Help me derive the relativistic equation of motion for the Universe

[QuantumX]

Guys,

My calculus is really rusty and I need help solving this equation using a time derivative (denoted with a dot) in order to get the relativistic equation of motion for the Universe

The equation is:

adot^2 = Λc^2a^2/3

where adot is the time derivative of the scale factor, lambda is a cosmological constant, and c is the speed of light.

The answer should be:

a(t) = a(t0)e^Ht

where H is Hubble's constant = sqrt(c^2Λ/3)

But I don't know how to get there. Any help is appreciated!

 

[tiny-tim]

Hi QuantumX! Welcome to PF!

(try using the X² button just above the Reply box )

adot² = Λc²a²/3

So da/dt = √(Λc²/3)a = Ha

 

[QuantumX]

Thanks Tiny-Tim,

So then the first equation is just the relationship between the scale factor and the Hubble constant...

So how do I then get a(t) = a(t0)e^Ht ? (that is for t>t0). That's what i need to ultimately arrive at. It's supposed to showcase that the Universe is accelerating in its expansion.

 

[tiny-tim]

So da/dt = √(Λc²/3)a = Ha

So how do I then get a(t) = a(t0)e^Ht ?

da/dt = Ha

so dt/da = 1/Ha, so t = loga/H + constant

so loga = Ht + constant

so a = e^{Ht + constant} = e^constante^Ht

and the e^constant is a(t_o)

 

[QuantumX]

Thanks!