Rapidly Rotating Neutron Star Collapse in Massive Scalar-Tensor Theories

Author(s)

M., José Carlos Olvera, Doneva, Daniela D., Cerdá-Durán, Pablo, Font, José A., Yazadjiev, Stoytcho S.

Abstract

We present a full 3D numerical evolution code to study neutron stars in massive-scalar-tensor theories. The code is embedded in the Einstein Toolkit framework and its implementation constitutes a modified version of the Baumgarte-Shapiro-Shibata-Nakamura formalism with an additional nonminimally coupled scalar field. The approach we follow preserves the standard hydrodynamic evolution for matter fields, allowing eventually for a straightforward inclusion of more microphysical effects and better flexibility. Using this code, we examine the gravitational collapse of rapidly rotating, scalarized neutron stars to a black hole by exploring the influence of the scalar field on the dynamical features of the process and on the gravitational-wave emission. We find that for the configurations studied in this work, there is an observational degeneracy in the tensorial gravitational-wave emission between collapsing scalarized stars and their counterparts in general relativity. However, this degeneracy can be broken through the emission of scalar radiation, which carries an energy of ~10^-3 M_sun c^2. This is orders of magnitude higher than the quadrupolar emission (~10^-7 M_sun c^2) and might be used as an observational probe of modified gravity. We also find that rapid rotation can enhance this signal, since fast rotating stars can sustain larger scalar field amplitudes.

Figures

Constant angular momentum sequences for rotating polytropic stars in MSTT, with $\alpha_0=10^{-3}$, $\beta_0=-7.5$ and $m_\Phi=1.33\times10^{-11}$ eV/c$^2$. This figure displays the degree of scalarization through the magnitude of the central scalar field (top panel), the circumferential equatorial radius of the star (second row panel), the rotational frequency (third row panel) and the spin parameter $J/M^2$ (bottom panel), where $J$ is the angular momentum and $M$ the mass of the star. In all panels solid lines represent the GR solution while dash-dotted lines show the scalarized solution. Stars and crosses represent the maximum mass solution for MSTT and GR, respectively.
Caption Constant angular momentum sequences for rotating polytropic stars in MSTT, with $\alpha_0=10^{-3}$, $\beta_0=-7.5$ and $m_\Phi=1.33\times10^{-11}$ eV/c$^2$. This figure displays the degree of scalarization through the magnitude of the central scalar field (top panel), the circumferential equatorial radius of the star (second row panel), the rotational frequency (third row panel) and the spin parameter $J/M^2$ (bottom panel), where $J$ is the angular momentum and $M$ the mass of the star. In all panels solid lines represent the GR solution while dash-dotted lines show the scalarized solution. Stars and crosses represent the maximum mass solution for MSTT and GR, respectively.
The figure shows the central scalar field (top panel) and baryonic mass of the stars (bottom panel) both in GR and MSTT for different sequences of constant angular momentum. The diamonds and dots represent the GR and MSTT solutions we study on this work, whose specific details are reported in Table \ref{tab:ID_Models}. These solutions are on the same constant mass line (horizontal lines) for each constant angular momentum sequence ($J=0.5M_\odot^2,1.8 M_\odot^2$), resulting in models that have the same baryon mass and angular momentum. We also show the spherically symmetric sequence ($J=0$) and the Keplerian one ($J=J_k$) for comparison. Stars and crosses indicate the maximum mass point for each branch.
Caption The figure shows the central scalar field (top panel) and baryonic mass of the stars (bottom panel) both in GR and MSTT for different sequences of constant angular momentum. The diamonds and dots represent the GR and MSTT solutions we study on this work, whose specific details are reported in Table \ref{tab:ID_Models}. These solutions are on the same constant mass line (horizontal lines) for each constant angular momentum sequence ($J=0.5M_\odot^2,1.8 M_\odot^2$), resulting in models that have the same baryon mass and angular momentum. We also show the spherically symmetric sequence ($J=0$) and the Keplerian one ($J=J_k$) for comparison. Stars and crosses indicate the maximum mass point for each branch.
