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 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)=V_{o}exp(√(2/p)* Φ/Μ_{p})
Φ(t)=√(2p)*M_{p}ln[√(V_{o}/24πp^{2}) *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_{o}exp(√(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}/24πp^{2}) *t/M_{p}) ] dt =
= √V_{o} ∫ [ exp (ln (V_{o}/24πp^{2}) *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}/24πp^{2}) *t/M_{p}) dt =
= √V_{o} * (V_{o}/24πp^{2})/ M_{p}) * ∫ t dt
and ∫ t dt= t^{2}/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}/24πp^{2})/ M_{p}) *t^{2}/2 =
V_{o}/ (2 M_{p}) * 1/√(24πp^{2}) * t^{2} ⇒
⇒ a(t) = exp [V_{o}/ (2 M_{p}) * 1/√(24πp^{2}) * t^{2} ]
which means that a(t) ~ t^{2} and not t^{8πp}
V(Φ)=V_{o}exp(√(2/p)* Φ/Μ_{p})
Φ(t)=√(2p)*M_{p}ln[√(V_{o}/24πp^{2}) *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_{o}exp(√(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}/24πp^{2}) *t/M_{p}) ] dt =
= √V_{o} ∫ [ exp (ln (V_{o}/24πp^{2}) *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}/24πp^{2}) *t/M_{p}) dt =
= √V_{o} * (V_{o}/24πp^{2})/ M_{p}) * ∫ t dt
and ∫ t dt= t^{2}/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}/24πp^{2})/ M_{p}) *t^{2}/2 =
V_{o}/ (2 M_{p}) * 1/√(24πp^{2}) * t^{2} ⇒
⇒ a(t) = exp [V_{o}/ (2 M_{p}) * 1/√(24πp^{2}) * t^{2} ]
which means that a(t) ~ t^{2} and not t^{8πp}