Core collapse in massive scalar-tensor gravity

This paper provides an extended exploration of the inverse-chirp gravitational-wave signals from stellar collapse in massive scalar-tensor gravity reported in [Phys. Rev. Lett. 119, 201103]. We systematically explore the parameter space that characterizes the progenitor stars, the equation of state, and the scalar-tensor theory of the core collapse events. We identify a remarkably simple and straightforward classification scheme of the resulting collapse events. For any given set of parameters, the collapse leads to one of three end states: a weakly scalarized neutron star, a strongly scalarized neutron star, or a black hole, possibly formed in multiple stages. The latter two end states can lead to strong gravitational-wave signals that may be detectable in present continuous-wave searches with ground-based detectors. We identify a very sharp boundary in the parameter space that separates events with strong gravitational-wave emission from those with negligible radiation.

Figure 1

The central density (top panel) and maximal scalarization (bottom panel) are shown for representative examples of the five collapse configurations summarized in Sec. 4a. On the left, we show three evolutions of progenitor s39 with EOS3 for different ST parameters corresponding to the NS formation scenarios 1 (single-stage low-compactness NS), 4 (multistage NS), and 5 (single-stage high-compactness NS), respectively. On the right, we show the evolution of progenitor z39 with EOS3 for different ST parameters corresponding to scenarios 2 (two-stage BH formation), 3 (multistage BH formation), 4 (multistage NS), and 5 (single-stage high-compactness NS), respectively. For comparison, we display with solid black curves the corresponding evolution of the progenitors in GR which result in a NS in the left “s39” case and a BH in the right “z39” case. All curves have been obtained for a scalar mass $\mu ={10}^{-14}\mathrm{eV}$.

Figure 2

A graphical illustration of the main collapse scenarios identified in our simulations. All stellar progenitors collapse into a weakly scalarized (GR-like) NS, a strongly scalarized NS, or a BH. As indicated in the diagram the outcomes may be reached promptly or in two or more stages. The two-stage formation of a strongly scalarized NS has been marked by a dashed arrow because this case appears in our set of simulations only a handful of times, and we suspect numerical noise to have impeded the occurrence of new stages. In the multistage NS category we count the simulations where all stages remain distinct, even if they happen on a short timescale.

Figure 3

We consider a fixed progenitor star with ZAMS mass $39M_{\odot }$, equation of state EOS3 of Table 1, and fix the scalar mass at $\mu ={10}^{-14}\mathrm{eV}$. The progenitor s39 in the left panel has solar metallicity $Z_{\odot }$, and the progenitor z39 in the right panel has primordial metallicity. Top row: For selected values of ${\alpha }_0$, we plot the maximal scalarization of the collapsing star as a function of ${\beta }_0$. The middle row provides a color (or “heat”) map of the same quantity in the $({\alpha }_0,{\beta }_0)$ plane: “Red” = strong scalarization, “Blue” = weak scalarization. The bottom row presents a color code of the five qualitatively different collapse scenarios listed in Sec. 4a. Note that the s39 progenitor exclusively collapses to a neutron star, whereas u39 collapses to a black hole for ${\beta }_0\gtrsim -8$ and to a neutron star for ${\beta }_0\lesssim -8$. We find that every progenitor model results in heat maps in the $({\alpha }_0,{\beta }_0)$ plane qualitatively equal to that on the left (the “neutron star” case) or that on the right (the “black hole” case).

Figure 4

Each panel shows the maximum scalarization of a core collapse as a function of the ZAMS progenitor mass for selected values of ${\beta }_0$ that bracket the transition from weak to strong scalarization. The left column represents EOSa and the right column EOS3. The rows represent a different metallicity as labeled in the panels. The right panels and the top-left panel exclusively contain collapse scenarios forming NSs. In the center-right and bottom-right panels, we distinguish NS cases from those forming BHs by using empty or filled symbols, respectively. Note that for primordial metallicity, the catalog of stellar progenitors contains models up to $M_{\mathrm{ZAMS}}=40M_{\odot }$ only.

Figure 5

A sketch of the coordinates used in the numerical evolution. The main axes show the standard $\{t,r\}$ coordinates and the inset arrows show the $\{u,r\}$ coordinates. The vertical blue line indicates the signal as a function of time at the extraction radius, $\sigma (t;r_{\mathrm{ex}})$, and the shading indicates the region where the signal propagates dispersively. A numerical grid based on the $\{u,r\}$ coordinates can cover the shaded region with less redundant space than one based on $\{t,r\}$.

