We find an explicit cosmological model which allows a special type of cosmological singularity which we call a $w$-singularity. This singularity has the scale factor finite, the energy density and pressure vanishing, and the only singular behaviour appears in a time-dependent barotropic index $w(t)$. It is different from the type IV cosmological singularity in that it does not exhibit the divergence of the higher derivatives of the Hubble parameter and from the big-brake since it does not fulfill the anti-Chaplygin gas equation of state. We also find an interesting duality between the $w$-singularities and the big-bang singularities. Physical examples of $w$-singularities appear in $f(R)$, scalar field and brane cosmologies.
Time-independent non-linear Schrödinger-type (NLS) formulation of FRW cosmology with canonical scalar field are considered in case of two barotropic fluids. We derived Friedmann formulation variables in terms of NLS variables. Seven exact solutions found by D'Ambroise \cite{DAmbroise:2010dgl} and one new found solution are explored and tested in cosmology. The result suggests that time-independent NLS formulation of cosmology case should be upgraded to the time-dependent case.
In this paper we characterize barotropic index singularities of homogeneous isotropic cosmological models. They are shown to appear in cosmologies for which the scale factor is analytical with a Taylor series in which the linear and quadratic terms are absent. Though the barotropic index of the perfect fluid is singular, the singularities are weak, as it happens for other models for which the density and the pressure are regular.
The field equations derived from the low energy string effective action with a matter tensor describing a perfect fluid with a barotropic equation of state are solved iteratively using the long-wavelength approximation, i.e. the field equations are expanded by the number of spatial gradients. In the zero order, a quasi-isotropic solution is presented and compared with the general solution of the pure dilaton gravity. Possible cosmological models are analyzed from the point of view of the pre-big bang scenario. The second order solutions are found and their growing and decaying parts are studie…
Recently a new type of cosmological singularity has been postulated for infinite barotropic index $w$ in the equation of state $p=w ρ$ of the cosmological fluid, but vanishing pressure and density at the singular event. Apparently the barotropic index $w$ would be the only physical quantity to blow up at the singularity. In this talk we would like to discuss the strength of such singularities and compare them with other types. We show that they are weak singularities.
We develop a technique for the reconstruction of the potential for a scalar field in cosmological models based on induced gravity. The potentials reproducing cosmological evolutions driven by barotropic perfect fluids, a cosmological constant, a Chaplygin gas and a modified Chaplygin gas are constructed explicitly.
We perform a thorough phase-plane analysis of the flow defined by the equations of motion of a FRW universe filled with a tachyonic fluid plus a barotropic one. The tachyon potential is assumed to be of inverse square form, thus allowing for a two-dimensional autonomous system of equations. The Friedmann constraint, combined with a convenient choice of coordinates, renders the physical state compact. We find the fixed-point solutions, and discuss whether they represent attractors or not. The way the two fluids contribute at late-times to the fractional energy density depends on how fast the ba…
We investigate barotropic perfect fluid cosmologies which admit an isotropic singularity. From the General Vorticity Result of Scott, it is known that these cosmologies must be irrotational. In this paper we prove, using two different methods, that if we make the additional assumption that the perfect fluid is shear-free, then the fluid flow must be geodesic. This then implies that the only shear-free, barotropic, perfect fluid cosmologies which admit an isotropic singularity are the FRW models.
