Confused about equations for the Big Bang in general relativity ad loop quantum gravity?

Question

I am trying to get a vague understanding of the mathematical equations for the Big Bang in GR and LQG. My understanding so far is that when the universe is assumed to be homogeneous and isotropic, which it practically is, then the Einstein Field Equations may be solved and you get the FLRW metric. I think the Friedman equations are based on the FLRW metric and they can be used to show that GR cannot deal with the big bang. I have the following questions about this:

1. In the Friedman equation $\frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$ what exactly is a?

2. From this equation I do not really see what it tells us about the big bang, mainly becuase of the constant. In this video at 10:10 https://www.youtube.com/watch?v=IFcQuEw0oY8 the equation given is $H^2=\frac{8\pi G}{3}\rho$ for GR where $H$ is $\left (\frac{\dot{a}}{a}\right)$, rather than the equation given above. I'm not sure what version I should be using.

3. If it is the case that the second one is a simplification that I shouldn't really be using, does anyone know where I can find the full equations for the LQG version of the equation above (in the video it is given as $H^2=\frac{8\pi G}{3}\rho\left(1-\frac{\rho}{\rho_{c}}\right)$ ).

Answer 1

I'll try to write this up into a quick answer. Maybe it will help.

What exactly is a?

$a$ is the universal scale factor. It is also written as a function of time: $a=a(t)$. As you can see, it features prominently in the Friedmann equations (and is featured in the FLRW metric), and is important when studying the expansion of the universe. As is the convention elsewhere in GR, the nth time derivative of $a$ is signified with n dots: $$\frac{da}{dt}=\dot{a}, \frac{d^2a}{dt^2}=\ddot{a},$$ etc.

$H$ is the Hubble parameter. It is defined as $$H\equiv\frac{\dot{a}}{a}$$ which is probably how you saw it.

As for LQG . . . I honestly don't know the equations involved there (I've been meaning to bone up on them), but since the Friedmann equations are directly from the equations of GR, I would think they are more important there.

Answer 2

In the Friedman equation $\frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$ what exactly is a?

$a=a(t)$ is called the "scale factor." It's part of the FLRW metric which describes the expanding universe. The scale factor determines spacial distances between points in space, and how those distances change over time.

From this equation I do not really see what it tells us about the big bang, mainly becuase of the constant.

The "Hubble constant" $H= \dot{a}/a$ is not actually a constant. It's a misnomer. It's more accurate to refer to it as the "Hubble parameter." As $\rho$ goes to infinity, $\ddot{a}/a$ goes to negative infinity.

In this video at 10:10 https://www.youtube.com/watch?v=IFcQuEw0oY8 the equation given is $H^2=\frac{8\pi G}{3}\rho$ for GR where $H$ is $\left (\frac{\dot{a}}{a}\right)$, rather than the equation given above. I'm not sure what version I should be using.

There are two Friedmann equations, both of which you have provided. Which one you should be using depends on what you're trying do. On the wiki page, the second equation you mention is listed as the first Friedmann equation. In the video they have simplified it by assuming $\, k=\Lambda=0.$ The first equation you mention is listed as the second Friedmann equation.

does anyone know where I can find the full equations for the LQG version of the equation above (in the video it is given as $H^2=\frac{8\pi G}{3}\rho\left(1-\frac{\rho}{\rho_{c}}\right)$ ).

This I can't help you with, because I'm not familiar with LQG.