Inflationary cosmology and the standard model Higgs with a small Hubble induced mass

We study the dynamics of the standard model Higgs field in the inflationary cosmology. Since metastability of our vacuum is indicated by the current experimental data of the Higgs boson and top quark, inflation models with a large Hubble parameter may have a problem: In such models, the Higgs field rolls down towards the unwanted true vacuum due to the large fluctuation in the inflationary background. However, this problem can be relaxed by supposing an additional mass term for the Higgs field generated during and after inflation. We point out that it does not have to be larger than the Hubble parameter if the number of $e$-folds during inflation is not too large. We demonstrate that a high reheating temperature is favored in such a relatively small mass case and it can be checked by future gravitational wave observations. Such an induced mass can be generated by, {\it e.g.,} a direct coupling to the inflaton field or nonminimal coupling to gravity.


I. INTRODUCTION
Recent results at the Large Hadron Collider (LHC) [1,2] are in excellent agreement with the Standard Model (SM) with a 125 GeV Higgs boson and thus far any significant deviation from the SM has not been reported. On the other hand, the current measurements of the Higgs and top quark masses [4] suggest the metastability of our vacuum [5][6][7]; the Higgs potential becomes negative typically at h 10 11 GeV [7]. It may be an important hint for high-energy physics.
One of the important ingredients in modern cosmology is inflation. It expands the primordial Universe at an accelerating rate. It solves the flatness and horizon problems and sows the seeds of the large scale structure of the present Universe. Within the current errors, there remains still a possibility that the SM Higgs drives inflation [3]. However, if the Higgs potential is negative at h 10 11 GeV, it is not possible unless there is a physics beyond the SM that makes the Higgs potential positive because the Higgs field value during inflation is required to be larger than 10 16−17 GeV in such Higgs inflation models. In this paper, we assume that the electroweak vacuum is metastable and inflation is driven by a scalar field other than the SM Higgs field, called inflaton.
The current data suggests that the lifetime of the electroweak vacuum is longer than the age of the Universe [8] and there is no constraint on the reheating temperature from the thermal-fluctuation-triggered electroweak vacuum decay [5,6,9]. However, the vacuum fluctuation in the quasi-de Sitter background of the Higgs field during inflation may also push it to the unwanted Anti de Sitter (AdS) vacuum if the Hubble parameter during inflation is large, e.g., as the recent BICEP2 result suggests [10]. Thus, it may spoil inflation or, at least, our Universe that sits in the metastable vacuum may be unlikely 1 . Therefore, low-energy scale inflation may be favoured in this viewpoint, contrary to the BICEP2 result [10], as discussed in other recent literatures [13].
As pointed out in Refs. [5,12,14], it can be avoided by supposing a coupling between inflaton and the SM Higgs field without any major effects on the dynamics of inflaton. This is because the coupling produces the "Hubble induced mass" during inflation, which makes the field value where the Higgs potential goes negative much larger. In the case where the induced mass is much larger than the Hubble parameter [14] and the Higgs potential remains positive up to the Planck scale, the Higgs field is quickly pushed to the origin and its fluctuation is suppressed. Thus, the unwanted vacuum decay can be avoided even if the initial field value of the Higgs field is relatively large, ∼ 0.1M Pl with M Pl being the reduced Planck mass. Consequently, the electroweak vacuum can be naturally selected.
On the other hand, if the induced mass is smaller than the Hubble parameter, it seems to be difficult to suppress the vacuum fluctuation and hence the vacuum decay cannot be avoided even if the Higgs field initially sets at the origin. In this paper, however, we point out that we can construct a scenario with a high-scale inflation in which most part of the Universe can avoid the vacuum decay while the induced mass is not so large, as is also recently suggested in Ref. [12]. This is because the expectation value of the Higgs field during inflation is smaller than the field value of the potential barrier if the Hubble induced In addition, if the reheating temperature is high enough, the present Universe can be safely realized. Note that after inflation the Higgs field still slow-rolls and the time-dependent potential barrier may catch up it. This makes the Higgs field fall into the unwanted AdS vacuum. If the Higgs field is thermalized before being caught up by the potential barrier, the Higgs field safely settles down to the electroweak vacuum. A relatively high reheating temperature makes an earlier thermalisation possible.
Here we give a rough estimate for such a healthy scenario. We also point out that it would be possible to verify such a high reheating temperature by the future gravitational wave experiments.