We show that cosmological sudden singularities that respect the energy conditions can occur at finite times in Brans-Dicke and more general scalar-tensor theories of gravity. We construct these explicitly in the Friedmann universes. Higher-order versions of these singularities are also possible, including those that arise with when scalar fields have a self-interaction potential of power-law form.7 pages, no figures
We demonstrate how holonomy corrections in loop quantum cosmology (LQC) prevent the Big Rip singularity by introducing a quadratic modification in terms of the energy density $ρ$ in the Friedmann equation in the Friedmann-Lemaître-Robertson-Walker (FLRW) space-time in a consistent and useful way. In addition, we investigate whether other kind of singularities like Type II,III and IV singularities survive or are avoided in LQC when the universe is filled by a barotropic fluid with the state equation $P=-ρ-f(ρ)$, where $P$ is the pressure and $f(ρ)$ a function of $ρ$. It is shown that the Little…
The dynamical systems observability properties of barotropic bosonic and fermionic FRW cosmological oscillators are investigated. Nonlinear techniques for dynamical analysis have been recently developed in many engineering areas but their application has not been extended beyond their standard field. This paper is a small contribution to an extension of this type of dynamical systems analysis to FRW barotropic cosmologies. We find that determining the Hubble parameter of barotropic FRW universes does not allow the observability, i.e., the determination of neither the barotropic FRW zero mode n…
Interest in cosmological singularities has remarkably grown in recent times, particularly on future singularities with the discovery of late-time acceleration of the universe and dark energy. Recent work has seen a proper classification of such singularities into strong and weak based on their strength, with weak singularities being the likes of sudden, w and big freeze singularities and strong singularities like the big rip. This has led to a classification of such singularities in various types like Big rip is Type 1, w-singularity is type V etc. While singularities of type I-type IV have be…
We study evolution of cosmological models filled with the scalar field and barotropic matter. We consider the scalar field minimally and non-minimally coupled to gravity. We demonstrated the growth of degree of complexity of evolutional scenario through the description of matter content in terms of the scalar field. In study of all evolutional paths for all initial conditions methods of dynamical systems are used. Using linearized solutions we present simple method of derivation corresponding form of the Hubble function of the scale factor $H(a)$.
We consider a spherically symmetric gravitational collapse of a tachyon field with an inverse square potential, which is coupled with a barotropic fluid. By employing an holonomy correction imported from loop quantum cosmology, we analyse the dynamics of the collapse within a semiclassical description. Using a dynamical system approach, we find that the stable fixed points given by the standard general relativistic setting turn into saddle points in the present context. This provides a new dynamics in contrast to the black hole and naked singularities solutions appearing in the classical model…
Multiple $Λ$CDM cosmology is studied in a way that is formally a classical analog of the Casimir effect. Such cosmology corresponds to a time-dependent dark fluid model or, alternatively, to its scalar field presentation, and it motivated by the string landscape picture. The future evolution of the several dark energy models constructed within the scheme is carefully investigated. It turns out to be almost always possible to choose the parameters in the models so that they match the most recent and accurate astronomical values. To this end, several universes are presented which mimick (multipl…
We study solutions of the Friedmann equations in case of the homogeneous isotropic Universe filled with a perfect fluid. The main points concern the monotony properties of the solutions, the possibility to extend the solutions on all times and occurrence of singularities. We present a qualitative classification of all possible solutions in case of the general smooth barotropic equation of state of the fluid, provided the speed of sound is finite. The list of possible scenarios includes analogs of the "Big Rip" in the future and/or in the past as well as singularity free solutions and oscillati…
We use one of the simplest forms of the K-essence theory and we apply it to the classical anisotropic Bianchi type I cosmological model, with a barotropic perfect fluid modeling the usual matter content and with cosmological constant. The classical solutions for any but the stiff fluid and without cosmological constant are found in closed form, using a time transformation. We also present the solution whith cosmological constant and some particular values of the barotropic parameter. We present the possible isotropization of the cosmological model, using the ratio between the anisotropic param…
In this talk we would like to analyse the appearance of singularities in FLRW cosmological models which evolve close to w=-1, where w is the barotropic index of the universe. We relate small terms in cosmological time around w=-1 with the correspondent scale factor of the universe and check for the formation of singularities.
We investigate the occurrence of various exotic spacelike singularities in the past and the future evolution of $k = \pm 1$ Friedmann-Robertson-Walker model and loop quantum cosmology using a sufficiently general phenomenological model for the equation of state. We highlight the non-trivial role played by the intrinsic curvature for these singularities and the new physics which emerges at the Planck scale. We show that quantum gravity effects generically resolve all strong curvature singularities including big rip and big freeze singularities. The weak singularities, which include sudden and b…
Scalar field cosmology is a model for dark energy and inflation. It has been recently found that the standard Friedmann formulation of the scalar field cosmology can be expressed in a non-linear Schrödinger-type equation. The new mathematical formulation is hence called non-linear Schrödinger (NLS) formulation which is suitable for a FRLW cosmological system with non-negligible barotropic fluid density. Its major features are reviewed here.
