Primordial black holes and induced gravitational waves from inflation in the Horndeski theory of gravity

Cosmic inflation, a period of accelerated expansion of the early universe, provides an excellent solution to cosmological problems, including the horizon, flatness, and monopole problems Guth:1980zm ; Linde:1981mu ; Albrecht:1982wi ; Sato:1980yn , and is a mechanism to generate the primordial seeds required to explain the observed large-scale structure in the universe Mukhanov:1981xt ; Hawking:1982cz ; Starobinsky:1982ee ; Guth:1982ec ; Bardeen:1983qw ; Kodama:1985bj . The most recent cosmic microwave background (CMB) data show that successful inflation models should predict a nearly scalar invariant and quasi-adiabatic Gaussian primordial curvature power spectrum with the amplitude of $\mathcal{P}_{S}(k_{\ast})=2.1\times 10^{-9}$ at the pivot scale $k_{\ast}=0.05~{}\text{Mpc}^{-1}$ and the spectral tilt of $n_{S}=0.965\pm 0.004$ Ade:2013zuv . Inflation models also predict that the second-order scalar perturbations during inflation can source the gravitational waves (GWs) Starobinsky:1979ty ; Allen:1987bk ; Sahni:1990tx ; Matarrese:1992rp ; Noh:2004bc ; Nakamura:2004rm ; Ananda:2006af ; Baumann:2007zm ; Alabidi:2012ex . Thus, GWs induced by the second-order scalar perturbations have attracted much attention in recent years Kohri:2018awv ; Cai:2018dig ; Fu:2019vqc ; Inomata:2018epa ; Inomata:2019zqy ; Domenech:2020kqm . Such scalar-induced GWs can be sizable if the curvature perturbations are enhanced at scales smaller than the CMB scale Ananda:2006af ; Baumann:2007zm ; Alabidi:2012ex .

The gravitational collapse of over-dense regions in the early universe can form black holes Zeldovich:1967lct ; Hawking:1971ei ; Carr:1974nx ; Carr:1975qj . In other words, the curvature perturbations exceeding a critical value collapses to form a primordial black hole (PBH) in the radiation-dominated (RD) era. The PBH formation is, therefore, a process of the threshold. In order to produce PBHs in the RD era, the primordial power spectrum for scalar modes at small scales should be enhanced by a factor of about $10^{7}$ with respect to its value at the CMB scale Carr:2009jm . Although there is no evidence, it was suggested that PBHs could be a natural candidate for dark matter (DM) Ivanov:1994pa ; Frampton:2010sw ; Belotsky:2014kca ; Carr:2016drx ; Inomata:2017okj ; Carr:2020xqk . PBHs with very light masses are anticipated to Hawking radiate energetically. If the evaporation of PBHs is halted at some point, there remain stable remnants with Planck mass and size, and such remnants of PBHs can also be a natural candidate for DM Adler:2001vs ; Chen:2002tu ; Chen:2004ft ; Scardigli:2010gm ; Dalianis:2019asr . The true nature of DM, however, remains unclear. Therefore, in this work, we focus on the idea of PBHs as DM, which has taken a new flight in recent years due to the absence of well-motivated particle candidates for DM and recent observations of GWs from the binary merger of black holes Bird:2016dcv ; Sasaki:2016jop . PBHs, if they are massive enough to avoid the Hawking evaporation, are considered to provide a significant fraction, if not all, of mysterious DM that dominate cosmic structures in the universe today.

The enhancement in the primordial curvature power spectrum is often associated with a local feature of inflaton potentials, particularly at small scales Ezquiaga:2017fvi ; Mishra:2019pzq ; Cheong:2019vzl . In other words, by specifying the form of inflaton potential, one can provoke a local feature like a peak in the curvature power spectrum on scales relevant for the production of PBHs and GWs. However, not every inflaton potentials induce such enhancement; hence a certain degree of fine-tuning is needed Hertzberg:2017dkh . On the other hand, non-standard models of inflation often introduce additional contributions to the primordial power spectra of scalar and tensor modes; therefore, they could, in principle, give rise to the production of PBHs and secondary GWs without adjusting the inflaton potential Lu:2019sti ; Yi:2020cut ; Gao:2020tsa ; Yi:2020kmq ; Lin:2020goi . Thus, to investigate the enhancement mechanism in the curvature perturbations, we employ a single field inflationary model proposed in Ref Tumurtushaa:2019bmc ; Bayarsaikhan:2020jww . The model acknowledges the effects of the derivative self-interaction of the scalar field and the kinetic coupling between the scalar field and gravity during inflation. However, it is challenging to achieve the correct values for the inflationary observable quantities, including the spectral indices and tensor-to-scalar ratio, on the CMB scale while enhancing the power spectrum on small scales such that the PBHs and GWs can be produced in the RD era. Still, this will be the focus of our work in this paper. The model we study is a subset of the so-called Horndeski theory of gravity Horndeski:1974wa , the most general four-dimensional scalar-tensor theory with equations of motion up to second order in derivatives of the scalar field, and its modern generalized Galileon formulation has been developed in Ref. Deffayet:2011gz ; Kobayashi:2011nu ; Gao:2011vs ; Gao:2012ib ; Deffayet:2013lga .

