Primordial Black Hole Production in Chiral Cosmological Inflationary Models ^†

Abstract

We consider a two-field chiral cosmological model of inflation connected with induced gravity. It is shown that the constructed inflationary model does not contradict recent observational data from the Atacama Cosmology Telescope (ACT) and the Dark Energy Spectroscopic Instrument (DESI). The corresponding values of the model parameters are determined. This model is suitable for the formation of primordial black holes (PBHs). The estimation of PBH masses allows us to consider these PBHs as dark matter candidates.

1. Introduction

The current observations of black holes with different masses suggest that it is difficult to explain the occurrence of some of these objects using models of stellar collapse [1,2,3]. This fact supports the hypothesis of primordial black hole (PBH) formation before the matter dominance epoch. The existence of overdensities exceeding a critical value during cosmological inflation is a necessary condition for PBH production during the radiation-dominated era [4,5,6,7,8]. Slow-roll inflation models are not suitable for generating such overdensities. In single-field inflation models, PBH formation is associated with an ultra-slow-roll stage [9,10,11,12,13]. In two-field models, if one scalar field plays the role of inflaton at the beginning of inflation and the other field plays the same role at the end, then large energy density perturbations may arise during the transition between these two inflationary stages. Consequently, such models are capable of describing PBH production [6,7,14,15,16,17,18]. Models with nonminimally coupled scalar fields are classically equivalent to General Relativity (GR) models with minimally coupled scalar fields. Using the conformal transformation of the metric, one can obtain the GR description known as the Einstein frame. The obtained GR model with two minimally coupled scalar fields has a non-standard kinetic part in the action; in other words, one gets a chiral cosmological model (CCM) [18,19,20,21,22,23,24,25]. Two-field CCMs are actively used to describe inflation with the PBH formation [7,14,15,17,18,25,26,27,28,29,30].

Most of the above-mentioned inflation models are in good agreement with the Planck/BICEP observations of the cosmic microwave background (CMB) radiation [31,32], which provide the following values of the spectral index $n_s$: $n_s=0.9651\pm 0.0044$.

The more recent observational data from ACT/DESI collaborations [33,34,35,36] slightly shift the tolerated range of values for $n_s$:

$$n_s=0.9743\pm 0.0034.$$

(1)

The ACT/DESI data are consistent with the previously obtained upper bounds on the tensor-to-scalar perturbation ratio r and value of the scalar power spectrum amplitude $A_s$ [37]:

$$A_s=(2.10\pm 0.03)\times {10}^{-9}\mathrm{and}r<0.028.$$

(2)

Some of two-field inflation models need to be modified in order to obtain values of $n_s$ that are consistent with the ACT/DESI data. For example, the Higgs-$R^2$ inflationary model [14,16,38,39,40,41,42,43,44,45], which corresponds to a two-field CCM, has been modified by the addition of an $R^3$ term [14,30,46]. Another $F(R,\chi )$ gravity inflation model, which fits the ACT/DESI data and is suitable for PBH production, has been proposed in Ref. [47].

For some inflationary models, it is sufficient to adjust the values of the model parameters. In this paper, we demonstrate that the induced gravity two-field model [28] proposed in 2024 is in good agreement with the ACT/DESI observational data [33,34,35,36] and determine the corresponding values of the model parameters. Using a conformal transformation of the metric, we obtain a CCM with two scalar fields and investigate the behavior of scalar fields in this model during inflation. We estimate the PBH masses in the model constructed to demonstrate that PBHs could be considered dark matter candidates.