The development of dark energy models has stimulated interest to cosmological singularities, which differ from the traditional Big Bang and Big Crunch singularities. We review a broad class of phenomena connected with soft cosmological singularities in classical and quantum cosmology. We discuss the classification of singularities from the geometrical point of view and from the point of view of the behaviour of finite size objects, crossing such singularities. We discuss in some detail quantum and classical cosmology of models based on perfect fluids (anti-Chaplygin gas and anti-Chaplygin gas …
We study a relation between the cosmological singularities in classical and quantum theory, comparing the classical and quantum dynamics in some models possessing the Big Brake singularity - the model based on a scalar field and two models based on a tachyon-pseudo-tachyon field . It is shown that the effect of quantum avoidance is absent for the soft singularities of the Big Brake type while it is present for the Big Bang and Big Crunch singularities. Thus, there is some kind of a classical - quantum correspondence, because soft singularities are traversable in classical cosmology, while the …
In this work we shall investigate the occurrence of future cosmological finite-time singularities in the dynamical system corresponding to two cosmological theories, namely that of vacuum $f(R)$ gravity and that of three fluids. The vacuum $f(R)$ gravity is an example for which the variables we will choose to quantify the phase space dynamics, do not necessarily blow-up near a cosmological singularity. After appropriately choosing the variables, we shall investigate the behavior of the corresponding dynamical system near some types of cosmological finite-time singularities, for some limiting c…
In this paper, we propose extended Chaplygin gas equation of state for which it recovers barotropic fluid with quadratic equation of state. We use numerical method to investigate the behavior of some cosmological parameters such as scale factor, Hubble expansion parameter, energy density and deceleration parameter. We also discuss about the resulting effective equation of state parameter. Using density perturbations we investigate the stability of the theory.
We present a new approach to the issues of spacetime singularities and cosmic censorship in general relativity. This is based on the idea that standard 4-dimensional spacetime is the conformal infinity of an ambient metric for the 5-dimensional Einstein equations with fluid sources. We then find that the existence of spacetime singularities in four dimensions is constrained by asymptotic properties of the ambient 5-metric, while the non-degeneracy of the latter crucially depends on cosmic censorship holding on the boundary.
We use the Sáez-Ballester theory on anisotropic Bianchi I cosmological model, with barotropic fluid and cosmological constant. We obtain the classical solution by using the Hamilton-Jacobi approach. Also the quantum regime is constructed and exact solutions to the Wheeler-DeWitt equation are found.
We consider the cosmological role of the scalar fields generated by the compactification of 11-dimensional Einstein gravity on a 7D elliptic twisted torus, which has the attractive features of giving rise to a positive semi-definite potential, and partially fixing the moduli. This compactification is therefore relevant for low energy M-theory, 11D supergravity. We find that slow-roll inflation with the moduli is not possible, but that there is a novel scaling solution in Friedmann cosmologies in which the massive moduli oscillate but maintain a constant energy density relative to the backgroun…
We show that simple scalar field models can give rise to curvature singularities in the effective Friedmann dynamics of Loop Quantum Cosmology (LQC). We find singular solutions for spatially flat Friedmann-Robertson-Walker cosmologies with a canonical scalar field and a negative exponential potential, or with a phantom scalar field and a positive potential. While LQC avoids big bang or big rip type singularities, we find sudden singularities where the Hubble rate is bounded, but the Ricci curvature scalar diverges. We conclude that the effective equations of LQC are not in themselves sufficien…
An isotropic and homogeneous cosmological model with a source of dark energy is studied. That source is simulated with a viscous relativistic fluid with minimal causal correction. In this model the restrictions on the parameters coming from the following conditions are analized: a) energy density without singularities along time, b) scale factor increasing with time, c) universe accelerated at present time, d) state equation for dark energy with "w" bounded and close to -1. It is found that those conditions are satified for the following two cases. i) When the transport coefficient (τ_{Π}), as…