The figure is organized into four panels. The top-left panel displays the initial spatial profile of the rest-mass density along the $x$- and $z$-axes (red and black lines, respectively), highlighting the polar radius (black vertical line) and equatorial radius (red vertical line). The bottom-left panel shows the corresponding profile for the scalar field, with the blue line indicating the Compton wavelength associated with the mass of the scalar field. The right panels depict the distribution of these quantities on the $x$-$z$ plane. In these views, the dashed white lines mark the stellar surface, while the dashed orange line represents the Compton wavelength.
Caption The figure is organized into four panels. The top-left panel displays the initial spatial profile of the rest-mass density along the $x$- and $z$-axes (red and black lines, respectively), highlighting the polar radius (black vertical line) and equatorial radius (red vertical line). The bottom-left panel shows the corresponding profile for the scalar field, with the blue line indicating the Compton wavelength associated with the mass of the scalar field. The right panels depict the distribution of these quantities on the $x$-$z$ plane. In these views, the dashed white lines mark the stellar surface, while the dashed orange line represents the Compton wavelength.
Time evolution of the central rest-mass density (top panel), central scalar field (middle panel), and of the central lapse (bottom panel). Each line is representative of the models shown in Table~\ref{tab:ID_Models} for the cold EoS evolution. Vertical lines mark the time of apparent horizon formation.
Caption Time evolution of the central rest-mass density (top panel), central scalar field (middle panel), and of the central lapse (bottom panel). Each line is representative of the models shown in Table~\ref{tab:ID_Models} for the cold EoS evolution. Vertical lines mark the time of apparent horizon formation.
(Top panel) The time evolution of the ratio between the polar and equatorial radius for all the models. The solid lines stop after all the material in the polar axis has been accreted onto the black hole. (Bottom panel) The time evolution of the maximum value of the scalar field for each model. The dotted lines represent the moment the apparent horizon is first found and the dashed lines show the instant when the maximum value of the scalar field has been reduced to $1\%$ of its initial value.
Caption (Top panel) The time evolution of the ratio between the polar and equatorial radius for all the models. The solid lines stop after all the material in the polar axis has been accreted onto the black hole. (Bottom panel) The time evolution of the maximum value of the scalar field for each model. The dotted lines represent the moment the apparent horizon is first found and the dashed lines show the instant when the maximum value of the scalar field has been reduced to $1\%$ of its initial value.
From top to bottom, the rest-mass density, the scalar field normalized by the initial central value, the lapse and the Hamiltonian constraint are shown. The left panels show the $x-$profiles while the right panels indicate the $z-$profiles. The dashed lines represent the corresponding profiles when the apparent horizon is first found. The model shown here is \texttt{MST\_F\_th}.
Caption From top to bottom, the rest-mass density, the scalar field normalized by the initial central value, the lapse and the Hamiltonian constraint are shown. The left panels show the $x-$profiles while the right panels indicate the $z-$profiles. The dashed lines represent the corresponding profiles when the apparent horizon is first found. The model shown here is \texttt{MST\_F\_th}.
Model \texttt{MST\_F\_th} immediately after horizon formation. Rest-mass density (top), Hamiltonian constraint (middle), and scalar field normalized by the central value at $t=0$ (bottom) are shown in the xy (left) and xz (right) planes. The stellar surface at $t=0$ is denoted by a white dashed line, and the apparent horizon by a solid green line. The dotted yellow line represents the Compton wavelength.
Caption Model \texttt{MST\_F\_th} immediately after horizon formation. Rest-mass density (top), Hamiltonian constraint (middle), and scalar field normalized by the central value at $t=0$ (bottom) are shown in the xy (left) and xz (right) planes. The stellar surface at $t=0$ is denoted by a white dashed line, and the apparent horizon by a solid green line. The dotted yellow line represents the Compton wavelength.
Model \texttt{GR\_F\_th} immediately after horizon formation. Rest-mass density (top) and Hamiltonian constraint (bottom) are shown in the xy (left) and xz (right) planes. The stellar surface at $t=0$ is denoted by a white dashed line, and the apparent horizon by a solid green line.
