Friedmann equations

Summary

The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density [[Rho (letter)|ρ]] and pressure p. The equations for negative spatial curvature were given by Friedmann in 1924. Assumptions Friedmann–Lemaître–Robertson–Walker metric The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, that is, the cosmological principle; empirically, this is justified on scales larger than the order of 100 Mpc. The cosmological principle implies that the metric of the universe must be of the form : -\mathrm{d}s^2 = a(t)^2 , {\mathrm{d}s_3}^2 - c^2 , \mathrm{d}t^2

Official source

https://en.wikipedia.org/wiki/Friedmann_equations

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Dark goo: bulk viscosity as an alternative to dark energy

Jean-Sébastien Gagnon, Julien Lesgourgues

We present a simple (microscopic) model in which bulk viscosity plays a role in explaining the present acceleration of the universe. The effect of bulk viscosity on the Friedmann equations is to turn the pressure into an "effective" pressure containing the bulk viscosity. For a sufficiently large bulk viscosity, the effective pressure becomes negative and could mimic a dark energy equation of state. Our microscopic model includes self-interacting spin-zero particles (for which the bulk viscosity is known) that are added to the usual energy content of the universe. We study both background equations and linear perturbations in this model. We show that a dark energy behavior is obtained for reasonable values of the two parameters of the model (i.e. the mass and coupling of the spin-zero particles) and that linear perturbations are well-behaved. There is no apparent fine tuning involved. We also discuss the conditions under which hydrodynamics holds, in particular that the spin-zero particles must be in local equilibrium today for viscous effects to be important.

2011

Empirical evolution equations

Victor Panaretos, Susan Wei

2018

We develop an extension of the basic inverse seesaw model which addresses simultaneously two of its drawbacks, namely, the lack of explanation of the tiny Majorana mass term for the TeV-scale singlet fermions and the difficulty in achieving successful leptogenesis. Firstly, we investigate systematically leptogenesis within the inverse (and the related linear) seesaw models and show that a successful scenario requires either small Yukawa couplings, implying loss of experimental signals, and/or quasi-degeneracy among singlets mass of different generations, suggesting extra structure must be invoked. Then we move to the analysis of our new framework, which we refer to as hybrid seesaw. This combines the TeV degrees of freedom of the inverse seesaw with those of a high-scale (M-N >> TeV) seesaw module in such a way as to retain the main features of both pictures: naturally small neutrino masses, successful leptogenesis, and accessible experimental signatures. We show how the required structure can arise from a more fundamental theory with a gauge symmetry or from warped extra dimensions/composite Higgs. We provide a detailed derivation of all the analytical formulae necessary to analyze leptogenesis in this new framework, and discuss the entire gamut of possibilities our scenario encompasses including scenarios with singlet masses in the enlarged range M-N approximate to 10(6) - 10(16) GeV. This idea of hybrid seesaw was proposed by us in arXiv:1804.06847; here, we substantially elaborate upon and extend earlier results.

Springer2019