<p>Suppose we assume that the gravitational constant is varying with time:
$$G=G_0 e^{-\frac{t}{t_0}}$$ where $t$ is the time from the Big Bang, and $G_0, t_0$ being some constants.</p>

<p>Assuming matter-only universe, which one of the expansion equation should we use to obtain the Hubble's parameter $H$?</p>

<p>Equation 1:</p>

<p>$$H^2=\left\{\frac{\dot R}{R}\right\}^2=\frac{8\pi G}{3}\rho_0\frac{R_0^3}{R^3}$$</p>

<p>Equation 2:</p>

<p>$$\frac{\partial (H^2R^2)}{\partial (R^2)}=\frac{\ddot R}{R}= -\frac{4\pi G}{3}\rho_0\frac{R_0^3}{R^3}$$</p>

CC BY-SA 3.0
2016-10-10T05:20:02.433Z
Making $G$ vary with time isn't as simple as modifying those equations of motion. You'll need to go all the way back to the [Einstein-Hilbert action](https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action) with the $G$ moved inside of the integral, then vary the action to derive new equations of motion. I'm unsure whether you should include varying the action with respect to $G$ to get an equation that determines how $G$ varies or not.
Sean Lake
CC BY-SA 3.0
2016-10-27T22:49:19.690Z
Side question: If G = G(t) instead of G = G(x, y, z, t) as in this question, only one equation would not be the same, is that correct? Since varying the EH action in tensor notation is just doing 4 equations at once.
Doug
<cosmology><expansion>
Cosmology: about the equations of expansions
NIkhil Reddy Ramolla
# Cosmology: about the equations of expansions

(Asked by: username_0 on 2016-10-10)

Suppose we assume that the gravitational constant is varying with time:
$$G=G\_0 e^{-\frac{t}{t\_0}}$$ where $t$ is the time from the Big Bang, and $G\_0, t\_0$ being some constants.

Assuming matter-only universe, which one of the expansion equation should we use to obtain the Hubble's parameter $H$?

Equation 1:

$$H^2=\left\{\frac{\dot R}{R}\right\}^2=\frac{8\pi G}{3}\rho\_0\frac{R\_0^3}{R^3}$$

Equation 2:

$$\frac{\partial (H^2R^2)}{\partial (R^2)}=\frac{\ddot R}{R}= -\frac{4\pi G}{3}\rho\_0\frac{R\_0^3}{R^3}$$

**Tags:** cosmology, expansion
> **Comments:**
> **username_1**: Making $G$ vary with time isn't as simple as modifying those equations of motion. You'll need to go all the way back to the [Einstein-Hilbert action](https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action) with the $G$ moved inside of the integral, then vary the action to derive new equations of motion. I'm unsure whether you should include varying the action with respect to $G$ to get an equation that determines how $G$ varies or not.  
> **username_2**: Side question: If G = G(t) instead of G = G(x, y, z, t) as in this question, only one equation would not be the same, is that correct? Since varying the EH action in tensor notation is just doing 4 equations at once.  

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## 0 answers

