Neutron - mirror neutron mixing and neutron stars

The oscillation of neutrons $n$ into mirror neutrons $n'$, their mass degenerate partners from dark mirror sector, can have interesting implications for neutron stars: an ordinary neutron star could gradually transform into a mixed star consisting in part of mirror dark matter. Mixed stars can be detectable as twin partners of ordinary neutron stars: namely, there can exist compact stars with the same masses but having different radii. For a given equation of state (identical between the ordinary and mirror components), the mass and radius of a mixed star depend on the proportion between the ordinary and mirror components in its interior which in turn depends on its age. If $50 \% - 50\%$ proportion between two fractions can be reached asymptotically in time, then the maximum mass of such"maximally mixed stars"should be $\sqrt2$ times smaller than that of ordinary neutron star while the stars exceeding a critical mass value $M^{\rm max}_{NS}/\sqrt2$ should collapse in black holes after certain time. We evaluate the evolution time and discuss the implications of $n-n'$ transition for the pulsar observations as well as for the gravitational waves from the neutron star mergers and associated electromagnetic signals.

The oscillation of neutrons n into mirror neutrons n , their mass degenerate partners from dark mirror sector, can have interesting implications for neutron stars: an ordinary neutron star could gradually transform into a mixed star consisting in part of mirror dark matter. Mixed stars can be detectable as twin partners of ordinary neutron stars: namely, there can exist compact stars with the same masses but having different radii. For a given equation of state (identical between the ordinary and mirror components), the mass and radius of a mixed star depend on the proportion between the ordinary and mirror components in its interior which in turn depends on its age. If 50% − 50% proportion between two fractions can be reached asymptotically in time, then the maximum mass of such "maximally mixed stars" should be √ 2 times smaller than that of ordinary neutron star while the stars exceeding a critical mass value M max N S / √ 2 should collapse in black holes after certain time. We evaluate the evolution time and discuss the implications of n − n transition for the pulsar observations as well as for the gravitational waves from the neutron star mergers and associated electromagnetic signals.

I. INTRODUCTION
The idea that there may exist a hidden particle sector consisting of mirror duplicates of the observed standard particles was introduced long time ago for restoring parity as a fundamental symmetry [1]. In this context, all known particles: electron e, proton p, neutron n, neutrinos ν etc. must have the mirror partners e , p , n , ν etc. which are supposed to be sterile to our Standard Model (SM) interactions SU (3) × SU (2) × U (1), but have their own SU (3) × SU (2) × U (1) gauge interactions (see e.g. [2][3][4] for reviews and [5] for a historical overview).
More generically, one can consider theories based on the direct product G × G of two identical gauge factors (SM or some its extension), with Lagrangian L tot = L + L + L mix (1) where L describes ordinary (O) particles, L describes their mirror (M) partners and L mix stands for possible cross-interactions between the O and M particles. The identical forms of the Lagrangians L and L is ensured by the Z 2 symmetry under the exchange G ↔ G of all O particles (fermions, Higgs and gauge fields) with their M twins ('primed' fermions, Higgs and gauge fields). In view of chiral character of the SM, this discrete symmetry can be imposed with or without chirality change between O and M fermions. Mirror parity corresponds to the former possibility, however this difference will not be important for our further considerations. M matter, invisible in terms of our photons, is gravitationally coupled to O matter and can contribute to cosmological dark matter, or perhaps could even represent its entire amount. It can form mirror nuclei and atoms with an abundance related to the mirror baryon asymmetry. The collisional and dissipative properties of M baryons can have specific implications for cosmology [6][7][8][9][10] as well as for the dark matter direct search [11].
If Z 2 symmetry is unbroken, then O particles and their M partners are degenerate in mass, and the interaction terms in L and L are exactly identical. Also the case of spontaneously broken Z 2 symmetry has been considered in the literature [7] when the weak scales in the two sectors become different, and thus mass splittings emerge between M particles and their O partners.
Lagrangian L mix in (1) may contain cross-terms that induce oscillation phenomena between O and M sectors. In fact, any neutral O particle, elementary (e.g. neutrinos) or composite (e.g. neutrons) can have mixing with its M twin and oscillate into the latter. In particular, "active-sterile" neutrino mixing between ordinary ν e,µ,τ and mirror ν e,µ,τ can be induced via effective operators 1 M φφ + h.c. with a large cutoff scale M , where and φ are lepton and Higgs doublets of O sector and and φ are their mirror partners [12,13]. These operators, violating both B − L and B − L symmetries, also suggest a co-leptogenesis mechanism which creates comparable baryon asymmetries in both O and M worlds, with Ω B ≥ Ω B [14][15][16].
In this paper we shall concentrate on the mixing between the neutron n and its mirror partner n [17,18]: In fact, the mixed Lagrangian L mix may include effective operators involving color-singlet combinations of ordinary u, d and mirror u , d quarks: (the gauge and Lorentz indices are omitted). These operators violate both O and M baryon numbers by one unit, ∆B = 1 and ∆B = −1, but the overall baryon number B = B + B is conserved. In UV complete theories they can be induced via a see-saw like mechanism involving new heavy particles, colored scalars and neutral fermions, with masses ∼ M [17,19]. Hence, for M at few TeV, the underlying theories can be testable at the LHC and future accelerators [19,20]. Operators (3) induce n − n mass mixing (2) with ε = C 2 Λ 6 QCD M 5 = C 2 10 TeV M 5 × 10 −15 eV (4) where C = O(1) is the operator dependent numerical factor in the determination of the matrix element 0|udd|n . The phenomenon of n − n oscillation is analogous to that of neutron-antineutron (n −n) oscillation [21] (for a review, see [22]), and in fact both phenomena can be related to the same new physics [19,23]. However, n −n oscillation is strongly restricted by experiment. Namely, the direct experimental limit on n−n oscillation is ε nn < 7.7 × 10 −24 eV while the nuclear stability bounds are yet stronger yielding ε nn < 2.5 × 10 −24 eV [22]. As for n − n oscillation, it is kinematically forbidden for neutrons bound in nuclei, simply by energy conservation [17], and so nuclear stability gives no limit on n − n mixing.
For free neutrons n − n oscillation can be effective, and it may even be much faster than the neutron decay [17,18]. Namely, n − n mixing mass can be as large as ε ∼ 10 −15 eV, corresponding to the characteristic oscillation time τ nn = ε −1 as small as 1 second or even smaller. This possibility is not excluded by existing astrophysical and cosmological limits [17], predicting observable effects for the ultra-high energy cosmic rays [24], for neutrons from solar flares [25] and for primordial nucleosynthesis [26]. Its search via the neutron disappearance (n → n ) and regeneration (n → n → n) experiments can be perfectly accessible at existing neutron facilities [17,27,28].
The reason why the disappearance of free neutrons so far skipped the experimental detection might be that in normal experimental conditions the n − n oscillation probability P nn is suppressed by environmental effects as e.g. mirror magnetic fields at the Earth [18]. Several dedicated experiments were performed searching n − n oscillations [29][30][31][32][33][34][35] and they still do not exclude small oscillation times if these environmental effects are strong enough. Moreover, some of these experiments show anomalous deviations from null hypothesis indicating to τ nn ∼ 10 s or so [34,36]. New experiments for testing these effects are underway [37,38].
Larger values of ε are also allowed if n and n are not exactly mass-degenerate. Moreover, n − n oscillation with ε ∼ 10 −10 eV or so can solve the neutron lifetime problem, the 4σ discrepancy between the neutron lifetimes measured via the bottle and beam experiments, provided that n and n have a mass splitting m n − m n ∼ 100 neV [39]. Such a small splitting can be naturally realized in models in which Z 2 parity is spontaneously broken [7] but with a rather small difference between the O and M Higgs VEVs φ and φ [40].
Although n − n transition is forbidden for neutrons bound in nuclei by nuclear forces, it is allowed in neutron stars (NS) in which neutrons are bound gravitationally. The transformation of nuclear matter into mirror matter should decrease the degeneracy pressure, thereby softening the equation of state (EoS) of the system. The gravitational binding energy increases and thus the process is energetically favorable. By n − n transitions, a NS born after supernova explosions gradually evolves in a mixed star (MS) partially consisting of mirror matter, with decreasing gravitational mass and radius. The softening of the EoS is particularly effective for the NS born with large masses. In this case, the conversion of O matter in M matter may produce a gravitationally unstable MS that collapses to a black hole (BH). On the other hand, a NS born with a mass below a threshold value would asymptotically evolve in maximally mixed star (MMS), with equal amounts of the O and M components.