2. CCM and Modified Gravity

In Ref. [28], we proposed the induced gravity model with the following action:

$$S=\int d^{4}x\sqrt{-\tilde{g}}\left[{\displaystyle \frac{\xi }{2}}{\sigma }^2\tilde{R}-{\displaystyle \frac{1}{2}}{\tilde{g}}^{\mu \nu }{\partial }_{\mu }\sigma {\partial }_{\nu }\sigma -{\displaystyle \frac{1}{2}}{\tilde{g}}^{\mu \nu }{\partial }_{\mu }\chi {\partial }_{\nu }\chi -\tilde{V}(\sigma ,\chi )\right],$$

(3)

where $\xi$ is a positive constant. The potential is

$$\tilde{V}(\sigma ,\chi )=\lambda {\sigma }^4\left(F_1\left(\chi \right)+F_2\left(\chi \right){\mathrm{e}}^{\gamma {\left[ln(\sigma /M_{\mathrm{Pl}})\right]}^{2\alpha }}\right),$$

(4)

where $M_{\mathrm{Pl}}$ is the reduced Planck mass,

$$F_1\left(\chi \right)={\left(1-{\displaystyle \frac{{\chi }^2}{{\chi }_0^2}}\right)}^2-d{\displaystyle \frac{\chi }{{\chi }_0}},F_2\left(\chi \right)={\displaystyle \frac{c_2{\chi }^2}{{\chi }_0^2}}+c_0,$$

(5)

$\alpha$, $\gamma$, $\lambda$, ${\chi }_0$, $c_0$, $c_2$, and d are constants.

In the Einstein frame, we have the corresponding CCM, described by the following action:

$$S_E=\int d^{4}x\sqrt{-g}\left[{\displaystyle \frac{M_{\mathrm{Pl}}^2}{2}}R-{\displaystyle \frac{1}{2}}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -{\displaystyle \frac{y}{2}}{g}^{\mu \nu }{\partial }_{\mu }\chi {\partial }_{\nu }\chi -V_E\right],$$

(6)

where

$$\varphi =M_{\mathrm{Pl}}\sqrt{6+{\displaystyle \frac{1}{\xi }}}ln\left({\displaystyle \frac{\sigma }{M_{\mathrm{Pl}}}}\right),y={\displaystyle \frac{M_{\mathrm{Pl}}^2}{\xi {\sigma }^2}}={\displaystyle \frac{1}{\xi }}exp\left(-2\sqrt{{\displaystyle \frac{\xi }{6\xi +1}}}{\displaystyle \frac{\varphi }{M_{\mathrm{Pl}}}}\right),$$

(7)

and the potential

$$V_E(\varphi ,\chi )=V_0\left(F_1\left(\chi \right)+F_2\left(\chi \right){\mathrm{e}}^{\beta {\left({\displaystyle \frac{{\varphi }^2}{M_{\mathrm{Pl}}^2}}\right)}^{\alpha }}\right),V_0={\displaystyle \frac{\lambda M_{\mathrm{Pl}}^4}{{\xi }^2}},\beta =\gamma {\left({\displaystyle \frac{\xi }{1+6\xi }}\right)}^{\alpha }.$$

(8)

3. Evolution Equations in the Einstein Frame

In the spatially flat Friedmann–Lemaître–Robertson–Walker metric with the interval

$${ds}^2=-{dt}^2+a^2\left(t\right)\left(d{x}_1^2+d{x}_2^2+d{x}_3^2\right),$$

the evolution equations have the following form [22]:

$$H^2={\displaystyle \frac{2V_E}{6M_{\mathrm{Pl}}^2-{{\varphi }^{\prime }}^2-y{{\chi }^{\prime }}^2}},H^{\prime }=-{\displaystyle \frac{H}{2M_{\mathrm{Pl}}^2}}\left[{{\varphi }^{\prime }}^2+y{{\chi }^{\prime }}^2\right],$$

(9)

where H is the Hubble parameter, primes denote derivatives with respect to the e-folding number $N=ln(a/a_0)$, and $a_0$ is a constant.

