Primordial black hole formation from transient $f(T)$ cosmology

Abstract

We study primordial black hole (PBH) formation in a minimally coupled $f(T)$ teleparallel cosmology that generates a transient departure from standard radiation domination. The model is constructed so that modified-gravity effects are negligible at early and late times, but become dynamically relevant over a finite epoch, during which an effective torsion component reduces the total equation-of-state parameter below 1/3.We show that this transient softening lowers the collapse threshold for overdensities at horizon re-entry, leading to an exponential enhancement of PBH formation. In addition, the modified background alters the relation between temperature and horizon mass, producing a localized feature in the PBH mass function. For representative parameters, PBHs with asteroid-scale masses can account for a significant fraction, or even the entirety, of dark matter for perturbation amplitudes $σ^2 \sim \mathcal{O}(10^{-3})$, while remaining consistent with current constraints. Our results demonstrate that modified gravity alone can efficiently generate PBHs, without requiring ad hoc modifications of the radiation sector.

Figures (7)

• Figure 1: Evolution of the total equation-of-state parameter $w_{\rm tot}(t)$ in the transient $f(T)$ model $f(T)=\lambda T_\star (T/T_\star)^3 e^{-T/T_\star}$ with $T=6H^2$ and $\lambda=\frac{0.2}{9}e^{3/2}$, as a function of the dimensionless shifted time $(t-t_{\mathrm{ref}})H_\star$. The reference epoch $t_{\mathrm{ref}}$ is defined by $x(t_{\mathrm{ref}})=T(t_{\mathrm{ref}})/T_\star=H(t_{\mathrm{ref}} )^2/H_\star^2=3/2$, at which $\Omega_f(t_{\mathrm{ref}})=0.1$ and $w_{\rm tot}(t_{\mathrm{ref}})=0.2$. The figure illustrates the transient softening of the cosmic background relative to radiation domination ($w_{\rm tot}=1/3$), induced by the effective torsion sector.
• Figure 2: Evolution of the effective torsion-fluid fraction $\Omega_f(t)$ for the transient $f(T)$ model, plotted as a function of the dimensionless shifted time $(t-t_{\mathrm{ref}})H_\star$, with $t_{\mathrm{ref}}$ defined by $x(t_{\mathrm{ref}})=3/2$. The coupling $\lambda=\frac{0.2}{9}e^{3/2}$ is fixed analytically so that $\Omega_f(t_{\mathrm{ref}})=0.1$. The figure clearly demonstrates the transient nature of the torsion sector, which becomes dynamically relevant around $t_{\mathrm{ref}}$, while being suppressed both at early times ($T/T_\star\to\infty$) and at late times ($T/T_\star\to 0$).
• Figure 3: Evolution of the radiation energy fraction $\Omega_r(t)$ as a function of the dimensionless shifted time $(t-t_{\mathrm{ref}})H_\star$ in the effective two-component (radiation + torsion) background. The evolution is obtained by integrating the background equations for $x\equiv H^2/H_\star^2$ with initial condition $x(t_{\mathrm{ref}})=3/2$. The figure illustrates the transient departure from pure radiation domination induced by the torsion sector, followed by a rapid restoration of $\Omega_r\to 1$ as the modification becomes negligible.
• Figure 4: Evolution of the effective torsion fraction $\Omega_f$ (blue), the radiation fraction $\Omega_r$ (red), and the total equation-of-state parameter $w_{\rm tot}$ (green), shown as functions of the primordial plasma temperature $T$. The transient $f(T)$ model is calibrated with $\lambda=e^{3/2}/45$, such that at the reference temperature $T_{\rm ref}=10^{5}\,\mathrm{GeV}$ (vertical dashed line) one has $x(T_{\rm ref})=3/2$, yielding $\Omega_f(T_{\rm ref})=0.1$ and $w_{\rm tot}(T_{\rm ref})=0.2$. The figure illustrates the time-localized impact of the torsion sector, which induces a transient reduction of the total EoS while the radiation component remains dominant, before the system returns to standard radiation-dominated behavior at both higher and lower temperatures.
• Figure 5: The PBH formation threshold $\delta_c$ as a function of the total equation-of-state parameter $w_{\rm tot}$, based on the analytic estimate of Harada-Kohri-Yoo. The curve illustrates the strong dependence of the collapse threshold on the background EoS, with larger pressure (higher $w_{\rm tot}$) leading to increased resistance to gravitational collapse. The markers indicate representative values for radiation domination ($w_{\rm tot}=\tfrac{1}{3}$) and for a softened background ($w_{\rm tot}=0.2$), highlighting how even a modest reduction of $w_{\rm tot}$ leads to a decrease of $\delta_c$ and thus enhances PBH formation.