Dynamical chameleon neutron stars: Stability, radial oscillations, and scalar radiation in spherical symmetry

Scalar-tensor theories whose phenomenology differs significantly from general relativity on large (e.g., cosmological) scales do not typically pass local experimental tests (e.g., in the Solar System) unless they present a suitable “screening mechanism.” An example is provided by chameleon screening, whereby the local general relativistic behavior is recovered in high-density environments, at least in weak-field and quasistatic configurations. Here, we test the validity of chameleon screening in strong-field and highly relativistic/dynamical conditions by performing fully nonlinear simulations of neutron stars subjected to initial perturbations that cause them to oscillate or even collapse to a black hole. We confirm that screened chameleon stars are stable to sufficiently small radial oscillations, but that the frequency spectrum of the latter shows deviations from the general relativistic predictions. We also calculate the scalar fluxes produced during collapse to a black hole, and we comment on their detectability with future gravitational-wave interferometers.

Figure 1

Mass-radius plots for varying chameleon energy scales. Dotted, dash-dotted, and dashed lines correspond to CNSs with ${\alpha }_0{M}_{\mathrm{pl}}=1$; darker lines correspond to lower-chameleon-energy scales. The darkest solid line corresponds to NS solutions in GR (${\alpha }_0=0$, $\mathrm{\Lambda }=0$, and no atmosphere). Red star tokens indicate the solutions with maximum mass. Cyan round tokens instead indicate the lightest star featuring a pressure-dominated ($T>0$) core: stable CNSs with a partially descreened core are those between the star and round tokens.

Figure 2

Partial descreening of CNSs. Gradient of the chameleon field around CNS solutions with varying central densities: less dense stars (blue end of the color scale) are screened in their interior, as can be seen from the suppression of the scalar field gradient; stellar solutions with higher central densities (red end of the color scale) feature a scalar field gradient (proportional to the chameleon-propagated fifth force) reactivated in their interior as a result of relativistic effects—i.e., the pressure-dominated cores. In the plot, the gradient of the scalar field is normalized by the radius of the star divided by the Planck mass, $R/M_{\mathrm{pl}}$, while the distance from the center is normalized by $R$.

Figure 3

Stability of CNS. Top panel: ${\rho }_c(t)\equiv \tilde{\rho }(t,r=0)$ (Jordan frame) vs time. Bottom panel: ${\varphi }_c(t)\equiv \varphi (t,r=0)$ vs time. The (magenta) solid line corresponds to a screened CNS with initial central density ${\rho }_c\simeq 1.38\times {10}^{15}\mathrm{g}/{\mathrm{cm}}^3$. The (cyan) dash-dotted line corresponds to a descreened CNS with initial central density ${\rho }_c\simeq 1.57\times {10}^{15}\mathrm{g}/{\mathrm{cm}}^3$. The matter and chameleon field configurations are stable against small perturbations given by truncation errors.

Figure 4

Migration test. The time evolution of the (Jordan-frame) central density ${\rho }_c(t)\equiv \tilde{\rho }(t,r=0)$ (red solid line) and central scalar field ${\varphi }_c(t)\equiv \varphi (t,r=0)$ (blue dashed line), for an unstable and partially descreened CNS with baryon mass $M_{\mathrm{bar}}=2.17M_{\odot }$ (gravitational mass $M=1.88M_{\odot }$) and ${\rho }_c\simeq 1.87\times {10}^{15}\mathrm{g}/{\mathrm{cm}}^3$. The star is expanding in volume and relaxes through large dampened oscillations to a stable descreened CNS with the same baryon mass and slightly lower gravitational mass $M=1.87M_{\odot }$, but lower central density ${\rho }_c\simeq 1.64\times {10}^{15}\mathrm{g}/{\mathrm{cm}}^3$.

Figure 5

Collapse of a descreened CNS. Time evolution of the lapse (solid red line), density (dashed green line), and chameleon field (dotted blue line) at the center of a collapsing descreened star with gravitational mass $M=1.88M_{\odot }$, normalized by their initial values. As matter collapses towards the center, the density and chameleon field values increase. The lapse function decreases to small values close to zero.

