Helmoltz problem for the Riccati equation from an analogous Friedmann equation

We report a solution of the inverse Lagrangian problem for the first order Riccati differential equation by means of an analogy with the Friedmann equation of a suitable Friedmann–Lemaître–Robertson–Walker universe in general relativity. This analogous universe has fine-tuned parameters and is unphysical, but it suggests a Lagrangian and a Hamiltonian for the Riccati equation and for the many physical systems described by it.

New full relativistic escape velocity and new Hubble related equation for the universe

The escape velocity derived from general relativity coincides with the Newtonian one. However, the Newtonian escape velocity can only be a good approximation when v ≪ c is sufficient to break free of the gravitational field of a massive body, as it ignores higher-order terms of the relativistic kinetic energy Taylor series expansion. Consequently, it does not work for a gravitational body with a radius at which v is close to c such as a black hole. To address this problem, we revisit the concept of relativistic mass, abandoned by Einstein, and derive what we call a full relativistic escape velocity. This approach leads to a new escape radius, where ve = c equal to a half of the Schwarzschild radius. Furthermore, we show that one can derive the Friedmann equation for a critical universe from the escape velocity formula from general relativity theory. We also derive a new equation for a flat universe based on our full relativistic escape velocity formula. Our alternative to the Friedmann formula predicts exactly twice the mass density in our (critical) universe as the Friedmann equation after it is calibrated to the observed cosmological redshift. Our full relativistic escape velocity formula also appears more consistent with the uniqueness of the Planck mass (particle) than the general relativity theory: whereas the general relativity theory predicts an escape velocity above c for the Planck mass at a radius equal to the Planck length, our model predicts an escape velocity c in this case.

Effective actions for loop quantum cosmology in fourth-order gravity

Loop quantum cosmology (LQC) is a theory which renders the Big Bang initial singularity into a quantum bounce, by means of short-range repulsive quantum effects at the Planck scale. In this work, we are interested in reproducing the effective Friedmann equation of LQC, by considering a generic f(R, P, Q) theory of gravity, where $$R=g^{\mu \nu }R_{\mu \nu }$$ R = g μ ν R μ ν is the Ricci scalar, $$P=R_{\mu \nu }R^{\mu \nu }$$ P = R μ ν R μ ν , and $$Q=R_{\alpha \beta \mu \nu }R^{\alpha \beta \mu \nu }$$ Q = R α β μ ν R α β μ ν is the Kretschmann scalar. An order reduction technique allows us to work in f(R, P, Q) theories which are perturbatively close to General Relativity, and to deduce a modified Friedmann equation in the reduced theory. Requiring that the modified Friedmann equation mimics the effective Friedmann equation of LQC, we are able to derive several functional forms of f(R, P, Q). We discuss the necessary conditions to obtain viable bouncing cosmologies for the proposed effective actions of f(R, P, Q) theory of gravity.

Warm constant-roll inflation in brane-world cosmology

In this paper, we study a constant-roll inflationary model in the context of brane-world cosmology caused by a quintessence scalar field for warm inflation with a constant dissipative parameter [Formula: see text]. We determine the analytical solution for the Friedmann equation coupled to the equation of motion of the scalar field. The evolution of the primordial scalar and tensor perturbations is also studied using the modified Langevin equation. To check the viability of the model, we use numerical approaches and plot some figures. Our results for the scalar spectral index and the tensor-to-scalar ratio show good consistency with observations.

Principles of local realism in a Friedmann universe

The problem of time is a statement of the inability to establish the standard model according to a consistent physical framework based on a valid starting point provided from either the concept of time in quantum mechanics (QM) or general relativity (GR). Using the deterministic local realism approach of Bell’s inequality experiment, a valid mathematical starting point incorporating both QM and GR can be established using the concept of energy conservation within a volume of spacetime. Because Friedmann established a system that correlates the energy level within the volume of spacetime with the proximity between energy mass, with two opposing universal forces that must act on the reconfiguration of particles when considering a realism-based definite position as they evolve in time independent of observation, it is possible to consider QM time evolution as a form of deterministic thermodynamic work. Considering this volume of spacetime in terms of the local realism interpretation allows one to consider the act of time evolution as a reconfiguration that occurs along with the expansion of volume which allows one to establish an energy conservation argument using only the particles that exist within the volume of spacetime to account for both the gravitational energy and the divergent energy usually attributed to the cosmological constant. With this argument time evolution must cost system energy. For energy to be conserved the use of system energy must be for the act of gravitation as a particle evolves in time. The definition of local realism allows Minkowski spacetime diagrams to pertain to the unseen intervals between measurements. This allows a center frame observer to serve as a background clock to measure time rates in correlation to scale factor expansion. This allows one to consider time rates in terms of work that must occur over an interval of a background clock. In the case of local realism, Minkowski mathematics allow a direct correlation between QM time evolution and the second Friedmann equation.

