Comment on ‘‘Is Dark Matter with Long-RangeInteractions a Solution to All Small-Scale Problems of� Cold Dark Matter Cosmology?’’

In a recent Letter [1], van den Aarssen et al. suggesteda general scenario for dark matter, where both the dark-matter particle � and ordinary neutrinos � interact with anMeV-mass vector boson V via L � g� ���

��V� þg� ���

��V� with g� and g� being the corresponding cou-

pling constants. Given a vector-boson mass 0:05 MeV &mV & 1 MeV, one needs 10�5 & g� & 0:1 to solve all thesmall-scale structure problems in the scenario of cold darkmatter [1].

Recently, Laha et al. [2] found that the �-V interactionmight lead to too large decay rates ofK� ! �� þ ��� þ V

and W� ! l� þ ��l þ V, indicating that the scenario pro-posed in Ref. [1] is severely constrained. However, suchexperimental bounds can be evaded if the longitudinalpolarization state of V is sterile or if V is coupled to sterileneutrinos rather than ordinary ones [1,2].

Now, we show that the constraints on g� and mV frombig bang nucleosynthesis (BBN) are very restrictive. In theearly Universe, V can be thermalized via the inverse decay�þ �� ! V and pair annihilation �þ �� ! V þ V and con-tribute to the energy density. If the inverse-decay rateexceeds the expansion rate H around the temperatureT ¼ 1 MeV, we obtain g� > 1:5� 10�10 MeV=mV formV &1MeV. For mV �1MeV, pair annihilation is moreefficient than inverse decay to thermalize V. Requiring theannihilation rate �pair / g4�T > H, one obtains g� > 3:4�10�5. For mV > 1 MeV, however, even if V is in thermalequilibrium, its number density will be suppressed by aBoltzmann factor. To derive the BBN bound, we first solvethe nonintegrated Boltzmann equation for the distributionfunction of V for given g� and mV , where only the decayand inverse-decay processes are included in the computa-tions [3–7]. Then, we calculate its energy density, andrequire the extra number of neutrino species �N� < 1 atT ¼ 1 MeV [8]. Thus, we can exclude a large region ofthe parameter space, as shown in Fig. 1. The contributionfrom V in thermal equilibrium reaches its maximum�N� � 1:71 in the relativistic limit.

Note that we have assumed �N� to be constant, but formV > 1 MeV it actually decreases during the BBN era, soour constraint should be somehow relaxed in the large-mass region. Since only the transverse polarizations of Vare involved in inverse decay in the limit of zero neutrinomasses, the BBN constraint does not depend on whetherthe longitudinal polarization is thermalized or not. If neu-trinos are Dirac particles, the right-handed neutrinos �R

can be in thermal equilibrium as well. Both V and �R

contribute to the energy density, so one or more speciesof �R is obviously ruled out. In addition, if V is coupled tosterile neutrinos, which are supposed to be thermalized, theBBN bound on g� and mV becomes more stringent.

We are grateful to Torsten Bringmann for valuablecommunication. Financial support from the SwedishResearch Council and the Goran Gustafsson Foundationis acknowledged.

Bjorn Ahlgren, Tommy Ohlsson, and Shun ZhouDepartment of Theoretical PhysicsKTH Royal Institute of Technology106 91 Stockholm, Sweden

Received 24 April 2013; revised manuscript received25 July 2013; published 6 November 2013DOI: 10.1103/PhysRevLett.111.199001PACS numbers: 95.35.+d, 98.35.Gi, 98.62.Gq, 98.80.Es

[1] L. G. van den Aarssen, T. Bringmann, and C. Pfrommer,Phys. Rev. Lett. 109, 231301 (2012).

[2] R. Laha, B. Dasgupta, and J. F. Beacom, arXiv:1304.3460.[3] E.W. Kolb and M. S. Turner, Front. Phys. 69, 1 (1990).[4] M. Kawasaki, G. Steigman, and H.-S. Kang, Nucl. Phys.

B403, 671 (1993).[5] A. Basbøll and S. Hannestad, J. Cosmol. Astropart. Phys.

01 (2007) 003.[6] J. Garayoa, S. Pastor, T. Pinto, N. Rius, and O. Vives,

J. Cosmol. Astropart. Phys. 09 (2009) 035.[7] F. Hahn-Woernle, M. Plumacher, and Y.Y.Y. Wong,

J. Cosmol. Astropart. Phys. 08 (2009) 028.[8] G. Mangano and P.D. Serpico, Phys. Lett. B 701, 296

(2011).[9] J. Hamann, S. Hannestad, G. G. Raffelt, and Y.Y.Y. Wong,

J. Cosmol. Astropart. Phys. 09 (2011) 034.

FIG. 1. Constraints on g� and mV from K decays (gray, solidline), W decays (gray, dashed line), and BBN. The two formerare reproduced from Ref. [2], and the sample region in Fig. 3of Ref. [1] is also presented (light gray). The hashed regionbounded by the thick solid curve is excluded by �N� < 1 at95% C.L. [8]. The excluded region will shrink for �N� < 1:5(see, e.g., Ref. [9]), and even further for the ‘‘maximally con-servative’’ limit �N� & 1:58 for mV > 0:05 MeV.

PRL 111, 199001 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

8 NOVEMBER 2013

0031-9007=13=111(19)=199001(1) 199001-1 � 2013 American Physical Society