Light millicharged particles and large scale cosmic magnetic fields
aa r X i v : . [ a s t r o - ph . C O ] A p r LIGHT MILLICHARGED PARTICLES AND LARGE SCALECOSMIC MAGNETIC FIELDS
Alexander D. Dolgov a Novosibirsk State University, Novosibirsk, 630090, RussiaInstitute of Theoretical and Experimental Physics, Moscow, 113259, RussiaDipartimento di Fisica, Universit`a degli Studi di Ferrara, I-44100 Ferrara, Italy
Abstract.
After a brief review of different types of scenarios suggested for the so-lution of the problem of galactic and intergalactic magnetic field generation, themechanism based on the electric current induction by hypothetical millichargedparticles interacting with electrons in cosmic medium is discussed. The proposedmodel successfully describes observational data. The new light millicharged par-ticles can contribute from a small fraction up to 100% to the cosmological darkmatter.
There are strong observational observational evidence for existence of galacticmagnetic fields with the field strength B gal of a few µG and the coherence lengthof several kiloparsecs. There are also some data in favor of weaker intergalacticmagnetic fields, B ∼ − G, but with much larger coherence scale of hundredskiloparseks. For a review on observations see refs. [1]. The origin of such largescale galactic and intergalactic magnetic fields remains a cosmological mysteryfor many years.There are a lot of proposals for mechanisms of large scale magnetic fieldgeneration. They can be roughly divided into two classes: either ones based onconventional physics, or those which invoke different new physics hypotheses.They are reviewed e.g. in papers [2]. Both types of mechanisms suffer fromthe similar problems. The magnitude of the generated field might be quitehigh but the coherence length happened to be much shorter than the necessaryone, or vice versa: the coherence length is large but the magnitude is tiny.The problems are milder in the case of new physics allowed but they stillremain vital. The case of tiny but the galactic scale seed magnetic field tosome extend could be cured by the galactic dynamo amplification. However,the seed fields are usually so weak that the galactic dynamo is not sufficientto amplify them up to the observed magnitude. As for intergalactic fields, thedynamo practically is not efficient there, so the existence of intergalactic fieldsputs strong restrictions on the mechanisms of generation of primordial (seed)magnetic fields.It is tempting to search for a possibility to generate primordial magneticfields during inflation, since the problem of large scale in this case is natu-rally and easily solved by exponential stretching. However, an electromagneticfield cannot be created during inflation because the classical electrodynamicsis conformally invariant and such fields are not produced in conformally flatFriedmann-Robertson-Walker metric [3]. However, quantum triangle anomaly a E-mail: [email protected] reaks conformal invariance and allows for photon production and thus formagnetic field generation [4]. This is a promising mechanism but more detailedinvestigation of renormalization in De Sitter space-time is necessary. Another,more exotic possibility, is an assumption of non-minimal coupling of electro-magnetic field to gravity through a new interaction, e.g. by term ξRA µ A µ [5],where R is the curvature scalar and ξ is a constant. Such an interaction whichnot only breaks conformal but the gauge invariance of electrodynamics as well.As a result the photon would acquire non-zero mass, due to one-loop graviton-photon diagram, of the order m γ ∼ ξ Λ /m P l , where Λ is an ultraviolet cut-offand m P l ≈ GeV is the Planck mass. An existence of the galactic magneticfields implies m γ < / kpc, which in turn demands Λ <
10 eV /ξ . So for a rea-sonable ultraviolet cutoff a tiny ξ is necessary which would drastically diminishthe effect of this new interaction on generation of primordial magnetic field.At later cosmological stages very strong magnetic fields could be createdat first order phase transitions but the coherence length in this case would bemicroscopically small and even with cosmological expansion it would remain byfar too short. This restriction is avoided if e.g. at electroweak phase transitiona condensate of electrically charged W-bosons were formed [6]. In this casethe primeval plasma could be spontaneously magnetized inside macroscopicallylarge domains and such magnetic fields may be the seeds for the observed todaygalactic and intergalactic fields.Next interesting period for generation of primordial magnetic fields might bethe epoch of big bang nucleosynthesis at t ≥ T ≤ l c <
100 pc in terms of the presentday units. In the case of big and inhomogeneous primordial lepton asymmetrythe magnitude of the the created magnetic field may be sufficiently large, suchthat after chaotic field line reconnection (analogous to the Brownian motion)the coherence scale might extend to galactic, but not to intergalactic scales [7].This process proceeds at the expense of a decrease of the field amplitude andgalactic dynamo is very much in order.During the last decade an attention was attracted to generation of cosmicmagnetic field during or around the epoch of recombination, z ∼ F ∼ σ ∼ /m , where σ is the cross-section of elastic γe or γp scattering. The difference between e and p accelerations is even bigger, because a ∼ F/m ∼ /m . So a circularelectric current, proportional to the rotational velocity of the protogalaxy, v rot ,must be induced. The force acting on electrons is given by the expression: ~F ∼ ~vσ eγ n γ ω γ . This force coherently acts on electrons during the collisiontime determined by ep -scattering, which is equal to: τ ep = m e h v e i πα h /v e i n e L e ≃ m / e T / e πα n e L e , (1)where the Coulomb logarithm is L e ∼
