CHAMP Cosmic Rays
David Dunsky, Lawrence J. Hall, Keisuke Harigaya
TL;DR
This paper investigates the behavior and detection prospects of cosmological relic particles with electric charge, analyzing their acceleration, flux, and experimental bounds, and identifying parameter regions accessible to future searches.
Contribution
It provides a comprehensive analysis of charged relics' acceleration, flux, and detection constraints, highlighting new parameter space regions for future experiments.
Findings
Accelerated relic flux extends to high energies up to ~5×10^4 q GeV.
Strong bounds on relics as dark matter exclude q > 10^{-9} for any mass.
Future experiments can probe specific (mass, charge) regions of relic parameter space.
Abstract
We study interactions of cosmological relics, , of mass and electric charge in the galaxy, including thermalization with the interstellar medium, diffusion through inhomogeneous magnetic fields and Fermi acceleration by supernova shock waves. We find that for , there is a large flux of accelerated in the disk today, with a momentum distribution extending to . Even though acceleration in supernova shocks is efficient, ejecting from the galaxy, are continually replenished by diffusion into the disk from the halo or confinement region. For , cannot be accelerated above the escape velocity within the lifetime of the shock. The accelerated form a…
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Figure 22| ISM Phase | Fractional Volume | |||
|---|---|---|---|---|
| Hot Ionized | ||||
| Warm Ionized | ||||
| Warm Neutral | ||||
| Cold Neutral | ||||
| Molecular |
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CHAMP Cosmic Rays
David Dunsky
Department of Physics, University of California, Berkeley, California 94720, USA
Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Lawrence J. Hall
Department of Physics, University of California, Berkeley, California 94720, USA
Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Keisuke Harigaya
School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, 08540
Department of Physics, University of California, Berkeley, California 94720, USA
Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Abstract
We study interactions of cosmological relics, , of mass and electric charge in the galaxy, including thermalization with the interstellar medium, diffusion through inhomogeneous magnetic fields and Fermi acceleration by supernova shock waves. We find that for , there is a large flux of accelerated in the disk today, with a momentum distribution extending to . Even though acceleration in supernova shocks is efficient, ejecting from the galaxy, are continually replenished by diffusion into the disk from the halo or confinement region. For , cannot be accelerated above the escape velocity within the lifetime of the shock. The accelerated form a component of cosmic rays that can easily reach underground detectors, as well as deposit energies above thresholds, enhancing signals in various experiments. We find that nuclear/electron recoil experiments place very stringent bounds on at low ; for example, as dark matter is excluded for and . For larger or , stringent bounds on the fraction of dark matter that can be are set by Cherenkov and ionization detectors. Nevertheless, very small is highly motivated by the kinetic mixing portal, and we identify regions of that can be probed by future experiments.
Contents
I Introduction
Theories Beyond the Standard Model may contain exotic stable CHArged Massive Particles, or CHAMPs, of mass and electric charge . They may arise in a variety of ways: for example, from exotic color-neutral matter added to the Standard Model, or even from exotic heavy colored states that bind with the known quarks Nomura and Tweedie (2005); Harigaya and Nomura (2016); De Luca et al. (2018); Dunsky et al. (2019). Another important possibility is that the gauge group is extended to a hidden sector where stable particles couple to a hidden photon that is kinetically mixed with our photon Holdom (1986).
In general, CHAMPs, , are produced in the early universe. The genesis mechanism, and hence the relic abundance, are extremely model-dependent so that, in addition to , we take the abundance of normalized by that of dark matter (DM), , to be a free parameter. We study cases where CHAMPs account for the entire DM, and where they form a sub-dominant component.
CHAMP DM has been considered for over 30 years. The kinetic mixing portal Goldberg and Hall (1986) allows DM from a dark sector to become visible by acquiring a small electric charge, . This charge may be suppressed by a loop factor, involving a heavy connector particle of mass that carries both charges, suggesting values of of order . However, in unified theories where hypercharge and hidden s are embedded in non-Abelian factors down to scales and much less than , the charge receives a power suppression by , and hence may naturally be very small. Power suppression of can also arise from an approximate symmetry. One example is a hidden U(1) embedded in a hidden SU(2) that is spontaneously broken by a triplet. A charge conjugation symmetry from SU(2) forbids kinetic mixing until higher-dimensional operators are added. In summary, it is well-motivated to examine a wide range of the plane.
A variety of constraints and signals of CHAMP DM with order unity charges were considered in De Rujula et al. (1990); Dimopoulos et al. (1990), and for CHAMPs with in Dobroliubov and Ignatiev (1990). An important cosmological bound on CHAMP DM, arising from the era of recombination from constraints on the CMB acoustic peaks and from damping of the density perturbations, requires for MeV Dubovsky et al. (2004); Burrage et al. (2009); McDermott et al. (2011); Dolgov et al. (2013). However, this bound disappears if contributes less than 1% of the DM. Chuzhoy and Kolb Chuzhoy and Kolb (2009) proposed that, over a certain range of , supernova (SN) shocks expel CHAMPs from the Milky Way, removing previous constraints on charged dark matter based on bounds from terrestrial observations for these .
In this paper, we investigate the evolution of the density and spectrum of CHAMPs in the galactic disk. We study rates for the three key processes: thermalization of with the InterStellar Medium (ISM), Fermi acceleration of by SN shocks, and diffusion of through magnetic irregularities. In general, diffusion allows to both enter and exit the disk. We find that the plane can be divided into three regions
- (I)
GeV The density and spectrum of in the disk are determined by virialization of the halo. Thermalization, Fermi acceleration and diffusion are negligible so that dark matter signals can be computed by ignoring them. 2. (II)
GeV X that are initially in the disk are efficiently ejected by SN shocks; however, there is a continual replenishment of by diffusion from the halo and confinement region to the disk. The number density and spectrum of in the disk today follows from a balance between accretion and ejection. 3. (III)
GeV collapse with baryons into the disk as it forms. Thermalization of with the ISM reduces the efficiency of ejection, leading to large densities of in the disk today.
In both (II) and (III) there are large fluxes of Fermi accelerated s, that we call CHAMP Cosmic Rays. These give signals in a variety of direct detection experiments deep underground as well as on the surface of the Earth, via nuclear recoil, electron recoil, ionization losses and Cherenkov radiation.
The parameter space where halo CHAMPs collapse into galactic disks is studied in Sec. II; determining the boundary between (II) and (III). After a halo virializes, CHAMPs that thermalize with the infalling baryons within a free-fall time are dragged along into the disk. CHAMPs that collapse cannot be halo dark matter. However, they may be a component of dark matter, and the accelerated CHAMP flux is enhanced by a factor of about , commensurate with the greater number of CHAMPs exposed to SN shocks in the disk.
In Sec. III, we examine three fundamental rates that determine the fate of CHAMPs in the galactic disk: their thermalization rate in the ISM, their encounter rate with SN shocks, and their escape rate from the disk. The three rates generally depend on the CHAMP speed as well as . The accelerated spectrum is determined by a competition between thermalization with the ISM plasma, SN shocks that accelerate to shock speeds or beyond, and the interstellar magnetic fields which confine to the disk.
We investigate the efficiency of the acceleration of CHAMPs and calculate the differential momentum spectrum of a distribution of shocked CHAMPs during their journey through the ISM in Sec. IV, taking into account thermalization, subsequent shock encounters, and escape losses, as depicted graphically by Figs. 4 and 20. Initially, a single SN shock transforms a batch of thermal CHAMPs into a distribution such that the CHAMP speed is the encountered shock speed and the amplitude is the relative probability of encountering a shock of that speed. Most CHAMPs thermalize quickly, but a small number do not and either escape or encounter additional SN shocks. In the Milky Way the latter dominates and produces a relativistic Fermi spectrum cutoff at , determined by the required acceleration time exceeding the lifetime of the shock. CHAMPs that obtain enough energy escape from the disk, with a high efficiency for GeV.
