We consider the Wheeler-DeWitt equation $Hψ=0$ in a suitable Hilbert space. It turns out that this equation has countably many solutions $ψ_i$ which can be considered as eigenfunctions of a Hamilton operator implicitly defined by $H$. We consider two models, a bounded one, $0<r<r_0$, and an unbounded, $0<r<\un$, which represent different eigenvalue problems. In the bounded model we look for eigenvalues $\Lam_i$, where the $\Lam_i$ are the values of the cosmological constant which we used in the Einstein-Hilbert functional, and in the unbounded model the eigenvalues are given by $(-\Lam_i)^{-\frac {n-1}{n}}$, where $\Lam_i<0$. The $ψ_i$ form a basis of the underlying Hilbert space.
All solutions have an initial singularity in $r=0$. Under certain circumstances a smooth transition from big crunch to big bang is possible.
Einstein's equation, in its standard form, breaks down at the Big Bang singularity. A new version, equivalent to Einstein's whenever the latter is defined, but applicable in wider situations, is proposed. The new equation remains smooth at the Big Bang singularity of the Friedmann-Lemaitre-Robertson-Walker model. It is a tensor equation defined in terms of the Ricci part of the Riemann curvature. It is obtained by taking the Kulkarni-Nomizu product between Einstein's equation and the metric tensor.
We study past-directed extendibility of Friedmann-Lemaître-Robertson-Walker (FLRW) and Bianchi type I spacetimes with a scale factor vanishing in the past. We give criteria for determining whether a boundary for past-directed incomplete geodesics is a parallelly propagated curvature singularity, which cannot necessarily be read off from scalar curvature invariants. It is clarified that, for incomplete FLRW spacetime to avoid the singularity, the spacetime necessarily reduces to the Milne universe or flat de Sitter universe toward the boundary. For incomplete Bianchi type I spacetime to be free…
We prove the existence of a spectral resolution of the Wheeler-DeWitt equation when the matter field is provided by a massive Yang-Mills field. The resolution is achieved by first solving the free eigenvalue problem for the gravitational field and then the constrained eigenvalue problem for the Yang-Mills field. In the latter case the mass of the Yang-Mills field assumes the role of the eigenvalue.
We prove the existence of a spectral resolution of the Wheeler-DeWitt equation when the matter field is provided by a Yang-Mills field, with or without mass term, if the spatial geometry of the underlying spacetime is homothetic to $\R[3]$. The energy levels of the resulting quantum model, i.e., the eigenvalues of the corresponding self-adjoint Hamiltonian with a pure point spectrum, are strictly positive.
The window is found in the space of the free parameters of the theory of viscoelastic matter for which the Friedmann singularity is stable. Under stability we mean that in the presence of the shear stresses the \textit{generic} solution of the equations of relativistic gravity possessing the isotropic, homogeneous and thermally equilibrated cosmological singularity exists.
The Friedman Universe with the remnant stochastic scalar field in the hybrid inflation model is examined. It is shown that the small effective cosmological constant appears which increases the cosmological expansion.
We completely classify the Friedmann-Lemaître-Robertson-Walker solutions with spatial curvature $K=0,\pm 1$ for perfect fluids with linear equation of state $p=wρ$, where $ρ$ and $p$ are the energy density and pressure, without assuming any energy conditions. We extend our previous work to include all geodesics and parallelly propagated curvature singularities, showing that no non-null geodesic emanates from or terminates at the null portion of conformal infinity and that the initial singularity for $K=0,-1$ and $-5/3<w<-1$ is a null non-scalar polynomial curvature singularity. We thus obtain …
We study the deep connections between the concepts of quantum information theory and cosmology. Employing Fermi normal coordinates and conformal Fermi coordinates, we construct a relation between Friedmann equations of Friedmann-Lemaitre-Robertson-Walker universe and entanglement. Friedmann equations are derived with the first law of entanglement under the assumption that entanglement entropy in a geodesic balls is maximized at fixed volume.
We consider an exactly solvable model for production of fermions in the Friedman flat universe with a scale factor linearly growing with time. Exact solution expressed through the special functions admit an analytical calculation of the number density of created particles. We also discuss in general the role of the phenomenon of the cosmological particle production in the history of universe.
In the Friedmann equations, an infinite initial density is avoided only when the universe is spatially flat. With such equations being then valid when the scale factor $a=0$, the universe must also be in the state of vacuum when $a$ is infinitesimal. Matter creation thus largely comes from nonlinearity of the Friedmann equations at a finite $a$. For the correspondence between such an initial vacuum and the Friedmann universe, therefore, a vacuum density is suggested to be gravitationally significant only when it could also have a matter phase. The effective cosmological term is then due to suc…
Cosmological singularity and asymptotic behaviour of scale factor of generalized cosmological models are analyzed in respect of their structural stability. It is shown, that cosmological singularity is structurally unstable for the majority of models with barotropic perfect fluid with strong energy condition. Inclusion of Lambda-term extends the set of structurally stable cosmological models.
