Dark Matter with Density-Dependent Interactions

University of Pennsylvania ScholarlyCommons

Department of Physics Papers Department of Physics

12-28-2012

Dark Matter with Density-dependent Interactions

Kimberly K. Boddy California Institute of Technology

Sean M. Carroll California Institute of Technology

Mark Trodden University of Pennsylvania, [email protected]

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Abstract The decay and annihilation cross sections of dark matter particles may depend on the value of a chameleonic scalar field that both evolves cosmologically and takes different values depending on the local matter density. This possibility introduces a separation between the physics relevant for freeze-out and that responsible for dynamics and detection in the late universe. We investigate how such dark sector interactions might be implemented in a particle physics Lagrangian and consider how current and upcoming observations and experiments bound such dark matter candidates. A specific simple model allows for an increase in the annihilation cross section by a factor of 106 between freeze-out and today, while more complicated models should also allow for scattering cross sections near the astrophysical bounds.

Disciplines Physical Sciences and Mathematics | Physics

Comments Boddy, K. K., Carroll, S. M., & Trodden, M. (2012). Dark Matter with Density-dependent Interactions. Physical Review D, 86(12), 123529. doi: 10.1103/PhysRevD.86.123529

This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/272 PHYSICAL REVIEW D 86, 123529 (2012) Dark matter with density-dependent interactions

Kimberly K. Boddy,1,* Sean M. Carroll,1,† and Mark Trodden2,‡ 1California Institute of Technology, Pasadena, California 91125, USA 2Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA (Received 31 August 2012; published 28 December 2012) The decay and annihilation cross sections of dark matter particles may depend on the value of a chameleonic scalar ﬁeld that both evolves cosmologically and takes different values depending on the local matter density. This possibility introduces a separation between the physics relevant for freeze-out and that responsible for dynamics and detection in the late universe. We investigate how such dark sector interactions might be implemented in a particle physics Lagrangian and consider how current and upcoming observations and experiments bound such dark matter candidates. A speciﬁc simple model allows for an increase in the annihilation cross section by a factor of 106 between freeze-out and today, while more complicated models should also allow for scattering cross sections near the astrophysical bounds.

DOI: 10.1103/PhysRevD.86.123529 PACS numbers: 95.35.+d

section might be much larger now than it was at freeze-out, I. INTRODUCTION due to the evolution of a background field. The particle physics properties of dark matter are impor- In a cosmological context, the evolution of background tant for three distinct aspects of its behavior: they determine fields can assert a significant influence on the properties of how the initial abundance of dark matter arose, they govern dark matter as a function of spatial location or cosmic how the dark matter distribution evolves and influences epoch [10–19]. A straightforward way to achieve such structure formation, and they delineate the possible ways effects is to invoke a light scalar field that interacts with in which dark matter may be detected. Of course, these three dark matter and/or ordinary matter as well as through its roles are not typically independent, since they all depend on own potential, and whose expectation value feeds into the the prescribed interactions between the dark matter parti- dark-matter properties. A popular scenario along these cles themselves and also between dark matter and the lines is the ‘‘chameleon mechanism,’’ which acts to screen Standard Model. These connections often provide a power- light, cosmologically relevant degrees of freedom to pro- ful motivation for particular dark matter candidates—for tect them from precision local tests of gravity [20–24]. example, the freeze-out abundance of weakly interacting In this paper we investigate dark matter that interacts massive particle points to new physics at the weak scale, through a gauge symmetry with a coupling constant that which in turn leads to an attractive connection between dark depends on a chameleonlike scalar field. (The effects of matter and proposed solutions to the hierarchy problem, chameleon vector bosons on laboratory experiments were such as weak-scale supersymmetry. considered in [25].) Just as the properties of a cosmologically The idea that dark matter could have interactions of relevant scalar can be drastically modified in the presence of astrophysically interesting magnitude has received a good local density inhomogeneities or after evolving over cosmic amount of attention [1–7], motivated in part by purported time, so the interactions of dark matter may be modified. We discrepancies between the standard CDM model and are able to find a model in which the late-time interaction observations of structure on small scales (as described in strength is considerably higher than that at freeze-out— [8], for example). While most approaches of this form although admittedly, this behavior does not seem generic. concentrate on giving an appreciable scattering cross- We begin by reexamining the conventional story of dark section to the dark matter, it is also interesting to consider matter freeze-out according to the Boltzmann equation, but enhanced annihilation cross sections [9]. with the additional ingredient that the dark matter proper- One obstacle to simple implementations of this idea is ties are evolving with time. We then look at specific models that the required cross section for a thermal relic to obtain featuring a Dirac dark matter particle and a U(1) gauge the right relic abundance is close to the weak scale, far too symmetry that is spontaneously broken, along with a cha- small to be relevant to dynamics in the late Universe. In meleon scalar field. We study the cosmological evolution this paper we explore the idea that the dark matter cross of this coupled system and calculate the dark matter properties, including annihilation and scattering cross sections. *[email protected] Finally, we exhibit numerical solutions to a specific model, †[email protected] showing that the annihilation cross section can increase ‡[email protected] substantially during cosmic evolution.