Caption Model \texttt{GR\_F\_th} immediately after horizon formation. Rest-mass density (top) and Hamiltonian constraint (bottom) are shown in the xy (left) and xz (right) planes. The stellar surface at $t=0$ is denoted by a white dashed line, and the apparent horizon by a solid green line.
Time evolution of the rest-mass density (top panel), the scalar field (middle panel) and the conformal factor (bottom panel). The different lines represent data extracted at different radii, as shown in the legend. The red line indicates the time when the apparent horizon is formed. The model shown here is \texttt{MST\_F\_th}.
Caption Time evolution of the rest-mass density (top panel), the scalar field (middle panel) and the conformal factor (bottom panel). The different lines represent data extracted at different radii, as shown in the legend. The red line indicates the time when the apparent horizon is formed. The model shown here is \texttt{MST\_F\_th}.
Top panels: evolution of the gravitational mass (left) and the relative errors with respect to the initial mass for each method (right). Solid lines indicate the mass calculated using Christodoulou formula, dotted lines show the ADM mass computed as a surface integral at infinity, and dashed lines indicate the mass calculated through Eq.~(\ref{eq:MassCeq}). Bottom panels: evolution of the angular momentum (left) and corresponding relative error (right). Vertical lines indicate the time when the apparent horizon is first detected.
Caption Top panels: evolution of the gravitational mass (left) and the relative errors with respect to the initial mass for each method (right). Solid lines indicate the mass calculated using Christodoulou formula, dotted lines show the ADM mass computed as a surface integral at infinity, and dashed lines indicate the mass calculated through Eq.~(\ref{eq:MassCeq}). Bottom panels: evolution of the angular momentum (left) and corresponding relative error (right). Vertical lines indicate the time when the apparent horizon is first detected.
$\Psi_4$ waveforms from a rotating neutron star collapsing to a black hole. The $(l,m)=(2,0)$ mode extraction for the models \texttt{MST\_F\_th} and \texttt{GR\_F\_th}. The extraction is performed at a radius of $r_{\rm ext}=443$ km.
Caption $\Psi_4$ waveforms from a rotating neutron star collapsing to a black hole. The $(l,m)=(2,0)$ mode extraction for the models \texttt{MST\_F\_th} and \texttt{GR\_F\_th}. The extraction is performed at a radius of $r_{\rm ext}=443$ km.
$r\Psi_4$ for the dominant $(l,m)=(2,0)$ mode. The upper panel displays the waveforms aligned at the time of horizon formation. The lower panel shows the modulus in logarithmic scale to highlight the black hole ringing before settling into a Kerr black hole. Shown are the simulations for the \texttt{MST\_F\_th} and \texttt{GR\_F\_th} models.
Caption $r\Psi_4$ for the dominant $(l,m)=(2,0)$ mode. The upper panel displays the waveforms aligned at the time of horizon formation. The lower panel shows the modulus in logarithmic scale to highlight the black hole ringing before settling into a Kerr black hole. Shown are the simulations for the \texttt{MST\_F\_th} and \texttt{GR\_F\_th} models.
The gravitational radiation from the $(l,m)=(2,0)$ mode extracted at different radii for the model MST\_F\_th.
Caption The gravitational radiation from the $(l,m)=(2,0)$ mode extracted at different radii for the model MST\_F\_th.
Integrated waveform $rh^{lm}(t)$ for the $(l,m)=(2,0)$ mode for the \texttt{GR\_F\_th} and \texttt{MST\_F\_th} models. The top panel displays the waveforms aligned at the time of horizon formation. The lower panels shows the absolute difference between both waveforms.
Caption Integrated waveform $rh^{lm}(t)$ for the $(l,m)=(2,0)$ mode for the \texttt{GR\_F\_th} and \texttt{MST\_F\_th} models. The top panel displays the waveforms aligned at the time of horizon formation. The lower panels shows the absolute difference between both waveforms.
