<p>In a homogeneous and isotropic Universe (even if recent observations challenge this hypothesis), you can derive the <a href="https://en.wikipedia.org/wiki/Friedmann_equations" rel="nofollow noreferrer">Friedmann equations</a>, which describe the evolution of the Hubble constant with time:
$\frac{\dot{a}}{a} = H(t) = \frac{8 \pi G}{3}\rho - \frac{k}{a^2} + \frac{\Lambda}{3}$   (with $c=1$) (Equation $1$)</p>

<p>where $a=a(t)$ is the scale factor, $\dot{a}$ its derivative, $G$ the gravitational constant, $\rho$ the matter density, $\frac{k}{a^2}$ the spatial curvature (a parameter that describes the metric of the Universe), and $\Lambda$ the cosmological constant (an integration constant added by Einstein).
It could be useful to rewrite the equation as:</p>

<p>$H^2 = \frac{8 \pi G}{3}(\rho + \rho_{\Lambda}) - \frac{k}{a^2}$</p>

<p>where $\rho_{\Lambda} = \frac{\Lambda}{8 \pi G}$ is the "density of cosmological constant".</p>

<p>We can also expand the matter density as $\rho = \rho_{matter} + \rho_{radiation}$.</p>

<p>So we have a "total" density $\rho_{tot} = \rho_{matter} + \rho_{radiation} + \rho_{\Lambda}$. The destiny of the Universe depends on this amount.</p>

<p>In case of $\rho_{tot} &gt; \rho_{crit}$, or equivalently a <em>closed</em> Universe ($k=+1$), the equation $(1)$ becomes:</p>

<p>$\dot a^2 = \frac{8 \pi G}{3}\rho a^2 -1$</p>

<p>Which points out that the scale factor must have an upper limit $a_{max}$ ($\dot a^2$ must be positive).
This in turn means that the second derivative $\ddot a$ of the scale factor must be negative, when approaching $a_{max}$, that is <em>the scale factor function inverts its behavior</em>:
<img src="https://i.sstatic.net/LI2GJ.jpg" alt="enter image description here"></p>

<p>Look at <a href="http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll8.html" rel="nofollow noreferrer">here</a> and <a href="http://www.ita.uni-heidelberg.de/~dullemond/lectures/cosmology_2011/Chapter_4.pdf" rel="nofollow noreferrer">here</a> if you want to go deeper.</p>

<p>@Bardathehobo This figure shows what I mean when I say that a currently accelerating Universe can still crunch. This is because we are basically ignorant upon the dark energy issue.</p>

CC BY-SA 3.0
2014-03-21T16:29:44.473Z
Can you explain this in simple terms? I am not familiar with the mathematical description of Friedman models.
Yashbhatt
CC BY-SA 3.0
2014-03-24T00:38:13.043Z
D'oh! What exactly you do not understand? Let's try to narrow the problem.
Py-ser
CC BY-SA 3.0
2014-03-24T10:15:25.203Z
What exactly is the cosmological constant?
Yashbhatt
CC BY-SA 3.0
2014-03-24T10:45:17.713Z
Did you try this? https://en.wikipedia.org/wiki/Cosmological_constant
Would you like some math or a more physical explanation?
Py-ser
CC BY-SA 3.0
2014-03-21T03:45:13.840Z
2014-03-21T03:45:13.840Z
Py-ser
<p>Well, the original idea of the Big Crunch came about when it was thought that gravity was slowing down the expansion of the universe and that one day it will stop expanding (a finite universe). At which point the gravitational pull of all the objects in the universe would make it contract, growing smaller into itself until eventually it is just a singularity.</p>

<p>However, today we see that the universe is expanding and accelerating due to dark energy's effect on gravity making it an infinite universe. This makes the original idea of the Big Crunch not possible (since the universe would be infinite), but a big crunch can still occur in an alternate way if dark energy becomes too weak to counteract gravity which would set in motion the Big Crunch. However there is still so much unknown about dark energy that it is also possible that it will never weaken and the universe will just keep expanding, which opens different end of the universe theories.  </p>

