# Standard Model False Vacuum Inflation: Correlating

the Tensor-to-Scalar Ratio to the Top Quark and Higgs Boson Masses

###### Abstract

For a narrow band of values of the top quark and Higgs boson masses, the Standard Model Higgs potential develops a false minimum at energies of about GeV, where primordial Inflation could have started in a cold metastable state. A graceful exit to a radiation-dominated era is provided, e.g., by scalar-tensor gravity models. We pointed out that if Inflation happened in this false minimum, the Higgs boson mass has to be in the range GeV, where ATLAS and CMS subsequently reported excesses of events. Here we show that for these values of the Higgs boson mass, the inflationary gravitational wave background has be discovered with a tensor-to-scalar ratio at hand of future experiments. We suggest that combining cosmological observations with measurements of the top quark and Higgs boson masses represents a further test of the hypothesis that the Standard Model false minimum was the source of Inflation in the Universe.

The fact that, for a narrow band of values of the top quark and Higgs boson masses, the Standard Model (SM) Higgs potential develops a local minimum CERN-TH-2683 ; hep-ph/0104016 ; strumia2 is nontrivial by itself, but it is even more suggestive that this happens at energy scales of about GeV, suitable for inflation in the early Universe.

Inflation from a local minimum is a viable scenario, provided a graceful exit to a radiation-dominated era can be obtained. Developing an explicit model with graceful exit in the framework of a scalar-tensor theory of gravity hep-ph/0511207 ; astro-ph/0511396 , in ref.Masina:2011aa we pointed out that the hypothesis that inflation took place in the SM false vacuum is consistent only with a narrow range of values of the Higgs boson mass, GeV, the error being mainly due to the theoretical uncertainty of the 2-loops Renormalization Group Equations (RGE) used in the calculation. This mass range is surprisingly compatible with the window GeV, were both ATLAS and CMS HCP11 recently reported excesses of events, in the di-photon as well as the 4-lepton Higgs decay channels. It is also compatible with preliminary results from Tevatron TEV .

These preliminary but very suggestive results provide a strong motivation to further investigate the scenario of SM false vacuum inflation, in particular by looking for complementary experimental tests. Inflation can generate tensor (gravity wave) modes as well as scalar (density perturbation) modes. It is most common to define the tensor contribution through , the ratio of tensor-to-scalar perturbation spectra at large scales. If inflation happened at a very high scale, as is the case for the SM false vacuum scenario, quantum fluctuations during inflation produced a background of gravitational waves with a relatively large amplitude.

In this Letter we argue that the tensor-to-scalar ratio, combined with the top quark and Higgs boson mass measurements, does represent a test of the hypothesis that inflation started from the SM false vacuum.

Let us consider the Higgs potential in the SM of particle physics. For very large values of the Higgs field , the quadratic term can be neglected and we are left with the quartic term, whose dimensionless coupling depends on the energy scale, which can be identified with the field itself:

(1) |

It is well known that, for some narrow band of the Higgs and top masses, the Higgs potential develops a new local minimum CERN-TH-2683 ; hep-ph/0104016 ; strumia2 .

If the Higgs field is trapped in a cold coherent state in the false minimum and dominates the energy density of the Universe, the standard Friedmann equation leads to a stage of inflationary expansion

(2) |

where is the scale factor, is the Hubble rate and is the Planck mass.

A nontrivial model-dependent ingredient is how to achieve a graceful exit from inflation, that is a transition to a radiation-dominated era, in a nearly flat Universe at a sufficiently high-temperature. In order to end inflation the Higgs field has to tunnel to the other side of the potential barrier by nucleating bubbles Coleman that eventually collide and percolate. Subsequently the Higgs field could roll down the potential, reheat the Universe and finally relax in the present true vacuum with GeV. Whether the tunneling event happens depends on and , the nucleation rate per unit time and volume.

