# The Higgs mass range from Standard Model false vacuum Inflation

in scalar-tensor gravity

###### Abstract

If the Standard Model is valid up to very high energies it is known that the Higgs potential can develop a local minimum at field values around GeV, for a narrow band of values of the top quark and Higgs masses. We show that in a scalar-tensor theory of gravity such Higgs false vacuum can give rise to viable inflation if the potential barrier is very shallow, allowing for tunneling and relaxation into the electroweak scale true vacuum. The amplitude of cosmological density perturbations from inflation is directly linked to the value of the Higgs potential at the false minimum. Requiring the top quark mass, the amplitude and spectral index of density perturbations to be compatible with observations, selects a narrow range of values for the Higgs mass, GeV, where the error is mostly due to the theoretical uncertainty of the 2-loop RGE. This prediction could be soon tested at the Large Hadron Collider. Our inflationary scenario could also be further checked by better constraining the spectral index and the tensor-to-scalar ratio.

## I Introduction

Despite the many experimental successes of the Standard Model (SM) of particle physics, its scalar sector is still to be confirmed by accelerator experiments. Extensions of the SM are intensively explored, motivated by the fact that the SM does not explain neutrino masses, dark matter, the baryon-antibaryon asymmetry, a primordial stage of inflation, etc. From a theoretical point of view, useful guides for extending the SM are represented by a possible unification of gauge couplings, by fine-tuning problems (such as the lightness of the Higgs mass) and by the inclusion of gravity.

We consider here the possibility that no major extensions of the SM Higgs sector are required. In this framework, we consider the possibility of realizing inflation by introducing only modifications in the gravitational sector.

The idea that inflation is realized in the SM Higgs sector has been proposed recently arXiv:0710.3755 , by allowing for a non-minimal coupling of the Higgs to gravity. This is an appealing point of view because it does not introduce new degrees of freedom. It is however controversial whether such scenario is viable at the quantum level, because radiative corrections could modify the potential arXiv:0809.2104 ; arXiv:0812.4946 .

In this paper we propose a slightly less minimalistic scenario, in which we do introduce a new scalar degree of freedom, but assume it to be decoupled from the SM: in such a way the SM is not altered at all and only gravitational physics is affected.

We use the fact that the Higgs potential has a local minimum between GeV, which is indeed known to exist for a narrow band of the top and Higgs mass values CERN-TH-2683 ; hep-ph/0104016 ; strumia2 . We will assume that the Universe started with the Higgs in this false vacuum, which leads to a stage of exponential inflation, where density perturbations are produced.

It is already a non-trivial fact that such a local minimum can exist in the Higgs potential, but is even more non-trivial that there exist an allowed parameter range in the Higgs and top masses, for which the energy density of this minimum has the right value (GUT scale) to give rise to the correct amplitude of density perturbations from inflation.

The additional scalar has the role of providing a graceful exit to inflation, which is possible in a Brans-Dicke scalar-tensor theory of gravity, as shown in ref. astro-ph/0511396 ; hep-ph/0511207 (and earlier in johri ; extended ; hyperextended for power-law inflation). After the stage of exponential inflation, the expansion is drastically slowed down by the Brans-Dicke scalar and the Higgs field can tunnel out efficiently, via production and collision of bubbles, and subsequently rolls down to its present vacuum expectation value, GeV. Moreover quantum fluctuations in the Brans-Dicke scalar generate a spectrum of density perturbations, which can be understood also as a tunneling which happens at slightly different times in different regions of space. In particular, we show that successful inflation can be achieved for a very broad class of couplings of the scalar-tensor theory and that the spectral index of density perturbations is largely independent on the particular choice of couplings.

We use the information from the amplitude and spectral index of cosmological density perturbations generated during inflation to predict the Higgs mass, finding

(1) |

where the error is mostly due to the theoretical uncertainty of the 2-loop RGE. This range is within the present experimental window set by direct searches, namely the GeV lower bound set by LEP PDG and the GeV upper bound set recently by LHC HCP11 , which restricted the previous GeV bound set by Tevatron arXiv:1107.5518 . We note that the range above is also compatible with the July 2011 global electroweak SM precision fit GFitter , which gave GeV at .

After the first release of this paper on the arXiv, LHC updated the Higgs boson exclusion limits last : as of 13 December 2011 ATLAS excludes at the CL masses outside GeV and CMS excludes at the CL masses outside GeV. Remarkably, the allowed band for SM false vacuum inflation includes the region GeV, where ATLAS and CMS recorded excesses of events in the di-photon as well as 4-leptons channels last . At present, the significance of this result is however still low.

Hopefully, our inflationary model prediction will be soon better tested by LHC. In the case of compatibility with LHC results, future more precise cosmological measurements of the scalar spectral index and the tensor-to-scalar ratio could provide further tests for the idea of Higgs false vacuum inflation Masina:2011un .

The paper is organized as follows. In section II we present our inflationary model, based on a scalar-tensor theory of gravity. First, we derive the expression for the slow roll parameters, the scalar spectral index and the tensor-to-scalar ratio. We then establish the relation between the amplitude of density perturbations and the Higgs potential at the false minimum. In section III we study values of the top and Higgs masses giving a SM false vacuum within the range consistent with inflation. In section IV we show that the set of values derived in this way are only in part consistent with the measurement of . This allows to predict a narrow range for the Higgs mass. In section V we provide details about the post-inflationary period, while in section VI we generalize the model by considering higher order terms. We finally draw our conclusions in section VII.

