journal of Mathematical Physics, 2010

We derive a transformation of the noncommutative geometry inspired Schwarzschild solution into new coordinates such that the apparent unphysical singularities of the metric are removed. Moreover, we give the maximal singularity-free atlas for the manifold with the metric under consideration. This atlas reveals many new features e.g. it turns out to describe an infinite lattice of asymptotically flat universes connected by black hole tunnels.

International Journal of Modern Physics D, 2013

An important issue in the study of continual gravitational collapse of a massive matter cloud in general relativity is whether shell-crossing singularities develop as the collapse evolves. We examine this here to show that for any spherically symmetric collapse in general, till arbitrarily close to the final singularity, there is always a finite neighborhood of the center in which there are no shell-crossings taking place. It follows that in order to study the visibility or otherwise of the ultra-dense region close to the final singularity of collapse where physical radius of the matter cloud shrinks to an arbitrarily small value, we can always consider without loss of generality a collapsing ball of finite comoving radius in which there are no shell-crossings.

Classical and Quantum Gravity, 2013

We obtain covariant expressions that generalize the growing and decaying density modes of linear perturbation theory of dust sources by means of the exact density perturbation from the formalism of quasi-local scalars associated to weighted proper volume averages in LTB dust models. The relation between these density modes and theoretical properties of generic LTB models is thoroughly studied by looking at the evolution of the models through a dynamical system whose phase space is parametrized by variables directly related to the modes themselves. The conditions for absence of shell crossings and sign conditions on the modes become interrelated fluid flow preserved constraints that define sub-cases of LTB models as phase space invariant subspaces. In the general case (both density modes being nonzero) the evolution of phase space trajectories exhibits the expected dominance of the decaying/growing in the early/late evolution times defined by past/future attractors characterized by asymptotic density inhomogeneity. In particular, the growing mode is also dominant for collapsing layers that terminate in a future attractor associated with a "Big Crunch" singularity, which is qualitatively different from the past attractor marking the "Big Bang". Suppression of the decaying mode modifies the early time evolution, with phase space trajectories emerging from an Einsteinde Sitter past attractor associated with homogeneous conditions. Suppression of the growing mode modifies the late time evolution as phase space trajectories terminate in future attractors associated with homogeneous states. General results are obtained relating the signs of the density modes and the type of asymptotic density profile (clump or void). A critical review is given of previous attempts in the literature to define these density modes for LTB models.

Physical Review D, 2016

Physical Review D, 2018

The Birkhoff theorem is a well-known result in general relativity and it is used in many applications. However, its most general version, due to Bona, is almost unknown and presented in a form less accessible to the relativist and cosmologist community. Moreover, many wield it mistakenly as a simple transposition of Newton's iron sphere theorem. In the present work, we propose a modern, dual null, presentation-useful in many explorations, including black holes-of the theorem that renders accessible most of the results of Bona's version. In addition, we discuss the fluid contents admissible for the application of the theorem, beyond a vacuum, and we demonstrate how the formalism greatly simplifies solving the dynamical equations and allows one to express the solution as a power expansion in r. We present a family of solutions that share the properties predicted by the Birkhoff theorem and discuss the existence of trapped and antitrapped regions. The formalism manifestly shows how the type of region-trapped or untrapped-determines the character of the Killing vector.

The European Physical Journal C, 2018

We examine the evolution of an inhomogeneous mixture of non-relativistic pressureless cold dark matter (CDM), coupled to dark energy (DE) characterised by the equation of state parameter w < −1/3, with the interaction term proportional to the DE density. This coupled mixture is the source of a spherically symmetric Lemaître-Tolman-Bondi (LTB) metric admitting an asymptotic Friedman-Lemaître-Robertson-Walker (FLRW) background. Einstein's equations reduce to a 5-dimensional autonomous dynamical system involving quasi-local variables related to suitable averages of covariant scalars and their fluctuations. The phase space evolution around the critical points (past/future attractors and five saddles) is examined in detail. For all parameter values and both directions of energy flow (CDM to DE and DE to CDM) the phase space trajectories are compatible with a physically plausible early cosmic times behaviour near the past attractor. This result compares favourably with mixtures with interaction driven by the CDM density, whose past evolution is unphysical for DE to CDM energy flow. Numerical examples are provided describing the evolution of an initial profile that can be associated with idealised structure formation scenarios.

