The relic density of shadow dark matter candidates

We present the results of relic density calculations for cold dark matter candidates coming from a model of dark energy and dark matter, which is described by an asymptotically free gauge group SU(2)_Z (QZD) with a coupling constant alpha_Z ~ 1 at very low scale of Lambda_Z ~ 10^(-3) eV while alpha_Z ~ weak coupling at high energies. The dark matter candidates of QZD are two fermions in the form of weakly interacting massive particles. Our results show that for masses between 50 and 285 GeV, they can account for either a considerable fraction or the entire dark matter of the Universe.


INTRODUCTION
It is almost universally accepted that the picture of the Universe made up of approximately 4% baryonic matter, 23% dark matter and 73% dark energy represents a realistic cosmological model. However, it is astounding that almost 96% of the energy density of the Universe resides in some as-yet-unknown form. What is "dark matter"? What is "dark energy"?
In Refs. [1,2], a model of dark energy and dark matter was proposed in which a new unbroken gauge group SU(2) Z -the shadow sector -grows strong at a scale ∼ 10 −3 eV. The gauge group SU(2) Z was nicknamed Quantum Zophodynamics, or QZD, in Refs. [1,2], where the subscript "Z" stands for the Greek word Zophos, meaning darkness. The model is described by an SU(2) Z instanton-induced potential of an axion-like particle, a Z , which possesses two degenerate minima. The degeneracy is lifted by a mechanism described in Refs. [2,3], yielding a false vacuum with energy density ∼ (10 −3 eV) 4 and a true vacuum with vanishing energy density. The present Universe is assumed to be trapped in the false vacuum [4], whose energy density mimics the cosmological constant. This is, in a nutshell, the dark energy model proposed in Ref. [2], which also computed various quantities of interest such as the tunneling rate to the true vacuum, etc. A Grand Unified Theory (GUT) involving the SM and SU(2) Z was considered by Ref. [5] (The models presented in Refs. [2,5] were later revisited by Refs. [6].).
As discussed in Ref. [2], the masses of the SU(2) Z triplet shadow fermions are found to be of the order of 100 -200 GeV for the SU(2) Z gauge coupling to grow strong at a scale ∼ 10 −3 eV, needed for the dark energy scenario. This coupling constant starts out at GUT-scale energy with a value comparable to that of the electroweak couplings, remains relatively flat until an energy comparable to the shadow fermion masses is reached, and then starts to grow after the shadow fermions drop out of the Renormalization Group (RG) equations. At that dropout point, the SU(2) Z gauge coupling becomes comparable to the weak SU(2) L coupling at the electroweak scale energy. These features have interesting consequences concerning the possibility of the shadow fermions being candidates for cold dark matter (CDM) in the form of weakly interacting massive particles (WIMP's) 1 .
The main reason is the fact that the annihilation cross sections for two shadow fermions into two SU(2) Z "shadow gluons" are of the order of the weak cross sections, a typical requirement for WIMP's. The estimates that were made in Ref. [2] showed that it was possible for shadow fermions to be candidates for CDM with the right relic density.
In this work, we would like to investigate this scenario in more details and by solving shadow fermions' evolution equations to determine the conditions under which they can be considered to be WIMP cold dark matter candidates. It will be seen that the mass range for the shadow fermions obtained by the requirement of having the right density fits in snugly with that used in the RG equations (i.e., the SU(2) Z gauge coupling grows strong at a scale ∼ 10 −3 eV).
The outline of the paper is as follows. First, we go over the QZD model as far as the issue of dark matter is concerned. Then, we derive the evolution equations for shadow fermions and consequently solve them numerically, to obtain their relic density. Finally, the results of our relic density calculations will be presented and discussed, in comparison with the observational values. The shadow fermions relic density, when computed, would only depend on their masses. Therefore, the parameter space is simply two dimensional.

THE SHADOW SECTOR AND ITS CANDIDATES FOR COLD DARK MATTER
In this work, we only concentrate on the potential candidates for cold dark matter that QZD provides in the form of fermions. However, as discussed in Refs. [1,2], the model offers a mechanism for leptogenesis through the decay of a messenger field, resulting in a net SM lepton surplus.
