- #1
supertramp87
- 1
- 0
Hi guys.
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.]
2. Relatively short gravity waves, created at inflation, after horizon re-entry at radiation
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
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.]
2. Relatively short gravity waves, created at inflation, after horizon re-entry at radiation
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
Last edited by a moderator:
