# (Solved): Please answer parts D, E and F The Lagrangian for a spatially-homogeneous scalar field, \phi (t), i ...

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)

.

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