Dark Matter Heating vs. Rotochemical Heating in Old Neutron Stars

TL;DR

This paper reevaluates dark matter heating in old neutron stars considering rotochemical heating, showing that DM heating signatures can be detected in old pulsars, but are concealed in millisecond pulsars, with implications for constraining dark matter models.

Contribution

It incorporates rotochemical heating into the analysis of dark matter heating effects in neutron stars, providing a more comprehensive understanding of their surface temperatures over time.

Findings

Dark matter heating can be detected in old ordinary pulsars.

Rotochemical heating conceals dark matter heating signals in old millisecond pulsars.

Observations of very cold neutron stars can tightly constrain dark matter models.

Abstract

Dark matter (DM) particles in the Universe accumulate in neutron stars (NSs) through their interactions with ordinary matter. It has been known that their annihilation inside the NS core causes late-time heating, with which the surface temperature becomes a constant value of $T_s \simeq (2-3) \times 10^3$ K for the NS age $t \gtrsim 10^{6-7}$ years. This conclusion is, however, drawn based on the assumption that the beta equilibrium is maintained in NSs throughout their life, which turns out to be invalid for rotating pulsars. The slowdown in the pulsar rotation drives the NS matter out of beta equilibrium, and the resultant imbalance in chemical potentials induces late-time heating, dubbed as rotochemical heating. This effect can heat a NS up to $T_s \simeq 10^6$ K for $t \simeq 10^{6-7}$ years. In fact, recent observations found several old NSs whose surface temperature is much higher than the prediction of the standard cooling scenario and is consistent with the rotochemical heating. Motivated by these observations, in this letter, we reevaluate the significance of the DM heating in NSs, including the effect of the rotochemical heating. We then show that the signature of DM heating can still be detected in old ordinary pulsars, while it is concealed by the rotochemical heating for old millisecond pulsars. To confirm the evidence for the DM heating, however, it is necessary to improve our knowledge on nucleon pairing gaps as well as to evaluate the initial period of the pulsars accurately. In any cases, a discovery of a very cold NS can give a robust constraint on the DM heating, and thus on DM models. To demonstrate this, as an example, we also discuss the case that the DM is the neutral component of an electroweak multiplet, and show that an observation of a NS with $T_s \lesssim 10^3$ K imposes a stringent constraint on such a DM candidate.

Figures

Equations

$$C\frac{d{T}^{\infty }}{dt}$$

$$n+N\to p+N+\ell +{\bar{\nu }}_{\ell },$$

$$p+N+\ell \to n+N+{\nu }_{\ell },$$

$$L_H^{\infty }{|}_{\mathrm{DM}}=e^{2\Phi (R)}\dot{N}{m}_{\mathrm{DM}}\left[\chi +(\gamma -1)\right],$$

Full text

††preprint: UT-19-10, IPMU 19-0072

I Introduction

Despite the firm evidences for dark matter (DM), its nature has not been unraveled yet. A promising possibility is that DM consists of a weakly-interacting massive particle (WIMP), as its thermal relic can naturally explain the observed DM density. WIMPs have interactions with ordinary matter other than the gravitational interactions, through which we may detect their signature. There have been various proposals for such experiments so far, e.g., the DM direct/indirect searches and the direct production of WIMPs at colliders.

Among other things, the detection of DM signature through the observation of neutron star (NS) surface temperature offers a distinct strategy for testing DM models Kouvaris (2008); Bertone and Fairbairn (2008); Kouvaris and Tinyakov (2010); de Lavallaz and Fairbairn (2010). DM particles which are trapped by the gravitational potential of a NS accumulate in the core after they have lost their kinetic energy through the scattering with the NS matter. These DM particles eventually annihilate and heat the NS. At late times, this heating effect balances with the energy loss due to the photon emission from the NS surface, and its surface temperature $T_{s}$ is kept constant at $T_{s}\simeq 2\times 10^{3}$ K. This consequence is in stark contrast to the prediction in the standard NS cooling theory Yakovlev et al. (1999a, 2001); Yakovlev and Pethick (2004), where isolated NSs cool down to $T_{s}<10^{3}$ K for the NS age $t\gtrsim 5\times 10^{6}$ years. This implies that we can in principle test this DM heating scenario by measuring the surface temperature of old NSs. See Refs. Bramante et al. (2017); Baryakhtar et al. (2017); Raj et al. (2018); Chen and Lin (2018); Bell et al. (2018); Camargo et al. (2019); Bell et al. (2019) for recent studies on the DM heating.

