Testing strong-field gravity with tidal Love numbers

TL;DR

This work investigates how tidal Love numbers encode the strong-field deformability of compact objects and uses them as probes of horizon-scale physics. It computes TLNs for exotic compact objects (ECOs) such as boson stars, wormholes, gravastars, and horizon-scale quantum-corrected models, revealing a universal logarithmic vanishing of TLNs in the BH limit, with a magnitude that can still be phenomenologically relevant. It extends the analysis to BHs in beyond-GR theories (Einstein-Maxwell, Brans-Dicke, and Chern-Simons gravity), finding vanishing TLNs in Einstein-Maxwell and Brans-Dicke but nonzero axial TLNs in CS gravity, and provides analytic and numerical results for the CS case. Through Fisher-matrix forecasts for LIGO, ET, and LISA, the paper demonstrates that TLN measurements can constrain ECOs and boson stars and that LISA, in particular, can access horizon-scale deviations, offering a powerful GW-based test of GR in the strong-field regime.

Abstract

The tidal Love numbers (TLNs) encode the deformability of a self-gravitating object immersed in a tidal environment and depend significantly both on the object's internal structure and on the dynamics of the gravitational field. An intriguing result in classical general relativity is the vanishing of the TLNs of black holes. We extend this result in three ways, aiming at testing the nature of compact objects: (i) we compute the TLNs of exotic compact objects, including different families of boson stars, gravastars, wormholes, and other toy models for quantum corrections at the horizon scale. In the black-hole limit, we find a universal logarithmic dependence of the TLNs on the location of the surface; (ii) we compute the TLNs of black holes beyond vacuum general relativity, including Einstein-Maxwell, Brans-Dicke and Chern-Simons gravity; (iii) We assess the ability of present and future gravitational-wave detectors to measure the TLNs of these objects, including the first analysis of TLNs with LISA. Both LIGO, ET and LISA can impose interesting constraints on boson stars, while LISA is able to probe even extremely compact objects. We argue that the TLNs provide a smoking gun of new physics at the horizon scale, and that future gravitational-wave measurements of the TLNs in a binary inspiral provide a novel way to test black holes and general relativity in the strong-field regime.

Figures (9)

• Figure 1: Relative percentage errors on the average tidal deformability $\Lambda$ for BS-BS binaries observed by AdLIGO (left panel), ET (middle panel), and LISA (right panel), as a function of the BS mass and for different BS models considered in this work (for each model, we considered the most compact configuration in the stable branch; see main text for details). For terrestrial interferometers we assume a prototype binary at $d=100\,{\rm Mpc}$, while for LISA the source is located at $d=500\,{\rm Mpc}$. The horizontal dashed line identifies the upper bound $\sigma_\Lambda/\Lambda=1$. Roughly speaking, a measurement of the TLNs for systems which lie below the threshold line would be incompatible with zero and, therefore, the corresponding BSs can be distinguished from BHs. Here $\Lambda$ is given by Eq. \ref{['deflambda']}, the two inspiralling objects have the same mass, and $\sigma_\Lambda/\Lambda\sim \sigma_{k_2^E}/{k_2^E}$.
• Figure 2: Polar (top panels) and axial (bottom panels) TLNs for minimal, massive and solitonic BSs. Left and right panels refers to $l=2$ and $l=3$, respectively. For massive and solitonic BSs we have considered $\alpha=10^4\mu^2$ and $\sigma_0=0.05$, respectively. With these values, the maximum mass scales approximately as shown in Table \ref{['tab:BSs']}. Numerical data are available online webpage. These plots include only stars in the stable branch.
• Figure 3: The $l=2$ and $l=3$, axial- and polar-type TLNs for a stiff wormhole constructed by patching two Schwarzschild spacetimes at the throat radius $r=r_0>2M$. The TLNs are negative and all vanish in the BH limit, $r_0\to 2M$. The latter is better displayed in the inset.
• Figure 4: TLNs for a toy model of Schwarzschild metric with a perfectly reflective surface at $r=r_0>2M$. The TLNs are all negative and vanish in the BH limit, $r_0\to 2M$. Close to the BH limit, the polar- and axial-type Love numbers for the same multipolar order are almost identical, as shown in the inset.
• Figure 5: TLNs for a thin-shell gravastar with zero energy density as a function of the compactness. More generic gravastar models are presented in Ref. Uchikata:2016qku. The TLNs are all negative and vanish in the BH limit, $r_0\to 2M$. Similar to the perfectly-reflective mirror case, the polar- and axial-type Love numbers for the same multipolar order coincide in the BH limit, as shown in the inset.