In the present paper we shall concentrate on the implications of n − n conversion for the NS. As was already noted in [17], for ε 10 −15 eV (i.e. τ nn ∼ 1 s or so) the transformation time is several orders of magnitude larger than the age of the Universe. Nevertheless, we show that the NS transformation into the MS with small mirror cores can have astrophysical signatures for the pulsar dynamics and for the gravitational mergers. On the other hand, as discussed above, larger values of ε are not excluded and the conversion process can be more rapid, so that the older NS could be already transformed in the MMS. The respective implications for NSs were briefly discussed in [41], in more details in [43], and were recently addressed also in [44,45]. Here we analyse conditions for the MS stability, derive the mass-radii scaling relations and discuss the possible observational effects.
Our work is organized as follows. In Sect. II we discuss effects of n − n mixing on the evolution as well as on the gravitational stability of compact stars, and derive the mass-radius scaling relations between the NS and MMS. In Sect. III we discuss n − n oscillations in dense nuclear matter and estimate the transformation time of neutron stars into mixed stars. In Sect. IV we discuss possible effects for the pulsar timing observations, and in Sect. V for the neutron star coalescences and associated signals. We draw our conclusions in Sec. VI.

II. NEUTRON STAR EVOLUTION IN MIXED STARS
Neutron stars are presumably born after the supernova explosions of massive stars. They are believed to have an onion-like structure that schematically consists of a thin rigid crust at the surface and a liquid nuclear matter in the core whose dominant component are neutrons, plus some fractions of protons and electrons and perhaps also heavier baryons and muons (for reviews, see e.g. [46,47]).
The supernova core-collapse should produce a NS consisting of ordinary nuclear matter. However, if n → n transitions are allowed they start to produce M neutrons, and the original NS will be gradually transformed into a MS with increasing fraction of mirror nuclear matter in its interior. We consider that this process conserves the total number of baryons, and assume that it is rather slow, with the effective n − n conversion rate much less than the typical cooling rate. Under these circumstances the transformation process is adiabatic and can be described by the Boltzmann equations where N O (t) and N M (t) are respectively the total numbers of O and M baryons in the star at the time t and Γ(t) is the n → n conversion rate (to be estimated in next section), whereas Γ Γ is the rate of the inverse process n → n. Starting at t = 0 from a newborn NS with N O = N and N M = 0, then N M (t) will increase and N O (t) will decrease in time, but with N O (t)+N M (t) = N at any stage since the overall number of O and M baryons in the MS is conserved. 1 Thus, neglecting the inverse reaction rate Γ , Eqs. (5) reduce to the single equation where Asymptotically in time the star can evolve (provided that it remains stable during the evolution) to the final equilibrium configuration with X = 1/2, corresponding to the MMS with equal amounts of the O and M components in its interior.

A. Structure of mixed neutron stars
As far as the evolution is adiabatic, we can use a static "two fluid" description in which the total energy density and pressure can be decomposed as the sum of ordinary and mirror components, i.e. ρ = ρ O + ρ M and p = p O + p M . Thus, the total energy-momentum tensor T µ ν = diag(ρ, −p, −p, −p) can be split as T µν = T O µν + T M µν , where T O µν and T M µν respectively are the energy momentum tensors of O and M matter. Assuming spherical symmetry, at any moment of time the MS can be described as a static configuration of two concentric O and M spheres (we neglect the star rotation which is known to have a small effect on the mass/radius relation) having the radii R O and R M , respectively corresponding to the positions where p O (r) and p M (r) vanish. Hence, we take the metric tensor in standard form with spherical symmetry, g µν = diag(−g tt , g rr , r 2 , r 2 sin 2 θ), where 2 g tt (r) = exp[−2φ(r)], g rr (r) = 1 1 − 2m(r)/r 1 This is because n − n oscillations conserve the combined baryon number B = B+B . In principle, the neutron could have mixings with both M neutron n and M antineutron n . In this case B would be violated [48], but here we do not discuss this possibility. 2 In this section we use geometrized units, c = 1 and G = 1.
where m(r) is the total gravitational mass within the radius r, and φ(r) is the gravitational potential. In hydrostatic equilibrium the density and pressure profiles in the star are determined by the Tolman-Openheimer-Volkoff (TOV) equations [49,50]: The first differential equation above is linear, and we can split it between two components, m(r) = m O (r)+m M (r): which give m α (r) = 4π r 0 ρ α (r)r 2 dr (α = O, M). Therefore, the total gravitational mass of the MS is the sum of gravitational masses of the two components, M MS = M O + M M . Since both components are in hydrostatic equilibrium,ρ α ,ṗ α = 0, the continuity equation for the energy-momentum tensor ∇ µ T µ(α) ν = 0 for each of them separately gives where the last equality follows from the first two. As for Eq. (10), it in fact couples the pressures and energy densities of the two fluids. Using Eqs. (11) and (12), it gives the coupled differential equations for the O and M components: The above system of differential equations can be solved once the EoS are given, and the appropriate boundary conditions are chosen. In our case, both components α = O,M should be the same EoS p α = F (ρ α ) by mirror symmetry. Then one can find their density profiles ρ α (r) by fixing the respective central densities, ρ O (0) = ρ cO and ρ M (0) = ρ cM . For initial configuration of NS composed exclusively of O baryons, the system of equations (11) and (13) reduces to the standard one-component TOV equations: Solving these equations with a given central density ρ O (0) = ρ c , one can find the NS density profile: where the function f (r) is normalized as f (0) = 1 and its shape depends on the EoS. The NS radius R NS ≡ R O corresponds to the distance at which f (r) vanishes, while the gravitational mass is In fact, the integration can be extended to infinity since f (r) = 0 for r > R NS . An important feature is that, for any EoS, there is no gravitationally stable solution if ρ c exceeds a certain critical value ρ max c , which determines the maximum mass of the neutron stars, M max NS = M NS (ρ max c ), also called the last stable configuration for a given EoS. Discovery of pulsars with gravitational masses M ≈ 2M [51][52][53] challenged several EoSs excluding the too soft ones. Moreover, observation of PSR J1748-2021B with mass estimated as M = (2.74 ± 0.21)M [54] indicates that the maximum mass may be considerably larger than 2M .
In this respect, in the following we consider two possible EoSs as examples of those which allow large enough maximum masses. One is the realistic SLy (Skyrme-Lyon) EoS [55], which gives both the energy density and pressure for discrete values of the baryon number density n O . The last stable configuration associated to this EoS has a mass M max As a second example we choose an EoS consisting of two joined polytropes: with γ 1 = 3 for the inner part of the NS and γ 2 = 4/3 for its outer part, with transition at half the nuclear saturation density ρ tr = 1.35 × 10 14 g/cm 3 , for details see Ref. [46]. This EoS allows the larger maximum mass M max NS = 2.57 M . For the typical NS with M 1.4 M both of these EoS imply a radius of about 12 km.
Although we shall restrict to these two EoSs, there is a large number of possible EoS (for reviews see e.g. Refs. [56,57].) For instance, the phenomenological EoS suggested in Ref. [58] is stiffer, predicting the last stable configuration with mass as large as 2.7 M , and the NS radii can be as large as 15 km. From a very fundamental point of view, without appealing to a particular EoS but only requiring the micro-stability and causality conditions 0 < dp/dρ < 1, one can put an absolute upper limit on the NS mass, M max NS 3.2 M , known as Rhoades-Ruffini bound [59].
The baryon number density n = n O is directly related to the gravitational density. The relation between the two densities, n = n(ρ), depends on the chosen EoS. (For relevant densities the ratio n/ρ is roughly constant but not exactly.) Therefore, the baryon density profile can be presented as n(r) = n[ρ(r)] and, integrating over the NS volume, we obtain its total baryon number: For fixed ρ c , the shapes of f (r) and g rr (r) depend on the EoS. E.g. for the Sly EoS ρ is explicitly given as a function of n in Ref. [55]. The total baryon number of the NS depends on its mass nearly linearly: where N = M /m n = 1.188 × 10 57 is the baryon amount in the sun. E.g. for the Sly EoS we have κ ≈ 1. It is useful to characterize the different MS configurations by the ratio of ordinary to mirror central densities, χ = ρ cM /ρ cO . As far as the evolution is adiabatic, and the total baryon number is conserved, for a star with a total baryon number N = N O + N M the value of χ = χ(X) is determined by the mirror baryon fraction X = N M /N at a given stage of its evolution. In our picture, a newborn NS has χ = 0, but χ increases with time when it evolves as a MS, and asymptotically approaches χ = 1 for a MMS configuration (X = 1/2) if the star remains stable during the evolution.