We present Friedmann flat spacetime uncertainty relations (STUR) together with some cosmological implications. An interesting link between the Principle of "gravitational stability against localization of events" (PGSL) and the holographic Bekenstein entropy bound (HEB) is also investigated. The same theorems leading to our STUR are used to calculate, thanks to the holographic principle, the entropy of the universe at its apparent horizon. The generalized entropy formula can be used to discuss interesting links with a quantum spacetime.
In this paper, we calculate the one-loop quantum cosmological corrections to the kink energy in the closed Friedmann-Robertson-Walker universe in which the fluctuation potential $V^{\prime\prime}$ has a shape invariance property. We use the generalized zeta function regularization method to implement our setup for describing quantum kink-like states. It is conjectured that the corrections lead to the renormalized gravitational constant.
First order rotational perturbations of the Friedmann-Robertson-Walker metric are considered in the framework of the brane-world cosmological models. A rotation equation, relating the perturbations of the metric tensor to the angular velocity of the matter on the brane is derived under the assumption of slow rotation. The mathematical structure of the rotation equation imposes strong restrictions on the temporal and spatial dependence of the brane matter angular velocity. The study of the integrable cases of the rotation equation leads to three distinct models, which are considered in detail. …
The evolution of spatially homogeneous and isotropic cosmological models containing a perfect fluid with equation of state p=wρ and a cosmological constant Λ is investigated for arbitrary combinations of w and Λ, using standard qualitative analysis borrowed from classical mechanics. This approach allows one to consider a large variety of situations, appreciating similarities and differences between models, without solving the Friedmann equation, and is suitable for an elementary course in cosmology.
We solve the source free electromagnetic wave equation in Friedmann-Robertson-Walker space-times for curvature $K=0$ and $K=-1$. Deriving a solution expression in the form of spherical means we deduce and compare two properties of the Maxwell propagator namely, decay rates, as well as continuity through the space-time singularity to that of the scalar wave equation presented by Abbasi and Craig [1].
A new approach for arbitrary dimension to the Friedmann cosmological models is presented. Taking suitable changes of the parameters of the spacetime the harmonic motion equations appear, where the curvature determines the angular frequency. Some physical interpretations are also explained. As a consequence of our approach, we give new exact solutions to the Einstein's equation when the observer is not comoving with the perfect fluid, as well as, when the pressure not vanishes, including a new procedure to acquire exact solutions for cases of universes at the dark energy dominated stage.
We discuss the holographic principle in a radiation-dominated, closed Friedmann-Robertson-Walker (FRW) universe with a positive cosmological constant. By introducing a cosmological D-bound on the entropy of matter in the universe, we can write the Friedmann equation governing the evolution of the universe in the form of the Cardy formula. When the cosmological D-bound is saturated, the Friedmann equation coincides with the Cardy-Verlinde formula describing the entropy of radiation in the universe. As a concrete model, we consider a brane universe in the background of Schwarzschild-de Sitter bl…
We completely classify Friedmann-Lemaître-Robertson-Walker solutions with spatial curvature $K=0,\pm 1$ and equation of state $p=wρ$, according to their conformal structure, singularities and trapping horizons. We do not assume any energy conditions and allow $ρ< 0$, thereby going beyond the usual well-known solutions. For each spatial curvature, there is an initial spacelike big-bang singularity for $w>-1/3$ and $ρ>0$, while no big-bang singularity for $w<-1$ and $ρ>0$. For $K=0$ or $-1$, $-1<w<-1/3$ and $ρ>0$, there is an initial null big-bang singularity. For each spatial curvature, there i…
A general class of nonlocal cosmological models is considered. A new method for solving nonlocal Friedmann equations is proposed, and solutions of the Friedmann equations with nonlocal operator are presented. The cosmological properties of these solutions are discussed. Especially indicated is $p$-adic cosmological model in which we have obtained nonsingular bouncing solution and string field theory tachyon model in which we have obtained full solution of nonlocal Friedmann equations with $w=-1$ at large times. The possibility of obtaining realistic value of cosmological constant from nonlocal…
We analyze the leading terms of the spectral action for a model of noncommutative geometry, which is a product of $4$-dimensional Riemannian manifold with a two-point space exploring the previously neglected case when the metrics over each sheet are different. Assuming the Friedmann-Lemaître-Robertson-Walker type of the metric for both sheets we obtain the action, which in addition to the the usual cosmological constant terms and the Einstein-Hilbert term involves a nonlinear interaction term. We study qualitative picture of potential consequences of such term in the basic cosmological models.
