White dwarf bounds on charged massive particles

White dwarfs (WD) effectively act as high-gain amplifiers for relatively small energy deposits within their volume via their supernova instability. In this paper, we consider the ways a galactic abundance of $\mathcal{O}(1)$-charged massive relics (i.e., CHAMPs) could trigger this instability, thereby destroying old WD. The dense central core structure formed inside the WD when heavy CHAMPs sink to its center can trigger a supernova via injection of energy during collapse phases, via direct density-enhanced (pycnonuclear) fusion processes of carbon nuclei dragged into the core by the CHAMPs, or via the formation of a black hole (BH) at the center of the WD. In the latter scenario, Hawking radiation from the BH can ignite the star if the BH forms with a sufficiently small mass; if the BH instead forms at large enough mass, heating of carbon nuclei that accrete onto the BH as it grows in size may be able to achieve the same outcome (with the conservative alternative being simply that the WD is devoured by the BH). The known existence of old WD that have not been destroyed by these mechanisms allows us to improve by many orders of magnitude on the existing CHAMP abundance constraints in the regime of large CHAMP mass, $m_X\sim {10}^{11}–{10}^{18}\mathrm{GeV}$. Additionally, in certain regions of parameter space, we speculate that this setup could provide a trigger mechanism for the calcium-rich gap transients: a class of anomalous, subluminous supernova events that occur far outside of a host galaxy.

Figure 1

upper panel: Energy deposition required with a trigger volume within a trigger time in order to initiate thermal runaway, as a function of the WD mass. We assume $T_{\mathrm{crit}.}\sim 0.5\mathrm{MeV}$ and that the BH forms near the center of the WD. lower panel: Diffusion time scale for the trigger volume as a function of the WD mass, under the same assumptions.

Figure 2

The CHAMP abundance that becomes gravitationally bound to a WD, $M_{X^{\pm }}^{\mathrm{accum}.}({\tau }_{\mathrm{accum}.})$ [Eqs. (c1)–(c3)], compared to the naïve estimate $M_{X^{\pm }}^{\mathrm{approx},\mathrm{III}}$ given at Eq. (34), when the CHAMP speed distribution and maximum stopping speed are taken into account. This plot assumes that the CHAMP passes through a distance $\sim R_{\mathrm{W}\mathrm{D}}$ of WD material with a density equal to the average WD density for a $M_{\mathrm{W}\mathrm{D}}\sim 1M_{\odot }$ WD with $R_{\mathrm{W}\mathrm{D}}\sim 0.01R_{\odot }$, and that the CHAMP velocity distribution far from the WD is given by a truncated Maxwellian distribution, Eq. (29). We assume that the WD moves with respect to the CHAMP distribution at the local circular speed in the MW, $v_{\mathrm{W}\mathrm{D}}\sim v_0\sim 220\mathrm{km}/\mathrm{s}$, and that the MW escape speed is $v_{\mathrm{esc},\mathrm{MW}}\sim 540\mathrm{km}/\mathrm{s}$. We further assume that $\sigma =200\mathrm{mb}$; see text. Note that the accumulated mass only begins to significantly deviate from its maximum value once $m_X\gtrsim 3\times {10}^{16}\mathrm{GeV}\sim 2m_X^{\max }$.; because the estimate at Eq. (33) was conservative in requiring all CHAMPs to accrete, a significant fraction still accrete even somewhat about this mass.