Although CHAMPs are ejected from the disk by Fermi acceleration, they are replenished by diffusion accretion from outside the disk. A balance between ejection and accretion leads to a steady state distribution of CHAMPs in the disk. These accelerated CHAMPs can hit the Earth before escaping from the disk, and we estimate the present flux of such cosmic ray CHAMPs in Sec. V
From the accelerated CHAMP spectrum impinging on the Earth, we calculate signal rates in nuclear recoil, electron recoil, ionization and Cherenkov detectors in Sec. VI. Since the accelerated CHAMPs are moving faster than typically assumed dark matter speeds (), there are new key features of these signals: (1) CHAMPs can reach underground detectors easily, even if their charges are large. (2) CHAMPs below can impart nuclear recoils above the threshold for direct detection experiments such as XENON1T and CDMS. (3) Similarly, CHAMPs below MeV impact electron recoils in direct detection experiments such as XENON10. (4) They produce ionization losses in detectors such as MAJORANA, MACRO and other monopole search experiments. (5) Relativistic CHAMPs, or electrons produced by recoils, emit Cherenkov light when traveling through water, leading to bounds from deep underground detectors such as Super Kamiokande and IceCube. These signals of the accelerated cosmic ray CHAMP flux lead to powerful constraints, and point to regions of parameter space where discoveries can be made at future experiments.
Conclusions are drawn in Sec. VII. Appendices consider CHAMP self interactions from hidden photon exchange, the CHAMP spectrum resulting from repeated shocks, and the diffuse extragalactic CHAMP flux from ejection from galaxies throughout the universe.
II Collapse of CHAMPs into the Galactic Disk
Halos that virialize at redshift have densities
[TABLE]
where is the background density Peacock (1999), and temperatures
[TABLE]
where is the halo mass and is the mean molecular weight of the baryonic gas.
The cosmological localities of CHAMPs and baryons can diverge during subsequent cooling of these halos. As noted by the authors in De Rujula et al. (1990), is dragged into the galactic disk with the baryons if the thermalization time of with the baryonic plasma is less than the dynamical or collapse time of the halo
[TABLE]
which is independent of halo mass.
Baryons with a cooling time shorter than are able to collapse into the disk on the time scale . Pre-reionization (), halos with virial temperatures above radiatively cool via bremsstrahlung and atomic line emission to , and then collapse isothermally at this temperature within a time Rees and Ostriker (1977). Note the gas in unable to cool further since the tail of the Boltzmann distribution becomes insufficient to collisionally excite atoms Loeb (2010).
However, post-reionization (), UV light from the first stars and galaxies permeate the intergalactic medium (IGM), heating up and ionizing halo atoms, making atomic line emission less effective and preventing plasma temperatures from dropping below , depending on the degree of self-shielding which is set by the plasma density Mo et al. (2010); Weinberg et al. (1997) . Consequently, post-reionization, only halos with virial temperatures above cool and collapse Loeb (2008).
Since thermalizes with the plasma through electromagnetic collisions via Rutherford scattering, the key difference between the pre- and post-reionization eras is the ionization fraction in virialized halos. Equating recombination and collisional ionization rates at implies the ionization fraction of the plasma is about pre-reionization, while it is near unity above post-reionization Draine (2011); Loeb (2008).
Specifically, the thermalization time between with mass and charge , and a background plasma of temperature is given by De Rujula et al. (1990); Spitzer (1956)
[TABLE]
where is the effective temperature of , is the density of protons or electrons with mass , and is the IR cutoff where electromagnetic shielding becomes effective beyond an inverse of the Debye momentum of the plasma, .
Initially, since the virial speeds of and protons are identical, being set by gravity. Because the proton and electron plasma quickly cools to a temperature for halos that virialize before reionization and for halos that virialize after, the second term in parenthesis of (4) quickly becomes . Likewise, the ion number density is given by
[TABLE]
where is the ionization fraction of the plasma and is before (after) reionization.
Inserting the plasma number density into (4), and demanding , yields the parameter space where collapses into the galactic disk with the baryons, as a function of halo mass and , as shown by the shaded regions of Fig. 1.
In general, the number of in the disk of a galaxy with halo mass is approximately
[TABLE]
where and the fraction of CHAMPs that actually end up in the disk, exposed to SN. Taking the disk formation efficiency to be similar for CHAMPs and baryons, observations set Mo et al. (1998) if collapse. The rest of reside outside the disk.
On the other hand, if do not collapse, the number of CHAMPs in the disk is suppressed, and is approximately
[TABLE]
where we have evaluated the dark matter mass fraction in the disk using an NFW profile Navarro et al. (1996). Here, () is the ratio of disk radius (disk height) to the halo virial radius, and the NFW halo concentration parameter Loeb (2008). Typical values of , , and Mo et al. (1998); Loeb (2008) imply (7) is a few in our galaxy.
The result we need for the rest of the paper is that for the Milky Way (, M , ), collapse into the disk for , and the number density of inside the disk is about 100 times larger than the naive scaling of the local dark matter density by . In this region of parameter space, cannot be the halo dark matter. From Fig. 1, the excluded range of for is somewhat larger, since other galaxies also have halo dark matter. Furthermore, we will discover that the resulting high density of in the disk leads to a large SN-accelerated CHAMP flux, giving a strong bound on .
One caveat is that when , thermalization increases the speed of relative to the proton thermal speed. To collapse fully, the orbital radius of must decrease by a factor , and hence with do not completely collapse for . However, with such small masses that do thermalize are already excluded by direct searches (see Sec. VI).
Finally, note that the thermalization time is always shorter than (4) during the collapse process because the plasma density increases while still maintaining a small, but non-neglible ion fraction. Self-shielding from the background UV light starts becoming effective when column densities exceed Noh and McQuinn (2014), but transition to neutrality () requires column densities two orders of magnitude greater Altay et al. (2011). Since the column density of a collapsing cloud corresponds to a characteristic number density by Schaye (2001); Noh and McQuinn (2014), densities of order are required for ionization fractions to drop below Mesinger (2016). Consequently, for lower mass halos which typically collapse at higher redshifts and densities, the collapse of into a disk may be partial. However, for the Milky Way, whose self-shielding density coincides with its post-collapse density, it is possible that some CHAMPs with may also thermalize and fall into the disk during the collapse. As we will see in Sec. IV, however, for GeV the ejection of is so efficient that most of which were initially inside the disk are ejected, and hence the accelerated CHAMPs which hit the Earth in the present universe are dominated by CHAMPs which were initially outside the disk and diffused into the disk later. Therefore for GeV, it does not matter whether collapses into the disk.
III Three Key Rates in the Galactic Disk
In this section we introduce three key rates, 1) the thermalization rate, 2) the supernova shock rate, and 3) the escape rate from galactic disks. The interplay of these rates determine the probability for to escape from galactic disks as well as the number density and spectrum of those that remain, as discussed in Sec. IV.
III.1 Thermalization Rate in the Interstellar Medium
The that do fall into the disk or happen to reside there are greatly influenced by the environment of the ISM. In our Milky Way, the ISM consists of hot, warm, and cool phases in pressure equilibrium ( constant), and a self-gravitating molecular phase Draine (2011). The cool phase is composed of small, atomic clouds and the warm and hot phases constitute the intercloud medium and essentially the entire ISM by volume. Typical properties of these phases for our Milky Way are shown in Table 1.
An moving through the ISM at a speed 111Initially, is set by the thermal speed if is dragged into the disk, or the virial speed if not; later, is determined by SN shocks. thermalizes at an expected rate
[TABLE]
where is the thermalization time (4) of in ISM phase , and its corresponding volumetric filling factor. The largest ambient electron density implies the shortest thermalization time, and hence the warm ionized medium (WIM) dominates the thermalization rate, as can be seen from Table 1. Thus, is most likely to be found in the warm medium and indeed, that is where about half the baryonic mass of the ISM lies Draine (2011). Eq. (8) assumes that is larger than the thermal speed of protons, km/s and electrons, km/s. If , in Eq. (8) should be replaced by .
A natural way to obtain small is to introduce a hidden under which is charged, and assume a small kinetic mixing between the hidden gauge field and the electromagnetic field. Then the interaction between s by the hidden also contributes to thermalization. As is shown in Appendix A, this interaction does not change the estimation of the efficiency of the evacuation if GeV or is produced before the onset of the Big-Bang Nucleosynthesis.