10 and h v e i = T e /m e . So we obtain thestandard expression for the conductivity: κ = e n e τ ep / m e ≃ T / e / παL e m / e . The conductivity does not depend on the density of the charge carriers, n e , un-less the latter is so small that the resistance is dominated by neutral particles.Thus the difference between rotational velocities of e and p is ∆ v e = τ ep F/ m e and the current j = en e ∆ v e . Naively estimating B by the Biot-Savart law as B ∼ πjR where R is the galaxy radius, we find that for a typical galaxy with R ∼
10 kpc v rot ∼
100 km/s: B ∼ µ G, very close to the observed value with-out any dynamo. However, this is incorrect since the time to reach stationary(Bio-Savart) limit is longer than the cosmological time.To proceed further we need to use the MHD equation modified by presenceof an external force: ∂ t ~B = ~ ∇× ~F /e + ~ ∇× ( ~v × ~B ) +(4 πκ ) − (∆ ~B − ∂ t ~B ) . (2)In the limit of high conductivity, the second term in the equation, the advectionterm, can lead to dynamo amplification of magnetic seed fields once the valueof the latter is non-zero.Assuming B = 0 at t = 0, we find ~B ( t ) = R t dt ~ ∇× ~F /e . The largest value ofthe magnetic seed is generated around the hydrogen recombination at z rec ∼ ,or t rec ∼ × yr. Earlier the plasma was strongly coupled and the relativemotion of electrons and protons was negligible.The seed field generated at this epoch with coherence length λ ∼ ∼ B λ ∼ Ω λ t rec B F ( t rec ) ≤ − G, (3)where Ω λ = | ~ ∇× ~v | λ ≤ ( δT /T ) /λ . The seeds with the coherence length ofa few kpc and B seed > − G are needed to fit the observations.ow we consider the case when instead of CMB photons new DM particles, X collide with protons and electrons and similarly generate an electric current.The current is proportional to the cross-section of Xe -elastic scattering, σ Xe ,to n X /n e ratio, and to X-particle momentum, p X = m X v rot . Therefore, toproduce stronger than CMB force on electrons, σ Xe should be large. This ispossible if X have long range interaction leading to an enhancement of σ Xe atlow momentum transfer. So we consider millicharged particles with the massfrom a few keV to several MeV.The bounds on the X-particle charge, e ′ = ǫe are summarized in ref. [9]. If m X < m e , then from ortho-positronium invisible decays follows ǫ < . · − .For m X = 1 MeV: ǫ < . × − . For m X = 100 MeV: ǫ < . × − . Weassume that m X >
10 keV to avoid strong limits on e ′ from the stellar evolution.The BBN bounds can be relaxed if the lepton asymmetry is non-zero [10].If X-particles were thermally produced, their abundance could be calculatedaccording to the Zeldovich [11] (a decade later called Lee-Weinberg) equation:Ω X h ≈ . x f g − / ∗ f (cid:18) vσ ann (cid:19) − , (4)where x f ≡ m X /T f = 10 + ln[( g X x / f m X ) / g ∗ f M eV )], g X is the numberof the spin states of X-particle, and g ∗ f is the effective number of particlespecies in the plasma at T = T f . If m X < m e , X-particles can annihilate onlyinto photons with vσ ( X ¯ X → γ ) = πα ′ /m X , where α ′ = e ′ / π = ǫ α .Thus Ω X h ≈ (cid:0) − m/ǫ keV (cid:1) . Hence X’s would be overproduced if ǫ < . · − . Additional annihilation into ¯ νν or dark photons could help,even if the CMB constraint is fulfilled: Ω X h < .
007 [12].If m X > m e , then the channel X ¯ X → e + e − is open and vσ ( X ¯ X → e + e − ) = παα ′ /m X . Correspondingly: Ω X h = 0 . (cid:0) − m/ǫ MeV (cid:1) and e.g. for m X = 10 MeV and ǫ = 3 · − , X-particles can make all DM. NeverthelessΩ X wlll be taken as free parameter.The force from X-particles on electrons is F = σ eX v rel n X m X v rot , where v rel σ eX = 4 παα ′ L/m X v rel , m X n X = 10 Ω X h κ ( z )(1 + z ) keV / cm , and κ ( z )is the dark matter overdensity in galactic halo with respect to the mean cos-mological density at redshift z .Before discussing the generation of B by X -particles let us comment ontheir role in the structure formation. Prior to recombination the characteristicscattering time of light X -particles is shorter than the universe age, τ Xe < t U ,so they are frozen in eγ -liquid. After recombination and till reionization theybehave as the usual WDM. After reionization τ Xe again becomes shorter than t U and the rotating ordinary matter in a protogalaxy would transfer a part ofangular momentum to X -particles and involve it in its turbulent motion. Soat this stage X-particles behave similarly to the usual matter.or an estimate of the magnitude of B generated by light X-particles weuse the obtained above equations but integrate them till reionization, z = 6 or t u = 1 Gyr. We take R = 100 kpc, κ = 100, v rot = 10 km/sec, and impose thelimit Ω x h = 0 .
