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Register a new userInflaton Effective Potential from Photons for General $epsilon$
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S. Katuwal
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We accurately approximate the contribution that photons make to the effective potential of a charged inflaton for inflationary geometries with an arbitrary first slow roll parameter $epsilon$. We find a small, nonlocal contribution and a numerically larger, local part. The local part involves first and second derivatives of $epsilon$, coming exclusively from the constrained part of the electromagnetic field which carries the long range interaction. This causes the effective potential induced by electromagnetism to respond more strongly to geometrical evolution than for either scalars, which have no derivatives, or spin one half particles, which have only one derivative. For $epsilon = 0$ our final result agrees with that of Allen on de Sitter background, while the flat space limit agrees with the classic result of Coleman and Weinberg.
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We develop an analytic approximation for the coincidence limit of a massive scalar propagator in an arbitrary spatially flat, homogeneous and isotropic geometry. We employ this to compute the one loop corrections to the inflaton effective potential from a quadratic coupling to a minimally coupled scalar. We also extend the Friedmann equations to cover potentials that depend locally on the Hubble parameter and the first slow roll parameter.
Scalar perturbations during inflation can be substantially amplified by tiny features in the inflaton potential. A bump-like feature behaves like a local speed-breaker and lowers the speed of the scalar field, thereby locally enhancing the scalar power spectrum. A bump-like feature emerges naturally if the base inflaton potential $V_b(phi)$ contains a local correction term such as $V_b(phi)left[1+varepsilon(phi)right]$ at $phi=phi_0$. The presence of such a localised correction term at $phi_0$ leads to a large peak in the curvature power spectrum and to an enhanced probability of black hole formation. Remarkably this does not significantly affect the scalar spectral index $n_{_S}$ and tensor to scalar ratio $r$ on CMB scales. Consequently such models can produce higher mass primordial black holes ($M_{rm PBH}geq 1 M_{odot}$) in contrast to models with `near inflection-point potentials in which generating higher mass black holes severely affects $n_{_S}$ and $r$. With a suitable choice of the base potential - such as the string theory based (KKLT) inflation or the $alpha$-attractor models - the amplification of primordial scalar power spectrum can be as large as $10^7$ which leads to a significant contribution of primordial black holes (PBHs) to the dark matter density today, $f_{rm PBH} = Omega_{0,rm PBH}/Omega_{0,rm DM} sim O(1)$. Interestingly, our results remain valid if the bump is replaced by a dip. In this case the base inflaton potential $V_b(phi)$ contains a negative local correction term such as $V_b(phi)left[1-varepsilon(phi)right]$ at $phi=phi_0$ which leads to an enhanced probability of PBH formation. We conclude that primordial black holes in the mass range $10^{-17} M_{odot} leq M_{rm PBH} leq 100, M_{odot}$ can easily form in single field inflation in the presence of small bump-like and dip-like features in the inflaton potential.
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