We analyze propagation equations for the polar modes of gravitational waves in cosmological space-times. We prove that polar gravitational waves must perturb the density and non-azimuthal components of the velocity of material medium of the Friedman-Lemaitre-Robertson-Walker spacetimes. Axial gravitational waves can influence only the azimuthal velocity, leading to local cosmological rotation. The whole gravitational dynamics reduces to the single "master equation" that has the same form for polar and axial modes. That allows us to conclude that the status of the Huygens principle is the same …
We consider the wave equation, $\square_gψ=0$, in fixed flat Friedmann-Lemaître-Robertson-Walker and Kasner spacetimes with topology $\mathbb{R}_+\times\mathbb{T}^3$. We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface $\{t=0\}$. The initial data sets are characterized by the condition that the Neumann data should domin…
We present the generalized Friedmann equations describing the cosmological evolution of a finite thick brane immeresed in a five-dimensional Schwarzschild Anti-de Sitter spacetime. A linear term in the density in addition to a quatratic one arises in the Friedmann equation, leading to the standard cosmological evolution at late times without introducing an ad hoc tension term for the brane. The effective four-dimensional cosmological constant is then uplifted similar to the KKLT effect and vanishes for a brane thickness equal to the AdS curvature size, up to the third order of the thickness. T…
In this note, we discuss how possible expansion histories of the universe can be inferred in a simple way, for arbitrary energy contents. No new physical results are obtained, but the goal is rather to discuss an alternative way of writing the Friedmann equation in order to facilitate an intuitive understanding of the possible solutions; for students and researchers alike. As has been noted in passing by others, this specific form of the Friedmann equation allows us to view the universal expansion as a particle rolling along a frictionless track. Specific examples depicted include the current …
The Generalized Uncertainty Principle (GUP) has emerged in numerous attempts to a theory of quantum gravity and predicts the existence of a minimum length in Nature. In this work, we consider two cosmological models arising from Friedmann equations modified by the GUP (in its linear and quadratic formulations) and compare them with observational data. Our aim is to derive constraints on the GUP parameter and discuss the viability and physical implications of such models. We find for the parameter in the quadratic formulation the constraint $α^{2}_{Q}<10^{59}$ (tighter than most of those obtain…
The Laser Interferometer Gravitational-Wave Observatory (LIGO) detection of gravitational waves that take away 5% of the total mass of two merging black holes points out on the importance of considering varying gravitational mass of a system. Using an assumption that the energy-momentum pseudo-tensor of gravitational waves is not considered as a source of gravitational field, we analyse a perturbation of the Friedmann-Robertson-Walker metric caused by the varying gravitational mass of a system. This perturbation leads to a modified Friedmann equation that contains a term similar to the 'cosmol…
We present our development of Zeldovich's ideas for the measurement of the cosmological angular diameter distance (ADD) in the Friedmann Universe. We derive the general differential equation for the ADD measurement which is valid for an open, spatially-flat and closed universe, and for any stress energy tensor. We solve the mentioned equations in terms of quadratures in a form suitable for further numerical investigations for the present universe filled by radiation, (baryonic and dark) matter and dark energy. We perform the numerical investigation in the absence of radiation, and show the str…
Different approaches to quantum cosmology are studied in order to deal with the future singularity avoidance problem. Our results show that these future singularities will persist but could take different forms. As an example we have studied the big rip which appear when one considers the state equation $P=ωρ$ with $ω<-1$, showing that it does not disappear in modified gravity. On the other hand, it is well-known that quantum geometric effects (holonomy corrections) in loop quantum cosmology introduce a quadratic modification, namely proportional to $ρ^2$, in Friedmann's equation that replace …
We study the cosmological solutions of the one loop corrected superstring effective action, in a Friedmann-Robertson-Walker background, and in the presence of the dilaton and modulus fields. A particularly interesting class of solutions is found which avoid the initial singularity and are consistent with the perturbative treatment of the effective action.
The singularity for the big bang state can be represented using the generalized anisotropic Friedmann equation, resulting in a system of differential equations in a central force field. We study the regularizability of this singularity as a function of a parameter, the equation of state, $w$. We prove that for $w >1$ it is regularizable only for $w$ satisfying relative prime number conditions, and for $w \leq 1$ it can always be regularized. This is done by using a McGehee transformation, usually applied in the three and four-body problems. This transformation blows up the singularity into an …
The talk at international conference in honor of Ya. B. Zeldovich 95th Anniversary, Minsk, Belarus, April 2009. The talk represents a review of the old results and contemporary development on the problem of cosmological singularity.