Classical and Quantum Gravity, 2020

We examine a novel class of toy models of cosmological inhomogeneities by smoothly matching along a suitable hypersurface an arbitrary number of sections of "quasi flat" inhomogeous and anisotropic Szekeres-II models to sections of any spatially flat cosmology that can be described by the Robertson-Waker metric (including de Sitter, anti de Sitter and Minkowski spacetimes). The resulting "pancake" models are quasi-flat analogues to the well known spherical "Swiss-cheese" models found in the literature. Since Szekeres-II models can be, in general, compatible with a wide range of sources (dissipative fluids, mixtures of non-comoving fluids, mixtures of fluids with scalar or magnetic fields or gravitational waves), the pancake configurations we present allow for a description of a wide collection of localized sources embedded in a Robertson-Waker geometry. We provide various simple examples of arbitrary numbers of Szekeres-II regions (whose sources are comoving dust and energy flux interpreted as a field of peculiar velocities) matched with Einstein de Sitter, ΛCDM and de Sitter backgrounds. We also prove that the Szekeres-II regions can be rigorously regarded as "exact" covariant perturbations on a background defined by the matching discussed above. We believe that these models can be useful to test ideas on averaging and backreaction and on the effect of inhomogeneities on cosmic evolution and observations.

The Astrophysical Journal, 2018

An exact solution of the Lemaître-Tolman-Bondi class is investigated as a possible model of the Schwarzschildlike black hole embedded in a nonstatic dust-filled universe for the three types of spatial curvature. The solution is obtained in comoving coordinates by means of the mass function method. It is shown that the central part of space contains a Schwarzschild-like black hole. The R-T structure of the resulting spacetime is built. It is shown that the solution includes both the Schwarzschild and Friedmann solutions as its natural limits. The geodesic equations for test particles are analyzed. The particle observable velocities are found. The trajectories of the test particles are built from the point of view of both comoving and distant observers. For the distant observer, the results coincide with the Schwarzschild picture within a second-order accuracy near the symmetry center.

Classical and Quantum Gravity, 2009

Previous papers dealt with the quantization of the Lemaître-Tolman-Bondi (LTB) model for vanishing cosmological constant Λ. Here we extend the analysis to the case Λ > 0. Our main goal is to present solutions of the Wheeler-DeWitt equation, to give their interpretation, and to derive Hawking radiation from them. We restrict ourselves to a discussion of those points that are different from the Λ = 0-case. These have mainly to do with the occurrence of two horizons.

2018

Hamiltonian formalisms provide powerful tools for the computation of approximate analytic solutions of the Einstein field equations. The post-Newtonian computations of the explicit analytic dynamics and motion of compact binaries are discussed within the most often applied Arnowitt-Deser-Misner formalism. The obtention of autonomous Hamiltonians is achieved by the transition to Routhians. Order reduction of higher derivative Hamiltonians results in standard Hamiltonians. Tetrad representation of general relativity is introduced for the tackling of compact binaries with spinning components. Configurations are treated where the absolute values of the spin vectors can be considered constant. Compact objects are modeled by use of Dirac delta functions and their derivatives. Consistency is achieved through transition to d-dimensional space and application of dimensional regularization. At the fourth post-Newtonian level, tail contributions to the binding energy show up. The conservative spin-dependent dynamics finds explicit presentation in Hamiltonian form through next-to-next-toleading-order spin-orbit and spin1-spin2 couplings and to leading-order in the cubic and quartic in spin interactions. The radiation reaction dynamics is presented explicitly through the third-and-half post-Newtonian order for spinless objects, and, for spinning bodies, to leading-order in the spin-orbit and spin1-spin2 couplings. The most important historical issues get pointed out. Keywords General relativity • Classical spin and gravity • Hamiltonian formalism • Compact binary systems • Canonical equations of motion • Radiation reaction and emission • Analytical and dimensional regularization B Gerhard Schäfer

Journal of Cosmology and Astroparticle Physics, 2009

In the context of bubble universes produced by a first-order phase transition with large nucleation rates compared to the inverse dynamical time scale of the parent bubble, we extend the usual analysis to non-vacuum backgrounds. In particular, we provide semi-analytic and numerical results for the modified nucleation rate in FLRW backgrounds, as well as a parameter study of bubble walls propagating into inhomogeneous (LTB) or FLRW spacetimes, both in the thin-wall approximation. We show that in our model, matter in the background often prevents bubbles from successful expansion and forces them to collapse. For cases where they do expand, we give arguments why the effects on the interior spacetime are small for a wide range of reasonable parameters and discuss the limitations of the employed approximations.

