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So theories of expansion of the universe (before dark matter and energy) said that the ultimate fate of the universe is based on the density of the universe and this being compared to the critical density.

But the density of the universe is always decreasing right?, with the expansion of the universe, so how can we attribute any significance to this idea?

Also how is the density of the universe related to curvature and if it is bounded/ infinite?

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    $\begingroup$ The critical density also changes with time, see the Friedmann equations $\endgroup$ – lemon Jun 30 '16 at 12:18
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The density of the 'stuff' that fills the universes determines the evolution of the universe. Begin with the Einstein Field Equations: \begin{equation} G_{\mu\nu}=8\pi G T_{\mu\nu} \end{equation} The left hand side is geometry (curvature of spacetime, e.g. gravity), the right hand side is the matter distribution of mass-energy in spacetime. The thing to see here is that the stuff that fills spacetime (the $T_{\mu\nu}$....also I choose to be ambiguous and say 'stuff' to be as general as possible) determines the curvature and evolution of the background spacetime (the $G_{\mu\nu}$).

If we want to be a bit more specific, we may assume an FLRW line element: $ds^2=-dt^2 + a(t)^2[dx^2+dy^2+dz^2]$. After solving for the Ricci Tensor and Ricci Scalar, and plugging these into the Einstein Field Equations, we arrive at the Friedmann equations: \begin{eqnarray} \left(\frac{\dot{a}}{a}\right)^2+\frac{k}{a^2} &=& \frac{8\pi G}{3}\rho \\ \frac{\ddot{a}}{a} &=& -\frac{4\pi G}{3}\left( \rho + 3p \right) \end{eqnarray}

From these equations, we see a direct link between the size of the universe (which is described by the scale factor a), the curvature parameter (k), and the density and pressure of the stuff that fills the universe ($\rho$ and p, which are found in $T_{\mu\nu}$).

Also the density of the universe is not necessarily always going to be decreasing. (For normal matter density will decrease as the universe gets bigger, but if you have (or want to invent) some weird/exoctic type of stuff with equation of state $p=w\rho$, and $w<-\frac{1}{3}$, the density of that weird stuff will actually increase as the universe gets bigger. (Although if you want to invent some weird form of stuff with such an equation of state, you better have a strong motivation to posit the existence of this stuff that violates the energy conditions.)

(I ignored the possible existence of a cosmological constant, but that only adds an extra term into the Friedmann Equations. The conclusion that density in closely connected to curvature and size and growth rate of the universe doesn't change.)

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  • $\begingroup$ You didn't ignore it, it's the case $w=-1/3$. It's probably the easiest example of a universe with $\dot{\rho} = 0$. $\endgroup$ – bapowell Jan 2 '18 at 20:54
  • $\begingroup$ Fair enough, I didn't specify the relation between $\rho$ and $p$ above. Although aesthetically, I prefer to keep $\Lambda$ out of $T_{\mu\nu}$ since $\Lambda$ is originally written in the EH action, and not the matter action. (But again, this is just personal aesthetics.) $\endgroup$ – Bob Jan 2 '18 at 20:57
  • $\begingroup$ @Bob Hi! I'm a beginner. Can you say what does a nonzero $a(t)$, $\dot{a}(t)$ and $\ddot{a}(t)$ mean? I think if $a$ is nonzero but constant so that $\dot{a}=0$, the universe is static. If $\dot{a}\neq 0$, but constant so that $\ddot{a}=0$, then the Universe is expanding but nota accelerating. When $\ddot{a}\neq 0$, then the expansion is accelerating. Do I get these right? $\endgroup$ – mithusengupta123 Jun 20 '18 at 15:42
  • $\begingroup$ @Bob Could you include role of critical density in your answer because I think the question asks for it. $\endgroup$ – mithusengupta123 Jun 20 '18 at 15:44

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