Top panel: scalar field extracted at a radius of $1.5\lambda_\Phi\sim22$ km for models \texttt{MST\_F\_th} and \texttt{MST\_S\_th}. Middle panel: corresponding scalar field luminosity. Bottom panel: corresponding total energy radiated away by the scalar field.
Caption Top panel: scalar field extracted at a radius of $1.5\lambda_\Phi\sim22$ km for models \texttt{MST\_F\_th} and \texttt{MST\_S\_th}. Middle panel: corresponding scalar field luminosity. Bottom panel: corresponding total energy radiated away by the scalar field.
Top panel: scalar field extracted at different multiples of the Compton wavelength (see legend in the bottom panel) for model \texttt{MST\_F\_th}. Middle panel: corresponding scalar field luminosity. Bottom panel: corresponding total energy radiated away by the scalar field.
Caption Top panel: scalar field extracted at different multiples of the Compton wavelength (see legend in the bottom panel) for model \texttt{MST\_F\_th}. Middle panel: corresponding scalar field luminosity. Bottom panel: corresponding total energy radiated away by the scalar field.
Instantaneous frequency for the scalar field at the different extraction radii for the \texttt{MST\_F\_th} model. The black horizontal line shows the fundamental frequency of the scalar field $f_\Phi$.
Caption Instantaneous frequency for the scalar field at the different extraction radii for the \texttt{MST\_F\_th} model. The black horizontal line shows the fundamental frequency of the scalar field $f_\Phi$.
Central rest mass density and scalar field evolution for a spherically symmetric star with gravitational mass of $1.7$ $M_\odot$. The scalar field is determined by a mass of $7.5\times10^{-2}\, M_\odot^{-1}$, with $\alpha_0=0.001$ and $\beta_0=-7$. The upper panel shows the evolution of the relative error with respect to the initial data for the central density, while the second row panel shows the convergence of different resolutions. The third row panel shows the evolution of the error for the central scalar field and the last panel shows the convergence of different resolutions. The grid spacing at the finest refinement level is 0.22, 0.295 and 0.369 km for the high, medium and low resolution runs respectively.
Caption Central rest mass density and scalar field evolution for a spherically symmetric star with gravitational mass of $1.7$ $M_\odot$. The scalar field is determined by a mass of $7.5\times10^{-2}\, M_\odot^{-1}$, with $\alpha_0=0.001$ and $\beta_0=-7$. The upper panel shows the evolution of the relative error with respect to the initial data for the central density, while the second row panel shows the convergence of different resolutions. The third row panel shows the evolution of the error for the central scalar field and the last panel shows the convergence of different resolutions. The grid spacing at the finest refinement level is 0.22, 0.295 and 0.369 km for the high, medium and low resolution runs respectively.
Central density and scalar field evolution for a rapidly rotating star with gravitational mass of $2.2725$ $M_\odot$ rotating at $\Omega=0.95\Omega_k$. The mass of the scalar field is of $10^{-1}\,M_\odot^{-1}$, with $\alpha_0=0.001$ and $\beta_0=-8$. The same panel description and resolutions as in Fig.~\ref{fig:convergenceTOV}.
Caption Central density and scalar field evolution for a rapidly rotating star with gravitational mass of $2.2725$ $M_\odot$ rotating at $\Omega=0.95\Omega_k$. The mass of the scalar field is of $10^{-1}\,M_\odot^{-1}$, with $\alpha_0=0.001$ and $\beta_0=-8$. The same panel description and resolutions as in Fig.~\ref{fig:convergenceTOV}.
A star undergoing spontaneous scalarization. The figure describes the behavior of the star at the center. The top panel shows the change in relative density, the second row panel shows the Hamiltonian constraint, the third panel shows the scalar field, the fourth row panel shows the lapse function and the last panel displays the relative error for the baryonic mass.
Caption A star undergoing spontaneous scalarization. The figure describes the behavior of the star at the center. The top panel shows the change in relative density, the second row panel shows the Hamiltonian constraint, the third panel shows the scalar field, the fourth row panel shows the lapse function and the last panel displays the relative error for the baryonic mass.
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