Physical Review D, 2014

It has been argued in the literature that in order to make a matter dominated Friedmann-Lemaitre-Robertson-Walker universe compatible with the generalized second law of thermodynamics, one must invoke dark energy, or modified gravity. In the present article we investigate if in a similar spirit, inhomogeneous cosmological models can be motivated on thermodynamic grounds. We examine a particular minimal void Lemaitre-Tolman-Bondi inhomogeneous model which agrees well with observations. While on the one hand we find that the entropy associated with the apparent horizon is not well-behaved thermodynamically, on the other hand the canonical Weyl curvature entropy shows satisfactory thermodynamic behavior. We suggest that evolution of canonical Weyl curvature entropy might be a useful way to evaluate the thermodynamic viability of inhomogeneous cosmologies.

Classical and Quantum Gravity, 2010

We give the necessary and sufficient (local) conditions for a metric tensor to be a non conformally flat spherically symmetric solution. These conditions exclusively involve explicit concomitants of the Riemann tensor. As a direct application we obtain the ideal labeling of the Schwarzschild, Reissner-Nordström and Lemaître-Tolman-Bondi solutions.

Classical and Quantum Gravity, 2013

We introduce a weighed scalar average formalism ("q-average") for the study of the theoretical properties and the dynamics of spherically symmetric Lemaître-Tolman-Bondi (LTB) dust models models. The "q-scalars" that emerge by applying the q-averages to the density, Hubble expansion and spatial curvature (which are common to FLRW models) are directly expressible in terms of curvature and kinematic invariants and identically satisfy FLRW evolution laws without the back-reaction terms that characterize Buchert's average. The local and non-local fluctuations and perturbations with respect to the q-average convey the effects of inhomogeneity through the ratio of curvature and kinematic invariants and the magnitude of radial gradients. All curvature and kinematic proper tensors that characterize the models are expressible as irreducible algebraic expansions on the metric and 4-velocity, whose coefficients are the q-scalars and their linear and quadratic local fluctuation. All invariant contractions of these tensors are quadratic fluctuations, whose q-averages are directly and exactly related to statistical correlation moments of the density and Hubble expansion scalar. We explore the application of this formalism to a definition of a gravitational entropy functional proposed by Hosoya et al (2004 Phys. Rev. Lett. 92 141302-14). We show that a positive entropy production follows from a negative correlation between fluctuations of the density and Hubble scalar, providing a brief outline on its fulfillment in various LTB models and regions. While the q-average formalism is specially suited for LTB (and Szekeres) models, it may provide a valuable theoretical insight on the properties of scalar averaging in inhomogeneous spacetimes in general.

Physical review, 2017

Light is the richest information retriever for most physical systems, particularly so for astronomy and cosmology, in which gravitation is of paramount importance, and also for solid state defects and metamaterials, in which some effects can be mimicked by non-Euclidean or even non-Riemannian geometries. Thus, it is expedient to probe light motion in geometrical backgrounds alternative to that of general relativity. Here we investigate this issue in generic metric-affine theories and derive (i) the expression, in the geometrical optics (eikonal) limit, for light trajectories, showing that they still are null (extremal) geodesics and thus, in general, no longer autoparallels, (ii) a generic formula to obtain the relation between source (galaxy) and reception (observer) angular size (area) distances, generalizing Etherington's original distance reciprocity relation (DRR), and then applying it to two particular representative non-Riemannian geometries. First in metric-compatible, completely antisymmetric torsion geometries, the generalized DRR is not changed at all, and then in Weyl integrable spacetimes, the generalized DRR assumes a specially simple expression.

