Asymmetric dark matter with a spontaneously broken $U(1)'$: self-interaction and gravitational waves

Motivated by the collisionless cold dark matter small scale structure problem, we propose an asymmetric dark matter model where dark matter particle interact with each other via a massive dark gauge boson. This model easily avoid the strong limits from cosmic microwave background (CMB) observation, and have a large parameter space to be consistent with small scale structure data. We focus on a special scenario where portals between dark sector and visible sector are too weak to be detected by traditional methods. We find that this scenario can increase the effective number of neutrinos ($N_{\text{eff}}$).In addition, the spontaneous $U(1)'$ symmetry breaking process, which makes dark gauge boson massive, can generate stochastic gravitational waves with peak frequency around $10^{-6} - 10^{-7} \text{ Hz}$.


I INTRODUCTION
There have been plenty of evidence for the existence of dark matter (DM) [1,2], but the nature of dark matter still remains to be revealed.Collisionless cold DM is consistent with the large scale structure of the Universe [3,4].However, N-body simulations of collisionless cold DM show some discrepancies between predictions and observations on the scale smaller than O(Mpc).Those discrepancies include core-cusp problem [5][6][7], diversity problem [8], missing satellites problem [9,10], and too-big-to-fail (TBTF) problem [11,12].Including baryon effects in the simulation helps to alleviate some tension [13,14], but it is still unclear whether baryon effects can solve all the small-scale problems.
Introducing a light mediator which couples to DM seems to be the easiest way to generate the velocity dependent inter-DM cross-section [32,.Such a scenario is favored in many aspects.For example, DM relic density can be realized via the so-called "secluded freeze-out" process [63], which means DM annihilate to light mediators instead of visible SM particles.And, because DM relic density and its coupling with SM are unbound, it is easier for such DM models to escape limits from direct detection or collider experiments [64][65][66][67][68][69][70].However, other studies pointed out that such a "self-interacting DM with light mediator" scenario is strongly constrained by Big Bang nucleosynthesis (BBN), cosmic microwave background (CMB), and indirect search results [71-76, 79, 80, 151, 152].This is because the Sommerfeld enhancement induced by the light mediator rapidly increase as DM velocity decreases in the expanding universe [81][82][83][84][85][86][87], and thus the energy injection from DM annihilation will affect observables (like BBN or CMB) even after DM freeze-out.Especially, the s-wave annihilation case (e.g.DM annihilate to dark gauge boson pair) has been fully excluded by CMB data [88].
A simple method to evade those constraints on DM annihilation is to consider the asymmetric dark matter (ADM).In the ADM scenario, DM is not neutral and self-conjugate, but instead DM is conjugated to anti-DM and the observed DM relic density is determined by the asymmetry between DM and anti-DM.See [89][90][91] for recent review.When the thermal bath temperature is much lower than the DM mass, the abundance of anti-DM has been reduced to negligible level.So, the annihilation between DM and anti-DM is much less constrained compared with the symmetric case [92,93].Another advantage of ADM model is that it helps to explain the "Ω DM 5Ω B " coincidence.See  for related studies.
In this work we study the DM self-interaction in a concise ADM model framework.Combing ADM and DM self-interaction is not a new idea, see e.g.[126][127][128][129][130][131].Compared with previous work, we only consider one flavor of DM (to be labeled as χ) which is charged under a dark U (1) .In addition, we introduce two dark Higgs bosons (to be labeled as S 1 and S 2 ) charged under the same dark U (1) .S 1 helps to generate the asymmetry in the dark sector and become dark radiation in the end, and S 2 is used to break U (1) and thus prohibit the long-range interaction between DMs.The reason for us to introduce two dark Higgs is that we do not want the troublesome Majorana dark matter mass to be induced by U (1) symmetry breaking.We will clarify this point in the next section.To simplify our analysis, we will consider a nearly independent dark sector, which means that the portal between dark sector and visible sector is too small to make these two sectors into thermal equilibrium.The portal between two sectors might be too feeble to be searched for via traditional methods like direct detection or collider experiment.However, the U (1) phase transition in the dark sector provides a possible method to detect the dark sector by gravitational waves (GWs), provided the phase transition is first order.In addition, dark radiation changes the value of effective number of neutrinos (N eff ) , which also make this model detectable in the near future.
This paper is organized as following.In the next section, we introduce the model framework we want to study.In section III we explain how to generate the asymmetry in the dark sector.We will also discuss the sequential thermal history and related constraints.Section III is dedicated to the DM self-interacting and its consistency with data.In section IV we discuss the possibility to detect this model via gravitational waves.We conclude this work in section V.

II MODEL FRAMEWORK
In this section we introduce the framework of our model, and specify the scenario we want to study.

A Model introduction
We consider the SM model extended by a dark U (1) gauged sector.Similar model framework see [111,112,130,132].The Lagrangian can be schematically expressed as: L Dark is the Lagrangian of a U (1) gauged dark sector.Dark sector includes a Dirac fermion χ (dark matter candidate charged under U (1) ), dark Higgs S 1 (has the same U (1) charge as χ), and dark Higgs S 2 (used to break U (1) later).The expression of L Dark is: Here D µ ≡ ∂ µ + ig Q i A µ (i = χ, S 1 , S 2 ) is the covariant derivative, with g and A µ being the dark gauge coupling and dark gauge boson respectively.U (1) charge {Q χ , Q S 1 , Q S 2 } are simply fixed to {+1, +1, +2}.F µν ≡ ∂ µ A ν − ∂ ν A µ is the field strength of dark gauge boson.And m χ is the mass of dark matter given by hand.Dark scalar potential V (S 1 , S 2 ) is: S 2 needs to obtain a vacuum expectation value (VEV) after dark phase transition, and thus we insert a minus mass square term −µ 2 2 for it.All possible triple and quartic dark Higgs interactions are given.
L Portal is the sector that connect visible sector and dark sector, including Higgs portal, Abelian gauge boson kinetic mixing, and right handed neutrino (RHN) portal.The general expression of L Portal is: Here λ S i H and are the coupling of Higgs portal and kinetic mixing parameter respectively.Two Majorana RHN, N 1 and N 2 , are introduced to generate the asymmetry in dark or visible sector, with the help of complex phases of y i and y i .L and H are the SM lepton doublet and Higgs doublet.Now we explain the reason to introduce two dark Higgs S 1 and S 2 .Assuming that there is no S 2 and U (1) is broken by VEV S 1 , then, by integrating out N 1 , a Majorana DM mass ( ∼ ) will be induced.This Majorana mass term makes DM oscillate to anti-DM in the late universe.Thus the asymmetry in the dark sector will be partly erased, and our model will be more limited [133,134].To make DM stable during the universe lifetime, M N 1 needs to be even higher than Planck scale.So, to forbid the annoying DM-anti-DM oscillation, in this work we introduce another dark Higgs S 2 to break U (1) and keep S 1 = 0 all the way.