CC BY-SA 3.0
2014-03-20T01:47:39.933Z
This is not correct. A Big Crunch scenario is still possible even in a presently accelerating Universe. It depends on the density of matter and cosmological constant (dark energy). Please edit your answer.
Py-ser
CC BY-SA 3.0
2014-03-20T13:51:01.283Z
Yeah you are right, I forgot about the idea of dark energy weakening, I edited it to include that the infinite universe made it so the original Big Crunch theory not possible but that a Big crunch could still occur if dark energy were to diminish in strength.
Bardathehobo
CC BY-SA 3.0
2014-03-21T00:21:50.477Z
I am sorry, but this is not correct anyway. We don't know the behavior of the $\Omega_{\Lambda}$ function with time. We don't how was in the past, why it is increasing now, and what it will do in future. This means that the Big Crunch scenario is still possible, and a presently accelerating Universe does NOT necessarily bring to an infinite Universe. We just don't know that much about the observed acceleration.
Py-ser
CC BY-SA 3.0
2014-03-21T16:33:33.587Z
So can we say that we cannot say anything for sure about the future of the Universe? And if the Universe is infinite then there's no way we could say that a thing called the Big Bang took place and the universe originated from a singularity.
Yashbhatt
CC BY-SA 3.0
2014-05-23T14:10:07.760Z
@Yashbhatt If the universe continues to expand for an *infinite* amount of time it will be expanding forever, however it can still have a beginning. Think of it in terms of the number scale: there are infinite positive numbers, but they all begin from zero.
christopherlovell
CC BY-SA 3.0
2014-05-24T06:38:14.737Z
I agree with you. But how do you define the beginning. If it's infinite how do you know at which point in time you are?
Yashbhatt
CC BY-SA 3.0
2014-06-09T08:42:26.830Z
@polyphant Yes, you are right. I thought you mentioned it being infinite in both directions.
Yashbhatt
CC BY-SA 3.0
2014-06-15T15:19:08.467Z
@polyphant You are right about it being infinite in time but it cannot be infinite in space. You cannot compress infinity into a point.
Yashbhatt
CC BY-SA 3.0
2014-03-19T19:11:12.253Z
2014-03-20T16:47:51.207Z
2014-03-20T16:47:51.207Z
Bardathehobo
<p>I've read that the further away a Galaxy is from us the faster it moves away. </p>

<p>By this logic how can scientists predict that there will eventually be a big crunch when every piece of matter is seemingly getting further and further apart?</p>

CC BY-SA 3.0
2014-03-21T09:09:42.690Z
Anyway, it is not that any galaxy move away at the speed of light. There are many discussions on SE that treat this argument.
Py-ser
<galaxy><universe>
How can we predict a Big Crunch when all galaxies are moving further apart?
Yashbhatt
# How can we predict a Big Crunch when all galaxies are moving further apart?

(Asked by: username_0 on 2014-03-19)

I've read that the further away a Galaxy is from us the faster it moves away.

By this logic how can scientists predict that there will eventually be a big crunch when every piece of matter is seemingly getting further and further apart?

**Tags:** galaxy, universe

> **Comments:**
> **username_1**: Anyway, it is not that any galaxy move away at the speed of light. There are many discussions on SE that treat this argument.  

---

## 2 answers

### Answer by username_1 on 2014-03-21. Score: 3

In a homogeneous and isotropic Universe (even if recent observations challenge this hypothesis), you can derive the [Friedmann equations](https://en.wikipedia.org/wiki/Friedmann_equations), which describe the evolution of the Hubble constant with time:
$\frac{\dot{a}}{a} = H(t) = \frac{8 \pi G}{3}\rho - \frac{k}{a^2} + \frac{\Lambda}{3}$ (with $c=1$) (Equation $1$)

where $a=a(t)$ is the scale factor, $\dot{a}$ its derivative, $G$ the gravitational constant, $\rho$ the matter density, $\frac{k}{a^2}$ the spatial curvature (a parameter that describes the metric of the Universe), and $\Lambda$ the cosmological constant (an integration constant added by Einstein).
It could be useful to rewrite the equation as:

$H^2 = \frac{8 \pi G}{3}(\rho + \rho\_{\Lambda}) - \frac{k}{a^2}$

where $\rho\_{\Lambda} = \frac{\Lambda}{8 \pi G}$ is the "density of cosmological constant".

