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What is the difference between an open and a flat accelerating expansion with regard to the FLRW equations?

Do we only change the density parameters and the equation of state for dark energy? Or is the FLRW equations modified for an open accelerating expansion?

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The homogeneity and isotropy hypotheses lead to the FLRW metric for the universe, namely : $$ds^2 = cdt^2 - a(t)^2\left [ \dfrac{dr^2}{1+kr^2/R^2}+r^2 d^2\Omega \right ]$$ where $R$ is the curvature radius of the universe and $k$ can take the values -1, 0, or 1 for a spherical, flat or hyperbolic universe. This result is a direct consequence of the cosmological principle.

In order to derive the relationship between these parameters and the universe content, one has to apply the Einstein equations. One thus obtains the so-called Friedmann equations : $$\left\{\begin{matrix} \dot{a}^2-\dfrac{8 \pi G}{3c^2} \displaystyle \sum_i \rho_i a^2 & = & \dfrac{kc^2}{R^2} \\ \dfrac{d}{dt}\left ( \rho_i a^3 \right ) & = & -P_i \dfrac{d}{dt} \left (a^3 \right) \\ \end{matrix}\right.$$ Where $\rho_i$ and $P_i$ denote the energy density and pressure of each component of the universe, whose behavior is determined by their state equation (e.g. $P = 0$ for dust-like matter, $P=\rho/3$ for radiation, $P=-\rho$ for dark-energy.)

The present value of $a$ is 1. Evaluating the first equation at the present time $t_0$, and using the definition of the Hubble constant $H_0 = \dot{a}(t_0)/a(t_0) = \dot{a}(t_0)$, we find : $$H_0^2 - \dfrac{8 \pi G}{3c^2} \displaystyle \sum_i \rho_i = \dfrac{kc^2}{R^2}$$

It is convenient to define the critical density as $\rho_c = 3c^2 H_0^2/(8\pi G)$, so that :

$$\rho_c - \displaystyle \sum_i \rho_i = \dfrac{3c^2}{8 \pi G} \dfrac{kc^2}{R^2}$$

This result means that the universe is spherical if the total energy density exceeds $\rho_c$, flat if they are equal, and hyperbolic if the total density is lesser. This is how the curvature is related to the universe content. You can see that this doesn't depend on the density distribution (i.e. how much is dark energy, how much is otherwise) One way to quantify departure of the universe from a flat geometry is to evaluate the curvature density parameter $\Omega_k = (\rho_c-\rho)/\rho_c$. The most stringent limits on $\Omega_k$ are obtained by the analysis of the CMB anisotropies and are compatible with a flat universe ($|\Omega_k| < 5 \times 10^{-3}$, arXiv:1502.01589) but there exists other ways to measure of this parameter using standard candles and standard rulers.

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  • $\begingroup$ The question is with regard to the results of computing Hogg Integration formula to obtain age of universe and size of universe. Is Hogg's numerical integration modified when dealing with an open universe, or do we change the density parameters and the equation of state only? @Lucas Gautheron $\endgroup$ – Vick Jun 29 '17 at 14:38

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