Cosmological Coleman-Weinberg potentials are induced when normal matter is coupled to the inflaton. It has long been known that the corrections from bosonic fields are positive whereas those from fermionic fields are negative. In flat space both take the form $\pm φ^4 \ln(φ)$, and they can be made to cancel by appropriately choosing the coupling constants. In an expanding universe the bosonic and fermionic results no longer take the same form, although their large field limits do. We choose the coupling constants so that the large field limits cancel, and then follow the deviations which result as inflation progresses. Although the result is not satisfactory we discuss how adding scalars with various conformal couplings likely solves the problem.
We extend the Coleman--Weinberg inflationary model where a scalar field $ϕ$ is non-minimally coupled to gravity with the addition of the $R^2$ term. We express the theory in terms of two scalar fields and going to the Einstein frame we employ the Gildener--Weinberg formalism, compute the one-loop effective potential and essentially reduce the problem to the case of single-field inflation. It turns out that there is only one free parameter, namely, the mixing angle between the scalars. For a wide range of this angle, we compute the inflationary observables which are in agreement with the latest…
We present the impact of non-minimal coupling $ξϕ^2 R$ on the inflationary parameters by taking into account the models of single-field inflation with the inflaton that has a non-zero vacuum expectation value ($v$) after the period of inflation in Palatini gravity. We discuss the well-known symmetry breaking type potentials, namely the Higgs potential and Coleman-Weinberg potential. We show the inflationary predictions of these potentials, for both $ϕ>v$ and $ϕ<v$ inflation, the regions in the $v-ξ$ plane for which the values of $n_s$ and $r$ are in agreement with the recent measurements. We a…
We apply the multiple point criticality principle to inflationary model building and study Coleman-Weinberg inflation when the scalar potential is quadratic in the logarithmic correction. We analyze also the impact of a non-minimal coupling to gravity under two possible gravity formulation: metric or Palatini. We compare the eventual compatibility of the results with the final data release of the Planck mission.
It has been previously shown that the linear inflation appears naturally as a solution of Coleman-Weinberg inflation, provided that the inflaton has a non-minimal coupling to gravity and the Planck scale is dynamically generated. We revisit the previous study by improving the discussion of reheating and by comparing the results of the metric and the Palatini formulations of non-minimal gravity. We find that both formulations predict linear inflation but a different number of $e$-folds. If the non-minimal coupling is larger than one, future experimental sensitivity can discriminate between the …
Coleman-Weinberg (CW) phenomena for the case of gravitons minimally coupled to massless scalar field is studied. The one loop effect completely vanishes if there is no self interaction term present in the matter sector. The one loop effective potential is shown to develop an instability in the form of acquiring an imaginary part, which can be traced to the tachyonic pole in the graviton propagator. The finite temperature counterpart of this CW potential is computed to study the behaviour of the potential in the high and low temperature regimes with respect to the typical energy scale of the th…
We consider an additional fine-tuning problem which afflicts scalar-driven models of inflation. The problem is that successful reheating requires the inflaton be coupled to ordinary matter, and quantum fluctuations of this matter induces Coleman-Weinberg potentials which are not Planck-suppressed. Unlike the flat space case, these potentials depend upon a still-unknown, nonlocal functional of the metric which reduces to the Hubble parameter for de Sitter. Such a potential cannot be completely subtracted off by any local action. In a simple model we numerically consider one possible subtraction…
We reconsider the fine-tuning problem of scalar-driven inflation arising from the need to couple the inflaton to ordinary matter in order to make reheating efficient. Quantum fluctuations of this matter induce Coleman-Weinberg corrections to the inflaton potential, depending (for de Sitter background) in a complex way on the ratio of the inflaton to the Hubble parameter. These corrections are not Planck-suppressed and cannot be completely subtracted because they are not even local for a general geometry. A previous study showed that it is not satisfactory to subtract a local function of just t…
We discuss how the non-minimal coupling $ξϕ^2 R$ between the inflaton and the Ricci scalar affects predictions of single field inflation models in Palatini formalism. To transition radiation dominated era, the inflaton field $ϕ$ must interact to matter fields at the end of inflation. Interactions of the inflaton with other fields lead to radiative corrections to the inflationary potential. These radiative corrections can be explained at leading order by Coleman-Weinberg (CW) one-loop corrections. In this work, using two different prescriptions debated in the literature, the effect of radiative…
The flat-space limit of the one-loop effective potential for SO(10) GUT theories in spatially flat Friedmann-Robertson-Walker cosmologies is applied to study the dynamics of the early universe. The numerical integration of the corresponding field equations shows that, for such grand unified theories, a sufficiently long inflationary stage is achieved for suitable choices of the initial conditions. However, a severe fine tuning of these initial conditions is necessary to obtain a large $e$-fold number. In the direction with residual symmetry $SU(4)_{PS} \otimes SU(2)_{L} \otimes SU(2)_{R}$, one…
We study non-minimal Coleman-Weinberg inflation in the Palatini formulation of gravity in the presence of an $R^2$ term. The Planck scale is dynamically generated by the vacuum expectation value of the inflaton via its non-minimal coupling to the curvature scalar $R$. We show that the addition of the $R^2$ term in Palatini gravity makes non-minimal Coleman-Weinberg inflation again compatible with observational data.
