Model-dependent boundaries of dark star formation and longevity: neutrinophilic mass thresholds, visible branching fractions, the full MSSM, and capture-driven lifetimes

Dark stars are a hypothesized phase of first-star evolution in which a Population III protostar is powered by dark matter annihilation rather than nuclear fusion. Gondolo, Huh, Kim and Scopel (JCAP 2010) showed that essentially any thermal dark matter candidate heavier than about 50 GeV whose velocity-averaged annihilation cross section is the same at freeze-out and inside the star forms a dark star, with a few exceptions: heavy neutralinos in coannihilation regions, annihilations resonant at freeze-out but not in the star, and light neutrinophilic dark matter annihilating only to neutrinos. Their analysis left the model dependence incomplete: the roughly 50 GeV neutrinophilic mass threshold was computed without internal bremsstrahlung of charged particles; the minimal visible annihilation branching ratio needed for dark star formation (of order $\sim10^{-5}$ by their estimate) was not systematically determined; the boundaries in the Minimal Supersymmetric Standard Model (MSSM) parameter space were mapped only in an effective 8-parameter parameterization; and the effect of dark matter capture on later evolution has been studied only with the WIMP-nucleon scattering cross section and the annihilation cross section fixed independently rather than derived from one underlying model. The question is open; the surviving open core is the model-dependent completion of the dark-star formation and longevity map.

Background

About 85% of the matter in the Universe is dark matter, and the leading particle candidates are weakly interacting massive particles (WIMPs). In the standard thermal-relic picture, WIMPs were kept in equilibrium with the hot early-Universe plasma by pair annihilation until production "froze out" at temperatures $T_{\rm fo}\sim m/20$; the observed relic abundance then fixes the velocity-averaged annihilation cross section at freeze-out, $\langle\sigma v\rangle_{\rm fo}\approx 3\times10^{-26}\ \mathrm{cm^3/s}$. How long a WIMP survives inside a star, and how fast it annihilates there, depends instead on its velocity distribution at stellar speeds ($v\sim 30$–$300$ km/s), which need not give the same thermal average $\langle\sigma v\rangle_{\rm ds}$.

Population III stars are the first generation of stars, formed in dark matter minihalos (the smallest gravitationally bound clumps of dark matter, which host the first star-forming gas clouds) at redshift $z\sim 10$–$50$. Spolyar, Freese and Gondolo (2008) proposed that before nuclear fusion ignites, the first protostars can pass through a "dark star" phase powered by dark matter annihilation: as the gas collapses, the deepening gravitational potential drags the surrounding dark matter along with it (adiabatic contraction, i.e. the slow compression of dark matter orbits in response to the changing gravitational potential), raising the dark matter density at the protostar. Dark matter annihilation then heats the gas at a rate per unit volume

$Q_{\rm DM}=f_Q\,\frac{\langle\sigma v\rangle_{\rm ds}}{m}\,\rho_\chi^2,$

where $\rho_\chi$ is the dark matter density in the protostar, $m$ is the WIMP mass, and $f_Q$ is the fraction of annihilation energy deposited in the gas rather than escaping as neutrinos. If this heating dominates all cooling along the protostar evolution track, collapse halts in a cool, puffy equilibrium supported by dark matter heating instead of fusion. Such dark stars could grow to $10^5$–$10^7\,M_\odot$, have been discussed as James Webb Space Telescope (JWST) targets and as seeds of early supermassive black holes, and remain unconfirmed.

Gondolo, Huh, Kim and Scopel (2010) asked which particle dark matter models actually satisfy the formation condition. Comparing heating to cooling along simulated protostar tracks, the condition is

$f_Q\,\frac{\langle\sigma v\rangle_{\rm ds}}{m}>1\times10^{-32}\ \mathrm{cm^3\,s^{-1}\,GeV^{-1}}.$

Their generic conclusion: any thermal relic heavier than $\simeq 50$ GeV with $\langle\sigma v\rangle_{\rm ds}=\langle\sigma v\rangle_{\rm fo}$ — as for an annihilation cross section with a non-vanishing s-wave contribution, meaning a non-zero limit as the relative velocity goes to zero, unlike p-wave annihilation which is velocity-suppressed at stellar speeds — forms a dark star. The exceptions they identified are: heavy neutralinos in the Minimal Supersymmetric Standard Model (MSSM, the minimal supersymmetric extension of the Standard Model whose lightest neutralino — a superposition of the neutral fermionic superpartners of the gauge and Higgs bosons, called gauginos and higgsinos — is a standard WIMP candidate) in coannihilation regions, where a nearly degenerate heavier superpartner present at freeze-out makes the relic density depend on the combined annihilation of the whole set rather than on the neutralino–neutralino annihilation that occurs in the cold star; annihilations that are resonant at freeze-out — amplified when the collision energy matches the Z-boson or Higgs-boson mass, a Breit–Wigner resonance — but not at the much lower dark-star energies; and light neutrinophilic dark matter.

