Primordial black holes from Higgs inflation with a Gauss-Bonnet coupling

If there were over-density regions in the early universe, primordial black holes (PBHs) may form as a result of the gravitational collapse during the radiation-dominated epoch Zel’dovich and Novikov (1967); Hawking (1971); Carr and Hawking (1974). Unlike astrophysical black holes, the PBHs can have a wide range of masses and can be the source for all or a part of dark matter (DM) Chapline (1975); Meszaros (1975) (see Refs. Khlopov (2010); Sasaki et al. (2018); Carr and Kuhnel (2020); Green and Kavanagh (2021); Villanueva-Domingo et al. (2021); Carr and Kuhnel (2022); Escrivà et al. (2022); Karam et al. (2022) for recent reviews). Although PBHs have not been observationally discovered yet, the detection of gravitational waves from binary black holes Abbott et al. (2016, 2019, 2021a, 2021b) suggested a possibility that they may arise from non-stellar origins Bird et al. (2016); Sasaki et al. (2016); Clesse and García-Bellido (2017); Wang et al. (2018). In addition, PBHs have been also considered as possible seeds of supermassive black holes in the center of galaxies Bean and Magueijo (2002). Various observations have given the upper limit to their abundance $f$ as a function of mass $M$. In particular, the mass window in which all DM can be explained by PBHs exists in the range $10^{-16}M_{\odot}\lesssim M\lesssim 10^{-11}M_{\odot}$, where $M_{\odot}$ is a solar mass (see Ref. Carr et al. (2021) for a recent study).

Inflation can provide a possible framework for generating the seed of PBHs on scales smaller than the observed cosmic microwave background (CMB) temperature anisotropies Ivanov et al. (1994); Garcia-Bellido et al. (1996); Bullock and Primack (1997); Yokoyama (1997, 1998); Kawasaki et al. (1998, 2006); Kohri et al. (2008); Saito et al. (2008); Bugaev and Klimai (2008); Alabidi and Kohri (2009); Drees and Erfani (2011, 2012); Martin et al. (2013); Kohri et al. (2013); Kawasaki et al. (2013); Clesse and García-Bellido (2015); Kawasaki and Tada (2016); Kawasaki et al. (2016); Pi et al. (2018); Garcia-Bellido and Ruiz Morales (2017); Kannike et al. (2017); Germani and Prokopec (2017); Ando et al. (2018); Ezquiaga et al. (2018); Motohashi and Hu (2017); Di and Gong (2018); Ballesteros and Taoso (2018); Garcia-Bellido et al. (2017); Hertzberg and Yamada (2018); Inomata et al. (2018); Cai et al. (2018a); Drees and Xu (2021); Atal et al. (2019, 2020); Mishra and Sahni (2020); Cheong et al. (2021); Fu et al. (2019); Dalianis et al. (2020); Ashoorioon et al. (2021); Lin et al. (2020); Yi et al. (2021); Palma et al. (2020); Braglia et al. (2020); Kefala et al. (2021); Ballesteros et al. (2020); Aldabergenov et al. (2020, 2021); Inomata et al. (2021, 2022a); Dalianis et al. (2021); Cai et al. (2022a); Lin et al. (2021); Kawai and Kim (2021a); Zhang (2022); Ahmed et al. (2022); Cai et al. (2022b); Pi and Wang (2022); Cheong et al. (2022); Kawai and Kim (2022). If there is an intermediate stage in which the velocity $\dot{\phi}$ of an inflaton field rapidly decreases toward 0 during inflation, it is possible to enhance curvature perturbations at particular scales. In the presence of an inflection point in the inflaton potential $V(\phi)$ around which the derivative ${\rm d}V/{\rm d}\phi$ is close to 0, the field velocity decreases as $\dot{\phi}\propto a^{-3}$ in the ultra-slow-roll (USR) regime, where $a$ is a scale factor Garcia-Bellido and Ruiz Morales (2017); Kannike et al. (2017); Germani and Prokopec (2017); Ezquiaga et al. (2018); Motohashi and Hu (2017); Di and Gong (2018); Ballesteros and Taoso (2018); Hertzberg and Yamada (2018); Drees and Xu (2021); Ballesteros et al. (2020). In many of these models, we require a tuning of model parameters to generate a plateau region of the potential. Moreover, if the number of e-foldings acquired in the USR regime exceeds the order 10, the scalar spectral index $n_{s}$ on CMB scales tends to be inconsistent with the value constrained by the Planck data Akrami et al. (2020).