In the previous studies of the DM heating, as well as in the standard NS cooling scenario, it is assumed that nucleons and charged leptons in NSs are in beta equilibrium throughout the time evolution of the NSs. For actual rotating NSs, however, this assumption is found to be invalid. A constant decrease in the rotational rate of a NS leads to a continuous reduction in the centrifugal force, and thus the NS keeps being contracted. This then changes the chemical equilibrium condition among nucleons and leptons all the time Reisenegger (1995). On the other hand, the rates of the beta processes are highly suppressed at late times, and thus the NS matter is unable to follow the change in the chemical equilibrium condition. As a result, the imbalance in the chemical potentials of nucleons and leptons increases constantly, which is to be partially dissipated as heat Reisenegger (1995); Haensel (1992); Gourgoulhon and Haensel (1993). This heating effect, called the rotochemical heating, can raise the NS surface temperature up to $T_{s}\simeq 10^{6}$ K for $t\simeq 10^{6-7}$ years Fernandez and Reisenegger (2005); Villain and Haensel (2005); Petrovich and Reisenegger (2010); Pi et al. (2010); González-Jiménez et al. (2015); Yanagi et al. (2019).

In fact, recent observations of old NSs suggest that such a heating mechanism indeed operates. For instance, the surface temperature of the millisecond pulsar (MSP) J0437-4715, whose age is estimated to be $t\simeq(6-7)\times 10^{9}\,\mathrm{yr}$, is measured to be $T_{s}^{\infty}\sim 3\times 10^{5}\,\mathrm{K}$ Kargaltsev et al. (2004); Durant et al. (2012); Gonzalez-Caniulef et al. (2019), where $T_{s}^{\infty}$ represents the red-shifted temperature at the infinite distance. Other examples of NSs whose surface temperature is higher than the prediction in the standard cooling theory include the MSP J2124-3358 Rangelov et al. (2017) and ordinary pulsars J0108-1431 Mignani et al. (2008) and B0950+08 Pavlov et al. (2017). On the other hand, for J2144-3933, there is only an upper bound on the surface temperature: $T_{s}^{\infty}<4.2\times 10^{4}\,\mathrm{K}$ Guillot et al. (2019). Quite interestingly, it is shown in Ref. Yanagi et al. (2019) that all of these observations can be explained by the effect of the non-equilibrium beta processes in a consistent manner.

It should be emphasized that the non-equilibrium beta effect mentioned above is an inevitable consequence of a rotating pulsar, not an ad-hoc assumption to invent a heating mechanism. Given this heating mechanism intrinsic to actual NSs, can we still expect to detect the signature of the DM heating in old NSs? This is the question we address in this letter.

II Minimal cooling

We first review the minimal cooling scenario Page et al. (2004); Gusakov et al. (2004); Page et al. (2009), which is the basis of the following discussions. In this paradigm, it is assumed that a NS is comprised of nucleons, electrons, and muons and they are in beta equilibrium. NSs cool via the emission of neutrinos from the core and photons from the surface. The photon emission dominates the neutrino emission at late times, $t\gtrsim 10^{5}$ years. It is found Yakovlev et al. (1999a, 2001); Yakovlev and Pethick (2004) that the thermal relaxation in a NS is completed in $t\lesssim 10^{2}$ years, and after that the red-shifted internal temperature defined by $T^{\infty}\equiv T(r)e^{\Phi(r)}$ becomes constant in the core, where $T(r)$ denotes the local temperature at the distance $r$ from the center and $e^{2\Phi(r)}=-g_{tt}(r)$ is the time component of the metric at the position. The time evolution of this red-shifted temperature is then determined by

[TABLE]

where $L_{\nu}^{\infty}$ and $L_{\gamma}^{\infty}$ are the red-shifted luminosities of the neutrino and photon emissions, respectively, and $C$ is the total heat capacity of the NS. $L_{H}^{\infty}$ represents the heating power, which vanishes in the minimal cooling.