Neutrinophilic dark matter couples dominantly or exclusively to neutrinos. If it annihilates only into $\nu\bar\nu$, then naively $f_Q=0$ and no dark star can form. However, electroweak bremsstrahlung — radiation of W or Z bosons off the annihilation products, first studied for dark matter indirect detection by Bell, Dent, Jacques and Weiler (2008) — produces visible decay products, so $f_Q>0$, growing with mass. Computing $f_Q(m)$ with the PYTHIA event generator, Gondolo et al. found that neutrinophilic dark matter forms dark stars only for masses above about 50 GeV. They explicitly left open that they had not included internal bremsstrahlung of charged particles (photon radiation off virtual charged particles in the annihilation process, such as the $\nu\bar\nu\gamma$ three-body final state), which would increase the annihilation cross section and could extend the mass limit below 50 GeV in a model-dependent way; that a purely neutrinophilic model is not natural, since realistic models have additional tree-level annihilation channels to visible particles, with a branching ratio at the $\sim10^{-5}$ level already sufficient, so the minimal visible fraction per model remained to be determined; and that their MSSM boundaries came from an effective MSSM with only eight electroweak-scale parameters, equal ratios among the trilinear scalar couplings (the parameters that set the strength of interactions among three scalar superpartners), focused on stau and sneutrino coannihilations (with the scalar superpartners of the tau lepton and of the neutrino), while in the general MSSM coannihilations with charginos (the charged analogues of the neutralino, mixtures of the charged gauginos and higgsinos) and the second-lightest neutralino are also possible.

Separately, the duration of the dark phase depends on whether capture — elastic scattering of ambient dark matter off nuclei in the star, with subsequent gravitational trapping — can replenish the annihilating supply. Capture is controlled by the WIMP–nucleon scattering cross section $\sigma_n$ (spin-dependent or spin-independent, according to whether the interaction couples to nucleon spin or to coherent nucleon number) and by the ambient dark matter density. Freese, Spolyar and Aguirre (2008) proposed capture as an additional power source for the first stars; Sivertsson and Gondolo (2011), in a Monte-Carlo treatment, found that for the first dark stars in standard halos the capturable WIMPs deplete within $\sim10^5$ yr so capture does not substantially prolong the dark star phase — but they fixed $\sigma_n$ and $\langle\sigma v\rangle$ independently rather than deriving both from one particle model. The review by Freese, Rindler-Daller, Spolyar and Valluri (2016) likewise parameterizes capture by the free product of scattering cross section and ambient density. Since in realistic models (e.g. the MSSM) the same couplings control annihilation and scattering, the model-correlated impact of capture on dark star lifetimes has not been determined.

Problem Statement

The common objective is to complete the model-dependent determination of when particle dark matter can form dark stars and for how long the dark-matter-fueled phase lasts — that is, to map the boundary in dark matter model space beyond the generic result that a thermal relic heavier than about 50 GeV with a velocity-independent (s-wave) annihilation cross section forms a dark star. All components below evaluate the same formation criterion $f_Q\,\langle\sigma v\rangle_{\rm ds}/m>10^{-32}\ \mathrm{cm^3\,s^{-1}\,GeV^{-1}}$ (and, for the last component, its time-dependent analogue as the dark matter supply evolves) in model classes where $f_Q$ and/or $\langle\sigma v\rangle_{\rm ds}$ deviate from the generic case:

(1) Neutrinophilic mass threshold: for dark matter annihilating exclusively into neutrinos, compute the visible energy fraction $f_Q(m)$ including internal bremsstrahlung of charged particles (the radiative corrections beyond the W and Z final-state bremsstrahlung already included, such as the $\nu\bar\nu\gamma$ three-body final state and radiation off virtual charged mediators), and determine the corrected threshold mass above which dark stars can form; the present threshold is about 50 GeV without these corrections, and the correction is expected to lower it in a model-dependent way.