## Ii The inflationary model

The model setup is the SM of particle physics in a scalar-tensor theory of gravity. We denote the Higgs field by . For very large field values, the quadratic term can be neglected in the Higgs potential which, at some scale becomes simply

(2) |

It is well known that the SM Higgs potential has a false minimum for some narrow band of the Higgs and top masses
CERN-TH-2683 ; hep-ph/0104016 ; strumia2 , as we also discuss in sections III and IV.
For such band, the coupling goes very close to zero at large field values and then rises again,
namely the Higgs potential can develop a new local minimum, which turns out to be well compatible with the GUT scale range, GeV.
We call the Higgs field at the false minimum^{1}^{1}1There is also an intermediate regime in which the Higgs can develop a flat region of the potential,
but it has been shown that inflation is not viable strumia2 , at least in standard Einstein gravity..

We are going to use this new minimum to drive cosmic inflation, since is a large potential energy which can be a source for an exponential expansion of the Universe. The non-trivial ingredient is to provide a graceful exit from inflation, that is a transition to a radiation dominated era, in a nearly flat Universe, at a sufficiently high-temperature. This is known to be impossible in standard gravity Guth . In fact, in order to end inflation the field would have to tunnel to the other side of the potential barrier by nucleating bubbles with a different value of the field Coleman and the bubbles would eventually collide with each other and reheat the Universe. However, the nucleation rate per unit time and volume (which has mass dimension four) has to be suppressed as compared to the fourth power of the Hubble rate , otherwise the Universe would tunnel quickly in a few Hubble times, without providing sufficient inflation. On the other hand, if the probability of tunneling is very small and the process is not efficient enough to produce a sufficient number of bubbles inside a Hubble horizon, which could collide producing a homogeneous radiation thermal bath. The possibility that we lived inside one single bubble, without collision with other bubbles, is also ruled out by the fact that the inner region of a bubble would have too large spatial curvature. In other words, a graceful exit would require to become larger than only after some time, but this is impossible because both quantities are time-independent.

Interestingly, such a behavior becomes possible in a scalar-tensor theory of gravity. This has been shown in earlier models, under the name of extended johri ; extended or hyperextended hyperextended inflation and more recently in astro-ph/0511396 ; hep-ph/0511207 . In earlier models only a power-law inflation (scale factor growing with time as ) was proposed, but subsequently this has been shown to be in tension with observations of the Cosmic Microwave Background, because it turns out to be difficult to produce a nearly flat spectrum of perturbations Liddle . In hep-ph/0511207 ; astro-ph/0511396 it was shown instead that a stage of exponential expansion is naturally incorporated in the model and then followed by a stage of power-law (even decelerated) expansion. In this way, it is possible to produce the correct spectrum of perturbations in the first stage of inflation and subsequently to slow down dramatically the expansion of the Universe, thereby allowing the field trapped in the false minimum to tunnel through percolation of bubbles.

A scalar-tensor theory of gravity is obtained by adding a scalar field (sometimes called Brans-Dicke scalar or dilaton), coupled to the Ricci scalar
via a non-minimal coupling. We will follow here very closely the scenario presented in astro-ph/0511396 ; hep-ph/0511207 .
The full action of the model^{2}^{2}2A potential term for the field could be added, and in fact it is likely to be needed for the post-inflationary
evolution. We assume that this potential term is subdominant before the tunneling.
is given by

(3) |

where includes all the SM of particle physics and where we require that is always positive. Here, for small values we assume that we can simply expand the function as:

(4) |

where plays the role of the Planck mass and , are dimensionless couplings. Indeed, the usual case of standard Einstein gravity corresponds to . In general we may view these models just as gravitational theories in which the effective Planck mass – given by – is fixed by the vacuum expectation value of . For small values of one then recovers the usual Einstein gravity with a Planck mass given by . As in hep-ph/0511207 , for large field values () we require to be a monotonic growing function, such that .

We start by assuming that the primordial Universe is initially in a cold state, where takes a very small value and the Higgs field is trapped in the false minimum of its potential, at a field value . The evolution equation astro-ph/0511396 ; espositofarese of the metric in an FLRW background is given by

(5) |

where ( is the scale factor) and the evolution equation for the scalar field is:

(6) |

Since is small we can approximate (5) with the standard equation

(7) |

leading to a stage of inflation with the Universe expanding almost exponentially with a scale factor .

In this minimum the Ricci scalar has a value:

(8) |

where the second equality follows from the fact that during inflation is almost constant.

In this false minimum the Higgs field can tunnel with a Coleman instanton Coleman , a bounce solution of the classical equations of motion, but it will do it with negligible probability if astro-ph/0511396 . However in such an inflationary background there is a time-dependent quantity, which is the value of the Brans-Dicke field , which sets dynamically the value for the Planck mass. It is easy to see combining eq. (6) and eq. (8) that, if the function is a monotonic increasing function, the presence of a nonzero background value for makes the additional field grow and at some point the field will reach values large enough so that the Planck mass, given by , starts becoming larger than . When we enter this regime, gravity becomes weaker and so the Hubble parameter starts decreasing with time. If we wait for a sufficiently long time, the tunneling via a Coleman transition will happen successfully when , since is a constant. Note however that the transition could happen also by quantum fluctuations due to the gravitational background, through the Hawking-Moss instanton, which would make also time dependent. We now disregard this possibility here and comment it later in section III.

