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I'm new here and I want to ask you if anybody of you have some ideas to solve this two problems:

1.Consider single field inflation in slow roll regime, at which the slow roll parameter ǫ

decreases in time. Take any inflaton potential consistent with the CMB and galaxy distri-

bution data. Show that during the period at inflation, which is responsible for generating

the adiabatic perturbations, the inflaton field rolls down at least by

\begin{equation} \Delta \phi \gtrsim 10rM_{planck}

\end{equation}

where r is the tensor-to-scalar ratio. [This means, in particular, that the discovery of tensor

modes with r ∼ 0.2, as originally claimed by BICEP-2, would imply that the variation of

the inflaton over the relevant period of time at inflationary epoch was super-Planckian.]

Relatively short gravity waves, created at inflation, after horizon re-entry at radiation

2.

domination can be viewed as a collection of gravitons (just like electromagnetic waves emitted

by antenna can be viewed as a collection of photons). Assuming that the Hubble parameter

H some 60 e-foldings before inflation end is known, calculate the average (over enesemble of

universes) number of gravitons <N(k,\Delta k)> in the present visible Universe in the interval of

momenta from k/a0 to (k +\Delta k)/a0, and relative variance of this number

\begin{equation} \frac{ \sqrt{ <N^{2}(k,\Delta k)> - <N(k,\Delta k)>^{2} }}{<N(k,\Delta k)>}

\end{equation}

Dropping the assumption about the value of the Hubble parameter, calculate these quantities

for the inflaton potential V = (m^2φ^2)/2. Give numerical estimates in the latter case for

k/a0 = 1 Mpc^(−1), \Delta k = k.

I thank you in advance for any kind of helps

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# A couple of inflationary problems

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