1550-7998=2012=86(12)=123529(15) 123529-1 Ó 2012 American Physical Society KIMBERLY K. BODDY, SEAN M. CARROLL, AND MARK TRODDEN PHYSICAL REVIEW D 86, 123529 (2012) II. THE GENERAL PICTURE: EVOLVING DARK Since the entropy in each sector is conserved indepen- MATTER IN THE EARLY UNIVERSE dently, the assumption that the two sectors were in equilibrium at some unification scale at time t allows us to Before discussing specific models, let us first consider u express the dark bath temperature in terms of the visible how the usual story of dark matter freeze-out might be bath temperature at some later time t via modified if the annihilation cross section depends on the dynamics of another field. In the next section, we will d ð Þ T3ðtÞ d ð Þ gS t d ¼ gS tu explore Lagrangians that couple the dark matter to a scalar 3 : (6) g ðtÞ T ðtÞ g ðt Þ field that affects its interaction cross sections. For simplic- S S u ity we work in a flat Friedmann-Robertson-Walker (FRW) All Standard Model particles contribute at tu to give 2 2 2 universe, described by the metric ds ¼dt þ a ðtÞ gSðtuÞ¼106:75, and all dark particles contribute to ð 2 þ 2 þ 2Þ ð Þ d ð Þ dx dy dz , with scale factor a t . gS tu . In what follows, we will use the temperature of The decoupling of dark matter takes place in the early the visible sector and convert Td to T as needed. For Universe in the radiation-dominated regime, in which par- convenience we write ticles with masses m T are the dominant component of d T ðtÞ g ðtÞ g ðt Þ 1=3 the cosmic energy budget. To a good approximation, we ðtÞ¼ d ¼ S S u : (7) ð Þ d ð Þ ð Þ may therefore ignore contributions from nonrelativistic T t gS t gS tu species in thermal equilibrium with the radiation and ap- The success of big bang nucleosynthesis (BBN) and the proximate the energy density as structure of the cosmic microwave background (CMB) 2 power spectrum place tight bounds on any new relativistic 4 ¼ gT (1) R 30 degrees of freedom in the dark sector. The limit on the effective number of light neutrino species is N ¼ 3:24 and the entropy density as 1:2 at the 95% confidence level [27], which gives 2 2 3 d 4 7 s ¼ g T ; (2) g ðt Þ¼ 2 ðN 3Þ2:52 ð95% confidenceÞ 45 S BBN 8 where, as usual, (8) X 4 X 4 for three light SM neutrino species [28]. The 5-year ¼ Ti þ 7 Ti g gi gi (3) WMAP data [29] also bounds the number of neutrino i¼bosons T 8 i¼fermions T species by N ¼ 4:4 1:5 at the 65% confidence level, and the 7-year WMAP data [30] places a tighter lower limit X 3 X 3 Ti 7 Ti of N > 2:7 at the 95% confidence level. gS ¼ gi þ gi (4) i¼bosons T 8 i¼fermions T A. The Boltzmann equation and g is the number of internal degrees of freedom for i c particle species i. Let us assume the dark matter is a stable particle that annihilates with a thermalized annihilation cross section For T * 300 GeV, g ¼ g ¼ 106:75, which includes S h i all particles in the Standard Model. When 100 MeV * v . The general Boltzmann equation governing the num- T * 1 MeV, the electron and positron are relativistic and ber density n of a particle of mass m is ¼ ¼ so gS g 10:75. At the temperature of the CMB n_ þ 3Hn þhviðn2 n2 Þ¼0; (9) ¼ ¼ ¼ EQ today, T0 2:725 K, gS;0 3:91, and g;0 3:36. Consider a dark sector that was in thermal equilibrium where H is the Hubble parameter with the visible sector at some very high temperature scale, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 tot below which they decouple effectively enough to consider a_ 8 4 Gg H ¼ ¼ G ¼ T2 (10) each sector separately to be in equilibrium. The visible a 3 R 45 sector is at temperature T with entropy density sðTÞ, while the dark sector is at temperature Td with entropy density and nEQ is the equilibrium number density sdðTdÞ. The expansion of the Universe is governed by both Z g g m 3 E=Td ¼ 2 sectors with nEQ d pe~ m TK2 ; (11) ð2Þ3 22 T T 4 totð Þ¼ ð Þþ dð Þ d g T g T g Td ; (5) where K is the modified Bessel function of the second T 2 kind of order two. Generalizing the traditional treatment, but quantities in the dark sector (for example, the dark we allow for the possibility that the mass of the dark matter matter annihilation cross section and number density) are m~ c ðÞ is a function of a real scalar chameleon field and determined by Td [26]. denote -dependent masses and couplings with a tilde.

123529-2 DARK MATTER WITH DENSITY-DEPENDENT INTERACTIONS PHYSICAL REVIEW D 86, 123529 (2012)

It is convenient to scale out the effects of the expansion where BðiÞ and m~ c ðiÞ are evaluated at the initial value of the universe by defining i ¼ ðxiÞ. ð Þ nc After the freeze-out value xf, Y x will asymptotically Y (12) s approach a constant value Y1. The energy density of nonrelativistic dark matter today is then (nc ðxÞ and YðxÞ are taken to be independent of ) and to ¼ ð Þ ð Þ¼ ð Þ use a new independent variable, related to the cosmic time 0 m~ c 0 nc x0 m~ c 0 Y1s0 t through 22 ¼ ð Þ 3 m m~ c 0 Y1 gS;0T0 : (22) xðtÞ T ; (13) 45 TðtÞ Having generalized the usual treatment of dark matter as where mT is some constant mass scale. In the usual deri- a fluid to the case in which there is a chameleon field vation, mT is chosen to coincide with the dark matter mass; determining the dark matter properties, we now turn to however, since our dark matter has varying mass, we use specific examples of particle physics models in which these this constant parameter instead. Defining phenomena might arise. sffiffiffiffiffiffiffiffiffiffiffiffi ¼ 45 1 b 3 (14) III. GAUGED DARK MATTER 4 G mT Consider dark matter to consist of a Dirac fermion c , allows us to write qffiffiffiffiffiffiffi charged under a dark U(1) gauge group with gauge boson dx mT tot A, and a dark Higgs field that spontaneously breaks the ¼ g ; (15) dt bx U(1). We also introduce a chameleonlike field that is a real scalar field with properties that depend on the dark which can be used to rewrite the Boltzmann equation for matter energy density. The chameleon couples to the other the dark matter as particles in the dark sector by entering into the dark matter 0 B mass m~ c ðÞ, the U(1) coupling f~ðÞ, and other couplings Y ðxÞþ ðY2 Y2 Þ¼0: (16) x2 EQ described below. We consider only an isolated dark sector Here a prime denotes a derivative with respect to x, and so that we may investigate the properties of this simple model without the complications of coupling to the visible 2 2 g sector. B ¼hvi pffiffiffiffiffiffiffiS bm2 ; (17) tot T 45 g A. A toy model for varying coupling which may depend implicitly on in our model via a dependence in the cross section. Note that, in terms of As a first step, let us consider the QED Lagrangian with these new variables, the equilibrium term is a real scalar field , but in which we allow the coupling constant e to vary as a function of spacetime [32]. 45g m~ c ðÞ 2 x m~ c ðÞ Specifically, it can vary as a function of . Let us write Y ¼ x K ; (18) EQ 2 2 2 ~ ð2 Þ gS mT mT the new coupling as fðÞ. Thus, ¼ 1 1 with g 2 for Dirac dark matter. L ¼ ð Þ ð Þ QED @@ V F F It remains, at this level, to specify Y xi , the initial 2 4f~2ðÞ condition for Y. We consider Y YEQ, the departure þ ic 6@c mc cc c c A ; (23) from equilibrium [31], which obeys where F ¼ @ A @ A . Making the redefinition 0 ¼ 0 B ð þ Þ Y 2YEQ : (19) ! ~ð Þ EQ x2 A f A, we obtain At early times (1