We show that both the Planck and electroweak mass scales can be generated from conformal gravity via the Coleman-Weinberg mechanism of dimensional transmutation. At the first step, the Planck scale is generated via the Coleman-Weinberg mechanism in the sector of conformal gravity, which means that radiative corrections associated with gravitons induce spontaneous symmetry breakdown of a local conformal symmetry. At the second step, the vacuum expectation value of a scalar field is transmitted to the sector of the standard model via a potential involving the conformally invariant part and the c…
We show that the Planck mass scale can be generated from conformal gravity in the Weyl conformal geometry via the Coleman-Weinberg mechanism of dimensional transmutation where quantum corrections stemming from the gravitational field and the Weyl gauge field trigger the symmetry breakdown of a local Weyl symmetry. It is also shown that the vacuum expectation value of a scalar field is transmitted to a sector of the standard model through a potential involving the scale invariant part and the contribution from the Coleman-Weinberg mechanism, thereby generating the electroweak scale.
We consider a coupling of dilaton gravity to the classically scale-invariant B-L extended standard model which has been recently proposed as a phenomenologically viable model realizing the Coleman-Weinberg mechanism of breakdown of the electroweak symmetry. It is shown in the present model that without recourse to the Coleman-Weinberg mechanism, the B-L gauge symmetry is broken in the process of spontaneous symmetry breakdown of scale invariance at the tree level and as a result the B-L gauge field becomes massive via the Higgs mechanism. Since the dimensionful parameter is only the Planck mas…
We consider a coupling of conformal gravity to the classically scale-invariant B-L extended standard model which has been recently proposed as a phenomenologically viable model realizing the Coleman-Weinberg mechanism of breakdown of the electro-weak symmetry. As in a globally scale-invariant dilaton gravity, it is also shown in a locally scale-invariant conformal gravity that without recourse to the Coleman-Weinberg mechanism, the B-L gauge symmetry is broken in the process of spontaneous symmetry breakdown of the local scale invariance (Weyl invariance) at the tree level and as a result the …
We have considered the quantum behavior of a conformally induced gravity in the minimal Riemann-Cartan space. The regularized one-loop effective potential considering the quantum fluctuations of the dilaton and the torsion fields in the Coleman-Weinberg sector gives a sensible phase transition for an inflationary phase in De Sitter space. For this effective potential, we have analyzed the semi-classical equation of motion of the dilaton field in the slow-rolling regime.
A canonical formulation of effective equations describes quantum corrections by the back-reaction of moments on the dynamics of expectation values of a state. As a first step toward an extension to quantum-field theory, these methods are applied here to the derivation of effective potentials around a homogeneous vacuum expectation value of scalar fields. A comparison with the standard Coleman-Weinberg potential shows that the new methods correctly include all relevant quantum corrections. At the same time, the effective potential is shown to be correct also for non-Fock and mixed states. Sever…
We propose a simple model that provides a dynamical cancellation mechanism of the vacuum energy density appearing either in the form of a bare cosmological constant, quantum fluctuations of matter fields or the result of phase transitions. This `conformal compensator model' is based on a conformal coupling $A(φ)$ between the Einstein and the Jordan frames. This couples a second scalar field $λ$ to the trace of the matter energy-momentum tensor, including the bare cosmological constant, and serves as a dynamical Lagrange multiplier. As a result, the scalar $λ$ relaxes to a value which cancels t…
We show that if the inflaton has a non-minimal coupling to gravity and the Planck scale is dynamically generated, the results of Coleman-Weinberg inflation are confined in between two attractor solutions: quadratic inflation, which is ruled out by the recent measurements, and linear inflation which, instead, is in the experimental allowed region. The minimal scenario has only one free parameter -- the inflaton's non-minimal coupling to gravity -- that determines all physical parameters such as the tensor-to-scalar ratio and the reheating temperature of the Universe. Should the more precise fut…
These lectures on the cosmological constant problem were prepared for the X Mexican School on Gravitation and Mathematical Physics. The problem itself is explained in detail, emphasising the importance of radiative instability and the need to repeatedly fine tune as we change our effective description. Weinberg's no go theorem is worked through in detail. I review a number of proposals including Linde's universe multiplication, Coleman's wormholes, the fat graviton, and SLED, to name a few. Large distance modifications of gravity are also discussed, with causality considerations pointing towar…
We briefly review the arising of an inflationary phase in the Universe evolution in order to discuss an inhomogeneous cosmological solution in presence of a real self interacting scalar field minimally coupled to gravity in the region of a slow rolling potential plateau. During the inhomogeneous de Sitter phase the scalar field dominant term is a function of the spatial coordinates only. We apply this generic solution to the Coleman-Weinberg potential and to the Lemaitre-Tolman metric. This framework specialized nearby the FLRW model allows a classical origin for the inhomogeneous perturbation…
We show a text-book potential for single-field inflation, namely, the Coleman-Weinberg model can induce double inflation and formation of primordial black holes (PBHs), because fluctuations that leave the horizon near the end of first inflation are anomalously enhanced at the onset of second inflation when the time-dependent mode turns to a growing mode rather than a decaying mode. The mass of PBHs produced in this mechanism lies in several discrete ranges depending on the model parameters. We also calculate the effects of non-Gaussian statistics due to higher-order interactions on the abundan…
