Solar luminosity bounds on mirror matter
aa r X i v : . [ a s t r o - ph . H E ] M a y MNRAS , 1–5 (2019) Preprint 31 May 2019 Compiled using MNRAS L A TEX style file v3.0
Solar luminosity bounds on mirror matter
Erez Michaely ⋆ , Itzhak Goldman , , and Shmuel Nussinov Astronomy Department, University of Maryland, College Park, MD School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel Department of Physics, Afeka Tel-Aviv Engineering College, Tel-Aviv, Israel
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present bounds on mirror dark matter scenario derived by using the effect of mirrormatter on the luminosity of the Sun. In the perturbative regime where the mirrormatter concentration is small relative to the ordinary matter we estimate the heattransfer from ordinary matter to the mirror sector by simple analytic consideration.That amount of heat transfer is radiated via mirror photons and increases the requiredenergy production in order to maintain the observed luminosity. We then present moredetailed numerical calculations of the total amount of this energy transfer.
Key words: dark matter – Sun: general
Finding the nature of Dark Matter (DM) which most likelycontributes ∼
25% of the energy density of the universe is anoutstanding challenge Feng (2010). Some types of DM whicharise beyond the standard model (BSM) of particle physicsare being experimentally searched and or are constrained byastrophysics. In particular DM particles such as axions, darkphotons and new types of neutrinos can be emitted from andlead to excessive cooling of neutron stars, white dwarfs redgiants etc. Raffelt (1996).The apparent inconsistency of the observed solar neu-trino fluxes with the predicted values was the first indicationBahcall (1989); Bahcall et al. (2004, 2006) for the only pieceof BSM physics (apart from DM!) known today, namelythat of neutrino masses and mixingsTanabashi et al. (2018).Early on it motivated an alternative suggestion that smallaccumulation in the sun of weakly interacting DM parti-cles (WIMPs) could cool the solar core and evade the ”solarneutrino problem” Spergel & Press (1985); Press & Spergel(1985); Faulkner & Gilliland (1985); Gilliland et al. (1986).Indeed at that time the main difficulty was the paucity ofthe energetic B neutrinos whose rate scales as T with T is the central temperature.In this note we again use the Sun to limit DM aris-ing in mirror models in which a hidden sector exists whereevery particle or parameter in the standart model - x ismirrored by an identical particle/ parameter x ′ Lee & Yang(1956); Foot (2014); Kobzarev & Okun’ (1968); for reviewsee Foot (2014) and references within. In much of the workthese models the symmetry between the mirror and ordi-nary sector is broken allowing different masses of particles ⋆ E-mail: [email protected] and their mirrors which helped address many astrophysi-cal and cosmological issues Berezhiani et al. (1996). Still al-most exact mirror models with minor changes made to allowΩ ( B ′ ) = 5Ω ( B ) ∼ Ω (DM) and a lower CMB tempera-ture in the mirror sector (which is required to avoid conflictwith BBN- Big Bang Nucleosynthesis) have been discussedat length Berezhiani et al. (2001); Foot (2014). It was sug-gested that this most restrictive framework with appropriateminimal weak mirror-ordinary matter interaction, can evadethe bullet cluster bound and the infall of DM into parallelgalactic disks expected for the strongly mutually interact-ing and dissipative DM made up of mirror atoms. Here wewill focus on the possible effect of the accumulation of mir-ror DM particles arising in the framework of ”almost exact”mirror symmetry in the Sun at a relative concentration η = M ′ M ⊙ (1)where M ′ is the overall mass of the mirror particles in theSun. We find that the resulting changes of the solar luminos-ity exclude η and mirror-ordinary matter cross-sections σ xx ′ values in a region allowed by all other constraints and whichwas strongly favoured by mirror DM models. This excludemost exact mirror DM variants. We start by describing the differences between earlier at-tempts to constrain massive DM particles by consideringthe consequences of their accumulation in the Sun and ourpresent discussion of almost exact mirror model particles.In the earliest works mentioned above Press & Spergel c (cid:13) E. Michaely et al. (1985); Spergel & Press (1985), only the mutual DM-nuclearcross-section σ xx ′ are used to trap in the sun DM particles ofmasses in the 5 − σ xx ′ & − cm was required in order to accumulate the minimal η ∼ − which