Figure 3

The number density of ions as a function of radius, normalized to the core number density ($n_{\mathrm{ion}}\approx 2.0\times {10}^{32}{\mathrm{cm}}^{-3}$), of an $M=M_{\odot }$ pure-C WD with fractional $X^{+}$ contamination $\eta =4\times {10}^{-10}$, for an $m_X={10}^5\mathrm{GeV}$ CHAMP (red, solid curve). We assume as input that the $X^{+}$ have all settled to the core of the star (complete stratification), forming an inner core of pure-$X^{+}$ with mass $M_{X^{+}}\approx 0.8M_{\mathrm{Chand}.,X}$, which is found to have radius $R_X\approx 5.4\times {10}^{-6}{R}_{\mathrm{WD}}$ (indicated by the vertical dotted black line); the whole star is found to have radius $R_{\mathrm{WD}}=8.2\times {10}^{-3}{R}_{\odot }$. We ignore thermal effects, which would presumably mildly smooth out the discontinuity in ion number density at $r=R_X$ (see text). Also shown are (a) the ion number density profile for an equal-mass pure-C WD without $X^{+}$ contamination (green, short-dashed line), which is found to have an almost identical radius to the contaminated star, and (b) the ion number density for a pure $X^{+}/e^{-}$ WD of mass equal to the core mass (blue, long-dashed line). Note that, for ease of comparison, the latter two profiles are shown here normalized to the central density and radius of the $X^{+}$-contaminated star, and not to their own central number densities. A comparison of these results indicates that, because $M_{X^{+}}\ll M_{\mathrm{W}\mathrm{D}}$, the $X^{+}$ have very little impact on the mechanical structure of the star outside the core (see inset). Moreover, because this particular core is not quite extremal, it does not well approximate the profile of the isolated, equal-mass, pure $X^{+}/e^{-}$ WD, as we expect a more extremal core would; see discussion in Appendix pp4.

Figure 4

Constraints on the abundance of CHAMPs $X^{+}$, ${\rho }_X/{\rho }_0$, as a function of CHAMP mass $m_X$ in GeV. This plot is drawn assuming that $f_{X^{+}}=1$ (see discussion in text). Here, ${\rho }_X$ is the local CHAMP mass density, and ${\rho }_0$ is taken to be the local halo DM abundance ${\rho }_0=0.3\mathrm{GeV}/{\mathrm{cm}}^3$. The dark red shaded region bounded by the thick dark red lines (annotated as “$M_X^{\mathrm{prim}.}+M_X^{\mathrm{accum}.}=M_{\mathrm{s}.\mathrm{g}.,X}$” at large $m_X$, and annotated as “$M_X^{\mathrm{prim}.}=M_{\text{Chand}.,X}$” at small $m_X$) denotes the region where (1) the sum of the primordial CHAMP abundance, $M_{X^{+}}^{\mathrm{prim}\mathrm{.}}$, and the CHAMP abundance accumulated onto the WD from the galactic CHAMP halo before the WD crystallization time, $M_{X^{+}}^{\text{accum.}}$, exceed the larger of the Chandrasekhar mass for $X^{+}$ or the self-gravitating mass Eq. (62), leading to BH formation; and (2) the subsequent BH evolutionary trajectory ends either with the BH accreting the entire WD mass (or triggering thermal runaway in the process of doing so) [see Sec. 7c2], or with the BH Hawking evaporating away, and triggering thermal runaway in the process [see Sec. 7c1]. This bound has a kink at $m_X^{\text{s.g}}\sim 6.4\times {10}^8\mathrm{GeV}$ owing to a crossover between the Chandrasekhar and self-gravitating masses (with $M_{\mathrm{s}.\mathrm{g}.,X}>M_{\text{Chand}.,X}$ above this value of $m_X$, and smaller below). Nevertheless, we also continue to show the parameters for which the Chandrasekhar mass is exceeded at larger $m_X$ by $M_X^{\mathrm{prim}.}+M_X^{\text{accum}.}$ with the faint red dotted line (annotated “$M_X^{\mathrm{prim}.}+M_X^{\mathrm{accum}.}=M_{\text{Chand}.,X}$” at large $m_X$), as this is the maximum extent to which the limits could improve if either (a) the conservatively high temperature we have assumed for the thermal structure ($T\sim \mathrm{keV}$) is lowered [$M_{\mathrm{s}.\mathrm{g}.,X}\propto T^{3/2}$], or (b) the thermal-pressure-supported structure does not actually form; see footnote 28. The shark-fin-shaped region around $m_X\sim {10}^9\mathrm{GeV}–{10}^{12}\mathrm{GeV}$ which is not shaded red and is annotated “${\tau }_{\text{destruction}}>{\tau }_{\mathrm{W}\mathrm{D}}$” is the region in which a sufficient CHAMP abundance is present (primordially) in the WD to cause BH formation, but the evolutionary time scales for the BH are too long to destroy the WD within its assumed lifetime. The green shaded part of this shark-fin-shaped region would however evolve rapidly enough to destroy the BH (by Hawking evaporation) if the initial primordial CHAMP accretion rate is assumed to be given by the refined estimates in Sec. 7a4, instead of by the naïve estimates discussed there. The thin red dashed line (annotated as “$M_X^{\mathrm{prim}.}=M_{\mathrm{s}.\mathrm{g}.,X}$” at large $m_X$) delineates the regions of parameter space where the primordial abundance alone is sufficient to exceed the self-gravitating mass (above the red dashed line), and where the primordial abundance must be augmented by the slow accretion of CHAMPs over the WD lifetime (until the crystallization time) in order to exceed a self-gravitating mass of CHAMPs in the star (below the red dashed line). These bounds all assume the WD parameters shown in Table. 4. Also shown are the NS-derived bounds reported by Ref. [50], with the region above the thick solid grey line excluded (annotated as “NS limit” and shaded grey except—for clarity—in the shaded red region). The region ${10}^5\mathrm{GeV}\lesssim m_X\lesssim {10}^{11}\mathrm{GeV}$, where the results of Ref. [52] call into question the presence of CHAMPs in the MW, is indicated by the thin long-dashed vertical black lines; in this region, we present all results as dotted lines (and the shading is lighter) to indicate the uncertainty as to their applicability; we do not claim robust bounds here (see discussion in text). CHAMPs lighter than $m_X\lesssim {10}^3\mathrm{GeV}$ have a sinking time scale in the WD estimated to approach or exceed the WD lifetime and/or crystallization time of $\sim \mathrm{Gyr}$, and we conservatively do not present limits here; see Sec. 6c.