Figure 6

Collapse of a descreened CNS. Snapshots of the radial profile of the lapse function (top panel) and scalar field (bottom panel) for the same collapsing star as in Fig. 5. Quantities are plotted against the Einstein-frame isotropic radius, $r$. Black dots indicate the position of the apparent horizon.

Figure 7

Spectra of radial oscillations. The two plots show the PSDs of the radial modes extracted from the time evolution of the central rest-mass density for three CNSs and one GR NS. Top panel: F, H1, and H2 modes. Bottom panel: higher overtones, $H_N$ with $N>2$, and the new family of scalar modes ($F_s$ and higher overtones). The results shown in this plot are valid for the parameter choice $(\mathrm{\Lambda },{\tilde{\epsilon }}_{\infty })=(175\mathrm{GeV}$, $6.5\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3$).

Figure 8

$F$-mode frequency vs average density. Linear fits of the fundamental mode ($F$) frequency as a function of the average density ($\overline{\rho }$) for NSs in GR (blue dotted line, cyan cross tokens) and in chameleon gravity (magenta dash-dotted line, red round tokens). The results shown in this plot are valid for the parameter choice $(\mathrm{\Lambda },{\tilde{\epsilon }}_{\infty })=(175\mathrm{GeV}$, $6.5\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3$).

Figure 9

Scalar radiation from oscillating stars. Top panel: ${\mathrm{\Phi }}_{22}$ vs retarded time. Bottom panel: strain amplitude vs retarded time. The results shown in these plots have been obtained for $(\mathrm{\Lambda },{\tilde{\epsilon }}_{\infty })=(175\mathrm{GeV}$, $6.5\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3$). The scalar radiation is extracted from simulations of oscillating CNSs with (dashed magenta lines) and without (continuous cyan lines) screening in the interior. The gravitational masses of the stars are, respectively, $M=1.02M_{\odot }$ (screened CNS) and $M=1.84M_{\odot }$ (descreened CNS). The distance of the detector from the source is set to $D_L=10\mathrm{kpc}$. To trigger the oscillations, an initial perturbation with amplitude $\delta {\xi }_0={10}^{-6}$ is employed.

Figure 10

Signal vs detector sensitivity curves—oscillating stars. The strain amplitude in the frequency domain, $S(f)=f|h(f){|}^2$, is compared to the design sensitivity curves, $S(f)=S_n(f)$, of Advanced LIGO (red dot-dashed line) and ET (blue dotted line). The source-detector distance is set to $D_L=10\mathrm{kpc}$. The signals correspond to the monopole GWs produced by screened (continuous magenta line) and descreened (dashed cyan line) stars. The gravitational masses of the stars are, respectively, $M=1.02M_{\odot }$ (screened CNS) and $M=1.84M_{\odot }$ (descreened CNS). The initial perturbations (in the specific internal energy) employed to trigger the oscillations and scalar GW emission have an amplitude of $\delta {\xi }_0=4\times {10}^{-3}$. The visible peaks in the signals correspond to the fundamental mode, $F$, of the characteristic radial oscillations of the CNSs. The results have been obtained for $(\mathrm{\Lambda },{\tilde{\epsilon }}_{\infty })=(175\mathrm{GeV}$, $6.5\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3$).

Figure 11

Scalar strain vs initial perturbation amplitude. The plots show the maximum amplitude of the monopole scalar radiation against the maximum amplitude of the initial perturbation of the specific internal energy, $\delta \xi$. The top and bottom panels correspond to a screened star with $M=1.02M_{\odot }$ and a descreened star with $M=1.84M_{\odot }$, respectively, located at a $D_L=10\mathrm{kpc}$ distance from the detector. The amplitudes of the initial perturbation (in the specific internal energy) that we consider are $\delta {\xi }_0=\{{10}^{-6},{10}^{-5},{10}^{-4},{10}^{-3},4\times {10}^{-3}\}$. The parameters of the model are set to $(\mathrm{\Lambda },{\tilde{\epsilon }}_{\infty })=(175\mathrm{GeV},6.5\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3)$. The black dashed lines show a power-law fit.