The field equations form the following dynamical system:

$$\begin{array}{cc}\hfill {\varphi }^{″}=& (\epsilon -3){\varphi }^{\prime }+{\displaystyle \frac{1}{2}}{\displaystyle \frac{dy}{d\varphi }}{{\chi }^{\prime }}^2-{\displaystyle \frac{1}{H^2}}{\displaystyle \frac{\partial V_E}{\partial \varphi }},\hfill \\ \hfill {\chi }^{″}=& (\epsilon -3){\chi }^{\prime }-{\displaystyle \frac{1}{y}}{\displaystyle \frac{dy}{d\varphi }}{\chi }^{\prime }{\varphi }^{\prime }-{\displaystyle \frac{1}{y{H}^2}}{\displaystyle \frac{\partial V_E}{\partial \chi }},\hfill \end{array}$$

(10)

where $\epsilon$ is the slow-roll parameter:

$$\epsilon =-{\displaystyle \frac{H^{\prime }}{H}}={\displaystyle \frac{1}{2M_{\mathrm{Pl}}^2}}\left[{{\varphi }^{\prime }}^2+y{{\chi }^{\prime }}^2\right].$$

(11)

Another slow-roll parameter is defined as

$$\eta =\epsilon -{\displaystyle \frac{{\epsilon }^{\prime }}{2\epsilon }}.$$

(12)

In the leading order of the slow-roll parameters, the inflationary parameters are

$$n_s=1-4\epsilon +2\eta ,r\approx 16\epsilon .$$

(13)

Using (10)–(12), we get

$${\epsilon }^{\prime }={\displaystyle \frac{(\epsilon -3)({\left({\varphi }^{\prime }\right)}^2+y{\left({\chi }^{\prime }\right)}^2)}{M_{\mathrm{Pl}}^2}}-{\displaystyle \frac{1}{M_{\mathrm{Pl}}^2{H}^2}}{\displaystyle \frac{d{V}_E}{dN}}$$

(14)

and

$$\eta =3+{\displaystyle \frac{1}{2M_{Pl}^2{H}^2\epsilon }}{\displaystyle \frac{d{V}_E}{dN}}.$$

(15)

In the saddle point ${\displaystyle \frac{d{V}_E}{dN}}=0$, we get $\eta =3$. Using this result and supposing $\epsilon \ll 1$ in (12), we get estimation $\epsilon \sim {\mathrm{e}}^{-6N}$. So, the slow-roll parameter $\epsilon$ becomes very small and the amplitude of scalar perturbations at the horizon crossing $A_s\approx {\displaystyle \frac{H^2}{8{\pi }^2{M}_{\mathrm{Pl}}^2\epsilon }}$ has a big jump.

4. Estimation of Inflationary Parameters

Numerical calculations show that the attractor solution corresponds to $\chi \approx 0$ during the first stage of inflation. Therefore, to estimate the inflationary parameters we can set $\chi =0$ and consider the slow-roll approximation for a single-field model with the potential

$${\tilde{V}}_E\left(\varphi \right)=V_0\left(1+c_0{\mathrm{e}}^{\beta {\left({\displaystyle \frac{{\varphi }^2}{M_{\mathrm{Pl}}^2}}\right)}^{\alpha }}\right)\approx V_0{c}_0{\mathrm{e}}^{\beta {\left({\displaystyle \frac{{\varphi }^2}{M_{\mathrm{Pl}}^2}}\right)}^{\alpha }},$$

(16)

where we choose such a value of $c_0$ that the first term in the brackets is negligibly small.

Using the standard slow-roll equations for single-field models [48],

$${\varphi }^{\prime }=-{\displaystyle \frac{M_{\mathrm{Pl}}^2{\tilde{V}}_{E,\varphi }}{{\tilde{V}}_E}},H^2={\displaystyle \frac{{\tilde{V}}_E}{3M_{\mathrm{Pl}}^2}},$$

(17)

we obtain

$$r=32{\alpha }^2{\beta }^2{\left[{\displaystyle \frac{\varphi }{M_{\mathrm{Pl}}}}\right]}^{4\alpha -2},n_s=1-4{\alpha }^2{\beta }^2{\left[{\displaystyle \frac{\varphi }{M_{\mathrm{Pl}}}}\right]}^{4\alpha -2}+2\alpha \beta \left(4\alpha -2\right){\left[{\displaystyle \frac{\varphi }{M_{\mathrm{Pl}}}}\right]}^{2\alpha -2}.$$

(18)

5. PBH Production in CCM

In the slow-roll regime, absolute values of both slow-roll parameters, $\epsilon$ and $\eta$, must be less than one. Cosmological inflation as an accelerated expansion of the Universe ends when $\epsilon ⩾1$; therefore, only the parameter $\eta$ can be greater than one during inflation.