123529-3 KIMBERLY K. BODDY, SEAN M. CARROLL, AND MARK TRODDEN PHYSICAL REVIEW D 86, 123529 (2012) c ! ei! c (26) replace by hi in the equations of motion. The vacuum expectation value (VEV) generates an additional mass term c ! eþi! c : (27) for c , but we can redefine m~ c ðÞ to absorb this term. We then have ~ ~ If we can neglect factors of ð@f=fÞ compared to all other mass scales in the theory (except the Planck mass), then the ðiD6 m~ c ðÞÞc 0 (34a) Lagrangian simplifies to the approximately gauge- 0 0 h V ðÞþm~ c ðÞcc 0: (34b) invariant form c 1 1 We calculate the energy-momentum tensor for by L ð Þ QED @@ V F F varying the action with respect to the metric. Taking care 2 4 to correctly handle the nontrivial metric dependence of the þ c 6 c cc ~ð Þc c i @ mc f A (28) covariant derivative [33], we have with U(1) current ðc Þ i T ¼ ½c ðrÞ c ðrð c ÞÞ c jðxÞ¼f~ðÞc c : (29) 2 ~ fðÞc ðAÞ c ; (35) B. The cosmological equations of motion where we have integrated by parts and used the field We now include gravity and a complex dark Higgs field equation of motion. Taking the trace, we obtain

to break the U(1) symmetry and give the dark gauge field ðc Þ i gT ¼ ½c r6 c c r6 c f~ðÞc A6 c a mass. We allow for a varying dark matter mass by using 2 ð Þ the effective mass parameter m~ c , and in the spirit of 1 effective field theory, we also allow all couplings [not just ¼ ½c iðrþ6 if~ðÞA6 Þc c iðr6 if~ðÞA6 Þc ~ð Þ 2 the U(1) coupling f ] to depend on . ¼ ð Þcc ~ ~ m~ c ; (36) Neglecting factors of ð@f=fÞ, the action is then Z pffiffiffiffiffiffiffi R where, again, we have used the Dirac equation for c and c 4 1 S d x g g rr VðÞ to obtain the last line. If we model the dark matter as 16G 2 nonrelativistic dust, its pressure is zero and so the trace 1 ð Þy ð Þ þ c 6 c of the stress tensor is approximately given by c . Thus, D D V0 F F i D 4 c ¼mc ðÞcc : y ~ (37) m~ c ðÞcc ~c ðÞð þ Þcc ; (30) As a final step in this section, we use this result to rewrite the equation of motion (34b)as where the gauge covariant derivative is D ¼r þ ~ 0 ifðÞA. The equations of motion for the fields then h ð Þ¼ Veff 0; (38) follow as y where the effective potential is ðiD6 m~ c ðÞ~c ðÞð þ ÞÞc ¼ 0 (31)

V ¼ VðÞþm~ c ðÞnc 0 0 ~0 eff h V ðÞþm~ c ðÞcc f ðÞc A6 c 2 3 0 y 2 mT ~ ðÞð þ Þcc ¼ 0; (32) ¼ VðÞþm~ c ðÞYðxÞ g : (39) c 45 S x where a prime denotes differentiation with respect to . Let us assume that the universe is dark-charge symmetric, so the average charge current density is negligible com- IV. CHAMELEON BEHAVIOR pared to the dark matter number density [see (37) below]. With a complete model in place, we now turn to a ~0 ~ Thus, the term proportional to f =f should be small com- detailed investigation of the dynamics. We first examine 0 0 0 pared to the one containing m~ =m~, given that f~ =f~ m~ =m~ the chameleon field, which is central to the effect we seek. to within a few orders of magnitude—a condition we will Assuming that is homogeneous and isotropic, so that we enforce later. We may write this last equation as can neglect spatial derivatives in h, the equation of 0 0 0 y motion becomes h V ðÞþm~ c ðÞcc ~c ðÞð þ Þcc 0: € þ _ þ 0ð Þþ 0 ð Þ ¼ (33) 3H V m~ c nc 0: (40) We will arrange for the dark Higgs to have a sufficiently It is convenient for seeking numerical solutions to work large mass that its perturbations are negligible and simply with a dimensionless variable,

123529-4 DARK MATTER WITH DENSITY-DEPENDENT INTERACTIONS PHYSICAL REVIEW D 86, 123529 (2012) The effective potential in (39)isnow P ; (41) mT ð Þ¼ 4 =m1 þ ð =m2 Þ Veff e mc 1 A2e 2 and to use x as our independent variable. The equation of 2 mT 3 YðxÞ g ; (44) motion becomes 45 S x