Figure 12

Scalar radiation from collapsing stars. Top panel: ${\mathrm{\Phi }}_{22}$ vs retarded time. Bottom panel: strain amplitude vs retarded time. Burst signals are extracted from simulations of collapsing CNSs, with (dashed magenta lines) and without (continuous cyan lines) screening in the interior. The source-detector distance is set to $D_L=10\mathrm{kpc}$.

Figure 13

Signal vs detector sensitivity curves—collapsing stars. The strain amplitude in the frequency domain, $S(f)=f|h(f){|}^2$, is compared to the design sensitivity curves, $S(f)=S_n(f)$, of Advanced LIGO (red dot-dashed line) and ET (blue dotted line). The source-detector distance is set to $D_L=10\mathrm{kpc}$. The signals correspond to the monopole GWs produced by screened and descreened stars undergoing gravitational collapse. The vertical dash-dotted black line corresponds to $f_{\infty }=m_{\infty }/2\pi$—i.e., the peak frequency of the burst, below which all frequencies are Yukawa-suppressed.

Figure 14

Scalar charge vs chameleon energy scale. The plot shows the scalar charge, $Q$, of CNSs with $M_{\mathrm{bar}}=1.75M_{\odot }$ against the chameleon energy scale, $\mathrm{\Lambda }$; the atmosphere density is kept constant, ${\tilde{\epsilon }}_{\infty }=6.5\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3$. Red dots represent data corresponding to $\mathrm{\Lambda }=\{175,122,73\}\mathrm{GeV}$. The black dashed line represents the power law $Q\sim {\mathrm{\Lambda }}^a$ (with $a\simeq 2$) fitting the data.

Figure 15

Scalar charge vs atmosphere density. The plot shows the scalar charge, $Q$, of CNSs with $M_{\mathrm{bar}}=1.75M_{\odot }$ against the atmosphere density, ${\tilde{\epsilon }}_{\infty }$; the asymptotic value of the chameleon field is fixed to ${\varphi }_{\infty }\simeq 0.17M_{\mathrm{pl}}$. Red dots represent data corresponding to ${\tilde{\epsilon }}_{\infty }=\{6.5,2.6,1.1,0.34,0.081\}\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3$. The black dashed line represents the power law $Q\sim {\tilde{\epsilon }}_{\infty }^b$ (with $b\simeq -3/5$) fitting the data.

Figure 16

Maximum burst amplitude vs chameleon energy scale. The plot shows the maximum amplitude, $h_{\max }$, of the burst produced by collapsing CNSs with $M_{\mathrm{bar}}=1.75M_{\odot }$ against the chameleon energy scale, $\mathrm{\Lambda }$; the atmosphere density is kept constant, ${\tilde{\epsilon }}_{\infty }=6.5\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3$. Red dots represent data corresponding to $\mathrm{\Lambda }=\{175,122,73\}\mathrm{GeV}$. The black dashed line represents the power law $h_{\max }\sim {\mathrm{\Lambda }}^c$ (with $c\simeq 5/2$) fitting the data.

Figure 17

Maximum burst amplitude vs atmosphere density. The plot shows the maximum amplitude, $h_{\max }$, of the scalar GW burst against the atmosphere density, ${\tilde{\epsilon }}_{\infty }$. The collapsing CNSs have a fixed baryon mass $M_{\mathrm{bar}}=1.75M_{\odot }$, and the asymptotic value of the chameleon field is fixed to ${\varphi }_{\infty }\simeq 0.17M_{\mathrm{pl}}$. Red dots represent data corresponding to ${\tilde{\epsilon }}_{\infty }=\{6.5,2.6,1.1\}\times {10}^{10}\mathrm{g}/{\mathrm{cm}}^3$. The black dashed line represents the power law $h_{\max }\sim {\tilde{\epsilon }}_{\infty }^d$ (with $d\simeq -7/10$) fitting the data.