Such a process can be studied by analyzing directly the above equations of motion, derived by the action (3). It is however simpler and more general to make a change of variables, as in hep-ph/0511207 , and go in the so-called Einstein frame (we use the bar to indicate a quantity in this frame), which is related to the original frame through the change of variable obtained via the conformal transformation of the metric . The action in this frame becomes (see hep-ph/0511207 ; espositofarese for further details)

(9) |

Let us now focus on the function , which can generically be written for any value of the field as:

(10) |

As shown below, it is required hep-ph/0511207 that for large field values grows faster than . This is achieved for instance if all coefficients of the higher dimensional operators are positive numbers. In order to keep the analysis simple, it is sufficient to focus on one single operator, and the simplest one is the operator with . As already anticipated in eq.(4), from now on we therefore consider only the and terms: this allows to have a model with only two parameters, and . We will show in section VI that for any coupling with , the predictions are very similar and converge to a single prediction at large . Note that for this reason in our scenario there is a very wide class of higher-dimensional operators which work well, and that the predictions are almost independent on the precise functional form of inside this class. This is different from the case proposed by arXiv:0710.3755 where only the quadratic term in works well (making the scenario possibly unstable under quantum corrections, because of the appearance of higher dimensional operators).

In terms of a canonically normalized field defined through , the action in eq.(9), can be further simplified to:

(11) |

Because of the conformal transformation to the Einstein frame, the Higgs potential becomes , so that the potential energy at the false Higgs minimum gives rise to a potential term for

(12) |

Now, in this frame, we can discuss in a very general way the dynamics, distinguishing between two stages: small and large . At small , considering in (10) only the terms as discussed, and assuming , the Higgs potential becomes

(13) |

This acts as a hill-top potential for the field. So, in this frame rolls down the potential from small to high values. If , the field rolls down slowly and the standard slow-roll approximation can be used.

For large field values, instead, it can be seen that, under the assumption that grows faster than quadratic, we have that . In this case in the numerator of in eq. (9), the second term dominates. Therefore, in this phase, the kinetic term can be approximated as

(14) |

We can now write the canonical variable in this regime simply as

(15) |

so that the potential for , eq.(12), becomes remarkably independent on the exact form of :

(16) |

Evolutions under such exponential potential corresponds to a power law phase, with decelerated expansion, given by It is also easy to see that grows and the kinetic energy is always proportional to (precisely it is ).

Now, the end of this phase is achieved when is equal to and at this point the Higgs field tunnels efficiently. Therefore the final field value at tunneling is given by:

(17) |

where the last equality has been derived using the fact that the dimensionful parameter rescales between the two frames as . In principle is calculable knowing the SM potential exactly, requiring an accurate numerical solution of the bounce equation Coleman . However, the quantity is exponentially sensitive to the SM parameters, and so we cannot compute it with the present known experimental errors. For this reason we will treat it as a free parameter, which leads to the conclusion that also is a free parameter. However, for practical purposes we only need to know that the transition is possible, leading to a radiation-dominated Universe. Knowing when the transition happens can change only the number of e-folds which correspond to our horizon scale today. For any practical purpose we leave this as a free parameter, as we discuss in section V, and focus now on the observational consequences of the slow-roll stage.

Assuming the initial value for to be of the order of the quantum fluctuations, given by , it is easy to check that the total number of e-folds is always huge.

### ii.1 Slow-roll parameters and the number of e-folds

We are now in the position to calculate the slow roll parameters in our model:

(18) |

(19) |

Inflation ends when one of these parameters becomes of order one. We call the value of the field at the end of inflation, for definiteness say when or become exactly equal to one. The number of e-folds corresponding to a smaller value of is

(20) |

For what concerns the small regime both couplings and could be important, but in order to have the simplest possible model we now focus on the case (we will analyze in more detail the case in section VI); this model has just one parameter, , in addition to those of the SM. The slow roll parameters become simply:

(21) |

For , becomes of order one at , before this happens to . On the contrary, for , becomes of order one at , before . The relation between and the associated number of e-folds becomes:

(22) |

where for , while for . The expression above allows to calculate the slow roll parameters of eq. (21) as functions of the number of e-folds.

In the case in which is non-vanishing but such that , it turns out (see section VI) that the expressions above specific to the case are marginally affected. This means that all the results that we are going to derive in the next section putting formally are slightly more general.

### ii.2 Tensor to scalar ratio and scalar spectral index

We call the number of e-folds corresponding to the present horizon of Mpc, which is expected to be in the range , as discussed in section V.

We are interested in , , since the observable quantities Peiris:2003ff are the tensor to scalar ratio and the scalar spectral index . When in eq.(22) is taken to be as large as , the term with in the denominator of the eq.(22) is negligible for both regimes provided , in which case we obtain:

(23) |

The fact that mildly depends on provides is shown in the left panel of fig.1. Considering for example , we obtain , precisely inside its experimentally preferred region Komatsu:2010fb , as depicted in the right panel. Clearly, future experiments with a better precision on could further check this model. As shown in section VI, if we consider in eq.(10) a term with , the prediction for leads to a slightly higher value which goes closer to the central measured value, .