If , the Universe tunnels quickly in a few Hubble times, without providing sufficient inflation. If , the tunneling probability is so small that the process does not produce a sufficient number of bubbles inside a Hubble horizon that could percolate. A graceful exit would thus require to become larger than only after some time, but this is impossible if both quantities are time-independent, as is the case for the pure SM embedded in standard gravity Guth .

A time-dependent necessarily requires the existence of an additional time-dependent order parameter. This can be realized in a scalar-tensor theory of gravity, where the value of the Planck mass is set by a scalar field , the Brans-Dicke scalar or dilaton. This allows coupling to the Ricci scalar via an interaction of the form , where thus sets the value of the Planck mass. The presence of such field makes the Planck mass time-dependent, and therefore also , naturally leading to an increase in .

This has been shown in early models johri ; extended ; hyperextended achieving power-law inflation, which later turned out to be in tension with observations of the Cosmic Microwave Background (CMB), since it is difficult to get a nearly flat spectrum of perturbations Liddle . In refs. hep-ph/0511207 ; astro-ph/0511396 , a stage of exponential expansion was naturally incorporated, later followed by a stage of power-law (even decelerated) expansion. In this way, it is possible to produce a flat nearly homogenous Universe during the exponential phase and, moreover, the quantum fluctuations in lead to the correct spectrum of perturbations. During the subsequent decelerated phase, decreases rapidly, allowing the field trapped in the false minimum to tunnel through percolation of bubbles. As discussed in Masina:2011aa , after tunneling we require the field to relax to zero, which allows us to identify the present Planck mass GeV with the Planck mass at inflation and, at the same time, to satisfy constraints from fifth-force experiments and time-dependence of the Newton constant Will:2005va .

An alternative to scalar-tensor theories is given by models Masina:2012yd with a direct coupling of the Higgs field to an additional scalar field, which induces a time-dependence directly into by flattening the barrier in the potential or it might be possible to achieve a graceful exit in other models with an additional coupling of the Higgs field to Masina:2011aa .

It is crucial to notice that a graceful exit can be generically realized only if at the end of inflation there is a very shallow false minimum,
otherwise the tunneling rate would be negligibly small,
since the probability is exponentially sensitive to the barrier Coleman .
So, the shape of the potential is very close to the case in which there is just an inflection point.
This leads to a powerful generic prediction^{1}^{1}1The relevant stage of inflation for predicting the amplitude of is the one at
e-folds before the end. In models in which the potential is time-independent as the ones in Masina:2011aa we necessarily have a shallow false minimum.
Models with time-dependent can also be constructed, but even in those models it is
likely that e-folds before the end of inflation the potential well is generically not deep, since a too rapid variation of the potential in the
last stages of inflation would probably be in conflict with observations of the spectral index, as in Masina:2012yd .
for the scale of inflation and therefore for .
So, if the false vacuum is very shallow, the specific model only affects the prediction for the spectral index of cosmological
density perturbations .
For instance, for a wide class of functions the models considered in hep-ph/0511207 lead to ,
in agreement with the central value subsequently measured by WMAP Spergel:2003cb .

Using 2-loop RGE and matching conditions, we studied the very specific values of the top and Higgs masses
allowing for the presence of a false minimum.
As an example, in fig.1 we display the Higgs potential for very specific values of and .
The extremely precise values shown in the caption are not to be taken sharply, because of a theoretical
uncertainty of about GeV on and about GeV on , which is intrinsic in the 2-loop RGE procedure hep-ph/0104016 ; strumia2 .
As mentioned above and discussed in Masina:2011aa , in order to have a sizable tunneling probability through the left side, the barrier must be very low,
as is the case for the middle curve.
For slightly smaller values of the second minimum becomes deeper and the tunneling probability essentially zero^{2}^{2}2For smaller
values of the potential turns negative, so that
the SM minimum at low energy becomes metastable hep-ph/0104016 ; strumia2 . .