The proposed model realizes a so-called two-stage inflationary scenario [7,18,28,29]. The slow-roll regime is violated when the first stage of the inflation ends. During a few e-folds, the absolute value of the parameter $\eta$ can be greater than one. After this, the slow-roll approximation recovers. In the ultra-slow-roll regime, $\eta \approx 3$. We use the supposition that the transition from the first stage of inflation to the second stage leads to growth of energy density perturbations, leading to PBH formation at the moment when perturbations with a wavenumber around $k_{*}$ re-enter the horizon $k_{*}=a_{*}{H}_{*}=a_{re}{H}_{re}=k_{re}$ [6,14].

The end of inflation occurs at $N_{tot}$, which is determined by the condition $\epsilon \left(N_{tot}\right)=1$.

To estimate the mass of PBHs, we apply the following formula [14,28,49]:

$$M_{PBH}\simeq {\displaystyle \frac{M_{\mathrm{Pl}}^2}{H_e}}exp\left(2(N_{tot}-N_{*})\right),$$

(19)

where $H_e$ is the value of the Hubble parameter at the end of inflation; $N_{*}$ is the minimal value of N, at which $\eta \left(N_{*}\right)=3$; and the difference $N_{tot}-N_{*}$ determines the duration of the second stage of inflation. As shown in Refs. [8,50], PBHs could be considered dark matter candidates if their masses belong to the interval

$${10}^{-17}{M}_{⨀}⩽M_{PBH}⩽{10}^{-12}{M}_{⨀},$$

(20)

where $M_{⨀}$ is the solar mass.

The results of the numerical integration of system (10) are presented in

Table 1. The dependence of the duration of inflation $N_{tot}$ and the PBH mass $M_{PBH}$ on the model parameter $\alpha$ at fixed $c_0=2$, $\beta =-1.3$, $d=0.005$, $c_2=147$, ${\chi }_0=3.5$, and $\xi =1$.

$\mathit{\alpha }$	$\mathit{\lambda }$	r	${\mathit{\varphi }}_0$	${\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}-{\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}$	${\mathit{M}}_{\mathbf{PBH}}/{\mathit{M}}_{⨀}$
$-0.25$	$1.80·{10}^{-10}$	$0.012$	$4.21$	$48.9$	$20.5$	$69.4$	$3.35·{10}^{-16}$
$-0.22$	$1.85·{10}^{-10}$	$0.012$	$4.00$	$47.6$	$20.3$	$67.9$	$2.38·{10}^{-16}$
$-0.21$	$1.86·{10}^{-10}$	$0.012$	$3.92$	$47.2$	$20.2$	$67.4$	$1.94·{10}^{-16}$
$-0.20$	$1.87·{10}^{-10}$	$0.012$	$5.02$	$46.8$	$20.7$	$67.5$	$5.28·{10}^{-16}$
$-0.18$	$1.86·{10}^{-10}$	$0.011$	$3.55$	$46.0$	$21.2$	$67.2$	$1.43·{10}^{-15}$
$-0.15$	$1.80·{10}^{-10}$	$0.010$	$3.34$	$44.2$	$21.0$	$65.2$	$9.61·{10}^{-16}$
$-0.13$	$1.58·{10}^{-10}$	$0.010$	$3.11$	$43.5$	$21.0$	$64.5$	$9.61·{10}^{-16}$
$-0.12$	$1.65·{10}^{-10}$	$0.009$	$2.99$	$43.4$	$20.0$	$63.4$	$7.54·{10}^{-17}$

,

Table 2. The dependence of the duration of inflation $N_{tot}$ and the PBH mass $M_{PBH}$ on the model parameter $c_0$ at fixed $\alpha =-0.18$, $\beta =-1.3$, $d=0.005$, $c_2=147$, ${\chi }_0=3.5$, and $\xi =1$.