Figure 5

As for Fig. 4, but showing the constraints on the abundance of CHAMPs $X^{-}$ immutably bound as $(NX)$, assuming $f_{X^{-}}=1$ and $Q_{(NX)}=+6$. The kink in the bound at $m_X^{\text{s.g}}\sim 8.3\times {10}^{11}\mathrm{GeV}$ is where the Chandrasekhar and self-gravitating masses cross for $Q_{(NX)}=+6$. The additional orange shaded region [annotated as ‘$(pX)$’] as compared to Fig. 4 indicates where, in the region ${10}^5\mathrm{GeV}\lesssim m_X\lesssim {10}^{11}\mathrm{GeV}$, an abundance of $(pX)$ at the level of ${10}^{-4}$ of the total number abundance of $X^{-}$ would accumulate to a super-Chandrasekhar mass of CHAMPs (larger than the self-gravitating mass for these $m_X$) in the WD within the crystallization time, assuming that these neutral particles are not evacuated from the galactic disk by supernova shockwaves (and magnetically inhibited from reentry) [52], but are efficiently captured by dense WD material owing to exchange reactions such as $(pX)+N\to (NX)+p$ [see discussion in Sec. 6b]. Of course, the actual limit would not jump discontinuously from the red line to the orange region at $m_X\sim {10}^{11}\mathrm{GeV}$; there would be some smooth crossover region. CHAMPs lighter than $m_X\lesssim {10}^3\mathrm{GeV}$ have a sinking time scale in the WD estimated to approach or exceed the WD lifetime and/or crystallization time of $\sim \mathrm{Gyr}$ and we conservatively do not present limits here; see Sec. 6c.

