The Higgs boson and cosmology

Abstract

I will discuss how the Higgs field of the Standard Model may have played an important role in cosmology, leading to the homogeneity, isotropy and flatness of the Universe; producing the quantum fluctuations that seed structure formation; triggering the radiation-dominated era of the hot Big Bang; and contributing to the processes of baryogenesis and dark matter production.

1. Introduction

A Higgs boson-like particle with mass ≃126 GeV has recently been discovered at CERN [1,2], and thus we now have a theory of the strong, weak, electromagnetic and gravitational interactions that may be a self-consistent effective field theory all the way up to the Planck scale E∼M_P≃2.44×10¹⁸ GeV; see a recent discussion in [3–7]. Of course, this does not mean that the Standard Model (SM) is a correct theory of Nature up to these energies, as this would be an enormous extrapolation of known physics into a region that is not accessible to present experiments. Nevertheless, since the assumption that the SM is valid up to the Planck scale is the most conservative option available to us, it has more predictive power than any other approach.

The Higgs boson is a very special particle in the SM. It provides a mechanism for including weakly interacting massive vector bosons in the SM, and for ‘giving’ masses to quarks and leptons. As will be discussed in this paper, the Higgs field may also have had an important role in cosmology: it could have made the Universe flat, homogeneous and isotropic, it could have produced the fluctuations that led to structure formation and it could also have enabled the radiation-dominated epoch of the hot Big Bang to occur [8–10]. Moreover, in the modest extension of the SM by three relatively light Majorana fermions—heavy neutral leptons (HNLs)—the Higgs field is important for baryogenesis, leading to the charge asymmetric Universe, and for dark matter production [11]. In addition, active neutrino masses and mixing are induced via Yukawa couplings of both HNLs and left-handed neutrinos to the Higgs field.

2. Higgs field and gravity

In order to embed the SM in the cosmological framework, one has to fix the coupling of the Higgs field to gravity. In addition to the replacement of the Minkowski metric η_μν by the generic curved space metric g_μν, one should add to the Lagrangian a non-minimal coupling of the Higgs field to gravity (e.g. [12]):

2.1

Here R is the scalar curvature, g is the determinant of the metric, H is the Higgs field and ξ is a new dimensionless coupling constant of the SM. As is the case with other couplings of the SM, the value of ξ cannot be fixed from within the model, but instead can be determined by specific experiments (cosmological observations in our case).

To elucidate the role of the non-minimal coupling ξ, let us consider large Higgs fields , (h²=H^†H/2), which may have existed in the early Universe. In this regime, the Higgs field not only gives masses ∝h to fermions and vector bosons, but also determines the gravity interaction strength, which is simply the inverse coefficient in front of the scalar curvature R: . In the limit of large Higgs fields, physical observables do not depend on h, as in all dimensionless ratios the magnitude of h cancels out; M_W/M^eff_P is one such example. In particular, the physical effective potential (obtained by transforming the theory from the so-called Jordan frame to the Einstein frame) does not depend on the Higgs field, as depicted in figure 1. It has been shown [13] that the form of the potential is not changed by perturbative higher order corrections, provided the mass of the Higgs boson obeys the requirement (see also [14]):

2.2

Here

2.3

where y_t(μ_t) is the top Yukawa coupling in the scheme taken at μ_t=173.2 GeV, and α_s(M_Z) is the strong coupling at the scale μ=M_Z. The theoretical uncertainty in M_crit is very small–approximately 70 MeV (see [6] and the discussion in [4,7]). The comparison of M_crit with experiment for ξ∼1 is presented in figure 2. There is 1–2 s.d. tension between the experimental values of the top and Higgs masses and the bound (2.2), with the main uncertainty coming from M_t; it is therefore imperative to obtain more precise measurements at the future e⁺e⁻ collider. In what follows I will assume that (2.2) is satisfied.

Figure 1.

Effective potential for a canonically normalized scalar field χ (related to the Higgs field in a way that is well understood) in the Einstein frame. For large h (or χ) the potential is flat.

Figure 2.

The shaded regions account for 1 and 2 s.d. experimental uncertainties in α_s (α_s=0.1184±0.0007) and the pole top quark mass M_t (Tevatron: M_t= 173.2±0.51±0.71 GeV), and also include theoretical errors in extraction of y_t from experiment. The thick straight line (blue in the online version) marks the relation between α_s and the pole top mass following from equation (2.3) if M_H is identified with M_crit. The enclosing shaded regions correspond to 1 and 2 s.d. experimental errors in the Higgs mass. Small ellipses (red in the online version) correspond to the accuracy achievable at the e⁺e⁻ collider [15]. (Online version in colour.)