There are also models containing one or more bumps/dips or steps in the potential Atal et al. (2019, 2020); Mishra and Sahni (2020); Kefala et al. (2021); Inomata et al. (2021, 2022a); Dalianis et al. (2021); Cai et al. (2022a, b); Pi and Wang (2022). In this case, the scalar field rapidly loses its kinetic energy around them, resulting in a strong enhancement of curvature perturbations. It is also known that oscillating features can appear in the scalar power spectrum especially for the step-type potential. One advantage of these models is that the number of e-foldings during transition can be as small as the order 1. This allows a possibility for the compatibility of models with the observed values of $n_{s}$ and tensor-to-scalar ratio $r$. There are also multi-field inflationary models leading to the enhancement of curvature perturbations at particular scales Yokoyama (1997); Garcia-Bellido et al. (1996); Kawasaki et al. (1998, 2013); Kohri et al. (2013); Ando et al. (2018); Aldabergenov et al. (2020, 2021); Palma et al. (2020); Braglia et al. (2020); Cheong et al. (2022); Kawai and Kim (2022). In this case, we need to address whether the presence of entropy perturbations does not contradict with CMB constraints on the isocurvature mode. The possibility for producing the seed of PBHs during preheating after inflation was also discussed in Refs. Green and Malik (2001); Bassett and Tsujikawa (2001); Suyama et al. (2005); Martin et al. (2020).

One of the advantages of PBHs as DM is that the origin of DM can be explained within the framework of Standard Model (SM) of particle physics. A minimal model of inflation without introducing additional scalar degrees of freedom to those appearing in SM is known as Higgs inflation, in which the Higgs field $\phi$ is nonminimally coupled to gravity Bezrukov and Shaposhnikov (2008); Bezrukov et al. (2009, 2011) (see Refs. Futamase and Maeda (1989); Fakir and Unruh (1990) for early related works). Indeed, this model is perfectly consistent with observational bounds on $n_{s}$ and $r$ constrained by the CMB data Komatsu and Futamase (1999); Tsujikawa and Gumjudpai (2004); Linde et al. (2011); Ade et al. (2014); Tsujikawa et al. (2013). If we allow the runnings of Higgs self-coupling $\lambda(\phi)$ and nonminimal coupling $\xi(\phi)$, then it is possible to have an inflection point in the Higgs potential. This gives rise to a plateau region in which the enhancement of curvature perturbations occurs to generate the seed of PBHs Ezquiaga et al. (2018); Drees and Xu (2021); Yi et al. (2021); Lin et al. (2021). In this scenario the typical number of e-foldings acquired during the USR regime is of order 10, which results in the values of CMB observables deviating from those in standard Higgs inflation. The model can be consistent with the current CMB observations, but it is typically outside the $1\sigma$ contour constrained by the Planck data Ezquiaga et al. (2018).

Recently, Kawai and Kim Kawai and Kim (2021a) proposed a single-field inflationary scenario in which the inflaton field is coupled to a Gauss-Bonnet (GB) curvature invariant $R_{\rm GB}^{2}$ of the form $\mu(\phi)R_{\rm GB}^{2}$. A scalar-field dependent GB coupling $\mu(\phi)$ can give rise to an inflection point $\phi=\phi_{c}$ in an effective potential of the inflaton. On the other hand, we have to caution that a large contribution from the scalar-GB coupling to the inflaton energy density modifies the primordial scalar power spectrum on CMB scales Hwang and Noh (2005); Guo et al. (2007); Satoh and Soda (2008); Guo and Schwarz (2009, 2010); Kawai and Kim (2021b). Moreover, the scalar-GB coupling leads to the propagation speeds of scalar and tensor perturbations different from the speed of light Kobayashi et al. (2011); Kase and Tsujikawa (2019). Since the Laplacian instabilities associated with negative squared propagation speeds may arise, we need to make sure whether the stability conditions are not violated during inflation.