For clarity, we list the particle content that is useful for our calculations, in particular the transformation of these particles under SU (3) • Two shadow fermions ψ (Z) (L,R),i with i = 1, 2, which transform as (1, 1, 0, 3).
• One singlet complex scalar field φ Z = (1, 1, 0, 1). The imaginary part a Z plays the role of the axion-like particle mentioned in section 1. The real part, σ Z , was used as the inflaton in a model of "low-scale" inflationary universe [9].
We now briefly review the relevant aspects of the shadow sector that would be used in our relic density calculations for shadow fermions.

The QZD Lagrangian
The Lagrangian of G SM ⊗ SU(2) Z is given by [2] where L SM is the SM Lagrangian and In the above Lagrangians, G (Z) µν 's are the field-strength tensors of SU(2) Z gauge bosons, the so-called shadow gluons, and the boldface typeset indicates the SU(2) Z triplet multiplicity. The sum over m is in fact over the number of SM families and the summation over i includes the number of shadow fermions. The coefficients gφ 1 m , gφ 2 m , and K i are complex. The covariant derivative in the Lagrangian can be written in the form whereT's are the generators of SU(2) Z , which ought to be in adjoint representation when acting on shadow fermions, and A (Z) µ 's are the shadow gluon fields. The QZD Lagrangian is invariant under a U(1) (Z) A global symmetry, which yields an instanton-induced axionlike potential driving the present accelerating Universe. The transformations of QZD and

SM particles under this U(1) (Z)
A global symmetry is given in detail in Ref. [2].

Masses and coupling constant
The masses of shadow fermions come from the spontaneous breakdown of U(1) (Z) A . Such a breakdown is made possible through the vacuum expectation value of φ Z . Therefore, in the Yukawa coupling of shadow fermions with φ Z , given in Eq. (2b), when φ Z attains vacuum expectation value, φ Z = v Z , shadow fermions receive masses The scalar messenger fields, on the other hand, are assumed to have zero vacuum expectation values to keep QZD symmetry unbroken. Their masses are non-trivially constrained by the evolution of QZD coupling, as explained in Ref. [2].
The QZD coupling constant, α Z = g 2 Z 4π, is close to the SM couplings at high energies, while it increases to α Z ∼ 1 at Λ Z ∼ 3 × 10 −3 eV. The RG analysis of α Z , conducted in Ref. [2], studies the evolution of α Z from M GUT to Λ Z through a two-loop β function for possible masses of QZD particles.
The RG analysis results indicate a direct correlation between the scale at which α Z (E) starts increasing promptly and the mass of the lighter shadow fermion, m 1 . At energies prior to m 1 , α Z (E) is mostly flat, but upon E ∼ m 1 it begins to grow toward its value at Λ Z , i.e., α Z (Λ Z ) ∼ 1.
Ref. [2] provides α Z (E) values for different conditions, i.e., masses, number of messenger fields, etc. However, a common thread among all analyses is that α Z does not change much from its value at M GUT until E ∼ m 1 , being almost scale independent in that interval.
At energies comparable to the masses of the shadow fermions, which themselves are of the order of he electroweak scale, α Z is comparable to the electroweak SU(2) L gauge coupling. This will partially qualify QZD's shadow fermions as WIMP's and their candidacy for CDM, as already explained.

Shadow fermions as candidates for cold dark matter
The two shadow fermions of QZD particle content meet the criteria for a WIMP, since • They interact very weakly with normal matter, i.e., through heavy scalar fields [2].
• They have cross sections of weak strength: masses in GeV and coupling constant in order of weak coupling [2].
• At least one is stable on cosmological scales: The lighter of the two shadow fermions is stable. The heavier one can decay into SM leptons and the lighter shadow fermion through the messenger scalar field (see Appendix B). However, if the shadow fermion masses are degenerate, both can be stable. Additionally, the shadow fermions can annihilate into shadow gluons or each other (if kinematically allowed).
The messenger fields do not qualify as CDM candidates since they are unstable. The relic densities of shadow fermions can be obtained reliably by solving their evolution equations. Solving the evolution equations will reveal the applicable masses, which would give meaningful relic densities and put the model's candidates for dark matter into the test.