3. Higgs boson, cosmological inflation and the hot Big Bang

It is well known that a number of important cosmological problems, such as the flatness, isotropy and homogeneity of the Universe, can be solved simultaneously by the accelerated expansion of the Universe in the distant past. This epoch of inflation is expected to be driven by some scalar field called the inflaton. The Higgs field is the only scalar field included within the framework of the SM, and in [8] it was shown that the Higgs field is a good inflaton candidate.

The evolution of the Universe with the Higgs field playing the role of the inflaton (Higgs inflation) proceeds as follows [9,10]. At the first stage, as is the case in any chaotic inflation scenario [16], the value of the Higgs field is large (), and it rolls slowly towards the minimum of the potential in figure 1. The potential energy of the Higgs field leads to the exponential expansion of the Universe, which then becomes flat, homogeneous and isotropic. The small-scale quantum fluctuations of the Higgs field are inflated and seed structure formation, thus leading to the creation of galaxies and clusters of galaxies.

After the Higgs field reaches the value , the slow roll ends, and the Higgs field starts to oscillate. The exponential expansion of the Universe becomes a power law, corresponding to matter domination. The Higgs field oscillations lead to the creation of the particles of the SM that couple most strongly to H, namely to intermediate vector bosons W and Z and the top quark. Eventually, W and Z thermalize through decays and inverse decays into fermions of the SM. Owing to all these processes the decay of the scalar field is completed sometime before the amplitude of the Higgs field reaches the value h≃M_P/ξ. As a result, the Universe is heated up to the temperature T_reh∼10¹⁴ GeV [9,10]. This is the start of the hot Big Bang stage of the Universe evolution, when the Universe is dominated by radiation.

The cosmological predictions of Higgs inflation can be compared with observations performed by the Planck satellite. The Higgs-inflaton potential depends on one unknown parameter, ξ. It can be fixed by the amplitude of the cosmic microwave background temperature fluctuations δT/T at the Wilkinson Microwave Anisotropy Probe normalization scale approximately 500 Mpc, with the use of precise knowledge of the top quark and Higgs masses, and with α_s. In general, ξ>600 [13]. The relatively large value of the non-minimal coupling ξ plays an essential role in the self-consistency of the Higgs inflation (see [14] and references therein). Since the Higgs mass lies near M_crit, the actual value of ξ may be close to the lower bound. In addition, the value of the spectral index, n_s, of scalar density perturbations and the amplitude of tensor perturbations r=δρ_s/δρ_t can be determined. The predictions, together with the Planck results, are presented in figure 3 and are well inside the 1 s.d. experimental contour. Moreover, as is the case for most single-field inflationary models, the perturbations are Gaussian, and therefore in excellent agreement with Planck [17].

Figure 3.

(a) Predictions for the inflationary indexes n_s and r for Higgs inflation. (b) Predictions for different inflationary models contrasted with the Planck results (adapted from [17]). The definitions of different models and measurements can be found in [17]. (Online version in colour.)

4. Minimal physics beyond the Standard Model

In order to continue this discussion, we need to consider physics beyond that described by the SM, since the SM cannot explain the matter–antimatter asymmetry of the Universe or the nature of dark matter. Towards this end, we refer to the minimal extension of the SM by three Majorana leptons: the νMSM (for neutrino minimal SM). It is based on exactly the same gauge group as the SM and does not introduce any new particle physics scale (figure 4). These three new particles, the HNLs or simply ‘Ns’—with masses in the range keV to GeV can explain simultaneously neutrino masses and oscillations, dark matter and the baryon asymmetry of the Universe [11] (for a review, see [18]).

Figure 4.

Particle content of the νMSM. In the SM the right-handed partners of neutrinos are absent. In the νMSM, all fermions have both left- and right-handed components and masses below the Fermi scale. (Online version in colour.)

The HNLs interact with the Higgs boson via Yukawa interactions in exactly the same way the other fermions do: (cf. with the name HNL), where the L_α are left leptonic doublets, ‘c’ is the sign of the charge conjugation and ‘T’ is the sign of the matrix transpose. These interactions lead to active neutrino masses via the GeV scale see-saw mechanism, the creation of matter–antimatter asymmetry at temperatures T∼100 GeV, and finally dark matter production at T∼100 MeV, as outlined below and in figure 5.

Figure 5.

A possible history of the Universe from inflation until the present time. (Online version in colour.)

(a). Baryogenesis

In the νMSM, nothing essentially interesting happens between temperatures 10³ and 10¹³ GeV, as all the SM elementary particles are nearly in thermal equilibrium. At the same time, the HNLs N_2,3 are out of equilibrium due to the tiny values of their Yukawa couplings. HNLs are created in interactions with real or virtual Higgs bosons in processes like , , and come into thermal equilibrium at T∼100 GeV. The charge–parity (CP) violation in these reactions leads to lepton asymmetry [19,11], which is then converted to baryon asymmetry of the Universe by SM sphalerons. The baryon number violating processes freezes out at T≃140 GeV.