DURING INFLATION
Let us start from the SM Higgs potential. At the large field values h v ≡ 246 GeV, it is well described by in the unitary gauge. The Higgs quartic coupling λ(h) runs logarithmically with respect to h from λ(M h ) 0.13 where M h is the Higgs mass. As is studied in Ref. [5][6][7]15], the Higgs quartic coupling becomes negative typically at h ∼ 10 11 GeV. Though the uncertainties of the Higgs and top mass data lead to the uncertainty in the point where the potential goes negative from 10 9 GeV to the Planck scale or higher, here we consider the case where the Higgs potential vanishes typically at 10 11 GeV. Then, the Higgs potential has also a maximum or a "barrier" at h = Λ 0 ∼ 10 11 GeV. If the Hubble parameter during inflation 2 H inf is larger than Λ 0 , the fluctuation of the Higgs field easily climbs up the potential barrier and falls down to the unwanted true vacuum during inflation even if it initially sits at the origin [5,[11][12][13]. It is claimed in Ref. [5]  As is pointed out in Refs. [5,12] and studied in detail in Ref. [14], the Higgs field can acquire a Hubble induced mass due to its interaction with the inflaton φ. For example, the "Higgs-portal" coupling with κ > 0 gives an effective positive mass squared κ φ 2 during and after inflation. Here the brackets represent the time average. In the case of massive chaotic inflation V (φ) = Note that in order for the quantum correction not to dominate the tree level potential, κ 10 −6 is required [14]. 2 The subscript "inf" represents that the variable is evaluated at the inflationary era. 3 See also the discussion in Ref. [12]. 4 Note that the kinetic energy and potential energy are equilibrated, m 2 φ 2 /2 = φ2 /2, at the oscillating phase.
A similar effect can be achieved by a non-minimal coupling of the Higgs field to gravity.
Suppose that the Einstein-Hilbert action is replaced by where g is the determinant of the metric, ξ is a negative parameter, and R is the scalar curvature. The effect of this term can be seen easily in the Einstein frame. By performing the conformal transformation and changing the frame to the Einstein frame, we get the effective Higgs potential as During inflation we have 3H 2 M 2 Pl V (φ), and during inflaton oscillation dominated era Here we assumed that the inflaton oscillates in the quadratic potential around its potential minimum. Thus, the Higgs field acquires positive mass squared −γξH 2 during and after inflation with γ being a parameter of order of O(1 − 10).
Motivated by the interactions discussed above, now we consider a simple modification of the Higgs potential during inflation, with c inf being a positive numerical parameter. Here we consider the case c inf O(1) and study vacuum fluctuation in this potential. For H inf Λ 0 , the Hubble-induced potential overwhelms the original potential around h ∼ Λ 0 and the potential barrier moves to a higher field value. In principle, we should calculate the running of the couplings to study the dynamics of the Higgs field. However, they vary only logarithmically with respect to h and hence we can treat them as constants, a negative quartic coupling λ(h) =λ −0.01 in the first approximation. Then, we obtain the field value at the potential barrier as which is roughly ten times larger than the Hubble parameter during inflation for c inf = O(1).
The Higgs field receives quantum fluctuations during inflation and acquires nonvanishing expectation value. If c inf is not too small, we can neglect the quartic term in the potential for the Higgs field whose initial field value is the origin and the expectation value of the Higgs field is evaluated as [16] where N is the number of e-folds. It is still under discussion what is the correct survival condition, and hence here we require h 2 < Λ 2 h as its representative. Then, we acquire the constraint on the Hubble-induced mass as regardless of the Hubble parameter during inflation. Here we have approximated 1 − Strictly speaking, we must calculate the probability distribution function (PDF) by using the stochastic approach or the Fokker-Planck approach [17,18] to estimate the survival probability, which cannot be solved analytically. We have instead performed numerical calculation to solve the Langevin equations [19] and find that the resultant 1-point PDF is if the above condition (Eq. (7)) is satisfied. Therefore, we conclude that for the parameter that satisfies Eq. (7), the probability for the Higgs field to sit inside the potential barrier during inflation is not suppressed exponentially and it gives a good condition for the survival of the electroweak vacuum.

III. DYNAMICS OF THE HIGGS FIELD AFTER INFLATION
In the previous section, we give a condition in which the Higgs field does not fall into the unwanted true vacuum during inflation in many regions in space in the presence of the Hubble-induced mass. Is this condition a sufficient condition for us to live in the electroweak vacuum likely? The answer is no. Since the expectation value of the Higgs field just after inflation can be larger than the zero-temperature barrier Λ 0 , we must consider the condition for the Higgs field to settle down to the electroweak vacuum through the dynamics after inflation.