III.2 Supernova Shock Rate
CHAMPs are accelerated by SN shocks in the same way a ball is accelerated by reflecting off a moving wall; the moving wall in this case is the moving magnetic field near the shock. When moving slower than the shock, the CHAMP is accelerated to the shock speed. When moving faster than the shock, the CHAMP may repeatedly reflect off the shock, resulting in an exponential momentum gain due to the change in momentum with each reflection. This latter process is known as first-order Fermi acceleration, and the rate at which CHAMPs are accelerated is thus intimately tied to the rate of encountering strong shocks in the ISM.
The expected rate of encountering a SN shock of speed is
[TABLE]
where is the volume of disk, is the volume of a SN remnant with shock speed , and the rate of SN in the galaxy. Note that the shock radius is a decreasing function of shock speed; that is, a CHAMP is much more likely to encounter a slower shock. The SN remnant size and shock speed depend on the medium to which it expands, and the theoretical evolution for a shock expanding into a homogenous ambient medium of density (the average intercloud density) is shown in Fig. 2 222While the average density of the ISM is about , the shock takes the path of least resistance, propagating primarily through the warm/hot intercloud medium and around the dense atomic clouds Martizzi et al. (2015); Slavin et al. (2015). As a result, the mass swept up by the shock is primarily the intercloud mass.
A SN shock begins life expanding at a constant speed near . Energy conservation then demands that the shock speed decreases proportional to the square root of mass swept up by the shock. For expansion into a homogenous medium with number density , the radius-velocity relation during this ‘Sedov-Taylor’ phase is Draine (2011)
[TABLE]
where is the SN energy output.
As the shell expands, radiative losses from the shock heated gas become comparable to the energy of the SN and the shell is propelled forward only by the pressure of the hot gas inside. The radius-velocity relation during this ‘snowplow’ phase is
[TABLE]
The shock continues to expand until reaching a maximum size before merging with the ISM, at which point its speed equals the thermal sound speed, around .
Now, as seen from (9), the largest shock size sets the shock encounter rate. However, the largest shocks are unable to Fermi accelerate . This is because the gas around the shock front must be fully ionized to maintain the strong turbulence necessary for efficient acceleration of , as is shown below. This condition begins to fail early in the snowplow phase, when neutrals begin forming near the shock Yamazaki et al. (2006). Thus we take the shock at the end of the Sedov phase to be the largest shock capable of Fermi-accelerating . This corresponds to a minimum shock speed slightly below , a maximum radius (10) of , and hence an expected SN shock rate of
[TABLE]
The strong turbulence is required so that the magnetic fields upstream and downstream of the shock are sufficiently tangled, making the mean free path of in the shock region as small as its gyroradius (Bohm diffusion), the minimum possible mean free path Bell (2013); Yamazaki et al. (2006). The necessity for such a small mean free path near the shock can be understood by calculating the maximum possible rigidity, (the ratio of momentum to charge, ), a SN shock can impart to a CHAMP. The maximum rigidity is set by spatial and temporal constraints. Spatially, the shock cannot accelerate a CHAMP anymore once its mean free path grows larger than the size of the shock region. For a SN of radius in the Sedov phase, hydrodynamic simulations show the thickness of the shock region is Draine (2011). A SN of radius , then cannot accelerate beyond
[TABLE]
where we have taken the shock magnetic field about three times the ambient ISM field due to shock compression Thompson et al. (2009). Temporally, the shock cannot accelerate a CHAMP for longer than the age of the remnant. The acceleration timecale to Fermi-accelerate a particle to rigidity and speed is approximately where the diffusion constant near the shock, and the mean free path Bell (2013). Equating the acceleration time with the age of the remnant, implies a SN of radius cannot accelerate beyond
[TABLE]
The maximum rigidities of (13) and (14) imply particles are unaffected by SN shocks in the limit . This condition must be true since SN shocks transfer momentum to encountered particles solely through electromagnetic scatterings.
Note the factor of in the denominator of (14) compared to (13). This is because first-order Fermi acceleration is more efficient at slower speeds since the momentum change upon reflection is greater for smaller . For CHAMPs with , the temporal constraint (14) dominates.333Ordinary cosmic rays are believed to be injected when the shock is young and the magnetic field is nearly a milligauss, which gives a maximum rigidity near , exactly where the proton ”knee” is observed in the cosmic ray spectrum. Further evidence for the validity of (14) is the iron knee, which drops at a momentum 26 times higher. CHAMPs which encounter young SN remnants can be accelerated above the rigidity (14), but we do not consider such a process.
Contours showing the largest possible for a given CHAMP mass and charge is shown in Fig. 3. CHAMPs with GeV cannot be ejected from the Milky Way as , and hence remain throughout the halo and disk with the virialized velocity distribution.
III.3 Escape Rate from the Disk
CHAMPs diffuse through the ISM by resonantly scattering off magnetic irregularities on the scale , where . This scattering leads to a mean free path , where is the magnetic rigidity, and is set by the magnetic field power spectrum Longair (1981); Jones et al. (2001). The observed steady-state cosmic ray secondary to primary spallation ratios at various rigidities implies and leads to a mean free path
[TABLE]
for most cosmic ray propagation models Strong et al. (2007); Jones et al. (2001) 444The mean free path becomes rigidity independent at rigidities below . However, the smallest CHAMP rigidities we consider, (see Eq. 18) are alway greater than a , except in parameter space already excluded by collider searches and .
For CHAMPs with speeds above the gravitational escape speed, , the typical rate to diffuse out of the disk is , where the diffusion constant . As with cosmic rays, diffusion has the effect of increasing the time it takes for CHAMPs to escape the disk. The resulting escape rate from the disk is 555The cosmic ray lifetime in the entire galaxy, not just the disk, is determined from the relative abundance of cosmic ray radioactive isotopes to their children, and is about times longer than the lifetime in the disk (16) Kulsrud (2005); Longair (1981). This is because diffusion continues above the disk into a high hot gas region, so called the confinement region. However, the lifetime in the stellar disk, where will encounter SN, is bounded by the grammage of matter traversed as observed from spallation products, and agrees well with (16) Kulsrud (2005).
[TABLE]
This escape rate breaks down when and hence is valid only for particles with rigidities . However, SN can only accelerate CHAMPs marginally beyond this rigidity anyway (13), so the ISM mean free path essentially always remains below .
IV Acceleration and Ejection from the Galaxy
IV.1 The Accelerated Spectrum
To understand the interplay between the thermalization rate in the ISM (8), the SN shock rate (12), and the escape rate from the disk (16), we define the parameter
[TABLE]
an independent-variable nearly mutual to each rate, and three values such that , , and . These critical points are given by
[TABLE]
We have normalized , , and to values for the Milky Way. Accidentally this leads to comparable prefactors when . However, for the hierarchy of speeds for the Milky Way is for any value of .
Galaxies with different disk and ISM properties will have different and ; however, there are only two possible orderings of these speeds corresponding to the two cases and , as shown in top panels of Figs. 4, 20. Equivalently, the two cases correspond to whether a CHAMP that surpasses the thermalization bottleneck is more likely to encounter another SN shock on the way out of the disk (and be Fermi-accelerated to relativistic speeds) or to escape the disk without meeting any further shocks (and remain non-relativistic upon escape). Our galaxy belongs to the first case for all which we investigate in the following.
Boom. A SN goes off and its shock expands in the ISM. We first consider the case that are thermal with speeds much less than . Since their speeds are also below the shock speed, when hit by a first shock they are accelerated only to the shock speed. Since the probability of a shock encounter goes as the shock radius cubed, we see from (10) and (11) that a SN remnant produces a CHAMP number density spectrum in the Sedov-Taylor phase (i.e. ) and in the snowplow phase. The latter thermalize so quickly that they are irrelevant to the following discussion.
We will now discuss the spectrum of this batch in terms of the differential momentum spectrum , since it is the momentum which is the fundamental quantity that describes the spectrum from non-relativistic to relativistic regimes. Because all occur at non-relativistic speeds, there is no loss in generality between , and its associated momentum .
