# Upper bound on the Inflation's e-foldings

• Trifis
In summary, the flatness problem in inflation theory requires the universe to be flat over the observable universe scale, which is determined by the comoving scale. This means that the universe must be flat at a certain time during inflation, and the number of e-foldings needed to achieve this is determined by the ratio of the scale factor at the end of inflation to the scale factor at this time. However, the number of e-foldings needed is not limited by the flatness condition alone, and the exact value may depend on the specific inflation model.
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.

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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.

Trifis said:
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.

Haelfix
@fzero

## What is an upper bound on the Inflation's e-foldings?

An upper bound on the Inflation's e-foldings is the maximum number of times the universe expands during the Inflationary period. It represents the limit of how much the universe can grow during this process.

## Why is an upper bound on the Inflation's e-foldings important?

Knowing the upper bound on the Inflation's e-foldings allows us to understand the extent of the universe's expansion during the Inflationary period and how it may have affected the formation of structures in the universe.

## How is the upper bound on the Inflation's e-foldings calculated?

The upper bound on the Inflation's e-foldings is calculated based on the duration of the Inflationary period, the speed of expansion, and the initial size of the universe. This calculation is based on theories and models of Inflation and is subject to ongoing research and refinement.

## What factors can affect the upper bound on the Inflation's e-foldings?

The upper bound on the Inflation's e-foldings can be affected by the type of Inflation model, the energy scale of the Inflationary period, and the presence of additional fields or particles during Inflation. These factors can alter the duration and speed of expansion, thus impacting the upper bound value.

## How does the upper bound on the Inflation's e-foldings relate to the overall theory of Inflation?

The upper bound on the Inflation's e-foldings is an important parameter in the theory of Inflation as it helps to constrain and validate different models and predictions. It also allows for comparisons between different Inflationary scenarios and helps to refine our understanding of the early universe.

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