B Our scenario
The model we introduced above is nearly the minimal model that can generate matter asymmetry and induce velocity dependent DM self-interaction.However, even for such a nearly minimal model, there are still a dozen parameters to be fixed.Diverse and complex phenomena can occur in different parameter spaces, which is difficult to be covered in a single paper.Thus, in this paper we choose a simplified scenario to analyze, instead of studying the entire allowed parameter space.
The first simplification we will perform is neglecting y i Ni LH † , which is used to generate visible matter asymmetry.The inclusion of y i Ni LH † inevitably entangle asymmetries in dark sector and visible sector [112], and force us to consider the limits from neutrino data [130].So, in order to focus on phenomena in the dark sector, we are temporarily agnostic to baryon asymmetry problem and neglect y i Ni LH † .
Secondly, we require all the other portals' couplings, i.e. λ S 1 H , λ S 2 H , and , to be small enough to avoid current limits from terrestrial experiments.Furthermore, we also require that these portals are too weak to keep dark sector and visible sector in the thermal equilibrium from reheating to current time.These requirements are made for simplicity.However, it is also important to study the detectability of this extreme scenario.As we will show later, stochastic GWs and the change of N eff are possible detection methods.

III THERMAL HISTORY OF THE DARK SECTOR AND ITS PARAMETER BOUNDS
Before the thermal history analysis, in Tab.I we present all the particles in the dark sector.Their mass range and the role they played are also given.The mass of dark matter (m χ ) and dark mediator (m A ) are chosen to be consistent with the small scale data.To generate asymmetry in the dark sector, the decay of N 1 needs to be out-of-equilibrium, and thus the mass of N 1 should be much larger than its decay products.The mass of s 2 (s 2 is the scalar component of S 2 after U (1) breaking) is chosen to be smaller than m A .As we will explain later, this is necessary if the U (1) symmetry breaking is a first order phase transition.Finally, the entropy in the dark sector should go to some nearly massless particles long after DM-anti-DM annihilation, otherwise there will be overclosure problem [121].So we require S 1 to be very light and serve as dark radiation.name mass range role Particle content in the dark sector, with their mass range and role given.
Furthermore, we define the ratio between dark sector temperature T and visible sector temperature T : The value of ξ will be different in different period.In this work we assume the dark sector and visible sector thermally decoupled very early, then these two sectors evolve independently.The temperature ratio ξ at the time when dark sector temperature T is lower than M N 2 and higher than M N 1 , is labeled by ξ ini , and we take it as an input parameter.
Co-moving entropy densities in each sector are conserved respectively.So the temperature ratio in different period will be rescaled by the effective numbers of relativistic degree of freedom (d.o.f.) in each sectors (to be labeled as g and g ) 1 at that time: In this work we assume g ,ini to be the SM value 106.75 [135].For the dark sector, g ,ini comes from N 1 , χ, S 1 , S 2 , and A .And so: Given two initial values g ,ini and g ,ini , temperature ratio ξ at a later time can be determined.
During the radiation dominant period, energy and entropy densities are given by: Here we define the effective d.o.f. for energy and entropy, g eff (T ) and h eff (T ), for later convenience.

A The generation of dark sector asymmetry
In this subsection we introduce the generation of Y ∆χ ≡ Y χ − Y χ.Here Y is the particle yield which equals to particle number density divided by entropy density.Before the U (1) symmetry breaking, U (1) charge is conserved and thus        Asymmetric yield Y ∆χ can be expressed as: Here Y N 1 is the yield of N 1 before it decays.Because N 1 is in the equilibrium with dark thermal bath initially, so the initial yield of N 1 is: is the CP asymmetry generated by N 1 decay: The expression of can be simplified when In this case, is approximately given by: η is the efficiency factor that reduce the final generated asymmetry.In the so-called "weak washout" case where the decay width of N 1 is smaller than the Hubble expansion rate (Γ N 1 < H(T = M N 1 )), the value of η can be close to 1.To simplify our analysis we will only consider "weak washout" case, and it leads to a constraint on the parameter space: Here M Pl 1.22 × 10 19 GeV is the Planck mass.So there is a large parameter space to satisfy the "weak washout" requirement.
For convenience, we define CP phase angle θ by: In Fig.
( 2) we show Y ∆χ as functions of CP phase angle θ with = 0.1 and ξ ini fixed to 1, 0.5, and 0.1 respectively.It can be seen that even for very small ξ ini , Y ∆χ can exceed 10 −9 .DM relic density can be estimated by: For dark matter mass m χ larger than 10 GeV, it is always possible to explain current observed relic density (Ω χ h 2 ≈ 0.12), provided ξ ini is not much smaller than 1.So we can take Y ∆χ as an input parameter, which should be consistent with Ω χ h 2 ≈ 0.12, in the following analysis.

B χ − χ annihilation
As we explained in the introduction, asymmetry DM helps to escape the limits from observations like CMB.To be more specific, compared with symmetric DM scenario, the energy injection rate in ADM scenario during recombination is suppressed by the asymptotic ratio: To obtain r ∞ (here "∞" correspond to recombination time), we need to solve following Boltzmann equations for yields Y χ and Y χ [140][141][142][143][144][145][146]: Here x ≡ m χ /T , and Y sym eq is the equilibrium yield of χ (or χ) with chemical potential being zero (correspond to the symmetric case): And: Ratio r(x) is a function of x, and the asymptotic ratio r ∞ is the value of r(x) when x → ∞: We follow the method proposed in [142,146] to calculate r ∞ .For later convenience, firstly we need to define equilibrium ratio r eq (x) as: Here, "sinh −1 Y ∆χ 2Y sym eq " is actually the ratio between chemical potential and temperature.Then Boltzmann equations ( 19) can be transferred to a differential equation for r(x) [142]: Before freeze-out (x < x F O ), χ and χ are in the thermal equilibrium and thus r = r eq .After freeze-out (x > x F O ), r eq decreases much faster than r and thus the Eq. ( 24) can be approximatively simplified to: Then we obtain the approximate expression of r ∞ : At the freeze-out temperature (x = x F O ), these is little difference between r and r eq .So Eq. ( 26) can be further simplified to: In this work, we consider dark matter within mass range 10 GeV -100 GeV.Previous numerical study [146] shows that, within this mass range, the inclusion of non-perturbative effects (i.e.Sommerfeld enhancement and bound state formation) is not important in the calculation of r ∞ 2 .
Thus we can approximately replace cross-section by its leading-order perturbative value: 2 However, for dark matter heavier than TeV, non-perturbative effects play key role in r∞ estimation.See Rf. [146] for more details.
Here α ≡ q 2 /4π is the dark fine structure constant.And Q χ has been fixed to 1 in this work.By this approximation, Eq. ( 27) is further simplified to: Finally, freeze-out temperature is determined by: With all the above information, we can estimate r ∞ numerically.
< l a t e x i t s h a 1 _ b a s e 6 4 = " In Fig.
(3) we present the value of r ∞ as functions of α .It clearly shows that r ∞ is very sensitive to the value of α .With α increasing from O(0.001) to O(0.01), r ∞ decreases by more than 10 orders.The decreasing of r ∞ decreasing becomes much quick for smaller dark matter mass.This trend is consistent with previous study [142].We also present the dependence of r ∞ on temperature ratio, and our results show that r ∞ will be smaller if dark sector is colder than the visible sector.This relationship can be understood by the enhanced √ g * during freeze-out when dark sector become colder.