The prediction for depends strongly on and, as will be discussed in more detail in section VI, very mildly on as far as . To make the connection with experiment even more direct, in the right panel of fig.1 we show as a function of for various values of and taking . The narrow shaded region is found by requiring to be in the range of interest, . One then realizes that for such range, only values of larger than allow both and to be in their experimentally preferred region Komatsu:2010fb . From now on we will therefore consider the parameter of our inflationary model to be in the range .

### ii.3 Amplitude of perturbations and Higgs potential at the false minimum

The amplitude of density perturbations in -space is specified by the power spectrum:

(24) |

where is the amplitude at some pivot point , predicted by inflation to be

(25) |

According to eq.(13), since we are in the small field regime for , the Higgs potential at the false minimum can be related to of eq.(23) by

(26) |

In the following we consider , taken as the best-fit value from Komatsu:2010fb for a pivot scale .

The value of the Higgs potential at the false minimum is shown in fig. 2 as a function of and for . This plot gives the window of values of the Higgs potential in the false minimum that are compatible with our inflationary model. Taking , where is the Newton constant, we obtain

(27) |

which is, by the way, the range where unification of couplings might take place. As we are going to discuss, in the SM this range of values for the Higgs potential at a false minimum is natural, even though it requires highly correlated values for the top and Higgs masses.

## Iii Finding a viable minimum

The fact that the Higgs field in the SM can develop a false minimum is non-trivial by itself and, as we are going to discuss, it is even more intriguing that this happens at the right energy scales required by our inflationary model.

The false minimum requires very specific values of the top and Higgs masses. Using 2-loop RGE and matching conditions as discussed e.g. in hep-ph/0104016 , we studied such values. Of course, the extremely precise values for and that we are going to present are not to be taken sharply, because of a theoretical uncertainty of about GeV on the Higgs mass and about GeV on the top mass, which is intrinsic in the 2-loop RGE running procedure (more on this later).

As an example, in fig. 3 we show the Higgs potential as a function of the Higgs field , by taking GeV and values of decreasing from down to GeV from top to bottom. The plot shows that it is possible to have a second minimum at high energy (magnified in the right plot), in addition to the usual SM minimum at low energy. Having fixed , this happens only for very specific values of . Increasing (decreasing) the top mass, the value of the second minimum increases (decreases), and larger (smaller) values of are required. The horizontal shaded band represents the range selected by the inflationary model discussed in the previous section, namely . The specific values considered in the plot are fine for our inflationary model.

As we are going to discuss, in order to have a sizable tunneling probability to the left side, the barrier must be very low, as is the case for the middle curve in the right panel of fig. 3, obtained for . For larger the potential has no false minimum, while for slightly smaller values of the second minimum becomes deeper and very soon negative. In this case it is the SM minimum at low energy that becomes metastable and it could catastrophically tunnel to minus infinity. (The points in the plane corresponding to the transition from stability to metastability are shown in fig.4 as dashed lines; the inner line is obtained when using the central value of , while the side ones represents its range.) For even smaller values of the tunneling probability increases so much to be inconsistent with the present age of the universe; the region of parameters is called instability region. For a discussion about the stability and metastability constraints of the Higgs potential see ref. hep-ph/0104016 .

The tunneling rate Coleman in our model is given by , where is the Euclidean classical action of the bounce solution which interpolates between and the value on the left side of the barrier, , and is a dimension four quantity of the order of the scale of the problem (GUT scale). We have numerically computed for several potentials finding that only if the potential is extremely shallow, such as the middle curve in the right panel of fig.3, we can obtain values of . If the potential well is instead deeper as the bottom curve, the exponent becomes rapidly extremely large and becomes essentially zero. In this case the transition would never happen, or it would happen at a too low energy, leading to a too small reheating temperature.

As mentioned earlier note that the transition could happen also through a Hawking-Moss instanton (see for instance HenryTye:2008xu ), which is due the presence of a gravitational background. In the case in which this is the dominant process the whole scenario would be different because would also be time-dependent, since the Planck mass and are varying with time. Such a transition is basically due to quantum jumps of the fields if its mass (the second derivative of the potential at the false minimum) is smaller than (which is about GeV), and so it could be important for extremely shallow barriers and very high (i.e. very large values of ). It may happen therefore that the scenario is viable only for low values of , which could translate on a further upper bound on the Higgs mass. In order to compare which instanton is the dominant one it is important for this analysis to have good control on the shape of the potential around the false minimum with a huge precision, of about 16 digits on . We postpone this analysis for future work. Note also that in the case in which such transition would not lead to sufficient inflation it is also possible to invoke an additional effect which would modify the Higgs field tunneling process, by considering a non-minimal coupling between the Higgs and gravity of the form . Since now is varying with time, this would introduce a time-dependence of the Higgs potential which could easily erase the potential barrier. We have in fact checked that a coupling of should be enough to erase the barrier, therefore opening another interesting possibility to implement inflation. However we leave also such a possibility for future work.

## Iv The Higgs mass range

As discussed, only with a restricted set of values of and it is possible for the false vacuum to be inside the band required by inflation. These very particular values for and are displayed as segments in fig. 4. The upper (lower) values of in the band correspond to the upper (lower) value of allowed in our inflationary model, namely GeV ( GeV). As can be seen in fig.3, this in turn corresponds to a scalar-to-tensor ratio close to (). The inner segment is obtained using the central value of , while the side ones mark its range.

