Asymmetric dark matter with a possible Bose-Einstein condensate

We investigate the properties of a Bose gas with a conserved charge as a dark matter candidate, taking into account the restrictions imposed by relic abundance, direct and indirect detection limits, big-bang nucleosynthesis and large scale structure formation constraints. We consider both the WIMP-like scenario of dark matter masses $ \gtrsim $ 1 GeV, and the small mass scenario, with masses $ \lesssim 10^{-11} $ eV. We determine that a Bose-Einstein condensate will be present at sufficiently early times, but only for the small-mass scenario it will remain at the present epoch.


Introduction
Understanding the nature of dark matter (DM) remains one of the most pressing contemporary issues in astroparticle physics and cosmology. To date, all DM properties have been inferred from its gravitational effects [1]; other probes, such as direct [2][3][4][5] and indirect [6][7][8] detection experiments and LHC measurements [9] have produced only limits. These constraints have led to a significant shrinkage of the allowed parameter space in many theoretically favored models [10][11][12], and this has spurred interest in alternative models involving dark sectors of varied complexity [13][14][15][16][17][18].
In this paper we will consider a DM model that involves a new continuous symmetry under which all SM particles are singlets. Given the appropriate conditions and particle content, this symmetry allows for a Bose-Einstein condensate (BEc) to form. This situation differs from most models involving BEc in that the symmetry involved is assumed to be exact, and so the presence of a condensate is not restricted to the case where DM is non-relativistic. We will assume a flat, homogeneous and isotropic universe; effects of fluctuations will be discussed in a future publication.
The simplest model that generates a Bose condensate involves a single complex scalar field χ: the associated U (1) symmetry, χ → e iα χ , (α = const.) (1.1) leads to the required conservation law. Models without an exact conservation law can of course also condense, but only in the non-relativistic regime, where particle number plays the role of a conserved charge; without a conserved charge the condensate will necessarily disappear as the temperature approaches the particle mass. In contrast, the presence or absence of a condensate in models with a conserved charge is determined by the temperature and density of the gas, in particular, relativistic gases of this sort can condense if the density is sufficiently high.
In this paper we will study several aspects of dark matter models that contain an exactly conserved charge. The thermodynamic parameters then will include the corresponding chemical potential µ that is bound by the particle mass |µ| ≤ m be ; a condensate will be present whenever the equality holds 1 . The condition µ = 0 presupposes the presence of a primordial charge whose possible origin we will not discuss in this paper. We will consider two mass regions for the DM: (i) m be ≥ 1 GeV where the behavior in many situations is WIMP-like; and (ii) m be 2 × 10 −11 eV where the gas can exhibit a condensate at the present epoch.
The model we consider has then the Lagrangian L = |∂χ| 2 − m be 2 |χ| 2 − 1 2 λ be |χ| 4 + |χ| 2 |φ| 2 + L sm , (1.2) where φ denotes the SM scalar isodoublet. This is a simple extension of the usual Higgs-portal models that involve a real scalar field. Various cosmological aspects of this type of model have been studied [27-40, 46, 49], with emphasis on the cosmological aspects of the theory and the low mass regime. Here we will be interested in a much wider range of masses, in various aspects of the direct detection of dark matter, and in studying the conditions under which a Bose-Einstein condensate can occur.
In the usual Higgs-portal models [61,62], for a given choice of DM mass, the relic abundance and direct detection constraints impose, respectively, lower and upper limits on the DM self coupling constant, and these limits are consistent only for a relatively small range of masses (55 GeV < m be < 62 GeV or m be > 400 GeV) [63]; in particular, light masses are excluded. The model eq. (1.2) sidesteps some of these difficulties because of the presence of a chemical potential µ: the relic abundance depends on the mass m be , the coupling and µ; the possibility of adjusting the latter relaxes the constraints on the first two parameters (the more severe restrictions found in the simplest Higgs-portal models reappear if one imposes the constraint µ = 0). We will assume that the self-coupling λ be in eq. (1.2) is sufficiently large to ensure that the gas remains in equilibrium and yet small enough to ensure that the theory remains perturbative. Under these circumstances these thermodynamic quantities take the naive expressions found in textbooks [64]. It is also worth noting that as a statistical system the BE gas may or may not be in equilibrium with the SM. This is determined by the strength of the coupling in eq. (1.2), as well as by the environment, specifically, by the rate of expansion of the universe. As long as the gas and the SM are in equilibrium, they will have the same temperature; when the gas and SM are not in equilibrium they can have different temperatures, but even then the gas will be in equilibrium with itself and behave as a regular statistical system.
In most publications the relic abundance is calculated using the Boltzmann equation to determine the DM abundance through the decoupling era and into the late universe. We will follow a different approach based on the Kubo formalism [65,66] that can be used to describe the decoupling of two statistical systems; since the Bose gas remains a statistical system after decoupling such an approach is desirable. For the relic abundance calculation we will use the naive criterion, where decoupling occurs when the interaction rate falls below the Hubble parameter. We do this for simplicity, but also because the presence of a chemical potential allows us to adjust the relic abundance to the experimentally required value, so the full calculation using the kinetics of a Bose gas is not warranted.
We assume throughout that the model is in the perturbative regime and that the BE field does not acquire a vacuum expectation value. If the Higgs potential takes the form λ sm (|φ| − v 2 ) 2 /2, then we assume (i) > − √ λ be λ sm to ensure (tree-level) stability; (ii) (m be /v) 2 > so that χ = 0, φ = 0; and (iii) 4π λ sm , λ be > 0, so that the model remains perturbative. As noted above, we will study the Bose gas in two mass regimes: the WIMP case where m be 1 GeV and the low mass scenario m be 2 × 10 −11 eV. The remaining model parameter, µ, is restricted by |µ| ≤ m be (to lowest order in λ be ).
The rest of the paper is organized as follows: in the next section we discuss the cosmology of a Bose gas to first order 2 in λ be and discuss some aspects of the conditions under which a condensate is present. We next consider relic abundance and the decoupling transition (section 4) and direct detection (section 5) in the WIMP regime. We discuss the low-mass scenario in section 6, including constraints from large scale structure formation and big-bang nucleosynthesis. Section 7 contains parting comments and conclusions while the appendices involve some formulae used in the text.