Figure 6

These plots indicate the regions (blue shaded) of $(m_X,{\widehat{\rho }}_X/{\rho }_0)$ parameter space where our speculations about a possible trigger mechanism for the CaRGTs are applicable, both for $X^{+}$ (left panel) and $X^{-}$ bound as $(NX)$ (right panel). These shaded regions are shown with very fuzzy edges to emphasize the large uncertainties associated with our estimates, and their highly speculative nature. Also shown for $m_X\gtrsim {10}^{11}\mathrm{GeV}$ are the bounds on $(m_X,{\rho }_X/{\rho }_0)$ from Figs. 4 and 5 [see captions there for explanations of the shading]. The limits from Figs. 4 and 5 in the region $m_X\lesssim {10}^{11}\mathrm{GeV}$ have been removed as, for our CaRGT picture to be applicable, the CHAMP expulsion mechanisms of Ref. [52] must be operative, and this invalidates the limits from Figs. 4 and 5 in that mass range (but see footnote 47 for an important caveat about the mass range of interest); we have nevertheless included very light shaded lines that follow the outlines of those limits in order to guide the eye in a comparison of the regions of parameter space. Note that the CaRGT regions are shown for ${\widehat{\rho }}_X/{\rho }_0$, while the limits from Figs. 4 and 5 are on ${\rho }_X/{\rho }_0$; in order to show both sets of results on the same plot and for the comparison to be meaningful, we have had to identify ${\widehat{\rho }}_X$ and ${\rho }_X$ [see Sec. 9 and footnote 50]. The blue shaded regions assume $M_{\mathrm{W}\mathrm{D}}=0.85M_{\odot }$ and ${\tau }_{\mathrm{accum}.}\sim \mathrm{Gyr}$; see Table 1 for other parameters.

Figure 7

The value of ${\alpha }_{\star }$ necessary to obtain the critical star assuming a pure $(NX)$ configuration with $Q_{(NX)}=+6$, as a function of the CHAMP mass $m_X$ (black circles). Over a wide range of masses, ${\alpha }_{\star }\propto m_X^{-1/3}$ (red line), as discussed in the text.

Figure 8

For an $X^{-}$ of varying mass as indicated in the legend, this plot shows a comparison of the central ion number densities in the central core of a fully stratified CO star containing some CHAMP contamination [denoted $n_{[NX+N]}$], with the central ion number densities in an isolated pure-($NX$) star of the same mass as the core [denoted $n_{[NX]}$], and the central ion number densities in the uncontaminated (CO) star [denoted $n_{[N]}$]. In all cases, the WD structure is computed in the mean-ion approach assuming the “mean ion” in the unperturbed CO star has charge $+7$ [resulting in a core of heavy charge $+6$ ($NX$) objects]. These results are obtained by fixing the total mass of the WD in the stratified and uncontaminated cases to be $M_{\mathrm{W}\mathrm{D}}=1.2M_{\odot }$. It is clear that the central core object in the stratified star is always more dense than that of either the uncontaminated star or the isolated completely contaminated star, as a result of the overburden in the stratified case. However, as the central density in the isolated completely contaminated star becomes much larger [respectively, smaller] than the central ion density in the uncontaminated star, the central number density in the equal-mass core in the partially contaminated stratified case behaves more and more like that of the isolated completely contaminated star (the black dotted line shows $n_{[NX+N]}=n_{[NX]}$) [respectively, like that of the uncontaminated star (the black dashed line shows $n_{[NX+N]}=n_{[N]}$)]. The value of the CHAMP mass $m_X$ appears to be irrelevant for this comparison, assuming it is large enough to guarantee complete stratification in the partially contaminated case case.

Figure 9

For an $X^{-}$ of varying mass $m_X$ as indicated in the legend, this plot shows a comparison of the radius of the central core of a fully stratified CO star containing some CHAMP contamination, with the radius of an equal-mass (denoted $M$) isolated pure-($NX$) star. In both cases, the WD structure is computed in the mean-ion approach assuming the “mean ion” in the unperturbed CO star has charge $+7$ [resulting in a core of heavy charge $+6$ ($NX$) objects]. These results are obtained by fixing the total mass of the WD in the stratified case to be $M_{\mathrm{W}\mathrm{D}}=1.2M_{\odot }$. It is clear that the central core object in the stratified star is always more compact than the isolated completely contaminated star, as a result of the overburden in the stratified case “squashing” the central core; however, as the mass of the core in the stratified case approaches the Chandrasekhar mass of the isolated completely contaminated star, its radius and that of the isolated completely contaminated star converge.