Novel very-high-frequency quasi-periodic oscillations of compact, non-singular objects

TL;DR

The paper examines very-high-frequency QPOs (VHFQPOs) that arise from inner, regulator-induced stable orbits in horizonless, non-singular compact objects described by Bardeen, Dymnikova, Hayward, and Simpson–Visser metrics. It derives general geodesic and frequency relations, identifies an inner L-ISCO at $r_ ext{L-ISCO}$ scaling with the regulator $\ell$ and demonstrates that horizonlessness ($L>L_\star$) permits these QPOs to escape to infinity, yielding frequencies up to $\sim$25 kHz for stellar-mass objects. The work combines numerical and analytical analyses to show the existence and scaling of L-ISCO, then computes VHFQPOs via the relativistic precession model, finding that for typical X-ray binaries these VHFQPOs lie above conventional HFQPO bands unless the mass is large (e.g., $M\sim50\,M_\odot$). A model-independent parametrization, supplemented by rescaling, clarifies how observed VHFQPOs could constrain the central geometry, offering a potential observational handle on horizon presence in compact objects.

Abstract

We report on a novel set of very-high-frequency quasi-periodic oscillations (VHFQPO's) in the context of compact, non-singular horizonless objects. Focussing on the static, spherically symmetric case we utilize metrics of non-singular black holes that are accompanied by a regulator length scale $L > 0$. The choice $L \gtrsim GM$ generically removes the horizon from these metrics leading to compact, horizonless but non-singular objects. This generically guarantees the existence of a stable orbit at small radii $r \ll r_\text{ISCO}$, independent of the angular momentum of the massive particle. Crucially, the absence of a horizon allows the resulting VHFQPO's to escape to infinity, spanning the range from 1kHz ($M = 10M_\odot$) to 25 kHz ($M = 2M_\odot$). Within the paradigm of non-singular spacetime geometries, the absence of such VHFQPO's from X-ray binary spectra implies the presence of a horizon around the central, compact object.

Figures (8)

• Figure 1: We sketch the effective potential of a massive particle in the Schwarzschild geometry (dotted line) and in the spacetime of a non-singular compact horizonless object (solid line). The Schwarzschild geometry features an innermost stable circular orbit (ISCO) at $r = r_\text{ISCO}$, and in presence of a regulator $L > 0$ its location is shifted slightly. More importantly, in case of the horizonless geometry, a new entirely $L$-induced innermost stable circular orbit (L-ISCO) at $r = r_\text{L-ISCO}$ arises. The existence of this L-ISCO is guaranteed by the regularity of the metric of the compact non-singular object, inducing an inflection point in the effective potential at small distances $r \sim L$. Notably, the L-ISCO is only connected to the entire spacetime if $L > L_\star$ such that the resulting metric has no horizon and describes a compact, non-singular object.
• Figure 2: We qualitatively compare the function $f(r)$ for the Schwarzschild, Bardeen, Dymnikova, Hayward, and Simpson--Visser metric in the black hole case (left) and the horizonless case (right). Notice the pronounced minimum in the metric functions of the Bardeen, Dymnikova, and Hayward metric at finite $r$, as opposed to the minimum at $r=0$ for the Simpson--Visser metric.
• Figure 3: We plot the observationally detected HFQPO's for five black hole X-ray binaries and two neutron star X-ray binaries (error bars magnified by a factor of 10 for the black hole systems and by a factor of 5 for 4U 1608--52) against the Hayward model for representative values for the mass $M$ and the regulator $L>0$, close to those reported elsewhere in the literature Boshkayev:2023rhr. Our methods reproduce a qualitatively good fit with the Cir X1 values, and different masses, regulators, and models can equivalently reproduce the other data points. We take this as a verification of our numerical setup.
• Figure 4: Very-high frequency quasi-periodic oscillations (VHFQPO's) for the horizonless versions of the Bardeen metric ($l=0.40$), Dymnikova metric ($l=0.70$), and Hayward metric ($l=0.55$). The lower VHFQPO's span up to a few kilohertz, whereas the upper frequency reaches the $\mathcal{O}(20)$kHz regime.
• Figure 5: We plot the VHFQPO's for the Hayward model for a physically somewhat unrealistic mass of $M = 50M_\odot$ for the regulator values $l=0.55$ and $l=0.75$, both corresponding to compact, non-singular horizonless objects. While the $l=0.55$ case (solid line) lies outside of the observed HFQPO's, the $l=0.75$ case (dashed line) has a small overlap with experimental data. Results for the Bardeen and Dymnikova cases are qualitatively similar and not displayed here.