The photon emission luminosity is given by $L_{\gamma}=4\pi R^{2}\sigma_{B}T_{s}^{4}$, where $\sigma_{B}$ is the Stefan-Boltzmann constant and $R$ is the NS radius. The surface temperature $T_{s}$ differs from the internal temperature due to the shielding effect of the envelope. To relate these temperatures, we use the formula given in Ref. Potekhin et al. (1997) for $T\gtrsim 10^{4}$ K. This relation depends on the amount of light elements in the envelope, which is parametrized by their total mass $\Delta M$. The relation in Ref. Potekhin et al. (1997) cannot be used for $T\lesssim 10^{4}$ K, for which we instead use the relation for the heavy-element envelope given in Ref. Gudmundsson et al. (1983).

There are multiple processes for the neutrino emission. Among them, the modified Urca process Yakovlev and Levenfish (1995); Gusakov (2002) and the pair breaking and formation (PBF) process Flowers et al. (1976); Voskresensky and Senatorov (1987); Senatorov and Voskresensky (1987); Yakovlev et al. (1999b); Kaminker et al. (1999); Leinson and Perez (2006) are the dominant processes in the minimal cooling. The modified Urca process consists of the reactions

[TABLE]

where $N=n$ or $p$ and $\ell=e,\mu$. The rate of this process goes as $\propto T^{8}$ at high temperatures, but is highly suppressed after the onset of nucleon superfluidity. In the NS core, protons form singlet pairings and neutrons form triplet pairings once the temperature becomes lower than the corresponding critical temperatures Page et al. (2013); Haskell and Sedrakian (2018); Sedrakian and Clark (2018). These nucleon parings yield an energy gap in the spectrum of nucleon quasi-particles, which gives a Boltzmann suppression factor to the reaction rates. The pairing gaps also suppress the contribution of nucleons to the heat capacity in Eq. (1) at low temperatures Yakovlev et al. (1999a).

The PBF process, on the other hand, operates only after the onset of nucleon superfluidity. This reaction proceeds along with the breaking of a nucleon pairing due to thermal fluctuation and its successive reformation, during which neutrinos are emitted. Since it requires the formation of nucleon pairings, it can occur only for $T<T_{C}^{(N)}$, where $T_{C}^{(N)}$ is the critical temperature of nucleon superfluidity. For $T\ll T_{C}^{(N)}$, the PBF process also suffers from the Boltzmann suppression due to the energy gap in the nucleon spectrum.

In summary, in the minimal cooling scenario, a NS cools via the emission of photons and neutrinos. For the neutrino emission, we consider the modified Urca and PBF processes, and neglect other subdominant processes for brevity. Notice that the fast cooling processes such as the direct Urca process are not included in the minimal cooling. It is known that the direct Urca process can occur only in heavy NSs Lattimer et al. (1991); for instance, in the case of the Akmal-Pandharipande-Ravenhall (APR) equation of state (EOS) Akmal et al. (1998), which we use in the following analysis, the direct Urca can occur for $M\gtrsim 1.97\,M_{\odot}$, where $M$ and $M_{\odot}$ are the NS mass and the solar mass, respectively. In this work, we follow the minimal cooling paradigm and do not consider the fast cooling processes.

