## Abstract

The discovery of the Higgs boson in 2012 and other results from the Large Hadron Collider have confirmed the standard model of particle physics as the correct theory of elementary particles and their interactions up to energies of several TeV. Remarkably, the theory may even remain valid all the way to the Planck scale of quantum gravity, and therefore it provides a solid theoretical basis for describing the early Universe. Furthermore, the Higgs field itself has unique properties that may have allowed it to play a central role in the evolution of the Universe, from inflation to cosmological phase transitions and the origin of both baryonic and dark matter, and possibly to determine its ultimate fate through the electroweak vacuum instability. These connections between particle physics and cosmology have given rise to a new and growing field of Higgs cosmology, which promises to shed new light on some of the most puzzling questions about the Universe as new data from particle physics experiments and cosmological observations become available.

This article is part of the Theo Murphy meeting issue ‘Higgs cosmology’.

## 1. Introduction

The Higgs field is a central piece in the standard model of particle physics. Its non-zero vacuum expectation value *ϕ*≈246 GeV breaks the electroweak SU(2)×U(1) symmetry into the U(1) symmetry of electromagnetism and makes the *W* and *Z* bosons massive. It also forms a bridge between the group representations of the left-handed and right-handed fermions, allowing them to acquire masses in the broken phase. The discovery of the Higgs boson in 2012 confirmed this picture, and together with other data from the Large Hadron Collider (LHC) demonstrated that the theory is accurate at least up to energies of a few *TeV* .

Before the start of the LHC, it was widely expected that it would discover not only the Higgs boson but also new particles beyond the standard model, perhaps supersymmetric partners of the known particles. The standard model would then be a low-energy effective theory valid only up to *TeV* -scale energies, and these new particles would give our first glimpse of the more fundamental theory that describes physics at higher energies. One reason for expecting this was that the Higgs self-interaction grows stronger at higher energies and eventually diverges at a finite energy, above which the theory cannot be valid.

However, no such new particles have been found so far. Furthermore, the mass of the Higgs boson, *m*_{H}=125.09±0.24 GeV, turned out to be in the relatively narrow range , for which the divergence is not reached until above the Planck scale *M*_{P}∼10^{19} GeV, where gravity needs to be included anyway [1,2]. This means that the standard model can potentially remain valid all the way to the scale of quantum gravity.

On the other hand, there are a number of cosmological puzzles that the standard model appears unable to explain, and which therefore provide evidence for new physics. These include baryogenesis, dark matter and inflation, which are all linked to the Higgs field. The dynamics of the Higgs field in the early Universe may also have potentially testable observable consequences, for example, in the form of a stochastic gravitational wave background, which could be detected and measured with future gravitational wave experiments such as the Laser Interferometer Space Antenna (LISA). Therefore, the Higgs field provides an interesting connection between cosmology and particle physics, and gives an opportunity to achieve a better understanding of both the fundamental laws of nature and the early Universe.

Cosmology also gives access to the last unknown renormalizable parameter of the standard model—the (non-minimal) coupling *ξ* between the Higgs field and the space–time curvature [1,3,4]. This parameter is required for the renormalizability of the standard model in curved space–time, but its value is practically impossible to measure in experiments because the space–time is almost perfectly flat today. By contrast, the space–time curvature was very high in the early Universe, and therefore the coupling *ξ* was important for the behaviour of the Higgs field. Cosmological observations can therefore provide a way to constrain or measure its value.

This theme issue of the *Philosophical Transactions of the Royal Society A* is based on the Theo Murphy discussion meeting ‘Higgs cosmology’ organized at the Royal Society Kavli Centre in March 2017. The articles present the current state of the art in this new and rapidly moving field of Higgs cosmology. They can be roughly divided into four general topics: baryogenesis, running coupling, inflation and gravitational effects. In this introductory article, the literature references point to the articles in the theme issue, in which extensive references to the original literature can be found.

## 2. Baryogenesis

The puzzle of the origin of baryonic matter, i.e. atoms, in the Universe has a long history. Empirically, particle interactions conserve the baryon number, which means that they always produce or destroy the same number of baryons and antibaryons. But if this is the case, why is there so much baryonic matter in the Universe but practically no antimatter? Furthermore, there is now strong evidence for a period of inflation in the early Universe, which would have diluted away any pre-existing baryonic matter, meaning that this matter–antimatter asymmetry must have been produced after inflation.

The conditions required for generating the asymmetry are known as the Sakharov conditions. First, the baryon number must be obviously violated. Second, the charge conjugation (C) and charge–parity (CP) symmetries must also be violated, because otherwise the baryon-producing and -destroying processes have the same rate and will therefore balance out. Finally, the process must take place out of equilibrium because the equilibrium state is symmetric under time reversal and therefore, by the charge–parity–time reversal (CPT) symmetry, also under CP. In practice, this means that baryogenesis had to take place out of equilibrium, and the baryon number-violating processes had to stop before the Universe reached equilibrium.