We study the expansion of the Universe using an effective Friedmann equation obtained from the dynamics of GFT (Group Field Theory) isotropic condensates. The evolution equations are classical, with quantum correction terms to the Friedmann equation given in the form of effective fluids coupled to the emergent classical background. The occurrence of a bounce, which resolves the initial spacetime singularity, is shown to be a general property of the model. A promising feature of this model is the occurrence of an era of accelerated expansion, without the need to introduce an inflaton field with…
We investigate Bianchi I cosmological model in the theory of a dilatonM field coupled to gravity through a Gauss-Bonnet term. Two type ofM cosmological singularity are distinguished. The former is analogous toM the Einstein gravity singularity, the latter (which does not appear inM classical General Relativity) occurs when the main determinant of theM system of field equations vanishes. An analogy between the latterM cosmological singularity and the singularity inside a black hole withM a dilatonic hair is discussed. Initial conditions, leading to theseM two types of cosmological singularity a…
Alternative scenarios to the Big Bang singularity have been subject of intense research for several decades by now. Most popular in this sense have been frameworks were such singularity is replaced by a bounce around some minimal cosmological volume or by some early quantum phase. This latter scenario was devised a long time ago and referred as an "emergent universe" (in the sense that our universe emerged from a constant volume quantum phase). We show here that within an improved framework of canonical quantum gravity (the so called Quantum Reduced Loop Gravity) the Friedmann equations for co…
We use a combination of numerical and analytical methods, exploiting the equations derived in a preceding paper, to classify all spherically symmetric self-similar solutions which are asymptotically Friedmann at large distances and contain a perfect fluid with equation of state $p=(γ-1)μ$ with $0<γ<2/3$. The expansion of the Friedmann universe is accelerated in this case. We find a one-parameter family of self-similar solutions representing a black hole embedded in a Friedmann background. This suggests that, in contrast to the positive pressure case, black holes in a universe with dark energy …
Based on the asymptotic analysis of ordinary differential equations, we classify all spherically symmetric self-similar solutions to the Einstein equations which are asymptotically Friedmann at large distances and contain a perfect fluid with equation of state $p=(γ-1)μ$ with $0<γ<2/3$. This corresponds to a ``dark energy'' fluid and the Friedmann solution is accelerated in this case due to anti-gravity. This extends the previous analysis of spherically symmetric self-similar solutions for fluids with positive pressure ($γ>1$). However, in the latter case there is an additional parameter assoc…
In the context of Brans-Dicke scalar tensor theory of gravitation, the cosmological Friedmann equation which relates the expansion rate $H$ of the universe to the various fractions of energy density is analyzed rigorously. It is shown that Brans-Dicke scalar tensor theory of gravitation brings a negligible correction to the matter density component of Friedmann equation. Besides, in addition to $Ω_Λ$ and $Ω_{M}$ in standard Einstein cosmology, another density parameter, $Ω_{_Δ}$, is expected by the theory. This implies that if $Ω_{_Δ}$ is found to be nonzero, data will favor this model instead…
We quantize three Friedmann-Robertson-Walker models in the presence of a negative cosmological constant and radiation. The models differ from each other by the constant curvature of the spatial sections, which may be positive, negative or zero. They give rise to Wheeler-DeWitt equations for the scale factor which have the form of the Schroedinger equation for the quartic anharmonic oscillator. We find their eigenvalues and eigenfunctions by using a method first developed by Chhajlany and Malnev, and use the eigenfunctions in order to construct wave packets for each case and evaluate the time-d…
The first-order loop quantum gravity correction of the simplest, classical general-relativistic Friedmann Hamiltonian constraint, emerging from a holomorphic spinfoam cosmological model peaked on homogeneous, isotropic geometries, is studied. The quantum Hamiltonian constraint, satisfied by the EPRL transition amplitude between the boundary cosmological coherent states, includes a contribution of the order of the Planck constant $\hbar$ that also appears in the corresponding semiclassical symplectic model. The analysis of this term gives a quantum-gravitational correction to the classical Frie…
Problem of cosmological singularity is discussed in the framework of gauge theories of gravitation. Generalizing cosmological Friedmann equations (GCFE) for homogeneous isotropic models including scalar fields and usual gravitating matter are introduced. It is shown that by certain restrictions on equation of state of gravitating matter and indefinite parameter of GCFE generic feature of inflationary cosmological models of flat, open and closed type is their regular bouncing character.