allows sufficient heat transport from the solar core soas to significantly reduce the B neutrino flux. Extensivedirect searches for DM in large cryogenic, underground de-tectors restrict by now Tanabashi et al. (2018) the He’ (thedominant mirror dark matter component in mirror models)nucleon cross-section, to be less than 10 − cm Foot (2014).The next class of DM particles considered in thiscontext was that of asymmetric, strongly self interactingDM Frandsen & Sarkar (2010); Cumberbatch et al. (2010);Taoso et al. (2010). In this case newly falling DM parti-cles can be captured by scattering on DM particles whichwere captured earlier in the Sun. This increases the capturerate until it reaches the ”Unitarity bound” when essentiallyevery DM particle hitting the Sun is captured. The inte-grated accumulation over the solar lifetime can then leadto the concentration of η ∼ − , the value mentionedabove. With the present understanding of neutrino mix-ing, extra heat convection from the very central region isno longer required to explain the solar neutrino ”Problem”.The resulting cooling of the core still has other more sub-tle yet observable effects on the standard solar model asdiscussed in Frandsen & Sarkar (2010); Cumberbatch et al.(2010); Taoso et al. (2010).The strong mutual x’-x’ scattering helps retain the cap-tured DM inside the Sun. Indeed in the absence of suchstrong scattering some x’s with energy of E ∼ few kT ∼ few KeV and a velocity v = (2 E/m x ′ ) / exceeding the es-cape velocity from the solar core of ∼ / Sec will bekicked from the solar core, if the DM particles are lighterthan ∼ m x ′ = m H ≈ GeV.The Mirror dark matter considered here differs fromthat in the above two cases. Thanks to the exchange ofthe mass-less mirror photon it is strongly interacting viaRuthdeford scattering σ x ′ x ′ ∼ α /E = σ xx (2)which is ∼ − cm for the relevant KeV energies. The other most important feature is that mirror nu-clei/electrons can emit the massless mirror photons andtherefore constitute dissipative DM.To make our argument as model independent as possi-ble we use the concentration of the mirror particles in the In the solar core the plasma (Debey) screening correction to theabove cross-section amounts to replacing the momentum transfersquared k in the photon momentum space propagator by k → k + k s where k s = k D = 4 παn ( e ) /T ∼ . Using thetemperature and electron density appropriate to the solar corewe find k s ≈ . This screening cuts-off the very forward,low momentum transfer scattering but only mildly affects therelevant transport cross-section as the average k ∼ m ∆ E ∼ mT ∼ even for the case of e-e or e’-e scattering with m ≈ / /k by 1 / (cid:0) k + k s (cid:1) isto reduce the cross-sections by just a factor of two Raffelt (1996). Sun η and the ordinary- mirror scattering cross-section σ xx ′ as the two independent parameters of the particle physicsmodel to be constrained by the astrophysical considera-tions. In generic almost exact mirror models not only thecross-sections for mirror- mirror interactions are fixed by theabove Ruthdeford scattering but also the coupling of mirrorcharged particles with the ordinary sector is via the ”PhotonPortal” - namely the kinetic mixing: ǫF µ,ν F ′ µ,ν of the mirrorphoton field strength tensor F ′ µ,ν = ∂ ν A ′ µ − ∂ µ A ′ ν and thefield strength tensor of the ordinary photon. Since both ourphoton γ and the mirror photon γ ′ are massless we shouldidentify our physical photon ˜ A with the superposition oforiginal fields: ˜ A µ = A µ + ǫA ′ µ . Indeed the two fields A and A ′ in this particular superposition are coherently emitted,propagated and absorbed by SM charges. This redefinitionthen subsumes all the ordinary matter mirror photon in-teraction and therefore ordinary particles no longer coupleto the mirror photon. Yet each mirror particle x ′ with amirror electric charge of e ′ ( x ′ ) = e ( x ) ≡ e couples to theordinary photon with a milli-charge of ǫe . The cross-sectionfor mirror- ordinary matter x’-x scattering generated by or-dinary photon exchange is then the standard Ruthdefordscattering above reduced by ǫ σ x ′ x = ǫ σ xx ≈ ǫ α E . (3)To address the apparent departure from exact mirrorsymmetry we note that the above definition is appropriatein regions of space which are dominated by ordinary mat-ter. The opposite scheme where the redefined physical mir-ror photon does not couple to ordinary matter and ordinarycharged particles are milli-charged with respect to A ′ is ap-propriate in regions dominated by mirror matter such as theinterior of mirror