Filtered Dark Matter at a First Order Phase Transition

We describe a new mechanism of dark matter production in the early Universe, based on the dynamics of a first order phase transition. We assume that dark matter particles acquire mass during the phase transition, making it energetically unfavourable for them to enter the expanding bubbles of the massive phase. Instead, most of them are reflected off the advancing bubble walls and quickly annihilate away in the massless phase. The bubbles eventually merge as the phase transition is completed, and only the dark matter particles which have entered the bubbles survive to constitute the observed dark matter today. This mechanism can produce dark matter with masses from the GeV scale to above the PeV scale, including a large region of viable parameter space beyond the Griest--Kamionkowski bound. Current and future direct detection and collider experiments can probe much of the viable parameter space.

(Received 23 January 2020; revised 21 April 2020; accepted 16 September 2020; published 9 October 2020) We describe a new mechanism of dark matter production. If dark matter particles acquire mass during a first order phase transition, it is energetically unfavorable for them to enter the expanding bubbles. Instead, most of them are reflected and quickly annihilate away. The bubbles eventually merge as the phase transition completes and only the dark matter particles that have entered the bubbles survive to constitute the observed dark matter today. This mechanism can produce dark matter with masses from the TeV scale to above the PeV scale, surpassing the Griest-Kamionkowski bound. DOI: 10.1103/PhysRevLett.125.151102 Introduction.-A wealth of observational evidence reveals that the Universe is permeated with a mysterious substance known as dark matter (DM) [1]. Very little, however, is known about the particle physics nature of DM or its origin in the early Universe. Historically, the favored scenario for DM production has been thermal relic production [2][3][4]. If a DM particle is thermalized with the standard model (SM) plasma in the early Universe then the cosmological expansion, which causes the plasma to cool adiabatically, will eventually make the DM's interactions with the SM inefficient, driving it out of equilibrium. Consequently, the DM relic abundance is determined when these interactions "freeze out," typically increasing with larger DM mass and decreasing with larger interaction strength. Above m DM ∼ 100 TeV the required interactions violate unitarity [5][6][7]. This places an upper bound on the mass of thermally produced DM, known as the Griest-Kamionkowski (GK) bound.
In this Letter, we propose a new mechanism for generating the DM relic abundance. We propose that DM freeze-out did not result from the gradual cooling of the cosmological plasma, but instead was triggered abruptly by a first order cosmological phase transition (FOPT). During the transition, DM particles acquired a mass and low-momentum particles were "filtered" out of the plasma. We will see that DM filtration provides a viable production mechanism, even for DM with masses above the GK bound.
Reference [12] recently studied a model where DM acquires mass during a strongly supercooled FOPT and its relic abundance was suppressed by the associated entropy injection. By contrast, our interest is in the dynamical interaction of DM particles with bubble walls and its impact on the relic abundance.
The mechanism.-Our proposed mechanism for DM filtration during a FOPT is illustrated in Fig. 1. DM particles χ initially have a small mass m out χ ∼ T and are in thermal equilibrium with SM particles and a new scalar particle ϕ. We imagine that ϕ undergoes the FOPT at temperature T n : its thermal expectation value is initially vanishing, hϕi ¼ 0, but jumps to a nonzero value, hϕi ¼ v in ϕ , during the FOPT. FOPTs proceed through the nucleation and growth of bubbles of the new hϕi ¼ v in ϕ phase [44]. These bubbles expand and merge until the whole Universe has transitioned. At the interface of the old and new phase there is a bubble wall where hϕi smoothly transitions from zero to v in ϕ . We assume that hϕi ≠ 0 generates a large mass for the DM particles, so light DM particles become heavy as they cross the wall into the bubble. Energy conservation implies that a DM particle can only penetrate the bubble wall if its kinetic energy E ≳ m in χ . Lower momentum modes are reflected by the advancing bubble walls. If m in χ ≫ T, then only an exponentially small fraction of the DM particles will have enough kinetic energy to enter the bubbles. As DM particles enter the bubble, their interactions are put abruptly out of equilibrium, preventing their annihilation. DM particles outside the bubble, in contrast, will continue to interact efficiently, so that the reflected particles quickly annihilate away into the thermal bath. Once the broken phase permeates the whole Universe, only the particles that have entered the bubbles remain and constitute the DM observed today.
A toy model.-To derive quantitative results, we introduce a toy model, which is a viable theory of DM in its own right. We augment the SM by a gauge-singlet real scalar field ϕðxÞ and a singlet Dirac spinor field χðxÞ. The Lagrangian defining this theory contains the terms where VðϕÞ is the scalar potential, y χ is a real Yukawa coupling, β is a real Higgs portal coupling, and HðxÞ is the SM Higgs field. We do not assume any particular form for VðϕÞ, only that it gives ϕ a mass m ϕ and causes a FOPT in which ϕ acquires a nonzero vacuum expectation value For simplicity, we assume that the mass of ϕ does not change appreciably during the FOPT. Note that χ enjoys a global U(1) symmetry that ensures its stability.
Before the FOPT, the Yukawa interaction leads to a thermal mass for χ, m out χ ¼ y χ T=4, while afterward it also induces a larger mass m in The Yukawa interaction allows χ to annihilate, chiefly via χχ → ϕϕ, while the thermal mass typically forbids the process χχ ↔ ϕ. In the following, we retain this condition but otherwise approximate m out χ ¼ 0. We treat v in ϕ =T n as a free parameter, since we do not specify the form of VðϕÞ, but we remark that large order parameters may arise from nearly conformal potentials [45][46][47] or models with heavy fermions (such as χ here) [48,49].
The Higgs portal interaction [50][51][52] in Eq. (1) allows the hidden sector to communicate with the SM, through reactions such as ϕϕ ↔ H † H if m ϕ is above the Higgs mass m h , and ϕϕ ↔ ff if not. We ensure that β is large enough to thermalize ϕ and the SM at a common temperature T n during the FOPT. At later times, the Higgs portal interaction allows ϕ particles to decay to SM particles. If m ϕ < m h =2 ≃ 62.5 GeV, the Higgs portal coupling is A relatively large m in χ ensures that χχ ↔ ϕϕ is out of equilibrium inside the bubble. If this were not the case, χ would remain in thermal equilibrium through the FOPT and its relic abundance would later be determined by standard thermal freeze-out. We therefore require the thermally averaged annihilation rate Γ to be smaller than the cosmological expansion rate H inside the bubble. This leads to the condition , with the thermally averaged annihilation cross section hσvi ≃ ð9y 4 χ T n Þ=½64πðm in χ Þ 3 [3] and the would-be equilibrium abundance n in;eq is the effective number of relativistic species, and M pl ≃ 2.43 × 10 18 GeV is the reduced Planck mass. Since m in (2) allows y χ ¼ Oð1Þ and v in ϕ =T n ¼ Oð10Þ; smaller y χ needs larger v in ϕ =T n . Analytic estimates.-We first estimate the DM relic abundance by employing a simplified description of the FOPT dynamics, treating the χ particles as they interact with the wall as if they were free particles. In other words, we assume that the thickness of the bubble wall l w is much smaller than the DM interaction length l int . Because of energy conservation, the mass increase of χ particles crossing the wall implies that only high-momentum particles can enter the bubble, while low-momentum ones will be reflected. After a distance l int these reflected particles will be absorbed back into the thermal bath, so low-momentum χ particles are filtered out of the plasma by the wall. Both reflected and penetrating particles transfer momentum to the bubble wall, leading to friction that limits the speed at which the wall advances v w [54,55].
Using energy and transverse momentum conservation, we find that a massless χ particle that is incident on the wall with momentum p ¼ ðp x ; p y ; p z Þ (in the plasma's rest frame) will only have sufficient energy to enter the bubble if γ w ðp z þ v w jpjÞ > m in χ [56], where γ w ¼ 1= is the wall's Lorentz factor and we have assumed the wall FIG. 1. "Filtered DM": only DM particles χ with kinetic energy E ≳ m in χ can penetrate the bubble; slower particles are reflected. In front of the bubble wall (pink region), DM is kept in thermal equilibrium through χχ ↔ ϕϕ, but this reaction is put abruptly out of equilibrium at the wall where χ obtains a mass (blue region). The field ϕ remains in equilibrium throughout. moves in the negative z direction. Once such a particle enters the bubble, it slows down to travel with a speed v in χ ¼ ½jpj 2 − ðm in χ Þ 2 1=2 =m in χ . We will be interested in nonrelativistic walls v w ≲ 0.1 because of the aforementioned friction effect. Moreover, if the wall moves relativistically, most χ particles enter the bubble.
If a thermal flux of χ particles is incident on the wall, the number density n in χ of χ particles that have entered the bubble is where the step function Θ enforces the kinematic condition above, n in;eq χ was defined below Eq. (2), and 1=v in χ accounts for the reduced speed of particles inside the bubble. The Boltzmann-like exponential factor is crucial in suppressing the abundance of DM inside the bubbles and therefore in setting the relic abundance. In front of the bubble wall, reflected DM annihilates χχ → ϕϕ, and ϕ remains in equilibrium. The associated entropy transfer and heating are negligible if g Ã ¼ Oð100Þ.
Since χχ ↔ ϕϕ is out of equilibrium inside the bubble, the χ andχ particles that enter during the phase transition will survive until today, where they constitute the relic population of DM. The corresponding relic abundance Ω DM is calculated by scaling n in χ þ n in χ with the entropy density s ¼ ð2π 2 =45Þg ÃS T 3 , where g ÃS ¼ g Ã at T n and g ÃS ¼ g ÃS0 ≡ 3.9 today (see also Ref. [57]). After normalizing to the critical density ρ c ¼ 3H 2 0 M 2 pl , we obtain where H 0 ¼ 100h km= sec =Mpc is the Hubble constant and T 0 ≃ 0.235 meV is the temperature of the cosmic microwave background today. In obtaining this estimate, we have neglected the heating of the SM bath by the annihilation of the reflected χ particles in front of the wall and by the eventual decay of ϕ. This is justified because the number of SM degrees of freedom at T n ≳ GeV is much larger than the number of dark sector degrees of freedom. The observed DM relic abundance, Ω obs DM h 2 ≃ 0.12 [58], is obtained if the DM mass increases to m in χ ∼ 30T n inside the bubble for T n ∼ 1 TeV. At higher (lower) phase transition temperatures, the required m in χ =T n becomes larger (smaller), but only logarithmically due to the exponential suppression. Comparing Eq. (4) against the standard thermal freeze-out calculation, we note that our predicted relic abundance only depends on the DM's interaction strength y χ through m in χ =T n ¼ y χ v in χ =T n , and consequently, there is a not a one-to-one mapping from the parameters that set the relic abundance to the parameters probed, for instance, by direct detection experiments.
Inside the bubbles, DM could be produced from freeze-in [12,59,60], but for typical parameters Ω χ h 2 ∼ 10 −6 ðy χ =2Þ 4 e −2ðm in χ =T n −32Þ , making this population negligible. Numerical solution of Boltzmann equation.-To obtain a more accurate estimate of the relic abundance, we numerically solve the Boltzmann equations describing the χ particles near the bubble wall (see Supplemental Material [61]).
Since the scattering and diffusion length scales are small compared to the curvature scale of a typical bubble, we assume that the bubble wall is planar and take the wall to be perpendicular to the z axis. Since the wall experiences a significant drag force from the scattering of χ particles, we assume a constant nonrelativistic (terminal) wall speed v w . We choose v w ¼ 0.01, but have checked that the final relic abundance is not strongly dependent on its precise value. We approximate the mass profile of DM particles across the wall with a smoothed step function, m χ ðzÞ ¼ 1 2 m in χ ½1 þ tanhð3z=l w Þ. Here and in the remainder of the Letter, we work in the wall's rest frame. We use a wall thickness l w ¼ 1=ð4T n Þ, but find that the final relic abundance does not depend strongly on the precise value.
Let f a ðt; x; pÞ be the phase space distribution functions for a ¼ χ,χ, and ϕ particles. We assume that the conserved χ-χ asymmetry is vanishing, thus fχ ¼ f χ , and that ϕ remains in equilibrium throughout the FOPT: f ϕ ¼ f eq ϕ is the Bose-Einstein distribution. This is justified provided that ϕ depletion is fast enough to keep up with ϕ production. Far in front of the wall (z → −∞), f χ ¼ f eq χ follows the Fermi-Dirac distribution. We adopt the ansatz motivated in the Supplemental Material [61]. The distribution f χ in the vicinity of the bubble wall can then be described by the Boltzmann equation The right-hand side includes the collision terms for the processes χχ → ϕϕ, χϕ → χϕ, χχ → χχ, and χχ → χχ. Note that we have integrated over p x and p y . Integrating over p z will then yield the number density at a position z. The first condition enforces an equilibrium phase space distribution for particles that have not yet interacted with the bubble wall, while the second condition is based on the assumption that at a large positive z the other side of the bubble is advancing with similar dynamics. We solve Eqs. (6) and (7) numerically using the method of characteristics, where the two-dimensional partial differential equation is rewritten as an infinite set of uncoupled ordinary differential equations. Each equation corresponds to a possible particle trajectory in the two-dimensional phase space spanned by z and p z , in the absence of collisions. A typical solution for Aðz; p z Þ is shown in Fig. 2, along with some of the aforementioned particle trajectories. Particles incident on the wall begin in equilibrium (upper-left quadrant), so Aðz; p z Þ ≈ 1. Those that started with a momentum larger than m in χ enter the bubble (upper right), with A ≈ 1.2. That is, with an abundance only slightly larger than the strongly Boltzmann-suppressed f eq χ ðz; pÞ. Particles that started with a momentum lower than m in χ are reflected by the wall (midleft). Particles that come from z → ∞ (lower right) receive a boost in momentum as they leave the bubble. These boosted particles and the reflected particles lead to an overdensity, which annihilates into the thermal bath as the particles travel away from the wall (bottom left).
We then integrate over p z deep inside the bubble to find the resulting DM relic abundance and present our results in Fig. 3. We assume m χ ≈ m in χ , implying a negligible change in the χ particle's mass between the FOPT and today. The observed relic abundance is obtained for m χ =T n ≈ (25,32,40) and T n ¼ (1 GeV, 1 TeV, 1 PeV), respectively. These parameters are consistent with the out-of-equilibrium condition (2) provided that y χ < ð0.2; ffiffiffiffiffi ffi 4π p ; ffiffiffiffiffi ffi 4π p Þ, respectively. The exponential sensitivity to m χ =T n ≈ m in χ =T n is clearly visible. Comparing the numerical result with the analytical estimate from Eq. (4), we find good agreement of the parametric dependences on T n and m in χ =T n , and the overall amplitude differs by a factor of ∼5.
Current and future probes.-Filtered DM is amenable to many of the same tests as thermal relic (weakly interacting massive particle) DM. Direct detection of χ particles is mediated, in this toy model, via exchange of ϕ particles and Higgs bosons (h), so the rate is suppressed by the tiny ϕ-h mixing [62]. In Fig. 4, the purple region shows the range of spin-independent χ-nucleon scattering cross sections σ SI χN . We impose the conditions that Ω χ ¼ Ω obs DM , couplings remain perturbative (y χ ; β < ffiffiffiffiffi ffi 4π p ), χ is in equilibrium outside the bubble and out of equilibrium inside the bubble, Eq. (2), and ϕ is in equilibrium throughout the FOPT. At m χ ≪ 100 GeV, the dark sector no longer stays in equilibrium outside the bubble because ϕϕ annihilation is suppressed by the Higgs mass and small SM Yukawa couplings. Around masses of several TeV, the value of β required to keep ϕ in equilibrium grows, making it impossible to obtain the correct Higgs mass from the scalar mass matrix. At even larger m χ , this problem disappears as new ϕ annihilation channels open up. We see that there is a large region of viable parameter space at masses above the Griest-Kamionkowski bound [5,6]. At current and future collider experiments, filtered DM can be tested through precision measurements of the Higgs boson's couplings to other SM particles [68][69][70][71]. These measurements already constrain the ϕ-h mixing for sub-TeV masses [72,73].
Annihilations of χ andχ to SM particles in the Milky Way's DM halo provide another avenue to indirectly detect filtered DM. Decays of the annihilation products may be a source of PeV-scale neutrinos. Detection prospects are, however, hampered by p-wave-suppressed annihilation cross sections.
The FOPT bubble dynamics produce a stochastic background of gravitational waves [74]. The frequency of this radiation is tied to the DM mass scale. However, we expect the signal strength to be suppressed by the small bubble wall speed and a dedicated analysis is required to determine if this signal is within reach of next-generation gravitational wave telescopes, e.g., Laser Interferometer Space Antenna [75]. FIG. 4. The predicted spin-independent DM-nucleon scattering cross section (purple shaded region) in comparison with various experimental exclusions limits (green shaded) [63][64][65][66], projected sensitivities of future experiments (green dashed) [67], and the neutrino floor (yellow shaded). Note that viable models of filtered DM are obtained even at DM masses above the Griest-Kamionkowski bound, m χ ∼ 100 TeV.