(2) Minimal visible branching fraction: in realistic models that are predominantly but not purely neutrinophilic — models whose dark matter annihilates dominantly into neutrinos while retaining subleading tree-level channels into visible particles — determine systematically, as a function of mass and of the underlying particle physics model, the minimum branching ratio of annihilation into visible final states required for dark star formation (order $10^{-5}$ by the source's estimate).

(3) Full MSSM boundary map: extend the computation of the dark-star formation boundaries from the effective eight-parameter MSSM (with trilinear-coupling ratios fixed equal and coannihilation focus on staus and sneutrinos) to the full MSSM, including coannihilations with charginos and with the second-lightest neutralino and the effect of freely varying trilinear couplings, and establish whether additional regions of parameter space exist in which dark stars cannot form.

(4) Capture-driven longevity with consistent microphysics: determine how the repopulation of dark matter inside an evolving dark star by capture — and hence the duration of the dark-matter-fueled phase and the star's total lifetime — depends on the WIMP–nucleon scattering cross section $\sigma_n$ (spin-dependent or spin-independent) when $\sigma_n$ and $\langle\sigma v\rangle$ are derived from the same underlying particle model rather than fixed independently, quantifying for which models capture can significantly prolong the dark star phase.

References

1. Douglas Spolyar, Katherine Freese, Paolo Gondolo, "Dark matter and the first stars: a new phase of stellar evolution", Physical Review Letters 100 (2008) 051101, arXiv:0705.0521, DOI: 10.1103/PhysRevLett.100.051101, https://arxiv.org/abs/0705.0521
2. Katherine Freese, Douglas Spolyar, Anthony Aguirre, "Dark Matter Capture in the First Stars: a Power Source and Limit on Stellar Mass", Journal of Cosmology and Astroparticle Physics 11 (2008) 014, arXiv:0802.1724, DOI: 10.1088/1475-7516/2008/11/014, https://arxiv.org/abs/0802.1724
3. Nicole F. Bell, James B. Dent, Thomas D. Jacques, Thomas J. Weiler, "Electroweak Bremsstrahlung in Dark Matter Annihilation", Physical Review D 78 (2008) 083540, arXiv:0805.3423, DOI: 10.1103/PhysRevD.78.083540, https://arxiv.org/abs/0805.3423
4. Paolo Gondolo, Ji-Haeng Huh, Hyung Do Kim, Stefano Scopel, "Dark Matter that can form Dark Stars", Journal of Cosmology and Astroparticle Physics 07 (2010) 026, arXiv:1004.1258, DOI: 10.1088/1475-7516/2010/07/026, https://arxiv.org/abs/1004.1258
5. Cosmin Ilie, Katherine Freese, Douglas Spolyar, "Dark Stars and Boosted Dark Matter Annihilation Rates", New Journal of Physics 13 (2011) 053050, arXiv:1008.0348, DOI: 10.1088/1367-2630/13/5/053050, https://arxiv.org/abs/1008.0348
6. Sofia Sivertsson, Paolo Gondolo, "The WIMP capture process for dark stars in the early universe", The Astrophysical Journal 729 (2011) 51, arXiv:1006.0025, DOI: 10.1088/0004-637X/729/1/51, https://arxiv.org/abs/1006.0025
7. Katherine Freese, Tanja Rindler-Daller, Douglas Spolyar, Monica Valluri, "Dark Stars: A Review", Reports on Progress in Physics 79 (2016) 066902, arXiv:1501.02394, DOI: 10.1088/0034-4885/79/6/066902, https://arxiv.org/abs/1501.02394
8. Cosmin Ilie, Saiyang Zhang, "Multiscatter capture of superheavy dark matter by Pop. III stars", Journal of Cosmology and Astroparticle Physics 12 (2019) 051, arXiv:1908.02700, DOI: 10.1088/1475-7516/2019/12/051, https://arxiv.org/abs/1908.02700
9. Cosmin Ilie, Jillian Paulin, Katherine Freese, "Supermassive Dark Star candidates seen by JWST?", Proceedings of the National Academy of Sciences 120 (2023) e2305762120, arXiv:2304.01173, DOI: 10.1073/pnas.2305762120, https://arxiv.org/abs/2304.01173
10. Lei Lei, Yi-Ying Wang, Guan-Wen Yuan, Tong-Lin Wang, Martin A. T. Groenewegen, Yi-Zhong Fan, "Can Dark Stars account for the star formation efficiency excess at very high redshifts?", The Astrophysical Journal 980 (2025) 249, arXiv:2501.07119, DOI: 10.3847/1538-4357/ada93b, https://arxiv.org/abs/2501.07119