stars discussed later.The effect of capturing mirror particles differs from thatin the previous case due to the fact that the mirror particlesradiate dark (mirror) photons. This provides yet anotherchannel for radiating the energy generated by nuclear reac-tions in the solar core- a channel which operates in parallelwith the usual ordinary photon radiation. Thus the mirrormatter not only transports heat, but just like the emissionof axions or massive dark photons that directly couple tothe solar nuclei/ electrons , it also changes the overall solarenergetics. This will allow us to derive limits on the mir-ror concentration η and the mirror ordinary sector particlescattering cross-section σ xx ′ which are more robust and lessmodel dependent than earlier limits.A second important difference between the almost ex-act mirror matter discussed here and the previous merelystrongly interacting DM, is that it’s concentration in the Sunis no longer restricted by the maximal capture rate duringthe lifetime of the Sun to be: η ∼ − .Both ordinary and mirror matter are disipative and mu-tually attract gravitationally. We therefore expect ordinarymatter to cluster in the gravitational wells of mini haloesgenerated in the mirror matter. Conversely, mirror mat-ter should cluster in the gravitational wells due to ordinarymatter galactic disks. This co-clustering or even co-collapsestends to mix the two types of matter. In particular it couldlead to an initial mirror matter concentration in the Sunwhich much exceeds the above limiting η ∼ − . Indeed ithas been estimated Foot (2014) that the original pre-solar MNRAS , 1–5 (2019) olar luminosity bounds on mirror matter cloud can efficiently accumulate mirror particles leading to η ∼ − .It is important to note that mirror matter is still arather subdominant component of the Sun. This justifiestreating the effect of this admixture perturbativly using theknown solar profiles of ordinary density, ρ ( r ), and temper-ature, T ( r ).For solar/stellar cooling by weakly interacting particlesemitted by nucleons and/or electrons it suffices to computethe volume emission of these particles which freely streamout. This is not the case here. First the mirror photons arenot emitted directly from the electron/protons in the Sunbut only by the mirror particles after kinetic/heat energyis transferred to them via collisions with ordinary core par-ticles at a total rate which we denote by dQ/dt . Thanksto their very strong mutual interactions the mirror parti-cles will then equilibrate and generate at each radius a localtemperature profile T ′ ( r ). In general this T ′ ( r ) is differ-ent from T ( r ), the temperature profile of ordinary protons/electrons. These mirror particles will then emit their energyvia mirror photons generated by bremsstrahlung in e ′ − p ′ or e ′ − α ′ collisions. Also unlike for the simple volume emissioncase mentioned above, the mirror photons will often scatteron their way out on the ambient mirror particles and willbe trapped for some time τ ′ . Both dQ/dt and τ ′ depend onthe, as yet unknown, profiles of the density ρ ′ ( r ) ∼ n ′ ( r )and of the temperature T ′ ( r ) of the mirror matter. Beforeembarking on the calcuation of these profiles and the result-ing dQ/dt we will first make some estimates using a simplerapproach. While we find that dQ/dt is comparable with the observedordinary solar luminosity L ⊙ for a range of allowed DMparameters σ xx ′ and η , the reverse heat flow to the origi-nal matter reservoir dQ ′ /dt is negligible . The reason for thisare the large self scattering of mirror matter σ x ′ x ′ whichexceeds by ǫ − > the mirror-ordinary particle collisioncross-section σ xx ′ and the large bremsstrahlung cross-sectionleading to γ ′ emission: σ x ′ x ′ → x ′ x ′ + γ ′ ∼ α × σ x ′ x ′ → x ′ x ′ ∼ − cm . Thus a mirror particle which has gained energyby a collision with an ordinary particle in the solar corewill collide and share it’s energy with other ambient mir-ror particles rather than collide again with an ordinary ionor electron. In turn bremsstrahlung quickly transfer this en-ergy to mirror photons. Since these mirror photons do notscatter at all from protons or electrons but only from mirrorparticles the energy transferred to the mirror sector, be itthe matter or radiation part, stays in that sector and even-tually is emitted as mirror photons. Thus to find the extraluminosity emitted via mirror photons we need only to find dQ/dt .We will mainly focus on the inner core, R ≈ . R ⊙ =1 . × cm which includes a total mass M ( R ) ≈ . M ⊙ and generates ∼