C Limit on χ − χ annihilation during recombination
As we explained in the introduction, the strong limit of CMB date on dark matter annihilation during recombination period can be greatly weakened by the asymmetry of dark matter.However, due to the scenario we chosen to study in this work, this problem is "over-solved".In our scenario, we let S 1 to be nearly massless and serve as dark radiation.So the dominant decay channel of mediator is γ → S † 1 S 1 , and the energy injection from χ − χ annihilation goes to dark thermal bath instead of visible sector.Thus the already bonded neutral hydrogen atoms will not be reionized by high energy electric shower process, and χ − χ annihilation in our scenario is save from the direct CMB limit.
But is still very interesting to see how the asymmetry helps to weaken the CMS limit.So in this subsection we will deviate from our scenario and assume that the mediator dominantly decay to electron.In this case, BBN might give a strong bound on the mass and lifetime on O(1) − O (10) MeV mediator (see e.g.Ref. [147][148][149] for detailed discussion).But here we will only focus on the CMB bound.
As we said in the last subsection, non-perturbative effects in annihilation process can be ignored in the calculation of r ∞ for dark matter lighter than 100 GeV.But in the study of energy injection during recombination, including the non-perturbative effects in annihilation is important.Here we perform an approximate analysis like [88], which only include the Sommerfeld enhancement in the estimation of annihilation cross-section during recombination period.
The annihilation cross-section can be written as the tree-level cross-section multiplied by a Sommerfeld enhancement factor [150]: Tree level annihilation cross section (σ ann v) 0 = πα 2 Q 4 χ /m 2 χ have been given in previous subsection.Sommerfeld enhancement factor S(v) is: with: Sommerfeld enhancement factor will reach it maximal value, or say saturates, when velocity v m γ /2m χ .During recombination period, this saturation condition is already satisfied [88], and thus the annihilation cross-section will generally be enhanced by several orders.
Many studies has been done on the CMB's constraints on dark matter annihilation [151][152][153][154].Recently study [155] proposes a slightly stronger constraint by using combined date from Planck [156], BAO [157,158], and DES [159].For electron final states and DM mass within 10 GeV to 100 GeV, limit on σ ann v rec /m χ is (for symmetric DM): Here we also given the limit in natural units.To illustrate how the CMB limit is, we can consider α = 0.01 and m χ = 100 GeV.In this case, < l a t e x i t s h a 1 _ b a s e 6 4 = " r m k Y G r 6 a 8  enhancement factor S(v rec ), this parameter choice can not avoid CMB constraint.In the next section we will show that α ∼ O(0.1) is generally required to solve the small scale problem.Thus it is very difficult for symmetric DM to be consistent with CMB data, provided the final state of DM annihilation is elections.
Different with the symmetric DM case, in our asymmetric DM case, this limit should be modified to: As we mentioned before, the energy injection from DM annihilation during recombination is reduced hugely by the small value of r ∞ .And thus the constrain from CMB to dark sector parameters become much looser.In Fig. 4 we present the allowed parameter region with m χ fixed to 100 GeV and 60 GeV, respectively.As we already shown in the last subsection, increasing α value lead to r ∞ exponential decreasing, and thus larger α is more easier to escape from CMB constrain.And for asymmetric DM mass within 10 GeV to 100 GeV, α 0.01 is large enough to escape CMB limit (even if the final state of DM-anti-DM annihilation are electrons).This is also favored by small scale data.
After the discussion of CMB constraint, we move back to our scenario with dark radiation.

D Change of N eff
As we explain before, in our scenario all the entropy in dark sector finally goes to nearly massless complex scalar S 2 , which is dark radiation.The presence of dark radiation will affect the measured value of the effective number of neutrino species N eff [121].N eff is defined by the measured radiative energy density in addition to photon energy density: Current constraint on N eff from joint Planck + BAO data analysis is [160]: On the other hand the SM prediction of N eff is [161]: Thus there is a room about ∆N eff < 0.29 for the existence of dark radiation (DR).
In this work we consider an independent dark sector, and hence T ν /T γ retain its SM value (4/11) 1/3 .Then ∆N eff can be expressed as: The temperature ratio T /T γ after the second equal sign should be estimated during recombination period by Eq. ( 6).Thus the limit on ∆N eff is transferred to the limit on ξ ini : It should be noted that this up-limit on ξ ini needs to be modified when the intensity of dark U (1) phase transition is large [162].We will discuss this point in the gravitational wave section.
The future CMB-S4 experiment will constrain the deviation from SM to ∆N eff < 0.06 at 95% C.L. [163].If the initial temperature ratio ξ ini is not too small, then we should observe an exceed of ∆N eff at CMB-S4.

E Dark acoustic oscillations and collisional damping
The presence of dark radiation (DR) cause another problem which might make our scenario constrained by current cosmology observations.In our scenario, DM χ and DR S 2 are both charged under the U (1) , and thus they can scatter with each other via the dark mediator γ .This DM-DR scattering may cause the so-called "dark acoustic oscillations" (DAO) and the collisional (Silk) damping between DM and DR [164,165], provided the kinetic equilibrium between DM and DR lasts long enough.DAO and the collisional (Silk) damping will modify the initial matter power spectrum, and then leave imprints on CMB anisotropy and large scale structure (LSS) [164,165].
Ref. [164] propose a parameter Σ DAO as the proxy of DAO effect.Σ DAO is related to the scattering cross-section between DM and DR (labeled as σ DM-DR ) via: where T dec is the DM kinetic decoupling temperature.σ DM-DR (T dec ) and ξ dec are scattering crosssection and temperature ratio at T dec , respectively.The bound on the value of Σ DAO is Σ DAO < 10 −4.15 (10 −3.6 ) for ξ = 0.5 (0.3) [164].
The kinetic decoupling temperature T dec is determined by: The left hand side of the above equation can be approximated by: Generally speaking, T dec is much smaller than m γ , so here we estimate σv DM-DR by a simple dimensional analysis.On the other hand, H(T dec ) ≈ 1.66 Combined with Eq. ( 41) we can induce a bound on coupling strength and spectrum (here we choose ξ dec = 0.5): So it can be seen that even for m χ = 10 GeV and m γ = 1 MeV, the bound on α is still very loose.