Remarkably, the standard model can, in principle, satisfy all of these conditions. Even though baryon number violation has never been observed, the theory predicts that baryons can turn into leptons through a quantum anomaly. In the current broken symmetry vacuum state, these baryon number-violating processes are extremely rare because of the high vacuum expectation value of the Higgs fields. However, at the high temperatures of the early Universe the electroweak SU(2)×*U*(1) symmetry was unbroken and therefore baryon number violation was unsuppressed. In the standard model, C is maximally violated by weak interactions and CP is violated through quark (and probably neutrino) mixing, and hence the second Sakharov condition is also satisfied.

In principle, the standard model could also satisfy the last condition. If the electroweak phase transition, in which the Higgs field acquired its non-zero vacuum expectation value and the electroweak SU(2)×U(1) became spontaneously broken to the U(1) of electrodynamics, was of first order, it would have proceeded through the formation of bubbles that grew and eventually filled the Universe. The motion of the bubble wall would have then provided the required non-equilibrium conditions, and if the transition was strong enough, baryon number violation would have been so strongly suppressed inside the bubbles that the generated matter–antimatter asymmetry would have survived until the present day [5,6].

However, detailed calculations show that this scenario, known as electroweak baryogenesis, does not work in the minimal standard model. For the actual measured values of the parameters, most importantly the Higgs boson mass, the electroweak phase transition is not of first order but rather a smooth crossover between the two phases, and the measured CP violation would be too weak anyway. Therefore, some new physics beyond the standard model is needed. It could be in the form of new TeV-scale particles that violate CP and modify the phase transition dynamics by making it stronger. In that case, it may be possible to find these new particles at the LHC in the near future.

Experimental data from the LHC have already provided increasingly strong constraints on models of electroweak baryogenesis, especially in the context of the minimally supersymmetric standard model. However, Servant [6] reports that a theory with varying Yukawa couplings is compatible with experiments, and Cline [5] shows that a singlet scalar model with a neutral Majorana fermion would also overcome these issues as well as provide a candidate for dark matter.

## 3. Running coupling and vacuum instability

In quantum field theories, the effective interaction strength depends on the energy scale of the process, and this is reflected in the way the renormalized parameters run with (i.e. depend on) the renormalization scale *μ*. In asymptotically free theories, such as quantum chromodynamics, the coupling approaches zero asymptotically at high energies, and therefore the theory can remain valid to arbitrarily high energies or arbitrarily short distances.

By contrast, theories that are not asymptotically free exhibit the opposite behaviour. The coupling becomes increasingly strong at high energies, and diverges at a finite energy known as the Landau pole. The theory cannot be applied to processes at energies higher than this. The theory is still perfectly applicable at lower energies, so this is simply an indication that some new physics must appear before the Landau pole to avoid the divergence. We already know that at energies around the Planck scale *M*_{Pl}∼10^{19} GeV the theory must be modified to incorporate quantum gravity, and therefore the Landau pole is only physically relevant if it is at a lower energy than this.

In the standard model, the running of the Higgs self-coupling λ(*μ*) gets a positive contribution from its interactions with bosonic particles and a negative contribution from fermionic particles. The strength of each contribution is proportional to the mass of the particle. Therefore, if the fermions are light and the bosons are heavy, the coupling increases and eventually diverges to . Correspondingly, if the fermions are heavy and the bosons are light, the coupling decreases and diverges to . Either way, the divergence would point towards some new physics beyond the standard model.

However, there is also the third possibility that the masses are balanced in such a way that the coupling remains finite up to the Planck scale. In practice, this depends mainly on the masses of the Higgs boson and the top quark. As discussed by Espinosa [1], the current best experimental estimates of their masses, *m*_{H}=125.09±0.24 GeV and *m*_{top}=173.34±0.76 GeV, indicate that we are in this intermediate area. Therefore, the standard model can actually remain valid all the way to the Planck scale. This can be seen as evidence against any theory of new physics beyond the standard model that would change the running significantly, because within such a theory there would be no reason for this observed fine balance between the top and Higgs masses.

Interestingly, though, the Higgs self-coupling appears to turn negative at high energies above 10^{10} GeV. This points to a possible true vacuum state with a high Higgs field value and negative vacuum energy. The current vacuum state would then be metastable, and eventually, after a long time, the Higgs field would tunnel through the potential barrier to the true vacuum. This would create a bubble that grows at the speed of light, inside which space collapses into a singularity because of the negative vacuum energy. The rate of this bubble nucleation process depends sensitively on the Higgs and top masses. If the rate was too high, this transition would have already happened. This is clearly not the case, so the theory would not be compatible with observations. For the central experimental values of the masses, the rate is, however, very slow and the predicted lifetime of our current vacuum state is many orders of magnitude longer than the age of the Universe. On the other hand, the rate would be changed by any modification of the standard model, and therefore vacuum stability provides a test that any such theory has to satisfy [1,2].

Furthermore, it is not enough that the vacuum is stable today, but it must have also been sufficiently stable throughout its history to survive until today [1,2,4]. Therefore, vacuum instability has to be analysed in the context of cosmology, and it can potentially constrain cosmological models.

