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Please answer parts D, E and F The Lagrangian for a spatially-homogeneous scalar field,

`\phi (t)`

, in an FRW spacetime with scale factor,

`a(t)`

, is given by:

`L_(FRW)=a^(3){(1)/(2)\phi ^(˙)^(2)-M^((10)/(3))\phi ^((2)/(3))}.`

where

`M`

is a constant. The Friedmann equation for the Hubble rate,

`H=(a^(˙))/(a)`

, in terms of the energy density

`\rho `

gives

`H^(2)=(8\pi G_(N))/(3)\rho .`

(a) Use the Euler-Lagrange equation to derive the Klein-Gordon equa- tion for the scalar field

`\phi ^(¨)+3H\phi ^(˙)+(2)/(3)M^((10)/(3))\phi ^(-(1)/(3))=0.`

(b) Show that the canonical momentum

`\Pi =a^(3)\phi ^(˙)`

and hence show that the energy density,

`\rho =(H_(FRW))/(a^(3))`

, is

`\rho =(1)/(2)\phi ^(˙)^(2)+M^((10)/(3))\phi ^((2)/(3)). `

where

`H_(FRW)`

is the Hamiltonian density. (c) Show that in the slow-roll approximation

`(\phi ^(2)&M^((10)/(3))\phi ^((2)/(3)))`

`H~~\sqrt((8\pi )/(3)(M^((5)/(3))\phi ^((1)/(3)))/(M_(P))).`

(d) Use the over-damped approximation for the Klein-Gordon equa- tion to show that

`9^(˙)prop\phi ^(-(2)/(3)).`

and give an expression for the slow-roll parameter,

`\epsi lon=-(H^(˙))/(H^(2))`

. as a function of the scalar field,

`\phi `

. (e) The power spectrum of primordial density perturbations from slow-roll inflation is given by

`P_(\zeta )(k)=((H^(2))/(2\pi (\phi ^(˙))))_(k)=\alpha H^(2)`

Show that the scale dependence,

`n-1=dln(P_(\zeta ))/(d)lnk`

, is

`n-1≃-(2)/(9\pi )((M_(P))/(\phi ))^(2),`

and hence becomes scale invariant for

`|\phi |>=M_(P)`

, where

`M_(P)=`

`G_(N)^(-(1)/(2))`

is the Planck mass. (f) Estimate the spectral tilt,

`n-1`

, at a scale that crosses outside the horizon,

`(k)/(a)=aH,60`

e-folds before the end of inflation where

`{:\epsi lon_(cad )=1)`

.