Thus, the relevant differential spectrum of this batch is initially proportional to , up to ,666For sufficiently large , cannot be accelerated to for reasons discussed in Sec. III.2. However, the galactic spectrum remains the same as long as , which we find always true. There also exists a momentum such that with diffusively catch up the same shock that initially accelerated . When this occurs, is Fermi-accelerated and the spectrum becomes above . This again does not change the galactic spectrum since we also confirm that . as shown by the dashed blue line in Fig 4. However, as time progresses, energy losses from thermalization even alter this spectrum, chipping away at the slower moving CHAMPs which thermalize first. The evolution of this differential spectrum due to thermalization obeys Longair (1981), whose solution implies for and unchanged for . Therefore, when this batch encounters a second SN shock with speed a time later, its spectrum is peaked at , dropping as for and for , as shown by the orange and dashed blue lines in Fig. 4.
The CHAMPs in the batch moving faster than the approaching shock can convectively and diffusively travel back and forth across the shock front. The expected momentum gain for each cycle as well as the probability of completing cycles can be calculated, which together yield the post-shock distribution Drury (1983). The above physics is encoded in a transformation of the original spectrum, , to a final spectrum, , by Bell (1978)
[TABLE]
Here, and is the theoretically predicted power dependence from Rankine-Hugoniot plasma boundary conditions. Performing the convolution (21) on the spectrum, we find the effect of the second shock is to leave unchanged the spectrum below but to change the spectrum above to a Fermi-accelerated spectrum, as shown by the green and orange lines in Fig. 4. Qualitatively, this is because the largest number of particles have initial momenta . Note the resulting spectrum now includes CHAMPs with relativistic speeds.
Those CHAMPs with momenta now above will quickly leave the disk and contribute to the extragalactic spectrum, as shown by the dotted green line in Fig. 4. Meanwhile, CHAMPs with momenta between and are more likely to stay in the disk and encounter additional SN shocks before escaping and reaching the momentum .777The escape probability for with is exponentially suppressed. Only for with is the escape proability non-negligible, with value . Consequently, the probability galactic CHAMPs with momenta less than escape before encountering repeated shocks is negligible. The evolution of the CHAMP spectrum by the repeated encounters is investigated in Appendix B, and it is shown that CHAMPs with momenta below eventually escape from the disk with a time scale .
The two-stage acceleration mechanism we consider becomes ineffective once is above , as almost all CHAMPs accelerated by the first shock are thermalized before encountering the next shock and hence cannot be ‘injected’, or Fermi-accelerated, at the second shock. This is the case if GeV, which is excluded by direct searches for CHAMPs (see Sec. VI). Note that nuclei belong to this parameter region and cannot be accelerated by the two-stage injection process described above. It is currently not understood how a very small fraction of thermal nuclei are directly injected from a single, young, shock (the so-called ‘injection-problem’ Kulsrud (2005); Draine (2011)) to become cosmic rays. It is likely too, that a very small fraction of thermal CHAMPs are also directly injected by a single shock, though large uncertainties exist since the process is not understood for even ordinary cosmic rays. Nevertheless, the authors of Hu et al. (2017) assume that CHAMPs are Fermi-accelerated in the same manner as nuclei, and obtain the spectrum of CHAMP cosmic rays from that of protons with the same rigidity. Since the efficiency of direct-injection is much less than two-stage injection, the resultant CHAMP cosmic ray abundance is much smaller than ours.
In the above discussion we assumed that are thermalized and have speeds below before encountering a SN remnant. However, if GeV, the thermalization does not occur and have velocities of , which is larger than the shock speed. On encountering the first SN shock, undergo Fermi acceleration. Hence, whether or not there is initial thermalization, the accelerated spectrum always has the form , cutoff at low speeds at
[TABLE]
IV.2 Efficiency of Expulsion
For now we ignore the diffusion of CHAMPs from outside the disk and compute the fraction of CHAMPs that escape the disk. The fraction is given by the probability to encounter a critical shock to overcome the thermalization bottleneck within the disk. As discussed in Sec III.2, for GeV it is enough to encounter a shock at the end of the Sedov-Taylor phase because of the inefficient thermalization. For GeV, encounter with a shock with a velocity is required. Shock speeds capable of reaching occur during the early Sedov-Taylor phase, where the shock radius-velocity relation (10) implies the expected encounter rate for a critical shock (9) is
[TABLE]
Since the chance of encountering a critical shock is rare, we expect the number of critical shocks encountered to be a Poisson random variable with an expected rate given by (23). Consequently, the fraction of CHAMPs that never encounter a critical shock and thus remain in the disk after a time is
[TABLE]
Assuming is independent of time, , and the shock expands into a homogenous medium of density , the fraction of CHAMPs that remain in the disk as a function of is shown in blue in Fig. 5. Note that while an order one fraction of the CHAMP population in the disk may be ejected after one folding-time , efficient removal from the disk requires many folding-times.
V Diffusion into the Disk and the Local CHAMP Flux
Although ejection from the disk is efficient for GeV, this does not imply the absence of CHAMPs in the disk. There is a continuous replenishing of CHAMPs in the disk by diffusive accretion from the halo and the confinement region. The balance between accretion and ejection leads to a (quasi-) steady state. Hence, even after years, there are CHAMPs that have been recently accelerated by SNe and hit the Earth before escaping from the disk. In this section we estimate the present flux of such accelerated CHAMPs.
V.1 CHAMPs that do not Collapse into the Disk
We first consider GeV, where CHAMPs do not collapse into the disk when it forms. Most galactic CHAMPs are outside the disk, so that their diffusion into the disk plays an important role. The disk of the Milky Way, which we take to have a width of pc, is surrounded by the confinement region, which we take to have a width of kpc Longair (1981); Strong et al. (2007). This region has a random magnetic field similar to that of the disk, so we take the CHAMPs to diffuse in this region with the same mean free path as in the disk. Diffusion through the confinement region plays a key role in determining the accelerated CHAMP flux hitting the Earth today.
We solve the following equations for , the number density inside the confinement region with the virial speed, and for , the number density of accelerated s in the disk
[TABLE]
where GeV/cm3 is the initial local number density, and the diffusion constant , with given in (15).
We take the escape rate of the accelerated CHAMPs to be the SN shock rate , as the number density of accelerated Xs is dominated by ones with low momenta, , and these typically escape by encountering SNe and are rapidly accelerated to momentum , where the escape rate is equal to the shock rate. Moreover, as shown in Appendix B, CHAMPs with momentum are repeatedly shocked and quickly evacuate the disk in a time as well.
Numerical results for the number density of the accelerated CHAMPs, , are shown in Fig. 6 as a function of . These results, including the slopes and the kink at GeV, can be understood from a simple analytic estimate. The acceleration of in the disk creates a gradient in the confinement region that drives a diffusion current of into the disk, from above and below
[TABLE]
where is the mean free path and the diffusion distance in time . For GeV, and we reach a steady state where from the halo diffuse through the confinement region to reach the disk. Even for the largest that lead to shock acceleration, this does not substantially alter the density of in the halo. For GeV, so that is time dependent; arises from transporting from the confinement zone a distance from the disk. Finally, note must always be less than which it is, since the mean free path .
In the disk, Eq. (25)-(27) then reduce to and leading to the (quasi-) steady state solutions , . Inserting from (28) gives
[TABLE]
where the differing powers of in (29) result from the different powers of in for the two cases. The continual accretion of CHAMPs onto the disk, followed by their acceleration and expulsion, has led, remarkably, to a large accelerated cosmic ray flux of CHAMPs today on Earth.