The European Physical Journal C, 2021

Scalar field cosmologies with a generalized harmonic potential and a matter fluid with a barotropic equation of state (EoS) with barotropic index $$\gamma $$ γ for the locally rotationally symmetric (LRS) Bianchi I and flat Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. Methods from the theory of averaging of nonlinear dynamical systems are used to prove that time-dependent systems and their corresponding time-averaged versions have the same late-time dynamics. Therefore, the simplest time-averaged system determines the future asymptotic behavior. Depending on the values of $$\gamma $$ γ , the late-time attractors of physical interests are flat quintessence dominated FLRW universe and Einstein-de Sitter solution. With this approach, the oscillations entering the system through the Klein–Gordon (KG) equation can be controlled and smoothed out as the Hubble parameter H – acting as time-dependent perturbation parameter – tends monotonically to zero. Numerical simu...

Physical Review D, 2019

A powerful result in theoretical cosmology states that a subset of anisotropic Bianchi models can be seen as the homogeneous limit of (standard) linear cosmological perturbations. Such models are precisely those leading to Friedmann spacetimes in the limit of zero anisotropy. Building on previous works, we give a comprehensive exposition of this result, and perform the detailed identification between anisotropic degrees of freedom and their corresponding scalar, vector, and tensor perturbations of standard perturbation theory. In particular, we find that anisotropic models very close to open (i.e., negatively curved) Friedmann spaces correspond to some type of super-curvature perturbations. As a consequence, provided anisotropy is mild, its effects on all types of cosmological observables can always be computed as simple extensions of the standard techniques used in relativistic perturbation theory around Friedmann models. This fact opens the possibility to consistently constrain, for all cosmological observables, the presence of large scale anisotropies on the top of the stochastic fluctuations.

The European Physical Journal Plus

Aims. This paper studies the cosmological power spectrum (PS) of the differential and integral galaxy volume number densities, respectively γ i and γ * i , constructed with the cosmological distances d i (i = A, G, L, Z), where d A is the angular diameter distance, d G is the galaxy area distance, d L is the luminosity distance and d z is the redshift distance. Theoretical and observational quantities were obtained in the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime with a non-vanishing cosmological constant. The radial correlation Ξ i , a quantity defined in the context of these densities, is also discussed in the wave number domain. All observational quantities were computed using luminosity function (LF) data obtained from the FORS Deep Field (FDF) galaxy survey. Methods. The theoretical and observational power spectra of γ i , γ * i , Ξ i and the ratio γ i /γ * i were calculated by performing Fourier transforms on values of these densities which were previously derived by Iribarrem et al. (2012) from the observational values [γ] obs and [γ * ] obs obtained by using the galactic absolute magnitudes and Schechter's parameters of the galaxy LF presented in Gabasch et al. (2004, 2006). These parameters were evaluated from a I-band selected dataset of the FDF in the redshift range 0.5 ≤ z ≤ 5.0 for its blue bands and 0.75 ≤ z ≤ 3.0 for its red ones. Results. The results show similar behavior of the power spectra obtained from γ and γ * using d L , d z and d G as distance measures. The PS of the densities defined with d A have a different and inconclusive behavior, as this cosmological distance reaches a maximum at z ≈ 1.6 in the adopted cosmological model. For the other distances, our results suggest that the PS of [γ i ] obs , [γ * i ] obs and [γ i /γ * i ] obs have a general behavior approximately similar to the power spectra obtained with the galaxy two-point correlation function and, by being sample size independent, they may be considered as alternative analytical tools to study the galaxy distribution.

General Relativity and Gravitation, 2012

Keyword Relativistic cosmology • Line element • Robertson-Walker metric • Stationary and non-stationary universes • Golden Oldie 1 The context H. P. Robertson's paper "Relativistic Cosmology" [48] was the first review paper on the subject of relativistic cosmological models: "This report deals with the attempts which have been made during the past decade and a half to solve the general problem of the structure of the universe as a whole." Such models had been initiated by Einstein's 1917 paper [12] on the Einstein static solution, 1 and then developed by a small group of pioneers: Einstein, de Sitter, Friedmann, Lanczos, Weyl, Lemaître, Robertson, Tolman, and Eddington. A second wave of pioneers-von Laue, Heckmann, McCrea, McVittie, Whittaker, Ward, Milne, and Walker 2 -joined in after the idea of an expanding universe became accepted in 1931, providing a sound theoretical basis for Hubble's observations of the magnitude-redshift relation for galaxies. Robertson's article was written 16 years after the first General Relativistic cosmological model and 11 years after the first expanding universe models