Reheating neutron stars with the annihilation of self-interacting dark matter

Compact stellar objects such as neutron stars (NS) are ideal places for capturing dark matter (DM) particles. We study the effect of self-interacting DM (SIDM) captured by nearby NS that can reheat it to an appreciated surface temperature through absorbing the energy released due to DM annihilation. When DM-nucleon cross section σχn is small enough, DM self-interaction will take over the capture process and make the number of captured DM particles increased as well as the DM annihilation rate. The corresponding NS surface temperature resulted from DM self-interaction is about hundreds of Kelvin and is potentially detectable by the future infrared telescopes. Such observations could act as the complementary probe on DM properties to the current DM direct searches.


Introduction
Dark matter (DM) composes one-fourth of the Universe, however, its essence is still elusive. Many terrestrial detectors are built to reveal the particle nature of DM either from measuring the coupling strength between DM and the Standard Model (SM) particles [1][2][3][4][5][6][7] or the indirect signal from DM annihilation in the space [8][9][10][11][12][13]. But the definitive evidence is yet to come.
A compact stellar object such as neutron star (NS) is a perfect place to capture DM particles even when DM-nucleon cross section σ χn is way smaller than the current direct search limits. Investigations on DM in compact stellar objects are studied recently in refs. [14][15][16][17][18][19][20][21][22][23][24][25]. Due to strong gravitational field, DM evaporation mass for NS is less than 10 keV [19]. Therefore, NS is sensitive to a broad spectrum of DM mass from 10 keV to PeV, sometimes it can be even extended to higher mass region. Unlike the Sun, it loses its sensitivity to DM when m χ 5 GeV as a consequence of evaporation [26,27]. In the later discussion, we will focus on the Weakly Interacting Massive Particle (WIMP) scenario with mass from MeV to hundreds of GeVs.
An old NS having age greater than billions of years could become a cold star after processing several cooling mechanism by emitting photons and neutrinos [28][29][30][31]. However, if the residing DM particles in the NS can annihilate to SM particles other than neutrinos, they will be absorbed by the host star and act like energy injections to heat the star up [14,15]. In addition, recent literature also suggests that the halo DM particles constantly bombard NS can deposit their kinetic energy to the star. This is called dark kinetic heating [32]. These two contributions might prevent NS from inevitable cooling as suggested by refs. [32,33].

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Besides the DM-nucleon interaction, inconsistencies in the small-scale structure between the observations and the N -body simulations [34][35][36][37][38][39][40][41][42][43][44] imply the existence of selfinteracting DM (SIDM) [45][46][47][48][49][50][51]. The constraint given in refs. [52,53] could mitigate such discrepancies as well as the diversity problem of the galactic rotation curves [54,55]. It brings us to where σ χχ is DM self-interaction cross section. The resulting effect of DM self-capture in NS was considered insignificantly due to it saturates quickly when the sum of the individual σ χχ exceeds the geometrical area over which DM is thermally distributed [19]. Its impact is unable to compete with the capture by DM-nucleon interaction when σ χn 10 −50 cm 2 . However, current direct searches have put more stringent limits on σ χn to test. If it is small enough, DM self-capture will eventually take over [20]. In this region, DM self-interaction will re-enhance the captured DM particles as well as the DM annihilation rate regardless how small σ χn is. The corresponding energy injection increases consequently. Hence, in the self-interaction dominant region, NS will experience a reheating effect with rising NS surface temperature. An old, isolated NS nearby the Solar System with surface temperature with hundreds of Kelvin emits infrared that is a very good candidate to pin down such reheating effects due to DM. The corresponding blackbody peak wavelength is potentially detectable in the future telescopes, e.g. the James Webb Space Telescope (JWST) [56], the Thirty Meter Telescope (TMT) [57] and the European Extremely Large Telescope (E-ELT). In the following context, we consider a nearby NS with age t NS 2 × 10 9 years, mass M = 1.44M where M ≈ 1.9 × 10 33 g is the Solar mass and radius R = 10.6 km. It also has the halo density ρ 0 = 0.3 GeV cm −3 , the DM velocity dispersionv = 270 km s −1 and the NS velocity relative to the Galactic Center (GC) v N = 220 km s −1 . For discussion convenience, we will use natural unit c = = k B = 1 and G = M −2 P in this paper. This paper is structured as follows: in section 2 and section 3, we briefly review the formalism of DM captured by NS and the cooling and heating mechanism respectively. In section 4, numerical results are presented as well as the discussion on the reheating effect. The implication of T sur for DM direct searches is discussed too. The work is summarized in section 5.