EVOLUTION EQUATIONS FOR SHADOW FERMIONS
The standard Boltzmann equation [10] describing the evolution of the number density n of a particle species ψ, is where H is the Hubble parameter, n eq is the equilibrium density, v is the relative velocity in the annihilation process ψψ → all, and σv denotes the thermal averaging of σv, with σ being the total cross section of the annihilation reaction. The equilibrium density n eq is given by where g is the species internal degrees of freedom and f x, p is the equilibrium distribution function. For particles that may play the role of CDM, the equilibrium number density in the nonrelativistic approximation is where T is the temperature, and m is the mass of the relic. The number density n satisfying Eq. (4) has two behaviors. In early times, n closely follows n eq but later when the temperature drops below m, the mass of the species, n eq starts to decrease exponentially until a"freeze-out" temperature is reached where the annihilation rate is not fast enough to maintain equilibrium. Below this temperature, n deviates substantially from n eq and eventually gives the present day abundance of the species. Equation (4) can be solved numerically in relativistic (hot relic) or nonrelativistic (cold relic) regime. Ref. [11] showed that the validity of Eq. (4) and its solution breaks down if the relic particle is the lightest of a set of particles whose masses are near-degenerate and can contribute to the density of the relic through annihilation or decay processes, the so-called coannihilation case.
For QZD's cold dark matter candidates, both shadow fermions can have present day abundances, if they have similar masses, which blocks the decay channel. For that reason, the evolution equations for the number densities of both species ought to be considered.
The trivial reduction of shadow fermions occurs through their annihilations into QZD gauge bosons and the decay of the heavier one. Parallel to that, shadow fermions can annihilate into each other as well, which is analogous to the coannihilation case of Ref. [11].
To summarize, the reactions entering into Boltzmann equations for densities of shadow fermions are • Annihilation of shadow fermions into shadow gluons: ψ • Annihilation of a pair of one species into a pair of another: • The decay of the heavier one into the lighter one and SM leptons: We assume negligible chemical potential for shadow fermions, which implies symmetry among the number densities for particle and antiparticle of each species. To be inclusive, there can be a particle-antiparticle asymmetry in the shadow sector originating from the decay mechanism of messenger fields. The decay of messenger fields induces a particle-antiparticle asymmetry in SM leptons (see the decay of a messenger boson in Fig. 11). The corresponding asymmetry in the shadow sector is expected to be as small as O 10 −7 and therefore negligible to be considered in our relic density calculations.
The evolution equations for number densities n 1 , n 2 of shadow fermions ψ where Γ 21 is the decay rate of the heavier shadow fermion, i.e., ψ (Z) 2 , σ ij (with i, j = 1, 2, A) refers to the total annihilation cross section for the processes and v ij is the relative velocity of the annihilating particles for each reaction. Also, with a Maxwell-Boltzmann distribution function 2 , n i,eq is given by where T Z is the temperature of QZD matter, m i is the mass of the species and K 2 is the modified Bessel function of second kind. The 1/2 factor on the right hand side of Eqs. (6) is to account for non-identical annihilating shadow fermions.
Equations (6) can be written in a more convenient form by considering the number of particles in a comoving volume which is the ratio of number density to entropy density, with the time derivative in the In the absence of entropy production (i.e., s = S R 3 with S = const.) which results in The evolution equations, then, can be reformulated in the form where Y i,eq = n i,eq /s. Additionally, it is convenient to use the QZD plasma temperature T Z as independent variable, in place of time t. The relation between T (the photon temperature) and T Z is easily found by the entropy conservation [1,2]. The technique is essentially the same as that for finding the neutrino temperature using entropy conservation [10].
For example, at temperatures higher than the mass of the lighter messenger field (i.e., ϕ (Z) 1 ) T > m ϕ 1 , the QZD matter is in thermal equilibrium with the rest of the Universe, i.e., T Z = T. When T falls below the mass of the lighter messenger field, T < m ϕ 1 , the QZD plasma conserves its own entropy separately and maintains its own temperature T Z T. The relation between T and T Z from there on can be found by entropy conservation anytime a particle decouples and transfers its entropy to the relativistic matter.
