Constraining decaying dark matter with neutron stars

The amount of decaying dark matter, accumulated in the central regions in neutron stars together with the energy deposition rate from decays, may set a limit on the neutron star survival rate against transitions to more compact objects provided nuclear matter is not the ultimate stable state of matter and that dark matter indeed is unstable. More generally, this limit sets constraints on the dark matter particle decay time, $\tau_{\chi}$. We find that in the range of uncertainties intrinsic to such a scenario, masses $(m_{\chi}/ \rm TeV) \gtrsim 9 \times 10^{-4}$ or $(m_{\chi}/ \rm TeV) \gtrsim 5 \times 10^{-2}$ and lifetimes ${\tau_{\chi}}\lesssim 10^{55}$ s and ${\tau_{\chi}}\lesssim 10^{53}$ s can be excluded in the bosonic or fermionic decay cases, respectively, in an optimistic estimate, while more conservatively, it decreases $\tau_{\chi}$ by a factor $\gtrsim10^{20}$. We discuss the validity under which these results may improve with other current constraints.

Disentangling the nature of dark matter (DM) is one of the greatest current challenges in physics. Whether it is realized through a stable or a decaying particle remains unknown to date. There is a vast literature with many well-motivated particle physics models containing unstable, long-lived, DM particle candidates, see e.g. [1] for a review. From the phenomenological side, there are results that constrain possible DM decay timescales, τ χ , from cosmic microwave background (CMB) anisotropies [2], galaxy cluster abundances [3], DM halo simulations [4], and the observed excess in the cosmic electron/positron flux [5]. In most of these works, it is usually assumed that the decay daughter particles are (nearly) massless although a more generic situation with arbitrary non-zero masses, m D , may well happen [6]. The spread of the current bounds on the DM lifetime τ χ or, equivalently, on the DM decay rate Γ χ = 1/τ χ is large. In light of the Pamela [7] and Fermi LAT [8] data, these can be interpreted in a scenario where a decaying χ-particle has a lifetime τ e + e − ∼ 10 26 s, for DM masses m χ > ∼ 300 GeV and well into the TeV range [9] (we use c = 1). Such lifetimes may appear in the context of supersymmetric grand unification theories through operators with mass dimension 6, τ GUT On the other hand, CMB data provide a constraint Γ −1 χ > ∼ 30 Gyr for massless daughter particles while for sufficiently heavy ones the decay time m D < ∼ m χ remains unrestricted [6]. This agrees with analysis on the stability of DM halos based on kick velocities of particles in the decay [4] and combined constraints based on Lyman-α forest, Planck and WMAP data [10,11].
In this work we consider a scenario where weakly interacting (WIMPy) scalar bosonic or fermionic metastable DM is gravitationally accreted on a neutron star (NS). These objects are compact, with typical radius R ≃ 10 km and mass M ≃ 1.5M ⊙ . In a simplified description, they are believed to have a central core region, which constitutes most of the star, where mass densities are supranuclear. Although there is a rich phenomenology on the possible internal core composition we consider it here as composed of nucleon fluid, with mass densities ρ n ∼ [1 − 10]ρ 0 (ρ 0 ≃ 2.4 × 10 14 g/cm 3 ). Under these conditions NSs are efficient DM accretors as they can effectively capture an incoming χ-particle passing through the star. In order to see this, let us recall that a WIMPy DM particle may have a mean free path much smaller than the typical NS radius λ χ ≃ 1 σχnnn where σ χn is the χ−nucleon elastic scattering cross-section and n n = ρ n /m n with m n the nucleon mass. Compilation of the latest results in direct detection searches [12] allows analysis to the level of σ χn ≃ 10 (−44÷−42) cm 2 in the m χ ∼ (10 − 10 4 ) GeV range. Inside the NS each DM particle will scatter a number of times given by However, accretion of DM will proceed not only during the NS lifetime, but also in the previous late stages of the progenitor star where the dense nuclear ash central core allows the build-up of a χ-distribution, n χ (r), over time.