We can also expand the matter density as $\rho = \rho\_{matter} + \rho\_{radiation}$.

So we have a "total" density $\rho\_{tot} = \rho\_{matter} + \rho\_{radiation} + \rho\_{\Lambda}$. The destiny of the Universe depends on this amount.

In case of $\rho\_{tot} > \rho\_{crit}$, or equivalently a *closed* Universe ($k=+1$), the equation $(1)$ becomes:

$\dot a^2 = \frac{8 \pi G}{3}\rho a^2 -1$

Which points out that the scale factor must have an upper limit $a\_{max}$ ($\dot a^2$ must be positive).
This in turn means that the second derivative $\ddot a$ of the scale factor must be negative, when approaching $a\_{max}$, that is *the scale factor function inverts its behavior*:
![enter image description here](https://i.sstatic.net/LI2GJ.jpg)

Look at [here](http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll8.html) and [here](http://www.ita.uni-heidelberg.de/~dullemond/lectures/cosmology_2011/Chapter_4.pdf) if you want to go deeper.

@username_2 This figure shows what I mean when I say that a currently accelerating Universe can still crunch. This is because we are basically ignorant upon the dark energy issue.

> **Comments:**
> **username_0**: Can you explain this in simple terms? I am not familiar with the mathematical description of Friedman models.  
> **username_1**: D'oh! What exactly you do not understand? Let's try to narrow the problem.  
> **username_0**: What exactly is the cosmological constant?  
> **username_1**: Did you try this? https://en.wikipedia.org/wiki/Cosmological_constant Would you like some math or a more physical explanation?  

---

### Answer by username_2 on 2014-03-19. Score: 1

Well, the original idea of the Big Crunch came about when it was thought that gravity was slowing down the expansion of the universe and that one day it will stop expanding (a finite universe). At which point the gravitational pull of all the objects in the universe would make it contract, growing smaller into itself until eventually it is just a singularity.

However, today we see that the universe is expanding and accelerating due to dark energy's effect on gravity making it an infinite universe. This makes the original idea of the Big Crunch not possible (since the universe would be infinite), but a big crunch can still occur in an alternate way if dark energy becomes too weak to counteract gravity which would set in motion the Big Crunch. However there is still so much unknown about dark energy that it is also possible that it will never weaken and the universe will just keep expanding, which opens different end of the universe theories.

> **Comments:**
> **username_1**: This is not correct. A Big Crunch scenario is still possible even in a presently accelerating Universe. It depends on the density of matter and cosmological constant (dark energy). Please edit your answer.  
> **username_2**: Yeah you are right, I forgot about the idea of dark energy weakening, I edited it to include that the infinite universe made it so the original Big Crunch theory not possible but that a Big crunch could still occur if dark energy were to diminish in strength.  
> **username_1**: I am sorry, but this is not correct anyway. We don't know the behavior of the $\Omega_{\Lambda}$ function with time. We don't how was in the past, why it is increasing now, and what it will do in future. This means that the Big Crunch scenario is still possible, and a presently accelerating Universe does NOT necessarily bring to an infinite Universe. We just don't know that much about the observed acceleration.  
> **username_0**: So can we say that we cannot say anything for sure about the future of the Universe? And if the Universe is infinite then there's no way we could say that a thing called the Big Bang took place and the universe originated from a singularity.  
> **username_3**: @username_0 If the universe continues to expand for an *infinite* amount of time it will be expanding forever, however it can still have a beginning. Think of it in terms of the number scale: there are infinite positive numbers, but they all begin from zero.  
> **username_0**: I agree with you. But how do you define the beginning. If it's infinite how do you know at which point in time you are?  
> **username_0**: @polyphant Yes, you are right. I thought you mentioned it being infinite in both directions.  
> **username_0**: @polyphant You are right about it being infinite in time but it cannot be infinite in space. You cannot compress infinity into a point.  

---