90% of the solar luminosity Paxton et al.(2011). The mainly ordinary matter densities therein of ∼ / cm corresponds to electron number density of n e ≈ . · cm − . The almost constant temperature ison average T ∼ . · − ergs / particle. Since theRuthdeford scattering depends only on the energy and not the mass of the colliding particles and in thermal equilib-rium electrons have the same energy of 3 / kT as protonsor He ions, the scattering of the faster moving e and e ′ willdominate the heat transfer process. Such scattering of twoequal mass particals tends to equalise their energy and onaverage an energy∆ E = 3 / (cid:0) kT ( r ) − kT ′ ( r ) (cid:1) ≈ kT ( r ) = 1 . × − erg (4)will be transferred to the mirror sector in each collision. Thedensity profile of mirror particles is given by the Boltzmanndistribution ∝ exp ( − V ( r ) /kT ′ ). V ( r ) ≈ π/ Gρ m He r isthe gravitational potential due to the ordinary roughly con-stant density ρ = 165gr / cm of ordinary matter. The mass m He ′ rather than m p ′ was used as mirror helium is the dom-inant component in the mirror sector, namely X ′ ≡ ρ ′ H ′ ρ ′ = 0 . Y ′ ≡ ρ ′ He ρ ′ = 0 . . (5)Foot (2014). ρ ′ ( r ) then is a Gaussian, exp − ( r/r ) with r = (cid:18) π kT ′ Gρ m He (cid:19) / ≈ cm . (6)Since this is less than R ≈ . R ⊙ most mirror particles areinside the above core. Each mirror electorn experiencesΓ ee ′ = n e v e σ ee ′ ≈ . × σ ee ′ sec − (7)collisions per second, where v e = (3 kT /m e ) / is the elec-tron velocity and n e ≈ . · cm − is the electron numberdensity.The total energy transferred to the mirror particles persecond then is: dQdt = N tot , e ′ Γ ee ′ ∆ E (8)where N tot , e ′ = N H ′ +2 N He ′ = ηM ⊙ m H (cid:18) X ′ + 12 Y ′ (cid:19) = 0 . ηM ⊙ m H ≈ η . × . (9)Hence the total energy transfer per second is dQdt = ησ ee ′ . × ergsec = ησ ee ′ . × L ⊙ . (10)Rewriting the previous equation in terms of σ − ≡ σ ee ′ / − and η − ≡ η/ − we find dQdt = η − σ − . L ⊙ . (11)The rather modest requirement that dQ/dt , the mirrorphoton luminosity will not exceed the 0 . L ⊙ then limitsthe region of allowed parameters by: η − σ − < . × − . (12)The rational for requiring dQ/dt < . L ⊙ is that theflux of pp solar neutrinoes which directrly reflects the nuclearenergy output is measured and understood at ∼
4% levelBergstr¨om et al. (2016). Originally considerations of energyloss were used e.g. by Raffelt (1996) in a conservative way,requiring only that the new extra luminosity will not ex-ceed the ordinary luminosity. During the past decades mea-surements of all types (pp, Berilium, Boron , etc) of solarneutrinos and the understanding of their apparent deficit vianeutrino mixing, the parameters of which was independentlymeasured in terrestrial experiments, have greatly improvedVissani (2017).
MNRAS , 1–5 (2019)
E. Michaely et al.
Figure 1.
The calculated log ( L ′ tot /L ⊙ ) for the 15X15 different values of η and σ xx ′ . For each pair of ( η, σ xx ′ ) we found the density andtemperature profile of the mirror particles (see next Sec. for details) and calculated the total energy transferred, L ′ tot . The black solidline corresponds to the estimated (subsection 2.1) energy transfer of 1 L ⊙ and the dot-dashed line corresponds for 0 . L ⊙ . Any pair of( η, σ xx ′ ) above this line is excluded. In this section we describe the numerical results and calcu-lation of dQ/dt , the rate of the total heat transferred fromthe ordinary matter to the mirror matter. The amount ofheat transferred to the mirror particles depends on theirdensity profile (number density), ρ ′ ( r ) ( n ′ ( r )) and temper-ature profile, T ′ ( r ). For a given η and σ ee ′ these functionsare unknown apriori, however they must satisfy the four wellknown stellar structure equations: dP ′ ( r ) dr = − GM ( r ) ρ ′ ( r ) r (13) dM ′ ( r ) dr = 4 πr ρ ′ ( r ) (14) dT ′ ( r ) dr = − L ′ ( r ) κ ′ ( r ) ρ ′ ( r )4 πr acT ′ ( r ) (15) dL ′ ( r ) dr = 4 πr ρ ′ ( r ) ǫ ′ ( r ) (16)where κ ′ is the opacity for Thomson scattering, a is theradiation constant and ǫ ′ is the energy source per unit mir-ror mass of the mirror particles. We identify the product ρ ′ ( r ) ǫ ′ ( r ) to be the transfers heat per unit mirror mass.For a specific pair of η and σ ee ′ we postulate an ansatzfor ρ ′ ( r ) = ηρ ( r ) and T ′ ( r ) = 0 . T ( r ), where ρ ( r )and T ( r ) are the density and temperature profiles of theSun taken from MESA, stellar evolution code Paxton et al.