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Register a new userThe Inflaton Effective Potential for General $epsilon$
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Authors
A. Kyriazis
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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.
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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.
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.
The interaction between two initially causally disconnected regions of the universe is studied using analogies of non-commutative quantum mechanics and deformation of Poisson manifolds. These causally disconnect regions are governed by two independent Friedmann-Lema^{i}tre-Robertson-Walker (FLRW) metrics with scale factors $a$ and $b$ and cosmological constants $Lambda_a$ and $Lambda_b$, respectively. The causality is turned on by positing a non-trivial Poisson bracket $[ {cal P}_{alpha}, {cal P}_{beta} ] =epsilon_{alpha beta}frac{kappa}{G}$, where $G$ is Newtons gravitational constant and $kappa $ is a dimensionless parameter. The posited deformed Poisson bracket has an interpretation in terms of 3-cocycles, anomalies and Poissonian manifolds. The modified FLRW equations acquire an energy-momentum tensor from which we explicitly obtain the equation of state parameter. The modified FLRW equations are solved numerically and the solutions are inflationary or oscillating depending on the values of $kappa$. In this model the accelerating and decelerating regime may be periodic. The analysis of the equation of state clearly shows the presence of dark energy. By completeness, the perturbative solution for $kappa ll1 $ is also studied.
The recent direct observation of gravitational waves (GW) from merging black holes opens up the possibility of exploring the theory of gravity in the strong regime at an unprecedented level. It is therefore interesting to explore which extensions to General Relativity (GR) could be detected. We construct an Effective Field Theory (EFT) satisfying the following requirements. It is testable with GW observations; it is consistent with other experiments, including short distance tests of GR; it agrees with widely accepted principles of physics, such as locality, causality and unitarity; and it does not involve new light degrees of freedom. The most general theory satisfying these requirements corresponds to adding to the GR Lagrangian operators constructed out of powers of the Riemann tensor, suppressed by a scale comparable to the curvature of the observed merging binaries. The presence of these operators modifies the gravitational potential between the compact objects, as well as their effective mass and current quadrupoles, ultimately correcting the waveform of the emitted GW.
The effective vacuum energy density contributed by the non-trivial contortion distribution and the bare vacuum energy density can be viewed as the energy density of the auxiliary quintessence field potential. We find that the negative bare vacuum energy density from string landscape leads to a monotonically decreasing quintessence potential while the positive one from swampland leads to the meta stable or stable de Sitter like potential. Moreover, the non-trivial Brans-Dicke like coupling between quintessence field and gravitation field is necessary in the latter case.
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