## 4. Inflation

Anisotropy of the cosmic microwave background (CMB) radiation provides strong evidence for a period inflation, accelerating expansion, in the early Universe. During this period, the energy density was apparently dominated by a slowly rolling scalar field known as the inflaton. The expansion stretched the quantum fluctuations of the inflaton field to super-horizon scales, where they acted as seeds for structure formation and are observed in the perturbations of the CMB.

In the same way, the accelerating expansion also stretched and amplified the fluctuations of any other light scalar field, including the Higgs field if the Hubble rate was higher than its mass, *H*>*m*_{H}. If the Hubble rate was higher than 10^{10} GeV, this could have thrown the field over the potential barrier, triggering vacuum decay [1,4]. However, interactions with the inflaton field or a non-minimal coupling to the space–time curvature may have given rise to a higher effective mass for the Higgs field, thereby preventing the growth of the fluctuations and preventing the vacuum decay.

The fluctuations of the Higgs field could also have other effects. In typical inflationary models, the inflaton field oscillates coherently about the minimum of its potential after the end of inflation, until it has transferred its energy to the standard model degrees of freedom. These oscillations can amplify the fluctuations of the Higgs field, either through a direct coupling or through their effect on the space–time curvature [4]. This can lead to observable effects such as gravitational wave production or even vacuum decay. If the Higgs field has a high amplitude at the end of inflation, it can also have a similar effect on other fields.

In fact, it has even been postulated that the Higgs field could be the inflaton [1,3]. To give rise to inflation, the inflaton potential has to be flat enough so that the energy density remains approximately constant in spite of the expansion of the Universe. As such, the Higgs potential is too steep for this, but if the Higgs field has a strong non-minimal coupling to space–time curvature *ξ*∼10^{4}, then the effective potential in the Einstein frame becomes flat at high field values [1,3]. In that case, inflation could actually be driven by the Higgs field and there would be no need for a separate inflaton field. This scenario is known as Higgs inflation. It makes definite predictions for the spectral index *n*_{s} and the tensor/scalar ratio *r*, which are in good agreement with the current CMB data. However, Espinosa [1] argues that the model suffers from unsolved theoretical problems, especially the unitarity problem associated with a low cut-off scale. Garcia-Bellido [3] discusses two potential ways around these problems: Higgs dilaton inflation and critical Higgs inflation.

## 5. Gravitational effects

The discovery of gravitational waves by LIGO and VIRGO interferometers opens a new way to observe the early Universe and, potentially, the dynamics of the Higgs field. The gravitational wave signals detected so far have been from black hole and neutron star mergers rather than cosmological sources, but even these can be important for Higgs cosmology. Indeed, the behaviour of the Higgs field in the presence of black holes is discussed by Traykova *et al.* [7].

The conventional assumption is that the merging black holes detected by LIGO/VIRGO were produced by stars that collapsed under gravity. However, there are scenarios according to which they were primordial; for example, critical Higgs inflation, as discussed by Garcia-Bellido [3]. In this theory, the running of the Higgs self-coupling has a critical point, leading to a flat plateau in the Higgs potential. This gives rise to a high peak in the power spectrum, and to production of solar-mass-scale primordial black holes, which would be a dark matter component and could be the black holes detected by LIGO and VIRGO.

Gravitational waves produced in the early Universe may be observable in the future in the form of a stochastic background. The frequency of the waves depends on the cosmological era in which they were produced and on the process that produced them. In most inflationary models, the frequency of gravitational waves produced at the end of inflation is in the MHz–GHz range, and therefore far too high for current or planned gravitational wave detectors. However, as discussed by Weir [8], the frequency of gravitational waves produced by a first-order phase transition at the electroweak scale would be in the range where LISA, the future space-based gravitational wave observatory, has its highest sensitivity. Using data from numerical simulations, Weir [8] reviews how the gravitational waves are produced in phase transitions and argues that if the electroweak phase transition was of first order, as required by electroweak baryogenesis, then these gravitational waves are a realistic candidate for detection by future gravitational wave detectors such as LISA.

## 6. Conclusion

As the articles in this theme issue demonstrate, Higgs cosmology has the potential to shed new light on many open questions in cosmology. It is likely to grow and become increasingly important as new results from the LHC provide more precise data on the properties of the Higgs field and other particles, and new data from cosmological observations and gravitational wave detectors place stronger constraints on cosmological scenarios. Hopefully, this theme issue will be a useful resource for physicists working in or entering this field.

## Data accessibility

This article has no additional data.

## Competing interests

I declare I have no competing interests.

## Funding

The author was supported by STFC grant nos. ST/L00044X/1 and ST/P000762/1.

## Acknowledgements

I thank Astrid Eichhorn, Malcolm Fairbairn and Tommi Markkanen as the co-organizers of the Theo Murphy discussion meeting ‘Higgs cosmology’, as well as the Royal Society staff, for making this very stimulating meeting possible.

## Footnotes

One contribution of 9 to a Theo Murphy meeting issue ‘Higgs cosmology’.

- Accepted November 27, 2017.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.