In Ref. Kawai and Kim (2021a), the scalar-GB coupling $\mu(\phi)=\mu_{0}\tanh[\mu_{1}(\phi-\phi_{c})]$ was proposed to generate the seed of PBHs around the inflection point $\phi=\phi_{c}$. Since $\mu(\phi)$ approaches constants in the two asymptotic regimes $\phi\ll\phi_{c}$ and $\phi\gg\phi_{c}$, the scalar-GB coupling is important only in the vicinity of $\phi=\phi_{c}$. A temporal USR region can arise from the balance between the GB term and the scalar potential. The enhancement of curvature perturbations in such a transient epoch was studied for natural inflation Kawai and Kim (2021a) and for $\alpha$-attractor Zhang (2022). In these papers, the authors mostly focused on the USR regime realized by a plateau-type effective potential. In this case, the CMB observables are subject to modifications by the presence of a plateau-region with the number of e-foldings of order 10. Hence it is nontrivial to produce the large amplitude of primordial scalar perturbations responsible for the seed of PBHs, while satisfying CMB constraints on $n_{s}$ and $r$.

In this paper, we will address this issue in Higgs inflation with a scalar-GB coupling mentioned above. We will not incorporate the runnings of Higgs and nonminimal couplings to focus on effects of the scalar-GB coupling on the background and perturbations. We show that, besides the plateau-type effective potential, it is possible to realize a bump- or step-type effective potential. In this latter case, the field velocity $\dot{\phi}$ around $\phi=\phi_{c}$ decreases faster in comparison to the USR regime with a smaller number of e-foldings of order 1. The primordial scalar power spectrum can also have a sharp feature with a peak amplitude enhanced by a factor of $10^{7}$. In such cases, PBHs can be the source for all DM in the mass range $10^{-16}M_{\odot}\lesssim M\lesssim 10^{-13}M_{\odot}$. Moreover, the bump-type effective potential can give rise to the values of $n_{s}$ and $r$ inside the $1\sigma$ observational contour constrained by the Planck CMB data. There are also intermediate-type potentials between plateau- and bump-types consistent with the CMB constraints, while generating the seed of PBHs. Thus, our inflationary scenario provides a versatile possibility for realizing various shapes of the effective scalar potential. We note that each shape of potentials was discussed separately in different contexts in the literature.

This paper is organized as follows. In Sec. II, we obtain the background equations of motion in Higgs inflation with a scalar-GB coupling $\mu(\phi)R_{\rm GB}^{2}$ and revisit the scalar and tensor power spectra generated in slow-roll Higgs inflation with $\mu(\phi)=0$. In Sec. III, we derive an effective potential $V_{\rm eff}(\phi)$ of the inflaton field and classify it into three classes: (1) plateau-type, (2) bump-type, and (3) intermediate-type. In Sec. IV, we compute the primordial scalar power spectra for three sets of model parameters with which there are neither ghost nor Laplacian instabilities. We show that the bump-type is favored over the plateau-type for the consistency with CMB observables. In Sec. V, we calculate the PBH abundance relative to the relic DM density and show that our model produces a sufficient amount of PBHs that can be the source for all DM. Sec. VI is devoted to conclusions. Throughout the paper, we use the natural units ($c=\hbar=1$).

We begin with theories given by the action

where $g$ is a determinant of the metric tensor $g_{\mu\nu}$, $M_{\rm Pl}$ is the reduced Planck mass, $\xi$ is a nonminimal coupling constant, $\phi$ is a scalar field with the covariant derivative operator $\nabla_{\mu}$, and $R$ is the Ricci scalar. The scalar field has a potential of the form

where $\lambda$ is a positive coupling constant. The dynamics of nonminimally coupled inflation with the potential (2) was originally addressed in Refs. Futamase and Maeda (1989); Fakir and Unruh (1990) (see also Refs. Salopek et al. (1989); Makino and Sasaki (1991); Kaiser (1995); Komatsu and Futamase (1999); Tsujikawa and Gumjudpai (2004)). It can also accommodate the Higgs potential $V(\phi)=\lambda(\phi^{2}-v^{2})^{2}/4$ in the large field regime $\phi^{2}\gg v^{2}$, where $v={\cal O}(10^{2})$ GeV Bezrukov and Shaposhnikov (2008); Bezrukov et al. (2009, 2011). Provided that the nonminimal coupling is in the range