We study an alternative geometrical approach on the problem of classical cosmological singularity. It is based on a generalized function $f (x, y) = x^{2} + y^{2} = (1 - z)z^{n}$ which consists of a cusped coupled isosurface. Such a geometry is computed and discussed into the context of Friedmann singularity-free cosmology where a pre-big bang scenario is considered. Assuming that the mechanism of cusp formation is described by non-linear oscillations of a pre-big bang extended very high energy density field ($> 3 \times 10^{94} kg/m^{3} $), we show that the action under the gravitational fiel…
The note presents a classification of the relevant distinct types of solutions of the general Friedmann equation without assuming a priori restrictions for the parameters occurring in this equation. The emphasis is on the case of a non-vanishing cosmological constant. The classification uses algebraic criteria. The result is: There are four distinct basic types of models. Explicit formulas for decisive terms are given. Characteristic mutual relations of cosmological constant, mass and radiation density to distinguish between the models are calculated.
The causal interpretation of quantum mechanics is applied to a homogeneous and isotropic quantum universe, whose matter content is composed by non interacting dust and radiation. For wave functions which are eigenstates of the total dust mass operator, we find some bouncing quantum universes which reachs the classical limit for scale factors much larger than its minimum size. However these wave functions do not have unitary evolution. For wave functions which are not eigenstates of the dust total mass operator but do have unitary evolution, we show that, for flat spatial sections, matter can b…
We examine the singularity resolution issue in quantum gravity by studying a new quantization of standard Friedmann-Robertson-Walker geometrodynamics. The quantization procedure is inspired by the loop quantum gravity programme, and is based on an alternative to the Schrödinger representation normally used in metric variable quantum cosmology. We show that in this representation for quantum geometrodynamics there exists a densely defined inverse scale factor operator, and that the Hamiltonian constraint acts as a difference operator on the basis states. We find that the cosmological singularit…
We consider the quantization of a Friedmann-Robertson-Walker universe. We derive a reduced square root Hamiltonian by choosing the scale factor as time variable and quantize the theory using the Dirac method. From the resulting spinor equation we show that there is no semiclassical wave packet that avoids the Big Bang singularity. Our work raises the question concerning the relationship between the choice of time and singularity avoidance.
The initial singularity problem in standard general relativity is treated on the light of a viewpoint asserting that this formulation of Einstein's theory and its conformal formulations are physically equivalent. We show that flat Friedmann-Robertson-Walker universes and open dust-filled and radiation-filled universes are singularity free when described in terms of the formulation of general relativity conformal to the canonical one.
We investigate the corrections to the inflationary cosmological dynamics due to a $R^2$ term in the Palatini formulation which may arise as quantum corrections to the effective Lagrangian in early universe. We found that the standard Friedmann equation will not be changed when the scalar field is in the potential energy dominated era. However, in the kinetic energy dominated era, the standard Friedmann equation will be modified and in the case of closed and flat universe, the Modified Friedmann equation will automatically require that the initial kinetic energy density of the scalar field must…
Our main aim is to apply the theory of regularly varying functions to the asymptotical analysis at infinity of solutions of Friedmann cosmological equations. A new constant $Γ$ is introduced related to the Friedmann cosmological equations. Determining the values of $Γ$ we obtain the asymptotical behavior of the solutions, i.e. of the expansion scale factor $a(t)$ of a universe. The instance $Γ< 1/4$ is appropriate for both cases, the spatially flat and open universe, and gives a sufficient and necessary condition for the solutions to be regularly varying. This property of Friedmann equations i…
It is proposed to consider dark matter as a perfect dilaton-spin fluid (with particles endowed with intrinsic spin and dilaton charge) in the framework of a gravitational theory with a Weyl-Cartan geometrical structure. The modified Friedmann-Lema\^ıtre equation (with a cosmological term) is obtained for the homogeneous and isotropic Universe filled with the dilaton-spin dark matter. On the basis of this equation, we develop a nonsingular cosmological model starting from an inflation-like stage (for super-stiff equation of state), passing radiation-dominated and matter-dominated decelerating s…
Within our thick brane approach previously used to obtain the cosmological evolution equations on a thick brane embedded in a five-dimensional Schwarzschild Anti-de Sitter spacetime it is explicitly shown that the consistency of these equations with the energy conservation equation requires that, in general, the thickness of the brane evolves in time. This varying brane thickness entails the possibility that both Newton's gravitational constant $G$ and the effective cosmological constant $Λ_4$ are time dependent.
In this note I discuss the problem of cosmological singularities within gauge theories of gravitation. Solutions of cosmological equations with the scalar field are considered.