Let us consider a case where the Higgs field still receives a positive Hubble-induced mass during inflaton oscillation dominated phase 6 , where c osc O(1) is a numerical parameter. It would have a relation to c inf but it is modeldependent, and hence we treat it as a different parameter. Note that the mechanism that induces the Hubble induced mass discussed in the previous section does not exactly give the effective interaction described by Eq. (9), since it induces an oscillating term coming from the inflaton oscillation, say, φ(t) = √ 2φ(t) cos mt whereφ is the slowly decreasing function and m is the inflaton mass around its potential minimum. However, if the time scale of inflaton oscillation is much smaller than that of the Higgs field dynamics, the dynamics of the Higgs field and inflaton is decoupled and it is valid to take the time average of the oscillating part of inflaton. As a result, Eq. (9) gives the sufficiently well-approximated solution. In our present case, the time scale of the Higgs field dynamics is given by (c This is the case when inflaton oscillates in the quadratic potential after inflation. During this phase before the complete reheating, partial decay of inflaton produces relativistic particles as a subdominant component of the Universe. If their scattering cross section is large enough, they are thermalized with a temperature [20] T (t) = 72 5π 2 g * (T ) where g * is the effective number of relativistic degrees of freedom, and T R is the reheating temperature. We here assume that at least the fields that do not have direct couplings to the Higgs such as gluons are thermalized just after inflation.
If the reheating temperature is not high enough, the Higgs field is not thermalized just after the end of the inflation since the fields that coupled to the Higgs field acquire large mass which prevents them from thermalization, and hence Higgs field itself also cannot be thermalized. The thermalization of the Higgs would take place when the particles that couples to Higgs field becomes light enough, h(t) < T (t), and the interaction rate is rapid enough, Γ ∼ T (t) > H(t). Thus, in the case when the following conditions and are satisfied, the Higgs field is thermalized just after the end of inflation. Otherwise, it takes some time for the Higgs field to be thermalized. (Or it is never thermalized as we will see.) One may wonder if the Higgs field is pushed to the unwanted AdS vacuum at the time of thernalization. It would be avoided if the Higgs field value is sufficiently small compared to the potential barrier Λ th T (t inf )/ √ −λ generated by the thermal potential V th T 2 h 2 .
From Eq. (12), we can easily see that if T R > T 1 R , the Higgs field value just after inflation is roughly ten times smaller 7 than the potential barrier, which would be sufficiently small to avoid the disaster. Here we take h 2 1/2 inf as a reference value of the Higgs field just after inflation and do not write the coupling constants of the order of the unity, such as top Yukawa coupling, explicitly.
Let us study the dynamics of the Higgs field before thermalization. The Higgs field evolves according to the potential This potential has a time-varying maximum at for Λ t > Λ 0 where λ −O(10 −2 ) is negative, Thus, for the healthy realization of the present Universe, h(t) < Λ t must be satisfied in the course of the evolution of the Higgs field 7 Note that √ −λ 10 −1 .
in substantial part of the Universe. Otherwise the Higgs field falls into the unwanted AdS vacuum, which may cause a cosmological disaster.
Now we evaluate the Higgs field dynamics taking h 2 1/2 inf as the initial condition. If the Higgs field does not fall into the unwanted AdS vacuum from this initial condition until its thermalization, the Higgs field successfully settles down to the electroweak vacuum in the many region of the Universe. In principle, it would be better to perform a lattice simulation taking into account the spatial distribution of the Higgs fields. However, the spatial derivative of the Higgs field is suppressed due to inflation, and hence we here only consider the homogeneously distributed Higgs field. Then, the equation of motion (EOM) is given byḧ As long as h(t) < Λ t is satisfied, we can neglect the last term in the EOM and get a solution, Here we consider the case where c osc < 9/16 and the Higgs field does not oscillate. Since the Higgs field value decreases slower than the potential barrier, Λ t , we must seek for the way to avoid for the Higgs field to be caught up by the potential barrier after inflation for the successful Universe, otherwise it falls into the unwanted AdS vacuum. This catching up would happen when Here we assumed that the approximations Eqs. (15) and (17) hold until that time.
