The tidal gap: causality bound on exotic compact objects with applications in the solar and sub-solar mass range

Abstract

In this work, we highlight the existence of a lower limit on the tidal deformability parameter $Λ$, determined by the requirement of relativistic causality. Additionally, by considering the upper bound set on compactness, we identify the region within the parameter space of compactness versus tidal deformability, where physically motivated exotic compact objects (ECOs) could potentially reside. Our analysis reveals the presence of a tidal gap between black holes, characterized by vanishing tidal deformability, and physically motivated ECOs. Prompted by this finding, we investigate the possibility that a population of maximally compact exotic objects, described by a linear equation of state (EoS), may simultaneously inhabit the lower mass gap and the sub-solar region, thus qualifying as (primordial) black hole mimickers while distinguishing themselves from the latter by their non-zero tidal deformability. Finally, considering the case of solitonic boson stars as proxies for ECOs described by a linear EoS, we discuss how it is possible to further reduce the lower limit on $Λ$, provided that the strong energy condition is violated (but not the dominant energy condition, and therefore causality).

Figures (29)

• Figure 1: Relationship between the linear EoS in eq. (\ref{['eq:LinEOS']}) and the energy conditions discussed in eqs. (\ref{['eq:NEC']}-\ref{['eq:SEC']}) in the energy density/pressure plane. We rewrite eq. (\ref{['eq:LinEOS']}) in the form $P/\epsilon_0 = \omega(\epsilon/\epsilon_0 -1)$, and consider both pressure and energy density in units of $\epsilon_0$. In the hatched region: horizontal lines indicate where the DEC is satisfied, vertical lines indicate where the NEC is satisfied, lines rotated counterclockwise by 45$^\circ$ indicate where the SEC is satisfied, and lines rotated clockwise by 45$^\circ$ indicate where the WEC is satisfied. In the yellow region, all energy conditions are satisfied. The linear EoS that satisfy the causality condition occupy this region. We also show the curve corresponding to a case of causality violation with $\omega = 10$. We observe that this corresponds to a violation of the DEC, although the WEC (and thus NEC) and SEC remain satisfied. The case of constant-density stars (a.k.a. incompressible stars) is indicated by a vertical magenta dot-dashed line
• Figure 2: Same as in fig. \ref{['fig:EnergyCondLinearEOS']} but considering realistic neutron star EoS. Note that in this case pressure and energy density are normalized with respect to the value $\epsilon_0 = m_n^4c^5/\pi^2\hbar^3 \simeq 1.64\times 10^{37}$ erg/cm$^3$. For each curve, the dashed portion indicates the values for which $P(\epsilon) > P(\epsilon_c^{\textrm{max}})$, see text for details.
• Figure 3: Same as in fig. \ref{['fig:EnergyCondLinearEOS']} but considering polytropic EoS, cf. eq. (\ref{['eq:PolyEoS']}). Pressure and energy density are normalized with respect to the quantity $\epsilon_0 \equiv 1/K^{\gamma}$. Each curve terminates at values of energy density and pressure beyond which no stable solutions exist.
• Figure 4: Top panel. Comparison between commonly employed equations of state for describing neutron star properties (red lines) and the linear EoS specified in eq. (\ref{['eq:LinEOS']}) with $\omega = 1$. The region shaded in green corresponds to $10^{35} \leqslant \epsilon_0\,\,[\textrm{erg}/\textrm{cm}^3] \leqslant 10^{36}$. Bottom panel. Speed of sound $c_s$ (cf. eq. (\ref{['eq:Speed']})) as function of the energy density $\epsilon$ for the same neutron star EoS shown in the top panel. We highlight in blue the EoS that violate the causality condition $c_s < 1$. The labels correspond to the EoS: ap2-ap4 Akmal:1998cf, wff1-wff2 Wiringa:1988tp.
• Figure 5: Relation between the average speed of sound squared (or, equivalently, the ratio of central pressure to central energy density, cf. eq. (\ref{['eq:AverageSoS']})) and the compactness $\mathcal{C}$ for the set of realistic neutron star EoS analyzed in fig. \ref{['fig:RealisticEoS']}.