IV DM SELF-INTERACTION AND SMALL SCALE STRUCTURE
In this section we investigate under which parameter settings the elastic scattering cross-section between DMs can be consistent with small-scale observations.Only {m χ , m γ , α } are relevant parameters in this section.
The calculation methods of DM scattering cross-section depend on the value of {m χ , m γ , α } and the relative velocity between DMs.Basically, there are four different regimes.In the Born regime ( α mχ m γ 1 ), one can do perturbative calculation and obtain analytic formula directly [38,39,43,166].In the classical regime ( α mχ m γ 1 and mχv rel m γ 1 ), numerical results can be fitted with analytical functions [38,39,167,168].In the quantum regime ( α mχ m γ 1 and mχv rel m γ 1 ), the cross-section can be estimated by using the Hulthén approximation [43].Recently, the analytic formulas in the semi-classical regime ( α mχ m γ 1 and mχv rel m γ 1) is also provided [169], which fills the gap between the quantum regime and classical regime.
< l a t e x i t s h a 1 _ b a s e 6 4 = " r m k Y G r 6 a 8  In the literature, momentum transfer cross section σ T ≡ dΩ(1 − cos θ) dσ dΩ is generally used as the proxy for DM elastic scattering.However, it is suggested to use viscosity cross section σ V ≡ dΩ sin 2 θ dσ dΩ instead of σ T as the proxy.Because σ V is more related to the heat conductivity and σ V is well defined for identical particles [43,169,170].Ref. [169] also suggest to use σ V = σ V v 3 rel /24 √ πv 3 0 as the velocity averaged cross section for this parameter is directly related to the energy transfer.All the above methods have been implemented in public code CLASSICS [169], and we will use this code to calculate the DM elastic scattering cross-section in our model.
We list the observations we considered to constrain DM scattering in Tab.II.Fitting results for galaxy groups and clusters come from Ref. [34].Firstly we perform a parameter scan with DM mass fixed to 100 GeV, 60 GeV, and 30 GeV, respectively.The scan results are present in Fig.In Fig. 5 (right) we present scattering cross section as functions of v for these three benchmark points.It shows a clear velocity dependence that fits the data.
V STOCHASTIC GRAVITATIONAL WAVES SIGNAL FROM DARK U (1) PHASE TRANSITION So for, we have built up a theory framework of ADM that is consistent with all the limits and can solve the small scale problems at the same time.In this section we discuss the detection of this scenario.Due to the nearly negligible portal between dark sector and visible sector in the scenario we chosen in this work, traditional methods are weak in detecting this scenario.But, if the spontaneous breaking of dark U (1) is induced by first order phase transition, then it is possible to detect the nearly independent dark sector by the stochastic gravitational wave signal [171][172][173][174][175][176][177][178][179][180][181][182][183][184][185][186][187].
Here we perform a brief analysis.
The sector related to the MeV scale dark U (1) symmetry breaking is generally called Abelian Higgs model in the literature [188,189].And Lattice simulation already shown that the phase transition of Abelian Higgs model is first order, provided the Higgs mass is smaller or much smaller than gauge boson mass [190,191].The corresponding Lagrangian is given by: Here we don not need to include χ and S 1 because their masses are far from MeV scale.The U (1) charge of S 2 has been fixed to +2 as we said in Sec.II.After S 2 got VEV, it can be expressed as: Here v ≡ 2µ 2 / √ λ 2 is the VEV of S 2 at zero temperature.s 2 and a are the scalar and pseudo-scalar components of S 2 respectively.In R ξ gauge, the gauge-fixing and ghost terms are: where c is the ghost field.Zero temperature spectrum are given by: At finite temperature, S 2 field value φ = v, and φ dependent spectrums are: In the rest of this section we will consider Landau gauge (ξ = 0) to decouple ghost fields.{m s 2 , m γ , α } are chosen as input parameters to induce other relevant parameters.

A Thermal effective potential
Free energy density of dark U (1) sector is the thermal effective potential.Thermal effective potential at temperature3 T can be schematically expressed as: Here V 0 is the tree-level potential, V 1-loop is the sum of one-loop Coleman-Weinberg potential and counter-terms, V T is the thermal correction, and V daisy is the correction from daisy resummation.Tree-level potential comes from the potential sector of Lagrangian ( 46) by replacing S 2 by φ/ √ 2: V 1-loop is composed by one-loop Coleman-Weinberg potential and counter-terms, where the Coleman-Weinberg potential under MS renormalization scheme is [192] : Here we need to emphasize that the potential parameter µ 2 2 and λ 2 is determined by input physical parameters {m s 2 , m γ , α } via tree-level relation Eq. (49).Thus, to prevent physical mass and VEV being shifted by one-loop correction, counter terms need to be added to obey following on-shell conditions: where ∆Σ ≡ Σ(m 2 s 2 ) − Σ(0) is the difference between scalar self-energy at different momentums.If all the involved particles are massive, it is harmless to ignore ∆Σ in Eq. (54).But Goldstone a in Landau gauge is massless, and it causes an infrared (IR) divergence when we perform on-shell conditions on Coleman-Weinberg potential.So we need the IR divergence in ∆Σ to make all IR divergences from Goldstone cancel out.See [196] for more detailed discussion.One-loop correction which satisfy ( 54) is [197]: Thermal correction is [193,194]: Here the bosonic thermal function J B is: To avoid the IR divergence when the mass of boson is much smaller than temperature, daisy resummation needs to be added for scalar and longitudinal component of γ [195] : + m 2 a (φ) + Π a (T ) + m 2 γ (φ) + Π γ (T ) where Π s 2 (T ) = λ 2 /6 + (2g ) 2 /4 T 2 , Π a (T ) = λ 2 /6 + (2g ) 2 /4 T 2 , and Π γ (T ) = (2g ) 2 /3 T 2 are thermal Debye mass squares.

B Nucleation temperature
When the temperature is below the critical temperature, false vacuum transfer to true vacuum via thermally fluctuation 4 .Transition rate per unit volume is given by [198][199][200]: Here A(T ) = ωT 4 with ω ∼ O(1), and S E (T ) is the 4-D Euclidean action.In the case of thermal transition, S E (T ) is the ratio between 3-D Euclidean action S 3 (T ) and temperature: And 3-D Euclidean action S 3 (T ) is given by: Due to the 3-D rotation invariance, φ only depend on radius r and thus S 3 can be rewritten as: Minimization condition of S 3 (T ) gives the equation of motion that φ should follow: Adding boundary conditions lim r→∞ φ(r) = 0 and dφ dr r=0 = 0, Eq. ( 63) can be solved numerically by overshoot/undershoot method [201].In this work we use public code CosmoTransitions [202] to do the calculation.Nucleation starts at the temperature where the transition rate within one Hubble volume approximates Hubble rate: ⇒A(T N )e −S 3 (T N )/T N ≈ 1.66 Here T N is the nucleation temperature, and we approximate ω to 1 in the third line.In our model, phase transition in the dark U (1) sector happens around MeV scale, and the temperature ratio ξ is generally not much smaller than 1.So the nucleation temperature T N is approximately determined by S 3 (T N )/T N ≈ 196.In Tab.III we present three benchmark points for illustration in this section.These three benchmark points are also consistent with small scale structure data.

C Phase Transition parameters
After nucleation, bubbles expand rapidly and after a while collide with each other and generate gravitational waves (GWs).There are three GWs generation mechanisms: bubble walls collision [203][204][205][206][207]233], sound waves [209][210][211]234], and magnetohydrodynamic turbulence [213-216, 218, 235].The generated gravitational waves stay in the universe and redshift in wavelength as the universe expands.To obtain current spectrum of these phase transition gravitational waves, firstly we need to calculate a set of parameters used to describe the phase transition dynamics: T * , A5 , A (A in dark sector), β/H * , v w , and κ b,s,t .Meaning of these parameters are given below.
T * is the characteristic temperature of GWs generation.Generally, T * can be chosen as percolation temperature, the temperature at which a large fraction of the space has been occupied by bubbles [219][220][221].But in the weak or mild supercooling case, using nucleation temperature T N as T * is also a good approximation.For the benchmark points we will study in this section, it is fine to approximates T * by T N because of their mild super cooling.
Strength parameter A is the change in the trace of energy-momentum tensor during phase transition divided by relativistic energy density: Here ρ * is the relativistic energy density at T * .∆V is the difference in free energy density between false vacuum and true vacuum.
It will be convenient to study dark sector dynamics if we define another strength parameter A by only considering the relativistic energy density in the dark sector [181]: Here ρ * the relativistic energy density in the dark sector at T * .β is the inverse of the duration of phase transition.Its ratio to Hubble expansion rate at T * is given by: v w is the velocity of bubble wall.κ b,s,t are the fractions of released vacuum energy that transferred to scalar-field gradients, sound waves, and turbulence, respectively.Before estimating these parameters, we need to judge whether the phase transition is "runaway" or "non-runaway".To do that, firstly we calculate the so-called threshold value of A , which is labeled as A ∞ [222]: Here c i = 1 (1/2) for bosons (fermions), n i is the number of degrees of freedom (absolute value), and ∆m 2 i is the difference in particle mass square between false vacuum and true vacuum.If A > A ∞ , the driving pressure will be larger than the friction from dark plasma and thus the bubble wall will eventually be accelerated to the maximal value, i.e. v w = 1.This is the so-called "runaway" case.In this case, fractions κ b,s,t are given by [223,234]: If A < A ∞ , bubble wall will eventually reach a subluminal velocity and this is called "nonrunaway" phase transition.In this case we simply choice v w = 0.9 for a fast and rough estimation of GWs signal 6 .For non-runaway phase transition, the main source of GWs will be sound waves and the contribution from bubble collision is negligible.Fractions κ b,s,t are given by [222]: In Tab.IV we present all the phase transition parameters for the three benchmark points.