${\mathit{c}}_0$	$\mathit{\lambda }$	r	${\mathit{\varphi }}_0$	${\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}-{\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}$	${\mathit{M}}_{\mathbf{PBH}}/{\mathit{M}}_{⨀}$
$0.6$	$1.53·{10}^{-10}$	$0.006$	$2.33$	$43.0$	$20.0$	$63.0$	$1.30·{10}^{-16}$
$1.0$	$1.81·{10}^{-10}$	$0.008$	$2.89$	$44.0$	$21.0$	$65.0$	$9.61·{10}^{-16}$
$1.5$	$1.90·{10}^{-10}$	$0.010$	$3.34$	$45.4$	$21.0$	$66.4$	$9.61·{10}^{-16}$
2	$1.86·{10}^{-10}$	$0.011$	$3.65$	$46.0$	$21.2$	$67.2$	$1.43·{10}^{-15}$
5	$1.35·{10}^{-10}$	$0.015$	$4.47$	$47.5$	$21.0$	$68.5$	$9.61·{10}^{-16}$
6	$1.22·{10}^{-10}$	$0.015$	$4.60$	$47.7$	$21.1$	$68.8$	$1.17·{10}^{-15}$

,

Table 3. The dependence of the duration of inflation $N_{tot}$ and the PBH mass $M_{PBH}$ on the model parameter d at $\alpha =-0.18$, $\beta =-1.3$, $\lambda =1.90·{10}^{-10}$, $c_0=1.5$, $c_2=147$, ${\chi }_0=3.5$, and $\xi =1$.

d	r	${\mathit{\varphi }}_0$	${\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}-{\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}$	${\mathit{M}}_{\mathbf{PBH}}/{\mathit{M}}_{⨀}$
$1·{10}^{-3}$	$0.010$	$3.34$	$45.4$	$26.0$	$71.4$	$2.12·{10}^{-11}$
$2·{10}^{-3}$	$0.010$	$3.34$	$45.3$	$23.8$	$69.1$	$2.59·{10}^{-13}$
$3·{10}^{-3}$	$0.010$	$3.34$	$45.4$	$22.0$	$67.4$	$7.15·{10}^{-15}$
$5·{10}^{-3}$	$0.010$	$3.34$	$45.4$	$21.0$	$66.4$	$9.61·{10}^{-16}$
$7·{10}^{-3}$	$0.010$	$3.34$	$45.4$	$19.0$	$64.4$	$1.76·{10}^{-17}$
$1·{10}^{-2}$	$0.010$	$3.34$	$45.4$	$18.0$	$63.4$	$2.38·{10}^{-18}$

,

Table 4. The dependence of the duration of inflation $N_{tot}$ and the PBH mass $M_{PBH}$ on the model parameter d at $\alpha =-0.18$, $\beta =-1.3$, $\lambda =1.53·{10}^{-10}$, $c_2=147$, ${\chi }_0=3.5$, $c_0=0.6$, and $\xi =1$.

d	r	${\mathit{\varphi }}_0$	${\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}-{\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}$	${\mathit{M}}_{\mathbf{PBH}}/{\mathit{M}}_{⨀}$
$1·{10}^{-3}$	$0.006$	$2.34$	$43.1$	$25.9$	$69.0$	$1.74·{10}^{-11}$
$2·{10}^{-3}$	$0.006$	$2.34$	$43.1$	$23.5$	$66.6$	$1.43·{10}^{-13}$
$3·{10}^{-3}$	$0.006$	$2.34$	$43.1$	$22.2$	$65.3$	$1.06·{10}^{-14}$
$5·{10}^{-3}$	$0.006$	$2.34$	$43.1$	$20.4$	$63.5$	$2.90·{10}^{-16}$
$7·{10}^{-3}$	$0.006$	$2.34$	$43.1$	$19.3$	$62.4$	$3.20·{10}^{-17}$
$1·{10}^{-2}$	$0.006$	$2.34$	$43.1$	$18.1$	$61.2$	$2.90·{10}^{-18}$