In An important aspect is that very massive neutron stars cannot evolve in MMS, since they collapse to a BH when the ratio of ordinary to mirror central densities reaches a critical value χ max . This is illustrated in Fig. 2, where we show the mass-radius relations for a mixed star with different O and M fractions (from X = 0, solid black lines, to X = 1/2, dashed green lines) using the two considered EoSs. The nearly horizontal black dashed curves show the evolution track with increasing values of χ at fixed total baryon number N = N O + N M . Hence, they trace the time evolution from the original NS (χ = 0, solid black lines), passing the intermediate MS stage and eventually reaching the MMS stage (χ = 1, dashed green lines). Along the evolution tracks both the gravitational masses and the radii decrease making the stellar object more compact. At a certain stage of their evolution, very massive stars may become gravitationally unstable forcing the collapse to BHs for some value of χ max < 1; the actual value of χ max depends on the initial NS mass and on the EoS used. In particular, for the Sly case only NSs with initial masses M NS 1.55 M or so can survive asymptotically in time and approach the MMS configuration with M MMS < 1.45 M ; more massive stars are doomed to collapse to BHs. For instance, a NS with a mass of about 1.8 M collapses to BH when χ 0.5.

B. Scaling relations between the neutron stars and maximally mixed stars
The masses and radii of mixed stars depend on central densities of the two components, ρ cO and ρ cM . In fact, any point in the mass-radius diagram is determined by the functions M MS = M (ρ cO , ρ cM ) and R MS = R(ρ cO , ρ cM ). The dependence on two parameters causes the mass/radius degeneracy in Fig. 2, meaning that compact stars with the same mass can have different radii, depending on the composition. The masses and radii along the dashed lines in Figs. 2, describing the MS evolution at fixed total baryon number N = N O + N M , are not related by a simple rule. However, one can find simple scaling relations between the two extreme configurations, meaning that the massradius trajectories corresponding to the MMS configurations (dashed green curves in Figs. 2) can be mapped onto those of the NS (black solid curves) by factor of √ 2 rescaling of masses and radii. In other words, the radii and masses (as well as total baryon numbers) of stars with ρ cO = ρ cM = ρ c are related to those of stars with ρ cO = ρ c , ρ cM = 0, as The MS configurations can be equivalently parametrized in terms of total central density ρ c = ρ cO + ρ cM and the ratio χ = ρ cM /ρ cO . Since the MMS and NS configurations correspond to χ = 1 and χ = 0 respectively, we can consider their radii as functions of respective total central densities, denoting R(ρ c , ρ c ) ≡ R MMS (2ρ c ) and R(ρ c , 0) ≡ R NS (ρ c ), and similarly for the masses and baryon numbers. Thus, in configurations related by √ 2 scaling (20) the central energy density and pressure of the MMS are twice the corresponding values of the NS.
The scaling relations can be derived as follows. Since MMS contains two fluids with the identical EoS and with equal central densities, we can set (11) and (13). Thus, both components satisfy identical differential equations: where α = O or M. By substituting these equations can be rewritten as which have exactly the same form as the one fluid TOV equations (14) for the ordinary NS. Therefore, Eqs. (14) and (23) should have identical solutions under the same boundary conditions. Namely, taking the central densi- with exactly the same shape function as f (r) in Eq. (15) but with the argument rescaled asr = √ 2r. The MMS radius corresponds to the distance at which f ( √ 2r) vanishes. Thus, we obtain that the radii of the MMS with central density ρ cO + ρ cM = 2ρ c and of the NS with the central density ρ cO = ρ c are related as On the other hand, by integrating the first equation (23) and taking into account the redefinitions (24) we get m α (r) = √ 8m α (r) = 4π r 0 ρ α (r)r 2 dr. Thus, for the MMS mass we have where the last equality follows from (16). In addition, Eqs. (25) and (26) show that the two configurations must have the same compactness: Analogously, for the moments of inetia one can obtain: The following remark is in order for avoiding the confusion. The NS with ρ cO = ρ c and the MMS with ρ cO = ρ cM = ρ c are not on the same evolutionary track with the conserved baryon number (dash lines in Figs. 2). In fact, the baryon numbers of these configurations also obey to the scaling law: which can be obtained in the analogous manner, by comparing the NS baryon number (18) with the total baryon number N O + N M = 2N O of the MMS, and taking into account that upon redefinitions (22) we have 2m α /r =m α /r in g rr component of the metric tensor (7). By this reason also the volumes of two configura-   Fig. 2). Thus, the binding energy increases during the evolution by 0.1 M . Stars with larger initial masses at some stage of the evolution will collapse to a BH, before reaching the MMS limit. E.g. star with initial mass M NS = 1.8 M (N NS ≈ 2.5 × 10 57 ) collapses when the ratio of central densities reaches The above scaling relations can be straightforwardly extended to scenarios with more than one mirror sector: if ordinary neutrons have mixings with k "mirror" neutrons n 1 , n 2 , . . . n k , and if all mirror matters have the same EoS as ordinary matter (E.g. such a scenario with k ∼ 10 32 mirror sectors was discussed in Ref. [60]), then the NS and MMS configurations can be related by a √ k-scaling rule analogous to Eqs. (25), (26) and (29). In particular, for the maximum masses we would have

III. THE TIME EVOLUTION OF MIXED STARS
A. Neutron-mirror neutron oscillation in nuclear medium The interior of neutron stars can be considered as degenerate nuclear matter dominantly consisting of the neutrons. 3 We assume n − n conversion time to be much larger than the cooling time, so that at any stage the temperature is much less than the chemical potential, T µ. (in this section we use natural units c = 1 and = 1.) Hence, one can neglect T and simply take µ as Fermi energy which in the approximation of ideal non-relativistic Fermi gas is E F ξ 2/3 × 60 MeV, where ξ = n O /n S is the ordinary baryon number density in units of nuclear density n S = 0.16 fm −3 = (107 MeV) 3 . However, this approximation is not sufficient for describing the observed NS (such "ideal" EoS would give the maximum NS mass , and one has to take into account the (repulsive) nuclear interactions which stiffen the EoS of the medium. The coherent scatterings modify the dispersion relation of the neutron which can be accounted by the in-medium optical potential where a is the neutron scattering length and a 3 = (a/3 fm). At supra-nuclear densities (ξ > 1) both V n and E F are ∼ 100 MeV, but V n > E F . The oscillation n − n in nuclear medium is described by the Schrödinger equation with effective Hamiltonian where E n = (m 2 n + p 2 ) 1/2 + V n , with p being the neutron momentum. 4 Analogously, we take E n with V n → V n , where V n is the mirror neutron potential obtained from Eq. (31) by substituting n O by the mirror baryon density n M . Hence, we have V n = βV n where β = n M /n O . The off-diagonal term ε comes from n − n mass mixing (2).
The eigenstates of Hamiltonian (32) are where c = cos θ, s = sin θ, and θ is the mixing angle: 3 In the following, for the sake of simplicity, we neglect subdominant fraction of protons and electrons whenever their role is not important. Let us also remark that n−n conversion is ineffective in outer crust where the neutrons are bound in heavy nuclei. 4 For simplicity, we consider the exact mirror model with m n = mn, however our results are applicable as well in the case of small mass splitting. We also neglect the Zeeman energy induced by the neutron magnetic moment µn since |µnB| 1 MeV even for magnetars where the magnetic field B can reach 10 15 G or so.
where ∆E = E n − E n = V n (1 − β). The probability of n − n oscillation for flight time t is For ∆E t 1 we have P nn = (εt) 2 , while for ∆E t 1 the time dependence can be averaged so that P nn = 1 2 sin 2 2θ. This can be simply interpreted also in following way. Creating the neutron as initial state is equivalent of creating the eigenstates n 1 and n 2 with the probabilities c 2 and s 2 respectively. The eigenstates do not oscillate among each other, and after freely propagating to some distance, one can detect n 1 and n 2 as n with probabilities s 2 and c 2 correspondingly. Thus, by combining the probabilities, we get P nn = 2c 2 s 2 = 1 2 sin 2 2θ. Once again, if the neutrons are considered as ideal (non-interacting) gas, n − n oscillation per sè cannot lead to the neutron star transformation into the mixed star. As far as ∆E ε, the mixing is extremely small: 5 where ε 15 = (ε/10 −15 eV). Hence, for the freely propagating neutrons the M neutron fraction remains negligibly small at any time, P nn 2θ 2 1. Therefore, one has to take into account the neutron interactions in the medium.