III Rotochemical heating

In the minimal cooling, nucleons and leptons in NSs are assumed to be in beta equilibrium. On the other hand, as discussed in the introduction, the local chemical-equilibrium condition in an actual NS changes continuously because of the constant reduction in the centrifugal force, which is caused by the slowdown in the NS rotation. It turns out that the NS system cannot follow this change at late times since the modified Urca process is strongly suppressed at low temperatures. Therefore, it is necessary to take account of the out-of-beta-equilibrium effect for the discussion of old NSs.

The deviation from the beta equilibrium is quantified by an imbalance in the chemical potentials of nucleons and leptons: $\eta_{\ell}\equiv\mu_{n}-\mu_{p}-\mu_{\ell}$. As discussed in Ref. Reisenegger (1997), the diffusion timescale of the chemical imbalance is short enough so that we can regard the red-shifted imbalance parameters $\eta_{\ell}^{\infty}\equiv\eta_{\ell}e^{\Phi(r)}$ as constant throughout the NS core. The time evolution of $\eta_{\ell}^{\infty}$ is then obtained by solving a couple of differential equations given in Ref. Fernandez and Reisenegger (2005), to which two classes of effects contribute competitively. One is the terms proportional to the difference between the reaction rates of the processes (2), $\Delta\Gamma_{M,N\ell}$, which reduce $|\eta_{\ell}^{\infty}|$, i.e., restore the system back to beta equilibrium. The other terms are proportional to $\Omega\dot{\Omega}$, where $\Omega$ is the angular velocity of the NS, which increase the value of $\eta_{\ell}^{\infty}$ as the NS is slowing down and thus drive the system out of beta equilibrium. Once the latter contribution dominates the former, $\eta_{\ell}^{\infty}$ keeps increasing and the system deviates far from beta equilibrium.

The energy stored in the chemical imbalance is partially released as heat. The heating rate per unit volume is given by $\eta_{\ell}\cdot\Delta\Gamma_{M,N\ell}$, and $L_{H}^{\infty}$ in Eq. (1) is obtained by integrating this quantity (with a red-shift factor) over the NS core. $\Delta\Gamma_{M,N\ell}$ is computed in the literature Fernandez and Reisenegger (2005); Villain and Haensel (2005); Petrovich and Reisenegger (2010); Pi et al. (2010); González-Jiménez et al. (2015); Yanagi et al. (2019), which is again suppressed in the presence of nucleon superfluidity due to the energy gap. A distinct feature of the non-equilibrium beta process is that it is strongly enhanced once $\eta_{\ell}$ exceeds a certain threshold value, $\Delta_{\mathrm{th}}=\mathrm{min}\{3\Delta_{n}+\Delta_{p},\Delta_{n}+3\Delta_{p}\}$ Reisenegger (1997), where $\Delta_{p}$ and $\Delta_{n}$ are the proton singlet and neutron triplet111We use the ${}^{3}P_{2}$ ($m_{J}=0$) gap for neutron, where the gap is given by $\delta_{n}=\Delta_{n}\sqrt{1+3\cos^{2}\theta}$ with $\theta$ the angle between the neutron momentum and the quantization axis. gaps, respectively. At early times, $\eta_{\ell}$ is negligibly small and thus the heating is ineffective. At later times, $\eta_{\ell}^{\infty}$ monotonically increases due to the spin-down of the NS, and once it exceeds the threshold $\Delta_{\mathrm{th}}$, $\Delta\Gamma_{M,N\ell}$ is strongly enhanced and the rotochemical heating becomes effective.