The first way to avoid the falling down to the unwanted AdS vacuum is that the Higgs field gets thermalized before being caught up. Let us take the criteria for the Higgs field thermalization as T (t) > h(t) and T (t) > H(t). Then, we have two cases for the successful Universe when the Higgs field is not thermalized just after inflation 8 ; 1. The case with T R > T 1 R and T R < T 2 R : H(t) = T (t) gets satisfied at a later time. For the successful Universe, both the conditions T (t) > h(t) and Λ t > αh(t) should be satisfied at H(t) = T (t). 8 In the case with T R < T 1 R and T R > T 2 R , h(t) < T (t) will never be satisfied because T (t) decreases much more rapidly than h(t) in this parameter region.
2. The case with T R < T 1 R and T R < T 2 R : For the successful Universe, the conditions T (t) > h(t), H(t) should be simultaneously satisfied before it gets Λ t < αh(t). Note that it must be also satisfied before reheating because there are no longer "Hubbleinduced mass" after inflaton decay.
One may wonder, again, if at the time when the Higgs field gets thermalized, thermal fluctuations may push the Higgs field to the unwanted AdS vacuum. At present we do not know how to calculate exactly the tunneling rate of a slow-rolling scalar field during the epoch when the system gets thermalized, unlike the case discussed in Ref. [9] where the Higgs field is at the potential minimum and the system is well-approximated to be in zero-temperature or fully thermalized. However, we may be allowed to guess it will be exponentially suppressed by using triangle approximation [21] if the scalar field value is far away enough from the potential barrier and it is high enough. In this reason, we introduced a numerical parameter α O(1) in order to take into account it, though a careful study would be required to determine its value exactly, strictly speaking, but it is beyond the scope of this paper.

Case 1: H(t) = T (t) is satisfied when
Then, the conditions T (H T 1 ) > h(H T 1 ) and Λ t (H T 1 ) > αh(H T 1 ) are rewritten in terms of the constraint on the reheating temperature as Case 2: In this case, for c osc > 27/64 9 . Then, we find that in the parameters we are interested in the falling down problem is avoided if the Hubble parameter becomes H T 2 before the catching up time and 9 In the case c osc < 27/64, h(t) will never catch up T (t).
reheating. In other words, the present Universe will be realized if both the conditions are satisfied, and are satisfied.
In summary, if one of the following conditions, are satisfied, the Higgs fields are thermalized before being caught up by the potential barrier and the present electroweak vacuum would be successfully selected.
The second way for the successful cosmic history is that the Higgs field value h(t) becomes smaller than the zero-temperature barrier Λ 0 and gradually its dynamics is dominated by λh 4 /4 term before being caught up by the potential barrier. The Higgs expectation value h(t) gets smaller than Λ 0 when Thus, if H Λ > H c , the present electroweak vacuum is successfully selected. This condition gives a constraint on the Hubble parameter during inflation as As an example, we show the allowed region in the T R -c inf plane in Fig. 1 for H inf = 10 12 , 10 13 and 10 14 GeV, respectively, with the parameters being chosen as Λ 0 = 10 11 GeV, and c osc = c inf /2. We also chose the value of α as α = 1.0, 10.0. We find the constraint in analytical expression for this case as We approximate the running of the Higgs quartic coupling near µ 10 11 GeV as [7] λ(µ) = −1.4 × 10 −3 ln µ 10 11 GeV , and evaluate it at µ = h(H T ) for T R = 10 10.5 , 10 9 , and 10 7.5 GeV for H inf = 10 14 , 10 13 , and 10 12 GeV, respectively, when we give the lower bounds on T R . The red coloured region is disfavoured due to the condition h 2 inf > Λ 2 h , which means the survival probability during inflation is exponentially suppressed. The blue coloured regions are excluded due to the condition for the Higgs field not to fall into the unwanted true vacuum. Note that there are no constraint for c inf > 10 −0.17 (10 0.04 ) for H inf = 10 12(13) GeV, in which the condition h(t) < Λ 0 is always satisfied before the falling down to the unwanted true vacuum. The running of the quartic coupling λ is not calculated strictly, but it does not change the result so much. We can see that the lower bound of the reheating temperature becomes severer as the Hubble parameter during inflation is larger. Increasing the parameter α makes the lower bound slightly higher, but it does not change it significantly. For the Hubble parameter H inf 10 14 GeV, which is suggested by the recent BICEP2 result, a relatively high reheating temperature, T R 10 12−13 GeV is required. This indicates that if the B-mode in the CMB polarization observation with r 0.2 is confirmed, the stochastic GW background must be detected in the gravitational detectors [22] such as DECIGO [23] or BBO [24] due to the relatively large reheating temperature. If not, it suggests that there is some physics beyond the SM to stabilize the Higgs potential [25] or the Hubble induced mass for the Higgs field during inflation is much larger than the Hubble parameter as is the case studied in Ref. [14]. Note that we here do not take into account the spatial derivative in this study. This would decrease the lower bound of the reheating temperature since it makes the Higgs field damp more rapidly. It would be effective for larger c inf . For smaller c inf , however, our result suggests that the Higgs field must be thermalized just after inflation, before the spatial derivative would get effective, and hence the constraint will not change significantly.

