Einstein-Friedmann Equations Research Articles

Analogies between Logistic Equation and Relativistic Cosmology

We develop several formal analogies between the logistic equation and the spatially homogeneous and isotropic relativistic cosmology described by the Einstein–Friedmann equations. These analogies produce an effective Lagrangian and Hamiltonian and new symmetries for the logistic equation.

Turning a Newtonian analogy for FLRW cosmology into a relativistic problem

A Newtonian uniform ball expanding in empty space constitutes a common heuristic analogy for FLRW cosmology. We discuss possible implementations of the corresponding general-relativistic problem and a variety of new cosmological analogies arising from them. We highlight essential ingredients of the Newtonian analogy, including that the quasilocal energy is always `Newtonian' in the sense that the magnetic part of the Weyl tensor does not contribute to it. A symmetry of the Einstein-Friedmann equations produces another one in the original Newtonian system.

A Symmetry of the Einstein–Friedmann Equations for Spatially Flat, Perfect Fluid, Universes

We report a symmetry property of the Einstein–Friedmann equations for spatially flat Friedmann–Lemaître–Robertson–Walker universes filled with a perfect fluid with any constant equation of state. The symmetry transformations form a one-parameter Abelian group.

A Solution to the Cosmological Constant Problem in Two Time Dimensions

For the last hundred years, the existence and the value of the cosmological constant Λ has been a great enigma. So far, any theoretical model has failed to predict the value of Λ by several orders of magnitude. We here offer a solution to the cosmological constant problem by extending the Einstein-Friedmann equations by one additional time dimension. Solving these equations, we find that the Universe is flat on a global scale and that the cosmological constant lies between 10-90 m-2 and 10-51 m-2 which is in range observed by experiments. It also proposes a mean to explain the Planck length and to mitigate the singularity at the Big Bang.

Expanding space-time and variable vacuum energy

The paper describes a cosmological model which contemplates the presence of a vacuum energy varying, very slightly (now), with time. The constant part of the vacuum energy generated, some 6 Gyr ago, a deceleration/acceleration transition of the metric expansion; so now, in an aged Universe, the expansion is inexorably accelerating. The vacuum energy varying part is instead assumed to be eventually responsible of an acceleration/deceleration transition, which occurred about 14 Gyr ago; this transition has a dynamic origin: it is a consequence of the general relativistic Einstein-Friedmann equations. Moreover, the vacuum energy (constant and variable) is here related to the zero-point energy of some quantum fields (scalar, vector, or spinor); these fields are necessarily described in a general relativistic way: their structure depends on the space-time metric, typically non-flat. More precisely, the commutators of the (quantum field) creation/annihilation operators are here assumed to depend on the local value of the space-time metric tensor (and eventually of its curvature); furthermore, these commutators rapidly decrease for high momentum values and they reduce to the standard ones for a flat metric. In this way, the theory is ”gravitationally” regularized; in particular, the zero-point (vacuum) energy density has a well defined value and, for a non static metric, depends on the (cosmic) time. Note that this varying vacuum energy can be negative (Fermi fields) and that a change of its sign typically leads to a minimum for the metric expansion factor (a ”bounce”).

Planckon Densely Piled Vacuum

Based on the vacuum microscopic quantum structure model: the planckon densely piled vacuum model and the principle of cosmology, with the Planck era as initial conditions, we have solved the Einstein-Friedmann equations to describe the evolution of the universe. The main results are: i) the solution of Einstein-Friedmann equations have yielded the observed result：the ratio of dark energy density to vacuum quantum fluctuation energy density ρde/ρvac ~ (tp/T0)2 ~ 10-122s (the Planck time tp = 10-43s and the universe age T0 = 1018s); ii) at the inflation time tinf = 10-35s, the calculated universe radiation energy density is ρ(tinf) ~ 10-16ρvac corresponding to the phase transition temperature Ec ~ 1015GeV consistent with the GUT theory; iii) the expanding universe with vacuum as its environment is a non-equilibrium open system constantly exchanging energy with vacuum; during its expansion, the planckons in the universe lose quantum fluctuation energy and create the cosmic expansion quanta-cosmons, the energy of the cosmons is the lost vacuum quantum fluctuation energy and contributes to the universe energy with the calculated value Ecosmos = 1022Msolarc2 (where Msolar is solar mass) consistent with astronomic data; iv) since all planckons in the vacuum of the expanding universe lose quantum fluctuation energy resulting in hole excitations as negative gravity energy and the lost energy of planckons is used to create cosmons which in turn convert into different kinds of universe energy, the negative gravity energy plus the positive universe energy is zero; v) the induced negative gravity potential and the gravity acceleration due to the creation of cosmons are derived with the nature of radially outwards repulsive force, indicating that the cosmon may be the candidate of the dark energy quantum; vi) both the initial solution (the Planck era solution or the planckon solution) and the infinite asymptotic solution of the Einstein-Friedmann equations are unstable: the former tends to expand and the latter tends to shrink, so that the Einstein-Friedman universe undergoes a cyclic evolution of successive expansion and shrinking.

Friedman versus Abel equations: A connection unraveled

We present an interesting connection between Einstein–Friedmann equations for the models of universe filled with scalar field and the special form of Abel equation of the first kind. This connection works in both ways: first, we show how, knowing the general solution of the Abel equation (corresponding to the given scalar field potential), one can obtain the general solution of the Friedman equation (and use the former for studying such problems as the existence of inflation with exit for particular models). On the other hand, one can invert the procedure and construct the Bäcklund autotransformations for the Abel equation.

COSMOLOGY FROM ANISOTROPIC DGP BRANEWORLD

We considered an anisotropic brane embedded in a five-dimensional bulk space–time. The Einstein equations and Friedmann equations are obtained. The corresponding Friedmann equations are modified with quadratic corrections to energy density. The anisotropic DGP model also admits a self-accelerating phase in the late universe. The acceleration conditions are obtained and they are consistent with the braneworld scenario. The acceleration condition for DGP model, in the late universe, is dependent on the crossover scale rc.

Mach's Principle and Minkowski Spacetime

It is shown that when the Minkowski metric is approached by a limiting process using two different static, spherically-symmetric, closed cosmological models, that although the energy-stress tensors for the Einstein-Friedmann field equations vanishes, their integral does not. Since part of this integral consists of the mass of the incoherent dust background, which is the same in both models, the Minkowski metric obtained by this limiting process cannot be regarded as “anti-Machian,” since there is an infinite amount of ponderable matter in the background, albeit at vanishing density. One of the models is the Einstein static universe with its cosmological term. The other model does not employ this term, but instead uses a tensor that has vanishing trace, negative energy density and negative pressure. Gravitational energy is also studied, and it is pointed out that for both models, this energy becomes infinitely negative in the Minkowski limit.