Our inflationary model works for a narrow band in the plane, which only partially overlaps with the top mass experimental range GeV, provided by the recent (July 2011) global electroweak precision fits of the SM GFitter . We are then left with the upper part of the band in the plane, the one for which GeV and . Such region is emphasized with a spot in fig.4 to remind that the 2-loop RGE running has a theoretical uncertainty of GeV in and of GeV in .

Consequently, the inflationary model based on the Higgs false minimum considered here gives a narrow prediction for the Higgs mass:

(28) |

This range is within the summer 2011 experimental window set by direct searches, namely the GeV lower bound set by LEP PDG and the GeV upper bound set by LHC HCP11 , which restricted the previous GeV bound set by Tevatron arXiv:1107.5518 . Interestingly, this range is perfectly compatible with – and even more precise than – the recent global electroweak precision SM fit GFitter , which gives GeV at .

Hopefully the prediction of our inflationary model could be tested soon by LHC, so the model can be experimentally supported or ruled out. Actually, immediately after the first version of this preprint appeared on the arXiv, the ATLAS and CMS collaborations summarized the status of the Higgs boson searches in a public seminar last . The main conclusion is that, at CL, the SM Higgs boson mass range outside GeV is excluded by the ATLAS experiment, and the one outside GeV by CMS. It is extremely interesting that both experiments found some excess of events in the region between GeV, mainly for the di-photon and four-lepton channels. Althought the significance of the excess is still too low, these preliminary results from LHC are very exciting for the idea of false SM vacuum inflation.

If these preliminary experimental results were to be confirmed, it would be important to reduce the theoretical error which affects the determination of the values of and allowing for the presence of a shallow SM false minimum. The dominant source of the uncertainty in the RGE at present arises from the matching of the quartic Higgs coupling, known only at 1-loop. Such error is usually estimated by varying the matching scale in some (somewhat arbitrary) range. Choosing for example such range from about GeV (close to ) and about GeV (close to ) one finds that the value of leading to a shallow false minimum at the GUT scale changes by GeV. In the literature one can find several different choices hep-ph/0104016 ; strumia2 and perhaps a conservative error of GeV can be assigned on , and of GeV on . This might overestimate the theoretical error, but in order to better understand it, one would need to know the 2-loop correction to the matching of the quartic Higgs coupling. Reasonably, one could expect that in this way the theoretical error on could be reduced down to GeV, which is comparable to the experimental precision on foreseen at LHC.

## V Post-Tunneling evolution

When many bubbles of the new phase quickly form, collide and rapidly give rise to a nearly homogenous state with the Higgs field on the other side of the barrier at a value smaller than , and some of its potential energy transformed into radiation. In this model, the fraction of energy converted into radiation is however very small, since the potential barrier has to be very shallow as can be seen in fig. 3. So, the two values and between which the bounce solution interpolates are very close and the difference in potential energy before and after the transition is small compared to .

It is well-known that a part of the energy density goes also into gravitational radiation through bubble collisions Turner , but for the same reason its energy density is going to be negligible compared to , unlike the case studied in hep-ph/0511207 where it was assumed that the entire energy density would be converted into radiation and gravity waves through bubble collisions.

After this rapid thermalization, the Higgs field is free to roll down its potential, quickly reach zero and undergo rapid oscillations around zero. This happens at a rate faster than the expansion, since the quartic potential is steep and the term in the Klein-Gordon equation wins against the Hubble friction. During these oscillations the field is expected to rapidly convert all of its energy into particles, via perturbative and non-perturbative decays (similarly to what is described in GarciaBellido:2008ab ). Since the Higgs couplings are large, it is expected that this process is very efficient, leading to reheating of the Universe, although probably through a very complicated sequence of processes. We may however assume that roughly all the energy density is converted into a bath of SM particles: equating the energy density of the produced radiation to the initial potential energy we get an estimate for the reheating temperature , where is the number of degrees of freedom of the SM, . After this, the Universe cools down as usual and finally the Higgs field settles down to its present electroweak vacuum expectation value .

Apart from the evolution of the Higgs field, it is also necessary to follow the evolution of the field. In fact, the field has to satisfy the fifth force constraints at late times, due to the fact that an additional light scalar can mediate a long-range force between matter bodies and there are strong constraints about this, especially from the solar system reviewPPN . Those constraints can be satisfied if the following quantity espositofarese

(29) |

is small today, reviewPPN , or if the field is massive enough , so that it does not mediate a long-range force reviewPPN . Moreover, it is necessary that at least after Big Bang Nucleosynthesis (BBN) the field do not evolve significantly, because the value of the Planck mass (set by ) is constrained to be close to its present value reviewPPN .

It is difficult to predict the evolution of the field during the oscillations of the Higgs, because this would require to compute the equation of state where is the pressure and is the energy density of the total amount of matter contained in the Universe. In the absence of dissipation and in the approximation in which the Higgs field has just a quartic potential this would be possible, since it is well-known that averaging oscillations on a quartic potential leads to an equation of state of . Moreover for the radiation produced the equation of state is also . However the fact that the energy gets dissipated decreases the kinetic energy of the field , leading probably to a slightly smaller than . Even when the oscillations are completed the equation of state of radiation is not exactly but it is slighly smaller because of quantum corrections which break the conformal invariance, due again to the running of couplings (mostly QCD corrections Peloso ). The value is critical because the field is driven to large values if or to smaller values if , and so we would conclude that the field after tunneling stays at a large value, close to the value that it takes at the tunneling epoch, , or slightly larger.

