Prove that a~t^(8pπ) (inflation)

[QuarkDecay]

Homework Statement

In a chaotic cosmological model during inflation, given the V(Φ) and Φ(t), we need to prove that a~t^(8pπ)

Relevant Equations

Friedmann's first equation; (da/adt)[SUP]2[/SUP]= 8πGV(Φ)/3
inflaton equation; 3HdΦ/dt= -dV/dΦ

Problem gives these for a chaotic model;

V(Φ)=V_oexp(-√(2/p)* Φ/Μ_p)

Φ(t)=√(2p)*M_pln[√(V_o/24πp²) *t/M_p]

There's a standard method to follow and find the a(t) by using Friedmann's and inflaton equations. I think my mistake is most likely on the math part, because in the physics aspect we always follow this method of using these two equations and solving them. Unless there's some approximation I have to make about the chaotic model and the Φ, a(t) etc

Starting from Friedmann's first equation
da/adt= √(8πGV(Φ)/3) ⇒ ∫da/a = ∫ √(8πGV(Φ)/3) dt = √(8πG) ∫√(V(Φ)) dt

Now for the integral I replace V's value I= ∫√(V(Φ)) dt = ∫√(V_oexp(-√(2/p)* Φ/Μ_p)) dt =
=√V_o* ∫ [exp(-√(2/p)* Φ/Μ_p)]^1/2 dt =

and multiple the 1/2 from the square root inside the exponential

=√V_o ∫ [exp (-√(2/p)* Φ/2Μ_p)] dt =

Now replacing the Φ value too

=√V_o ∫ [exp (-√(2/p) * 1/(2Μ_p) * √(2p) * Μ_p * ln( √(V_o/24πp²) *t/M_p) ] dt =

= √V_o ∫ [ exp (-ln (V_o/24πp²) *t/M_p) ] dt =

now I wasn't too sure how to proceed at this point. Not sure if exp(lnx)= x , although I read this is correct, so I used this

= √V_o ∫- (V_o/24πp²) *t/M_p) dt =

= √V_o * (V_o/24πp²)/ M_p) * ∫ t dt

and ∫ t dt= t²/2

Also going back in the beginning, the first part of the equation; ∫da/a =lna

So the final one is

lna =√V_o * (V_o/24πp²)/ M_p) *t²/2 =
-V_o/ (2 M_p) * 1/√(24πp²) * t² ⇒

⇒ a(t) = exp [-V_o/ (2 M_p) * 1/√(24πp²) * t² ]

which means that a(t) ~ t² and not t^8πp

 

[Nate810]

2 as the problem asked. The mistake might be in the math part while I was solving the integral, because I wasn't so sure how to treat the exponential and ln in the integral. Also if it is not correct, any suggestions on how to make this more clear?