B. Neutron-mirror neutron conversion processes
Although the ordinary and mirror nucleons have separate strong interactions, the mixed interactions emerge in the basis of the Hamiltonian eigenstates (33). E.g. the couplings π 0 nγ 5 n + π 0 n γ 5 n with ordinary and mirror neutral pions (the coupling constant g πN N is omitted) give rise to non-diagonal terms between n 1 and n 2 states: s π 0 n 2 γ 5 n 1 + π 0 n 2 γ 5 n 1 + h.c.
The same occurs for the analogous couplings with ρ mesons, etc. More generically, strong interactions of the neutron with the target nucleons (N = n, p) via single or multi-pion and ρ-meson exchanges etc. can be described by the effective couplings (nλn)(N λN ) where λ = γ 5 , γ µ etc. stand for the possible Lorentz structures. Since n = c n 1 + s n 2 , the above interactions in the basis of eigenstates have the form c 2 n 1 λn 1 +s 2 n 2 λn 2 +cs n 2 λn 1 +cs n 1 λn 2 N λN (38) which contains also mixed entries between n 1 and n 2 . The mirror neutron production rate can be estimated as follows. The stationary neutron state in the nuclear 5 In real situations the condition ∆E ε is always satisfied as far as ε is very small. In the limit ∆E → 0 (i.e. β = 0) θ has no singularity since Eq. (34) implies maximal mixing θ = π/4. medium can be viewed as the eigenstate n 1 (barring a tiny (≈ θ 2 ) fraction of n 2 ). The processes n 1 N → n 1 N , N = n, p are Pauli blocked, but n 1 N → n 2 N are not blocked. Therefore, every n 1 n 1 collision can produce n 2 ≈ n with a cross section 2θ 2 σ nn where σ = 4πa 2 is nn scattering cross section. 6 Taking the mean relative velocity as v = p F /m n , the rate of n → n transformation can be estimated as where for θ we take Eq. (36) with ∆E = V n − V n = 2πa(1 − β)n O /m n . Thus, the dependence on the poorly known scattering length a in fact cancels out in Eq. (39). The Pauli blocking factor η takes into account that the final state n 1 cannot have a momentum below the Fermi momentum p F = (2m n E F ) 1/2 ξ 1/3 × 340 MeV while the momentum of the produced n 2 ≈ n state should be above p F (βξ) 1/3 × 340 MeV. The blocking factor η(β) as a function of E F /E F = β 2/3 is estimated by a Montecarlo simulation of hard sphere scatterings, and the resulting dependence is reported in Fig. 3. This function has a maximum η(0) ≈ 0.18 at E F = 0, then it decreases and, as expected, asymptotically vanishes when the star approaches the MMS state, E F → E F . 7 For the younger MS, with n M n O , we can set β = 0 and Eq. (39) gives This rate depends on the baryon density, meaning that n − n conversion should proceed somewhat faster in peripheral low density regions of the star rather than in central regions where ξ > 1. 6 The process n 1 n 1 → n 2 n 2 has no Pauli suppression but its rate is proportional to θ 4 and thus it is negligible. The processes n 1 p → n 2 p involving protons and heavier baryons contribute with the rate ∝ θ 2 , but for simplicity we neglect them because of small (∼ 10 %) fraction of protons in the NS interior. 7 Let us remark that the reverse process n 2 n 2 → n 2 n 1 is Pauli blocked as far as E F > E F . This justifies why we have neglected the inverse reaction rate Γ in Eq. (5) to obtain Eq. (6).
Since the process n 1 n 1 → n 1 n 2 (hereafter we call it simply as nn → nn process) takes place at the momenta close to the Fermi surface, the mirror neutron is produced with typical energy E F = ξ 2/3 × 60 MeV. This energy will be radiated away by cooling and produced mirror neutrons start to form degenerate core with E F < E F . (In fact, mirror neutrons produced at very initial stages will decay as n → p e ν and cooling can be related to p and e components which then undergo the 'neutronization' in the core.) Therefore, the energy production rate per baryon can be roughly estimated as Notice that the ξ-dependence is cancelled. Mirror neutrons can be produced also by other processes. In normal conditions the mixed interaction terms (37) cannot induce decays n 1 → n 2 π(π ) with the emission of neutral (ordinary and mirror) pions, simply by kinematical reason. However, such decays become possible in dense nuclear medium. This effect is similar to the matter induced neutrino decay [61], and we shall simply call it n → n decay. Namely, considering the large enough densities at which the threshold condition ∆E = V n > m π is satisfied, the matter-induced decays n 1 → n 2 π 0 (π 0 ) can proceed with no Pauli blocking. The decay rates (taking again β = 0) read (42) where the Goldberger-Treiman relation g πN = g A m n /F π is used for the pion-nucleon coupling constant, with F π = 93 MeV being the pion decay constant and g A = 1.27 the axial coupling constant. The decay n → p π into mirror proton and "negative" mirror pion is also possible. However, the rate (42) can be competitive with (40) only in central regions of the heavy NS where the density can be as large as ξ = 8 or so. Therefore, in the following our estimations will be based on the rate given by Eq. (40). The heavy eigenstate can also decay into the lighter one with emission of the ordinary or mirror photons, n 1 → n 2 γ(γ ), via the transitional magnetic moment (TMM) µ 12 between the two eigenstates. The decay rates read In particular, the mass mixing (2) induces the TMM µ 12 = θµ n between n 1 and n 2 states right from the magnetic moments of n and n , µ n = µ n = −1.91µ N , where µ N = e/2m N is the nuclear magneton. Then we get Γ n→n γ (γ) = ε 2 µ 2 n π V n = ε 2 15 a 3 ξ × 3 · 10 −51 GeV (44) which is four orders of magnitude below (40). Hence, n → n γ decay cannot be the dominant effect if the TMM is induced solely by n − n mass mixing (2). However, the situation changes if there is a direct TMM µ nn between ordinary and mirror neutrons, in which case we would have µ 12 = µ nn . In fact, while the TMM between the neutron and antineutron is excluded by fundamental symmetry reasons, it is allowed between n and n [62], and can be induced in some models of n − n mixing [42]. Present experimental limits imply the upper bound µ nn < 10 −5 µ n or so [63]. Clearly, for the large enough TMM, µ nn θµ n , the rate (43) can be larger than (40). However, here we shall not discuss this situation.
The role of weak interactions is negligible. The Lagrangian describing mirror neutron β-decay n → p e ν e , induces n 1 → p e ν e decay via the admixture of n 1 in n . It produces mirror protons and electrons with the rate (45) which is vanishingly small as compared to (40).

C. Estimating the evolution time
Let us consider the evolution track of a star with a given overall baryon number N = N O +N M , starting from its initial (NS) configuration with N O = N and N M = 0. Since n−n transition rate (39) depends on ξ and β, these values should be averaged over the density distributions in the star. As far as the adiabatic evolution is assumed, at any time t the profiles of the number densities n O (r) = ξ(r)n S and n M (r) = β(r)ξ(r)n S , as the functions of radial coordinate r, are fully determined by the mirror baryon fraction X(t) = N M (t)/N . Hence, we have where with the parentheses meaning the average over the density distributions in the star at given M fraction X, and is the 'starting' conversion rate obtained from Eq. (40) by averaging over ξ(r) distribution in the initial NS with X = 0 (i.e. β = 0). Considering a NS containing N baryons and having a radius R, we have  (49) where for the last step we used the relation (19). For a given EoS, the total baryon number determines it the mass and radius. For massive stars with M 2 M the Sly EoS implies κ = 1.2 ands R = 10 km, and the respective evolution time can be estimated as Other EoS can give somewhat different results, but within factor of two or so. Integrating Eq. (6) with Γ given by (46), we obtain the age of the MS in which the mirror baryon fraction has reached a value X < 1/2: Since F → 1 when x → 0, for younger MS we obtain meaning that X linearly increases in time until it remains small enough, X 1. But the evolution gradually slows down with growing X, since the transformation rate (47) decreases due to increase of ξ and β and respective decrease of Pauli factor η(β).