We take the CHAMP velocity to be the virial velocity to estimate the diffusion constant . Since are efficiently ejected from the disk, refilling by diffusion from outside the disk determines the present number density of CHAMPs. The diffusion of unaccelerated CHAMPs inside the disk is not important. Outside the disk, thermalization is ineffective on cosmological time scales for GeV and we may safely take the virial speed for the above estimation. For GeV, the thermalization occurs and their velocity decreases down to the thermal velocity. If the velocity is below the Alfven velocity km/s, the scattering by the turbulent magnetic field accelerates CHAMPs up to the Alfven velocity with the rate as large as the gyrofrequency Wu et al. (1997), and hence the CHAMP velocity is at the smallest the Alfven velocity. The diffusion constant for the Alfven velocity is about 8 times smaller than that for the virial velocity. For GeV, GeV and is proportional to . The accelerated number density decreases at the most by a factor of three because of the thermalization. We neglect the small suppression.
Finally, it is worthwhile to mention that the steady-state accelerated spectrum is fairly insensitive to whether or not collapses. This is because either way, since the baryon disk formation efficiency is only . Moreover, since acceleration out of the disk is efficient for CHAMPs that do not collapse, , the same steady-state spectrum is quickly reached regardless of the initial disk density. Similarly, since acceleration out of the disk and diffusion into the disk are less efficient for CHAMPs that do collapse, , the same steady-state spectrum is reached regardless of the density at the confinement-halo interface. We find that, even if we assume the collapse of CHAMPs (see below), is enhanced by a factor of few for just above .
V.2 CHAMPs that do Collapse into the Disk
For GeV, CHAMPs collapse into the disk. We solve Eqs. (25, 27) with the initial and boundary conditions
[TABLE]
where is the initial concentration from collapse at disk formation and the concentration that remain in the halo near the confinement interface Gaensler et al. (2008). For GeV, even CHAMPs outside the disk but inside the confinement region are thermalized. We take the velocity of the unaccelerated CHAMPs to be the maximal of the thermal velocity and the Alfven velocity. We find that and in the present universe are approximated by the following semi-empirical formulae,
[TABLE]
This result can be understood as follows. The large charge and the low velocity implies that diffusion is ineffective, so that the number density inside the disk is basically given by , which is corrected by the second factor in the exponent taking into account the suppression of the ejection by small diffusion out from the disk. The number density of accelerated CHAMPs is then determined by the quasi-steady state solution with .
V.3 The Local CHAMP Flux and Spectrum
The accelerated CHAMPs initially have the spectrum . CHAMPs with momentum above have an escape rate larger than by a factor of for non-relativistic and for relativistic . Taking account the larger escape rate, the flux of the accelerated CHAMPs is given by
[TABLE]
We note that the spectrum in Eq. (V.3) may be further modified during the diffusion between the acceleration site and the Earth. In Fig. 7, we show the typical distance with a velocity can travel before encountering another shock, . In the shaded region, the distance is smaller than the typical distance between the acceleration site and the Earth, pc, and is likely to encounter multiple shocks before hitting the Earth. We define the momentum of above which the encounter typically does not occur as , which is at the most as large as . Then the spectrum is the one with in Eq. (V.3) replaced by , with a subdominant spectrum in . This does not weaken the constraints derived in the next section, since the stopping by the Earth crust and/or the energy threshold of the searches require the momentum of detectable to be above in the parameter region with inefficient diffusion. Rather, the signal rates may be enhanced by a factor of (which is at the most ). Since we are not able to determine in a reliable manner, we do not consider this possible enhancement in this paper. One should take care of this issue if momenta are important for searches.
CHAMPs that escape from other galaxies and reach our galaxy also contribute to the accelerated CHAMP spectrum. The spectrum of the extragalactic CHAMP background is estimated in Appendix C and is found to be subdominant.
VI Direct Detection of Accelerated CHAMP Cosmic Rays
In this section we discuss direct detections of CHAMPs on the Earth. We first investigate two possible barriers for the detection: the solar wind and stopping in the Earth before reaching detectors. Then we compute signal rates in experiments sensitive to nuclear recoil, ionization and Cherenkov radiation. We assume that couples to nucleons, electrons and photons dominantly through the charge . If feels the strong interaction, as is the case with heavy colored states that bind with the known quarks, the constraint is altered.
VI.1 The Solar Wind and Stopping by the Earth
The direct detection of CHAMPs on Earth can dramatically be affected by the solar wind, an outflow of charged particles and associated magnetic fields from the sun which suppress the flux of interstellar charged particles that reach the earth.
The net flux of CHAMPs a distance from the Sun is given by a convection-diffusion equation Longair (1981)
[TABLE]
where is the number density of CHAMPs, the solar wind speed, and the rigidity-dependent diffusion constant of charged particles in the interplanetary magnetic field. The net flux (35) is zero in the steady-state regime and leads to the solution
[TABLE]
Observations of the low-energy cosmic ray flux on Earth find (36) to be well fit by Longair (2011); Meyer (1969)
[TABLE]
where parameterizes the modulation of the solar wind and the interplanetary magnetic field, the rigidity dependence of the particle mean free path, and () the number density of CHAMPs on Earth (far away in the ISM).
During the -year solar cycle minimum, when the solar wind suppression on the cosmic ray flux is weakest, measurements indicate , and Longair (2011); Meyer (1969)888The change in the rigidity dependence of the mean free path can be explained by a change in the interplanetary magnetic field power spectrum. Measurements from the Mariner 4 spacecraft indicate the power spectrum changes its power dependence at wavenumbers near Jokipii (1971), corresponding to a scattering gyroradius of Longair (2011). Since the solar wind magnetic field is around , the critical rigidity occurs at , in excellent agreement with (38).
[TABLE]
CHAMPs with scatter frequently enough with the magnetic fields carried by the solar wind that they cannot travel ‘upstream’ from the outskirts of the heliosphere to the Earth. The parameter space where the solar wind suppression is significant is shown by the shaded region of Fig. 8.
CHAMPs that penetrate the solar wind must also penetrate the Earth’s atmosphere/ crust to the depth of the detector. CHAMPs with passing through matter slow down chiefly from electron ionization. The stopping power is well described by the Bethe equation for ,
[TABLE]
For CHAMPs slower than the Fermi-velocity ), energy losses from collisions with electrons are proportional to the CHAMP velocity Ziegler (1988). Unlike ions, which are partially ionized in this velocity regime and must be assigned an effective nuclear charge as described by the Lindhard-Scharff equation, the effective CHAMP charge remains and hence its stopping power through a material is just times the proton stopping power Ziegler (1988), which is given in the NIST Database Berger et al. (2017). We use the tabulated stopping power for and the Bethe equation for .
Contours in the plane of the minimum to reach underground detectors 500 m below the Earth’s surface are shown in Fig. 9. Contours for penetrating the Earth’s atmosphere may be obtained by shifting these contours up above by a factor of 10 in charge . At high radiative losses dominate over electron ionization, but we find radiative losses are not important for values of that are allowed by direct searches and astrophysics.
VI.2 Nuclear Recoil at Deep Underground Detectors
The scattering cross section between and a nucleus of mass and charge is
[TABLE]
where is the reduced mass, is the speed of , is the momentum transfer and is the nuclear form factor. The recoil energy of the nucleus is
[TABLE]
and the minimum speed to obtain such a recoil energy is
[TABLE]
The integrated cross section above a threshold for fixed speed is
[TABLE]
where is evaluated at the threshold recoil energy and takes into account the suppression of the scattering by the form factor
[TABLE]
Assuming the Helm form factor Helm (1956); Lewin and Smith (1996), we find . The signal rate in a given detector with a number of target nuclei is
[TABLE]
where is the minimum detectable speed.
Using Eqs. (45), we compute the signal event rate at XENON1T Aprile et al. (2018) with keV, and require fewer than 16 expected events for a 1 ton-year exposure, putting an upper bound on the fraction of as dark matter, as shown in Fig. 10. In the analysis of Aprile et al. (2018), events with extra ionization electrons are vetoed. Thus we conservatively require that the ionization energy loss of passing through 1m of liquid Xenon is below eV, so that typically no electron recoils occur. The minimal velocity is determined by this requirement through the dependence of the ionization energy loss on the velocity, the threshold energy, the minimal velocity to reach the detector, and .
