Unification of inflation and dark matter in the Higgs–Starobinsky model
Abstract
In this work, we propose unified picture of inflation and dark matter in the Higgs–Starobinsky (HS) model. As pointed out in the literature, Starobinsky \(R^2\) inflation is induced by quantum correction effect from the large Higgscurvature (graviton) coupling. We start with nonminimal coupling HS action in Jordan frame. We then transform the Jordan frame action into the Einstein one using the conformal transformation. The inflation potential is derived from the gravitational action of nonminimalHiggs coupling and Starobinsky term in Einstein frame where the \(R^2\) term is dominated in the inflationary phase of the universe. For model of inflation, we compute the inflationary parameters and confront them with Planck 2015 data. We discover that the predictions of the model are in excellent agreement with the Planck analysis. In addition, we find that the HS model is equivalent to a scalar singlet dark matter (SSDM) or Higgsportal model. The renormalization group equations (RGEs) of HS scenario with standard model at oneloop level is qualitatively analyzed. By using the solutions of parameter spaces from RGE analysis, the coupling constants of the HS model will be verified and can be used to constrain the SSDM using dark matter relic abundance.
1 Introduction
An inflationary scenario is a wellestablished paradigm describing an early universe and posts an indispensable ingredient of modern cosmology. Regarding degrees of freedom contained in the SM of particle physics or quantum general relativity, Higgs inflation [1] and Starobinsky model [2, 3] have received much attraction in the recent years. The preceding scenario technically requires a large nonminimal coupling \((\xi )\) between a Higgs boson H and the Ricci scalar R, i.e. \(\xi H^{\dagger }H R\) which leads to successful inflation and produces the spectrum of primordial fluctuations in good agreement with the observational data. The formulation of the later model is based on \(R^{2}\) gravity. It is worth noting that these two models are minimalistic and nicely compatible with the latest Planck data [4]. As pointed out in Ref. [5], both operators \(\xi H^{\dagger }H R\) and \(R^{2}\) are expected to be generated when the SM of particle physics is coupled to general relativity. More importantly, due to a large nonminimal coupling of the Higgs boson and the Ricci scalar, Starobinky inflation can be generated by quantum effects [5, 6]. In this situation, the Higgs boson need not to start at a high field value at inflation. In addition, in the HS model, a Higgs potential can be stabilized. Notice that the simplest modification of the EinsteinHilbert action, \(R^{2}\)gravity was able to explain the dark matter [7].

(i) There is no need of physics beyond the SM of particle physics since the operators here are expected to be generated when general relativity is coupled to the SM of particle physics.

(ii) We propose a cosmological scenario that unifies comic inflation and dark matter to a single framework. The model is minimalistic.

(ii) Regarding this twofield scenario, the model dose not suffer from the unitarity problem as that of the Higgs inflation.
2 Model setup
2.1 Inducing Starobinsky, \(R^2\) term by nonminimal Higgs coupling
2.2 Quantum corrections and renomalization group equations
In this subsection, we will briefly review and discuss the nonminimal Higgs coupling induced the Starobinsky \(R^2\) term. After that we will close this subsection with the renormalization group equations at oneloop of the standard model parameters in the presence of curved spacetime [21, 22]. The nonminimal Higgs coupling induced \(R^2\) has been shown in Refs. [5, 6]. In this work, we follow the HS mechanism and briefly review that how the \(R^2\) term is induced as shown in [5, 23].
2.2.1 Pure gravitational terms and Higgs field
2.2.2 Oneloop renormalization group equations for standard model
The Starobinki inflation generated from nonminimal Higgs coupling term has been discussed and demonstrated in the previous subsection. More completely, we will extend our study to the renormalization group equations for the standard model of particle physics in the presence of the curved spacetime. Results in this section discussing below will be very useful for the study of dark matter in the HS model.
3 Inflationary implication from the HS model
3.1 Slowroll approximation
3.2 Contact with observational constraints
4 Dark matter from the HS model
In this section, we solve the renormalization group equations for demonstrating residual effect of the inflation to dark matter in the HS model. In addition, we will demonstrate how the HS is equivalent to the SSDM. The results of the SSDM from thermal relic abundance constraints are discussed in terms of the couplings of the HS framework.
4.1 The HS model as the SSDM
In general, one may expect that scalaron and Higgs boson are unstable and decay quickly. This implies that the scalaron may not work as dark matter candidate for our proposal in this model. However, in any case, there are other possibilities to overcome this problem. For instance, Ref. [7], here the author has introduced new mode of the scalaron from higher order gravity (i.e., \(R^2\) term) and the new scalar mode oscillates around the minimum of its potential. Then, the oscillations might correspond to nonzeromomentum condensate and be able to associate with standard nonrelativistic matter. This process is a socalled “vacuum misalignment” mechanism which mimics a famous mechanism of the axion production as a cold DM candidate [62, 63, 64]. For the details about the mechanism of the scalaron stability and its main decay channels and how the scalaron can work as DM we refer to Ref. [7]. In addition, similar approaches for the scalaron of the higher order gravity as DM candidate have been investigated in the literature [65, 66, 67].
4.2 Relic abundance
5 Conclusion
In this work, we presented a unified description of inflation and dark matter in the context of the HS model. The salient feature of this work is to demonstrate that the HS scenario can simultaneously describe inflation and dark matter without introducing new physics beyond standard model. We considered the original action describing the HS model. We started with nonminimal coupling HS action in Jordan frame and transformed it to the Einstein frame using the conformal transformation. We also derived the inflation potential from the gravitational action of nonminimalHiggs coupling and Starobinsky term in Einstein frame where the \(R^2\) term is dominated in the inflationary phase of the universe. For model of inflation, we computed the inflationary parameters and confronted them with Planck 2015 data. We discovered that the predictions of the model are in excellent agreement with the Planck analysis.
In addition, we considered the HS model as a candidate for dark matter. We analyzed the renormalization group equations (RGEs) of HS scenario with the standard model at oneloop level. More importantly, we discovered that the HS model is equivalent to the SSDM. We made qualitative discussions to identify the coupling constants from dark matter relic abundance constraints. Employing results of the SSDM with GAMBIT collaboration and composite NJL model, the DM mass in the HS model flavors the light mass around \(M_{DM} = m_\sigma /2\) in order to reproduce the reasonable values of the Higgs selfinteraction coupling.
However, there are some limitations in the present work  for example, one should complete the RGEs for all scales and solve them numerically. Moreover, regarding this single framework, another crucial issue for successful models of inflation is the (pre)reheating mechanism. We plan to investigate this mechanism, within our framework, by following a composite inflationary scenario [61]. As mentioned in the previous section, the stable properties of the scalaron DM are worth investigating. As a result, the cosmological consequences of the present work can be further considered. These also include, e.g., preheating and reheating processes. In addition, cosmic history of chameleon mechanism of the DM is also interesting to be investigated, see for example [65]. We hope to address these issues with future investigations.
Notes
Acknowledgements
We are graceful to the anonymous referees for valuable comments and intuitive suggestions that helped us improve our manuscript. DS is supported by Rachadapisek Sompote Fund for Postdoctoral Fellewship, Chulalongkorn University and Thailand Research Fund (TRF) under contract No. TRG6180014. PC is financially supported by the Institute for the Promotion of Teaching Science and Technology (IPST) under the project of the “Research Fund for DPST Graduate with First Placement”, under Grant no. 033/2557.
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