IV. SUMMARY
In this article, we studied the evolution of the SM Higgs field in the inflationary cosmology in the light of recent collider experiments, which suggests the metastability of the electroweak vacuum. If the electroweak vacuum is metastable, high-scale inflation may be problematic since the Higgs field falls into the unwanted AdS vacuum and the probability for the Higgs field remains in the electroweak vacuum is exponentially suppressed, though it is still under discussion if it is a real catastrophe for our Universe or not. We found that the Hubble-induced mass can avoid the exponentially suppressed survival probability of the electroweak vacuum during inflation while it is not necessarily larger than the Hubble parameter during inflation. We also found that the present Universe can be successfully realized even in the case of the relatively small Hubble-induced mass if the reheating temperature is high enough. This is because the Higgs field is thermalised before being caught up by the time-dependent potential barrier and the fall into the unwanted AdS vacuum. As a result, relatively high-energy scale inflation is allowed, and hence we can expect for the detection of GW background in the future experiments without any physics beyond the SM that is introduced to stabilize the Higgs potential. We also pointed out that the direct GW background detection will give us the clue to study the physics beyond the SM.

Acknowledgments
The author is grateful to M. Asano, O. Lebedev, and A. Westphal for collaboration at the early stage of this project. The author also thanks M. Shaposhnikov for useful comments.
This work has been supported in part by the JSPS Postdoctoral Fellowships for Research Abroad.
Appendix A: The validity of the approximation for the Hubble induced mass during inflaton oscillation dominated era In our analytic calculation, we integrate out the inflaton dynamics and treat the effect as the "Hubble induced mass" term in the Higgs potential during inflaton oscillation dominated era. Here, we calculate the time evolution of the Higgs field h numerically without integrating out of the inflaton field φ in order to demonstrate the validity of our approximation.
In this appendix, we consider the massive chaotic inflation model with a h 2 φ 2 interaction term as a simple example which induces the Hubble-induced mass, with m = 10 13 GeV. For simplicity, we assume an approximate formula, to estimate the scale dependence of the quartic term of the Higgs potential. From the potential in Eq.(A1), the basic equations are given bÿ where we neglect the spatial derivatives, ∇φ, ∇h.

Approximate calculation
First, we estimate the time evolution of the Higgs field with an approximation in which we assume the Higgs field dynamics does not affect on the inflaton dynamics and cosmic expansion. In this case, the inflaton oscillation and the Hubble parameter after inflation are given as respectively. By averaging the inflaton oscillation in time, the "effective mass" of the Higgs field can be obtained from the (κ/2)φ 2 h 2 coupling as, where c osc = 3κ(M Pl /m) 2 . The averaging the inflaton oscillation can be justified when the time scale of the inflaton field evolution is much shorter than the Higgs field evolution.
Neglecting the quartic term, the dynamics of the Higgs field after inflation is then described by the following equation of motion, Unspecifying the initial condition for the Higgs field, the solution of this equation of motion is given by, for c osc < 9/16 which we are now interested in. The quartic coupling can be neglected only where we neglect the h dependence of λ. In other words, if h(t) becomes less than that value, the above approximation no longer valid and it falls quickly into the unwanted true The Higgs field starts slow-rolling at mt = 1 and goes to the true vacuum, finally, when the negative quartic term gets effective. The time when the Higgs field falls into the true vacuum is a little later than the approximated result (Eq. (A12)), but this is because it takes time for the Higgs field really to fall down to the unwanted vacuum after it starts to feel the negative quartic term. Thus, our approximate calculation can be useful to estimate the time evolution of the Higgs field approximately as shown in Fig.2.