The organization of the paper is as follows. In Sec. II, we review the single-field potential-driven inflationary models with the derivative self-coupling of the scalar field and the kinetic coupling between the scalar field and gravity. In Sec. III, we discuss the enhancement mechanism of primordial curvature perturbations for our model and present the formulae for the PBHs abundance as DM in Sec. III.1 and the energy spectrum of GWs in Sec. III.2 produced in the RD era. In Sec. IV, we reconstruct the inflaton potential and self-coupling functions from the power spectrum and its spectral tilt. Finally, Sec. V is devoted to conclusion and discussion.

The action of the cosmological models we investigate in this work is given as Tumurtushaa:2019bmc ; Bayarsaikhan:2020jww

$\displaystyle S=\int d^{4}x\sqrt{-g}\left[\frac{M_{pl}^{2}}{2}R-\frac{1}{2}\left(g^{\mu\nu}-\frac{\alpha}{M^{3}}\xi(\phi)g^{\mu\nu}\partial_{\rho}\partial^{\rho}\phi+\frac{\beta}{M^{2}}G^{\mu\nu}\right)\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]\,,$

(1)

where the $\alpha$ and $\beta$ are dimensionless constants. Thus, by rescaling the $\alpha$ and $\beta$ to replace the mass scale $M$ with the reduced Planck mass $M_{pl}$, one can decrease the number of free parameters. The second term inside the parentheses reflects the derivative self-interaction of the scalar field $\phi$, while the third term indicates the kinetic coupling between the scalar field and gravity. $V(\phi)$ and $\xi(\phi)$ are the potential and self-coupling functions of the $\phi$, respectively.

Varying Eq. (1) with respect to spacetime metric $g_{\mu\nu}$, we obtain the Einstein equation

$\displaystyle G_{\mu\nu}=M_{pl}^{-2}T_{\mu\nu}^{\phi}\,,$

(2)

where the energy-momentum tensor for the scalar field is given by

	$\displaystyle T_{\mu\nu}^{\phi}$	$\displaystyle=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(\partial_{\alpha}\phi\partial^{\alpha}\phi+2V\right)\,$
		$\displaystyle+\frac{\alpha}{2M^{3}}\left[(\xi\nabla_{\alpha}\phi\nabla^{\alpha}\phi)_{(\mu}\nabla_{\nu)}\phi-\xi\square\phi\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}(\xi\nabla_{\alpha}\phi\nabla^{\alpha}\phi)_{\beta}\nabla^{\beta}\phi\right]\,$		(3)
		$\displaystyle+\frac{\beta}{M^{2}}\left[-\frac{1}{2}\nabla_{\mu}\phi\nabla_{\nu}\phi R+2\nabla_{\alpha}\phi\nabla_{(\mu}\phi R^{\alpha}\,_{\nu)}+\nabla^{\alpha}\phi\nabla^{\beta}\phi R_{\mu\alpha\nu\beta}+\nabla_{\mu}\nabla^{\alpha}\phi\nabla_{\nu}\nabla_{\alpha}\phi\right.$
		$\displaystyle\left.-\nabla_{\mu}\nabla_{\nu}\phi\square\phi-\frac{1}{2}G_{\mu\nu}\nabla_{\alpha}\phi\nabla^{\alpha}\phi+g_{\mu\nu}\left(-\frac{1}{2}\nabla^{\alpha}\nabla^{\beta}\phi\nabla_{\alpha}\nabla_{\beta}\phi+\frac{1}{2}\left(\square\phi\right)^{2}-\nabla_{\alpha}\phi\nabla_{\beta}\phi R^{\alpha\beta}\right)\right]\,.$

Using the Bianchi identity $\nabla^{\mu}G_{\mu\nu}=0$ in Eq. (2), we obtain the evolution equation for the scalar field as

