How do I convert the number of efolds to conformal time during inflation?

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In summary, the conversation discusses the need to convert the number of efolds to conformal time during inflation in order to perform numerical integrations. There is a question about the relationship between N (the number of efolds) and τ (conformal time) and a proposed relationship of τ = -1/(H*e^N). It is also mentioned that the value of the scale factor at a specific time is not important, but rather the ratio of scale factors at different times is what matters in cosmology. Additionally, N is defined as the number of efolds before the end of inflation, with dN = -Hdt.
  • #1
Xepto
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Hi everyone,
I need to convert the number of efolds to confromal time during inflation in order to do numerical integrations. Suppose expansion is De Sitter and you have to calculate the integral $$\int^0_{-\infty} d\tau F(N)$$ where F is a function of the efolds number N (from the beginning of inflation). All issues arise at the integration limits. I know that $$a = - \frac{1}{H \tau} $$ but I cannot obtain the correct relationship between N and τ.
(My guess is $$\tau = - \frac{1}{H e^N} $$ but this doesn't seem to be correct. In fact for ##N\rightarrow \infty## we get ##\tau \rightarrow 0##, but for ##N\rightarrow 0## we get ##\tau\rightarrow - \frac{1}{H}##)
Can anyone help me?
Thanks
 
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  • #2
What is the relationship between a and N? Hint: N is not defined with respect to the start of inflation, but the end.
 
  • #3
bapowell said:
What is the relationship between a and N? Hint: N is not defined with respect to the start of inflation, but the end.
From ##dN = H dt##, we get $$N = \text{ln} \frac{a(t_{end})}{a(t)}$$. Tthen ##a(t) = a(t_{end}) e^{-N}##, that can be rewritten in terms of the number of efolds as ##a(N(t)) = a(N_{end}) e^{-N}.## If expansion is DeSitter, ##H## is a constant and $$ \tau = -\frac{1}{a(\tau) H} = - \frac{1}{a(N_{end}) e^{-N} H},$$ that can be inverted $$ N=\text{ln} \Biggl( - \frac{1}{\tau H a(N=0) } \Biggl). $$ If this is correct, how can I calculate the scale factor ##a## at the end of inflation?
 
  • #4
If the universe grows by [itex]N[/itex] e-folds of expansion during inflation, then [itex]a_{end} = e^N a_i[/itex]. What's important is not the value of the scale factor at a particular time, because it can always be renormalized (e.g. the scale factor is often defined to be equal to 1 today). What's generally important in cosmology is the ratio of scale factors at two different times because this gives the amount of expansion.

Also, [itex]N[/itex] is defined as the number of e-folds before the end of inflation. This means that [itex]dN = -Hdt[/itex] -- the number [itex]N[/itex] gets smaller as inflation progresses, and becomes [itex]N=0[/itex] at the end.
 

FAQ: How do I convert the number of efolds to conformal time during inflation?

What is conformal time and how is it related to the concept of efolds?

Conformal time is a measure of distance in the universe that takes into account the expansion of space. It is defined as the time it takes for light to travel through a given distance in an expanding universe. Efolds, on the other hand, are a measure of the amount of expansion that has occurred since the beginning of the universe. They are related to conformal time through the formula: N = ln(a/ai), where N is the number of efolds, a is the scale factor, and ai is the scale factor at the beginning of inflation.

How is conformal time different from regular time?

Conformal time is different from regular time in that it takes into account the expansion of space, while regular time does not. Regular time is measured by clocks, while conformal time is measured by the distance light has traveled. In a non-expanding universe, conformal time and regular time are the same.

What is the significance of conformal time in cosmology?

Conformal time is significant in cosmology because it allows us to compare distances and times at different points in the history of the universe. This is particularly useful when studying the early universe, as it allows us to calculate the amount of expansion that has occurred and make predictions about the future of the universe.

How is conformal time related to the concept of the cosmic microwave background (CMB) radiation?

The cosmic microwave background (CMB) radiation is the leftover thermal radiation from the Big Bang. It is closely related to conformal time in that the CMB radiation was emitted at a specific conformal time in the early universe, known as the surface of last scattering. By studying the CMB radiation, we can gain insights into the properties of the universe at that specific conformal time.

Can conformal time be measured directly?

No, conformal time cannot be measured directly. It is a theoretical concept that is used in cosmology to make calculations and predictions about the universe. However, it can be indirectly measured through observations of the CMB radiation, which provides a snapshot of the universe at a specific conformal time in its early history.

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