DM evolution equation
When the halo DM particles scatter with NS and lose significant amount of energies, they will be gravitationally bounded in the star. The evolution of DM number N χ in NS can be characterized by the differential equation where C c is the capture rate due to DM-nucleon interaction, C s the DM self-capture rate due to DM self-interaction and C a the DM annihilation rate. A general solution to N χ is -2 -

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given by where τ = 1/ C c C a + C 2 s /4 is the equilibrium timescale. In the case of t τ , dN χ /dt = 0 where N χ reaches the steady state. Hence we have Thus, R signifies how crucial that the DM self-capture is in the DM evolution in NS. Additionally, we can obtain two solutions to N χ when dN χ /dt = 0, by examining eq. (2.1), That means, either the capture is dominated by C c or C s that could accumulate the same amount of DM particles in NS.

Rates of DM capture and annihilation
The capture rate due to DM scattering with target neutrons in NS is given by [19] where ρ 0 is the DM density,v the DM velocity dispersion, N n = M/m n the total number of target neutrons in NS, and M and R are the mass and radius of NS respectively. The suppression factor ξ = δp/p F is due to the neutron degeneracy effect. The momentum transfer in each scattering is δp √ 2m r v esc where m r = m χ m n /(m χ + m n ) the reduced mass and v esc 1.8 × 10 5 km s −1 . Since the DM-nucleon cross section σ χn cannot exceed the geometric limit that is given by N n σ c = πR 2 where σ c 2 × 10 −45 cm 2 is the critical cross section for DM-nucleon in NS. Thus, σ eff χn ≡ min(σ χn , σ c ) is the effective DM-nucleon cross section. The last factor Unless m χ 10 TeV, the term in the parentheses is roughly unity.
Another way of capture is due to the halo DM particle scatters with the trapped DM particle. This is DM self-capture and is given by [20,58] where v esc (R) is the escape velocity at the surface of NS. For a rather conservative calculations, we take φ χ = 1 [20]. The quantity η = 3/2(v N /v) where v N = 220 km s −1 is the NS velocity relative to GC.

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Usually the term C s N χ in eq. (2.1) is proportional to N χ σ χχ , but it cannot grow arbitrarily as N χ increases. The sum of individual σ χχ never surpasses the geometric area over the DM particles are thermally distributed in the NS. The geometric area is characterized by the thermal radius r th [19]: where T χ is the DM temperature and such limitation is called the geometric limit for DM self-capture in NS. Therefore, for any σ χχ range given in eq. (1.1), the term N χ σ χχ must not larger than πr 2 th . Qualitative, we take eq. (1.1) as our initial input for σ χχ in the numerical calculation. However, it only serves the purpose of how fast N χ σ χχ approaching πr 2 th . After reaching this saturation, the initial σ χχ input is irrelevant. Though N χ could grow thenceforth, the overall quantity N χ σ χχ remains πr 2 th . This procedure is carried out by our numerical program. As a remark, such saturation for DM self-capture always happens. If it is not considered in the calculation, N χ will be highly overestimated.
When more and more DM particles accumulate in the NS, the chance of DM annihilation becomes appreciated. Thus, and the total annihilation rate is In the later discussion, we will use thermal relic annihilation cross section σv = 3 × 10 −26 cm 3 s −1 .

Neutron star cooling and energy injection due to DM annihilation
After the birth of NS, it undergoes the cooling mechanism due to neutrino and photon emissions [28,29]. Nonetheless, if the residing DM particles in NS can annihilate, the annihilation products will be absorbed and act like energy injections to heat the host star up. The NS interior temperature T int can be described by the following differential equation where ν,γ,χ are the emissivities due to neutrino emission, photon emission and DM respectively. They are given by [14,28] where n 3.3×10 38 cm −3 is the NS baryon number density and n 0 0.17 fm −3 the baryon density for the nuclear matter [14]. It is therefore n/n 0 2.3.

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Since the NS outer envelope shields us from observing T int directly. We can only observe the luminosity L γ emitted from this envelope. The corresponding temperature can be inferred from Stefan-Boltzmann's law that is defined as the NS surface temperature T sur where L γ = 4πR 2 σ SB T 4 sur . A relation that connects T int and T sur is given by [29,59,60] T sur = 0.87 × 10 6 K g s 10 14 cm s −2 where g s = GM/R 2 = 1.85×10 14 cm s −2 is the surface gravity. In general, T sur is lower than T int . However, when T int 3700 K, the distinction between the two becomes negligible [29]. Applying eq. (3.3) to obtain T sur from T int is unnecessary when T int 3700 K.
On the other hand, L γ is also responsible for the energy loss due to photon emission. Hence we have the effective photon emissivity where eq. (3.4a) is obtained from ref. [14] and eq. (3.4b) is the expression for γ when T int 3700 K. In addition, NS heating comes from the contributions of DM annihilation and dark kinetic heating. They are given by for annihilation and for dark kinetic heating. The factor f χ characterizes the energy absorption efficiency which runs from 0 to 1. The term E s = m χ (γ − 1) is the DM kinetic energy deposited in NS and γ 1.35 [32]. Therefore, we have for DM emissivity. The last quantity c V is the NS heat capacity of NS and is expressed as [14] c For calculation convenience, we have expressed all these quantities in terms of natural unit.