During the evolution the ratio of the central densities χ = ρ cM /ρ cO increases along with the M fraction X. For the lighter stars that can reach the MMS stage with with χ = 1, Eq. (51) can be integrated up to X = 1/2. However, the heavier MS at a certain moment should collapse into BHs and so Eq. (51) should be integrated up to some value X < 1/2 at which χ(X) reaches the critical value χ max , which in turn depends on the initial mass of the progenitor NS as well as on the EoS. For example, for the case of the Sly EoS the NS with initial mass M ≈ 1.8 M will collapse at χ ≈ 0.5 (with its gravitational mass reduced to 1.7 M ) which corresponds to the evolution time t ≈ 0.1τ ε . The stars with smaller mass can have larger lifetimes: e.g. a NS born with M = 1.6 M can survive for time t τ ε , and will collapse with mass reduced to 1.5 M or so. And finally, stars with initial masses M ≤ 1.55 M do not collapse, and asymptotically in time can reach the MMS configuration with masses rescaled down to M ≤ 1.45 M , gaining the gravitational binding energy of about 0.1 M . Fig. 4 shows the time evolution of the mirror fraction X in a MS in units of the characteristic time highlighting the maximal possible mass of a MS at the age t, after which time they collapse. The curves on Fig. 4, corresponding to our two examples of EoS, Sly and joined polytrope, are obtained by numerical calculations of factors F(X) (47) by averaging over the density profiles of mixed stars containing a fraction X of mirror baryons and respective values of χ(X).
The "starting" rate of the NS energy loss due to n − n conversion is obtained by multiplying the energy production rate per baryon (41) on the baryon amount (19):  which is applicable for the evolution times t τ ε . For larger t becoming comparable with τ ε the energy loss rate decreases. Eq. (53) can be compared to the energy radiation rate due to the pulsar magnetic dipole fielḋ E magn = − 1 6 B 2 α R 6 Ω 4 B 2 9 R 6 12 P −4 10 × 3 · 10 33 erg/s (54) Here Ω = 2π/P is the pulsar rotation angular frequency (P 10 = P/10 ms) and B α = B sin α, where B is the magnetic field strength at the pole and α is the angle between the pulsar magnetic moment and its rotation axis (B 9 = B α /10 9 G). Hence, for ε = 10 −15 eV or so, for the typical pulsars one expects thatĖ nn Ė magn . But for larger values of ε the two rates can be comparable and one can even haveĖ nn >Ė magn . Let us note thatĖ nn (53) is a rather regular quantity essentially depending only on the inferred value of ε, whileĖ magn (54) depends on the pulsar magnetic field and its rotation frequency and thus can vary by orders of magnitude between different pulsars, achieving very large values for magnetars.

IV. COMPARISON WITH ASTROPHYSICAL OBSERVATIONS
Let us discuss now the possible astrophysical implications of n − n conversion in the neutron stars and derive limits on the transformation time τ ε (49) which in turn can be translated into the bounds on ε. Most of these effects are model dependent, and one should be careful in their interpretation. Namely, they can depend on the EoS as well as on the specific environmental conditions for some stars. For the sake of simplicity, we shall discuss two possible situations. First we discuss the case when the transformation time τ ε is larger than the age of the universe t U = 14 Gyr. In this case no star can reach the MMS configurations: all NS should still be neutron dominated hosting in in their interior only a small mirror fraction proportional to their age t, X = t/τ ε 1. We shall discuss effects of n − n transformation for the timing, mass loss and heating of pulsars. Then we shall concentrate on the possibility of τ ε being much smaller than t U . In this case all stars older than τ ε should be already transformed into the MMSs, and thus should be more compact than the younger stars.

A. Effects of slow NS to MS transformation
The pulsar ages can be estimated assuming that the spin-down rate of the pulsar rotation is dominated by the energy radiated by a rotating dipole magnetic moment. If this is the only braking mechanism, the pulsar age t is given by the relation where P (t) is the rotation period at the time t and P (0) is the rotation period at the NS birth, t = 0. The measurable value τ c = P/(2Ṗ ) = −Ω/(2Ω), usually called the pulsar characteristic (or spin-down) age, coincides with the true age if P (0) P (t), meaning that the pulsar rotation period at birth was much smaller than at present time, and if the magnetic dipole emission has always been dominant over other slowing down processes. For most of known pulsars, more prominently for those which were observed with large spin-down ratesṖ /P , the true age t can be rather close to their characteristic age τ c .
The heaviest recycled pulsars observed up to now and their characteristic ages are listed in Table I. The masses of first three pulsars in this Table are compatible with 2 M , (within 1.5σ error bars for PSR J0740+6620). Thus in the context of the Sly EoS, which admits the standard NS with masses up to M max χ=0 ≈ 2.05M , these pulsars would collapse due to n − n within the time t < τ ε /200 or so as its is shown on upper Fig. 4. All these pulsars have spin-down ages τ c of few Gyr. Thus, assuming that their true ages correspond to the respective τ c values, the very existence of these pulsars in the context of the Sly EoS would imply a conservative limit τ ε > 200τ c ∼ 10 12 yr or so. Using Eq. (50), this in turn can be translated into the upper bound ε < 2 × 10 −14 eV or so.
The case of the globular cluster pulsar J1748−2021B (or NGC 6440B) is more interesting. In spite of large  Fig. 4 we see that with this mass the star had to collapse within the time t = τ ε /20 or so. But the true age of this object cannot be determined since the measured value of itsṖ is negative: instead of spinning-down, this pulsar seemingly spins-up.
As we discuss below, in the presence of n−n conversion the NS true age and its characteristic age are no more related via Eq. (55). In fact, this equation follows from integration of the pulsar evolution differential equatioṅ which is obtained by equating the energy radiation rate (54) with the time derivative of the pulsar kinetic rotational energy E rot = IΩ 2 /2 and assuming that the moment of inertia I remains constant in time, so thaṫ E rot = IΩΩ. Then, using the measured values of Ω anḋ Ω, from this equation one can also determine the value of the pulsar magnetic field and the respective rate of the pulsar energy loss (54). Let us remark, however, that these are derived parameters obtained under assumption that the pulsar losses the rotational energy dominantly by the radiation due to its magnetic dipole. However, in the presence of n − n conversion the moment of inertia is not constant, and we haveĖ rot = IΩΩ +İΩ 2 /2. Hence, the pulsar evolution differential equation should be modified aṡ The valueİ/I =Ṁ /M +2Ṙ/R is negative as far as in the process of n−n transformation both the mass and radius of the star decrease. If this term is significantly large, then Eq. (55) becomes invalid and the pulsar age t cannot be related to its spin-down time τ c . In fact, the heavy pulsars in Table I could have small spin-down rates not because they are very old and posses very small magnetic fields, 9 but due to partial cancellation between the first (negative) and second (positive) terms in (57). In this way, the actual values of magnetic fields can be larger than the derived ones, and the true ages of these pulsars can be much less than their spin-down ages. Therefore, the naive limits on n − n mixing, as ε < 3 × 10 −14 eV obtained by estimating the collapse time of 2 M stars in the context of the Sly EoS, are not applicable. Moreover, if in Eq. (57) the positive contribution −(İ/2I) Ω is dominant, then we would haveΩ > 0 which means that the pulsar spins-up. In particular, it is tempting to propose that the negativeṖ of PSR J1748−2021B in Table I is originated from this effect, meaning that this pulsar is spinning-up instead of slowing down (though this effect could be related also to the gravitational acceleration in globular cluster). Interestingly, in NGC6440 and NGC6441 there are other pulsars withṖ < 0, two isolated ones as J1748−2021C and J1750−3703C, and one with a light companion, J1748−2021F [54]. If future observations will find some other spinning-up pulsars, with the intrinsic value ofṖ /P being confidently negative after subtracting the acceleration effects, then this could be interpreted as the effect of the star contraction due to n − n transformation.
The interesting possibility that the neutron star mass loss during its transformation into mixed star can affect the orbital period in the the NS binaries was discussed in Ref. [44]. If the system losses the mass, the orbital period P b should increases with the rate which is related to the mass loss rate: where M is the NS mass and M c is the mass of its companion in the binary system which can be e.g. a white dwarf of another neutron star. Assuming again that τ ε > t U and using Eq. (53), the mass loss rate by the neutron star can be estimated aṡ In this way, the observational data on the orbital period decayṖ b /P b can be used to obtain the limits on ε.