In previous work, we have considered the effect of a selfannihilating DM particle on the internal NS dynamics [13][14][15][16] but here we will focus on the possibility that the only process depleting DM is decay. We will assume that DM has remained in the universe with an abundance such as to give the local abundance we measure today.
The DM accretion process onto NSs has been previously estimated [17] by means of the DM particle capture rate, C χ , given an equation of state for regular standardmodel matter in the interior of the NS at a given galactic location and with a corresponding ambient DM density. Taking as reference a local value for DM density ρ ambient χ,0 ≃ 0.3 GeV/cm 3 , the DM capture rate is approximated as Therefore, in the NS, the DM particle number, N χ , can be written through a differential equation considering competing processes, namely DM capture and decay, the latter via a generic decay rate Γ as resulting in a DM population at time t The solution takes into account the possibility of an existing DM distribution in the progenitor star before the time of the collapse, t col , producing the supernova explosion.
Depending on the χ-mass and thermodynamical conditions inside the star, it may be possible to thermally stabilize a DM internal distribution. In this case the DM particle density takes the form with n 0, χ the DM particle density at the NS center. Φ(r) is the gravitational potential Φ(r) = r 0 . Assuming a constant baryonic density in the core M (r) = r 0 ρ n 4πr 2 dr. Finally we obtain with a thermal radius r th = ( 3kT 2πGρnmχ ) 1/2 . The progenitor star may indeed have accumulated enough DM at the time the supernova explodes. In order to estimate this amount, we will consider a 15M ⊙ progenitor. After the He burning stage for t He→CO ≃ 2 × 10 6 yr a CO mass ∼ 2.4M ⊙ sits in the core with a radius R ∼ particles. In this case, a coherence factor relates the nucleus (N) and nucleon (n) scatterings, i.e. σ χN ≃ A 2 µ mn 2 σ χn where A is the baryonic number and µ the reduced mass for the χ−N system. Since the later stages proceed rapidly, this expression gives the main contribution to the progenitor. As the fusion reactions happen at higher densities and temperatures, the DM thermal radius contracts. In this way, for example, for m χ = 1 TeV in the He → CO, r th ≃ 470 km, while for Si → FeNi, r th ≃ 70 km. The thermalization time t th is accordingly small where n N = ρN mN . For the two cases mentioned above, both are small compared to the dynamical time t th /t He→CO ≃ 10 −5 , t th /t Si→FeNi ≃ 10 −7 . However during the core collapse, the dynamical timescale involved is ∆t dyn col ≃ 3 8πρG ≃ 10 −3 s whereρ is an average matter density. Assuming a proto-NS forms with T ≃ 10 MeV, central density n n = 5n 0 and a neutron-rich frac- 3π 2 , thermalization time in this phase takes longer to be achieved [18] The core collapse may thus affect the DM population inside the star as just a fraction will be gravitationally retained. Due to the lack of gravitational binding and possibly high initial velocity kicks, the remaining DM particles outside the proto-NS may evaporate. The number of DM particles in the star interior, r < R * , is written as N χ = R * 0 n 0, χ e −(r/r th ) 2 dV . It is a dynamical quantity since r th is temperature (time) dependent. As long as R * >> r th , we obtain N χ = n 0, χ (πr th ) 3 . The retained fraction is so that for a R PNS ≃ 10 km, f χ ≃ 2 × 10 −3 . The retained DM population in the PNS after the collapse is Let us note that the central DM density in the newly formed PNS n 0, χ ≃ 3 × 10 23 cm −3 is much smaller than that in the baryonic medium ∼ 10 38 cm −3 . Although rare, and in a similar way to proton decay experiments, we can estimate the number of DM decays in the NS phase within a time interval ∆t = t − t col << Γ −1 using a linear approximation in Eq. (4) and with aid of Eq. (8), For a time interval comparable to known ages of ancients pulsars (like that estimated for the isolated pulsar PSR J0108-1431) ∆t ≃ τ old NS = 2 × 10 8 yr decays may have profound implications. Neglecting any phase space blocking effects, the number of decays assuming decay times similar to those in cosmic positron/electron anomaly τ e + e − , is given by and the time-averaged rate of decays in this case iṡ N D,χ = 6.7 × 10 10 fχ 2×10 −3 1 TeV mχ At this point, we should check that the DM population number indeed does not exceed the Chandrasekar limiting mass for the star to survive. If this was the case, it may lead to gravitational collapse of the star (see [19,20]). Therefore, for fermionic DM, we expect N with M Pl the Planck mass, and for the bosonic case N Ch ∼ (M Pl /m χ ) 2 ∼ 1.5 × 10 32 (1 TeV/m χ ) 2 . In case a Bose-Einstein condensate is considered [21] N BEC ≃ 10 36 (T /10 5 K) 5 and the condition is N χ (t) < N Ch + N BEC . As described, in the fermionic case, DM remains at all times below the limiting mass, but this may not be the case in the cooling path of the PNS if a Bose-Einstein condensate is formed for a DM particle in the ∼TeV mass range. We can see that the scenario described here may be at the border of the collapse case, however we will restrict our discussion to the precollapsed state, leaving the possibility of additional complexity for further investigation.