The recently proposed Horava-Lifshitz (HL) theory of gravity is analyzed from the quantum cosmology point of view. By employing usual quantum cosmology techniques, we study the quantum Friedmann-Lemaitre-Robertson-Walker (FLRW) universe filled with radiation in the context of HL gravity. We find that this universe is quantum mechanically nonsingular in two different ways: the expectation value of the scale factor $<a>(t)$ never vanishes and, if we abandon the detailed balance condition suggested by Horava, the quantum dynamics of the universe is uniquely determined by the initial wave packet a…
We discuss three separate types of infinities that occur in non-supersymmetric canonical gravity and string theory. We consider UV perturbative singularities in non-supersymmetric gravity coupled to matter and how these are related to loop corrections to beta functions for non-supersymmetric strings. Next we consider classical cosmological singularities that occur in these equations and discuss a specific singular cosmology of Dudas and Mourad for non-supersymmetric string theory. Finally we discuss the infinities that occur in quantum cosmology associated with topology change and discuss how …
The present work is related to anisotropic cosmological evolution in metric $f(R)$ theory of gravity. The initial part of the paper develops the general cosmological dynamics of homogeneous anisotropic Bianchi-I spacetime in $f(R)$ cosmology. The anisotropic spacetime is pervaded by a barotropic fluid which has isotropic pressure. The paper predicts nonlinear growth of anisotropy in such spacetimes. In the later part of the paper we display the predictive power of the nonlinear differential equation responsible for the cosmological anisotropy growth in various relevant cases. We present the ex…
We consider a very general scenario of our universe where its geometry is characterized by the Finslerian structure on the underlying spacetime manifold, a generalization of the Riemannian geometry. Now considering a general energy-momentum tensor for matter sector, we derive the gravitational field equations in such spacetime. Further, to depict the cosmological dynamics in such spacetime proposing an interesting equation of state identified by a sole parameter $γ$ which for isotropic limit is simply the barotropic equation of state $p= (γ- 1) ρ$ ($γ\in \mathbb{R}$ being the barotropic index)…
In a universe whose elementary constituents are point particles there does not seem to be any obvious mechanism for avoiding the initial singularities in physical quantities in the standard model of cosmology. In contrast in string theory these singularities can be absent even at the level where spacetime is treated classically. This is a consequence of the basic degrees of freedom of strings in compact spaces, which necessitate a reinterpretation of what one means by a very small universe. We discuss the basic degrees of freedom of a string at the classical and quantum level, the minimum size…
We study the cosmology of a general scalar field and barotropic fluid during the early stage of a brane-world where the Friedmann constraint is dominated by the square of the energy density. Assuming both the scalar field and fluid are confined to the brane, we find a range of behaviour depending on the form of the potential. Generalising an approach developed for a standard Friedmann cosmology, in \cite{delaMacorra:1999ff}, we show that the potential dependence $V(ϕ)$ can be described through a parameter $λ\equiv -\sqrt{2} m_5^{3/2} V'/(\sqrt{H}V)$, where $m_5$ is the 5-dimensional Planck mas…
In this work we study the occurrence of finite-time cosmological singularities in a cosmological system comprising from three fluids. Particularly, the system contains two dark fluids, namely that of dark energy and dark matter, which are interacting, and of a non-interacting baryonic fluid. For the study we adopt the phase space approach by constructing the cosmological dynamical system in such a way so that it rendered to be an autonomous polynomial dynamical system, and in order to achieve this, we appropriately choose the variables of the dynamical system. By employing a rigid mathematical…
Interest in cosmological singularities has remarkably grown in recent times, particularly on future singularities with the discovery of late-time acceleration of the universe and dark energy. While such explorations have previously been done in various modified gravity and quantum gravitationally inspired cosmologies (besides standard general relativistic cosmology), no such an endeavour has been taken up till now in the realms of renormalization group approaches to cosmology and we have hence took up on this journey. In this work, we consider the formation of cosmological singularities in an …
The familiar Bang/Crunch singularities of classical cosmology have recently been augmented by new varieties: rips, sudden singularities, and so on. These tend to be associated with final states. Here we consider an alternative possibility for the initial state: a singularity which has the novel property of being inaccessible to physically well-defined probes. These singularities arise naturally in cosmologies with toral spatial sections.
