Two numerical questions about inflation and the flatness problem

[KayLepton]

Good day to you all
I just started studying cosmology recently using several book and especially using
TASI Lectures on Inflation(https://arxiv.org/abs/0907.5424)

On page 25 the fine tuning problem of the flatness is introduced with conditions as
|Ω(BBN)-1|≤ ο(10^-16)
|Ω(GUT)-1|≤ ο(10^-55)
|Ω(pl)-1|≤ ο(10^-61)
My first question is how to derive this condition

The second would be page 34 where the number of inflationary e-folds were introduced and simply put that it must be more than 60 to solve the flatness and horizon problems
I was unable to derive this also

Furthermore I suppose that these conditions are derived from observations
It would be great if I could calculate these conditions a bit more exact using recent data...

Thank you for your time

 

[AndyW]

and for sharing your thoughts on cosmology!

Hello! It's great to hear that you are studying cosmology and using resources like the TASI Lectures on Inflation to deepen your understanding. The conditions you mentioned on page 25 are derived from the equations of general relativity and the Friedmann equations, which describe the evolution of the universe. These equations relate the energy density of the universe to its curvature and expansion rate.

In order for the universe to have the observed flatness, the energy density must be very close to a critical value. This is where the condition |Ω-1|≤ ο(10-16) comes from. Similarly, the conditions for the GUT and Planck eras come from the energy densities at those times, which can be calculated using the equations of general relativity.

As for the number of inflationary e-folds, this is a result of the inflationary model itself. Inflation is a period of rapid expansion in the early universe that solves the flatness and horizon problems. In order for inflation to have a significant impact, it must last for at least 60 e-folds, which is where that number comes from.

These conditions are indeed derived from observations and can be calculated more accurately using recent data. In fact, cosmologists are constantly refining and updating our understanding of these conditions as new data becomes available. Keep up the good work in your studies and don't hesitate to ask for help or clarification if needed. Cosmology is a complex and fascinating field, and there is always more to learn!