Decaying vacuum and evolution from early inflation to late acceleration

Decaying vacuum models are a class of models that incorporate a time-dependent vacuum energy density that can explain the entire evolution of the universe in a unified framework. A general solution to the Friedmann equation is obtained by considering vacuum energy density as a function of the Hubble parameter. We have obtained the asymptotic solution by choosing the equation of state for matter, [Formula: see text] and radiation, [Formula: see text]. Finite boundaries in the early and late de Sitter epoch are defined by considering the evolution of primordial perturbation wavelength. An epoch invariant number [Formula: see text] determines the number of primordial perturbation modes that cross the Hubble radius during each epoch.

About the Origin of the Dark Energy and the Zero Mass/Energy Universe

Based on the Gravito-Electro-Magnetic (GEM) equations as another form (for low fields) of Einstein's Equations of General Relativity Theory (GRT) an equation is derived for the total energy density in the universe, including the gravitational fields, the contribution thereof is always negative and so it seems to represents the Dark Energy (DE). When calculating the total energy of the universe from this equation, the result is near to zero because of negative contributions from gravitational fields, depending a little on the available parameters of the universe as e.g. it's baryonic mass. Thus the assumption is given a high amount of probability, that the total energy (mass) in the universe is really zero and very likely is always zero. This would mean, that the universe developed from empty space-time or from nothing (may be by quantum fluctuations). Looking on the development it could be that the average energy density is zero for each sufficient large part of the universe at any time, except for very local deviations (e.g. galaxies, black holes etc.). As a consequence the expansion of the universe is probably not retarded by gravity (thus the Friedmann equation and others do not apply). The expansion of the universe can be considered as driven by the pressure of a gas-like medium with positive masses as by intergalactic gas, dust, stars and galaxies. Conclusions are drawn as to the interpretation of the formation of voids in the universe, flat space etc.

A note on supergravity inflation in braneworld

We discuss supergravity inflation in braneworld cosmology for the class of potentials [Formula: see text] with [Formula: see text]. These minimal SUGRA models evade the [Formula: see text] problem due to a broken shift symmetry and can easily accommodate the observational constraints. In the high energy regime [Formula: see text], the numerical predictions and approximate analytic formulas are given for the scalar spectral index [Formula: see text] and tensor-to-scalar ratio [Formula: see text]. The models with smaller [Formula: see text] are preferred while the models with larger [Formula: see text] are out of the [Formula: see text] region. Remarkably, the [Formula: see text] correction to the energy density in Friedmann equation results in sub-Planckian inflaton excursions [Formula: see text].

Thermodynamics in new model of loop quantum cosmology

The thermodynamic properties of loop quantum cosmology (LQC) without considering the Lorentz term were established in Li and Zhu (Adv High Energy Phys 2009:905705, 2009). In this paper, we extend this result to the recent proposed new model of LQC with the Lorentz term. We investigate the thermodynamics of LQC on the apparent horizon of the Friedmann–Lematre–Robertson–Walker universe. The result shows that the effective density and effective pressure in the modified Friedmann equation of LQC not only determines the evolution of the universe but can also serve as the thermodynamic quantities. Moreover, with the help of the Misner–Sharp energy, the first law of thermodynamics of the LQC is still valid as expected. This in turn endows precise physical meaning to the effective matter density $$\rho _{eff}$$ ρ eff and the effective pressure $$P_{eff}$$ P eff .

Variable Planck’s Constant: Treated As A Dynamical Field And Path Integral

The constant ħ is elevated to a dynamical field, coupling to other fields, and itself, through the Lagrangian density derivative terms. The spatial and temporal dependence of ħ falls directly out of the field equations themselves. Three solutions are found: a free field with a tadpole term; a standing-wave non-propagating mode; a non-oscillating non-propagating mode. The first two could be quantized. The third corresponds to a zero-momentum classical field that naturally decays spatially to a constant with no ad-hoc terms added to the Lagrangian. An attempt is made to calibrate the constants in the third solution based on experimental data. The three fields are referred to as actons. It is tentatively concluded that the acton origin coincides with a massive body, or point of infinite density, though is not mass dependent. An expression for the positional dependence of Planck’s constant is derived from a field theory in this work that matches in functional form that of one derived from considerations of Local Position Invariance violation in GR in another paper by this author. Astrophysical and Cosmological interpretations are provided. A derivation is shown for how the integrand in the path integral exponent becomes Lc/ħ(r), where Lc is the classical action. The path that makes stationary the integral in the exponent is termed the “dominant” path, and deviates from the classical path systematically due to the position dependence of ħ. The meaning of variable ħ is seen to be related to the rate of time passage along each path increment. The changes resulting in the Euler-Lagrange equation, Newton’s first and second laws, Newtonian gravity, Friedmann equation with a Cosmological Constant, and the impact on gravitational radiation for the dominant path are shown and discussed.