SIDM implication for neutron star temperature
Before presenting the numerical results, we briefly introduce the setups of our calculations. We let the time starts at t 0 = 100 years after the birth of NS and the age of NS in our study is t NS = 2 × 10 9 years. The beginning temperature is 10 9 K for both T int and T χ . In the presence of DM self-interaction, it might come to a time t s that DM selfinteraction cross section reaches its geometric limit σ c χχ = πr 2 th /N χ (t s ) where σ c χχ is some variable for our program to discriminate the DM self-capture rate attaining this limit or not. If the program detects that the initial σ χχ input is larger than σ c χχ at any time t > t s , it will automatically return σ c χχ to avoid overestimating the effect of DM self-interaction. Moreover, σ c χχ depends on T χ due to its dependence on r th , eq. (2.8). If DM is in thermal equilibrium with NS, then T χ = T int and T int can be used to identify σ c χχ . However, it costs some time to thermalize with NS. The thermalization timescale is given by [ If such timescale is longer than the age of NS, then T χ is unable to thermalize with T int . Thus, once our program detects t th < t NS in solving the coupled differential equations eqs. (2.1) and (3.1) at some time step t i , it will return T χ (t i ) = T (t i ) and use it in the next step t i+1 of calculation. On the other hand, if the program finds that t th > t NS at t i , it will not only return T χ (t i ) = T (t i ) but also identify T χ (t i ) as the decouple temperature T dec χ . No matter how time evolves, the program will recognize T χ (t i+1 ) = T χ (t i+2 ) = T χ (t i+··· ) = · · · = T dec χ . The DM temperature T χ decoupled from the NS cooling curve and always stays at T dec χ . This phenomenon is depicted in figure 1. The explanation will be given in the following subsection.

Numerical results
In each panel of figure 1, the red solid line in is the NS cooling curve of T int , and the dashed lines are T χ . The DM masses m χ are indicated by different colors. The initial temperature is 10 9 K for both T int and T χ . The NS cooling curve is generally insensitive to this initial condition. After the first few decades, the effect of initial temperature becomes negligible and this agrees with ref. [14].
Taking m χ = 1 TeV in the left panel for instance, when t 10 5 years, T χ is able to thermalize with NS for a given σ χp = 10 −56 cm 2 . After t 10 5 years, the corresponding t th is longer than t NS and DM is unable to thermalize with NS. Thus T χ decoupled from T int since then. Because t th depends on m χ as well, T dec χ is not the same for different m χ . For the right panel of figure 1, it is easily seen that m χ = 1 TeV is never in thermal equilibrium with T int from the beginning. This is due to σ χp given in this panel is too weak to have t th smaller than t NS even with such high initial temperature 10 9 K. Hence, the initial temperature is the maximum T dec χ that DM can have. 1 To determine t th one needs to solve the DM energy loss rate: dE/dt = −ξnnσχnvδE. The detail discussion is beyond the scope of this work. It can be found in refs. [19,20,61] and references therein. σ χn = 10 -58 cm 2 Figure 1. The evolution of T int (solid) and T χ (dashed). DM masses m χ are marked with different colors. When T χ is cold enough at some time, the corresponding t th is longer than t NS . Therefore, T χ will decouple from T int since then (flat dashed lines). We use σ χχ /m χ = 4 cm 2 g −1 and σv = 3 × 10 −26 cm 3 s −1 in the calculation. See main text for detail.
Once NS cooling due to ν and γ emissions are balanced by DM heating χ , the NS interior temperature T int stops dropping. The associated NS surface temperature T sur for an isolated NS with age t NS = 2 × 10 9 years is shown in figure 2. The resulting T int for obtaining T sur in figure 2 are all smaller than 3700 K. Thus, there is no distinction between T int and T sur below this threshold. But for convenience, we will still use T sur in the following discussion. See the discussion in section 3.
Without DM heating effect, a two-billion-year old isolated NS has T sur ≈ 120 K predicted by standard NS cooling mechanism. For T sur < 120 K would be impossible. It is indicated by the pink shaded region in figure 2. When σ χn 10 −50 cm 2 , DM-nucleon interaction dominates the capture process and is mainly responsible for the DM heating and the deviation of T sur from 120 K. While 10 −50 cm 2 σ χn 10 −57 cm 2 , neither N χ captured through DM-nucleon interaction nor DM self-interaction can trigger enough DM heating. The energy loss due to NS cooling, particularly from γ , overwhelms the energy injection from DM annihilation. The contribution from χ is negligible.
When σ χn < 10 −57 cm 2 , the capture process is dominated by DM self-interaction. For smaller σ χn , the thermalization timescale becomes longer than t NS hence T χ decoupled from T int in the earlier time with higher decouple temperature T dec χ . See the dashed lines in figure 1. The decouple temperature T dec χ decides how much N χ will be captured inside the host star ultimately. Since T dec χ does not change after decouple, it portrays the size of r th , eq. (2.8). The weaker σ χn is, the higher T dec χ and the larger r th . The DM selfcapture rate is stronger. More N χ will be captured and results in greater energy injections. This explains the rising T sur like a reheating when σ χn < 10 −57 cm 2 in figure 2. One can also notice that the rising of T sur is not unlimited. When σ χn is small enough, T dec χ will not grow into larger value as its maximum is the initial temperature 10 9 K. Such also corresponds to the maximum DM heating when the capture process is dominated by the DM self-interaction. 2 The plateaus for each m χ in figure 2 shows the maximum T sur caused by DM in this region.
A broad scan of T sur over m χ − σ χn plane is displayed in figure 3. The region below the green dashed line indicates the plateaus to each m χ shown in figure 2. The NS surface temperature T sur does not change regardless of any smaller σ χn . The dark purple region on the right-top is where DM heating is negligible. The corresponding T sur ≈ 120 K as indicated by standard NS cooling.  Figure 4. The purple shaded area is where standard NS cooling overwhelms the DM heating. The corresponding T sur is about 120 K. DM-nucleon interaction and DM self-interaction are responsible for the heating on T sur above and below the purple shaded region in the middle of this figure respectively. We use σ χχ /m χ = 4 cm 2 g −1 and σv = 3 × 10 −26 cm 3 s −1 in the calculation. See main text for detail. The constraints on σ χn from different DM direct searches such as DARWIN [3], LUX [4] and XENON1T [7] are shown in the plot as well.

