Upper bound on the Inflation's e-foldings

[Trifis]

It is not clear to me, why textbooks do not mention an upper bound for the e-foldings of the basic inflation theory.

To my knowledge, in order to deal with the flatness problem, we require:

$$\frac{Ω^{-1}(t_0)-1}{Ω^{-1}(t_i)-1} = \frac{Ω^{-1}(t_0)-1}{Ω^{-1}(t_e)-1} \frac{Ω^{-1}(t_e)-1}{Ω^{-1}(t_i)-1} = \left( \frac{a(t_i)}{a(t_e)} \right)^2 \left[ \left( \frac{T_{eq}}{T_0} \right) \left( \frac{T_e}{T_{eq}} \right)^2 \right] ≈ 1$$

where $0, i, e, eq$ are the indices, which denote respectively the current time, the beginning of the inflation, the ending of the inflation (GUT scale) and the time of radiation/matter equality. The values $T_0 ≈ 10^{-13}GeV, T_{eq} ≈ 10^{-9}GeV$ are more or less "certain" and the value for $T_e$ is constrained and considered to be at the GUT scale ($~ 10^{15}GeV$). As a result, we get a constraint for $\frac{a(t_i)}{a(t_e)}$ and using the above values, we get: $\frac{a(t_i)}{a(t_e)} ≈ 10^{-52}$, which corresponds to the classical 60 e-foldings.

Prof. Susskind states here:

that 1000 e-foldings could work equally well. I cannot understand, why this is the case, since $T_e$ cannot get any arbitrary values.

 

[bapowell]

The number of efolds supported by inflation depends on the potential. Very flat potentials can support lots and lots of inflation, because the field does not roll very far in a Hubble time.

 

[fzero]

It is not clear to me, why textbooks do not mention an upper bound for the e-foldings of the basic inflation theory.

To my knowledge, in order to deal with the flatness problem, we require:

$$\frac{Ω^{-1}(t_0)-1}{Ω^{-1}(t_i)-1} = \frac{Ω^{-1}(t_0)-1}{Ω^{-1}(t_e)-1} \frac{Ω^{-1}(t_e)-1}{Ω^{-1}(t_i)-1} = \left( \frac{a(t_i)}{a(t_e)} \right)^2 \left[ \left( \frac{T_{eq}}{T_0} \right) \left( \frac{T_e}{T_{eq}} \right)^2 \right] ≈ 1$$

where $0, i, e, eq$ are the indices, which denote respectively the current time, the beginning of the inflation, the ending of the inflation (GUT scale) and the time of radiation/matter equality. The values $T_0 ≈ 10^{-13}GeV, T_{eq} ≈ 10^{-9}GeV$ are more or less "certain" and the value for $T_e$ is constrained and considered to be at the GUT scale ($~ 10^{15}GeV$). As a result, we get a constraint for $\frac{a(t_i)}{a(t_e)}$ and using the above values, we get: $\frac{a(t_i)}{a(t_e)} ≈ 10^{-52}$, which corresponds to the classical 60 e-foldings.

The condition that you write is actually far more restrictive than what is required. As discussed in, e.g., Liddle and Leach, we should really require flatness over the observable universe, which means roughly over the comoving scale $k_\text{hor} = a_0 H_0$, which is the proper distance to the particle horizon. Whether or not the total universe is flat over longer scales is completely irrelevant to us.

In practice, what this means is that if we trace the evolution of the universe backwards into the inflationary stage, then there is some time $t_k$ such that $t_i< t_k < t_e$ and $a(t_k)H(t_k) = k_\text{hor}$ (see fig 1 in the cited paper for a clear picture). Whatever the curvature of the universe was at $t=t_k$, we need enough e-foldings $e^{N_k} = a_e/a_k$ by the end of inflation in order to wash it away. So we should really be considering

$$ \frac{Ω^{-1}(t_0)-1}{Ω^{-1}(t_k)-1}\approx 1$$

or similar quantities.

Whatever was happening during $t_i \rightarrow t_k$ depends on the details of the inflation model and might influence observable perturbations, but flatness does not put a limit on $t_i$ by itself.

 

[Trifis]

@fzero
Excellent reply, many thanks.