# Calculating the lifetime of the universe

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.

• 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? Commented Jun 22, 2021 at 18:41
• 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? Commented Jun 22, 2021 at 19:13
• 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? Commented Jun 22, 2021 at 19:16
• 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. Commented Jun 22, 2021 at 19:23

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.

• 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". Commented Aug 12, 2021 at 19:55