Figure 6

A sketch plot showing the time-frequency structure of the “inverse chirp” in Eq. (53). The frequency decays over time; the high frequency components (traveling at almost the speed of light) arrive first, followed by the slower low frequency components. Frequencies below ${\omega }_{*}$ are exponentially suppressed and never reach large radii.

Figure 7

GW signal $\sigma (t,r_{\mathrm{ex}})$ extracted from the collapse of an s12 (i.e., $12M_{\odot }$, solar metallicity) progenitor model with ${\alpha }_0={10}^{-2}$, ${\beta }_0=20$, using EOS5 at ${\omega }_{*}{r}_{\mathrm{ex}}=5.07$ for different values of the scalar mass $\mu \in [2\times {10}^{-15}\mathrm{eV},{10}^{-12}\mathrm{eV}]$. The overall amplitude increases monotonically with decreasing $\mu$. For reference, we also show the wave signal obtained for $\mu =0$ (dashed curve). In this case, we cannot rescale the time with ${\omega }_{*}$ and instead measure time in seconds as labeled on the upper horizontal axis.

Figure 8

The wave signals of Fig. 7 propagated according to Eqs. (53) and (57) to ${\omega }_{*}r=1.56\times {10}^{13}=(\mu /{10}^{-14}\mathrm{eV}{)}^{-1}10\mathrm{kpc}$. As expected, the curves for the rescaled frequency $\mathrm{\Omega }/{\omega }_{*}$ overlap in the upper panel. The amplitude in the lower panel shows a mild increase as we decrease the scalar mass $\mu$.

Figure 9

Signal amplitudes $\sqrt{S_o}$ for quasimonochromatic GWs emitted during stellar collapse for several $({\alpha }_0,{\beta }_0)$ values with $\mu ={10}^{-14}\mathrm{eV}$ using progenitor $s39$ with EOS1 (top) and EOS3 (bottom). These are compared against the expected noise curves $\sqrt{S_n(f)}$ of LIGO [82], the Einstein Telescope, and the Cosmic Explorer [83]. The ratio of $\sqrt{S_o}$ to $\sqrt{S_n(f)}$ gives the SNR which is calculated assuming a 2 month period of observation. Results are shown for observations performed different times after the original supernova: $t=1$, 3, 10, 30, 100, 250, 500, and 1000 years increasing from right to left on the plot. The signal frequency decreases slowly with $t$ (inverse chirp) while the amplitude remains at the same order of magnitude for up to $t=1000\mathrm{years}$. These results were computed for a galactic supernova at a distance $D=10\mathrm{kpc}$ from the Earth.

Figure 10

The central baryon density ${\rho }_c$ (left), the central scalar field value ${\phi }_c$ (middle), and the wave signal $\sigma =r_{\mathrm{ex}}\phi$ extracted at $r_{\mathrm{ex}}=3\times {10}^4\mathrm{km}$ (right column) are shown as a function of time for three configurations. Top: Progenitor s39 with EOS3 and scalar parameters ${\alpha }_0={10}^{-3}$, ${\beta }_0=-2$ promptly forms a weakly scalarized NS. Center: Progenitor s39 with EOS1 and ${\alpha }_0={10}^{-1}$, ${\beta }_0=-7$ undergoes a multistage collapse to a strongly scalarized NS. Bottom: Progenitor z39 with EOS1 and ${\alpha }_0={10}^{-3}$, ${\beta }_0=-20$ promptly collapses into a strongly scalarized NS. The Roman numerals label separate stages in the time evolution.

Figure 11

The central baryon density ${\rho }_c$ (left), the central scalar field value ${\phi }_c$ (middle), and the wave signal $\sigma =r_{\mathrm{ex}}\phi$ extracted at $r_{\mathrm{ex}}=3\times {10}^4\mathrm{km}$ (right column) are shown as a function of time for two configurations. Top: The progenitor u39 with EOS1 and scalar parameters ${\alpha }_0={10}^{-3}$, ${\beta }_0=-2$ temporarily forms a weakly scalarized NS before it collapses to BH at $t\approx 0.35\mathrm{s}$. Bottom: The same configuration with ${\beta }_0=-5$ also collapses into a BH eventually, but not before briefly settling down into a strongly scalarized NS phase between $t\approx 0.35\mathrm{s}$ and $t\approx 0.54\mathrm{s}$. For comparison, the scalar field and wave signal of the second configuration are also displayed as dashed lines in the upper panels. Note how the additional strongly scalarized NS stage leads to an increase in the GW signal by several orders of magnitude.