At present, i.e., after e ± decoupling, T Z = [(43/583)/(11/18)] 1/3 T. Ref. [1,2] discusses the relation between T Z and T in more detail. Let us define where the time derivative of T Z satisfies Considering all this, we can rewrite Eqs. (13) in their final forms Equations (16) where s 0 is the present-day entropy density of the shadow sector and ρ crit = 3H 2 0 /8πG. Finally, with H 0 = 100h km sec −1 Mpc −1 and s 0 = 12π 2 T 0 3 Z /45, Eq. (17) can be written in the from where h is the Hubble constant in units of 100 km sec −1 Mpc −1 . Since ψ (Z) 2 decays, the relevant relic density is that of ψ (Z) 1 . If m 1 = m 2 , however, both shadow fermions can have present day abundances and only in such case, may we speak of two relic densities.

COMPUTATIONAL METHOD
Equations (16) include thermal averages σv 's, Hubble parameter H, and the derivative of entropy density ds/dx i , all of which need to be determined for numerical integration.
The annihilation cross sections and the decay rate Γ 21 can be calculated analytically.
They are derived in Appendixes A and B and are given in closed forms, to leading order.
The thermal averages σv 's were then computed numerically using the compact integral form of Ref. [13]. In Appendix C, the relativistic thermal averages are provided in closed integral forms, expressed in terms of x i .
On the other hand, the Hubble parameter in a radiation-dominated Universe is given by where G is the gravitational constant and ρ is the total energy density of the Universe, written as where g eff (T) is the effective number of relativistic degrees of freedom. Ref. [13] provides do not depend on the choice of T QCD , mainly because the freeze-out temperatures for shadow fermions are always much higher than T QCD , due to their large masses. As we already discussed, the relation between T and T Z can be easily determined by entropy conservation. As a result, the Hubble parameter in evolution equations was evaluated in terms of T Z and consequently x i , consistently.
The entropy density s , in Eqs. (16), is mostly the entropy of the shadow sector. For temperatures T > m ϕ 1 , the QZD matter is in thermal equilibrium with normal matter and s is where g * s = 459/4, and T = T Z . However, for most of the time T < m ϕ 1 and s is the entropy of the shadow sector, which is conserved separately, given by In both cases s is easily evaluated in terms of x i , providing values for ds/dx i of Eqs. (16).
The numerical integration of the density evolution equations, Eqs. (16), was carried out using an implicit trapezoidal scheme 3 . We integrate from x i = 0 to x i = m i /T 0 Z , where T 0 Z = 1.346 K is the present-day temperature of the QZD matter corresponding to T 0 = 2.725 K, the photon temperature of the Universe today.
Equations (16) were integrated for different sets of masses of shadow fermions varying between 30 and 300 GeV. The QZD coupling constant, α Z (E), values at energies Λ Z < E < 10 23 GeV are given for different sets of m 1 and m 2 in Ref. [2]. Within the mass range we perform our relic density calculations, α Z varies so slowly and continuously that it can be obtained for any set of m 1 and m 2 by simple interpolation and extrapolation of the values provided in Ref. [2]. In this work, we have taken α Z dependence on m 1 , m 2 , and E into account in our relic density calculations. Nevertheless, it is worth mentioning that for a fixed m 2 and at a given E, α Z does not vary much as m 1 changes. For example, from which demonstrate how α Z varies for 1 GeV m 1 50 GeV . The α Z variation within such range (and similar m 1 ranges) is even less noticeable in relic density calculations, since we are dealing with α 2 Z in the annihilation cross sections. Our calculations showed that one could safely use an average α 2 Z value over a wide range of m 1 values without any sensible loss of accuracy. For instance, an α 2 Z = 3.49961 × 10 −2 for the above range works just fine.