2 b2x2 dV possessing a critical point at P00ðxÞþ P0ðxÞþ 3 tot 3 x m g d ¼Pm m m m mc m Y T T ¼ 1 2 ln A 1 T ; (45) 2 2 min 2 4 3 dm~ c m m m x þ 2 b gS ð Þ¼ 2 1 2 tot Y x 0: (42) ¼ 45x g d PmT which is real and finite. In order to generate a mass for the excitations of , we require this critical point to be a We choose the initial conditions for to begin at the minimum, which holds for minimum of its effective potential and to move with the same initial velocity as the changing minimum. The mini- m2 >m1: (46) 0 ð Þ¼ mum min solves the equation Veff min 0, so one of The minimum moves with a speed the initial conditions for this equation can be obtained by m m dY 3 evaluating this expression at xi, using the relevant value for dmin ¼ 1 2 ð Þ Y ; (47) Y xi from (21). Furthermore, since min is a function of x, dx m2 m1 dx x the initial velocity is found simply by taking a derivative which is positive ( increases with x). Finally, we and using the Boltzmann equation to obtain the relevant min 0 identify the initial conditions for , value for Y ðxiÞ. 3 m m m mc m Yðx Þ ðx Þ¼ 1 2 ln A 1 T i ; (48) A. Exponential potentials i 2 4 3 m2 m1 m2 xi Our goal here is to work out a single example model that exhibits the effects we are investigating, while at the same d m1m2 dY 3 ðxiÞ¼ YðxiÞ ðxiÞ time remaining compatible with experimental constraints. dx m2 m1 dx xi For simplicity we will choose exponential functions, which 2 m m 3 m~ c Y ðx Þþðx m~ c Þ=ð4Bm Þ also have the nice feature that observables approach a fixed ¼ 1 2 þ EQ i i T : ð Þþð 2 Þ ð Þ asymptotic value at late times. m2 m1 xi mT YEQ xi xi m~ c = 2BmT With these comments in mind, we therefore choose the (49) form of the effective potential and U(1) coupling to be In order to ensure m~ c > 0, we require 4 =m VðÞ¼ e 1 (43a) ð Þ >m2 ln A2 (50) =m m~ c ðÞ¼mc ð1 A e 2 Þ (43b) 2 for all relevant for our calculation. ~ð Þ¼ ð þ =m3 Þ 3 f e 1 A3e ; (43c) B. An attractor solution where and mc are constants with dimensions of mass, A particularly interesting and simple possible evolution and e and A2, A3 > 0 are dimensionless. The term with A2 is necessary to incorporate the properties of c into the for the chameleon field is for it to begin at the minimum ¼ of the effective potential and then to adiabatically track equation of motion for . The possibility for A3 0 (constant gauge coupling) is viable, but we are specifically this minimum as it evolves cosmologically. This attractor interested in increasing the cross section for c as the solution [34] is achieved if the physical mass of the cha- ~ meleon satisfies Universe expands. We choose this form for f so that both qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the annihilation and scattering cross sections, which we m ¼ V00 ð Þ H: (51) calculate below, increase with time. ;ph eff min The largest energies of the particles in our theory are of If (51) holds during radiation dominance, when order mc for nonrelativistic dark matter, since all other qffiffiffiffiffiffiffi particles should be lighter than the dark matter to allow for mT tot 2 HR ¼ g x ; (52) annihilation. We, therefore, require m2, m3 mc to sup- b press higher-dimensional operators involving derivatives then we can avoid solving the coupled differential of m~ c and f~ when we expand the action. Additionally, we equations (16) and (40) and simply use the expression for * need m1 to suppress higher-dimensional operators in min for the evolution of . Similarly, if (51) holds during the self-couplings of . matter domination, when

123529-5 KIMBERLY K. BODDY, SEAN M. CARROLL, AND MARK TRODDEN PHYSICAL REVIEW D 86, 123529 (2012) x 3=2 term generates a contribution to the mass of c , but since H ¼ H 0 ; (53) M 0 x c already has a Dirac mass, we need not rely on the dark Higgs to be the sole source of the c mass. We, therefore, then we can easily determine 0, the value of today, absorb the dark Higgs contribution into the definition of which is needed to calculate the values of the -dependent m~ c and retain the freedom to choose this mass scale and parameters today. the coupling ~c ðÞ separately. Under the approximation that m2 m1, A typical choice for the pure dark Higgs potential ð Þ 3 V0 is 22 mc m 1=2 T 1=2 1=2 3=2 m~ ;ph A2 Y gS x : (54) 45 m1m2 1 1 2 V ðÞ¼ ~ ðÞ y v2 ; (59) 0 4 h 2 It follows that HR decreases more rapidly than m~;ph with time, whereas during matter domination, HM and m~;ph have the same x dependence. We shall verify later that which, when expanded about the VEV, yields these attractor solutions exist by numerically solving all 1 1 1 the relevant equations of motion. V ðhÞ¼ ~ ðÞv2h2 þ ~ ðÞvh3 þ ~ ðÞh4 (60) 0 4 h 4 h 16 h V. PARTICLE PHYSICS INTERACTIONS The mass of the physical dark Higgs particle h is therefore AND CONSTRAINTS sffiffiffiffiffiffiffiffiffiffiffiffiffi In the adiabatic regime described above, we now have ~ hðÞ all the ingredients necessary to understand the cosmologi- m~ hðÞ¼ v; (61) cal evolutions of the fields. We next turn to the particle 2 physics phenomenology of the model. To do this, we rewrite the action (30) without gravity to give the and we see that the masses of the A and h fields are then Lagrangian related by 1 sffiffiffiffiffiffiffiffiffiffiffiffiffi L @ @ VðÞðD ÞyðDÞV ðÞ 2 2 0 M~ ðÞ¼f~ðÞ m~ ðÞ: (62) A ~ h hðÞ 1 F F þ ic 6@c m~ c ðÞcc 4 y Since the relative sizes of f~ðÞ and ~ ðÞ are unrestricted, f~ðÞc A6 c ~c ðÞð þ Þcc ; (55) h in principle the relative masses of A and h are not fixed. ~ However, in order to simplify the analysis, we will impose with D ¼ @ þ ifðÞA. the hierarchy m~hðÞ > 2M~ AðÞ for all relevant so that h has a tree-level decay channel to A. Uð1Þ A. Breaking the dark symmetry Our Lagrangian at this stage is then The potential of the dark Higgs field is chosen so that this field acquires a vacuum expectation value (VEV) 1 1 1 ~ 2 2 L ¼ @@ VðÞ @h@ h hðÞv h v 2 2 4 h0jðxÞj0i¼pffiffiffi : (56) 1 1 1 2 ~ ðÞvh3 ~ ðÞh4 FvF 4 h 16 h 4 v

Decomposing into two real scalar fields via 1 2 M~ ðÞA A þ ic 6@c m~ c ðÞcc 2 A 1 ðxÞ¼pffiffiffi ðv þ hðxÞÞeiðxÞ=v; (57) 1 f~ðÞc A c ½2f~2ðÞh2AA 2 4 1 pffiffiffi we can then use unitary gauge ðxÞ¼0 to rewrite the ~ ~ ½2fðÞM~ AðÞhA A 2c ðÞhcc: (63) kinetic term for as 2 1 1 ðD ÞyD ¼ @h@ h f~2ðÞðv þ hÞ2AA : What remains is to incorporate the fact that is adia- 2 2 batically tracking the minimum of its effective potential. (58) To achieve this, we expand ðxÞ¼cðtÞþðxÞ around its classical value and recall that m2 and m3 are sufficiently Thus, the Goldstone boson is eaten to give the dark Uð1Þ large to suppress nonrelevant terms of OðÞ or higher. The ~ gauge boson A a mass M~ AðÞ¼fðÞv. The Yukawa Lagrangian (63) then becomes