If we now focus on the typical decay final states of interest for fermionic or bosonic (neutral) DM, we can estimate the energy deposition in the medium. Strictly speaking, injection and deposition are related by an injection fraction that remains unknown since we do not know the preferred decay channels. In this work and in order to size of the effect, we will consider the photon contribution to decays by two-body channels with intermediate (massive) state daughter particles, quarks, leptons, weak bosons Φ w or, more generic Φ bosons and photons. Reactions include χ → Φ w Φ w , l + l − , q + q − , 2Φ, Φγ, χ → Φ w l, keeping in mind more generic decay final states [22] may well happen. Using the photon spectrum dNγ dE from [23,24], we estimate the rate of particles injected per unit volume and unit energy in the ith-channel with corresponding decay rate Γ i at stellar radial location r Then the energy rate injected in the prompt decay channel is written as Energy release from DM decay is injected in a typical volume V th = 4 3 πr 3 th , where heating and cooling processes compete. At this point we must note that although there are indeed energy losses due to photon cooling and gravitational contraction, these effects do not significantly change the picture. As a result, over a time interval ∆t < τ old NS the DM decay has a local net heating effect yielding an additional average energy density u decay ≃ ∆t EQ(E, r)dEdV.
DM decay may be regarded as a spark-seeding mechanism in similar fashion to modern versions of other nucleation experiments such as COUPP [25] based on hot spike models. In the present case this may allow further changes induced in the NS as a result of possible quark bubble nucleation. A thermally induced quark bubble nucleation, has been already suggested [26] and some studies [27] conclude that quark matter bubbles may nucleate if the temperature exceeds a few MeV, provided the MIT model bag constant is B 1/4 = 150 ± 5 MeV. In the scenario depicted here, the energy release in decays may provide the injection of energy to create a bubble.
In order to see this we estimate the minimum critical work needed to nucleate a neutral stable spherical quark bubble in the core of the cold NS. It is given by [27] W c = 16π 3 where ∆P = P q − P n is the pressure difference and P q (P n ) is the quark (nucleon) pressure. For a two-flavour ud-quark system this is given by and assuming a neutron-rich system P n ≃ µ 2 n −m 2 n 15π 2 mn and all pressure will effectively be provided by neutrons. γ = i=u,d µ 2 i 8π 2 , is the curvature coefficient and µ i (µ n ) is the quark (nucleon) chemical potential related to the Fermi momentum of the degenerate system µ i = p F i (µ n = m 2 n + p 2 F n ). Electrical charge neutrality requires for the ud matter n d = 2n u and n n = nu+n d 3 with n i = µ 3 i π 2 in the light quark massless limit. Note that we do not include further refinements due to quark masses, in-medium effects, Coulomb or surface droplet tension since they do not change the global picture as we want to keep a compact meaningful description of the nucleation process. Bubbles have a radius R c = 2γ/∆P and their stability is granted as they reach the minimum baryonic number A min ∼ 10 when A ∼ R 3 c n n > A min . The energy density necessary to create a quark bubble with volume V d ≃ 4 3 πR 3 c is therefore u bub ≃ W c /V d ≃ 5.4 × 10 35 erg/cm 3 . This estimate is in agreement with similar and more detailed calculations [28]. Then to allow the quark bubbles to nucleate, the average energy densities must be of the same order, i.e., u decay > ∼ u bub . For a cold and old NS, the central temperature is T ∼ 10 5 K and if a quark deconfinement transition in a bubble size comparable to the DM thermal volume takes place, it most likely will produce a macroscopic transition. Some attemps to model this computationally have been recently performed in [29].