Ref. [2] carries out RG analysis of QZD's coupling constant considering a messenger field mass scale (mass ofφ (Z) 1 the lighter messenger field) m ϕ 1 = 300 GeV and higher, which points to when the QZD plasma decouples from the rest of the Universe. The decay of ψ (Z) 2 into a pair of SM leptons and ψ (Z) 1 happens through a messenger field (see Fig. 11 of Appendix B). When the mass difference ∆m = m 2 − m 1 is not very large, the decay rate for one of the possible decays can be given in an approximate form where α ϕ 1 = g 2 ϕ 1 /4π, Γ ϕ 1 is the decay width of the messenger field and x = m 2 1 /m 2 2 . As already said, we concentrate on the messenger field being sufficiently heavier than ψ (Z) 2 where the "singularity" in the decay rate is not present, which can be seen from the 4 The thermal contact between the shadow and visible sectors may still be in effect through virtual exchange of a messenger boson for some temperatures below the mass of the lighter messenger field. With m ϕ 1 being sufficiently larger than m 2 , the QZD plasma is ensured to have decoupled from the rest of the Universe before ψ (Z) 2 enters its nonrelativistic epoch, decoupling from an isolated QZD matter. approximate from of Γ 21 , Eq. (26). We shall explain the interesting case of m 2 = m ϕ 1 when we present our results in the next section. It is worth mentioning, nevertheless, that such mass degeneracy poses no computational difficulty due to the presence of the messenger field's decay width Γ ϕ 1 .
On the other hand, α ϕ 1 is constrained for the model to predict the observed baryon asymmetry through an initial lepton asymmetry produced in the decay of messenger fields [2]. That requirement sets α ϕ 1 ≈ 2.9 × 10 −17 , which will consequently correspond to a long lifetime for ψ (Z) 2 (not less than 10 7 sec). For that reason, the decay rate of ψ (Z) 2 does not effectively enter the relic density calculations 5 , where the evolution equations are dominated by the annihilation processes. The remnant ψ (Z) 2 's (after the freeze-out) decay into SM leptons and ψ (Z) 1 's anyway and we end up with no relic for ψ (Z) 2 if the shadow fermion masses are not degenerate.

RESULTS
The relic density of shadow fermions depend on two parameters: their masses, m 1 , and m 2 . The masses affect the annihilation cross sections and consequently the dynamics of the evolution equations. Our relic density calculation results, therefore, are displayed either in terms of masses or mass difference.
Suppose there were only one shadow fermion; in that case, the corresponding evolution equation would be administered by shadow fermion's annihilation process and the expansion of the Universe. Since the annihilation cross section into shadow gluons and its thermal average σv are inversely proportional to the mass squared, a heavier shadow fermion would freeze out earlier than a lighter one, as it could not sustain a rate larger than the Hubble rate for as long. That would allow less time (at temperatures below the mass of the sole shadow fermion) for the Boltzmann factor to diminish the density, which would result in a higher relic density compared to a light shadow fermion's. This can be seen from the behavior shown by the dashed line in Figs. 1, 2, and 3, which describes the density of ψ (Z) 1 or ψ (Z) 2 if they were the sole fermion in the QZD particle content. From those graphs, one sees that a heavy sole shadow fermion would have a higher relic than 5 That means the decay of ψ  Of those mechanisms, the decay of ψ (Z) 2 plays no role in the early dynamics of evolution equations. Briefly, that is because the lifetime of ψ (Z) 2 , which depends on m 1 , m 2 , and m ϕ 1 , turns out either too long or too short to be a factor in the determination of freezeout temperatures. For a well-separated set of m 2 , and m ϕ 1 , the lifetime of ψ (Z) 2 is within 10 7 sec τ 2 10 13 sec when m 1 m 2 , i.e, a nondegenerate case. That roughly corresponds to a temperature 1 keV T 1 eV, which is well after a typical freeze-out for ψ (Z) 2 . That means, the remainder of ψ (Z) 2 will decay into ψ (Z) 1 and SM leptons after the freeze-out, which leaves no present day abundance for ψ (Z) 2 . The decay of an unstable shadow fermion at such low temperature into SM leptons can potentially disturb the cosmic microwave background (CMB). That, as we shall see, will place a bound on the mass of ψ (Z) 2 which determines the density of ψ (Z) 2 at the time of its decay. With a mass degeneracy, i.e., 2 is stable and decay is irrelevant. In that case, since the annihilation channel into another is also closed, we end up with two one-species cases: one for ψ (Z) 1 and one for ψ (Z) 2 . When m 2 = m ϕ 1 , the decay width of the messenger field determines the lifetime of ψ (Z) 2 . As discussed in Ref. [2], the requirement for the lightest messenger field to decouple before decaying yields Γ ϕ 1 ≈ m ϕ 1 α ϕ 1 , which is less than the expansion rate of the Universe, at T = m ϕ 1 . Since α ϕ 1 is of the order ∼ 2 's will decay anyway, there will be no relic for ψ (Z) 2 if m 1 m 2 , which is reflective in Fig. 1, where ψ (Z) 2 's relic densities are displayed in solid lines for different m 2 's as m 1 varies. The relic density of ψ (Z) 2 falls down rapidly when the mass difference between the two shadow fermions is enough to allow the decay before our time and therefore to deplete the phase space from ψ (Z) 2 pairs. The maximum relic density, however, is always at m 1 = m 2 , where the annihilation cross section, σ ij , and the decay rate Γ 21 are vanishing and it is essentially the one-species case.