NS surface temperature as a complementary probe for DM properties
Here we display our final result in figure 4 as well as the constraints on σ χn from different DM direct searches. The purple shaded region is where DM heating has no contribution. Thus, standard NS cooling mechanism predicts T sur ≈ 120 K for an isolated two-billionyear old NS. Above the shaded region, the larger σ χn , the more N χ will be captured as well as the stronger DM heating from annihilation. Below the shaded region, T sur is reheated as a consequence of DM self-interaction and the color portion is the maximum T sur can be obtained in the DM self-interaction dominant region. This color portion in the bottom indicates the plateaus in figure 2 and the region below the green dashed line in figure 3.
Interestingly, T sur above the purple shaded region coincides with T sur in some parameter space below the purple shaded region. This phenomenon is indicated in eq. (2.5). For instance, there are two lines show T sur = 300 K in figure 4. It could be helpful as an extra information for determining DM properties along with current DM direct searches. If we observed T sur = 300 K for an isolated two-billion-year old NS and concur that the heating is purely from DM. But 10 GeV m χ 100 GeV with T sur = 300 K is already disfavored by DARWIN. Thus, DM could be either lighter than a few GeV or σ χn is very small. Future DM direct searches could reveal more constraint on sub-GeV DM, together with the astrophysical observations on T sur , we can further unravel more information about DM.

Summary
In this work, we found that when DM self-interaction dominates DM capture process, T sur increases as σ χn decreases. This is contrary to DM-nucleon interaction dominant case where T sur becomes colder as σ χn diminishes. In addition, when 10 −50 cm 2 σ χn 10 −57 cm 2 , the heating from DM is negligible. The energy loss from photon emission overwhelms the energy deposition from DM annihilation. Thus, standard NS cooling is the dominant process. For a two-billion-year old isolated NS, standard NS cooling predicts a lower bound T sur ≈ 120 K. If DM properties such as m χ and σ χn lie within this parameter space, they are unable to probe from the measurement of T sur . However, the precise value for such lower bound depends on the knowledge of NS cooling mechanism. Once we have more constraints from the astrophysical observations on T sur for isolated old NSs, this lower bound could be subject to some correction.
The parameter space in the capture processes dominated by DM-nucleon interaction and by DM self-interaction can generate the same T sur as shown in figure 4. Together with the current DM direct searches, it could improve our knowledge on DM properties in various ways.
In closing, the NS surface temperature T sur induced by DM self-interaction roughly ranges from 120 K to 700 K. The corresponding blackbody peak wavelength is infrared and could be detected by the forthcoming telescopes such as JWST, TMT and E-ELT. The corresponding observations on T sur could act as the complementary probe to DM direct searches in the future.