Below the thick solid line of Fig. 10, the maximum speed can gain from SNe is below the escape velocity, and the standard constraint is applicable. Above the dashed line collapse into the disk. Note that a bound exists even if GeV, where XENON1T is insensitive to dark matter with a virial speed due to the threshold. The accelerated CHAMPs have speeds much larger than the virial speed, and easily deposit energies above the threshold. The larger velocity also help CHAMPS to reach the underground detector, strengthening the constraint at larger values of . For few , electron recoils typically occur while pass through the detector, and hence scattering events may be vetoed.
We also show bounds on the parameter space from direct searches Davidson et al. (1991); Prinz et al. (1998); Davidson et al. (2000); Chatrchyan et al. (2013); Magill et al. (2018), SN cooling Chang et al. (2018) and from the dark radiation abundance in the universe. The constraint from dark radiation is weaker than the one in Vogel and Redondo (2014), as entropy production could occur near the MeV scale for MeV.
We compute the signal rate at CDMS-II Ahmed et al. (2010) with keV, and require fewer than 10 expected events for a 600 kg-day exposure. The constraint is shown in Fig. 11. Signal regions are defined by a small ionization yield, below 30% of the recoil energy. Thus we require that the ionization energy loss of passing through cm of germanium is below keV. A muon veto is also imposed, but we find that as long as the energy loss in the germanium is below keV signals evade the veto. Although the constraint is weaker then that from XENON1T, CDMS-II constrains a region with larger values of , up to .
VI.3 Electron Recoil at Deep Underground Detectors
Nuclear recoil experiments cannot probe the region with small and , since the maximum velocity of CHAMPs are still below the threshold. Such a region can be probed by searches for electron recoils at deep underground detectors with low thresholds. The estimation of the precise signal rate requires a computation involving an atomic form factor and is beyond the scope of the paper. Instead, we obtain a rough estimation of the constraint by scaling the constraint in Essig et al. (2017) in the following way. Essig et al. (2017) defines a DM-free electron scattering cross section with the matrix element artificially evaluated at the momentum transfer of as . For CHAMPs, it is given by
[TABLE]
where is the reduced mass between an electron and a CHAMP. Assuming that the CHAMP is the dominant component of dark matter and has a virial velocity, the upper bound is for MeV. The bound becomes rapidly weaker for MeV as the kinetic energy of CHAMPs becomes smaller than the typical electron binding energy. We adopt the following as a rough estimation of the bound,
[TABLE]
where is the minimal detectable speed. It is the maximum of , , for sufficient kinetic energy, and the minimal velocity to deposit less than 10 eV by ionization while passing 40 cm of liquid Xenon. The constraint is shown in Fig. 12, which covers the small and region.
VI.4 Relativistic Electron Recoil and Subsequent Cherenkov Light
Relativistic CHAMPs passing through water may deposit enough energy to accelerate electrons to relativistic speeds. If the speed of these recoiling electrons is , they emit detectable Cherenkov light. Such events are detected by Super-Kamiokande for deposition energies above the threshold of 100 MeV Kachulis et al. (2018). The main target of the search is dark matter coming from the center of the galaxy, and constraints are put on events within a cone with a certain opening angle measured from the center of the galaxy. The accelerated CHAMPs come isotropically, and hence we use the bound on the signal rate for the largest cone, giving limits on shown in Fig. 13. Below the dashed line, the maximum momentum of accelerated CHAMPs is below the threshold, . For , photomultiplier tubes (PMTs) in the outer detector typically receive more than one photon from the Cherenkov radiation of CHAMPs, giving events that are vetoed in the analysis of Kachulis et al. (2018).
VI.5 Cherenkov Light from Relativistic CHAMPs
Relativistic with speeds above produce Cherenkov light when traveling through water. For , the intensity of Cherenkov light is typically too low to observe individual tracks of in the ice. Nevertheless, the parameter space can be constrained when the total number of Cherenkov photons emitted from relativistic surpasses the observed background count rate of the IceCube PMTs.999Radioactive decays are dominantly responsible for the remaining dark counts Abbasi et al. (2011).
The integrated flux of atmospheric muons below the Antarctic ice is Patrignani (2018) and contributes Abbasi et al. (2011) of the background rate. The number of Cherenkov photons emitted per unit wavelength and unit pathlength of is proportional to Jackson (1998). Requiring CHAMPs to give a signal below the observed PMT dark count rate constrains the integrated CHAMP flux above
[TABLE]
The constraint (48) is shown in Fig. 14. It is generally weaker than constraints from nuclear recoils in XENON1T or from energy deposition in MAJORANA.
VI.6 Ionizing Particle Searches
As grows, CHAMPs yield significant ionization. The MAJORANA experiment searches for such ionizing particles with a threshold of keV, and puts an upper bound on the flux, Alvis et al. (2018). Taking , where is the minimum velocity to exceed the threshold, as the minimum speed that can reach MAJORANA and yield signals, we find the upper bound on the fraction of as dark matter as shown in Figure 15. The bound complements that from nuclear recoil experiments. There is no constraint for , since even maximally ionizing , with a velocity , cannot deposit an energy above the threshold. Here we use the NIST Database Berger et al. (2017) to calculate the typical energy deposit on germanium. In the region close to the solid line, and must be larger to deposit enough energy. Similarly, for small , the required value of becomes larger, as must have a larger velocity to reach MAJORANA, and is less ionizing.
We expect that larger parameter regions are actually constrained. Even if the typical energy deposit is below a keV, there is a probability for to deposit an energy above the threshold, as computed in Alvis et al. (2018) for a minimally ionizing speed. This effect will lead to constraints in broader parameter regions, but is beyond the scope of our paper.
Relativistic CHAMPs with charges between produce visible tracks in MACRO’s scintillation and streamer detectors that can be distinguished from integer charged cosmic rays through the dependence of‘ . The upper bound on the flux of CHAMPs with is Ambrosio et al. (2004) for and weaker outside this range. Taking the lowest speed that MACRO can detect to be , we find the upper bound on the fraction of as dark matter as shown in Figure 16. For GeV, and hence the constraint is absent.
While the flux constraints from MACRO are strong, the trigger efficiency of the MACRO hardware is sensitive only to relativistic with . Moreover, the MACRO experiment is over one km underground which prevents slower moving CHAMPs from reaching the detector (see Fig. 9). An experiment at the Institute for Cosmic Ray Research (ICRR), designed to look for the scintillation light of slow, penetrating, and highly ionizing particles on the surface of the Earth, constrains the flux of CHAMPs with to Kajino et al. (1984). The ICRR experiment is sensitive to ionization deposits greater than the minimum ionization Barish et al. (1987). As discussed in Sec VI.1, the ionization losses of with can be read from the experimental stopping power of protons, scaled by , and imply charges as low as can be detected. From the observed stopping power of protons through plastic scintillators Berger et al. (2017), we find the upper bound on the fraction of as dark matter as shown in Figure 17. The lower edge of the constrained region is determined by ionization losses. To the left of the dotted lines, the velocity of CHAMPs which reach the detector is larger than . To compute , we use the stopping power of air, shown in the NIST Database Berger et al. (2017) and a column depth of g/cm2.
Last, the Baksan experiment Alexeyev et al. (1983, 1985), an underground scintillation detector searching for slowly-moving ionizing particles, complements MACRO by providing comparable flux constraints to non-relativistic CHAMPs with . The upper bound on the flux of CHAMPs with is . The Baksan experiment is sensitive to ionization deposits greater than Alexeyev et al. (1983), and implies as low as can be constrained for . CHAMPs with cannot traverse the length of the detector within one integration time and thus must have greater (that is, greater ) to be detected. Taking into account the stopping by the Earth and the stopping power of protons through liquid scintillators,101010Liquid scintillators have similar to plastic-based ones since both are organic compounds. we find the upper bound on the fraction of as dark matter as shown in Figure 17. Note while Baksan and ICRR both probe similar velocity ranges, Baksan has stronger flux constraints but is not as sensitive to small as ICRR, nor can it detect the slowest CHAMPs which stop in the Earth before reaching it.
