Very low scale Coleman-Weinberg inflation with non-minimal coupling

We study viable small-field Coleman-Weinberg (CW) inflation models with the help of non-minimal coupling to gravity. The simplest small-field CW inflation model (with a low-scale potential minimum) is incompatible with the cosmological constraint on the scalar spectral index. However, there are possibilities to make the model realistic. First, we revisit the CW inflation model supplemented with a linear potential term. We next consider the CW inflation model with a logarithmic non-minimal coupling and illustrate that the model can open a new viable parameter space that includes the model with a linear potential term. We also show parameter spaces where the Hubble scale during the inflation can be as small as $10^{-4} $ GeV, $1$ GeV, $10^4 $ GeV, and $10^8$ GeV for the number of $e$-folds of $40,~45,~50$, and $55$, respectively, with other cosmological constraints being satisfied.


Introduction
Inflation is one of the successful paradigms in modern cosmology that can address various cosmological issues [1,2,3] and generate primordial perturbations [4,5,6,7]. The underlying particle physics is, however, still unclear, and it is indispensable in understanding physics in the early Universe. To this end, in particular, it is legitimate to ask what is a consistent inflationary scenario for a specific physics model beyond the standard model of particle physics.
Two categories are often used to classify various inflationary models: large-field and smallfield inflation, according to whether the inflaton field excursion during inflation exceeds the Planck scale or not. Each class of models has its own virtues. For instance, the large-field models have an advantage in the initial condition of inflation [8,9], whereas in the small-field models, inflation can take place with the inflaton field value well below the Planck scale, and hence, its field theoretical description is verified and well understood.
In small-field models with a symmetry-breaking-type potential, inflation takes place at the vicinity of the origin, and the inflaton field slowly rolls down toward the potential minimum located below the Planck scale. From the normalization of temperature anisotropy of cosmic microwave background (CMB) radiation of O(10 −5 ), the energy scale of small-field inflation models, which is equivalently the Hubble parameter during inflation, turns out to be rather small. Thus, a small-field inflation generally leads to a rather low reheating temperature. Such a low-scale inflation and its resultant low reheating temperature are attractive from several viewpoints. Here, we note several examples and those motivations. First, in a Peccei-Quinn (PQ) extended model to solve the strong CP problem in the standard model of particle physics [10,11], if the PQ symmetry is broken before or during inflation, axion fluctuations on the order of the Hubble parameter during inflation are generated and induce axion isocurvature perturbations [12,13,14,15]. To satisfy the stringent bound on axion isocurvature perturbation by the CMB temperature anisotropy, a small Hubble parameter during inflation, 10 7 GeV, is required [16]. Second, one of the most promising scenarios for generation of baryon asymmetry is the Affleck-Dine mechanism with a flat direction [17]. An appropriate amount of baryon asymmetry can be generated by a flat direction lifted by a dimension-six operator for a low reheating temperature T R of about 100 GeV [18]. Affleck-Dine baryogenesis by such flat directions is interesting because it provides a solution to the coincidence of energy densities between baryon and dark matter with the formation of Q-balls [19,20,21]. Third, an issue of supersymmetric models in cosmology is the overproduction of the gravitino [22,23,24]. Because gravitino abundance produced through thermal scatterings is proportional to the reheating temperature after inflation, in order to avoid overproduction, the upper bound on the reheating temperature is imposed. For a recent estimation, see, e.g., Refs. [25,26]. Finally, the recently proposed relaxion mechanism [27], as a solution to the hierarchy problem of Higgs boson by utilizing a slowly rolling scalar field in the context of inflationary cosmology, also requires a very long period and a very low energy scale of inflation for a phase transition by QCD(-like) strong dynamics to take place during inflation, not only in the minimal model [27,28,29] but also in some extended models [30,31,32,33]. (However, for other extensions where the relaxion mechanism can work at relatively high scale, see e.g., Refs. [34,35,36].) In this paper, we pursue a possible realization of viable small-field inflationary models based on the Coleman-Weinberg (CW) model [37]. In particular, we discuss how small inflation scale can be achieved in the CW model with some possible modifications. The CW inflation model is a typical model of low-scale and small-field inflation [38,39,40]. However, the original CW inflation model is doomed by the observed scalar spectral index [41], which is significantly larger than that of the model predictions. 1 Iso et al. have proposed simple extensions to ameliorate this discrepancy [49]. In this paper, we revisit known examples of such extension, and explore further possibilities by considering other promising extensions.
The rest of this paper is organized as follows. In Sec. 2, we first go over the models discussed in Ref. [49] and move onto other possible extensions based on a non-minimal coupling to gravity. We devote Sec. 3 to discussions and conclusions.

