# Prove that a~t^(8pπ) (inflation)

#### QuarkDecay

Problem 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)=Voexp(-√(2/p)* Φ/Μp)

Φ(t)=√(2p)*Mpln[√(Vo/24πp2) *t/Mp]

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 = ∫√(Voexp(-√(2/p)* Φ/Μp)) dt =
=√Vo* ∫ [exp(-√(2/p)* Φ/Μp)]1/2 dt =

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

=√Vo ∫ [exp (-√(2/p)* Φ/2Μp)] dt =

Now replacing the Φ value too

=√Vo ∫ [exp (-√(2/p) * 1/(2Μp) * √(2p) * Μp * ln( √(Vo/24πp2) *t/Mp) ] dt =

= √Vo ∫ [ exp (-ln (Vo/24πp2) *t/Mp) ] 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

= √Vo ∫- (Vo/24πp2) *t/Mp) dt =

= √Vo * (Vo/24πp2)/ Mp) * ∫ 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 =√Vo * (Vo/24πp2)/ Mp) *t2/2 =
-Vo/ (2 Mp) * 1/√(24πp2) * t2

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

which means that a(t) ~ t2 and not t8πp

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