Figure 12

Snapshots of the baryon density and scalar field profiles in the collapse of the progenitor u39 with EOS1 and ST parameters ${\alpha }_0={10}^{-3}$, ${\beta }_0=-5$. The dashed curves show the initial data. A first outgoing shock results from the core bounce at $t\approx 0.086\mathrm{s}$. The second contraction leads to a second core bounce at $t\approx 0.356\mathrm{s}$, and this time the scalar field also increases in amplitude (left panel), signaling the temporary formation of a strongly scalarized NS. At $t\approx 0.537\mathrm{s}$, the baryon density once again starts increasing sharply, this time leading to the formation of a BH and the corresponding descalarization.

Figure 13

Snapshots of the baryon density and scalar field profiles in the collapse of the progenitor s39 with EOS1 and ST parameters ${\alpha }_0={10}^{-1},{\beta }_0=-7$. The dashed curves show the initial data. A first outgoing shock results from the core bounce at $t\approx 0.072\mathrm{s}$. The second contraction leads to a second core bounce at $t\approx 0.116\mathrm{s}$, and this time the scalar field also increases in amplitude (left panel), signaling the temporary formation of a strongly scalarized NS. At $t\approx 0.345\mathrm{s}$, both the baryon density and the scalar field amplitude once again jump, but by lesser margins. Close inspection of the data shows a mild shock that is barely perceptible in the density profile at $t=0.345\mathrm{s}$ around $r\approx 15\mathrm{km}$.

Figure 14

Snapshots of the baryon density (left) and scalar field (right) profiles during the simulation of a single-stage high-compactness NS. The progenitor is $z39$ with EOS1, and the scalar parameters are ${\alpha }_0={10}^{-3}$, ${\beta }_0=-20$, and $\mu ={10}^{-14}\mathrm{eV}$. The dotted black line represents the initial profile. In this case, core bounce occurs at $t\approx 0.087\mathrm{s}$ which leads to a shock propagating outwards. Over the remaining duration of the simulation no further shocks appear and the central density barely changes.

Figure 15

Similar to Fig. 3, for each panel we consider a fixed progenitor star with ZAMS mass $12M_{\odot }$, solar metallicity, and several equations of state of Table 1. Top rows: For selected values of ${\alpha }_0$, we plot the maximal scalarization of the star as a function of ${\beta }_0$. The middle rows provide a color (or “heat”) map of the same quantity in the $({\alpha }_0,{\beta }_0)$ plane: “Red” = strong scalarization, and “Blue” = weak scalarization. The bottom rows present a color code of the five qualitatively different collapse scenarios listed in Sec. 4a. Note that all progenitor models displayed here result in heat maps in the $({\alpha }_0,{\beta }_0)$ plane qualitatively equal to that on the left side of Fig. 3 (the neutron star case).

Figure 16

Similar to Fig. 3, for each panel we consider a fixed progenitor star with ZAMS mass $39M_{\odot }$, all three metallicities and equations of state EOS1 and EOS3 of Table 1. Top rows: For selected values of ${\alpha }_0$, we plot the maximal scalarization of the star as a function of ${\beta }_0$. The middle rows provides a color (or “heat”) map of the same quantity in the $({\alpha }_0,{\beta }_0)$ plane: “Red”=strong scalarization, and “Blue”=weak scalarization. The bottom rows present a color code of the five qualitatively different collapse scenarios listed in Sec. 4a. Note that all progenitor models displayed here result in heat maps in the $({\alpha }_0,{\beta }_0)$ plane qualitatively equal to that on the right side of Fig. 3 (the black hole case).

Figure 17

The top panel shows the results of evolving an initially sin-Gaussian waveform out to radii $r_1=r_{\mathrm{ex}}+500cT$ and $r_2=r_{\mathrm{ex}}+1200cT$ using the time domain numerical evolution of the wave equation (see Sec. 5a). The evolution to large radii was also performed using the analytic Fourier domain approach (see Sec. 5b), and the bottom panel shows the differences, or residuals, between the two methods.

Figure 18

Similar to Fig. 9, but for stellar collapse of the $u39$ progenitor model.

Figure 19

Similar to Fig. 9, but for stellar collapse of the $z39$ progenitor model.