007 to find B = 10 − G ǫ /m keV . B can rise by factor 100,becoming 10 − G, when the protogalaxy shrinks from 100 kpc to 10 kpc, byfar larger then the minimal necessary strength of the seed.For heavier X, m X > m e , a larger charge is allowed, ǫ > − , and X-particlescan make all dark matter. After reionization, electron scatterings would notforce X-particles into the galaxy rotation and thus the effective integration timecan be longer and magnetic fields as large as 10 − G can be generated.To conclude, we have shown that an existence of millicharged particles withmass in keV - MeV range allows to:1. Explain the origin of galactic and intergalactic magnetic fields.2. Introduce DM with time dependent interaction with normal matter.3. To solve or smooth down the problems of galactic satellites, angular mo-mentum, and cusps in galactic centres inherent to ΛCDM-cosmology.4. The model can be tested in direct experiment.
Acknowledgment
The author acknowledge the support of the Russian Federation GovernmentGrant No. 11.G34.31.0047.[1] P.P. Kronberg, Rept. Prog. Phys. , 325 (1994);J.L. Han, R. Wielebinski, Chinese Journal of Astronomy and Astro-physics, , 293 (2002);F. Govoni, L. Feretti, Int. J. Mod. Phys. D , 1549 (2004);R. Beck, AIP Conf. Proc. , 83 (2009);A. Neronov, I. Vovk, Science , 523 (2010);F. Tavecchio et al. , MNRAS Lett. L70 (2010);K. Dolag et al, Astrophys. J. Lett.
L4 (2011);A. M. Taylor, I. Vovk, A. Neronov, Astron. Astrophys.
A144 (2011);K. Takahashi, M. Mori, K. Ichiki, S. Inoue, Astrophys. J. Lett, Volume , L7 (2012);K. Takahashi, M. Mori, K. Ichiki, S. Inoue, H. Takami, Astrophys. J.Lett, Volume , L42 (2013).[2] D. Grasso, H.R. Rubinstein, Phys. Repts. , 163 (2001);A.D. Dolgov, in
From integrable models to gauge theories , eds. V.G.Gurzadyan et al. , pp. 143-154 [hep-ph/0110293];L.M. Widrow, Rev. Mod. Phys. , 725 (2002);A. Brandenburg, K. Subramanian, Phys. Repts. , 1 (2005) ;R.M. Kulsrud, E.G. Zweibel; Rept. Prog. Phys. , 0046091 (2008);A.D. Dolgov, astro-ph/0306443;. Giovannini, Int. J. Mod. Phys. D , 391 (2004);A. Kandus, K.E. Kunze, C.G. Tsagas, Phys. Repts. , 1 (2011);L.M. Widrow et al. , arXiv:1109.4052.[3] L. Parker, Phys. Rev. Lett. , 562 (1968).[4] A.D. Dolgov, Sov. Phys. JETP, bf 54, 223 (1981) [ZETF. , 417 (1981);A.D. Dolgov, Phys. Rev. D , 2499 (1993).[5] M.S. Turner, L.M. Widrow, Phys. Rev. D , 2743 (1988).[6] A.D. Dolgov, A. Lepidi, G. Piccinelli, JCAP , 031 (2010).[7] A.D. Dolgov, D. Grasso, Phys. Rev. Lett. , 011301 (2002).[8] Z. Berezhiani, A.D. Dolgov, Astropart. Phys. , 59 (2004);S. Matarrese, S. Mollerach, A. Notari, A. Riotto, Phys. Rev. D ,043502 (2005)[9] Z. Berezhiani, A.D. Dolgov, I.I. Tkachev, Eur.Phys.J. C , 2620 (2013).[10] Z. Berezhiani, A. Dolgov and I. Tkachev, JCAP , 010 (2013).[11] Ya. B. Zeldovich, Adv. Astron. Ap. , 42 (1965).[12] S.L. Dubovsky, D.S. Gorbunov and G.I. Rubtsov, JETP Lett. (2004);A.D. Dolgov, S.L. Dubovsky, G.I. Rubtsov, I.I. Tkachev, Phys.Rev. D88