In Fig (1), we can see the logarithm of the DM particle decay time versus its mass. The colored regions represent exclusion regions for the decaying particle phase space since they would produce NS transitions over ages below those assumed for regular NS. Darker colored regions represent more efficient energy deposit processes, corresponding gradually to bosonic (fermionic) decay channels for more (less) efficient energy injection. Data points correspond to the required lifetimes from [5] and [11]. Since NS can effectively test decaying DM, there is thus a natural scale constrained by its lifetime τ old NS . We can thus use this result to set exclusion regions for τ χ complementary to those shown in other works [4] [11][10] [6]. We can see that for τ χ < ∼ 6.3 × 10 15 s, masses (m χ /TeV) > ∼ 8 × 10 2 are excluded. Since the bubble nucleation may involve complex dynamics, we do not attempt to model the details here and we estimate that the number of bubbles being created by spark seeding due to DM decay is given byṄ bub ≃ dN bub dE dE dt , and over NS lifetime N bub ≃Ṅ bub t. If this was indeed the scenario, a possible catastrophic event of NS to quark star transition could happen when the macroscopic deconfinement proceeds via detonation modes to rapidly consume the star. The GRB signal emitted has been estimated in [15] and subsequent emission in the cosmic ray channels is also expected [16].
In our galaxy, the supernova rate is about R = 10 −2 yr −1 so that an average rate of NS formation over the age of the universe τ U ∼ 4.34 × 10 17 s yields N N S ∼ Rτ U ∼ 10 8÷9 . Given this population, one should be able to set a lower limit for τ χ on the age of the oldest NS known.
In order to further compare with other analyses in the literature, we consider a generic decay process where a decaying DM particle produces a stable DM (SDM) particle Φ SDM and a lighter particle L in a reaction χ → Φ SDM L. The mass loss fraction f = mχ−mΦ SDM mχ and the recoil kick velocity of the SDM is v k = f c, assuming non-relativistic momenta. We assume the lighter particle is injected into the medium. The spectrum in this channel allows us to plot the exlusion phase space as shown in Fig. (2). We depict DM decay time (in Gyr) versus recoil kick velocity (in km/s). Excluded (bluecolored) regions based on previous work by Wang et al. [10] on combined CMB and Ly-α analysis are compared with our constraints (green regions). We assume again ambient χ-densities of ∼ 0.3 GeV/cm 3 . We can see that the low (1 < ∼ v k < ∼ 10) km/s unrestricted region in [10] is effectively constrained since in those cases there would be efficient production of NS transitions over NS lifetimes. In conclusion, we have shown that the current population of NS in the galaxy may have the capability of further constraining the nature of a possibly decaying bosonic or fermionic DM particle with mass in the > ∼ TeV range. In this case, DM particles with lifetimes τ χ < ∼ 6.3 × 10 15 s exclude masses (m χ /TeV) > ∼ 5 × 10 1 or (m χ /TeV) > ∼ 8 × 10 2 in the bosonic or fermionic cases, respectively. These results are obtained from the prior of avoiding nucleation of quark bubbles in a NS core due to efficient energy injection by spark seeding. If this was the case, a conversion from NS into quark star would be triggered, thereby reducing the population of regular NS in the galaxy. Our results provide complementary constraints in the low recoil kick velocity v k region of the m χ −τ χ phase space for a weakly interacting DM particle candidate.
MAPG would like to thank useful discussions with R.