123529-6 DARK MATTER WITH DENSITY-DEPENDENT INTERACTIONS PHYSICAL REVIEW D 86, 123529 (2012) 1 1 1 1 1 00 2 3 ~ 2 2 L ¼ @@ @h@ h F F þ ic 6@c VðcÞþ V ðcÞ þ Oð Þ ½hðcÞþOðÞv h 2 2 4 2 4 1 3 1 3 1 ~ ð ÞþOð Þ v 3 ~ ð ÞþOð Þ 4 ½ ~ 2 ð ÞþOð Þ ½~ð Þ h c h h c h MA c A A f c 6 2 24 2 2 pffiffiffi 0 2 ~ þ OðÞc A c ½m~ c ðcÞþm~ c ðcÞ þ Oð Þcc 2½c ðcÞþOðÞhcc 1 1 ½2f~2ð ÞþOðÞh2AA ½2f~ð ÞM~ ð ÞþOðÞhAA : (64) 4 c 2 c A c

B. The dark matter annihilation cross section subdominant to the other processes, which are s wave. M Our central goal is to understand how the dependence of Also, the diagrams involving exchanges of h in 1 and M dark matter cross sections on the chameleon field changes 2 do not significantly contribute. Thus, the dark Higgs the standard dark matter creation, evolution, and detection has the opportunity to influence dark matter annihilations ~ story. To this end, we next turn to the calculation of the only via 3. Let us insist that the Yukawa coupling c is dark matter annihilation cross section. The relevant small enough (recall that the dark matter does not rely on Feynman rules can be found in the Appendix. this coupling to obtain a mass) such that 3 can be safely We assume that the dark matter is the heaviest particle in ignored. In this case only 1 remains and, since it is an the dark sector, such that m~ c m~h, M~ A. Then, the lowest s-wave cross section, it is a simple task to carry out the order, tree-level processes for 2 ! 2 dark matter annihila- thermal averaging required in the Boltzmann equation. tion are shown in Fig. 1, and their amplitudes are Note, however, that if thermal averaging is needed (following Ref. [35]), we must use the dark sector temperature Td M ¼ ½~2ð Þð ð 0 Þ 1 i 0 0 v2 f c c p1 k in the expression 1 2 pffiffiffi 1 þ ð 0 Þ Þþ ~ð Þ ~ ð Þ ~ ð Þ c p1 k2 2f c c c MA c 2 ð þ Þ 1 g h p1 p2 g u1 (65) hvi¼ ðnEQðT ÞÞ2 2ð2Þ4 c d Z pffiffiffi 2 0 0 1 pffiffiffi s M ¼ iv 2~c ð Þðc ðp k Þþc ðp k ÞÞ ð 2 Þ ð Þ 2 2 c 1 1 1 2 ds sTdK1 s 4m~ c s : (71) 2 4m~ c Td þ p3ffiffiffi ~ ð Þ ~ ð Þv ð þ Þ c c h c h p1 p2 u1 (66) 2 pffiffiffi 0 M ¼ 0 ½ ~ð Þ ~ ð Þð ð Þ 3 i1 v2 2f c c c c p1 k1 þ ð 0 ÞÞ þ ~2ð Þ ~ ð Þ c p1 k1 f c MA c ð þ Þ A p1 p2 u1: (67) Working in the center-of-mass frame and in the nonrelativistic limit we then obtain ~4ð Þ f c 1v 2 (68) 16m~ c ðcÞ

15~4 ð Þ c c 2 2v 2 v (69) 128m~ c ðcÞ

2 2 2 f~ ð Þ~c ð Þ ~c ð Þ v c c ¼ c ; (70) 3 ~ 2 ð Þ v2 8MA c 8 where v is the relative velocity of the incoming particles. The interaction of main interest is between the dark FIG. 1. Tree-level c annihilation diagrams. The massive vec- matter and the gauge boson mediator. The dark Higgs’s tor boson A is a wavy line, and the scalar h is a dashed line. primary role is to break the U(1) symmetry to give the Annihilations to A þ A and h þ h via exchange and annihi- mediator a mass, and most of its particle interactions can be lations to ﬁnal-state particles are suppressed by large-mass neglected. The contribution 2 is p wave and thus factors.

123529-7 KIMBERLY K. BODDY, SEAN M. CARROLL, AND MARK TRODDEN PHYSICAL REVIEW D 86, 123529 (2012) the dark matter becomes comparable to the range of the 2 interaction. At lower velocities with A v, the Yukawa potential cannot be ignored. As v ! 0, the de Broglie wavelength increases to a value larger than the interaction range, and thus the enhancement saturates at

1 ~M~ A Sk : (75) m~ c FIG. 2. Ladder diagrams for dark matter annihilation (left) and A scattering (right). Furthermore, zero-energy bound states may form for cer- tain values of A, giving resonance regions with larger 2 C. Corrections to the cross section enhancements A=v until they are cut off by finite width effects. In the early Universe, freeze-out typically occurs at We are interested in nonrelativistic dark matter, for velocities v 0:3, so that > 1 and the Sommerfeld which the relative velocities are much less than the speed f v enhancement can be ignored. Note that there are no of light. It is well known that for sufficiently low velocities, enhancements for > 1. nonperturbative effects can have a large impact on the A To find the thermally averaged cross section, taking into annihilation and scattering cross sections; and ladder dia- account the Sommerfeld enhancement, we integrate S grams, such as the ones shown in Fig. 2, must be included k using a Maxwellian distribution in the calculation.