Cosmology with a Bose gas
As mentioned in the introduction, we will consider the behavior of a Bose gas in an expanding universe, including the possibility that a Bose-Einstein condensate (BEc) may be present at some time. We will assume that the rate of expansion of the universe is sufficiently slow that the gas will be in local thermodynamic equilibrium 3 . To order λ 0 be (defined in eq. (1.2)) the thermodynamics quantities correspond to the well-know expressions for an ideal Bose gas [64].
The O(λ be ) can be obtained using standard perturbative methods; we summarize the results in appendix A. In the calculations below we neglect the O( ) contributions (cf. eq. (A.5)) since they are subdominant for the range of parameters being considered in this section: m be m H and | | λ be (see appendix A).
The occupation numbers for particles and antiparticles are given by where E = p 2 + m 2 and u = |p|/m be . Defining (see eq. (A.7)) the phase transition line is given by A condensate will not form if µ 2 < m be 2 + λ be F; when λ be = 0 this reduces to the well-known result that a condensate is present only if |µ| = m be . The conserved charge associated with the symmetry eq. (1.1) is given by where q (e,c) be are the charge densities in the excited states and in the condensate (if present). Without loss of generality we will assume q (c) be ≥ 0; if there is a condensate then µ > 0. The entropy and energy densities for the Bose gas are given by The O(λ be ) corrections are given in eq. (A.10) and eq. (A.12), and though we will use them in the calculations below, they are not displayed so as not to clutter the above expressions.
The Standard Model energy and entropy densities are approximately given by [68] ρ sm = π 2 30 where g i denotes the number of internal degrees of freedom, and T i the temperature for each particle; we assumed a zero chemical potential for the SM particles.
In the discussion below we repeatedly use the fact that when the SM and Bose gas are in equilibrium the ratio q be /s tot is conserved, where s tot = s be + s sm is the total entropy. When the SM and Bose gas are not in equilibrium the ratios q be /s sm and s be /s sm are separately conserved (in this case q be /s tot is also conserved, but it is not independent of these quantities).

The Bose-Einstein condensate
As noted above, whether the SM and gas are in equilibrium or not, the ratio Y is conserved (though the (e) and (c) contributions in general are not). A condensate will be present whenever the total charge cannot be accommodated in the excited states, that is, (3.2) Now, since s sm > 0 we have the following inequality: Therefore, a condensate will be always present if Y > 0.78. The behavior of Y (e) for various choices of m be and λ be is given in figure 1. For large temperatures 4 and λ be = 0, ν be /s be ∼ 1/T (cf. eq. (A.14)) (since the leading particle and antiparticle contributions to ν be in eq. (2.4) cancel); it follows that Y (e) (λ be = 0) → 0 as T → ∞, in particular, a condensate will always be present at sufficiently high temperatures 5 [70]. This behavior changes when λ be = 0: Y (e) has an m be -dependent minimum 6 , so that a self-interacting BE gas with a sufficiently small Y will never condense.
To clarify this behavior note that T → ∞ corresponds to a → 0, where a denotes the distance scale in the Robertson-Walker metric: a contracting co-moving volume accompanies an increasing temperature. There are then two competing effects on the Bose gas: the reduction of volume favors the formation of the condensate, while the increase in temperature tends to destroy it; the above results indicate that the volume effect dominates. When λ be = 0 a third effects comes into play: the repulsive force generated by the Bose gas self-interactions, which gives rise to the non-monotonic behavior of Y (e) . Because of the exact U (1) symmetry of the dark sector, the presence of this condensate does not require the gas to be non-relativistic (in which case particle number is conserved). We will see later (see section 6) that experimental constraints allow for the condensate to persist to the present day only if m be is in the pico-eV range; for WIMP scenarios (m be 1GeV) the condensate disappears already in the very early universe.

Conditions for a BEc at decoupling
We will show below that for WIMP-like masses (m be 1 GeV) the gas will be non-relativistic at T d ; it then follows that it will also be non-relativistic at present. Then where we used the known value of the SM entropy now, and the fact that for a non-relativistic gas ρ DM = m be q be ; as noted in section 2, the left hand side of eq. (3.4) is conserved below T d . This can be used to determine whether a Bose condensate would have been present at the decoupling temperature. The condition for the presence of a condensate is Next, using eq. (3.4) to eliminate q be (T d ) and eq. (2.6) for the SM entropy, we find Finally, since for a non-relativistic gas m be > T d and g s < 106.75 we find (using 3σ errors) A condensate can occur at decoupling only for light Bose particles. In the non-relativistic limit the O(λ be ) corrections to the above expressions are smaller than the sub-leading tree-level effects; see the eq. (A.13) and surrounding discussion in appendix A.

Conditions for a BEc to exist at present
Before proceeding with the calculation of the cross section relevant for direct detection, we study the possibility that the Bose gas supports a condensate at present. To this end we note first that a non-relativistic Bose gas will have a condensate provided q be (m be T ) −3/2 > ζ 3/2 (2π) −3/2 , see eq. (3.5); denoting the current gas temperature by T now it follows that a condensate will be currently present if We now use the fact that the conservation of s be /s sm allows us to obtain a relation between T now and the decoupling temperature T d . Noting that the gas is non relativistic at T d , and that a condensate at T now implies a condensate was also present at T d (see section 2), we find where we used eq. (2.5) and eq. (2.6). Combining eq. (3.8) and using eq. (A.15) and eq. (3.9), and since m be > T d , this gives It follows that a WIMP-like Bose gas will not exhibit a condensate at the present era 7 (nonetheless, for completeness we include in Appendix B the expressions for the cross section when a condensate does occur). The case of a light Bose gas with a condensate will be considered in section 6.

The BEc transition temperature:
For WIMP-like masses we will show (section 4) that the SM and Bose gas will be in equilibrium down to a decoupling temperature T d , below T d the ratios q be /s sm and s be /s sm will be separately conserved; above T d only q be /s tot is conserved; we will also show that in this case the gas was non-relativistic at T = T d and that the relic abundance constraint reduces to the simple relation q be = 0.4 eV(s sm /m be ) (cf. eq. (3.4)). Combining these results we find that the temperature T BEC at which the condensate forms (the same for the gas and SM since T BEC > T d ) is given by x], and we used the fact that | ln z| m be /(0.4eV) for all cases being considered. As noted previously, the O(λ be ) corrections can be ignored in these calculations.
For example, T BEC ∼ 10 7 GeV if g (T BEC ) ∼ 100 and m be ∼ 1 GeV (though, of course, the number of relativistic degrees of freedom at these high temperatures may be much higher); while T BEC ∼ 1.75 TeV if g (T BEC ) = 106.75 and m be ∼ 10MeV. It is worth noting that for the WIMP-like scenario, the condensate, when it forms, will hold a small fraction of the total energy density of the gas: using eq. (A.14) and eq. (A.15) and the above conservation laws we find, be /ρ be → 1: the charge is mainly in the condensate, but the energy is mainly in the excited states.
For an ultra-light DM (m be ∼ 10 −12 eV) the situation is completely different. We discuss this in section 6.

Relic abundance
In obtaining the relic abundance we will follow an approximate method that will not involve solving the Boltzmann equation. Instead we imagine the Bose gas and the SM to be in equilibrium at some early time and describe their decoupling using the Kubo formalism [65]. As we see below, the BE gas will be non-relativistic, so that the O(λ be ) corrections can be ignored (see appendix A).
The total Hamiltonian for the system is of the form where O sm = |φ| 2 O be = |χ| 2 and is defined in eq. (1.2). Using the same arguments as in [66] (for a different situation), we find that a possible temperature difference (and hence a lack of equilibrium) between the SM and Bose gas obeyṡ where H is the Hubble parameter. This expression is valid when the temperature difference is small, so the width Γ can be evaluated at the (almost) common temperature T . We use this expression to define the temperature T d at which the SM and Bose gas decouple by the condition Explicitly we have [66] (for a different situation), where c sm , c be denote the heat capacities per unit volume, T the common temperature, and The heat capacities are given by where Li denotes the Poly-logarithmic function and z = exp[(µ − m be )/T ].

Evaluation of G
In the presence of a condensate we follow [69] and write χ = [(A 1 + C) + iA 2 ]/ √ 2, A 1,2 denote the fields and C the condensate amplitude. We also assume that decoupling occurs below the electroweak phase transition so that |φ| 2 = (v + H) 2 /2, where v is the SM vacuum expectation value, and H the Higgs field. Substituting in eq. (4.5) we find, after an appropriate renormalization, where In the absence of a condensate we have evaluated at a chemical potential |µ| < m be We evaluate the G n−m using the standard Feynman rules for the real-time formalism of finite-temperature field theory (see for example [67]) and the propagators derived in appendix A. The calculation is straightforward but tedious; to simplify the expressions we use the following shortcuts: and and m H denotes the Higgs mass. Then the G n−m (for arbitrary µ) are given by where the 4 terms represent the processes HH ↔ χχ † , Hχ → Hχ and Hχ † → Hχ † respectively; the factors of 1/2 are due to Bose statistics.
these 4 terms represent the processes HH ↔ Cχ † and HC ↔ Hχ, where C corresponds to a particle in the condensate (mass m be and zero momentum); the factors of 1/2 are due to Bose statistics.
these 2 terms represent the processes H ↔ χχ † .
these 2 terms represent the processes H ↔ Cχ † .
In the non-relativistic limit, where m be , m H T we find 9 where K 1 , ζ 3 and Li denote the usual Bessel, zeta and Poly-logarithmic functions, and we defined Before continuing it is worth pointing out a slight difference between the expression for Γ derived from eq. (4.5) and eq. (4.4) , and the corresponding expression usually found in the literature (see e.g. [68]). The expression eq. (4.4) describes the energy transfer between the SM and the Bose gas, as indicated by the factors of (E ± E ) 2 in eqs. (4.13) to (4.16). As a result Γ in eq. (4.4) has a factor ∼ (mass/T ) 2 compared to the usual expressions, which calculate the change in the DM particle number. As a consequence the decoupling temperature obtained from eq. (4.3) will be somewhat higher than usual; this difference, however, is not significant given that the criterion eq. (4.3) itself is not sharply defined.

The decoupling temperature
For a non-relativistic at T = T d , we have from eq. (3.4) We will use this expression to eliminate µ in eq. (4.3); in doing this we implement the requirement that the Bose gas generates the correct DM relic abundance 10 Using then eq. (4.19) to eliminate µ, the condition Γ = H in eq. (4.3) provides a relation between T d , m be and , which we plot in Fig. 2. We see that, as we assumed, the Bose gas is non-relativistic at T d for a wide range of couplings . The resonance effects are broadened below m H /2 due to the effects of the non-resonant term in G 4−2 that are proportional to θ(r − 2). The rapid change in curvature observed for m be ∼ 100 GeV is produced by G 4−4 , which dominates Γ for large masses. We also see that, for the range of couplings being considered, T d m be /10 so that the gas is non-relativistic at decoupling, as was assumed above.

Direct detection
We first calculate the cross section for the process ηχ → ηχ, where η denotes a neutral scalar coupled to the Bose gas via an interaction The interesting case of nucleon scattering will reduce to the expressions obtained for η in the non-relativistic limit, for an appropriate choice of g, except for a spin multiplicity factor. The transition probability is given by where the initial state consists of an η particle with momentum p and the Bose gas in state I: |i in = a in † p |0; I (where 0 denotes the perturbative vacuum for the η); the final state has an η of momentum q and the Bose gas in a state F: |f out = a out † q |0; F . We require p = q, since we are looking for non-trivial interactions.
Using the standard LSZ reduction formula we find 11 out f |i in = 0; F |Θ p,q | 0; I , where T is the time-ordering operator and we ignored a wave-function renormalization factor (we will be working to lowest non-trivial order where this factor is one). In order to obtain the cross section, we sum over the final gas states (F) and thermally average over initial gas states (I); this gives where . . . β indicates a thermal average at temperature 1/β. W i→f β can be evaluated using standard techniques of the real-time formulation of finite-temperature field theory 12 [67], while the optical theorem relates this quantity to the desired cross section: where E q is the energy of the outgoing η, q be the number density of Bose gas particles, and V denotes the volume of space-time; the prime indicates that the region p q is to be excluded. To lowest order in λ (see eq. (5.1)) we have where the propagators are given in eq. (A.22) and eq. (A.24), and C = 0 implements the absence of a condensate. The evaluation of this expression is straightforward, we find 11 We work to O(g) and assume a non-relativistic gas, so the (λbe) corrections can be ignored. 12 In particular, under T, the complex times in eq. (5.4) are later than the real ones.
where n (±) be are defined in eq. (4.10), and f in eq. (A.22); the second expression is valid in the non-relativistic limit. Substituting this into eq. (5.5) gives where σ 0 is the T = 0 non-relativistic cross section, and we used n = 2 m be T 2π in eq. (5.5). The above expression for W i→f β holds also for non-relativistic nucleons, except for a factor of 2m 2 N , where m N is the nucleon mass. Also, since for the direct-detection reactions the momentum transfer for this process is very small, the coupling g will be given by where v denotes the SM vacuum expectation value, m N the nucleon mass, and g N−H 0.0034 the Higgs-nucleon coupling [12,71,72]. For the range of parameters we consider, the temperature of the Bose gas at present is very small, so that where r is defined in eq. (4.18), and v 10 −3 is the nucleon-dark matter relative velocity and, as above, r = m H /m be . These results can be compared to the most recent XENON [4] and CDMSLite [73] constraints, we present the results in Fig.3. We find that the leading temperature correction in eq. (5.11) is negligible except for very small m be , in this case, however the cross section itself is very small.
The graphs in Fig. 3 represent the strongest constraints on the model parameters. If the parameters are allowed by the direct-detection constraint the model will satisfy the relic abundance requirement for an appropriate choice of µ.

Bose condensate in the small mass region
As noted above, a condensate can occur when the gas has sub-eV masses. In this case, however, there are additional constraints stemming form the possible effects of such light particles on large scale structure (LSS) formation and on big-bang nucleosynthesis (BBN). In this section we will investigate the regions in parameter space allowed by these constraints assuming that the gas is currently condensed; as noted in section 2 this ensures the presence of a condensate in earlier times 13 .
For the small masses needed to ensure the presence of a BEc now (see below) the condition H = Γ used in section 4 (eqs. (4.3) and (4.4)) would require a coupling orders of magnitude above the perturbativity limit 14 (see sect. 1), hence in this case the gas is decoupled from the SM during the BBN and LSS epochs.
LSS formation occurred at redshift z LSS ∼ 3400, when the matter-dominated era began [68]. To ensure that the Bose gas does not interfere with the formation of structure we require it to be non-relativistic at that time; in addition, since we assume the presence of a BEc at present, a BEc was also present at the LSS epoch (sect. 3). Then the conservation of a 3 s be gives, using eq. (A.15), a 3 x −3/2 = constant (a denotes the scale factor in the Robertson-Walker metric); equivalently, Since the gas must be non-relativistic during the LSS epoch, x LSS > 3, so we have x now > 3.5 × 10 7 .  In addition, the requirement that a BEc be present now implies where we used the fact that the gas is currently non-relativistic 15 .
The regions in the m be − T and m be − x planes allowed by eq. (6.2) and eq. (6.3) are given in Figure 4 (here T refers to the gas temperature). It is worth noting that if these conditions occur at present, most of the gas will be in the condensate: using eq. (3.4) and eq. (6.2) the gas fraction in the excited states is given by which is negligible in view of the range of masses being here considered (see figure 4).
We now turn to the BBN constraints. We write the contributions from the gas to the energy density in the form of an effective number of neutrino species ∆N ν : where T γ 0.06 MeV denotes the photon temperature during BBN [74]. Imposing the relicabundance constraint eq. (3.4) we find, using eq. (2.4) and eq. (2.5), where r be − ν be corresponds to the energy outside the condensate.
The limit (see [75]) −0.7 < ∆N ν < 0.4 shows that the first contribution to ∆N ν can be ignored. Also, the LSS constraint m be < 2 × 10 −11 eV (see Fig. 4), implies (m be /T γ ) 10 −62 , so that the second contribution to ∆N ν is also small except if the gas was ultra-relativistic during BBN. In this case so the BBN constraint is significant only in the extreme ultra-relativistic case where x BBN < 10 −62 .  Figure 5. Region in the x BBN − x now plane consistent with the conservation laws, and with the assumption that a BEc is currently present. We used the expressions in appendix A and s sm | now = 2889.2/cm 3 , s sm | BBN = 4.82 × 10 28 /cm 3 and took λ be = 0.5. When λ be = 0 the allowed region collapses to dark line in the figure.
To examine this possibility we first obtain in figure 5 the regions in the x BBN − x now plane consistent with the fact that s be /s sm and q be /s sm are conserved, together with the assumption that a BEc is currently present. The lower bound in this region corresponds to x BBN ≥ 4.9/ √ x now ; using this, and the BBN constraint ∆N ν < 0.4 in eq. (6.7), we obtain The parameter region where the gas exhibits a BEc now and satisfies both the LSS and BBN constraints are determined by eq. (6.8), eq. (6.2) and the allowed x BBN − x now and m be − T now regions in figures 4 and 5, respectively. It is worth noting that when λ be = 0 the allowed region in the x BBN − x now plane reduces to the dark line in figure 5, in which case the BBN constraint does not impose new restrictions.
It remains to see whether a gas satisfying eq. (6.2) can be in equilibrium with the SM (at an epoch earlier than that of BBN). Given the small range for m be and the large values of x now , such equilibrium could have occurred only when the gas was ultra-relativistic, in which environment the presence or absence of a condensate will have no effect. The situation then reduces to that of a standard Higgs-portal model with DM masses in the pico-eV range. Concerning direct detection experiments it is clear that for the very small masses being considered in this section the cross sections will be negligible. We will not consider these points further.

Comments and conclusions
In this paper we considered a complex scalar model of dark matter and studied the possible presence of a Bose condensate which can occur even in the relativistic regime due to the presence of a conserved quantum number, associated with the "dark" U (1) symmetry We showed that a Bose condensate will be present at sufficiently early times provided the abundance is above a λ be and m be -dependent minimum (when m be > m H this minimum will also depend on ); for λ be = 0 a condensate will always form in the early universe. The condensate will persist until the present only if the dark matter mass is in the pico-eV range if the constraints from large scale structure formation are imposed.
The model can meet the relic-density constraint for all masses in the cold dark-matter regime (m be 1 GeV) provided the portal coupling ≤ 0.1 and for a wide range of masses; for larger values of the mass range is somewhat restricted, see Figure 2). The limits derived from direct-detection experiments are much more restrictive allowing only small couplings and/or small masses (figure 3), still the allowed region in parameter space is considerably extended compared to the usual Higgs-portal model [63] because of the presence of a chemical potential that can be adjusted to ensure the correct relic density.
For WIMP-like masses we have shown above that there is no condensate for T < T d but that a condensate forms in the early universe; at very high temperatures the condensate then carries the net charge of the gas, but most of the energy density is carried by the excited states (section 2). In contrast, for very small masses, m be ∼ 10 −12 eV the gas can form a condensate even at present temperatures, while also satisfying the relic abundance requirement. In this case, however, the Bose gas and the SM are never in equilibrium (assuming natural values of the portal coupling ).
Most of the radiative effects in this model are small, being suppressed not only by powers of λ be , but, in the non-relativistic limit, by inverse powers of m be /T . We found two exceptions: first, the above-mentioned condition on the formation of a condensate in the early universe. Second, the constraint in eq. (6.8) derived from BBN.
We have not discussed indirect detection constraints because, for WIMP-like masses they will be identical to those derived for the standard Higgs portal models [76].
A Appendix: Thermodynamics of a Bose gas In this appendix we provide for completeness a summary of the Bose gas thermodynamics. We begin with the Lagrangian Then the Hamiltonian and total conserved charge Q be are given by where π i is the canonical momentum conjugate to A i . To include the possibility of a Bose condensate we replace A 1 → A 1 + C; using then standard techniques of finite-temperature field theory (we use the Matsubara formalism) [77] we find that to O(λ be ) the pressure P be is given by [69,70] When one adds the coupling |φ| 2 |χ| 2 to the Standard Model (see eq. (1.2)) there is an additional contribution where F H is generated by the φ. We have assumed that the φ does not acquire an expectation value, if it does then F H → v 2 + F H /4. This term is subdominant when m H > m be as we will assume for the most part of this paper; note also that stability conditions (see section 1) do not allow to be too large and negative. The total pressure has additional terms, generated by the standard model; these terms, however, do not involve C.
Before proceeding we remark on the type of perturbative expansion we will use. We assume that C is independent of λ be , which requires µ to have a λ be dependence 16 ; we believe this to be reasonable because when λ be = 0 one must have µ = m be , and µ > m be when λ be = 0 (see below); we will then take µ = m be + O(λ be ) when a condensate is present.
The zero-momentum component C is determined by the condition that it minimizes the thermodynamic potential −P be (C, µ, T ): where (F ± are defined in eq. (A.4)) µ 2 = m be 2 + λ be δ ; So there are two cases: 1. δ < F: then there's a single extremum, C = 0, which is a maximum and corresponds to the stable state; there is no BEc.
2. δ > F: then there are two extrema, C = 0 which is now a minimum, and does not correspond to the stable state, and which is a maximum and corresponds to the stable (BEc) configuration.
The transition occurs when δ = F; approximating F F(m be = 0) we find that the critical temperature is that is a known result [69,70]. From P be we find the expressions for the charge density q be and entropy density s be to O(λ be ): • δ < F: ± (n ± be + 1) ln(n ± be + 1) − n ± be ln n ± be , (A.10) where K 2 = 4 dp F + . 16 If, on the other hand µ is assumed to be independent of λbe, then C ∝ 1/ √ λbe diverges as λbe → 0.
• δ = F: • δ > F: The O(λ be ) corrections to q be in the BEc phase are obtained from the O(λ 2 be ) terms in P be , fortunately these are not needed.
In the non-relativistic limit (x 1) the O(λ be ) can be ignored in the phase where there is no condensate. To see this, consider, for example the expression for P be : 13) which shows that the leading O(λ be ) corrections are smaller than the subdominant O(λ 0 be ) contributions. This behavior is reproduced in all thermodynamic quantities in when x 1 and there is no BEc.
We also need the behavior of the thermodynamic quantities at the transition (when δ = F) in the ultra-relativistic (x 1) and non-relativistic (x 1) limits: ρ be = π 2 m be 4 15 x 4 1 + 5λ be 16π 2 + · · · (A.14) x 1 : P be = m be 4 ζ 5/2 (2π) 3/2 x 5/2 1 + where ρ be is the energy density. In particular, for small x, which has a minimum when x min = λ be 12 + 3λ be 8π 2 + · · · (A.17) The above minimum occurs when the O(λ be ) corrections to q be are of the same size as the O(λ 0 be ) contributions, so the validity of the expressions for such values of x should be examined. The leading expression for q be is ∝ d 3 pF − and behaves as x −2 , instead of x −3 as might be expected on dimensional grounds; such a suppression is not present in the O(λ be ) corrections. We argue that a reasonable estimate of the region where perturbation theory is valid is obtained by comparing the O(λ be ) corrections to q be with a quantity that does not exhibit the above suppression, such as d 3 pF + . Using this we obtain d 3 p (2π) 3F + > m be 3 λ be 36 x 4 1 − 3 π 2 x ln x + · · · ⇒ x 1 − (3/π 2 )x ln x > λ be 8.8 (A.18) as the restriction on x for our perturbative expression to be trustworthy. Since x min satisfies this condition, the expression for q be /s be can be trusted near the minimum.
The above Hamiltonian and charge operators can be used to derive the propagator and Feynman rules in the real-time formalism, which we use in our calculations. Defining, as usual 17 Then if, A straightforward (though tedious) calculation yields This has the expected form when µ = 0. For the calculations in this paper we only need the expression when λ be = 0: (1 ± τ 2 )ε(k 0 ∓ µ)δ((k 0 ∓ µ) 2 −Ē 2 k ) , (A. 24) whereĒ k = √ m be 2 + k 2 . This expression is also valid in the presence of a condensate, when µ = m be .

A.2 Higgs propagator and resonant contributions
When the SM and the Bose gas are in thermal equilibrium a similar expression can be derived for the Higgs propagator, however, this approach misses an important resonant contribution which can occur when m H = 2m be ; to include it we replace 25) in D ≷ H , where Γ H denotes the Higgs width.

B Appendix: Cross section in the presence of a condensate
In this case, writing again χ → [(A 1 + C) + iA 2 ]/ √ 2 we find, to lowest order, where W i→f is defined in eq. (5.4), V denotes the volume of space time, and we assumed that the incoming momentum p of the SM particle is different form its outoging momentum q. Now, using eq. (A.22) and eq. (A.24) we find where n (±) be are defined in eq. (4.10), E ± in eq. (5.7), and P = p − q. Then σ = σ (1) + σ (2) , where E q is the energy of the outgoing η, q be the number density of Bose gas particles, and we used q (c) be = m be C 2 for the number density in the condensate; the prime indicates that the region p = q should be excluded.
The evaluation of σ (2) is more involved. We begin with the non-relativistic expression for E ± : (B.7) where = β|p| 2 /(8m be ). This must be evaluated numerically for moderate values of , while for → ∞, it gives eq. (5.11).