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I am working on the following exercise for my class on general relativity:

Suppose that the spatial volume of a closed, matter dominated FLRW universe with spherical space sections and vanishing cosmological constant is $10^{12} \, \text{Mpc}^3$ at the moment of maximum expansion. What is the duration of this universe from big bang to big crunch in years?

In a previous exercise, we had to calculate a solution of the Friedmann equation in a closed, matter-dominated FLRW universe with vanishing cosmological constant. It was of the form

$$ R = C(1-\cos \eta) \quad , \qquad t = C(\eta - \sin \eta)$$

where $R$ is the scale factor and $d\eta = dt/R$. The constant $C$ is given as

$$ C = \frac{4\pi G}{3c^4} \, R^3 \, \varrho $$

with $\varrho$ being the matter density of the universe. Note that the Friedman equation implies that $R^3 \, \varrho = \text{const}$.

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Now, I thought that the time from big bang to big crunch should finally be

$$ t(2\pi) = C \cdot 2 \pi$$

However, I don't think this approach is correct, because I am neither using the fact that the space sections are spherical, nor do I use the exact value of the volume. I also do not know the matter density $\varrho$ of the universe and therefore can't calculate the time.

Any ideas on how to solve this problem/on what is the problem with my approach?

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UPDATE 1:

I added more information on the constant $C$ and clarified the problem.

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    $\begingroup$ You're definitely using the fact that the spatial sections are spherical. After all, what happens to $R(\eta)$ if the spatial sections are not spherical? $\endgroup$ Commented Jun 22, 2021 at 18:41
  • $\begingroup$ Are you suggesting that if the spatial section were not spherical, then $R$ would not be isotropic and therefore a function of the spatial coordinates (or the direction) as well? $\endgroup$
    – Octavius
    Commented Jun 22, 2021 at 19:13
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    $\begingroup$ more simply, what is the fate of the universe for the flat and hyperbolic models? Knowing that, could the above possibly describe either of those universes? $\endgroup$ Commented Jun 22, 2021 at 19:16
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    $\begingroup$ Their fate is the big chill, i.e. never ending expansion, i guess. In that case the exercise above would certainly not make any sense.Ok, so I am using the fact that the spatial sections are spherical - granted. However, I am afraid, I am still a bit lost. $\endgroup$
    – Octavius
    Commented Jun 22, 2021 at 19:23

1 Answer 1

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I now found the solution.

Given the time $t(2\pi) = C \cdot 2\pi$ from big bang to big crunch, one uses an equation that relates the scale factor $R$ to the volume $V$ by

$$V = 2\pi^2 \cdot R^3$$

Since it is known that

$$V(\pi) = 2 \pi^2 \cdot R^3(\pi) = 10^{12} \, \mathrm{Mpc^3}$$

as well as

$$ R(\pi) = C \cdot (1 - \cos \pi )$$

it follows that

$$ 10^{12} \, \mathrm{Mpc}^3 = 2 \pi^2 \cdot C^3$$

which can be solved for $C$. This determines the time $t(2\pi)$ and therefore the lifetime of the universe.

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  • $\begingroup$ The main reason for lack of any favorable response to the question as posted may be the possibility that the universe (as opposed to "local universes", within a "multiverse", whose causally-separate individuality's usually indicated by capitalizing "universe" [i.e., typing "Universe"] in reference to any one of them) may be eternal to the past as well as to the future: Recent cosmological models allowing that possibility include the 2020 Nobel winner Roger Penrose's "Conformal cyclic cosmology, Nikodem Poplawski's "Cosmology with torsion", and Aguirre & Deutsche's "State-to-state cosmology". $\endgroup$
    – Edouard
    Commented Aug 12, 2021 at 19:55

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