Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 55 / 5

Differentiate the scalar-field Friedmann equation on the expanding branch $H>0$:The scalar equation of motion gives $V^{\prime }+\ddot{\Phi }=-3H\dot{\Phi }$. Canceling $H$ yieldsThus the Hubble parameter cannot increase for a canonical homogeneous scalar field. The mass scale here is the reduced Planck mass.

The inflaton has quantum fluctuations while their wavelengths are shorter than the Hubble radius. Choosing the Bunch-Davies vacuum fixes the short-wavelength positive-frequency modes. During near-de Sitter slow-roll inflation, expansion carries each mode through $k=aH$, leaving a field fluctuation of amplitude approximately $H/(2\pi )$ on spatially flat slices.

The inflaton clock-shift origin of curvature perturbations converts this field displacement into a local time shift of magnitude $|\delta \Phi /\dot{\Phi }|$, and hence into a comoving curvature perturbation of magnitude $H|\delta \Phi /\dot{\Phi }|$. For the single-field adiabatic growing mode on an attractor, the superhorizon conservation of single-field comoving curvature makes this perturbation approximately constant at $k\ll aH$. These vacuum-generated perturbations seed the later density fluctuations.

The slow-roll equations give $H^2\simeq V/(3M_{\mathrm{Pl}}^2)$ and $\dot{\Phi }\simeq -V^{\prime }/(3H)$. Substituting in the stated slow-roll curvature power spectrum yieldsWith the potential slow-roll parameter ${ϵ}_V=M_{\mathrm{Pl}}^2{V}^{\prime 2}/(2V^2)$,All quantities are evaluated at horizon exit. The two forms have the same normalization because $(\sqrt{8\pi }{)}^4=64{\pi }^2$.

At horizon exit $k=aH$, $d\log k/dt=H(1-{ϵ}_H)$, where ${ϵ}_H=-\dot{H}/H^2$ is the Hubble slow-roll parameter. To leading slow-roll order,The slow-roll curvature power spectrum is proportional to $V^3/V^{\prime 2}$. Therefore the scalar spectral index in potential slow-roll parameters isUsing the potential slow-roll parameter and the second potential slow-roll parameter givesThe neglected horizon-exit derivative correction contributes at higher slow-roll order.

Apply the horizon-exit derivative from part (d) to the potential slow-roll parameter and the second potential slow-roll parameter. The potential slow-roll flow equations arewhere the third potential slow-roll parameter is ${\xi }_V^2=M_{\mathrm{Pl}}^4{V}^{\prime }{V}^{\prime \prime \prime }/V^2$. For example, differentiating ${\eta }_V$ gives $-M_{\mathrm{Pl}}^4(V^{\prime }/V)(V^{\prime \prime \prime }/V-V^{\prime \prime }{V}^{\prime }/V^2)$, which is the second expression. Differentiating $n_s-1\simeq -6{ϵ}_V+2{\eta }_V$ now gives the running of the scalar spectral indexThis is the leading second-order slow-roll running. The notation ${\xi }_V^2$ denotes a signed derivative combination and does not force its value to be nonnegative.