We consider different aspects of the problem of cosmological singularity such as the BKL oscillatory approach to singularity, the new features of the cosmological dynamics in the neighbourhood of the singularity in multidimensional and superstring cosmological models and their connections with such a modern branch of mathematics as infinite-dimensional Lie algebras. Besides, we consider some new types of cosmological singularities which were widely discussed during last decade after the discovery of the phenomenon of cosmic acceleration.
We argue that the reasonings that underlie a recent comment by Gotay and Demaret [1, gr-qc/9605025] are defective. We maintain that, contrary to what they assert, our previous papers [2,3,4] are correct and indeed disprove their conjecture that quantum cosmological singularities are predetermined on the classical level by the choice of time.
We consider the fate of future singularities in the effective dynamics of loop quantum cosmology. Non-perturbative quantum geometric effects which lead to $ρ^2$ modification of the Friedmann equation at high energies result in generic resolution of singularities whenever energy density $ρ$ diverges at future singularities of Friedmann dynamics. Such quantum effects lead to the avoidance of a Big Rip, which is followed by a recollapsing universe stable against perturbations. Resolution of sudden singularity, the case when pressure diverges but energy density approaches a finite value depends on…
We review the variety of new singularities in homogeneous and isotropic FRW cosmology which differ from standard Big-Bang and Big-Crunch singularities and suggest how the nature of these singularities can be influenced by the varying fundamental constants.
In this paper, we study a projectable Hořava-Lifshitz cosmology without the detailed balance condition minimally coupled to a non-linear self-coupling scalar field. In the minisuperspace framework, the super Hamiltonian of the presented model is constructed by means of which, some classical solutions for scale factor and scalar field are obtained. Since these solutions exhibit various types of singularities, we came up with the quantization of the model in the context of the Wheeler-DeWitt approach of quantum cosmology. The resulting quantum wave functions are then used to investigate the poss…
We refer to the classic definition of a singularity in Einstein's general relativity (based on geodesic incompletness) as well as to some other criteria to evaluate the nature of singularities in cosmology. We review what different (non-Big-Bang) types of singularities are possible even in the simplest cosmological framework of Friedmann cosmology. We also show that various cosmological singularities may be removed or changed due to the variability of physical constants.
The cosmological singularities of the Bianchi I universe are analyzed in the setting of loop geometry underlying the loop quantum cosmology. We solve the Hamiltonian constraint of the theory and find the Lie algebra of elementary observables. Physical compound observables are defined in terms of elementary ones. Modification of classical theory by holonomy around a loop removes the singularities. However, our model has a free parameter that cannot be determined within our method. Testing the model by the data of observational cosmology may be possible after quantization of our modified classic…
We discuss various types of exotic (non-standard) singularities in the Universe: a Big-Rip (BR or type I), a Sudden Future Singularity (SFS or type II), a Generalized Sudden Future Singularity, a Finite Scale Factor singularity (FSF or type III), a Big-Separation (BS or type IV) and a $w$-singularity. They are characterized by violation of all or some of the energy conditions which results in a blow-up of all or some of the physical quantities: the scale factor, the energy density, the pressure, and the barotropic index. We relate the emergence of these singularities with physical theories (su…
In this paper we present the noncommutative Bianchi Class A cosmological models coupled to barotropic perfect fluid. The commutative and noncommutative quantum solution to the Wheeler-DeWitt equation for any factor ordering, to the anisotropic Bianchi type II cosmological model are found, using a stiff fluid ($γ=1$). In our toy model, we introduce noncommutative scale factors, is say, we consider that all minisuperspace variables $\rm q^i$ does not commute, so the simplectic structure was modified.