4 2 2 FðvÞ¼ pffiffiffiffi v2ev =v ; (76) 1. The annihilation cross section v3 In the case of annihilation, performing this summation is where v is the characteristic velocity of the astrophysical equivalent to solving the Schro¨dinger equation in quantum system of interest. Thus, mechanical scattering theory [36]. This yields the so-called h i¼ð Þ h i ‘‘Sommerfeld enhancement’’ [37] of the annihilation cross v v s-wave Sk (77) section (for detailed reviews in the context of dark matter, Z 1 see, for example, [38–40]). We consider the annihilation hS i¼ dvFðvÞS : (78) ¼ k k cross section 0 for a pointlike interaction near r 0 in 0 perturbative field theory. For small velocities, the attractive For the purposes of this paper, we choose to work in the Yukawa potential A > 1 regime. This has two consequences. Practically, the ~ ~ calculation becomes much simpler, since we need not VðrÞ¼ e MAr; (72) r worry about the Sommerfeld enhancement at all. In addi- tion, by deemphasizing the Sommerfeld enhancement, we ¼ ~2ð Þ where ~ f c =4, distorts the wave function at the clarify the extent to which the novel effects developed in origin and cannot be ignored. Including the potential will this paper can alone increase the cross section over time in ¼ enhance the annihilation cross section to 0Sk by the areas of parameter space that the Sommerfeld enhance- Sommerfeld enhancement factor Sk. Let us define the ment cannot reach. dimensionless parameters v ¼ (73) 2. The scattering cross section v ~ To find the scattering cross section, we can use nonrelativistic quantum mechanics and sum over partial M~ A A ¼ ; (74) waves. The total cross section is ~m~ c 4 X1 ¼ ð2l þ 1Þsin2 ; (79) where v is the velocity of each annihilating particle in k2 l the center of mass frame. In the case of a massless gauge l¼0 boson with a Coulomb potential, it is possible to solve although a more useful quantity to compare to observatio- the Schro¨dinger equation analytically to obtain the nal constraints is the transfer cross section Sommerfeld enhancement. Z For a massive gauge boson, the situation is more com- ¼ ð Þ d tr d 1 cos plicated, since the attractive potential has a finite range that d limits the enhancement from being arbitrarily large for 4 X ¼ ½ð2l þ 1Þsin2 2ðl þ 1Þ very low velocities. In the regime 2 , we recover 2 l A v k l the Coulomb case. At the crossover point v A (or sinl sinlþ cosðlþ lÞ; (80) equivalently m~ c v M~ A), the de Broglie wavelength of 1 1

123529-8 DARK MATTER WITH DENSITY-DEPENDENT INTERACTIONS PHYSICAL REVIEW D 86, 123529 (2012) directly applied to bounds on physics at freeze-out or before. It is important to remember here that, in our model, the evolution of the chameleon ﬁeld means that such a connection is far less direct, and such bounds typically do not apply in the early Universe.

FIG. 3. Tree-level h decay. D. Dark decays The dark Higgs h and the dark gauge boson A are which controls the rate at which energy is transferred allowed to decay. As mentioned earlier, we assume m~ ð Þ > M~ ð Þ h between colliding particles. Following [41], analytic esti- h c 2 A c so that has a tree-level decay chan- A mates for the cross section are nel to , as shown in Fig. 3. Its decay width is then uvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ~2ð Þ 3ð Þ u ~ 2 ð Þ ¼ ð þ Þ2 f c m~h c t 4MA c 2 2 1 L (81) ¼ 1 v h ~ 2 ð Þ 2ð Þ rel 32 MA c m~h c 2 4 4M~ ðcÞ M~ ðcÞ ¼ 4 ð þ Þ 1 A þ 12 A : (86) tr 1 L ; (82) 2 4 2 2 m~ ðcÞ m~ ðcÞ vrel h h ¼ where L vrelbmax is the largest angular momentum Although the A particle is allowed to decay to parti- needed to describe the interaction between two particles cles, which are substantially smaller in mass, this occurs of reduced mass ¼ m~ c =2 that travel with a relative through a 1-fermion-loop process, as shown in Fig. 4. The velocity v and maximum relevant impact parameter amplitude is also suppressed by two factors of m2 from the rel 2 b . Note that these estimates are only valid for L * 1. Oð Þ term in the expansion of m~ c ðÞ. The nonzero max ~ We estimate the impact parameter by solving amplitude in the limit of m~ MA, m~ c is

~2 2 0 2 1 f =4 ~ 4i f~ðm~ c Þ v2 ¼ e MAbmax : (83) M ¼ 0 ð Þ½ ~ ½ 2 ð 0 Þ2 rel k2 p 4m~ c MAC0 p ; p k2 ; 2 bmax ~ 2 MA 2 þð 2 þ ~ 2 Þ ½ 2 If we work in the A > 1 regime to avoid Sommerfeld k2; m~ c ; m~ c ; m~ c 8m~ c MA B0 p ; m~ c ; m~ c enhancements, then we will also tend to avoid enhance- 2 ½ 02 ments to the scattering cross section and can expect to be 8m~ c B0 k2 ; m~ c ; m~ c ; (87) working in the Born limit. Simply taking the nonrelativistic where B and C are scalar Passarino-Veltman functions limit of the perturbative cross section gives 0 0 [43–46], defined via 4 2 2 f~ ð Þm~ c ð Þ m~ c ð Þ Z ¼ c c ¼ c 1 1 : (84) ½ 2 2 2¼ 4 ~ 4 ð Þ 8v4 B0 p ;m ;m d l 8MA c i2 ðl2 þ m2Þ½ðl þ pÞ2 þ m2 Assuming that dark matter self-interactions are not needed (88) to explain the structure of dwarf galaxies [41], we use a conservative bound [42] (see also [6,7]), ½ 2 ð Þ2 2 C0 p ; p p1 ;p1;m;m;m 2 Z =m~ c < 0:1cm=g; (85) 1 1 ¼ d4l : 2 ð 2 þ 2Þ½ð þ Þ2 þ 2½ð þ Þ2 þ 2 for characteristic velocities of 10 km=s. As we mention i l m l p m l p1 m below, it would not be difficult to find parameters that (89) violate this bound. In the usual treatment of dark matter, constraints such as The C0 integral is finite and, in the approximation m~ c this one, obtained from present-day observations, can be M~ A m~, reduces to

FIG. 4. 1-loop A decay. Only the ﬁrst diagram is nonzero.