The situation for ψ (Z) 1 is more complicated. The relic density of ψ (Z) 1 is shown through a solid line in Figs. 2, and 3 for different m 2 's as m 1 varies. For an extremely heavy ψ (Z) 1 , i.e. m 1 = m 2 , ψ (Z) 1 's relic density coincides with the one-species case, as expected. As ∆m deviates from zero ψ (Z) 2 starts to dispense ψ (Z) 1 pairs into the phase space (by annihilation earlier, and decay later) and thus Ω 1 increases. Prior to freeze-out, this positive contribution comes from the pair annihilation of ψ (Z) 2 into ψ (Z) 1 , which will face a growing competition from ψ (Z) 1 's annihilation channel into shadow gluons, as m 1 declines. Since the annihilation cross section into shadow gluons grows for small masses, it will start to contend the rate of the extra ψ (Z) 1 pairs coming from ψ (Z) 2 's annihilation. For that reason, as m 1 decreases, the annihilation channel into shadow gluons depletes the phase space from ψ (Z) 1 pairs more effectively and therefore ψ (Z) 1 's density before the freeze-out, which consequently diminishes its relic Ω 1 . After the freeze-out, the remnant of ψ (Z) 2 will decay into ψ For a nondegenerate mass case, ψ (Z) 1 's relic density is what remains of shadow fermions. It is only at m 1 = m 2 that the relic consists of both shadow fermions (equally so). To be inclusive of the degenerate case, the total relic density of shadow fermions Ω T = Ω 1 + Ω 2 is presented in Fig. 4 against both masses and in Figs. 5, 6 against m 1 at fixed m 2 's, where the one-species case is also presented. The gray areas in Figs. 5, and 6 indicate the current bounds on the dark matter density from WMAP3 and all data sets [15].
It can be seen in Figs. 5, and 6 that the total relic density Ω T increases as m 1 does, attaining a sharp maximum for the degenerate case, as if there were two "one-species" shadow fermions. On the other hand, Ω T also increases with m 2 , which means for staying in the cosmologically allowed region a larger and larger mass difference would be needed. This can also be seen by looking at Fig. 8, in which the total relic density is displayed versus ∆m and the bounds are shown with two white dashed lines. We conclude that for m 2 < 50 GeV, the total relic density is not enough to account for the total dark matter, even though the shadow fermions would still be relic particles taking on a fraction of the dark matter in the Universe.
On the other hand, for m 2 320 GeV, the total density of shadow fermions go beyond the upper bound and give unacceptable values even if we extend the mass difference to an extreme where m 1 = 1 GeV. That is shown in Fig. 7, where the total relic density at m 2 = 318 GeV is only viable for a large mass difference of about 317 GeV and at m 2 = 400 GeV is no longer relevant. By going to such an extreme mass difference, we place a naive bound on the mass of the heavier shadow fermion, i.e., m 2 ≅ 320 GeV, above which the total relic density is no longer viable.
There are, however, more restrictive bounds on m 2 coming from the decay of ψ (Z) 2 into SM leptons at low temperature, and its potential disturbance of the CMB of the Universe.
We demand that 1. The density of ψ (Z) 2 at the time of decay could not exceed that of the SM particles.