In fact, several pulsar binaries have positive measured values ofṖ b . However, apart of positive contribution of n − n effect, the observed value should be corrected for the dynamical effects related to the system acceleration in the galactic gravitational potential, kinematic Shklovskii effect of apparent acceleration and, for compact systems, also for the quadrupolar emission of gravitational waves (GW): 9 E.g., for millisecond pulsar J1614−2230 the derived values of surface magnetic field and spin-down luminosity were evaluated respectively as B 2 × 10 8 G andĖmagn 10 34 erg/s [51].
For example, the binary system containing one of the most stable pulsars J0437−4715, with mass M ≈ 1.4 M and companion mass M c ≈ 0.25 M , one has P b = 5.7410 days andṖ obs b = (3.73 ± 0.06) × 10 −12 [64]. However, after subtracting the above effects, also taking into account that the GW emission is negligible for this system, one obtainṡ (60) which is compatible with vanishing effect. Therefore, within 1σ error, we haveṖ b /P b < 2.8 × 10 −11 /yr. Thus, assumingṀ c = 0 (meaning that the companion mass loss is negligible), from Eq. (58) we obtain |Ṁ /M | < 1.65 × 10 −11 /yr. Then from Eq. (59) we can derive an upper limit ε < 4.3 × 10 −13 eV, which nicely coincides with the result of Ref. [44]. The pulsars J1141−6545 and J1952−2630 discussed in [44] imply comparable bounds. Somewhat stronger limit can be obtained from the Hulse-Taylor pulsar B1913+16 known as a perfect binary system for testing General Relativity. The masses of the pulsar and its companion (presumably another NS) were determined with a great precision: M = 1.4398(2) M and M c = 1.3886(2) M , as well as the orbital period and its derivative: P b = 0.322997 days andṖ b = −2.423(1) × 10 −12 [65]. For this compact binary, the predicted GW contribution isṖ GW b = −2.402531(14) × 10 −12 which almost saturates the observed value. After subtracting also the galactic correctionṖ gal b = (−0.027 ± 0.005) × 10 −12 , one obtainṡ This, assuming that both NS suffer mass loss, from (58) we get (Ṁ +Ṁ c )/(M + M c ) = −(3.7 ± 2.8) × 10 −13 /yr which can be interpreted as an upper limit Energy loss due to n−n transition can have interesting implications also for the NS surface temperatures. As we discussed in Sect. III, per every process nn → nn in the NS the mirror neutron n , due to the Pauli blocking, is produced with with typical energy E F = ξ 2/3 × 60 MeV, and the energy production rate per baryon can be estimated as (41). According to Eq. (52), for τ ε t U we have X 1, so that all neutron stars can be considered as young, and the energy production rateĖ nn in their interior is given by Eq. (53). Since the M matter density is much smaller than the neutron density. the produced mirror neutrons will preferably decay as n → p e ν producing a hot plasma of mirror protons and electrons gravitationally trapped inside the neutron star, though also various nucleosynthesis processes as should take place. Then, taking that the fraction x of the produced energy is radiated via the thermal spectrum of mirror photons γ , the "mirror photosphere" temperature T γ of the NS can be estimated as simply by equating xĖ nn = 4πR 2 · σT 4 γ , with σ being the Stefan-Boltzmann constant. 10 On the other hand, some part of energy produced in the NS will be emitted in terms of ordinary photons and neutrinos. Namely, disappearance of the neutron in the reaction nn → nn leaves the "empty" level in the Fermi see which will be filled by transition of the neutron from the higher level. Once again, since nn → nn reactions take place close to the Fermi surface, this transition energy should be smaller than the Fermi energy E F by a factor of 50 or so as we obtained by the MC simulation. Thus, the ordinary component should have less heating than the mirror one, and the NS surface temperature in terms of ordinary photons can be roughly estimated as The standard cooling mechanisms predict sharp drop of the temperature with the age of the star, leading to the surface temperature below 10 4 K after 10 7 yr, and below 10 3 K after 10 8 yr. However, observations of some old pulsars detect that they are still warm, with the surface temperatures 10 5 K or larger. For example, above discussed PSR J0437−4715 is the brightest millisecond pulsar in UV and X-rays. Its characteristic age is τ c = 3.2 × 10 9 yr but its UV spectral shape suggests a thermal emission with the surface temperature T γ = (1.5 ÷ 3.5) × 10 5 K [66]. Also PSR J2124−3358, a solitary 5 ms pulsar of the age τ c = 3.8 × 10 9 yr shows a thermal spectrum with T γ = (0.5 ÷ 2) × 10 5 K [67]. On the other hand, an younger pulsar B0950+08 (τ c = 1.8 × 10 7 yr) also has a surface temperature T γ = (1 ÷ 3) × 10 5 K [68]. These temperatures are much higher than predicted by cooling models, which means that some heating mechanisms operate in the NS. Namely, in the context of our model such surface temperatures can be explained by n − n transformation with ε ∼ 10 −15 eV. It should be noted, however, that the determination and interpretation of the NS surface temperatures are model dependent, namely they depend on interstellar extinction and the models of pulsar magnetosphere, accretion from the partner and non-thermal emission, and there are other heating scenarios related to Urca processes with vortex friction and rotochemical reactions.
Let us also note that observations of isolated slow pulsar J2144−3933 (P = 8.5 s) with τ c = 3.3 × 10 8 yr imply solely an upper bound T γ < 4.2 × 10 4 K [69] which disfavors some of heating mechanisms but still remains compatible with n − n transformation with ε < 2 × 10 −16 eV or so. On the other hand, the suppressed thermal spectrum in PSR J2144−3933 can be related also to environmental factors, and it would be premature to derive any serious conclusion just on the basis of one non-detection.
Finally, let us remark that for τ ε t U the mirror neutron stars can be in fact "visible" in the UV diapason, with the surface temperature given by Eq. (63). The energy produced in the mirror NS due to n − n transformation will be emitted in ordinary far UV photons with the rate (53). For e.g. ε = 10 −14 eV this rate will be ∼ 10 33 erg/s which is equivalent to the solar luminosity.

B. Effects of fast NS to MS transformation
Let us discuss now the situation when the NS transformation time is rather small, say τ ε < 10 5 yr or so. In view of Eq. (49), this would correspond to n − n mixing mass ε > 10 −10 eV or so. Then the stars with the age larger than τ ε should be already transformed in the MMS, with (almost) equal amounts of ordinary and mirror baryons inside. Correspondingly, these stars will have no more energy losses due to n production and would not show the evolution effects discussed in previous subsection. In addition, also the compact objects initially born as mirror NS can be visible for us in this mixed form, and in addition they can be detected as ordinary pulsars provided that by some mechanism they acquire also the ordinary magnetic field. This can naturally be naturally induced by kinetic mixing term between ordinary and mirror photons ( /2)F µν F µν [70] which effectively makes M protons and electrons mini-charged (with ordinary electric charges ∼ ). The cosmological bounds imply < 5 × 10 −9 or so [71] while the experimental limit from the positronium decays is yet weaker: < 5 × 10 −8 [72]. As we noted above, the neutrons produced via n −n transition at first stages of the mirror NS evolution decay and produce electrons and protons which then enter in nuclear reactions with the neutrons forming nuclei. In any case, the mirror NS rotation, via Rutherford-like scatterings due to the photon kinetic mixing, will drag the electrons rather than protons and ions, inducing circular electric currents which can give rise to substantially large magnetic fields by the mechanism suggested in Ref. [73]. In this way, it can become a complicated task how to distinguish between the old pulsars initially originated from ordinary and mirror NS since they should also have comparable surface temperatures. One possibility can be that "mirror-born" pulsars should dominantly accrete mirror matter, since their companions in binaries should be M stars, and thus they should be dominantly active in terms of mirror X-rays rather than in ordinary X-rays.