The increase rate of $\eta_{\ell}^{\infty}$ depends on the slowdown factor $\Omega\dot{\Omega}$. We assume that the spin-down rate of NSs is given by the energy loss due to the magnetic dipole radiation and thus follows the power-law deceleration: $\dot{\Omega}(t)=-k\Omega(t)^{3}$ with $k$ a positive constant. By solving this equation, we obtain $\Omega(t)=2\pi/\sqrt{P_{0}^{2}+2P\dot{P}\,t}$, where $P$ and $\dot{P}$ are the rotational period of the NS and its time derivative at the time $t$, respectively, $P_{0}$ denotes the initial period, and $k=P\dot{P}/(4\pi^{2})$. $P\dot{P}$ is related to the surface magnetic field by $B_{s}\simeq 3.2\times 10^{19}\,(P\dot{P}/\mathrm{s})^{1/2}\,\mathrm{G}$ for a NS of radius $R=10\,\mathrm{km}$ and moment of inertia $I=10^{45}\,\mathrm{g}\,\mathrm{cm^{2}}$. These expressions show that for a larger $P_{0}$, $|\Omega\dot{\Omega}|$ becomes smaller, which results in a suppression in the increase rate of $\eta_{\ell}^{\infty}$.

To see this in more detail, in Fig. 1, we show the time evolution of $\eta_{\ell}^{\infty}$ for different values of the initial period $P_{0}$. Throughout this work, we follow the analysis given in Ref. Yanagi et al. (2019) for the calculation of the non-equilibrium beta processes. The horizontal line indicates the minimal value of $\Delta_{\mathrm{th}}$ in the NS core multiplied by a red-shift factor at the position, which we denote by $\Delta_{\mathrm{th}}^{\infty}$. For nucleon pairing gaps, we use the CCDK model Chen et al. (1993) for the proton singlet gap and the “a2” model Page et al. (2013) for the neutron triplet gap, which is used in Ref. Page et al. (2011).222This choice of the pairing gaps is found to be compatible with the observed rapid cooling of the NS in Cassiopeia A Page et al. (2011); Shternin et al. (2011); Ho et al. (2015); Hamaguchi et al. (2018). This figure shows that as $P_{0}$ gets larger the increase rate of $\eta_{\ell}$ gets smaller. It is also found that for $P_{0}>7$ ms, $\eta_{\ell}^{\infty}$ never exceeds the rotochemical threshold; in this case, we expect that the rotochemical heating is fairly suppressed, as we will confirm in the following analysis.

IV Dark matter heating

The accretion and annihilation of WIMP DM in a NS can be another heating source Kouvaris (2008); Bertone and Fairbairn (2008); Kouvaris and Tinyakov (2010); de Lavallaz and Fairbairn (2010). DM falling toward a NS hits the NS if its impact parameter is smaller than $b_{\mathrm{max}}=R(v_{\mathrm{esc}}/v_{\mathrm{DM}})e^{-\Phi(R)}$ Goldman and Nussinov (1989), where $G$ is the gravitational constant, $v_{\mathrm{DM}}$ is the DM velocity distant from the NS, and $v_{\mathrm{esc}}=(2GM/R)^{1/2}$ is the escape velocity. The rate of DM particles hitting the NS is then obtained as $\dot{N}\simeq\pi b_{\mathrm{max}}^{2}v_{\mathrm{DM}}(\rho_{\mathrm{DM}}/m_{\mathrm{DM}})$, where $\rho_{\mathrm{DM}}$ and $m_{\mathrm{DM}}$ are the local energy density and mass of DM, respectively. We use $\rho_{\mathrm{DM}}=0.42~{}\mathrm{GeV}\cdot\mathrm{cm}^{-3}$ and $v_{\mathrm{DM}}=230~{}\mathrm{km}\cdot\mathrm{s}^{-1}$ Pato et al. (2015) in what follows. A more accurate expression of $\dot{N}$ is given in Ref. Kouvaris (2008), which we use in the following analysis.

