Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 70 / 1

Write the reduced Planck mass as $M=(8\pi G{)}^{-1/2}=m_{\mathrm{pl}}/\sqrt{8\pi }$, and use natural units. Near the flat maximum, the quartic hilltop inflation trajectory has $V\simeq \alpha {\sigma }^4$ and $V^{\prime }\simeq -12\alpha {\varphi }^3$. The slow-roll approximation gives $H^2\simeq V/(3M^2)$ and $3H\dot{\varphi }\simeq -V^{\prime }$. Therefore the number of e-folds isThe lower field value dominates this integral. ThusThis is far below $\sigma$ for the assumed small symmetry-breaking scale. Strict slow roll actually fails before the minimum: $|M^2{V}^{\prime \prime }/V|\sim 1$ at ${\varphi }^2\sim {\sigma }^4/(36M^2)$. The last approach to the minimum is not described by slow roll. Using that earlier upper limit changes the dominant fifty-e-fold estimate by an order-one number of e-folds, not its leading scale dependence. No extrapolation of slow roll all the way to the minimum is needed.

A comoving wavenumber crosses the inflationary Hubble scale at $k=aH$. Consequently $d\ln k=d\ln a+d\ln H\simeq Hdt$ when $H$ changes slowly. Combining with the slow-roll approximation givesBoth $k$ and $\varphi$ increase along the rolling branch. If $N$ denotes the remaining number of e-folds rather than elapsed expansion, then $d\ln k\simeq -dN$. More precisely $d\ln k=(1-{ϵ}_H)d\ln a$; the correction is beyond the constant-Hubble approximation used here.

At leading order the scalar curvature power and total tensor power areApplying the preceding $d/d\ln k$ relation gives the scalar spectral index $n-1=-6{ϵ}_V+2{\eta }_V$ and the tensor spectral index $n_T=-2{ϵ}_V$. For this hilltop,Thus, fifty e-folds before the end,The tiny $-6{ϵ}_V$ contribution and finite-endpoint corrections have been omitted in the quoted leading scalar tilt. The tensor tilt is nearly zero but slightly negative; treating $H$ as exactly constant in its spectrum would incorrectly discard its leading nonzero value.

The given large-angle temperature transfer relation gives the low-multipole tensor fraction in quartic hilltop inflationThis is not the primordial tensor-to-scalar ratio $r=16{ϵ}_V$, because the scalar and tensor contributions to low CMB multipoles have different transfer functions. With $\sigma \ll m_{\mathrm{pl}}$, the tensor contribution is far below the scalar temperature signal and its cosmic variance, and the corresponding primordial gravitational-wave or polarization signal is extremely small.

For COBE normalization of quartic hilltop inflation, the historical scalar horizon amplitude is of order ${\delta }_H\simeq 2\times 10^{-5}$, or ${\mathcal{P}}_S\simeq 25{\delta }_H^2/4\sim 2.5\times 10^{-9}$ in the usual matter-era convention. The slow-roll spectrum simplifies toHenceLate-time transfer and tilt corrections to the COBE normalization change the order-one coefficient, not the conclusion that the dimensionless coupling must be extraordinarily small. The leading normalization is independent of $\sigma$: reducing the height of the potential also changes the slope and field value at fixed $N$, causing the scale factors to cancel.