VI.7 Constraints on DM or Thermally Produced CHAMPs
Fig. 18 summarizes the constraint on assuming all of dark matter is . In the orange-shaded region indicated as “Coupled around recombination”, couples to baryons around the era of recombination, and the fluctuations of the cosmic microwave background are altered Dubovsky et al. (2004); Dolgov et al. (2013). We also show the prospected sensitivity of the LZ experiment Mount et al. (2017) assuming a 15 ton-years exposure with a threshold energy of keV. The purple dotted line shows the prediction for the charge from the Freeze-In production of dark matter Chu et al. (2012); see below for a rough estimation. Nuclear recoil experiments have just begun to reach the sensitivity to probe Freeze-In production.
We take a closer look at the thermally produced . We assume that initially only the Standard Model sector is thermalized. The Freeze-In abundance of from pair production by s-channel photon exchange is
[TABLE]
where is the energy density of , is the entropy density, and is the effective number of degrees of freedom. It is almost independent of but grows with . This gives for . We use the precise estimation of Chu et al. (2012) in the following. For large enough this abundance becomes sufficient for pair-annihilation of to occurs, so that the final yield is then given by Freeze-Out
[TABLE]
where is the annihilation cross section, and is the temperature at Freeze-Out. If Freeze-Out occurs then it determines the final abundance, otherwise it is determined by Freeze-In.
If is taken to be the only addition to the Standard Model then almost the entire region of interest having larger than is excluded because . Hence we add a massless dark photon so that can pair-annihilate into dark photons. This simple scheme for dark matter has been studied for general values of the gauge coupling in some depth Chu et al. (2012). Here, for simplicity, we fix the gauge coupling of to the dark photon to be the same as between an electron and photon. Fig. 19 shows the constraints on resulting from this thermal abundance of . The abundance exceeds the dark matter abundance in the purple-shaded region at the right of the figure. At the edge of this region , with production from Freeze-In along the bottom edge and from Freeze-Out along the left edge. Moving to the left, drops as . The constraint from fluctuations of the cosmic microwave background, the orange-shaded region, is applicable if comprises more than 1% of dark matter Dolgov et al. (2013). The constraint from dark radiation is taken from Vogel and Redondo (2014), with the latest bound Aghanim et al. (2018).
Last, we comment on the possible effect of CHAMPs on the structure of the halo for . For GeV, the mean free path and the gyro-radius of with is smaller than the height of the confinement region. This means that the dynamics of are not only governed by the gravitational force but are also affected by the magnetic field in the confinement region, which may change the distribution of CHAMP dark matter in the inner part of the halo, and possibly lead to further limits/signals. The ejection of CHAMP by SNe may further affect the halo structure. We do not pursue this possibility further in this paper.
VII Summary and discussion
Cosmological relics, whether comprising all of dark matter or just a component, are almost always considered to be electrically neutral. However, charged relics may have escaped detection either because their electric charge is very small, or because their mass, , is very large. While CHAMPs may arise as simple additions to the Standard Model, kinetic mixing provides a window to dark sectors that contain a gauge group, greatly enhancing the importance of CHAMP searches, and strongly motivating searches over a wide range of the plane.
For GeV, form a virialized halo that is not disturbed on cosmological time scales by interactions with the interstellar medium or by magnetic fields. However, for smaller values of , Fermi acceleration by shock waves of supernova remnants, diffusion through magnetic inhomogeneities, and thermalization via Rutherford scattering with ionized interstellar matter play crucial roles in determining the number density and spectrum of hitting the solar system today. We have discovered that, over a wide region of , a steady state is established balancing efficient ejection of from the galaxy by SN shocks with continuous diffusion of into the disk from the halo. The resulting accelerated cosmic ray flux at the solar system today is shown in Eq. (V.3). It has a spectrum to a maximum momentum determined by the size and the lifetime of the shock-wave accelerator, as is shown in Fig. 3. The corresponding local number density of the accelerated is very large, of the halo density for GeV. Hence, in this region of , limits from direct detection experiments are very powerful, whereas previously, neglecting diffusion in from the halo, they were believed to be absent.
For GeV, collapse into the disk as it forms. Clearly cannot be halo dark matter; however, they may still provide a window to the dark sector. The inefficient diffusion in from the halo because of the thermalization or the small rigidity of suggests that constraints on a component of might be weak. This is incorrect: are strongly coupled with the ISM giving a thermalization bottleneck, inhibiting Fermi acceleration and ejection from the galaxy. Although most are ejected by today, there remains a local flux of accelerated , and for MeV the charge is sufficiently large that direct detection limits are extremely powerful.
Over the entire plane with GeV, the accelerated CHAMPs have speeds larger than typically assumed for dark matter, opening up new signals and regions of parameter space to be probed by experiments. We have derived constraints from XENON1T, CDMS II, XENON10, Super Kamiokande, IceCube, MAJORANA, MACRO, ICRR as well as Baksan. Over a large part of the plane, the most powerful constraints on , the fraction of dark matter that can be , arise from direct detection limits from nuclear/electron recoil. Indeed, for , the only limits come from nuclear/electron recoil. Limits on from the XENON1T, CDMS II and XENON10 experiments are shown in Figures 10, 11 and 12. At larger the most powerful bounds on arise from signals from Cherenkov light and ionization; frequently these bounds are extremely powerful, although they often apply to only a small region of the plane, as shown in Figures 13-17.
We briefly comment on the EDGES detection of an enhanced absorption feature in the 21-cm line at Bowman et al. (2018). Such an anomaly can be explained if a fraction of DM are CHAMPs with mass and charge Berlin et al. (2018). However, such a scenario is ruled out by XENON10 and Super-K experiments by orders of magnitude according to Figs. 12 and 13.
Constraints on CHAMPs comprising all of dark matter, no matter what the production mechanism, are severe, as shown in Figure 18. is excluded for any . It will be exciting to see how much of the Freeze-In region can be reached by future experiments.
There are two clear signal regions for thermally produced CHAMPs that contribute all of dark matter in theories with a dark photon. In Figure 19, these are along the edges of the purple region that is excluded by overproduction of dark matter. The first arises from Freeze-In production of the CHAMPs from the Standard Model sector, and has and TeV. The second arises from Freeze-Out annihilation to dark photons and has TeV, and in the fairly narrow range of , with larger being excluded by XENON1T. Future nuclear recoil experiments will continue to probe the Freeze-Out region and may eventually reach the Freeze-In region. For TeV, Freeze-out gives , and future nuclear recoil experiments will probe significant regions of the plane where is a sub-dominant component of dark matter.
Acknowledgement
The authors thank Simon Knapen, Daniel McKinsey, Eliot Quataert, and Nick Rodd for useful discussion. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contract DE-AC02-05CH11231 and DE-SC0009988 (KH), as well as by the National Science Foundation under grants PHY-1316783 and PHY-1521446.
Appendix A Interaction between CHAMPs
In computing the ejection of from the disk, we have ignored any scattering between CHAMPs. In particular, after particles are accelerated by a SN shock, they could be slowed down by scattering from ambient in the disk. In this appendix we consider such scattering to arise from massless hidden photon exchange, and derive the condition such that the scattering does not change our estimation of the accelerated CHAMPs. We show that our previous results are not affected if is produced before Big-Bang Nucleosynthesis, or GeV.
For a hidden charge , the thermalization rate of via hidden photon exchange is
[TABLE]
The velocity of above which the thermalization rate is smaller than the encounter rate with SNe is
[TABLE]
which is analogous to derived in the main text based on the -baryons scattering. In order for our previous results not to be affected, should be smaller than ;
[TABLE]
If this condition is violated, in Eq. (V.3) is replaced with .
The scattering can also affect the estimation of . The encounter rate with SNe with a shock velocity above is
[TABLE]
For GeV, is not affected as long as this encounter rate is larger than , which requires that
[TABLE]
If this condition is violated, in Eq. (32) is replaced with . For GeV, is insensitive to and hence to . This is however not the case if becomes close to 1 because of inefficient acceleration. Requiring that , we obtain
[TABLE]
If this condition is violated, , and .
In the parameter space of our interest, GeV, all of the above conditions are weaker than the restriction if GeV.