It is found that an electroweak/TeV-scale WIMP DM is captured in NSs after one scattering if the DM-nucleon scattering cross section is larger than $\sigma_{\mathrm{crit}}\simeq R^{2}m_{N}/M$, with $m_{N}$ the nucleon mass Kouvaris (2008). After the DM is trapped, the rest of its kinetic energy is soon lost by successive scatterings with the NS matter. DM particles then accumulate in the NS core and eventually annihilate. As shown in Ref. Kouvaris and Tinyakov (2010), for a typical WIMP, its annihilation and capture rates become in equilibrium in old NSs. As a result, the contribution of the DM heating to the luminosity $L_{H}^{\infty}$ in Eq. (1) is computed as

[TABLE]

where $\gamma=1/\sqrt{1-v_{\mathrm{esc}}^{2}}$ and $\chi$ is the fraction of the annihilation energy transferred to heat Kouvaris and Tinyakov (2010). In what follows, we take $\chi=1$ unless otherwise noted. The first term in Eq. (3) represents the heat from the DM annihilation, while the second term corresponds to the deposit of the kinetic energy of the incoming WIMP DM Baryakhtar et al. (2017).

If we neglect the rotochemical heating, the DM heating balances with the cooling due to the photon emission at late times, i.e., $L_{H}^{\infty}|_{\mathrm{DM}}\simeq L_{\gamma}^{\infty}$. This condition fixes the NS surface temperature to be a few thousand K, which has been regarded as a smoking-gun signature of the DM heating Kouvaris (2008); Bertone and Fairbairn (2008); Kouvaris and Tinyakov (2010); de Lavallaz and Fairbairn (2010). Below, we study if this signature can still be seen even in the presence of the rotochemical heating.

V Results

Now we examine the time evolution of the NS temperature by including all of the effects discussed above. We first consider a NS which models a typical ordinary pulsar, where we fix $M=1.4M_{\odot}$, $P=1$ s, $\dot{P}=1\times 10^{-15}$, and $\Delta M/M=1\times 10^{-15}$. The initial values of $T^{\infty}$ and $\eta_{\ell}^{\infty}$ are taken to be $T^{\infty}=10^{10}$ K and $\eta_{\ell}^{\infty}=0$, respectively. We find that the following results have little dependence on the choice of these parameters.

In Fig. 2, we show the time evolution of $T_{s}^{\infty}$ for different values of $P_{0}$ in the black solid lines. For $P_{0}=1$, 5, and 6 ms, the time evolutions with and without DM heating are indistinguishable and the lines overlap. For $P_{0}>7$ ms, the solid (dashed) line represents the case with (without) DM heating. We again use the CCDK Chen et al. (1993) and “a2” Page et al. (2013) models for the proton and neutron gaps, respectively. We also show the observed temperatures of old ordinary pulsars with blue crosses, where the lines indicate the uncertainties; we take this data from Ref. Yanagi et al. (2019). This figure shows that for $P_{0}=1$ ms the surface temperature remains as high as ${\cal O}(10^{5})$ K for $t\gtrsim 10^{6}$ years since the rotochemical heating is quite effective. The temperature curve in this case is consistent with most of the observed temperatures, but the DM heating effect is completely hidden by the rotochemical heating effect. For a larger $P_{0}$, $T_{s}^{\infty}$ at late times gets lower, and for $P_{0}>7$ ms, it becomes independent of the initial period. In this case, the rotochemical heating is ineffective since $\eta_{\ell}$ does not exceed the rotochemical threshold, as we have seen in Fig. 1. Thus, the late-time temperature is determined by the DM heating, with $T_{s}^{\infty}\simeq 2\times 10^{3}$ K. Notice that a NS cools down to this temperature before it reaches the conventional death-line Ruderman and Sutherland (1975); Bhattacharya et al. (1992),333We however note that the theoretical estimation of the death-line suffers from huge uncertainty, and thus one should not take this bound too seriously. Indeed, as can be seen from Fig. 2, J2144-3933, e.g., is located beyond the conventional death-line, though its pulsation is detected Young et al. (1999). For more discussions on the death-line, see Refs. Chen and Ruderman (1993); Zhang et al. (2000); Zhang (2003); Zhou et al. (2017). $B_{s}/P^{2}=0.17\times 10^{12}~{}\mathrm{G}\cdot\mathrm{s}^{-2}$, shown by the red dashed line in Fig. 2. Therefore, it is possible to detect the DM heating effect via the temperature observation of ordinary pulsars if their initial period is sufficiently large.