On the other hand, in order for our model to be predictive we need to identify with the present value of the Planck mass , otherwise we would not be able to tell what is the value of the scale needed from fig. 3, which leads to our prediction on the Higgs mass. This can be achieved if the post-tunneling evolution of drives it back to zero, and in this case it automatically follows that the solar-system constraints are satisfied, since is very close to zero. As we have just discussed this could happen if or, alternatively, upon introduction of a potential in (3). Since we do not want to alter the SM significantly, it seems that the latter option is to be taken.

In this case, we can analyze the evolution again in the Einstein frame, remembering that a potential term becomes now a potential . Since we want to go to zero, we have to require that is a function which grows more rapidly than for large . On the other hand we also want to be negligible before tunneling, so the choice of requires some care, but it can be shown that such functions can be constructed.

An additional relevant issue is that after tunneling the field may lead to some additional inflation, because the potential effectively can introduce a slow-roll phase, since the field is at values , similarly to what happens in chaotic inflation models Linde . This would shift the needed number of e-folds by some model-dependent number , which has to be not too large in order not to erase the predictions of our model discussed in the previous section.

Although these are relevant and interesting issues, they are however very model-dependent, and it is sufficient to say for our purposes that a mechanism to drive back the field to zero has to be implemented, most likely using a potential , and this has to be done without introducing too many e-folds of an additional inflationary phase in order for our predictions to be valid. Under the assumption that is zero or negligible we can compute the value of which corresponds to our present horizon as follows.

First of all, let us compute when a particular comoving scale went outside the horizon during inflation. We count the number of e-folds starting from the end of exponential inflation (whose scale factor we call ), going backwards in time. In general a scale leaves the horizon at some e-folding number if:

(30) |

The reheating temperature is given by and the redshift during the power-law phase is given by . Here we have assumed that the transition between the exponential and the power-law phase is very quick and this is true for most functions, . Now, the largest scale observed today is the horizon scale, which leads to:

(31) |

where we have used , Mpc, and we have taken the extreme value .

Also, if the field is driven to zero, it would start oscillating around the zero and it could overclose the Universe moduli , unless it can decay. Since we do not want in principle to introduce new couplings, this can be achieved either by decay into quanta, through self-interactions, or via decay into gravitons. The latter can be estimated to have a decay rate of the order of , and the decay has to happen before BBN, which has a temperature of . This leads to the requirement that the decay rate is larger than the expansion rate: which leads to .

### v.1 Further issues

Finally, let us comment on the possibility that other cosmological puzzles have to be solved in this minimalistic model with SM and tensor-scalar gravity. It is well-known for example that other ingredients need to be introduced beyond the SM in order to satisfy cosmological observations, such as baryogenesis and dark matter.

For what concerns baryogenesis it is interesting to note that the tunneling event provides an out-of-equilibrium event in the early Universe which could be used for generating the baryon asymmetry. This would require only the addition of some source of CP-violation. While in the usual SM electroweak phase transition the CP-violation is considered to be too small it would need to be investigated if this remains true in such a scenario. In any case, it is always possible to add right-handed neutrinos and achieve a new-source of CP-violation in the lepton sector of the SM without changing significantly the running of (which depends mostly on the gauge and top Yukawa couplings) and therefore without affecting at all our scenario. So baryogenesis could proceed in a new non-trivial way during reheating and oscillations of the Higgs field, or it could be also obtained via the usual leptogenesis scenario Yanagida , where right-handed neutrinos are thermally produced and decay out-of-equilibrium, since the reheating temperature in this model can be large enough.

For the dark matter problem many solutions could be possibly incorporated in our model. For instance, there could simply be an additional weakly-interacting stable particle, which again would not change significantly the running of the SM. Alternatively, it could be worth considering the hypothesis that quanta of the field itself are left as a remnant of the post-tunneling evolution of . Finally, in the same spirit of the proposal of the present paper, it is possible also to add another particle completely decoupled from the SM, which could perhaps be produced gravitationally or via decay of the field during or after inflation.

## Vi Exponential inflation with quadratic and higher oder terms

### vi.1 Effect of

We study now the inclusion of the quadratic coupling in eq.(10) and in order to do this we focus on the case , so that

(32) |

We study the evolution equations both analytically and numerically.

For the analytical calculation, we work directly in the Einstein frame, computing by solving the differential equation via a power series expansion in the parameter . We thus obtain an expression for the potential which generalizes the one in eq.(13)

(33) |

The number of e-folds is calculated using the first approximation of eq.(20). By selecting a value for the number of e-folds , one obtains the associated predictions for and , as displayed in fig.5. As already anticipated, as far as , the prediction for does not change significantly with respect to the case in which . This can be seen from the left plot of fig. 5. For higher values of there is a slight increase in , which then falls down in the experimentally excluded region at . The dependence of on is less pronounced, see the right plot of fig. 5. When , falls down rapidly.