and

Table 5. The dependence of the duration of inflation $N_{tot}$ and the PBH mass $M_{PBH}$ on the model parameter d at $\alpha =-0.15$, $\beta =-1$, $\lambda =1.98·{10}^{-10}$, $c_0=1.5,$$c_2=147$, ${\chi }_0=3.5$, and $\xi =1$.

d	r	${\mathit{\varphi }}_0$	${\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}-{\mathit{N}}_{*}$	${\mathit{N}}_{\mathbf{tot}}$	${\mathit{M}}_{\mathbf{PBH}}/{\mathit{M}}_{⨀}$
$2·{10}^{-3}$	$0.0078$	$2.914$	$44.2$	$24.3$	$68.5$	$7.08·{10}^{-13}$
$5·{10}^{-3}$	$0.0078$	$2.914$	$44.2$	$21.2$	$65.4$	$1.43·{10}^{-15}$
$7·{10}^{-3}$	$0.0078$	$2.914$	$44.2$	$20.1$	$64.3$	$1.59·{10}^{-16}$
$9·{10}^{-3}$	$0.0078$	$2.914$	$44.2$	$19.2$	$63.4$	$2.62·{10}^{-17}$

. We demonstrate that the proposed model is in good agreement with the ACT/DESI data for the model parameters that belong to appropriate intervals.

The value of $\varphi$ at the beginning of inflation, ${\varphi }_0$, is defined by the observation restrictions (1) and (2). The value of ${\varphi }_0$ can be estimated using equation (18). To determine ${\varphi }_0$ more precisely, we fix the model parameters, numerically solve system (10), and apply expressions (11)–(13). We choose ${\varphi }_0$ so that the spectral index $n_s=0.974$ and confirm that the value of r satisfies constraint (2). Note that the inflationary parameters $n_s$ and r are independent of $\lambda$. We fix the parameter $\lambda$ by imposing $A_s=2.1·{10}^{-9}$ at $\varphi ={\varphi }_0$. We set $a=a_0$ at $\varphi ={\varphi }_0$, in other words, $N=0$ at this moment. At the end of inflation $\varphi \approx 0$; hence, $\sigma \approx M_{\mathrm{Pl}}$. We fix $\xi =1$, so the induced gravity model (3) tends to a GR model at the end of inflation. We show how the duration of inflation depends on $\alpha$ (Table 1) and $c_0$ (Table 2). Other parameters are fixed, so the duration of the second stage of inflation and the black hole mass $M_{PBH}$ only slightly change.

By varying the parameter d, we can generate inflationary scenarios that produce PBHs with any mass within the interval (20). This is demonstrated in Table 3, Table 4 and Table 5.

6. Conclusions

Recent ACT/DESI observations motivate the search for modifications to known inflationary models in order to achieve the required increase of the scalar spectral index $n_s$. In some cases, it is sufficient to change the values of parameters in the existing models. In this paper, we have considered the induced gravity two-field model proposed in Ref. [28] and shown that it is in good agreement with the ACT/DESI observational data at specific values of the model parameters.

The possibility that a significant part or even all of dark matter is not a new type of matter but consists of PBHs is currently under active investigation [43,50,51,52,53,54,55]. PBH formation results from a violation of the slow-roll regime during inflation. Observational constraints on the black hole mass range imposed by the study of dark matter allow us to accurately determine the number of e-folds between the moment of slow-roll violation and the end of inflation. As shown in Table 1, Table 2, Table 3, Table 4 and Table 5, the proposed model gives PBH masses in a range that allows them to play the role of dark matter.