2. The CMB density disturbance caused by the late decay of ψ (Z) 2 would not violate the CMB fluctuation, which has been observed to be at 10 −5 level [15].  2 masses. The gray band represents the allowed density from WMAP3 and all data sets [15]. Figure 9 illustrates these two conditions in graphs versus the mass of ψ (Z) 2 . In Fig. 9 a, ρ 2 ρ SM , i.e., the density of ψ (Z) 2 to the density of the SM matter -right before the decay -is plotted, which shows that the density of ψ (Z) 2 remains less than that of the SM particles for m 2 285 GeV. The possible CMB density disturbance, δρ γ ρ γ , that the late decay of ψ (Z) 2 can create is shown in Fig. 9 b. The CMB density disturbance goes above the 10 −5 order for ψ (Z) 2 s heavier than 245 GeV. The two above conditions, therefore, place a strong bound of 245 GeV on ψ (Z) 2 's mass. As we discussed, the lifetime of ψ (Z) 2 could be very short if ψ (Z) 2 and the messenger field were degenerate in mass. In that case, the total relic density of shadow fermions is simply that of the one-species case and it yields the right density for masses between 190 and 210 GeV.
For 50 GeV m 2 245 GeV, the total relic density of shadow fermions can account for the amount of the dark matter in the Universe, depending on the mass difference. The total relic density lies within the observational bounds with small and even zero mass difference for light ψ (Z) 2 's and with large mass differences when ψ (Z) 2 is heavy. dashed lines indicate the bounds from WMAP3 and all data sets [15].

SUMMARY
We solved evolution equations for number densities of shadow fermions and obtained their total present-day density. The heavier shadow fermion turned out to be long lived if its mass differs from that of the messenger field. In that case, our results revealed an upper bound on the mass of the heavier shadow fermion, i.e., m 2 ≈ 245 GeV, above which its late decay can potentially disturb the CMB density of the Universe beyond the measured fluctuation level of 10 −5 .
For lighter shadow fermions, the total relic density can account for the entire dark matter of the Universe depending on the mass combination of shadow fermions. When the total density falls short of the observationally suggested density, it still, for most of masses, provides significant fraction of the dark matter of the Universe.
Our results showed that if the heavier shadow fermion's mass is large, considerable mass differences would be needed to comply with experimental bounds. On the other hand, if the heavier shadow fermion's mass is small, little or even no mass differences suffice to give the right relic density. In that sense, degenerate and near-degenerate mass cases become relevant at low mass scales, but not for less than 50 GeV.
A very short lifetime is expected for the heavier shadow fermion if its mass is the same as that of the messenger field. In that case, the calculations reduce to a one-species case.
Our results suggest that a sole shadow fermion must have a mass of about 190 -210 GeV to account for the whole dark matter of the Universe. Last but not least, possible detections of the shadow fermion CDM candidates are briefly discussed in Ref. [2]. Needless to say, more work along this line is warranted for displayed in Fig. 10, where the former process happens through three diagrams in t, u, and s channels and the latter in s channel. In those diagrams, p, p ′ , k, k ′ , p i , p ′ i , p j , p ′ j are momenta, l, l ′ , n, n ′ and a, b are the QZD colors of shadow fermions and shadow gluons, s, s ′ , s i , s ′ i , s j , s ′ j and λ, λ ′ are the spins of fermions and final polarizations of shadow gluons, and q t , q u , q s , q are momentum transfers. ones summed over, which corresponds to the averaged squared amplitude

APPENDIX B: THE HEAVIER SHADOW FERMION'S DECAY
The decay of ψ (Z) 2 → ll ′ ψ (Z) 1 is possible through the lighter messenger fieldφ (Z) 1 (either real or virtual, depending on masses) and can yield any pair of SM leptons. Due to considerably small leptonic masses, when compared to shadow fermions', we carry out the decay rate calculation in the limit of massless SM leptons. In that sense, the decay rate for ψ (Z) 2 throughφ (Z) 1 with mass m ϕ 1 and a Yukawa coupling g ϕ 1 , representing any of g iφ 1 m , can be computed. The process, to leading order, occurs through the diagram of Fig. 11.
The covariant amplitude M of the diagram is where k = p 2 − p, Γ ϕ 1 is the decay width of the messenger field and momenta p 1 , p 2 refer to those of shadow fermions, while p, p ′ are the momenta of the SM leptons. Similar to previous cases, we are looking for an unpolarized decay rate with the initial degrees of freedom averaged over and the final ones summed over, which corresponds to the averaged squared amplitude There is not much of γ-matrix algebra involved in computing |M| 2 , which easily gives Finally, the decay rate, in the rest frame of ψ (Z) 2 , can be found after the usual three-body decay kinematical considerations, which yields the integral form