the self-coupling of order $\lambda=0.01\sim 0.1$ can be consistent with the amplitude of observed CMB temperature anisotropies³³3If we consider quantum corrections arising from the renormalization group running of the standard model, the Higgs self-coupling $\lambda$ can be much smaller than 0.01 or even negative at inflationary energy scales De Simone et al. (2009); Hamada et al. (2014, 2015); Bezrukov et al. (2015, 2018). In this paper, we do not consider the runnings of coupling constants $\lambda$ or $\xi$..

The scalar field is coupled to a GB curvature invariant defined by

with a $\phi$-dependent coupling function $\mu(\phi)$, where $R_{\mu\nu}$ and $R_{\mu\nu\rho\sigma}$ are the Ricci and Riemann tensors, respectively. The action (1) belongs to a subclass of Horndeski theories with second-order field equations of motion Horndeski (1974); Kobayashi et al. (2011); Deffayet et al. (2011); Charmousis et al. (2012) (see Appendix. A). In this case, there is only one propagating scalar degree of freedom besides two tensor polarizations. As we will study in Sec. III, it is possible to enhance scalar perturbations at particular scales for a specific choice of $\mu(\phi)$. The action (1) corresponds to Higgs inflation corrected by the Higgs-GB coupling. This allows a possibility for generating the seed of PBHs as the source for all DM within the framework of SM of particle physics.

We note that the $\xi\to 0$ limit in the action (1) with the potential of natural inflation corresponds to the model studied by Kawai and Kim Kawai and Kim (2021a). In our model, the basic mechanism for the generation of seeds of PBHs is similar to that advocated in Ref. Kawai and Kim (2021a). However, natural inflation is in tension with the observation of CMB temperature anisotropies Akrami et al. (2020). Instead, we would like to construct an explicit inflationary model consistent with CMB observations, while enhancing curvature perturbations on scales relevant to PBHs. As we will show later, this is indeed possible for Higgs inflation with $\xi\gg 1$ in the presence of the scalar-GB coupling.

For the background, we consider a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) line element given by

where $a(t)$ is a time-dependent scale factor. On this background, the Friedmann and scalar-field equations of motion are

where a dot represents a derivative with respect to $t$, $H=\dot{a}/a$ is the Hubble expansion rate, and we use the notations $\mu_{,\phi}={\rm d}\mu/{\rm d}\phi$ and $V_{,\phi}={\rm d}V/{\rm d}\phi$. Taking the time derivative of Eq. (6) and using Eq. (7), we obtain the closed-form differential equations

where

Notice that $V_{\rm eff}$ is an effective potential of the scalar field. Numerically, we solve Eqs. (8) and (9) for $H$ and $\phi$ with the initial conditions of $H$, $\phi$, and $\dot{\phi}$ consistent with the Hamiltonian constraint (6).

Let us proceed to the case in which the scalar-GB coupling $\mu(\phi)R_{\rm GB}^{2}$ is present. If $\mu(\phi)$ is a smooth function whose time variation is small during inflation, the inflaton field $\phi$ can slowly evolve along the potential. In such a case, the primordial power spectra of scalar and tensor perturbations are given by Eq. (24). Since $Q_{s}$ is proportional to $\dot{\phi}^{2}$, a smaller inflaton velocity generally leads to a larger amplitude of ${\cal P}_{\zeta}$. In the context of slow-roll inflation, however, this enhancement of ${\cal P}_{\zeta}$ is limited by a small time variation of $\dot{\phi}$.