We investigate non-linear Schrödinger-type formulation of cosmology of which our cosmological system is a general relativistic FRLW universe containing canonical scalar field under arbitrary potential and a barotropic fluid with arbitrary spatial curvatures. We extend the formulation to include phantom field case and we have found that Schrödinger wave function in this formulation is generally non-normalizable. Assuming power-law expansion, $a \sim t^q$, we obtain scalar field potential as function of time. The corresponding quantities in Schrödinger-type formulation such as Schrödinger total …
We review recent work on the existence and nature of cosmological singularities that can be formed during the evolution of generic as well as specific cosmological spacetimes in general relativity. We first discuss necessary and sufficient conditions for the existence of geodesically incomplete spacetimes based on a tensorial analysis of the geodesic equations. We then classify the possible singularities of isotropic globally hyperbolic universes using the Bel-Robinson slice energy that closely monitors the asymptotic properties of fields near the singularity. This classification includes all …
The Bianchi IX model has been used often to investigate the structure close to singularities of general relativity. Its classical chaos is expected to have, via the BKL scenario, implications even for the approach to general inhomogeneous singularities. Thus, it is a popular model to test consequences of modifications to general relativity suggested by quantum theories of gravity. This paper presents a detailed proof that modifications coming from loop quantum gravity lead to a non-chaotic effective behavior. The way this is realized, independently of quantization ambiguities, suggests a new l…
The fundamental singularity theorem of FLRW cosmologies assumes that the matter content in the cosmological model obeys the strong energy condition along with a nonpositive cosmological constant which gives rise to an irrotational geodesic singularity. In this paper, we show that the important case of a spatially flat Friedmann-Lemaître-Robertson-Walker universe with barotropic matter obeying only the \emph{weak} energy condition with a nonnegative cosmological constant also contains a past singularity. We accomplish this using topological methods from dynamical systems theory. The methods emp…
We give a brief review of the non-minimal derivative coupling (NMDC) scalar field theory in which there is non-minimal coupling between the scalar field derivative term and the Einstein tensor. We assume that the expansion is of power-law type or super-acceleration type for small redshift. The Lagrangian includes the NMDC term, a free kinetic term, a cosmological constant term and a barotropic matter term. For a value of the coupling constant that is compatible with inflation, we use the combined WMAP9 (WMAP9+eCMB+BAO+ $H_0$) dataset, the PLANCK+WP dataset, and the PLANCK $TT,TE,EE$+lowP+Lensi…
Different regularizations of the Hamiltonian constraint in loop quantum cosmology yield modified loop quantum cosmologies, namely mLQC-I and mLQC-II, which lead to qualitatively different Planck scale physics. We perform a comprehensive analysis of resolution of various singularities in these modified loop cosmologies using effective spacetime description and compare with earlier results in standard loop quantum cosmology. We show that the volume remains non-zero and finite in finite time evolution for all considered loop cosmological models. Interestingly, even though expansion scalar and ene…
We discuss the fate of classical type IV singularities in quantum cosmology. The framework is Wheeler-DeWitt quantization applied to homogeneous and isotropic universes with a perfect fluid described by a generalized Chaplygin gas. Such a fluid can be dynamically realized by a scalar field. We treat the cases of a standard scalar field with positive kinetic energy and of a scalar field with negative energy (phantom field). We first present the classical solutions. We then discuss in detail the Wheeler-DeWitt equation for these models. We are able to give analytic solutions for a special case a…
Loop quantum gravity is believed to eliminate singularities such as the big bang and big crunch singularity. This belief is based on studies of so-called loop quantum cosmology which concerns symmetry-reduced models of quantum gravity. In this paper, the problem of singularities is analysed in the context of the Bohmian formulation of loop quantum cosmology. In this formulation there is an actual metric in addition to the wave function, which evolves stochastically (rather than deterministically as the case of the particle evolution in non-relativistic Bohmian mechanics). Thus a singularity oc…