$\displaystyle\nabla^{\mu}\,T_{\mu\nu}^{\phi}=0\,.$

(4)

In a spatially flat Friedman-Robertson-Walker universe with metric

$\displaystyle ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}\,,$

(5)

where $a(t)$ is a scale factor, the background equations of motion are obtained as

	$\displaystyle 3M_{pl}^{2}H^{2}=\frac{1}{2}\dot{\phi}^{2}+V+\frac{3\alpha}{M^{3}}H\xi\dot{\phi}^{3}\left(1-\frac{\dot{\xi}}{6H\xi}\right)-\frac{9\beta}{2M^{2}}\dot{\phi}^{2}H^{2}\,,$		(6)
	$\displaystyle M_{pl}^{2}\left(2\dot{H}+3H^{2}\right)=-\frac{1}{2}\dot{\phi}^{2}+V-\frac{\alpha}{M^{3}}\xi\dot{\phi}^{3}\left(\frac{\ddot{\phi}}{\dot{\phi}}+\frac{\dot{\xi}}{2\xi}\right)+\frac{\beta\dot{\phi}^{2}}{2M^{2}}\left(2\dot{H}+3H^{2}+4H\frac{\ddot{\phi}}{\dot{\phi}}\right)\,,$		(7)
	$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}-\frac{\alpha}{2M^{3}}\dot{\phi}\left[\ddot{\xi}\dot{\phi}+3\dot{\xi}\ddot{\phi}-6\xi\dot{\phi}\left(\dot{H}+3H^{2}+2H\frac{\ddot{\phi}}{\dot{\phi}}\right)\right]-\frac{3\beta}{M^{2}}H\dot{\phi}\left(2\dot{H}+3H^{2}+H\frac{\ddot{\phi}}{\dot{\phi}}\right)=0\,,$		(8)

where $V_{,\phi}\equiv dV/d\phi$ and the dot denotes the derivative with respect to time.

In a slow-roll scenario, inflation occurs as the scalar (or inflaton) field rolls from some initial $\phi_{i}$ value to another $\phi_{e}$ value. $\phi_{i}$ and $\phi_{e}$ denote the beginning and end of inflation, respectively. For inflation to be successful, the rolling process must happen very slowly; therefore, during inflation, the potential energy of the scalar field is much larger than its kinetic energy, $V\gg\dot{\phi}^{2}$, and the friction is much larger than its acceleration, $3H\dot{\phi}\gg\ddot{\phi}$. To reflect the slow-roll inflation, we define the following new parameters,

$\displaystyle\epsilon_{1}\equiv-\frac{\dot{H}}{H^{2}}\,,\quad\epsilon_{2}\equiv-\frac{\ddot{\phi}}{H\dot{\phi}}\,,\quad\epsilon_{3}\equiv\frac{\xi_{,\phi}\dot{\phi}}{\xi H}\,,\quad\epsilon_{4}\equiv\frac{\xi_{,\phi\phi}\dot{\phi}^{4}}{V}\,,\quad\epsilon_{5}\equiv\frac{\dot{\phi}^{2}}{2M_{pl}^{2}H^{2}}\,,$

(9)

and require $|\epsilon_{i}|\ll 1$, where $i=1,2,3,4,5$. Then, Eq. (8) can be rewritten in terms of these newly defined parameters as

$\displaystyle 3H\dot{\phi}\left[1-\frac{\epsilon_{2}}{3}-\frac{3\beta H^{2}}{M^{2}}\left(1-\frac{2\epsilon_{1}}{3}-\frac{\epsilon_{2}}{3}\right)+\frac{3\alpha\xi H\dot{\phi}}{M^{3}}\left(1-\frac{\epsilon_{1}}{3}-\frac{2\epsilon_{2}}{3}+\frac{2\epsilon_{2}\epsilon_{3}}{9}\right)\right]=-V_{,\phi}\left(1-\frac{\alpha V\epsilon_{4}}{2M^{3}V_{,\phi}}\right)\,.$

(10)

In the limit $(\alpha,\beta)\rightarrow 0$ during inflation, the above equation reduces to $3H\dot{\phi}\simeq-V_{,\phi}$.

In addition to Eq. (9), we introduce the following relation between the second and third terms inside the parenthesis in Eq. (1),

$\displaystyle\gamma\equiv\frac{\alpha\xi g^{\mu\nu}\square\phi/M^{3}}{\beta G^{\mu\nu}/M^{2}}\,,$

(11)

which weighs the contributions of each term. In other words, if the kinetic coupling between the scalar field and gravity is much stronger (or weaker) than the derivative self-interaction of the scalar field, then we get $\gamma\ll 1$ ($\gamma\gg 1$). Consequently, the $\gamma\sim\mathcal{O}(1)$ if both the second and third terms contribute equally during inflation. However, these terms cancel each other if the $\gamma=1$ because of their opposite signs in Eq. (1), and results, in that case, should be that of the standard slow-roll inflation. We are interested in the $\gamma\sim\mathcal{O}(1)$ limit in this work. Eq. (11) is rewritten as

$\displaystyle\gamma=\frac{\alpha}{\beta M}\frac{\xi\dot{\phi}}{H}\,,$

(12)

and its implication is still the same. The importance of introducing the $\gamma\sim\mathcal{O}(1)$ parameter is to determine the shape of $\xi(\phi)$ using the observational constraints, which we will discuss in Sec. IV. Requiring the slow-roll conditions ($V\gg\dot{\phi}^{2}$ and $3H\dot{\phi}\gg\ddot{\phi}$) to be satisfied during inflation, we rewrite Eqs. (6) and (7) as

$\displaystyle 3M_{pl}^{2}H^{2}\simeq V\,,\qquad 3H\dot{\phi}\left(1+\mathcal{A}\right)\simeq-V_{,\phi}\,,$

(13)

where

$\displaystyle\mathcal{A}\equiv\frac{3\alpha}{M^{3}}\xi\dot{\phi}H-\frac{3\beta}{M^{2}}H^{2}.$

(14)

The equations in Einstein gravity are recovered for $|\mathcal{A}|\ll 1$ in Eqs. (13), and the limit $|\mathcal{A}|\gg 1$ is known as the high friction limit Germani:2010gm ; Germani:2010ux ; Tsujikawa:2012mk ; Yang:2015zgh ; Yang:2015pga ; Sato:2017qau . Deviation from Einstein’s gravity is, therefore, reflected in $|\mathcal{A}|\sim 1$. Using Eq. (13), we rewrite Eq. (9) in the following form,

$\displaystyle\epsilon_{1}\simeq\frac{\epsilon_{V}}{1+\mathcal{A}}\,,\quad\epsilon_{2}\simeq\frac{\eta_{V}-3\epsilon_{V}}{1+\mathcal{A}}+\frac{2\epsilon_{V}}{(1+\mathcal{A})^{2}}\,,\quad\epsilon_{3}\simeq\frac{\eta_{V}-4\epsilon_{V}}{1+\mathcal{A}}+\frac{2\epsilon_{V}}{(1+\mathcal{A})^{2}}\,,$

(15)

where

$\displaystyle\epsilon_{V}\equiv\frac{M_{pl}^{2}}{2}\left(\frac{V_{,\phi}}{V}\right)^{2}\,,\qquad\eta_{V}\equiv M_{pl}^{2}\frac{V_{,\phi\phi}}{V}\,.$

(16)

Inflation ends when the slope of the potential gets steep enough such that the first slow-roll condition is violated. Thus, the field value at the end of inflation is estimated by solving $\epsilon_{1}(\phi_{e})=1$. The amount of inflation is quantified by the number of $e$-folds expressed as

$\displaystyle N=\int^{\phi_{e}}_{\phi_{i}}\frac{H}{\dot{\phi}}d\tilde{\phi}\simeq\frac{1}{M_{pl}^{2}}\int^{\phi_{e}}_{\phi_{i}}\frac{V}{V_{,\tilde{\phi}}}(1+\mathcal{A})d\tilde{\phi}\,.$

(17)

The kinetic coupling between the scalar field and gravity and the inflaton self-interaction affect the dynamical evolution of the universe both at the background and perturbation level. Thus, their presence is imprinted in the observed power spectra of the scalar and tensor modes. Following Kobayashi:2011nu ; Gao:2011vs ; Gao:2012ib ; Tumurtushaa:2019bmc , we perform the perturbation analyses of both the scalar and tensor modes in the uniform field gauge, where $\delta\phi(t,\vec{x})=0$.

Let us begin with the perturbations of tensor modes characterized by the tensor part of the metric perturbations $h_{ij}$. Then, the quadratic action for the tensor perturbations reads Kobayashi:2011nu ; Gao:2011vs ; Gao:2012ib ; Tumurtushaa:2019bmc

$\displaystyle S_{T}^{2}=\frac{M_{pl}^{2}}{8}\int dtd^{3}xa^{3}\left[G_{T}\dot{h_{ij}}^{2}-\frac{F_{T}}{a^{2}}\left(\partial_{k}h_{ij}\right)^{2}\right]\,,$

(18)

where

$\displaystyle G_{T}=1-\frac{\beta}{2M^{2}M_{pl}^{2}}\dot{\phi}^{2}\,,\qquad F_{T}=1+\frac{\beta}{2M^{2}M_{pl}^{2}}\dot{\phi}^{2}\,.$

(19)

The evolution equation of the tensor perturbation modes is written as

$\displaystyle u_{\lambda,k}^{\prime\prime}+\left(c_{T}^{2}k^{2}-\frac{\mu_{T}^{2}-1/4}{\tau^{2}}\right)u_{\lambda,k}=0\,,$

(20)

where $u_{\lambda,k}\equiv z_{T}h_{\lambda,k}$ with $z_{T}=(a/2)\left(G_{T}F_{T}\right)^{1/4}$, $c_{T}^{2}\equiv F_{T}/G_{T}$, and $\mu_{T}\simeq 3/2+\epsilon_{1}$. Here, $\tau$ is conformal time and $\lambda=\times,+$ denotes the two polarization modes of the tensor perturbations. If one assumes the Bunch-Davies vacuum for the initial fluctuation modes at $c_{T}k|\tau|\gg 1$ and the constant slow-roll parameters, the solution to the above equation is

$\displaystyle u_{\lambda,k}=2^{\mu_{T}-\frac{3}{2}}\frac{\Gamma(\mu_{T})}{\Gamma(3/2)}\frac{(-c_{T}k\tau)^{\frac{1}{2}-\mu_{T}}}{\sqrt{2c_{T}k}}e^{i\left(\mu_{T}-\frac{1}{2}\right)\frac{\pi}{2}}\,.$

(21)

The power spectrum of the tensor modes on the large scale, $c_{T}k|\tau|\ll 1$, is obtained as

$\displaystyle\mathcal{P}_{T}=\frac{k^{3}}{\pi^{2}}\sum_{\lambda}\left|u_{\lambda,k}{z_{T}}\right|^{2}\simeq\frac{H^{2}}{2\pi^{2}M_{pl}^{2}c_{T}^{3}}\,.$

(22)

The spectral tilt of the tensor power spectrum is computed at the time of horizon crossing as

$\displaystyle n_{T}\equiv\left.\frac{d\ln\mathcal{P}_{T}}{d\ln k}\right|_{c_{T}k=aH}=3-2\mu_{T}\simeq-\frac{2\epsilon_{V}}{1+\mathcal{A}}\,.$

(23)

Now, let us consider the perturbations for scalar modes $\zeta$. The quadratic action for the scalar perturbation is written as Kobayashi:2011nu ; Gao:2011vs ; Gao:2012ib ; Tumurtushaa:2019bmc

$\displaystyle S_{S}^{2}=M_{pl}^{2}\int dtd^{3}xa^{3}\left[G_{S}\dot{\zeta}^{2}-\frac{F_{S}}{a^{2}}\left(\partial_{i}\zeta\right)^{2}\right]\,,$

(24)

where

$\displaystyle G_{S}=\frac{\Sigma}{\Delta^{2}}G_{T}^{2}+3G_{T}\,,\quad F_{S}=\frac{1}{a}\frac{d}{dt}\left(\frac{a}{\Delta}G_{T}^{2}\right)-F_{T}\,,$

(25)

with

	$\displaystyle\Sigma=\frac{1}{2}\dot{\phi}^{2}-3M_{pl}^{2}H^{2}+\frac{9\beta}{M^{2}}H^{2}\dot{\phi}^{2}+\frac{\alpha}{M^{3}}\xi\dot{\phi}^{3}H\left(6-\frac{\dot{\xi}}{\xi H}\right)\,,$
	$\displaystyle\Delta=M_{pl}^{2}H\left(1-\frac{3\beta}{2M^{2}M_{pl}^{2}}\dot{\phi}^{2}\right)-\frac{\alpha}{2M^{3}}\xi\dot{\phi}^{2}\,.$		(26)

The perturbation equations for the scalar modes reads

$\displaystyle v_{k}^{\prime\prime}+\left(c_{S}^{2}-\frac{\mu_{S}^{2}-1/4}{\tau^{2}}\right)v_{k}=0\,,$

(27)

where $v_{k}\equiv z_{S}\zeta_{k}$ with $z_{S}=\sqrt{2}a(G_{S}F_{S})^{1/4}$, $c_{S}^{2}\equiv F_{S}/G_{S}$, and $\mu_{S}\simeq 3/2+\epsilon_{1}-\epsilon_{2}$. Taking the Bunch-Davies vacuum for the initial fluctuations into account, we obtain the solution to the scalar perturbation equation as

$\displaystyle v_{k}=2^{\mu_{S}-\frac{3}{2}}\frac{\Gamma(\mu_{S})}{\Gamma(3/2)}\frac{(-c_{S}k\tau)^{\frac{1}{2}-\mu_{S}}}{\sqrt{2c_{S}k}}e^{i\left(\mu_{S}-\frac{1}{2}\right)\frac{\pi}{2}}\,.$

(28)

The power spectrum of the scalar perturbations on large scales, $c_{T}k|\tau|\ll 1$, is computed as

$\displaystyle\mathcal{P}_{S}=\frac{k^{3}}{2\pi^{2}}\left|\frac{v_{k}}{z_{S}}\right|^{2}$

$\displaystyle\simeq\frac{H^{2}}{8\pi^{2}M_{pl}^{2}c_{S}^{2}\epsilon_{V}}(1+\mathcal{A})=\frac{V^{3}}{12\pi^{2}M_{pl}^{6}V_{,\phi}^{2}}(1+\mathcal{A})\,,$

(29)

where $\mathcal{A}$ is given in Eq. (14). An implication of Eq. (29) is that the scalar power spectrum can be enhanced if the function $\mathcal{A}$ is large enough, $\mathcal{A}\gg 1$. Therefore, if the power spectrum is enhanced on scales smaller than the CMB scale due to the presence of $\mathcal{A}$, (i.) the PBHs can be formed, and (ii.) the induced GWs can also be generated. We devote the following sections to such possibilities.

The spectral tilt of the curvature perturbations at the time of horizon crossing gets

$\displaystyle n_{S}-1=\left.\frac{d\ln\mathcal{P}_{S}}{d\ln k}\right|_{c_{S}k=aH}=3-2\mu_{S}=\frac{2}{1+\mathcal{A}}\left[\eta_{V}-\frac{3+4\mathcal{A}}{1+\mathcal{A}}\epsilon_{V}\right]\,,$

(30)

and the tensor-to-scalar ratio reads

$\displaystyle r\equiv\frac{\mathcal{P}_{T}}{\mathcal{P}_{S}}\simeq\frac{16\epsilon_{V}}{1+\mathcal{A}}\,.$

(31)

As we mentioned earlier that the $|\mathcal{A}|\ll 1$ limit is equivalent to Einstein gravity. Thus, the above results reduce to that of the standard single-filed inflation in Einstein gravity: $n_{S}-1=-6\epsilon_{V}+2\eta_{V}$ and $r=16\epsilon_{V}$. The deviation from GR is implied for the opposite $|\mathcal{A}|\gg 1$ case, and the predictions on $n_{S}$, $n_{T}$, and $r$ are suppressed by a factor of $\mathcal{A}$: $n_{S}-1=(-8\epsilon_{V}+2\eta_{V})/\mathcal{A}$, $n_{T}=-2\epsilon_{V}/\mathcal{A}$, and $r=16\epsilon_{V}/\mathcal{A}$ Tumurtushaa:2019bmc .

Eq. (29) shows that the power spectrum of curvature perturbations can be enhanced if the function $\mathcal{A}$ is large enough. On the other hand, sufficiently large curvature fluctuations can form PBHs due to the gravitational collapse when they reenter the horizon in the RD era Zeldovich:1967lct ; Hawking:1971ei ; Carr:1974nx ; Carr:1975qj , which consequently implies that the formation of PBH may be possible for our model. Since the CMB temperature and polarization measurements constrain the primordial perturbations to be very small at large scales, we are interested in enhancing the curvature power spectrum on scales smaller than the CMB scale, i.e., $k_{\text{PBH}}\gg k_{\ast}$. Besides, the sufficiently large density fluctuations generated during inflation can also simultaneously produce a substantial amount of GWs in the RD era Ananda:2006af ; Baumann:2007zm . We investigate such possibilities of the PBH and GW productions in this section.

The enhancement mechanism of the power spectrum on small scales is often associated with a local feature of the inflaton potential Ezquiaga:2017fvi ; Mishra:2019pzq . However, as we mentioned in the Introduction, not every inflaton potentials induce such enhancement, which is a common drawback in these approaches. Therefore, to investigate the formation of PBHs and GWs, we first specify the primordial power spectrum instead of the inflaton potential. Then, construct the inflaton potential from the power spectrum. We propose the following parameterization of the primordial power spectrum Cai:2018dig ; Inomata:2018epa ; Kohri:2020qqd

$\displaystyle\mathcal{P}_{S}(k)=\mathcal{P}_{S}(k_{\ast})\left(\frac{k}{k_{\ast}}\right)^{n_{S}-1}\left[1+\frac{A_{S}}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{1}{2\sigma^{2}}\left(\ln\frac{k}{k_{\text{PBH}}}\right)^{2}}\right]\,,$

(32)

where $\mathcal{P}_{S}(k_{\ast})$ is the amplitude of the CMB spectrum at the pivot scale $k_{\ast}$ and $n_{S}$ is the spectral tilt given in Eq. (30). The $A_{S}$ and $\sigma$ are free parameters determining the height and width of the Gaussian peak, respectively. To form PBHs on scales smaller than the CMB scale ($k_{\text{PBH}}\gg k_{\ast}$), the $\mathcal{P}_{S}(k_{\text{PBH}})$ should be enhanced by a factor of about $10^{7}$ with respect to its value at the CMB scale, which is possible mainly due to the peak amplitude $A_{S}/\sqrt{2\pi\sigma^{2}}$. In Fig. 1, we plot the $k$ dependence of Eq. (32) in light of the several observational limits and constraints adopted from Ref. Inomata:2018epa . In particular, the orange line at $\mathcal{P}_{S}(k_{\text{PBH}})\sim 10^{-2}$ indicates the required amplitude of power spectra for the PBHs formation Green:2020jor . The figure also implies that the power spectra that have peaks at either $k\sim\mathcal{O}(10^{6})\,\text{Mpc}^{-1}$ or $k\sim\mathcal{O}(10^{12})\,\text{Mpc}^{-1}$ lead to the production of stochastic GW background in the frequency band targeted by the pulsar timing array (PTA) experiments and the space-based interferometers Inomata:2018epa .

Then, to explain the small scale enhancement as parameterized in Eq. (32), we will reconstruct the inflaton potential and the self-coupling functions in Sec. IV from the power spectrum in Eq. (32) and its spectral tilt in Eq. (30).

In Sec. II, we discussed that the primordial curvature power spectrum could be enhanced due to the self-interaction of the inflaton field and the kinetic coupling between the inflaton and gravity. Then, by proposing the power spectrum with a local Gaussian peak, we estimated the abundance of PBHs and the energy spectrum of induced GWs in Sec. III. Therefore, it is imperative for us to investigate the configuration of inflaton potential and self-coupling function that yields the power spectrum given in Eq. (32). In order words, in this section, we reconstruct the inflaton potential and self-coupling function for our model from the power spectrum and its spectral tilt. Our reconstruction method closely follows Refs. Chiba:2015zpa . We start from Eq. (17) to obtain

$\displaystyle\epsilon_{V}=\frac{1}{2}(1+\mathcal{A})\frac{V_{,N}}{V}\,,\qquad\eta_{V}=\frac{1}{2}\left(1+\mathcal{A}\right)\left(\frac{V_{,N}^{2}+VV_{,NN}}{VV_{,N}}+\frac{\left(1+\mathcal{A}\right)_{,N}}{1+\mathcal{A}}\right)\,,$

(51)

where $V_{,N}>0$ is required. Consequently, we derive from Eqs. (30)–(31)

$\displaystyle n_{S}-1$

$\displaystyle\simeq\ln\left[\frac{V_{,N}}{V^{2}}\left(1+\mathcal{A}\right)\right]_{,N}\,,\qquad r=\frac{8V_{,N}}{V}\,.$

(52)

Since the potential and couplings are functions of the scalar field, we need a relation between $\phi$ and $N$, which we obtain from Eq. (17) as

$\displaystyle\int_{\phi_{e}}^{\phi_{\ast}}d\phi$

$\displaystyle=M_{pl}\int\sqrt{\frac{V_{,\tilde{N}}}{V(1+\mathcal{A})}}d\tilde{N}\,,$

(53)

where $\phi_{e}$ and $\phi_{\ast}$ are scalar field values at the end of inflation and the horizon exit of the CMB mode, respectively. Thus, Eqs. (52)–(53) are the key equations for reconstructing $V(\phi)$ and $\xi(\phi)$. In this work, we adopt the following form of $n_{S}(N)$,

$\displaystyle n_{S}-1=-\frac{2}{N}\,,$

(54)

which is in good agreement with the CMB measurement Ade:2013zuv for $N_{\ast}\simeq 60$ and is extensively studied in the literature Chiba:2015zpa . Applying Eq. (54) to Eq. (52), we obtain