Also for the numerical analysis we work directly in the Einstein frame, computing by solving the differential equation , which gives us the potential of eq.(12). We solve then the equations of motion for and the Friedmann equation in order to find the evolution of the scale factor . The number of e-folds is computed as , where corresponds to the end of inflation, and the slow roll parameters and have been computed as usual by computing the first and second derivatives of . The result for is given in fig.6 for and . Here is here defined as the time in which becomes negative. Note that close to the slow-roll parameters are already . This leads to a slight difference compared to the approximate expression of eq.(20) where the end of inflation has been defined there as the moment when either or are equal to , and which is valid only when : all this leads to a shift in of about 5. Note also that if is large enough the tunneling event could happen before the transition to the decelerated phase, because is already decreasing before that the asymptotic behavior is reached. In this case there can be another shift in , which makes the spectral index slightly higher and slightly smaller: for instance for the case and the numerical calculation can make increase by and decrease by in the numerical result. Anyway remind that is subject to other uncertainties due to post-inflationary evolution, as discussed in the previous section.

### vi.2 Higher order terms

We derive in this section the predictions for the case in which

(34) |

where , further elaborating some results already derived in hep-ph/0511207 . Again, for , the canonical scalar field variable and almost coincide () and to lowest order in we get:

(35) |

In the slow-roll approximation,

(36) |

We call the field at the end of inflation, for definiteness say when or become equal to one. For , one has , otherwise . The number of e-foldings as a function of is given by

(37) |

which can be rewritten as

(38) |

where for , and otherwise.

This expression can be substituted again in eqs. (36) in order to obtain observable quantities as the tensor to scalar ratio and the scalar spectral index , for a fixed number of e-folds and as a function of , as shown in fig.7 for . One can see that, when is large enough so that is negligible in the denominator of eq.(38), becomes nearly constant and it is approximately given by:

(39) |

Hence, increases with and eventually reaches the maximum value , which is close to for . The prediction for is well inside its present observational range for reasonable values of . On the other hand, the tensor-to-scalar ratio is shown in the right plot of fig.7 for and can be approximated by

(40) |

Observationally , which implies the lower bounds: , , .

In fig.8 we show the (dashed) curves of constant in the plane , for from top to bottom. The plot also show the (solid) curves of constant , considering as reference values . Due to the proportionality between and , it also possible to display for each value of the values of the Higgs and top masses giving rise to a shallow false minimum; this is represented graphically via the vertical arrow. The lower bound on is about GeV GFitter but, as already explained, in the determination of the shallow minimum there is theoretical uncertainty of about GeV in and of about GeV in ; we have thus displayed the exclusion region for by means of the shaded region below GeV. The plots show that the three models with are well compatible with the present observed values of and Komatsu:2010fb at and , displayed via the shaded red regions. They also show that, due to the lower bound on , in these models should be found above roughly

## Vii Conclusions

We proposed to exploit the possible presence of a false minimum in the SM Higgs potential at very high-energies, GeV, to provide a large amount of potential energy which can drive primordial inflation. In the framework of scalar-tensor theories of gravity, a graceful exit from inflation can be achieved through tunneling of the Higgs field and subsequent relaxation down to its present vacuum expectation value GeV.

Requiring the amplitude and spectral index of cosmological density perturbations from inflation and the top quark mass to be compatible with observations, we showed that this possibility is realized only within a small region of values for the Higgs mass, see fig.4, leading to the prediction

where the error is mostly due to the present theoretical uncertainty of the 2-loop RGE. This prediction can be tested soon by LHC, in particular for the decay mode . The inflationary model proposed here could thus meet experimental support or be ruled out. It is exciting that preliminary results from LHC last show an excess of events in the range GeV.

If these results will be confirmed, further checks of the model will be offered by better determinations of the scalar spectral index , predicted in this model to be within . Using the constraints coming from the top quark mass, the scalar-to-tensor ratio in this model is constrained to be within . Actually, it can be shown Masina:2011un that any model in which the false vacuum is very shallow, the relation between the potential at the false minimum and the amplitude of perturbations, eq. (25), implies . With forthcoming more precise cosmological measurements, such as the Planck satellite mission, one can test the region of large values of , while improving the top quark mass measurement can further constrain from below.

Moreover, we showed that our scenario, and in particular the prediction for , can be obtained in a wide class of scalar-tensor theories. In particular, higher dimensional operators can be safely present.

As a completely general remark, we point out that discovering a Higgs with mass close to GeV is a very suggestive hint in favor of the existence of a false minimum in the SM Higgs potential at energies close to GeV, which could be the starting point for inflation in our Universe. Indeed, it would lead to a period of exponential expansion producing density perturbations with the right amplitude. We have shown here that in a scalar-tensor theory of gravity this inflationary stage can end, allowing the Higgs field to tunnel out of the false minimum and subsequently to relax down to its present electroweak scale value.

Acknowledgements We thank Tirthabir Biswas for discussions in the early stages of this project and Pietro Dalpiaz for interesting suggestions. We also thank Jaume Garriga and Giovanni Villadoro for useful comments.

