Torsional oscillations of neutron stars in scalar-tensor theory of gravity

We study torsional oscillations of neutron stars in the scalar-tensor theory of gravity using the relativistic Cowling approximation. We compute unperturbed neutron star models adopting realistic equations of state for the neutron star’s core and crust. For scalar-tensor theories that allow for spontaneous scalarization, the crust thickness can be significantly smaller than in general relativity. We derive the perturbation equation describing torsional oscillations in scalar-tensor theory, and we solve the corresponding eigenvalue problem to find the oscillation frequencies. The fundamental mode (overtone) frequencies become smaller (larger) than in general relativity for scalarized stellar models. Torsional oscillation frequencies may yield information on the crust microphysics if microphysics effects are not degenerate with strong-gravity effects, such as those due to scalarization. To address this issue, we consider two different models for the equation of state of the crust and we look at the effects of electron screening. The effect of scalarization on torsional oscillation frequencies turns out to be smaller than uncertainties in the microphysics for all spontaneous scalarization models allowed by binary pulsar observations. Our study shows that the observation of quasiperiodic oscillations following giant flares can be used to constrain the microphysics of neutron star crusts, whether spontaneous scalarization occurs or not.

Figure 1

Pressure $\tilde{p}$ versus energy density $\widetilde{ϵ}$ for the crust EOSs considered in this work: EOS DH (solid line) and EOS KP (dashed line).

Figure 2

Properties of our stellar models in scalar-tensor theory. From left to right we show the mass-radius relation, the scalar field at the center of the star ${\phi }_c$ as a function of the central density ${\widetilde{ϵ}}_c$, the dimensionless ratio $-\alpha =Q/M$ as a function of the compactness $\tilde{\mathcal{C}}$ and the fractional crust thickness $\tilde{\mathcal{R}}$ as a function of $\tilde{\mathcal{C}}$. The choice of crustal EOS does not sensibly affect the crust thickness and the onset of scalarization. In all panels, curves with various line styles correspond to stellar models using EOS DH for the NS crust: solid lines correspond to $\beta =0.0$, dashed lines to $\beta =-4.5$, and dotted lines to $\beta =-6.0$. Different symbols correspond to stellar models using EOS KP for the crust: circles for $\beta =0.0$, squares for $\beta =-4.5$ and triangles for $\beta =-6.0$.

Figure 3

Comparison between Eq. (25) and the numerical results for $\beta =-6.0$, using $\sigma =0.02326$. The GR expression (24) is also shown. Since the integration of Eqs. (12)–(15), gives us $\phi$ in the Einstein-frame radial coordinate $r$, the compactness and fractional crust thickness are evaluated in this frame. Notice, however, that even for $\beta =-4.5$ (a value marginally excluded by binary pulsars observations [34]) the percent difference between the compactnesses and fractional crust thicknesses in the two frames is less than 1.0%, and therefore Eq. (25) is accurate for all physically sensible values of $\beta$.

Figure 4

Shear velocity profile ${\tilde{v}}_s(r)$ in the NS crust in the following cases: (i) GR without electron screening (solid line); (ii) GR with electron screening (dashed line); (iii) scalar-tensor theory ($\beta =-6.0$) without electron screening (dashed-dotted line); (iv) scalar-tensor theory ($\beta =-6.0$) with electron screening (dotted line). The top panel refers to EOS DH, and the bottom panel to EOS KP. The sharp peaks occur near the neutron drip density $\widetilde{ϵ}\approx 3\times {10}^{11}\mathrm{g}/{\mathrm{cm}}^3$ [68].

Figure 5

Frequencies of the torsional modes in scalar-tensor theory as a function of $M/M_{\odot }$. Top panels: The fundamental torsional mode ${}_0{t}_2$ without (left) and with (right) electron screening. Lower panels: The first overtone ${}_1{t}_{\ell }$ without (left) and with (right) electron screening.

Figure 6

Frequencies of the torsional modes in scalar-tensor theory as a function of $\beta$ for stellar models with mass $M=1.8M_{\odot }$. Circles and dotted lines correspond to $\mathrm{APR}+\mathrm{DH}$; squares and dashed lines correspond to $\mathrm{APR}+\mathrm{KP}$. In the right panel we plot the mode frequencies ${}_0{t}_{\ell }$ for $\ell =2,3,4$ and 5. In the left panel we show the frequencies of the first overtone ${}_1{t}_{\ell }$.

Figure 7

This plot compares modifications in torsional oscillation frequencies due to the underlying gravitational theory with crustal EOS uncertainties for models constructed using EOS APR in the core. Regions bounded by dashed lines correspond to oscillation frequencies in GR with different crustal EOSs; regions bounded by solid lines correspond to oscillation frequencies in scalar-tensor theory with different crustal EOSs. The degeneracy between modified gravity and crustal EOS is broken when the two regions do not overlap. Left panels refer to a scalar-tensor theory with $\beta =-4.5$, and right panels to a theory with $\beta =-6.0$ (a value already excluded by binary pulsar experiments [34]).

Figure 8

The ratio $\eta$ defined in Eq. (51) for all stellar models considered in this work. Values of $\eta >1$ mean that the effect of scalarization is larger than that of electron screening. This would only be possible for values of $\beta$ that are already ruled out by binary pulsar experiments.