(b). Electroweak crossover

Yet another event, though without observable consequences, happens around temperatures T∼100 GeV. The Higgs field expectation value grows from small values up to the zero temperature value 250 GeV. For the experimentally determined value of the Higgs mass, there is no phase transition [20], but instead a smooth crossover with the temperature given in the lowest order of perturbation theory by

4.1

and shifted by non-perturbative effects to T_c≃160 GeV [21]. The phase diagram of the electroweak theory on temperature versus Higgs mass plane is similar to the phase diagram of the vapour–liquid system on the pressure–temperature plane: one can transfer from the symmetric ‘phase’ (vapour phase) to the Higgs ‘phase’ (liquid phase) continuously, without crossing any phase transition boundary (figure 6).

Figure 6.

The phase diagram of the electroweak theory and of the vapour–liquid system. The critical point (end point of the line of the first-order phase transition) has coordinates T^crit=109.2±0.8 GeV, M^crit_H=72.3±0.7 GeV. (Online version in colour.)

(c). Dark matter

The N₁ HNL can be sufficiently stable to play the role of the dark matter particle [22–29]. It is produced in the early Universe via processes like (figure 7) at a temperature in the region of:

4.2

and never come into thermal equilibrium since their interaction strength is very small. The Higgs field is essential for these processes, as it induces the mixing between HNLs and active neutrinos via Yukawa couplings. For a review of the various constraints on N₁ properties, see [18].

Figure 7.

Production of dark matter particles via process . The Higgs vacuum expectation value leads to N−ν mixing. (Online version in colour.)

5. Experiments

Of course, the picture of the Universe evolution presented above is based on the extrapolation of known physics up to high-energy scales and on specific hypotheses about physics beyond the SM. Since the model of new physics—the νMSM—is extremely minimalistic, it is highly predictive and thus can be tested in future experiments.

In order to check the self-consistency of the SM and of the νMSM up to the inflationary scale, we should determine the mass of the Higgs boson, the top Yukawa coupling and strong coupling constant with the highest possible accuracy (figure 2). This is one of the arguments in favour of the future e⁺e⁻ collider: a top quark factory in which a precision measurement of the top quark mass would be possible. In order to verify the cosmological predictions of Higgs inflation, and to distinguish it from other models, such as Starobinsky R² inflation [30], we need to have measurements of the spectral index n_s of scalar perturbations at the level of 10⁻³ (figure 8). In addition, the determination of the tensor-to-scalar ratio should be performed down to values of r≃0.003 and that of the running of the spectral index down to 5×10⁻⁴. These measurements are thought to be possible at COrE (http://www.core-mission.org/science.php), PRISM (http://www.prism-mission.org/) and SKA (http://www.skatelescope.org/).

Figure 8.

Predictions of the Higgs inflation versus other models. The accuracy of the Polarized Radiation Imaging and Spectroscopy Mission (PRISM) is indicated. (Online version in colour.)

In order to verify the mechanism for baryogenesis and neutrino mass generation, one should experimentally look for new particles—the HNLs N_2,3. This is a challenging task, since they must be very weakly coupled to satisfy the Sakharov condition of out of equilibrium. The expression of interest to search for HNLs [31] has recently been submitted to the CERN SPS Committee; see also http://ship.web.cern.ch/ship/.

The nature of the dark matter HNL, N₁, is very different from that of weakly interacting massive particles (WIMPs). This dark matter candidate can be observed with the help of high-resolution X-ray telescopes (for a review, see [32]), which can look for HNL decays N₁→νγ by detecting monochromatic photons from regions where dark matter is concentrated. Some indications for the existence of an unidentified X-ray line which may correspond to the HNL dark matter particle with mass 7.1 keV have recently been reported in [33,34].

6. Conclusion

The SM Higgs field could play an important role in cosmology:

• — it could help generate a universe that is flat, homogeneous and isotropic;

• — quantum fluctuations of the Higgs field could lead to structure formation;

• — coherent oscillations of the Higgs field could make the hot Big Bang and produce all the matter in the Universe;

• — real and virtual Higgs bosons could play a crucial role in baryogenesis, leading to a charge asymmetric Universe;

• — dark matter production may come about as an effect of mixing between neutrinos and HNLs, induced by the Higgs field; and

• — several new experiments are needed to reveal the ‘secret’ couplings of the Higgs boson.

Funding statement

This work was supported by the Swiss National Science Foundation. I am grateful to Mark Lovell for careful reading of the manuscript and important suggestions.