## References

- (1) F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B 659 (2008) 703 [arXiv:0710.3755 [hep-th]]. F. L. Bezrukov, A. Magnin and M. Shaposhnikov, Phys. Lett. B 675 (2009) 88 [arXiv:0812.4950 [hep-ph]]. F. Bezrukov and M. Shaposhnikov, JHEP 0907 (2009) 089 [arXiv:0904.1537 [hep-ph]]. F. Bezrukov, A. Magnin, M. Shaposhnikov and S. Sibiryakov, JHEP 1101, 016 (2011) [arXiv:1008.5157 [hep-ph]].
- (2) A. O. Barvinsky, A. Y. .Kamenshchik and A. A. Starobinsky, JCAP 0811 (2008) 021 [arXiv:0809.2104 [hep-ph]]. A. O. Barvinsky, A. Y. .Kamenshchik, C. Kiefer, A. A. Starobinsky and C. F. Steinwachs, arXiv:0910.1041 [hep-ph].
- (3) A. De Simone, M. P. Hertzberg and F. Wilczek, Phys. Lett. B 678 (2009) 1 [arXiv:0812.4946 [hep-ph]]. M. P. Hertzberg, JHEP 1011 (2010) 023 [arXiv:1002.2995 [hep-ph]].
- (4) N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B 158 (1979) 295. P. Q. Hung, Phys. Rev. Lett. 42 (1979) 873. M. Lindner, Z. Phys. C 31 (1986) 295. M. Lindner, M. Sher and H. W. Zaglauer, Phys. Lett. B 228 (1989) 139. M. Sher, Phys. Rept. 179 (1989) 273. B. Schrempp and M. Wimmer, Prog. Part. Nucl. Phys. 37 (1996) 1 [hep-ph/9606386].
- (5) J. A. Casas, J. R. Espinosa and M. Quiros, Phys. Lett. B 382 (1996) 374 [hep-ph/9603227]. G. Isidori, G. Ridolfi and A. Strumia, Nucl. Phys. B 609 (2001) 387 [hep-ph/0104016]. J. R. Espinosa, G. F. Giudice and A. Riotto, JCAP 0805 (2008) 002 [arXiv:0710.2484 [hep-ph]]. N. Arkani-Hamed, S. Dubovsky, L. Senatore and G. Villadoro, JHEP 0803 (2008) 075 [arXiv:0801.2399 [hep-ph]].
- (6) G. Isidori, V. S. Rychkov, A. Strumia and N. Tetradis, Phys. Rev. D 77 (2008) 025034 [arXiv:0712.0242 [hep-ph]].
- (7) T. Biswas and A. Notari, Phys. Rev. D 74 (2006) 043508 [hep-ph/0511207].
- (8) F. Di Marco and A. Notari, Phys. Rev. D 73 (2006) 063514 [astro-ph/0511396].
- (9) C. Mathiazhagan, V. B. Johri, Class. Quant. Grav. 1 L29-L32 (1984).
- (10) D. La and P. J. Steinhardt, Phys. Rev. Lett. 62, 376 (1989) [Erratum-ibid. 62, 1066 (1989)].
- (11) P. J. Steinhardt and F. S. Accetta, Phys. Rev. Lett. 64, 2740 (1990).
- (12) K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010) and 2011 partial update for the 2012 edition.
- (13) The ATLAS and CMS experiments presented their combined results at the Hadron Collider Physics Symposium 2011 in Paris, France (14-18 November). See, e.g. the ATLAS website, http://atlas.ch/
- (14) [TEVNPH (Tevatron New Phenomena and Higgs Working Group) and CDF and D0 Collaborations], arXiv:1107.5518 [hep-ex].
- (15) See the GFitter webpage, http://gfitter. desy.de/Standard_Model
- (16) ATLAS Collaboration, arXiv:1202.1408 [hep-ex]. CMS Collaboration, arXiv:1202.1488 [hep-ex].
- (17) I. Masina and A. Notari, arXiv:1112.5430 [hep-ph].
- (18) A. H. Guth, Phys. Rev. D 23, 347 (1981).
- (19) S. R. Coleman and F. De Luccia, Phys. Rev. D 21, 3305 (1980).
- (20) A. R. Liddle and D. H. Lyth, Phys. Lett. B 291, 391 (1992).
- (21) G. Esposito-Farese and D. Polarski, Phys. Rev. D 63, 063504 (2001) [arXiv:gr-qc/0009034].
- (22) ÊS. W. Hawking and I. G. Moss, ÊÊPhys. Lett. B 110, 35 (1982). Ê
- (23) H. V. Peiris et al. [WMAP Collaboration], Astrophys. J. Suppl. 148, 213 (2003) [arXiv:astro-ph/0302225].
- (24) E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011) [arXiv:1001.4538 [astro-ph.CO]].
- (25) M. S. Turner, E. J. Weinberg and L. M. Widrow, Phys. Rev. D 46, 2384 (1992).
- (26) J. Garcia-Bellido, D. G. Figueroa and J. Rubio, Phys. Rev. D 79, 063531 (2009) [arXiv:0812.4624 [hep-ph]].
- (27) B. Bertotti, L. Iess and P. Tortora, Nature 425 (2003) 374. C. M. Will, Living Rev. Rel. 9, 3 (2005) [arXiv:gr-qc/0510072].
- (28) J. A. R. Cembranos, K. A. Olive, M. Peloso and J. P. Uzan, JCAP 0907, 025 (2009) [arXiv:0905.1989 [astro-ph.CO]].
- (29) A. D. Linde, Phys. Lett. B 129 (1983) 177.
- (30) G. D. Coughlan, W. Fischler, E. W. Kolb, S. Raby and G. G. Ross, Phys. Lett. B 131, 59 (1983); J. R. Ellis, D. V. Nanopoulos and M. Quiros, Phys. Lett. B 174, 176 (1986); B. de Carlos, J. A. Casas, F. Quevedo and E. Roulet, Phys. Lett. B 318, 447 (1993) [arXiv:hep-ph/9308325]; T. Banks, D. B. Kaplan and A. E. Nelson, Phys. Rev. D 49, 779 (1994) [arXiv:hep-ph/9308292].
- (31) M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986).