123529-9 KIMBERLY K. BODDY, SEAN M. CARROLL, AND MARK TRODDEN PHYSICAL REVIEW D 86, 123529 (2012) 1 ½ 2 ð 0 Þ2 02 VI. NUMERICAL SOLUTIONS C0 p ; p k2 ;k2 ; m~ c ; m~ c ; m~ c 2 : (90) 4m~ c While we have described a number of ways to understand the evolution of the fields analytically, including, for The B integral diverges, so we cut off the loop-momentum 0 example, the adiabatic approximation in which the chame- integral at some large scale. Using m for this purpose, 3 leon tracks the minimum of its effective potential, ulti- since we will often find it numerically to be the largest mately we are able to numerically solve the relevant mass-suppression scale in our theory, we have equations of motion completely. To do so, of course, m we must make sensible choices for our parameters to ½ 2 ½ 02 3 B0 p ; m~ c ; m~ c B0 k2 ; m~ c ; m~ c 2ln : (91) satisfy the various bounds and simplifying inequalities m~ c we have specified. We need to implement the correct relationship between Putting everything together, the decay width of A is then the dark sector temperature Td and that in the photon sector given by T, which in turn requires us to correctly enumerate the massless degrees of freedom at the relevant scales. At 4 2 1 mc A2 m3 the unification scale, all the dark particles ðc ;A;h;Þ 2f~2 e 4c=m2 : A ln (92) d ð Þ¼ 6M~ A m2 m~ c are relativistic, so gS tu 8:5. Around the epoch of dark matter freeze-out, only c is nonrelativistic, so d ð Þ¼ ¼ ð Þ¼ The A bosons must decay efficiently enough not to con- gS tf 5. Thus, at freeze-out, f 1:19 for gS tf tribute significantly to the energy density budget today. 106:75 or f ¼ 0:56 for gSðtfÞ¼10:75. With these num- Though the decaying exponential makes meeting this bers, the bound on the number of effective neutrino species criterion more difficult, there is still a small sample of in (8) is easily satisfied. parameter space for which the A energy density does not The model is insensitive to M~ AðÞ and m~hðÞ at lowest pose a problem. order. We choose v such that cc ! AA is kinematically

3.5 10 20

3.0 10 21 2.5 s 10 22 3 2.0 cm

e 23

0 10

1.5 x v 24 1.0 10

0.5 10 25

0 100 200 300 400 500 10 7 0.001 10 105 2 m GeV mx0 cm g

10 6 105 g f 2 x 10 4 10 v cm 0 0 x 0.001 v 100 mx

10 7

1 0 100 200 300 400 500 0 100 200 300 400 500 m GeV m GeV

FIG. 5 (color online). Scan of parameter space. Blue points indicate sets of parameters that satisfy all constraints, except (for most 2 points) for having a negligible A energy density. Red points do not satisfy the scattering cross section bound =m~ c < 0:1cm=g. Green points do not satisfy the adiabatic condition in (51) and should be solved with the coupled differential equations.

123529-10 DARK MATTER WITH DENSITY-DEPENDENT INTERACTIONS PHYSICAL REVIEW D 86, 123529 (2012) today. For this ﬁgure we use the bounds from the 7-year 0.01 WMAP data [30], assuming a CDM cosmology,

10 11 2 ¼ 0 ¼ DMh 0:1109 0:0056: (93) c0 10 20 Parent Given these constraints, we numerically solve the 10 29

Decays Boltzmann equation and show a sample of parameter space in Fig. 5, resulting from a random, uniform scan 10 38 2½ 2½ 5 7 2 over mc 0:1; 500 GeV; m1 10 ; 10 GeV; m2 ½ 5 8 2½ 5 8 10 47 5 10 ; 5 10 GeV; m3 5 10 ; 5 10 GeV; 0 100 200 300 400 500 2½10; 103 GeV; A 2½0:1; 9:9; A 2½0:1; 10; and m pffiffiffiffiffiffiffi 2 3 e 2½0:01; 4. The upper-left panel shows the coupling FIG. 6 (color online). The number of A decays per particle parameter e vs the dark matter mass parameter mc . The between freeze-out and BBN. Points above the horizontal line at upper-right panel shows the annihilation cross section v 1 indicate that all A particles should have decayed and thus do vs the scattering cross section =m~ c , both evaluated at x0 not contribute significantly to the energy budget of the Universe. today. The bottom panels show the boost in annihilation cross section from freeze-out to today and the scattering allowed today, while ensuring A > 1 and <~ 1. We must cross section today vs the mass parameter mc . Again, there ~ ð Þ v then check that MA 0 satisfies scattering cross-section is flexibility when choosing without affecting the evolu- bounds. It is simplest to assume the attractor solution for tion of and Y at lowest order, so it is possible to obtain and then later verify that it is in fact adhered to. The A valid models for a scaled value of =m~ c . Here, we show gauge bosons need to decay away before BBN so that their the largest possible scattering cross sections, while staying energy density is negligible. Finally, we must ensure that within the bound A > 1. As demonstrated in Fig. 6, only a the evolution ends with the observed density of dark matter small portion of the sampled parameter space fulfills the

0.01

106 10 4

105

10 6 104 Y GeV

10 8 1000

10 100 10

Yeq

10 10 12 10 0.1 0.001 10 5 10 7 10 9 10 11 100 10 1 0.1 TGeV TGeV

100

0.1 f GeV

m 0.01

0.001

10 0.1 0.001 10 5 10 7 10 9 10 11 10 0.1 0.001 10 5 10 7 10 9 10 11 TGeV TGeV

FIG. 7 (color online). The evolution of (upper left), Y (upper right), m~ c (lower left), and f~ (lower right) as a function of T in GeV for the model with mc ¼ 123 GeV. The red, dotted line indicates the approximate dark matter freeze-out temperature.