Increasing (decreasing) , one has also to increase (decrease) in order to develop the shallow false minimum; accordingly, the value of both and increase (decrease). The solid line in fig.2 shows the points in the plane where the shallow SM false minima exists. We recall however that the line has a (vertical) uncertainty of GeV in and a (horizontal) one of GeV in . The shallow false minima are just at the right of the dashed line marking the transition from stability to metastability. Ticks along the solid line display the associated values of in units of GeV.

The measurement of implies the lower bounds GeV and GeV, considering the theoretical errors intrinsic in the RGE. On the other hand, as it is well-known and explained below, the scale of inflation cannot be too high, GeV, which leads to the constraint GeV. Remarkably, the allowed band for SM false vacuum inflation includes the region GeV, where ATLAS and CMS HCP11 recorded excesses of events in the di-photon as well as 4-leptons channels. Moreover, now CMS has set the upper bound GeV at CL, which further restricts the allowed region.

This striking coincidence of values deserves further exploration both experimentally,
by reducing the error on and , and theoretically by improving the RGE^{3}^{3}3Note that additional particles
could exist at high scales, modifying the running of . If unification of gauge couplings is realized in nature,
the modification is however probably constrained to be small..

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. By varying the matching scale 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. Clearly the range of where to vary the matching scale is somewhat arbitrary, and in the literature one can find different choices. As in many other papers hep-ph/0104016 ; strumia2 using the same accuracy for the RGE as ours, we considered it conservative to assign an error of GeV on , and an error 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. By varying within a range of GeV, one obtains in our scenario a prediction for within a range of about GeV, call it . The experimental precision on needed to falsify our scenario depends on the difference between and the experimental central value of . The smaller this difference is, the more precision is needed on .

A complementary way of testing the possibility that inflation started from the SM false vacuum at high energy is to look at the tensor-to-scalar ratio of cosmological perturbations. The amplitude of density fluctuations in the observed Universe as seen by the CMB and Large-Scale structure data is parametrized by the power spectrum in -space

(3) |

where is the amplitude at some pivot point . We consider the best-fit value from Komatsu:2010fb , at .

In any inflationary model that can be analyzed through the slow-roll approximation Komatsu:2010fb , there is a relationship between the scale of inflation, the amplitude of density perturbations, and the amount of gravity waves that can be produced:

(4) |

If inflation actually started from a SM shallow false minimum, then each point in the plane has to be associated with a specific value of , as shown in fig.2 via the lower row of ticks. The upper limit on is at present about Komatsu:2010fb , which gives rise to the above mentioned upper bound GeV. The lower bound on instead implies , partially at hand of future experimental sensitivity for various experiments such as Planck Planck , EPIC Bock:2009xw and COrE Collaboration:2011ck . Improving the top quark mass measurement and/or discovering the Higgs mass close to GeV could further constrain from below.

The relationship between , and is completely general in any model with a SM shallow false vacuum. As we have mentioned, it is conceivable that models could be constructed in which the false vacuum initially is not shallow, but somehow the Higgs potential becomes time-dependent and is lifted up, making large and leading inflation to an end. In such models only at the end of inflation the final shape of the potential is given by the SM Higgs potential with a shallow minimum. In this case we can make two important statements. First, the height of the minimum during the observationally relevant stage of inflation (i.e. e-folds before the end) is always at most as high as the one in the shallow case: as a consequence, our prediction on always applies strictly as an upper bound for any model which uses the Higgs false minimum to source inflation. So, if future experiments will measure and accurately and if the RGE theoretical error is reduced, measuring above the corresponding value displayed in fig. 2 would rule out all such models. Second, even in the case of a time-dependent Higgs potential, it is very likely that at e-folds before the end of inflation the barrier is still close to the shallow case, so fig. 2 would probably apply even in such models.

Summarizing, we argue that precision measurements of , , , together with theoretical improvements of the SM RGE, will represent a test of the hypothesis Masina:2011aa that inflation occurred in the SM false vacuum at about GeV.

## Acknowledgements

We thank T. Hambye, A. Strumia and G. Villadoro for useful discussions.

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