D Gravitational waves
In this subsection we present the calculation of today's GWs signal in our model.formulas used in this subsection can be found in the literature [231][232][233][234][235].
The total GWs signal is the linear superposition of spectrums from bubble collisions, sound waves, and turbulence: The three indivisual contributions can be further divided into peak amplitudes (Ω peak b,s,t ) and spectral shape functions (S b,s,t ): The calculation of vw in a concrete model is still quite difficult.See Ref. [224][225][226][227][228][229][230] for previous studies.
Peak amplitudes are determined by all the phase transition parameters we obtained before: where " * " and "0" correspond to GWs producing time and current time, respectively.These formulas look different from expressions commonly found in the literature, because we need to recalculate the redshift factor for MeV scale dark phase transition [180].For our benchmark points, the factor inside above expressions can be approximated to: Spectral shape functions are given by:  For our three benchmark points, factor β/H * vw ∼ O(1000).Thus their peak frequencies are around 10 −6 − 10 −7 Hz, which is not favored by either SKA telescope [236] or space-based LISA interferometer [237].In Fig. 6 we present the GWs spectrums of our three benchmark points.As we expected, these signals are barely detectable by the SAK or LISA.However, this result depends significantly on our choice of benchmark point.In order not to change the limit about N eff we got in Sec.III D, for all the benchmark points we consider, the strength of phase transitions are quite weak (i.e. the value of A and A are quite small).If we increase the intensity of phase transition and make a strong supercooling, then the generated GWs signal can be easily detected by SKA.The reason is twofold.Firstly, h 2 Ω GW is approximately proportional to A 2 .So increasing A by an order of magnitude, we can increase h 2 Ω GW by roughly two orders of magnitude.Secondly, strong supercooling makes T * far below MeV scale, and thus make the peak frequency of h 2 Ω GW closer to the detection region of SKA.Certainly, in the strong supercooling case we need to revisit the limit from N eff .Detailed analysis is left for a future study.

VI CONCLUSION
In this work we propose an asymmetry DM model with massive mediator to explain DM small scale structure data and to avoid the limit from CMB.In our model, the DM candidate is a vectorlike fermion charged under a dark U (1) , and the mediator is the U (1) gauge boson that gain mass from the spontaneous symmetry breaking.The asymmetry between DM and anti-DM is generated by the CP violated and out-of-equilibrium decay of a neutral heavy fermion.The model is consistent with cosmology observations like CMB and LLS.The existence of dark radiation increases the value of N eff , and it makes this model to be detectable by the future measurement of N eff .Finally, the MeV scale U (1) symmetry breaking generate GWs signal with peak frequency around 10 −6 − 10 −7 Hz.It also possible to make the GWs from U (1) symmetry breaking to be detected by SKA, if we consider a strong supercooling phase transition.
Similar to the vanilla leptogenesis [136-139], non-zero Y ∆χ is generated by the CP violated and out-of-equilibrium decay of N 1 .See Fig.(1) for illustration.< l a t e x i t s h a 1 _ b a s e 6 4 = " 2 n r e 2 d 8 q 7 l b 3 9 g 8 O j 6 v F J V 8 e p I r R D Y h 6 r f o A 1 5 U z S j m G G 0 3 6 i K B Y B p 7 1 g e p f 7 v S e q N I v l o 5 k l 1 B d 4 I l n I C D a 5 N C r e 2 d 8 q 7 l b 3 9 g 8 O j 6 v F J V 8 e p I r R D Y h 6 r f o A 1 5 U z S j m G G 0 3 6 i K B Y B p 7 1 g e p f 7 v S e q N I v l o 5 k l 1 B d 4 I l n I C D a 5 N C

1 <
9 9 C G D h C I 4 B l e 4 c 0 R z o v z 7 n w s W 0 t O M X M K f + B 8 / g A B C I 4 2 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 nJ G z z o V t Z O M + N / 8 k U w o o O 4 S u z A = " > A A A B 6 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l q Q Y 9 F L x 4 r 2 A 9 o Q 9 l s N 8 3 S 3 U 3 Y 3 Q g l 9 C 9 4 8 a C I V / + Q N / + N m z Y H b X 0 w 8 H h v h p l 5 Q c K Z N q 7 7 7 Z Q 2 N r e 2 d 8 q 7 l b 3 9 g 8 O j 6 v F J V 8 e p I r R D Y h 6 r f o A 1 5 U z S j m G G 0 3 6 i K B Y B p 7 1 g e p f 7 v S e q N I v l o 5 k l 1 B d 4 I l n I C D a 5 N C Q R G 1 V r b t 1 d A K 0 T r y A 1 K N A e V b + G 4 5 i k g k p D O N Z 6 4 L m J 8 T O s D C O c z i v D V N M E k y m e 0 I G l E g u q / W x x 6 x x d W G W M w l j Z k g Y t 1 N 8 T G R Z a z 0 R g O w U 2 k V 7 1 c v E /b 5 C a 8 M b P m E x S Q y V Z L g p T j k y M 8 s f R m C l K D J 9 Z g o l i 9 l Z E I q w w M T a e i g 3 B W 3 1 5 n X Q b d e + q 3 n h o 1 l q 3 R R x l O I N z u A Q P r q E F 9 9 C G D h C I 4 B l e 4 c 0 R z o v z 7 n w s W 0 t O M X M K f + B 8 / g A B C I 4 2 < / l a t e x i t > N l a t e x i t s h a 1 _ b a s e 6 4 = " F D D L L / Y W E Y U J m s J / C F u o 2 W W H 7 C g = " > A A A B 6 n i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k V 9 F j 0 4 k k q 2 g 9 o Q 9 l s J + 3 S z S b s b o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d l Z W 1 9 Y 3 N g t b x e 2 d 3 b 3 9 0 s F h U 8 e p Y t h g s Y h V O 6 A a B Z f Y M N w I b C c K a R Q I b A W j m 6 n f e k K l e S w f z T h B P 6 I D y U P O q L H S w 1 3 P 6 5 X K b s W d g S w T L y d l y F H v l b 6 6 / Z i l E U r D B N W 6 4 7 m J 8 T O q D

