Evolution of the scale factor with a variable cosmological term

2021

The relative expansion of the Universe is parametrized by a dimensionless scale factor R. This is a key parameter in Friedman equations and also known as the cosmic scale factor or Robertson Walker scale factor. In the early stages of the Big Bang, most of the energy was in the form of radiation and that radiation has a dominant influence on the expansion of the Universe. Later, with cooling from the expansion, the roles of matter and radiation changed and the Universe entered into a matter dominated era. Recent observational results suggest that we have already entered an era dominated by dark energy (DE). But investigation of the roles of matter and radiation is most important for a good understanding of the early Universe. One should note that, the effective energy density of the Universe is usually expressed in terms of the scale factor. Also, the dynamics of the Universe is assessed through an equation of state parameter ω usually

Journal of Cosmology and Astroparticle Physics, 2020

We argue that the discrepancy between the Planck mass scale and the observed value of the cosmological constant can be largely attenuated if those quantities are understood as a result of effective, and thus scale-dependent, couplings. We exemplify this mechanism for the early inflationary epoch of the universe by solving the corresponding effective gap equations, subject to an energy condition. Several non-trivial checks and extensions are discussed. A comparison of our results to the renormalization group flow, obtained within the asymptotic safety program reveals a stunning agreement. Contents I. Introduction 1 A. Classical Λ dominated universe 2 B. The cosmological constant problem in the very early universe 2 II. Scale-Dependence in Gravity 2 A. Scale setting 3 B. Modified Friedmann equations 3 C. Null energy condition (NEC) 3 III. Results and discussion 3 A. Equation of state of scale-dependent G(t) 4 B. Deflation of the cosmological constant problem 4 C. Comparison with the functional renormalization group 5 IV. Conclusions 6 References 6

arXiv (Cornell University), 2014

A new class of gravity-matter models defined in terms of two independent non-Riemannian volume forms (alternative generally covariant integration measure densities) on the space-time manifold are studied in some detail. These models involve an additional R^2 (square of the scalar curvature) term as well as scalar matter field potentials of appropriate form so that the pertinent action is invariant under global Weyl-scale symmetry. Scale invariance is spontaneously broken upon integration of the equations of motion. After performing transition to the physical Einstein frame we obtain: (i) An effective potential for the scalar field with two flat regions which allows for a unified description of both early universe inflation as well as of present dark energy epoch; (ii) For a definite parameter range the model possesses a non-singular "emergent universe" solution which describes an initial phase of evolution that precedes the inflationary phase.

Modern Physics Letters A, 1999

Realizations of scale invariance are studied in the context of a gravitational theory where the action (in the first-order formalism) is of the form [Formula: see text] where Φ is a density built out of degrees of freedom, the "measure fields" independent of gμν and matter fields appearing in L1, L2. If L1 contains the curvature, scalar potential V(ϕ) and kinetic term for ϕ, L2 another potential for ϕ, U(ϕ), then the true vacuum state has zero energy density, when theory is analyzed in the conformal Einstein frame (CEF), where the equations assume the Einstein form. Global scale invariance is realized when V(ϕ)=f1eαϕ and U(ϕ)=f2e2αϕ. In the CEF the scalar field potential energy V eff (ϕ) has, in addition to a minimum at zero, a flat region for αϕ→∞, with nonzero vacuum energy, which is suitable for either a new inflationary scenario for the early universe or for a slowly rolling decaying Λ-scenario for the late universe, where the smallness of the vacuum energy can be unde...

International Journal of Theoretical Physics, 2009

Choosing the three phenomenological models of the dynamical cosmological term Λ, viz., , and Λ∼ρ where a is the cosmic scale factor, it has been shown by the method of numerical analysis for the considered non-linear differential equations that the three models are equivalent for the flat Universe k=0 and for arbitrary non-linear equation of state. The evolution plots for