We note, however, that the lower limit on $P_{0}$ for the condition that the DM heating effect is detectable highly depends on the nucleon pairing gaps. To see this, in Fig. 3, we show the time evolution of $T_{s}^{\infty}$ for different choices of nucleon pairing gaps, with $P_{0}=7$ ms. The blue lines correspond to the “a2” model Page et al. (2013), while the green and red lines are for the “a” and “b” models in Ref. Page et al. (2004) for the neutron gap. The solid and dashed lines show the cases for the AO Amundsen and Ostgaard (1985) and CCDK Chen et al. (1993) models of the proton gap, respectively. As we see, for the AO model, whose gap is smaller than that of the CCDK model, the late-time temperature is predicted to be higher than $T_{s}^{\infty}\simeq 2\times 10^{3}$ K; in this case, $\Delta_{\mathrm{th}}^{\infty}$ is rather small, and thus $\eta_{\ell}$ can overcome the rotochemical threshold at late times even for $P_{0}=7$ ms, making the rotochemical heating operative. On the other hand, for the CCDK proton and ”a2” or ”b” neutron pairings, the rotochemical heating is ineffective and thus we can see the DM heating effect at late times. The results shown in Fig. 3 demonstrate that it is crucial to take account of both proton and neutron pairings in order to evaluate the effect of non-equilibrium beta processes appropriately Yanagi et al. (2019).

We also found that the rotochemical heating does not operate for any choice of pairing gaps if the initial period is as large as 100 ms444Since the pairing gap has density dependence, the critical value of $P_{0}$, above which the rotochemical heating is ineffective, depends also on other star parameters such as NS mass. We find the critical $P_{0}$ is at most $\mathop{\mathcal{O}}(10)\,\mathrm{ms}$.—in this case, the late-time surface temperature is always determined by the DM heating. It is intriguing that recent studies suggest that the initial period distribution extends to well beyond 100 ms; it is independently estimated from the kinematic age of several tens of observed NSs Popov and Turolla (2012); Noutsos et al. (2013); Igoshev and Popov (2013), the population synthesis of pulsars Faucher-Giguere and Kaspi (2006); Popov et al. (2010); Gullón et al. (2014, 2015), or the supernova simulation for proto-NSs Müller et al. (2019). Hence, we expect that there are quite a few ordinary pulsars that can be a probe of the DM heating in future observations.

Finally, let us consider MSPs. In this case, $|\Omega\dot{\Omega}|$ is much larger than that for ordinary pulsars, and thus $\eta_{\ell}$ can always exceed the rotochemical threshold at late times. Therefore, the rotochemical heating is highly effective for MSPs Yanagi et al. (2019). Although this feature is advantageous for explaining the old warm MSPs such as J0437-4715 Kargaltsev et al. (2004); Durant et al. (2012); Gonzalez-Caniulef et al. (2019) and J2124-3358 Rangelov et al. (2017), this makes MSPs inappropriate for testing the DM heating scenario.

VI Conclusion and discussion

We have studied the time evolution of NS surface temperature, taking account of both the rotochemical and DM heating effects. We have found that for ordinary pulsars the DM heating effect can still be observed even with the rotochemical heating if the initial period of NSs is relatively large, since in this case the chemical imbalance does not overcome the threshold $\Delta_{\mathrm{th}}^{\infty}$ and thus the rotochemical heating is ineffective. The rotochemical heating operates if the initial period is as small as $\mathop{\mathcal{O}}(1)\,\mathrm{ms}$. Thus for MSPs, the DM heating is always concealed by the rotochemical heating.

The surface temperature at late times depends not only on the initial period but also on the choice of the nucleon pairing gaps, as shown in Fig. 3. Depending on these unknown quantities, the rotochemical heating effect may mimic the DM heating effect in old ordinary pulsars. For instance, the late-time temperature for the proton AO and neutron ”b” gaps in Fig. 3 is kept at a few thousand K due to the rotochemical heating. To distinguish these two heating effects, therefore, it is necessary to improve our knowledge on nucleon pairing gaps as well as to evaluate the initial period of pulsars accurately. We note in passing that it is possible to estimate the initial period of a pulsar if, for instance, the pulsar is associated with a supernova and its age is computed from the motion of the supernova remnant, as is performed in Ref. Popov and Turolla (2012),

In any case, in the presence of both the rotochemical and DM heating effects, the late-time temperature is bounded below, i.e., $T_{s}^{\infty}\gtrsim 2\times 10^{3}$ K, which is determined by the DM heating and thus independent of the initial period and pairing gaps. As a consequence, an observation of a NS with a surface temperature that is sufficiently below this lower bound readily excludes the DM heating caused by typical WIMPs, and thus can severely constrain such DM models. To see this significance, as an example, we consider the case where DM is comprised of the neutral component of an electroweak multiplet Cirelli et al. (2007); Cirelli and Strumia (2009); Farina et al. (2010, 2013); Nagata and Shirai (2015). This class of DM candidates includes the pure wino/higgsino in the supersymmetric models. Although the elastic scattering cross section of such a DM candidate with a nucleon is generically small Hisano et al. (2011, 2015), it can still be trapped by NSs through the inelastic scattering with the NS matter. For this class of DM candidates, the charged components of the DM multiplet are degenerate with the DM component in mass, with mass differences of $\mathcal{O}(100)$ MeV, which are smaller than the energy transfer of the DM scattering in NSs, $\Delta E\lesssim 1$ GeV Baryakhtar et al. (2017). As a result, the inelastic scattering accompanied with the charged component can occur in NSs. Since it is induced by the tree-level exchange of the $W$ boson, its cross section is much larger than the critical value $\sigma_{\mathrm{crit}}\simeq R^{2}m_{N}/M\sim 10^{-45}~{}\mathrm{cm}^{2}$ Baryakhtar et al. (2017). Hence, we can directly use the above results for this class of DM candidates,555There is a small difference since the annihilation of these DM candidates can generate neutrinos in the final state and thus the parameter $\chi$ in Eq. (3) is smaller than unity. This difference only results in an $\mathcal{O}(1)$% change in the late-time temperature, which is in effect negligible in the present discussion. and in particular we conclude that an observation of an old NS with $T_{s}^{\infty}\lesssim 10^{3}$ K can exclude all of these DM candidates. Notice that this constraint is independent of the DM mass as long as it is $\lesssim 1$ PeV, which is the case for these DM candidates, whose masses are predicted to be 1–10 TeV Farina et al. (2013).

Finally, we note that there are other heating mechanisms proposed in the literature Gonzalez and Reisenegger (2010), such as the vortex creep heating Alpar et al. (1984); Shibazaki and Lamb (1989); van Riper et al. (1991); Umeda et al. (1993); Van Riper et al. (1995); Larson and Link (1999) and rotationally-induced deep crustal heating Gusakov et al. (2015). These heating mechanisms may also compete with the rotochemical and DM heating effects, and therefore the consequence drawn in this letter may be altered if they are also included. We will study the implications of these heating mechanisms for the DM heating on another occasion Hamaguchi et al. .

Acknowledgements.

Acknowledgments

We thank Teruaki Enoto, Kenji Fukushima, Kazuhiro Nakazawa, Hideyuki Umeda, and Satoshi Yamamoto for valuable discussions and suggestions. We are also grateful to Jiaming Zheng for useful comments on the manuscript. This work is supported in part by the Grant-in-Aid for Scientific Research A (No.16H02189 [KH]), Young Scientists B (No.17K14270 [NN]), Innovative Areas (No.19H04612 [KH], No.18H05542 [NN]). The work of KY was supported by JSPS KAKENHI Grant Number JP18J10202.