1 <
X + o l y 7 z u M o w D G c w B l 4 c A k 1 u I U 6 N I D B A J 7 h F d 4 c 4 b w 4 7 8 7 H v H X F y W e O 4 A + c z x / M 8 4 1 6 < / l a t e x i t > N l a t e x i t s h a 1 _ b a s e 6 4 = " F D D LL / Y W E Y U J m s J / C F u o 2 W W H 7 C g = " > A A A B 6 n i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k V 9 F j 0 4 k k q 2 g 9 o Q 9 l s J + 3 S zS b s b o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d l Z W 1 9 Y 3 N g t b x e 2 d 3 b 3 9 0 s F h U 8 e p Y t h g s Y h V O 6 A a B Z f Y M N w I b C c K a R Q I b A W j m 6 n f e k K l e S w f z T h B P 6 I D y U P O q L H S w 1 3 P 6 5 X K b s W d g S w T L y d l y F H v l b 6 6 / Z i l E U r D B N W 6 4 7 m J 8 T O q D

1 <
X + o l y 7 z u M o w D G c w B l 4 c A k 1 u I U 6 N I D B A J 7 h F d 4 c 4 b w 4 7 8 7 H v H X F y W e O 4 A + c z x / M 8 4 1 6 < / l a t e x i t > N l a t e x i t s h a 1 _ b a s e 6 4 = " F D D LL / Y W E Y U J m s J / C F u o 2 W W H 7 C g = " > A A A B 6 n i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k V 9 F j 0 4 k k q 2 g 9 o Q 9 l s J + 3 S zS b s b o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d l Z W 1 9 Y 3 N g t b x e 2 d 3 b 3 9 0 s F h U 8 e p Y t h g s Y h V O 6 A a B Z f Y M N w I b C c K a R Q I b A W j m 6 n f e k K l e S w f z T h B P 6 I D y U P O q L H S w 1 3 P 6 5 X K b s W d g S w T L y d l y F H v l b 6 6 / Z i l E U r D B N W 6 4 7 m J 8 T O q D

2 < 2 <
X + o l y 7 z u M o w D G c w B l 4 c A k 1 u I U 6 N I D B A J 7 h F d 4 c 4 b w 4 7 8 7 H v H X F y W e O 4 A + c z x / M 8 4 1 6 < / l a t e x i t > N l a t e x i t s h a 1 _ b a s e 6 4 = "V S R i S 1 i j + A f g d h M K + d m l 5 R x p X F k = " > A A A B 6 n i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k V 9 F j 0 4 k k q 2 g 9 o Q 9 l s N + 3 S z S b s T o Q S +h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I J H C o O t + O y u r a + s b m 4 W t 4 v b O 7 t 5 + 6 e C w a e J U M 9 5 g s Y x 1 O 6C G S 6 F 4 A w V K 3 k 4 0 p 1 E g e S s Y 3 U z 9 1 h P X R s T q E c c J 9 y M 6 U C I U j K K V H u 5 6 1 V 6 p 7 F b c G c g y 8 X J S h h z 1 X u m r 2 4 9 Z G n G F T F J j O p 6 b o J 9 R j Y J J P i l 2 U 8 M T y k Z 0 w D u W K h p x 4 2 e z U y f k 1 C p 9 E s b a l k I y U 3 9 P Z D Q y Z h w F t j O i O D S L 3 l T 8 z + u k G F 7 5 m V B J i l y x + a I w l Q R j M v 2 b 9 I X m D O X Y E s q 0 s L c S N q S a M r T p F G 0 I 3 u L L y 6 R Zr X j n l e r 9 R b l 2 n c d R g G M 4 g T P w 4 B J q c A t 1 a A C D A T z D K 7 w 5 0 n l x 3 p 2 P e e u K k 8 8 c w R 8 4 n z / O d 4 1 7 < / l a t e x i t > N l a t e x i t s h a 1 _ b a s e 6 4 = " V S R i S 1 i j + A f g d h M K + d m l 5 R x p X F k = " > A A A B 6 n i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k V 9 F j 0 4 k k q 2 g 9 o Q 9 l s N + 3 S z S b s T o Q S + h O 8 e F D E q 7 / I m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I J H C o O t + O y u r a + s b m 4 W t 4 v b O 7 t 5 + 6 e C w a e J U M 9 5 g s Y x 1 O 6 C G S 6 F 4 A w V K 3 k 4 0 p 1 E g e S s Y 3 U z 9 1 h P X R s T q E c c J 9 y M 6 U C I U j K K V H u 5 6 1 V 6 p 7 F b c G c g y 8 X J S h h z 1 X u m r 2 4 9 Z G n G F T F J j O p 6 b o J 9 R j Y J J P i l 2 U 8 M T y k Z 0 w D u W K h p x 4 2 e z U y f k 1 C p 9 E s b a l k I y U 3 9 P Z D Q y Z h w F t j O i O D S L 3 l T 8 z + u k G F 7 5 m V B J i l y x + a I w l Q R j M v 2 b 9 I X m D O X Y E s q 0 s L c S N q S a M r T p F G 0 I 3 u L L y 6 R Z r X j n l e r 9 R b l 2 n c d R g G M 4 g T P w 4 B J q c A t 1 a A C D A T z D K 7 w 5 0 n l x 3 p 2 P e e u K k 8 8 c w R 8 4 n z / O d 4 1 7 < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " H F b 7 FIG.1.CP violated process in the dark sector that generates the asymmetry between DM and anti-DM.