If the scalar-GB coupling generates a period in which the field velocity $\dot{\phi}$ temporally approaches 0, it is possible to realize the large enhancement of ${\cal P}_{\zeta}$ for scales smaller than those of the observed CMB temperature anisotropies. One possible choice of $\mu(\phi)$ is a dilatonic coupling of the form $\mu(\phi)=\mu_{0}e^{-\lambda\phi}$ Gross and Sloan (1987); Metsaev and Tseytlin (1987); Gasperini et al. (1997); Kawai et al. (1998); Cartier et al. (2001); Calcagni et al. (2005); Guo et al. (2007). However, this type of continuously varying functions affects not only the scalar perturbation on particular scales but also that on other scales including CMB. Moreover, it can happen that the dominance of the scalar-GB coupling over the potential and nonminimal couplings leads to the violation of stability conditions (18).

Instead, we consider the step-like coupling given by Kawai and Kim (2021a); Zhang (2022); Khan and Yogesh (2022)

where $\mu_{0}$, $\mu_{1}$, and $\phi_{c}$ are constants. Around the field value $\phi=\phi_{c}$, this coupling rapidly changes from the asymptotic constant $\mu(\phi)=-\mu_{0}$ (for $\phi\ll\phi_{c}$) to the other asymptotic constant $\mu(\phi)=+\mu_{0}$ (for $\phi\gg\phi_{c}$). Since the GB curvature invariant is a topological term, the scalar-GB coupling does not affect the cosmological dynamics in the two asymptotic regimes with constant $\mu(\phi)$. Provided that $\phi_{c}$ is in the range $\phi_{f}<\phi_{c}<\phi_{\rm CMB}$, where $\phi_{\rm CMB}$ is the field value about 60 e-foldings before the end of inflation (with the field value $\phi_{f}$), it should be possible to enhance the scalar power spectrum for scales smaller than those of observed CMB temperature anisotropies. In Sec. IV, we will study whether the sufficient generation of seeds for PBHs is possible, while satisfying observational constraints on $n_{s}$ and $r$ on CMB scales.

Provided that the field kinetic term is sufficiently small during inflation, we can ignore the $\dot{\phi}$-dependent terms in Eq. (10). Then, the $\phi$-derivative of the effective potential $V_{\rm eff}$ is approximately given by

From Eq. (6), the Hubble parameter is approximately given by

Substituting Eq. (31) into Eq. (30) and integrating it with respect to $\phi$, the effective potential $V_{\rm eff}$ can be numerically known as a function of $\phi$.

In the following, we classify the effective potential into three classes: (1) plateau-type, (2) bump-type, and (3) intermediate-type. The set 1, 2, 3 model parameters shown in Table 1 are the typical examples of plateau-, bump-, and intermediate-types, respectively. The nonminimal coupling constant is fixed to be $\xi=5000$ in all cases.

Since PBHs are treated as the main component of DM in this paper, we consider the PBH mass range $10^{-16}M_{\odot}\lesssim M\lesssim 10^{-11}M_{\odot}$. This gives a constraint on the value of $\phi_{c}$. The two constants $\mu_{0}$ and $\mu_{1}$ determine the types of $V_{\rm eff}(\phi)$ mentioned above. Instead of the parameter $\mu_{0}$, we will use the combination

which appears later in Eq. (36). The values of $\tilde{\mu}_{0}$ are chosen to be close or not far away from the right hand side (RHS) of Eq. (36), see the last two columns in Table 1. The Higgs self-coupling $\lambda$ is determined by the observed amplitude of primordial curvature perturbations on CMB scales. As we see in Table 1, $\lambda$ is of order 0.01 for three sets of model parameters.

In this section, we study how the power spectrum of curvature perturbations is enhanced by the presence of the scalar-GB coupling (29). As we discussed in Sec. III, the inflaton effective potential $V_{\rm eff}(\phi)$ can be classified into three classes: (1) plateau-type, (2) bump-type, and (3) intermediate-type. Examples of the model parameters corresponding to each type of $V_{\rm eff}(\phi)$ are given in Table 1 as Sets 1, 2, 3, respectively. In Sec. IV.1, we first discuss whether each model satisfies the stability conditions (18). In Sec. IV.2, we compute the primordial scalar power spectra by paying particular attention to the enhancement of curvature perturbations around $\phi=\phi_{c}$. In Sec. IV.3, we confront our models with the observed scalar spectral index and tensor-to-scalar ratio on CMB scales.