Friedmann equation of motion w.r.t Number of E-folds

Question

So the number of e-folds is given by $N=ln(a)$ which can act as a substitute for time variable. So I would like to use the fact that $H = dot{a}/a$ in order to derive the equation of motions for $phi$, using $N=ln(a)$ as a time variable. This means that in the differential equation
$$ddotphi + 3 H dotphi + V'(phi)=0$$
and
$$3 H^2 M_p^2 = dots$$
I want the equation to have no time derivatives anymore, but only derivatives with respect to $N$.
Can anyone help me in doing this or point out any papers/resourced which have done this before?

bapowell · Answer

You need to write the time derivatives as derivatives with respect to N.  From the definition,
$$a(t) = e^{-N(t)}$$
we have $dN = -dln a = -Hdt$.  Now just use this to write $d/dt$ in terms of $d/dN$:
$$frac{d}{dt} = -H frac{d}{dN}$$
and
$$frac{d^2}{dt^2} = -Hfrac{d}{dN}left(-Hfrac{d}{dN}right) = H frac{dH}{dN}frac{d}{dN} + H^2frac{d^2}{dN^2}$$

Friedmann equation of motion w.r.t Number of E-folds

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