1 < 0 FIG. 3 .
FIG. 3. r ∞ as functions of dark fine structure constant α , with different dark matter mass m χ and different initial temperature ratio ξ ini .
H a 6 j D P T S g C Q T G 8 A y v 8 O Z I 5 8 V 5 d z 4 W r Q U n n z m G P 3 A + f w D w 2 4 9 S < / l a t e x i t > ↵ 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " m v o I / e z G I 0 R 0 O + z 2 K Y g A k m 9 F Z 7 c = " > A A A C A n i c b V D L S g N B E J z 1 G e M r 6 k m 8 D A b R U 9 i V o B 6 D H v Q Y w T w g W c L s p J M M m d l d Z n r F s A Q v / o o X D 4 p 4 9 S u 8 + T d O H g d N L G g o q r r p 7 g p i K Q y 6 7 r e z s L i 0 v L K a W c u u b 2 x u b e d 2 d q s m S j S H C o 9 k p O s B M yB F C B U U K K E e a 2 A q k F A L + l c j v 3 Y P 2 o g o v M N B D L 5 i 3 V B 0 B G d o p V Z u X 7 X S Z p c p x Y 6 H t I n w g C l t X E P V H 7 Z y e b f g j k H n i T c l e T J F u Z X 7 a r Y j n i g I k U t m T M N z Y / R T p l F w C c N s M z E Q M 9 5 n X W h Y G j I F x k / H L w z p k V X a t B N p W y H S s f p 7 I m X K m I E K b K d i 2 D O z 3 k j 8 z 2 s k 2 L n w U x H G C U L I J 4 s 6 i a Q Y 0 V E e t C 0 0 c J Q D S xj X w t 5 K e Y 9 p x t G m l r U h e L M v z 5 P q a c E 7 K x R v i / n S 5 T S O D D k g h + S E e O S c l M g N K Z M K 4 e S R P J N X 8 u Y 8 O S / O u / M x a V 1 w p j N 7 5 A + c z x / H 3 p c L < / l a t e x i t > m 0 [GeV] < l a t e x i t s h a 1 _ b a s e 6 4 = " m v o I / e z G I 0R 0 O + z 2 K Y g A k m 9 F Z 7 c = " > A A A C A n i c b V D L S g N B E J z 1 G e M r 6 k m 8 D A b R U 9 i V o B 6 D H v Q Y w T w g W c L s p J M M m d l d Z n r F s A Q v / o o X D 4 p 4 9 S u 8 + T d O H g d N L G g o q r r p 7 g p i K Q y 6 7 r e z s L i 0 v L K a W c u u b 2 x u b e d 2 d q s m S j S H C o 9 k p O s B M y B F C B U U K K E e a 2 A q k F A L + l c j v 3 Y P 2 o g o v M N B D L 5 i 3 V B 0 B G d o p V Z u X 7 X S Z p c p x Y 6 H t I n w g C l t X E P V H 7 Z y e b f g j k H n i T c l e T J F u Z X 7 a r Y j n i g I k U t m T M N z Y / R T p l F w C c N s M z E Q M 9 5 n X W h Y G j I F x k / H L w z p k V X a t B N p W y H S s f p 7 I m X K m I E K b K d i 2 D O z 3 k j 8 z 2 s k 2 L n w U x H G C U L I J 4 s 6 i a Q Y 0 V E e t C 0 0 c J Q D S x j X w t 5 K e Y 9 p x t G m l r U h e L M v z 5 P q a c E 7 K x R v i / n S 5 T S O D D k g h + S E e O S c l M g N K Z M K 4 e S R P J N X 8 u Y 8 O S / O u / M x a V 1 w p j N 7 5 A + c z x / H 3 p c L < / l a t e x i t > m 0 [GeV]< l a t e x i t s h a 1 _ b a s e 6 4 = " w + V o 4 j q 6 S q A 4 p 6M O q k B X / A g 3 N G o = " > A A A C P X i c b V B N a x s x E N U m T e M 6 a e s k x 1 5 E T M G 5 m N 0 S 0 h x C C O 2 l R w d s J + A 1 R i v P 2 s K S d p F m Q 8 y i P 5 Z L / k N v u e X S Q 0 P p N d f K H 4 X U 7 g P B m / d m G M 1 L c i k s h u F D s L H 5 a u v 1 d u V N d W f 3 7 b v 3 t b 3 9 r s 0 K w 6 H D M 5 m Z 6 4 R Z k E J D B w V K u M 4 N M J V I u E o m X 2 f + 1 Q 0 Y K z L d x m k O f c V G W q S C M / T S o N a O p a a x h B Q b N E 4 N 4 + W 8 O I u t G C k 2 K G O E W y y Z 1 s 7 R G 0 p j I 0 Z j P P + r G + D O u V L 5 m o + F c w v 7 a F C r h 8 1 w D r p O o i W p k y V a g 9 r 3 e J j x Q o F G L p m 1 v S j M s V 8 y g 4 J L c N W 4 s J A z P m E j 6 H m q m Q L b L + f X O / r R K 0 O a Z s Y / j X S u v p w o m b J 2 q h L f q R i O7 a o 3 E / / n 9 Q p M T / u l 0 H m B o P l i U V p I i h m d R U m H w t + P c u o J 4 0 b 4 v 1 I + Z j 5 D 9 I F X f Q j R 6 s n r p P u p G Z 0 0 j y + P 6 x d f l n F U y A d y S B o k I p / J B f l G W q R D O L k j j + Q n e Q r u g x / B r + D 3 o n U j W M 4 c k H 8 Q P P 8 B K 3 q x F Q = = < / l a t e x i t > ln ✓ h annvirec m ◆ < l a t e x i t s h a 1 _ b a s e 6 4 = " w + V o 4 j q 6 S q A 4 p 6 M O q k B X / A g 3 N G o = " > A A A C P X i c b V B N a x s x E N U m T e M 6 a e s k x 1 5 E T M G 5 m N 0 S 0 h x C C O 2 l R w d s J + A 1 R i v P 2 s K S d p F m Q 8 y i P 5 Z L / k N v u e X S Q 0 P p N d f K H 4 X U 7 g P B m / d m G M 1 L c i k s h u F D s L H 5 a u v 1 d u V N d W f 3 7 b v 3 t b 3 9 r s 0 K w 6 H D M 5 m Z 64 R Z k E J D B w V K u M 4 N M J V I u E o m X 2 f + 1 Q 0 Y K z L d x m k O f c V G W q S C M / T S o N a O p a a x h B Q b N E 4 N 4 + W 8 O I u t G C k 2 K G O E W y y Z 1 s 7 R G 0 p j I 0 Z j P P + r G + D O u V L 5 m o + F c w v 7 a F C r h 8 1 w D r p O o i W p k y V a g 9 r 3 e J j x Q o F G L p m 1 v S j M s V 8 y g 4 J L c N W 4 s J A z P m E j 6 H m q m Q L b L + f X O / r R K 0 O a Z s Y / j X S u v p w o m b J 2 q h L f q R i O7 a o 3 E / / n 9 Q p M T / u l 0 H m B o P l i U V p I i h m d R U m H w t + P c u o J 4 0 b 4 v 1 I + Z j 5 D 9 I F X f Q j R 6 s n r p P u p G Z 0 0 j y + P 6 x d f l n F U y A d y S B o k I p / J B f l G W q R D O L k j j + Q n e Q r u g x / B r + D 3 o n U j W M 4 c k H 8 Q P P 8 B K 3 q x F Q = = < / l a t e x i t > ln ✓ h annvirec m ◆ < l a t e x i t s h a 1 _ b a s e 6 4 = " l g b 8 M 1 o M q m + r 1 + G G V p M T w + h i R R s = " > A A A C A 3 i c b V D L S g N B E J z 1 G e M r 6 k 0 v g 0 H w F H Y l q B c h 6 E G P E c w D k h B m J 7 3 J k N k H M 7 1 i W B a 8 + C t e P C j i 1 Z / w 5 t 8 4 S f a g i Q U N R V U 3 3 V 1 u J I V G 2 / 6 2 F h a X l l d W c 2 v 5 9 Y 3 N r e 3 C z m 5 d h 7 H i U O O h D F X T Z R q k C K C G A i U 0 I w X M d y U 0 3 O H V 2 G / c g 9 I i D O 5 w F E H H Z / 1 A e I I z N F K 3 s O 9 3 k z Y f i J R e U M e 2 a R v h A R N 6 D f W 0 W y j a J X s C O k + c j B R J h m q 3 8 N X u h T z 2 I U A u m d Y t x 4 6 w k z C F g k t I 8 + 1 Y Q 8 T 4 k P W h Z W j A f N C d Z P J D S o + M 0 q N e q E w F S C f q 7 4 m E + V q P f N d 0 + g w H e t Y b i / 9 5 r R i 9 8 0 4 i g i h G C P h 0 k R d L i i E d B 0 J 7 Q g F H O T K E c S X M r Z Q P m G I c T W x 5 E 4 I z + / I 8 q Z + U n N N S + b Z c r F x m c e T I A T k k x 8 Q h Z 6 R C b k i V 1 A g n j + S Z v J I 3 6 8 l 6 s d 6 t j 2 n r g p X N 7 J E / s D 5 / A P X 0 l n U = < / l a t e x i t > m = 100 GeV < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 O 1 K 7 p 9 9 B W Z I y a C 5 X v c E 0 i n Y B 3 8 = " > A A A C A n i c b V D J S g N B E O 2 J W 4 x b 1 J N 4 a Q y C p z A j I X o R g h 7 0 G M E s k A m h p 1 N J m v Q s d N e I Y R i 8 + C t e P C j i 1 a / w 5 t / Y W Q 6 FIG. 4. CMB limit on α -m γ plane with m χ fixed to different values.In this plot we assume that the χ − χ annihilation mainly go to electron final states.The color indicate the natural logarithmic value of σannv rec mχ in units of GeV −3 .The point with ln σannv rec mχ > −23.88 (blank region) have been excluded by current data.