Applied Mathematics, 2013

We once again reference Theorem 6.1.2 of the book by Ellis, Maartens, and MacCallum in order to argue that if there is a non zero initial scale factor, that there is a partial breakdown of the Fundamental Singularity theorem which is due to the Raychaudhuri equation. Afterwards, we review a construction of what could happen if we put in what Ellis, Maartens, and MacCallum call the measured effective cosmological constant and substitute Effective    in the Friedman equation. I.e. there are two ways to look at the problem, i.e. after Effective    , set Vac  as equal to zero, and have the left over  as scaled to background cosmological temperature, as was postulated by Park (2002) or else have Vac  as proportional to 38 2 10 Vac GeV  which then would imply using what we call a 5 dimensional contribution to  as proportional to 5~c onst/ T D     . We find that both these models do not work for generating an initial singularity.  removal as a non zero cosmological constant is most easily dealt with by a Bianchi I universe version of the generalized Friedman equation. The Bianchi I universe case almost allows for use of Theorem 6.1.2. But this Bianchi 1 Universe model almost in fidelity with Theorem 6.1.2. requires a constant non zero shear for initial fluid flow at the start of inflation which we think is highly unlikely.

2010

We study cosmological solutions for the very early universe beginning at the Planck scale for a universe containing radiation, curvature and, as a simplification of a possible scalar field potential, a cosmological constant term. The solutions are the natural counterpart of the well known results for a post-inflationary universe of non-relativistic matter, curvature and a cosmological constant. Contrary to the common belief that inflation arises independently of the initial curvature we show that in the positive curvature case the universe collapses again into a Big Crunch without allowing the cosmological term to dominate and to produce inflation. There is a critical value for the cosmological constant which divide the regions where inflation is allowed from those where inflation cannot occur. One can also have loitering solutions where the scale factor remains almost constant growing to produce inflation (or decreasing to a Big Crunch) after a time which depends on the amount of energy above (or below) the critical energy. At the critical energy the solution approaches asymptotically a particular value for the scale factor (Einstein's static pre-inflationary universe). The cases where the cosmological term vanishes or becomes negative are also studied providing a complete discussion of Friedmann models.

We present a nonlinear post-Friedmann framework for structure formation, generalizing to cosmology the weak-field (post-Minkowskian) approximation, unifying the treatment of small and large scales. We consider a universe filled with a pressureless fluid and a cosmological constant Λ, the theory of gravity is Einstein's general relativity and the background is the standard flat ΛCDM cosmological model. We expand the metric and the energy-momentum tensor in powers of 1/c, keeping the matter density and peculiar velocity as exact fundamental variables. We assume the Poisson gauge, including scalar and tensor modes up to 1/c 4 order and vector modes up to 1/c 5 terms. Through a redefinition of the scalar potentials as a resummation of the metric contributions at different orders, we obtain a complete set of nonlinear equations, providing a unified framework to study structure formation from small to superhorizon scales, from the nonlinear Newtonian to the linear relativistic regime. We explicitly show the validity of our scheme in the two limits: at leading order we recover the fully nonlinear equations of Newtonian cosmology; when linearized, our equations become those for scalar and vector modes of first-order relativistic perturbation theory in the Poisson gauge. Tensor modes are nondynamical at the 1/c 4 order we consider (gravitational waves only appear at higher order): they are purely nonlinear and describe a distortion of the spatial slices determined at this order by a constraint, quadratic in the scalar and vector variables.

Though predictions of the simplest inflationary cosmological models with cold dark matter, flat space and approximately flat initial spectrum of adiabatic perturbations are remarkably close to observational data, we have to go beyond them and to introduce new physics not yet discovered in laboratories to account for all data. Two extensions of these models which seem to be the most actual at present time are discussed. The first one is the possibility that we are living at the beginning of a new inflation-like era. Then classical cosmological tests, like the luminosity distance or the angular size of distant objects as functions of redshift, as well as the behaviour of density perturbations in a dustlike matter component including baryons as a function of redshift, can provide information sufficient for the unambiguous determination of an effective potential of a corresponding present inflaton scalar field. The second, unrelated extension is a possibility of brokenscale-invariant cosmological models which have localized steps or spikes in the primordial perturbation spectrum. These features can be produced by fast phase transitions in physical fields other than an inflaton field in the early Universe during inflation and not far from the end of it. At present, it seems that the only scale in the spectrum around which we might see something of this type is k = 0.05 h Mpc −1 .

Physics Letters B, 1987

We discuss cosmologies where the cosmological constant X depends on time. The requirements of realistic cosmology impose restrictions on the functional dependence of J~ on the Hubble parameter H. We show that for a wide class of functions with X of the order H 3 the system of field equations leads to a stable fix-point behaviour with h naturally very small today. The age of the universe, critical matter density and deceleration parameter may be modified.