Small-field Coleman-Weinberg inflation model
We study a class of small-field CW inflation where the inflaton starts to roll down from the vicinity of the origin to the potential minimum [38,39,40,49]. The scalar potential for the inflaton φ is given by with a scale M(< M pl ) and M pl being the reduced Planck mass. V 0 is determined by the vanishing cosmological constant at the minimum. Derivatives of the potential with respect to φ are We find that the vacuum expectation value at the minimum is given by φ = M and V 0 is obtained by V (M) = 0 as shown above. Thus, the slow roll parameters are calculated as with field value φ * where the pivot scale k * exits from the Hubble radius. φ end denotes the field value at the end of inflation. Therefore, the low-scale CW inflation model must be modified to be consistent with cosmological observations. In most studies, N * is taken to be about 50 or 60. In fact, N * weakly depends on the energy scale of inflation and delay of reheating after inflation [51,52,53] as with the energy density at the reheating ρ R and the energy density at the moment of the pivot scale horizon crossing during inflation V * for the standard thermal history after inflation in which the Universe becomes the matter dominated with the equation of state w = 0 during the coherent oscillation of inflaton after inflation, followed by the radiation-dominated Universe. Here, we used V * ≃ V (φ end ). Now, N * is a function of V * and ρ R . In the following analysis, because we are interested in very low scale CW inflation, we vary N * from 40 to 55 under the condition

Fermion condensates
A possibility to increase n s is the introduction of a linear term, which can be generated by a fermion condensation in the inflaton potential discussed in Ref. [49]. In the work, two examples that induce a linear term have been shown. One is the condensation of right-handed neutrinos N, which couples to φ through a Yukawa interaction y N φN c N. The other one is the chiral condensation, which generates a linear term as C h h in the Higgs (h) potential. Then, the mixing between the Higgs and inflaton induces a linear term in the inflaton potential. In both cases, a linear term, Cφ, in the inflaton potential can be induced from a fermion condensate. The potential (1) is changed to V ′ and ǫ are also modified to V ′′ is unchanged, but N * is modified as Thus, the relation between n s and N * changes from the original CW inflation. H inf is approximated as H inf ≃ V 0 /3M 2 pl in the model. N max could be an interesting quantity from the viewpoint of the relaxion scenario as stated in the Introduction. N max is defined as

Non-minimal coupling to gravity
Let us now discuss another possible realization of a viable small-scale CW inflation, where we introduce a non-minimal coupling of the inflaton to gravity. In Ref. [49], a non-minimal coupling to gravity of L ξ = −ξφ 2 R/2 with R being the Ricci scalar andφ being the Jordan frame inflaton field, has been discussed, but it was concluded that this term cannot make the original CW inflation viable. 2 Instead of utilizing the quadratic coupling to gravity, we introduce a logarithmic term of non-minimal coupling to gravity. Such a form of the non-minimal coupling may be obtained by incorporating quantum corrections to theφ 2 R term [54] 3 , and here we parametrize it as in the Jordan frame. In this case, the potential (1) is changed to in the Einstein frame where Ω 2 ≡ 1 − ξMφ M 2 pl ln(φ/M − c), and φ is a canonically normalized inflaton field in this frame. Although one can safely approximate Ω 2 ≃ 1 and dφ/dφ ≃ 1 with V (φ) ≃ V E (φ(φ)) ≃ V 0 in a small-field region, the additional term induced from the logarithmic form of non-minimal coupling to gravity cannot be negligible for V ′ E and V ′′ E in a certain parameter space. In particular, the model with a larger c gives similar predictions from the CW model with a linear term discussed in the previous subsection.
Taking derivatives with respect to φ, we have where V E ≃ V 0 is utilized in the small-field region. N * becomes whereas N max is also changed to The resultant n s in the model with a logarithmic form of non-minimal coupling to gravity is shown in Fig. 3. Left and right panels correspond to ξ = 10 −16 and 10 −8 , respectively. Lines and regions in the figures represent the same meaning as those in Fig. 1. Similar to the case of the CW model with a linear term, we definẽ so that this parametrization leads to the −Cφ term in Eq. (14). One can approximate Eq. (14) as with the dimension-full parameter C defined in Eq. (22). The condition | ln(φ/M)| ≪ c corresponds to

Conclusions
The CW potential for an inflaton realizes a small-field inflation but the current bound on n s from cosmological observation rules out the simplest small-field CW inflation model with smaller M. Thus, some modifications are necessary for such models to be consistent with cosmological observations. An introduction of a linear term in the inflaton potential, which can be induced from fermion condensate, has been proposed to make the model realistic. In this work, first we have revisited this model with a linear term. In particular, we have investigated the inflation scale, that is the In summary, the logarithmic non-minimal coupling can help make the small-scale CW inflation viable by increasing n s . The non-minimal coupling can also realize a small inflation scale. In addition, motivated by the relaxion scenario, we have estimated the maximal number of e-folds, N max , which turns out to be O(100) and cannot be so enormous as required in relaxion models.
The summary of possible additional terms to make the original CW model realistic is given in Tab. 1.