(2011). Using this ansatz one can calculate the left hand side (lhs) and right hand side (rhs) of equations (13-16). The ra-tios of the lhs and the rhs, q i , where i runs over the abovefour stellar structure equations, is a measure of the qual-ity of the initial guess. We repeatedly altered the functions ρ ′ ( r ) and T ′ ( r ) in order to minimize | q i − | . We are ableto find the profiles that satisfy the stellar structure equa-tions within the tiny errors so that the computed integratedluminosity satisfies R L ′ n+1 dr − R L ′ n dr R L ′ n dr < × − (17)where n indicated that n-th iteration. In figure 2 we presenta representative example, the black solid line is the Suntemperature profile, T ( r ), while the red dashed line is theansatz, T ′ ( r ) = 0 . T ( r ). After many iterations that min-imize q i , the caclulated profile satisfied eq. (17), we foundthe blue dotted line. The same mechanism is done for thedensity profile, ρ ′ ( r ).Once we find the mirror density, ρ ′ ( r ) and tempera-ture profile, T ′ ( r ) for a pair of η and σ ee ′ we calculate thetotal energy transferred and hence emitted by mirror pho-ton and record it. We calculated dQ/dt ≡ L ′ tot = R L ′ dr for the following parameters: 15 values equally spaced in logof the mirror matter- ordinary mirror cross-section σ xx ′ = (cid:8) − − − (cid:9) and 15 values equally spaced in log of η = (cid:8) − − − (cid:9) . Figure 1 present the results of our cal-culation on the above 15X15 grid. The results are presentedin terms of log ( L ′ tot /L ⊙ ). Our results agree well with ourestimate from subsection 2.1 for 1 L ⊙ (black solid line) and0 . L ⊙ (black dot-dashed line). Any pair of values ( η, σ xx ′ )which is above the 0 . L ⊙ is therefore excluded. MNRAS , 1–5 (2019) olar luminosity bounds on mirror matter Figure 2.
Black solid line, is the Sun temperature profile, T ( r ).Red dashed line is the initial ansatz, T ′ ( r ) = 0 . T ( r ). Blue dot-ted line is the calculate profile that satisfies equation (17). We start by pointing that: • Since our analysis above was essentially perturbative innature it cannot directly apply to the cases where the com-puted mirror photon luminosity considerably exceeds the4% of the solar luminusity, as the ordinary solar parame-ters would need then to be modified as well. Still it is quitesafe to assume that the very large mirror luminosity arisingwhen ησ xx ′ > − will be indicative of some fatal difficul-ties with the observed Sun. • In order to evade the Bullet cluster upper bound onthe dark dark (here mirror-mirror scattering cross-section)we need that only some fraction, say 15% , of the mirrormatter in the haloes will stay unclussterd and that majorityform collisonless stars. The argument for η ∼ − can thenbe ”mirrored ” to suggest a similar admixture of ordinarymatter within the mirror stars. If the latter are still activethen even a small fraction of their nuclear energy productionchanneled into ordinary radiation, analogous to that foundabove, will make theses stars visible and no allow them tobe DM in the first place.To summarize we note that the most relevant difference be-tween our and previous limits steming from DM capturedin the Sun is the fact that we use the solar luminosityrather than more subtle aspects like Helio-seismography.This in turn limits the product of η the solar concentration ofDM and σ xx ′ the dark-ordinary matter cross-sections, ratherthan each of these separately. This is particularly relevantfor the case of (almost) symmetric mirror models which pro-vided the framework of the present analysis. The point isthat in these models both σ xx ′ and η are fixed by the samesingle dimensionless kinetic mixing parameter ǫ of the pho-ton and mirror photon. Specifically ǫ appears in the mirror-ordinary matter Ruthdeford like scattering (3) above. In or-der to evade the apparent difficulties associated with mirrormatter forming a disc overlaping the ordinary Milky Waydisc one needs a minimal σ xx ′ corresponding to a high ep-silon value of ∼ − . The parameter ǫ also controls theexpected fraction of mirror matter η ∼ ǫ which is mixedinto the presolar cloud. This preferred optimal value yieldsthe η ∼ − and σ xx ′ ∼ − . The product ησ xx ′ ∼ − will then exceed the maximum value we found from the 4%limit of the solar luminosity emitted via mirror photons to be ησ xx ′ ∼ − , by a factor of 10 ! Thus our new limitstends to most strongly exclude the above optimal ǫ value andthe large class of almost exactly symmetric mirror modelswhich depend on it. ACKNOWLEDGEMENTS
EM thanks Nathan Roth, Richard Mushotzky and ColemanMiller for stimulating discussions.
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