123529-11 KIMBERLY K. BODDY, SEAN M. CARROLL, AND MARK TRODDEN PHYSICAL REVIEW D 86, 123529 (2012) v m 10 6 10 23 10 7 10 24 0.001 10 8 10 25 10 9 10 26 10 10 10 27 10 11 28 7 2 10 3

s 10 g 3 10 12 10 29 2 cm cm GeV 10 13 10 30 GeV 10 14 10 31 10 15 10 32 10 16 10 33 10 17 10 34 4 10 18 10 7 9 11 10 0.1 0.001 10 5 10 10 10 10 0.1 0.001 10 5 10 7 10 9 10 11 T GeV TGeV

FIG. 8 (color online). The evolution of the annihilation cross section hvi (left) and the scattering cross section =m~ c (right) as a function of T in GeV with mc ¼ 123 GeV. The red, dotted line indicates the approximate dark matter freeze-out temperature. requirement that the A gauge boson energy density is now. This difference is due to the scattering cross section, negligible by the time of BBN. While finding a set of =m~ c / m~ c =v 4, and the annihilation cross section, v / parameters that satisfies all constraints is certainly pos- 4 2 f~ =m~ c , depending differently on via m~ c and f~.We sible, the effect of having very large increases in the choose the form of f~ to force the annihilation cross section annihilation cross section does not seem to be a general to grow more slowly, whereas the scattering cross section feature of the model. has no such term countering its growth. With these par- As a concrete example, we show a specific model with ticular choice of parameters, the scattering cross section is the following parameter choices: mc ¼ 123 GeV; mT ¼ ¼ ¼ ¼ ¼ too small to have interesting astrophysically observable mc ; m1 38 TeV; m2 500 TeV; m3 500 TeV; consequences. ¼ ¼ ¼ v ¼ 18 GeV; A2 0:6; A3 9:2; e 0:96; and 1 GeV. As shown in Fig. 5, there are other choices of parameters This comprises an optimistic set of parameter choices that will still give a boost to the annihilation cross section that satisfies all our bounds and provides a large change while yielding a larger scattering cross section to match 6 of order 10 in the annihilation cross section over observational bounds [6,7]; however, again, most of the v the history of the Universe. Our choice for the value of plotted parameter space is restricted from the A energy ¼ gives A 1:07 today, and we can ignore the Sommerfeld density requirement. One option for increasing the viable v enhancement. Larger values of work equally well; parameter space is to relax the requirement that > 1 and v v A increasing increases A and decreases =m~ c to work in the regime of Sommerfeld enhancements; our 2 v 4. The dark matter relic density is c h ¼ 0:1097, model would still provide significant increases in the cross within a standard deviation of the observed value. The sections, and Sommerfeld enhancements would serve to scattering cross section today is 4:9 104 cm2=g, well further increase the boosts. Another clear option is to open below the conservative limit in (85). We must also check an alternative decay channel for A. that these parameters satisfy the assumptions we have made in writing down the model. For example, we VII. CONCLUSIONS ~ ~ ~_ ~ neglected terms with @f=f, and here we note that f=f 9 6 In this paper we have investigated the possibility that the 10 –10 GeV, which is much smaller than other mass properties of dark matter depend crucially on the dynamics terms in the perturbative expansion. The adiabatic approxi- of a chameleon field—a scalar field whose cosmological 11 mation is satisfied with H=m;ph 10 throughout the evolution depends not only on its bare potential but also on evolution of . Finally, we use the decay width of the A the local density of other matter (such as dark matter itself) particles to determine that they have decayed away in the in the Universe. We have shown that such a coupling time from freeze-out to BBN, so they do not contribute to allows the annihilation cross section (for example) of the the energy budget we observe from the CMB. dark matter particles to change by several orders of mag- The results for the evolution of , Y, the dark matter nitude between freeze-out and today, while remaining ~ ¼ mass m~ c , and the coupling f as a function of T mT=x consistent with all observational constraints. We have pre- are shown in Fig. 7. We also show the annihilation and sented a general formalism to describe how this might scattering cross sections in Fig. 8. The scattering cross happen and have provided a specific particle physics ex- section quickly approaches its asymptotic value by the ample, in which all relevant quantities can be calculated. time of dark matter freeze-out, while the annihilation cross While there are significant observational and theoretical section still grows orders of magnitude from freeze-out to constraints on models of this type, it is nevertheless

123529-12 DARK MATTER WITH DENSITY-DEPENDENT INTERACTIONS PHYSICAL REVIEW D 86, 123529 (2012)

FIG. 9. Feynman rules for h (dashed line), c (solid line), A (wavy line), and (dotted line). We include the Yukawa interaction with c and , which is relevant for the 1-loop A-decay amplitude in (87), but other -interaction vertices are not shown. All parameters labeled by a tilde are evaluated at c. possible for the cross section to evolve in such a way that [48,49]. This extension of our model should be able to there may be interesting implications for the detection of easily accommodate the relevant particle physics con- dark matter and for its dynamical effects on late-universe straints [50–53], while easily allowing for decays of the astrophysics. dark gauge boson to Standard Model particles well before There are, of course, other possible complications to this BBN. The dark matter annihilations would still be domi- idea that are beyond the scope of the current paper but that nated by the channel cc ! AA, since annihilation to provide interesting avenues for future study. One natural Standard Model particles would be suppressed by the small step is to couple our model directly to the Standard Model. coupling parameter for the U(1) mixing. However, it is a One way to achieve this is to directly add the dark U(1) to more delicate issue to decide what a natural route would be the current SM gauge group [47]. Another possibility is to to couple the visible and dark scalar sectors, particularly couple to the Standard Model through Uð1Þ kinetic mixing with regards to coupling the chameleon to normal matter.

123529-13 KIMBERLY K. BODDY, SEAN M. CARROLL, AND MARK TRODDEN PHYSICAL REVIEW D 86, 123529 (2012) Finally, we did not attempt a careful analysis of the the 1-loop A decays. We would like to acknowledge helpful effect of late-universe inhomogeneities on the chameleon discussions with Matt Buckley, Katie Freese, Koji ﬁeld or the dark matter properties on which it depends. In Ishiwata, Justin Khoury, and Mark Wise. This work was the speciﬁc models we considered, it seems as if such supported in part by the Department of Energy, the effects would be small, but a more careful examination is National Science Foundation, NASA, and the Gordon warranted. and Betty Moore Foundation.

ACKNOWLEDGMENTS APPENDIX: FEYNMAN RULES

We use MATHEMATICA 8 to numerically solve the The Feynman rules are shown in Fig. 9. All of these Boltzmann equation, JAXODRAW 2.1-0 [54,55] to create diagrams have higher-order corrections that involve the Feynman diagrams, and FeynCalc [56] to calculate particles.

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