.09 cm 2 /g TABLE II.Small scale data we considered to constrain DM scattering.hvi [km/s] < l a t e x i t s h a 1 _ b a s e 6 4 = " d w p W + j 3 8 E x O A T o X Z 2 5 E y / q n 9 k J A = " > A A A C C X i c b V A 9 S w N B E N 3 z 2 / h 1 a m m z G A S r e K e C F i J B G 0 s F k w h 3 R 9 j b z C V L 9 j 7 Y n Q u G I 6 2 N f 8 X G Q h F b / 4 G d / 8 Z N v M K v B w O P 9 2 a Y m R d m U m h 0 n A 9 r a n p m d m 5 + Y b G y t L y y u m a v b z R 1 m i s O D Z 7 K V N 2 E T I M U C T R Q o I S b T A G L Q w m t s H 8 + 9 l s D U F q k y T U O M w h i 1 k 1 E J D h D I 7 V t 6 k u I 8 I Q O q K 9 E t 4 e n 1 K e e j 3 C L R T / e 0 6 O g b V e d m j M B / U v c k l R J i c u 2 / e 5 3 U p 7 H k C C X T G v P d T I M C q Z Q c A m j i p 9 r y B j v s y 5 4 h i Y s B h 0 U k 0 9 G d M c o H R q l y l S C d K J + n y h Y r P U w D k 1 n z L C n f 3 t j 8 T / P y z E 6 D g q R Z D l C w r 8 W R b m k m N J x L L Q j F H C U Q 0 M Y V 8 L c S n m P K c b R h F c x I b i / X / 5 L m v s 1 9 6 C 2 f 3 V Y r Z + V c S y Q L b J N d o l L j k i d X J B L 0 i C c 3 J E H 8 k S e r X v r 0 X q x X r 9 a p 6 x y Z p P 8 g P X 2 C W i I m Y c = < / l a t e x i t > C o r e -c u s p p r o b le m < l a t e x i t s h a 1 _ b a s e 6 4 = " B p b S E d x G 3 C O s G 5 a d Q U / a P O L t n G Y = " > A A A C A X i c b V C 7 S g N B F J 3 1 G e M r a i P Y D A b B x r A b B S 2 D a S w j m A c k I c x O b p I h s z v L z F 0 x L L H x V 2 w s F L H 1 L + z 8 G y f J F p p 4 Y O B w z j 3 c u c e P p D D o u t / O 0 v L K 6 t p 6 Z i O 7 u b W 9 s 5 v b 2 6 8 Z F W s O V a 6 k 0 g 2 f G Z A i h C o K l N C I N L D A l 1 D 3 h + W J X 7 8 H b Y Q K 7 3 A U Q T t g / V D 0 B G d o p U 7 u s I X w g E l Z a T j j s Y l o p J X N B u N O L u 8 W 3 C n o I v F S k i c p K p 3 c V 6 u r e B x A i F w y Y 5 q e G 2 E 7 Y R o F l z D O t m I D E e N D 1 o e m p S E L w L S T 6 Q V j e m K V L u 0 p b V + I d K r + T i Q s M G Y U + H Y y Y D g w 8 9 5 E / M 9 r x t i 7 a i c i j G K E k M 8 W 9 W J J U d F J H b Q r N H C U I 0 s Y 1 8 L + l f I B 0 4 y j L S 1 r S / D m T 1 4 k t W L B O y 8 U b y / y p e u 0 j g w 5 I s f k l H j k k p T I D a m Q K u H k k T y T V / L m P D k v z r v z M R t d c t L M A f k D 5 / M H + u 2 X O g = = < / l a t e x i t > 1 0 c m 2 / g < l a t e x i t s h a 1 _ b a s e 6 4 = " I B / L X D d g u s o p L b Y W 0 c X E m + k t e j 0 = " > A A A C A X i c b Z D L S s N A F I Y n X m u 9 R d 0 I b g a L 4 K o m V d B l 0 Y 3 L C v Y C T S y T 6 a Q d O p O E m R O x h L j x V d y 4 U M S t b + H O t 3 F 6 W W j r D w M f / z m H M + c P E s E 1 O M 6 3 t b C 4 t L y y W l g r r m 9 s b m 3 b O 7 s N H a e K s j q N R a x a A d F M 8 I j V g Y N g r U Q x I g P B m s H g a l R v 3 j O l e R z d w j B h v i S 9 i I e c E j B W x 9 5 3 H Q 9 7 w B 4 g o z K / q 5 x M u J d 3 7 J J T d s b C 8 + B O o Y S m q n X s L 6 8 b 0 1 S y C K g g W r d d J w E / I w o 4 F S w v e q l m C a E D 0 m N t g x G R T P v Z + I I c H x m n i 8 N Y m R c B H r u / J z 3 9 r s 7 Y I o S w y e W I F H M 3 u q S E S o k x i Z U s i H 4 y y + v k t Z l 1 b + q 1 h 5 q l f p t H k c R T u A U L s C H a 6 j D P T S g C Q T G 8 A y v 8 O Z I 5 8 V 5 d z 4 W r Q U n n z m G P 3 A + f w D w 2 4 9 S < / l a t e x i t > ↵ 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " m v o I / e z G I 0 R 0 O + z 2 K Y g A k m 9 F Z 7 c = " > A A A C A n i c b V D L S g N B E J z 1 G e M r 6 k m 8 D A b R U 9 i V o B 6 D H v Q Y w T w g W c L s p J M M m d l d Z n r F s A Q v / o o X D 4 p 4 9 S u 8 + T d O H g d N L G g o q r r p 7 g p i K Q y 6 7 r e z s L i 0 v L K a W c u u b 2 x u b e d 2 d q s m S j S H C o 9 k p O s B M y B F C B U U K K E e a 2 A q k F A L + l c j v 3 Y P 2 o g o v M N B D L 5 i 3 V B 0 B G d o p V Z u X 7 X S Z p c p x Y 6 H t I n w g C l t X E P V H 7 Z y e b f g j k H n i T c l e T J F u Z X 7 a r Y j n i g I k U t m T M N z Y / R T p l F w C c N s M z E Q M 9 5 n X W h Y G j I F x k / H L w z p k V X a t B N p W y H S s f p 7 I m X K m I E K b K d i 2 D O z 3 k j 8 z 2 s k 2 L n w U x H G C U L I J 4 s 6 i a Q Y 0 V E e t C 0 0 c J Q D S x j X w t 5 K e Y 9 p x t G m l r U h e L M v z 5 P q a c E 7 K x R v i / n S 5 T S O D D k g h + S E e O S c l M g N K Z M K 4 e S R P J N X 8 u Y 8 O S / O u / M x a V 1 w p j N 7 5 A + c z x / H 3 p c L < / l a t e x i t > m 0 [GeV] < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 9 y I 3 3 1 / I + r B x V y u v S y a a e u p R C FIG. 5.Left: parameter region consistent with small scale data with DM mass fixed to different values.Right: averaged elastic DM scattering cross-section as functions of DM scattering velocity for three benchmark points.

TABLE III .
Benchmark points we considered for first order phase transition study.
is current degree of freedom for entropy in the visible sector.For the MeV scale dark phase transition in our model, the value of h * can be approximated to: