Early Universe cosmology in the light of the mirror dark matter interpretation of the DAMA/Libra signal

Mirror dark matter provides a simple framework for which to explain the DAMA/Libra annual modulation signal consistently with the null results of the other direct detection experiments. The simplest possibility involves ordinary matter interacting with mirror dark matter via photon-mirror photon kinetic mixing of strength epsilon ~ 10^(-9). We confirm that photon-mirror photon mixing of this magnitude is consistent with constraints from ordinary Big Bang nucleosynthesis as well as the more stringent constraints from cosmic microwave background measurements and large scale structure considerations.

A mirror sector of particles and forces can be well motivated from fundamental considerations in particle physics, since its existence allows for improper Lorentz symmetries, such as space-time parity and time reversal, to be exact unbroken microscopic symmetries [1]. The idea is to introduce a hidden (mirror) sector of particles and forces, exactly duplicating the known particles and forces, except that in the mirror sector the roles of left and right chiral fields are interchanged. We shall denote the mirror particles with a prime ( ′ ). In such a theory, the mirror protons and nuclei are naturally dark, stable and massive, and provide an excellent candidate for dark matter consistent with all observations and experiments [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. For a review, see e.g. ref. [17]. Dark matter from a generic hidden sector is also possible, see e.g. ref. [18] for a recent study.
It has been shown in ref. [19], up-dating earlier studies [20], that the mirror dark matter candidate is capable of explaining the positive dark matter signal obtained in the DAMA/Libra experiment [21], while also being consistent with the null results of the other direct detection experiments. The simplest possibility sees the mirror particles coupling to the ordinary particles via renormalizable photon-mirror photon kinetic mixing [22] (such mixing can also be induced radiatively if heavy particles exist charged under both ordinary and mirror U(1) em [23]): This mixing enables mirror charged particles to couple to ordinary photons with charge ǫqe, where q = −1 for e ′ , q = +1 for p ′ etc. The mirror dark matter interpretation of the DAMA/Libra experiment requires [19] ǫ ∼ 10 −9 , which is consistent with laboratory and astrophysical constraints [24]. The purpose of this note is to study the implications of such mixing for the early Universe. In particular, we will check that this kinetic mixing is consistent with constraints from ordinary Big Bang nucleosynthesis (BBN) as well as more stringent constraints from cosmic microwave background (CMB) and large scale structure (LSS) considerations.
In the mirror dark matter scenario, it is assumed there is a temperature asymmetry (T ′ < T ) between the ordinary and mirror radiation sectors in the early Universe due to some physics at early times (for specific models, see e.g. [25]). This is required in order to explain ordinary BBN, which suggests that T ′ /T < ∼ 0.6. In addition, several analyses [7,8] based on numerical simulations of CMB and LSS suggest T ′ /T < ∼ 0.3. However, if photonmirror photon kinetic mixing exists, it can potentially thermally populate the mirror sector. For example, Carlson and Glashow [26] derived the approximate bound of ǫ < ∼ 3 × 10 −8 from requiring that the mirror sector does not come into thermal equilibrium with the ordinary sector, prior to BBN. The inferred value of ǫ ∼ 10 −9 is consistent with this bound, so that we expect the kinetic mixing to populate the mirror sector, but with T ′ < T . Assuming an effective initial condition T ′ ≪ T , we can estimate the evolution of T ′ /T in the early Universe as a function of ǫ, and thereby check the compatibility of the theory with the BBN and CMB/LSS constraints on T ′ /T . Photon-mirror photon kinetic mixing can populate the mirror sector in the early Universe via the process e + e − → e ′+ e ′− . This leads to the generation of energy density in the mirror sector of: where E is the energy transferred in the process, v møl is the Møller velocity (see e.g. ref. [27]), and n e − ≃ n e + ≃ 3ζ(3) 2π 2 T 3 . It is useful to consider the quantity: ρ ′ /ρ, in order to cancel the time dependence due to the expansion of the Universe [recall ρ = π 2 gT 4 /30]. Using the time temperature relation: with g = 10.75 and M P l ≃ 1.22 × 10 22 MeV, we find that: Let us focus on σv M øl E . This quantity is: where we have neglected Pauli blocking effects. If one makes the simplifying assumption of using Maxwellian statistics instead of Fermi-Dirac statistics then one can show (see appendix) that in the massless electron limit: and Eq.(4) reduces to: where Note that the e ′± will thermalize with γ ′ . However, because most of the e ′± are produced in the low T ′ < ∼ 5 MeV region, mirror weak interactions are too weak to significantly populate the ν ′ e,µ,τ [i.e. one can easily verify a posteriori that the evolution of T ′ /T for the parameter space of interest is such that Thus to a good approximation the radiation content of the mirror sector consists of e ′± , γ ′ leading to g ′ = 11/2 and hence ρ ′ /ρ = (g ′ /g)(T ′4 /T 4 ), with g ′ /g ≈ 22/43. Eq. (7) has the analytic solution: where we have assumed the initial condition T ′ = 0 at T = T i . Let us now include the effects of the electron mass. With non-zero electron mass, the evolution of T ′ /T cannot be solved analytically, but Eq.(4) can be solved numerically. Note that the number density is: and, as we discuss in the appendix, where the cross section is: Numerically solving Eq.(4) with the above inputs (i.e. numerically solving the integrals Eq.(10) and Eq.(11) at each Temperature step), we find that 3 where x f is the final value (T → 0) of x = T ′ /T . In figure 1, we plot the evolution of T ′ /T , for ǫ = 8.5 × 10 −10 .
In deriving this result we have made several simplifying approximations. The most significant of these are the following: a) Using Maxwellian statistics instead of Fermi-Dirac statistics to simplify the estimate of σv M øl E . Using Fermi-Dirac statistics should decrease the interaction rate by around 8% as discussed in the appendix. b) We have neglected Pauli blocking effects. Including Pauli blocking effects will slightly reduce the interaction rate since some of the e ′± states are filled thereby reducing the available phase space. We estimate that the effect of the reduction of the interaction rate due to Pauli blocking will be around < ∼ 10%. c) We have assumed that negligible ν ′ e,µ,τ are produced via mirror weak interactions from the e ′± . Production of ν ′ e,µ,τ will slightly decrease the T ′ /T ratio. The effect of this is equivalent to reducing the interaction rate by around < ∼ 10%. Taking these effects into account, we revise Eq.(13) to: Successful large scale structure studies [7,8]  In conclusion, previous work has shown that the mirror dark matter candidate can explain the DAMA/Libra annual modulation signal consistently with the null results of the other direct detection experiments provided that there exists photon-mirror photon kinetic mixing of strength ǫ ∼ 10 −9 . Here we have examined the implications of this kinetic mixing for early Universe cosmology, where we showed that it is consistent with constraints from ordinary BBN and CMB/LSS data.

Appendix
Here we shall examine the quantity σv M øl E and derive Eq.(6) and Eq.(11) used in our analysis. Following ref. [27], we have: where p 1 and p 2 are the three-momenta and E 1 and E 2 the energies of the colliding particles in the cosmic comoving frame. Recall that E = E 1 + E 2 is the energy transfer per collision. As elaborated in ref. [27], evaluation of these integrals can be facilitated by changing variables to E ± ≡ E 1 ± E 2 and s = 2m 2 e + 2E 1 E 2 − 2p 1 p 2 cos θ. In terms of these variables the volume element becomes and Performing the E − integration, we have: where σF = σv M øl E 1 E 2 and F = 1 2 s(s − 4m 2 e ). Also, as discussed in ref. [27] e where K 2 is the modified Bessel function of order 2. Hence we see that In the m e → 0 limit, where σ = 4πα 2 ǫ 2 3s , and using the dimensionless variable z ≡ E + / √ s, we find: where Also, e −E 1 /T e −E 2 /T d 3 p 1 d 3 p 2 = 4πm 2 e T K 2 (m e /T ) 2 = 64π 2 T 6 in the m e → 0 limit.
Thus we find: Our results for σv M øl E , Eq. (20) [or Eq. (24) for the m e → 0 limit], have assumed Maxwellian distributions for the fermions to simplify the integrals. In the m e → 0 limit, it is possible to evaluate the integrals for the realistic case of Fermi-Dirac distributions. In which case, one finds: where We find numerically that: Thus, we see that the approximation of using Maxwellian statistics overestimates σv M øl E by around 8%.