# Higgs Starobinsky inflation

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## Abstract

In this paper we point out that Starobinsky inflation could be induced by quantum effects due to a large non-minimal coupling of the Higgs boson to the Ricci scalar. The Higgs Starobinsky mechanism provides a solution to issues attached to large Higgs field values in the early universe which in a metastable universe would not be a viable option. We verify explicitly that these large quantum corrections do not destabilize Starobinsky’s potential.

### Keywords

Higgs Boson Early Universe Ricci Scalar Effective Field Theory Renormalization Group EquationThe idea that inflation may be due to degrees of freedom already present in the standard model of particle physics or quantum general relativity is extremely attractive and has received much attention in the recent years. In particular two models stand out by their simplicity and elegance. Higgs inflation [1, 2, 3] with a large non-minimal coupling of the Higgs boson *H* to the Ricci scalar (\(\xi H^\dagger H R\)) and Starobinsky’s inflation model [4] based on \(R^2\) gravity are both minimalistic and perfectly compatible with the latest Planck data.

These two models should not be considered as physics beyond the standard model but rather both operators \(\xi H^\dagger H R\) and \(R^2\) are expected to be generated when general relativity is coupled to the standard model of particle physics. We will come back to that point shortly. The aim of this paper is to point out an intriguing distinct possibility, namely that Starobinsky inflation is generated by quantum effects due to a large non-minimal coupling of the Higgs boson to the Ricci scalar. In that framework, we do not need to posit that the Higgs boson starts at a high field value in the early universe which would alleviate constraints coming from the requirement of having a stable Higgs potential even for large Higgs field values [5, 6, 7].

*R*stands for the Ricci scalar, \(R^{\mu \nu }\) for the Ricci tensor, \(E=R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }- 4 R_{\mu \nu }R^{\mu \nu }+R^2\), \(C^2=E+2R_{\mu \nu }R^{\mu \nu } -2/3 R^2\), the dimensionless \(\xi \) is the non-minimal coupling of the Higgs boson

*H*to the Ricci scalar, the coefficients \(c_i\) are dimensionless free parameters, the cosmological constant \(\Lambda _C\) is of order of \(10^{-3}\) eV, the Higgs boson vacuum expectation value, \(v=246\) GeV contributes to the value of the Planck scale,

Besides describing all of particle physics and late time cosmology, the action given in Eq. (1) can also describe inflation if some of its parameters take specific values and if some of its fields fulfil specific initial conditions in the early universe. This action, depending on the initial conditions can describe either Higgs inflation if \(\xi \sim 10^4\) and the Higgs field is chosen to take large values in the early universe or Starobinsky inflation if \(c_1 \sim 10^9\) and the corresponding scalar extra degree of freedom, which can be made more visible by going to the Einstein frame, takes large values in the early universe.

These two models are very attractive because they do not necessitate physics beyond the standard model. Furthermore, they are compatible with current cosmological observations which favor small tensor perturbations that so far have not been observed. It has actually been pointed out that both models are phenomenologically very similar [16, 17]. However, while Starobinsky’s inflation model does not suffer from any obvious problem, it has recently been pointed out that in the case of Higgs inflation, which necessitate the Higgs field to take very large field values, our universe will not end up in the standard model Higgs vacuum if it is metastable as suggested by the latest measurement of the top quark mass, but rather in the real vacuum of the theory which does not correspond to the world we observe. In this paper we point out that there is an alternative possibility. Namely when quantum corrections are taken into account, a large non-minimal coupling of the Higgs boson can generate Starobinsky inflation by generating a large coefficient for the coefficient of \(R^2\) in the early universe. While the model corresponds to Starobinsky’s model, the Higgs boson plays a fundamental role as it triggers inflation by generating a large coefficient for \(R^2\).

*g*the SU(2) gauge coupling, and \(g^\prime \) the U(1) gauge coupling. Quantum gravitational corrections will be suppressed by powers of the Planck mass and can thus be safely ignored as long as we are at energies below the Planck mass.

Note that the coefficients of *E* and of \(C^2\) do not depend on the non-minimal coupling of the Higgs boson to the Ricci scalar. Furthermore in 4 dimensions, *E* does not contribute to the equations of motion. The coefficient of the term \(C^2\) is assumed before renormalization to be of the same order as that of \(R^2\), i.e. of order 1. However, after renormalization the coefficient of \(R^2\) is tuned to be very large and of the order of \(10^9\) while the coefficient of \(C^2\) remains small compared to the renormalized coefficient of \(R^2\). \(C^2\) is thus negligible.

*F*(

*R*) gravity with \(F(R)=R + \alpha R^2 + \beta R \log \frac{-\Box }{\mu ^2} R\). There is a well established procedure to map such models from the Jordan frame to the Einstein frame; see e.g. [19]. The potential for the inflaton in the Einstein frame is given by

*F*(

*R*) term of the potential (13) is also suppressed by \(\frac{\beta }{2 \alpha }\) compared to the usual Starobinsky’s potential.

We conclude that the large quantum corrections induced by the large Higgs boson non-minimal coupling do not affect the flatness of Starobinsky’s potential. Let us add a few remarks. The model discussed above is not a new model. Physics (including reheating or preheating and all of particle physics) is identical to that predict in Starobinsky’s model. We merely identify a new connection between the Higgs boson and inflation. As in the case of the standard Starobinsky model, a coupling \(\phi ^2 h^2\) will be generated. It is, however, suppressed by factors of \(m_{\mathrm{Higgs}}^2/M_P^2\), which is a small number, particle physics will thus not be affected and the Higgs boson behaves as the standard model Higgs boson. Furthermore, the Higgs field does not take large values in the early universe, we can thus safely ignore the term \(H^\dagger H R\) when studying the inflationary potential. Note that there are subtleties when considering the equivalence of quantum corrections in different parameterizations/representations of the theory (i.e. when going from the Jordan frame to the Einstein frame). Here we are avoiding this problem: we renormalized the theory in the Jordan frame where the model is defined and then map the effective action to the Einstein frame. When proceeding this way, there are no ambiguities or risks to mix up the orders in perturbation theory and the expansion in the conformal factor (see e.g. [21, 22, 23]).

In this paper, we have identified a new connection between the Higgs boson and inflation. In the model envisaged here, the Higgs boson is not the inflaton but it generates inflation by creating a large Wilson coefficient for the \(R^2\) operator and it is thus at the origin of Starobinsky’s inflation. This mechanism is interesting as it does not require physics beyond the standard model. The Higgs boson does not need to take large field values in the early universe and we could thus be living in a metastable potential.

## Notes

### Acknowledgments

This work is supported in part by the Science and Technology Facilities Council (Grant Number ST/L000504/1) and by the National Council for Scientific and Technological Development (CNPq—Brazil).

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