A clear phenomenological implication of the gravita-tional mass scaling (30) is that for any EoS, the last stable MMS mass is √ 2 times smaller then the last stable mass of the ordinary NS. Applying this relation to our first example ( 2M and characteristic ages τ c > 10 9 yr demand rather stiff EoS allowing M max NS 2.7 M or so. As already discussed, the scaling relations do not hold for stars with the same total baryonic number. Therefore the configurations related by the √ 2-scaling of Eq. (20) and preserving the stellar compactness do not correspond to different evolution stages of the same star. A young neutron star with a mass M NS exclusively composed of O component with central density ρ cO , due to n → n conversion slowly evolves to a more compact MS, with ρ cM (t) = 0 and χ(t) = ρ cM (t)/ρ cO (t) adiabatically increasing in time. If this way, the star continuously converts gravitational energy in heat, which should be emitted presumably in terms of ordinary and mirror neutrinos and photons. As depicted in Fig. 2, during the conversion the star becomes more compact, and its mass becomes somewhat smaller due to the gain in the gravitational binding energy. The mass difference between the initial and final states is about 0.1 M for both considered EoSs. This means, that e.g. for the case of the SLy EoS, a NS with the initial mass M NS = 1.5 M and radius of about 12 km, can evolve into an asymptotic final configuration of a MMS (χ = 1) with a slightly smaller mass (M 1.4 M ) but considerably more compact, with the radius of about 8 km or so. Therefore, if the observations will find two compact stars with masses of e.g. 1.4 M but having very different radii, such objects can be interpreted as two MS with different "mixtures" of the O and M components (i.e. with different values of χ) and thus with different ages. Unfortunately, for the compact stars whose masses are known with a high precision the radii remain practically unknown, and few cases when both masses and radii can be both determined have very poor precision [56]. However, if the precise radius measurements by the NICER mission, see for instance [74], yield very different radii irrespectively of the objects mass, one could interpret this result as two MSs in different stages of the evolution: the more compact star should be older than the less compact one.
Quite interestingly, recent analysis of NS masses in binary systems suggest a bimodal distribution [75]. A possible interpretation is that the two distributions correspond to different type of stars, with one (lighter) component corresponding to standard NS and another (heavier) component corresponding to twin configurations made of hybrid stars (HS) with a deconfined quark matter core or entirely quark stars (QS). However, the quark stars can only cover a restricted region of the mass-radius diagram. In our picture, instead, the MS evolution allows to span a larger region of the mass-radius diagram.
Remarkably, the n → n conversion process could af-fect the total distribution of the NS masses. For definiteness, let us focus on the SLy EoS. First, we note that NS born with mass M NS < 1.55 M continuously decreases its gravitational mass and become the MMS after a time t > τ ε . Second, NS with larger initial masses, M NS > 1.6 M , cannot become MMS. In this case the star is doomed to collapse to a black hole. A neutron star with initial mass M = 1.8 M will become unstable and collapse to a BH in a time t 0.1 τ ε after the birth, while the NS born very close to the last stable configuration would collapse much faster. Both effects should alter the NS mass distribution predicted by the supernova explosion mechanism, reducing the number of compact stars with large masses while increasing the number of stars with low masses, as was discussed in [43]. Unfortunately, the situation is more complex because one cannot exclude the possibility that due to the matter fall-back after supernova explosion the NS can transformed into the HS or QS [76] (see also discussion in [77]). In general, standard evolution processes may play an important role, leading to a (0.2−0.3) M increase of masses due to matter fallback after the supernova explosion or accretion in recycled NS [78]. Which final state, HS or QS, will be reached in this transition depends on the details of the quark matter EoS and on the amount of accreted mass. All these mechanisms lead to the mass increase, thus it is generally believed that the mass of NS after their birth can only increase, while the ordinary to mirror matter conversion should reduce the gravitational mass. 11 The NSs in double neutron star (DNS) binaries represent an interesting case for studying the effect of the n → n conversion on the mass distribution of compact stars. These NSs are thought to have received little or no accretion, thereby reflecting the NS stellar mass at birth with a direct link to the supernova mechanism. The observed DNS masses can be nicely fitted with a gaussian with a central value at ≈ 1.4 M and rather small dispersion, σ ≈ 0.05 M [56], though it is not quite clear why a general supernova explosion should lead to such a central mass and to such a peaked mass distribution. Indeed, the gravitational mass for collapsed cores could be less than 1.3 M , but since in a DNS accretion can only happen by matter fallback, a large increase in the dispersion of NS masses is expected [79].
The effect of n → n conversion process is to shift the peak of the mass distribution at birth towards lower values [43]. Therefore, if one could prove that the NS mass distribution at birth is wider and extends to values significantly above than 1.5 M , then the narrow distribution of masses in the DNS systems could evidence of the effect of the n → n conversion process. Once again, in the context of the Sly EoS, this distribution in fact should be non-gaussian with a cutoff at maximum mass 11 On the other hand, also the NS with estimated masses 1M have been observed [78], which challenges the present understanding of core-collapse neutron star formation [46]. M max 1.45 M for older stars, with ages exceeding τ ε .
As for younger stars, they still can be in the transformation process, and thus can have different radii. In addition, the evolution can manifest in observational phenomena as e.g. pulsar "glitches", sudden increase of the rotation frequency caused by irregular transfer from the NS interior to the crust and by the after "star quake" rearrangement of the crust. In fact, n − n conversion can proceed only in the NS interior which adiabatically shrinks the neutron liquid while this shrinking is not adiabatically followed by the shrinking of the rigid crust which then ruptures in discrete events. Depending on the situation, such effects can cause also pulsar "antiglicthes", events of a sudden spin-down. It is tempting to think that also the phenomena of soft gamma repeaters or intermittent pulsars can be related to the effects of n − n conversion during the evolution.
Finally, let us comment on the combined effect of conversion to mirror matter and formation of strange quark matter at high nuclear densities. According to the Bodmer-Witten hypothesis [80] strange quark matter can be the energetically favored ground state at large densities. Thus, the NS whose masses can strongly increase by accretion reaching a critical value, say M th 1.6 M , can be promptly transformed into a star made at least in part of deconfined quark matter [76]. Which final state, HS or QS, will be reached in this transition depends on the amount of accreted mass as well as on the details of the quark matter EoS. In quark matter the transformation to mirror matter should be suppressed for two reasons, first because there are not much neutrons to transform, and second because quark matter is self-bound (as standard nuclei), and therefore the transition to mirror nuclear matter should give no energy gain, in particular if quark matter is in a color superconducting phase [86,87]. Therefore, in QSs almost entirely consisting of quark matter will not be transformed into mixed stars. The case of HS is more interesting since n − n transition will be still effective in the part of star consisting of the neutron liquid. Therefore, in a time τ ε it will be transformed into a QS with a core of M neutrons in fact forming a mirror NS inside the QS. For rather heavy stars, in which also the density of the mirror core can reach the threshold value, the mirror neutrons core can in turn transform into quark matter, thus forming a mixed quark star. Therefore, also the strange QS can have some M cores consisting of mirror neutrons or deconfined mirror quarks, depending in their mass and evolution history. Reciprocally mirror QS can have ordinary matter cores which can be detectable by their electromagnetic radiation. Once again, mirror stars with small enough initial masses should consist entirely of M neutrons and thus they should evolve into the MMS.

V. NEUTRON STAR MERGERS AND ASSOCIATED SIGNALS
Let us briefly discuss the implications on n − n transitions for gravitational wave (GW) bursts from the NS mergers and the associated electromagnetic signals as gamma ray burst (GRB) and kilonova events which are also known as the main source of production of heavy (trans-iron) elements in the Universe.
LIGO Collaboration detected two candidates. The first event GW170817 [81] is considered as a clear signal of the ordinary NS merger, with masses of two stars M 1,2 compatible with 1.4 M and their total mass M tot = (2.75 ± 0.02) M typical for NS binaries. Remarkably, the GW signal was accompanied also by a weak GRB as well as by electromagnetic afterglows in different diapasons.
The second candidate GW190425 [82] is more unusual. While the best fit masses of individual components (M 1 1.8 M and M 2 1.6 M ) are within the mass range of the observed NS, both the source-frame chirp mass (1.44 ± 0.02) M and the total mass (3.4 ± 0.2) M of this system are significantly larger than those of any other known NS binaries. In addition, no confirmed electromagnetic event has been identified in association with this GW signal which suggests that this event could be originated by the merger of mirror NS [83].
The possibility of n − n conversion adds new features to the picture. Namely, if the conversion time τ ε is short, then the old neutron stars should exist today in the MMS form, with equal amounts of the O and M components inside. Therefore, the MMS mergers should have potentially observable electromagnetic counterparts irrespectively of their origin (ordinary or mirror). In fact, one cannot exclude the possibility that GW170817 was induced by coalescence of stars which were initially born as the mirror NS and then evolved into the MMS.
As for GW190425, the location of this merger is practically unknown since the GW was essentially detected only by the LIGO Livingston interferometer: the LIGO Hanford at this moment was off-line while the signal of Virgo was at the level of noise. Thus, the non-detection of the associated electromagnetic counterpart does not really exclude that it was a merger of the two MMS. In addition, one can also speculate about the unusual mass parameters of this system assuming that it was born as a mirror NS binary. As far as mirror world should be helium dominated [8] and so the evolution of M stars and their pre-collapse conditions can be different from those of ordinary matter stars [10], then core-collapse of mirror progenitors could produce the NS with somewhat larger birth masses. One can also hypothesise that GW190425 was a coalescence of M quark stars with null or small amount of ordinary matter inside, and so without potentially detectable electromagnetic association.
However, the enhanced compactness of the MMS with respect to NS can have interesting implications. Namely, the GW signal from the NS coalescence is sensitive to the tidal deformations that each components gravitational field induces on its companion, and thus it can give relevant information about the EoS describing the NS and their radii. In particular, the analysis of GW170817 waveform [84] favors the softer EoS as Sly [55] rather than the stiffer ones as e.g. [58]. For the Sly EoS, assuming that the component masses are 1.4 M , the limits on tidal deformability obtained from GW170817 waveform analysis implies the limit on their radii R 1,2 > 10 km or so [84]. This is somewhat larger than the MMS radii R MMS = 8.5 km predicted by the Sly in view of scaling relation (25). However, it should be premature to make strong conclusions from this discrepancy before solid statistics is achieved on the GW signals, moreover that other interpretations were also discussed which allow rather small radii [85].
If the transition time is larger than the age of the universe, τ ε t U , then only a small fraction of ordinary nuclear matter can be transformed into M matter in the NS. The produced mirror matter should form a small core inside the original star. This core should only be bound by gravity with no material friction with the dominant ordinary component. Therefore, every sudden collision with external bodies or fast accretion of a large chunk of matter could cause relative vibrations between the two components, which may manifest as some sort of glitches.
In any case, this should contribute to the deformability of the compact star, which can be tested by analyzing the GW waveforms from the NS mergers. Further, when the dominant O components of a merging binary system hit each other and coalesce, the smaller mirror cores should continue their rotation for some time and merge with some delay, eventually producing some non-trivial perturbations in the GW waveform. However, if the mirror cores have masses smaller than the evaporation limit (0.1M or so) and moreover if their orbiting ellipse has a high eccentricity, they can be thrown away from the merger site by a sling-like effect and then explode due to decompression. This can giverise to the kilonova-type events producing a hot mirror neutron rich cloud which can be at the origin of r-processes producing heavy elements in the mirror sector.
A different and intriguing possibility is related to the mergers of mirror neutron stars. in the absence of n − n mixing, the mirror NS will suffer no conversions into ordinary matter and thus the mirror NS mergers would produce gravitational wave signals not accompanied by any standard electromagnetic counterpart [83]. In this case the mirror NS mergers will look as 'invisible but not silent'. But If the mirror star hosts an ordinary nuclear matter, meaning that it is a MS with dominant in mass (and larger in size) M component having a small core of standard nuclear matter, then after the merger of dominant M components the subdominant ordinary cores could continue the orbiting for a while and then, via their collision or decompression, give rise to a hot neutron-rich cloud around the coalescence site. In other words, we suggest that the observed kilonova events as well as gamma ray bursts, or at least some of them, could originate from the merging of MSs with a dominant mirror component. The existence of MSs with a dominant M component may have a number of additional phenomenological effects, in particular if their ordinary cores produced by n −n transitions develop substantial magnetic fields, they could be observed as ordinary pulsars.
Another intriguing possibility is that mirror matter has the baryon asymmetry of the opposite sign to ordinary matter, so that mirror neutrons inside transform into the standard anti-neutrons rather than into the neutrons. Such a situation can be naturally realized in cobaryogenesis models between O and M sectors discussed in Refs. [16,89]. In this case a mirror anti-neutron star would develop in its interior a core consisting of ordinary antimatter. The gravitational merging of such mirror star binaries can be at the origin of anti-r processes which would produce ordinary anti-nuclei, and anti gold in particular. Electromagnetic signals of such anti-kilonovae cannot be distinguished from ordinary kilonova events, but the produced anti-nuclei can be hunted by the AMS2 Collaboration in the spectrum of cosmic rays. In addition, if reasonably large magnetic fields can be transferred to the rotating ordinary anti-core, such a star can be seen as a pulsar.

VI. CONCLUSIONS
We have discussed the possibility that the ordinary neutron stars, via n − n conversion, can develop the mirror matter cores in their interior. These cores gradually increase in time and only the stars with masses than some (the EoS dependent) critical value can survive asymptotically in time reaching the maximally mixed configuration while the heavier ones should collapse into black holes.
To distinguish from other works, let us remark that possibility of the neutron stars with small mirror cores formed by the dark matter accretion was discussed in Refs. [90]. However, dark matter accretion rate cannot be very effective and it can destabilize only heaviest pulsars with masses already very close to maximum mass allowed by the given EoS. In fact, the neutron star with a bigger rate would accrete the normal matter in which interior it was born, as in is the case for recycled neutron stars.
The neutron star conversion into dark neutron star via the transitional magnetic moment induced by the neutron-dark neutron mixing was discussed in Refs. [91]. The dark neutron was considered as an elementary particle with mass closely degenerate to the neutron mass, within 1 MeV or so, and without significant selfinteraction. It was shown that such dark stars cannot have masses larger than 0.7 M . For stabilizing these objects, the dark neutron self-interactions were ad hoc introduced in Ref. [92], which possibility for compact dark matter stars was previously studied in Ref. [93].
In our case, once the concept of mirror matter is adopted, there is no need for complementary hypothe-ses for maintaining the stability of the mixed stars, since the mirror nuclear matter should have exactly the same EoS as the ordinary one. Therefore, the existence of the maximally mixed stars, with evenly distributed O and M components, implies only the upper limit on their maximum mass (30). However, this limit also depends on the chosen EoS (and it can also be avoided by assuming that the heavy pulsars are in fact the quark stars, since n − n transition should be ineffective in quark matter).
We have discussed various astrophysical implications of this scenario. First we discussed the situation when the NS transformation time into the MS is larger than the universe age, τ ε > t U in which case all existing NS should still be in the evolution processes. We have shown that no astrophysical limits related to the pulsar characteristic ages, orbital period change in binary pulsars or the pulsar surface temperatures, exclude the possibility of n − n mixing (2) with ε < 10 −15 eV (corresponding to oscillation times τ nn = ε −1 > 1 s or so) which is the target of several ongoing and planned experiments for searching n − n oscillations via the neutron disappearance (n → n ) or regeneration (n → n → n) in the minimal picture of mirror world with exact Z 2 parity. In this case n − n oscillation can have also interesting implications for the ultra-high energy cosmic rays.
Neither the possibility of ε > 10 −10 can be excluded. Also this parameter area has phenomenological interest since n − n oscillation could solve the neutron lifetime problem provided that O and M neutrons have a mass splitting of few hundred neV [39], and experiments are underway for testing it. It implies the NS transformation times τ ε < 10 5 yr or so in which case the observed NS with typical ages 10 6 ÷ 10 10 yr should be already transformed to the MMS configuration. In this case no evolution effects can be manifested by old pulsars while for pulsars younger than 10 5 yr are still hot, with intrinsic temperatures larger than 10 7 K and thus compatible with the limits on heating produced by n − n conversion.
The intermediate range ε = (10 −15 ÷10 −10 ) eV is more subtle. Namely, for ε ≤ 10 −13 eV we have τ ε ≥ t U the interval ε = (10 −14 ÷ 10 −13 ) eV is disfavored by the limits on the pulsar and pulsar binary timings and on the NS surface temperatures. As for the interval ε = (10 −13 ÷ 10 −10 ) eV, the effects of n − n transition will be dependent on the NS age and a careful analysis is needed to determine the upper edge of the excluded area.
We have also briefly discussed the effects of n − n conversion in hybrid quark stars and for the gravitational mergers in binary systems. Interesting possibility is that also the coalescence of mirror-born neutron stars could give rise to the weak GRB and associated kilonova events.