- #1

JD_PM

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- Homework Statement
- While studying Cosmology I came across a particular differential equation I do not see how to solve. Please read below for details

- Relevant Equations
- N/A

This question arose while studying Cosmology (section 38.2 in Lecture Notes in GR) but it is purely mathematical, that is why I ask it here.

I do not see why the equation

$$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$

Has the following solution

$$a(t) = a_0 \left( \frac{(\Omega_M)_0}{(\Omega_{\Lambda})_0}\right)^{1/3} \left(\sinh \left[(3/2)\sqrt{(\Omega_{\Lambda})_0}H_0t\right]\right)^{2/3}$$

Where (and ##\dot a## represents the derivative of ##a## wrt time)

$$H:=\left( \frac{\dot a}{a}\right)^2, \ \ \ \ H_0:=\left( \frac{\dot a_0}{a_0}\right)^2$$I found a similar problem here. I suspect that, to start off, we should find a smart change of variables but which one?

Might you please guide me towards the solution?

Any help is appreciated.

Thank you

I do not see why the equation

$$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$

Has the following solution

$$a(t) = a_0 \left( \frac{(\Omega_M)_0}{(\Omega_{\Lambda})_0}\right)^{1/3} \left(\sinh \left[(3/2)\sqrt{(\Omega_{\Lambda})_0}H_0t\right]\right)^{2/3}$$

Where (and ##\dot a## represents the derivative of ##a## wrt time)

$$H:=\left( \frac{\dot a}{a}\right)^2, \ \ \ \ H_0:=\left( \frac{\dot a_0}{a_0}\right)^2$$I found a similar problem here. I suspect that, to start off, we should find a smart change of variables but which one?

Might you please guide me towards the solution?

Any help is appreciated.

Thank you