$\displaystyle-\frac{2}{N}=\ln\left[\frac{V_{,N}}{V^{2}}\left(1+\mathcal{A}\right)\right]_{,N}\,,$

(55)

which can also be integrated to give

$\displaystyle\frac{V_{,N}}{V^{2}}\left(1+\mathcal{A}\right)=\frac{c_{1}}{N^{2}}\,,$

(56)

where $c_{1}$ is a positive constant of integration since $V_{,N}>0$. By comparing Eq. (29) with Eq. (32) in the $k\gg k_{\ast}$ limit, we obtain

$\displaystyle\mathcal{A}(N)$

$\displaystyle=\frac{A_{S}}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{\bar{N}^{2}}{2\sigma^{2}}}\,,$

(57)

where $\bar{N}\equiv N-N_{p}$ is defined from the end of inflation. The $N$ counts the $e$-folds between the horizon exit of a mode $k$ during inflation and the end of inflation. The difference $N_{p}=N_{\ast}-N_{\text{PBH}}$ indicates how long after the CMB mode $k_{\ast}$ exit the horizon, the mode $k_{\text{PBH}}$ corresponding to PBHs would exit the horizon during inflation. Thus, the $N$ varies within an interval $0\leq N\leq N_{\ast}$ as the $k$ runs between $k_{\ast}\leq k\leq k_{\text{max}}$, where the $k_{\text{max}}$ indicates the smallest scale that exits the horizon during inflation. The CMB data favor $N_{\ast}\simeq 50-60$ Ade:2013zuv . We solve Eq. (56) for $V(N)$ using Eq. (57), and the solution is given by

$\displaystyle V(N)=\left[c_{0}+\frac{c_{1}}{N}-\frac{2c_{1}\sigma}{\left(1+\frac{A_{S}}{\sqrt{2\pi\sigma^{2}}}\right)N_{p}^{2}}\right]^{-1}\,,$

(58)

where $c_{0}$ is an integration constant. The last term in Eq. (58) is negligible compared to the second term $c_{1}/N$ if we consider $A_{S}\sim\mathcal{O}(10^{6}-10^{7})$, $\sigma\sim\mathcal{O}(0.1-1)$ and $N_{p}\sim\mathcal{O}(10)$. Consequently, from Eq. (12), we obtain

$$\xi(N)=\gamma\frac{\beta M}{\alpha M_{pl}}\sqrt{\frac{V}{V_{,N}}\left(1+\mathcal{A}\right)}\,.$$

(59)

After using Eq. (53), which allows us to interchange $N$ with $\phi$, we get

	$\displaystyle V(\phi)$	$\displaystyle=\frac{1}{c_{0}}\tanh^{2}\left(\frac{c_{2}}{2}\frac{\phi}{M_{pl}}\right)\,,$		(60)
	$\displaystyle\xi(\phi)$	$\displaystyle=\gamma\frac{\beta M}{2\alpha c_{2}M_{pl}}\sinh\left(c_{2}\frac{\phi}{M_{pl}}\right)\sqrt{1+\mathcal{A}(\phi)}\,,$		(61)

where

$\displaystyle\mathcal{A}(\phi)=\frac{A_{S}}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{1}{2\sigma^{2}}\left[\frac{1}{c_{2}^{2}}\sinh^{2}\left(\frac{c_{2}}{2}\frac{\phi}{M_{pl}}\right)-\phi_{p}\right]^{2}}\,,$

(62)

and $c_{2}=\sqrt{c_{0}/c_{1}}$. The peak in the power spectrum Eq. (32) is, therefore, explained by a local feature of the self-coupling function at $\phi_{p}=2/c_{2}\,\text{arcsinh}\left(\sqrt{c_{2}^{2}N_{p}}\right)$. The Eqs. (60) and (61) are our key results of this section and describe the shape of the inflaton potential and self-coupling functions, respectively. It is worth noting here that the potential obtained in Eq. (60) is the same as that in the so-called “T-model” of inflation proposed in Ref. Kallosh:2013maa . In Fig. 3, we plot Eq (61) as functions of the scalar field $\phi$ for positive and negative values of $\beta/\alpha$. In conclusion, if the sign of $\beta/\alpha$ is positive (negative), there appears a bump (dip) in the self-coupling function $\xi(\phi)$ at $\phi_{p}$. The value of $\phi_{p}$ determines the location of a peak in the power spectra. In other words, the peak scale at which the scalar power spectrum is enhanced can be easily adjusted by the value of $\phi_{p}$.