Note that the fraction is bounded from above for a given and the production temperature. We assume that is produced at the temperature of . The number density of is bounded by the annihilation of into hidden photons,
[TABLE]
The fraction and the charge is bounded as
[TABLE]
Let us assume that is above 4 MeV, as required if the production mechanism also produces Standard Model particles with an energy density comparable to that of . For CHAMPs satisfying the condition (58), all off the conditions above are satisfied.
Appendix B Repeated Shock Encounters
Here we consider the effect of repeated shocks on the galactic CHAMP spectrum, which occurs for CHAMPs residing in case 1 galaxies. We show that with a momentum below is ejected from the disk within a time . Consequently, using as the escape rate in (27) is a good approximation.
First note the spectrum of a batch of CHAMPs after one shock can be written as a transformation on the original spectrum Bell (1978)
[TABLE]
where is the original spectrum and the theoretical value from Rankine-Hugeniot plasma boundary conditions at the shock discontinuity. Without loss of generality, let so that the spectrum after the one shock is
[TABLE]
which is the standard Fermi spectrum. Now, the CHAMPs in this spectrum with momentum above are more likely to escape the disk before encountering another shock, while those with momentum below more likely to encounter another shock before escaping. Thus, the spectrum after one shock approximately bifurcates into a galactic and extragalactic spectrum:
[TABLE]
After the next shock, the galactic spectrum is
[TABLE]
Which again bifurcates into a galactic (where ) and extragalactic (where ) spectrum and so on. After shocks, the galactic spectrum is
[TABLE]
which quickly approaches [math] for , which is in the parameter space of the interest. As a result CHAMPs with an initial momentum escape from the disk with the time scale . Meanwhile, the extragalactic spectrum is
[TABLE]
which approaches the Fermi accelerated spectrum of single shock with an input spectrum . Note the extragalactic spectrum is independent of initial conditions, given that initially reaches a speed greater than or equal to for case 1.
Appendix C The Diffuse Extragalactic CHAMP Background
In this appendix we investigate the ejection of CHAMPs in galaxies and estimate the extragalactic CHAMPs spectrum.
In case one galaxies, with rate orderings of Fig. 4, CHAMPs with momentum repeatedly encounter shock and are accelerated. Once the momenta are above , they escape from the disks before encountering another shock, ultimately producing an extragalactic spectrum
[TABLE]
where is the total number of ejected CHAMPs from that galaxy.
In case two galaxies, with rate orderings of Fig. 20, CHAMPs with momentum generally enconter one shock before escape. Just as before, when a SN explodes, its Sedov-Taylor shock produces a batch of CHAMPs with a differential spectrum as shown by the dashed blue line of Fig. 20. As time progresses, the numerous slower CHAMPs at the far left of the spectrum again thermalize first, and are converted to a spectrum which moves to the right with time. Concurrently, the scarce number of faster CHAMPs at the far right of the spectrum escape first, and are converted to a non-relativistic extragalactic spectrum which moves to the left with time. The last, and greatest number of CHAMPs to escape have momentum , the momentum when the escape and thermalization rates are equal. When this occurs at time , the galactic and extragalactic spectrums are peaked at , with the galactic spectrum dropping as for and the extragalactic spectrum dropping as for , as shown by the orange and dotted red lines in Fig. 20. Finally, note that a fraction, of the escaping CHAMPs will encounter a second SN shock and be Fermi-accelerated before escaping. The probability is peaked at , which leads to a subdominant extragalactic spectrum, as shown by the dotted green line in Fig. 20. Consequently, the final extragalactic spectrum produced from galaxies in case is given by the sum of the following two contributions,
[TABLE]
The key difference between galaxies of case 1 (68) and galaxies of case 2 (69) is the resulting relativistic extragalactic CHAMP spectrum. In case 1, CHAMPs faster than the thermalization bottleneck generally encounter an additional SN shock before escaping, resulting in a dominant relativistic extragalactic spectrum. However, in case 2, CHAMPs faster than the thermalization bottleneck generally escape before encountering an additional SN shock, resulting in a dominant non-relativistic extragalactic spectrum and a subdominant relativistic spectrum.
As can be seen from (18), (19), and (20), the critical points depend on various ratios of the disk volume, disk height, SN rate, and SN volume. Consequently, for a fixed , the momenta , , and may vary between galaxies, implying each galaxy produces a potentially different extragalactic spectrum. However, galaxies, unlike snowflakes, are not completely unique. Scaling relations allow us to estimate how the three fundamental rates depend on a given halo mass.
First, let us consider the disk radius. In current theories of disk formation, the baryonic mass in the halo which cools and falls cannot collapse to a central point due to angular momentum conservation. Instead, the baryons pancake into a disk with a radius proportional to the halo virial radius Loeb and Furlanetto (2013); Mo et al. (1998). Thus,
[TABLE]
which is supported observationally Loeb and Furlanetto (2013).
Similarly, energy conservation dictates that the disk height is proportional to , where is the velocity dispersion of gas in the disk and the mass surface density. If the centrifugal force cannot balance the gravitational force, the disk is subject to fragmentation and collapse. This instability occurs when the Toomre parameter Toomre (1964); Hopkins et al. (2012), where is the angular frequency. Amazingly, via self-regulation, star formation and SN feedback maintain over the disk Hopkins et al. (2012).111111Disk regions where do not initially collapse but eventually cool and then collapse; hence drops below . Regions where collapse and form stars which inject energy so that increases and collapse is halted; hence rises above . Combining these two conditions imply
[TABLE]
where we assume the asymptotic circular speed of the disk is proportional to the virial speed.
To determine how the SN rate depends on galactic parameters, we first note that is dominated by core-collapse SN so that on cosmological timescales, there is little delay between the star formation rate and the SN rate and hence they are proportional. The global star formation rate per area is empirically observed to be proportional to Draine (2011), where is the gas surface density of the disk,121212This relation is known as the Schmidt-Konneticut Law and is observed over a wide-range of galactic environments. One theoretical motivation for the power is the star formation rate should be proportional , where is the mass of the gas in the disk. which implies
[TABLE]
Last, since the SN radius goes as , the max SN volume goes as
[TABLE]
Putting this altogether, we find
[TABLE]
To simplify these rates further, we note:
Simulations indicate that the number densities, temperatures, and filling factors of the ISM in other star-forming galaxies are very similar to our own (Tab. 1), being self-regulated by SN shocks Hopkins et al. (2012). Thus , and are independent of galaxy and (76) is constant. 2. 2.
Simulations show the turbulent magnetic field in our disk is similar to those in other galaxies and forms very early in the formation of the disk. Thus we assume is universal in other galaxies. 3. 3.
Observations and simulations show that for halo masses , but falls much more steeply ( power) for lower mass halos Stinson et al. (2015).131313Since the size and escape speed of these galaxies are small, one idea to explain their lack of gas (and hence stars and low luminosity) is the SN remnants from their first stars blew out of the disk and into the halo, expelling or disrupting the disk and halo gas and severely hampering subsequent star formation Dekel and Silk (1986); Loeb and Furlanetto (2013). Therefore, in this lower mass halo regime, the SN encounter rate mildly drops as while the escape rate sharply rises as , implying CHAMPs in these disks are typically far in the case 2 regime and easily escape non-relativistically, just as the baryons do. Moreover, even CHAMPs with a very small charge that remain in the case 1 regime for small mass galaxies still have an extragalactic spectrum dominated by high mass galaxies since is greater for heavier mass galaxies (and is nearly constant). Thus, we will only consider the extragalactic flux of CHAMPs from disks with halo masses , which at worst, underestimates the diffuse CHAMP flux. Lighter galaxies can dominate . 4. 4.
Likewise, we do not consider halo masses since such massive halos have difficulty cooling and cannot form disks within Rees and Ostriker (1977). 5. 5.
Since the relevant halo mass range to consider is , which typically virialize near (see Fig. 1), we have a constant.
Putting this all together, the three fundamental rates scale as
[TABLE]
so that
[TABLE]
After escaping from a galaxy, the CHAMPs diffuse into the intergalactic medium (IGM). If the CHAMPs traverse intergalactic distances () within the lifetime of the universe, the extragalactic fluxes from different galaxies overlap and produce a steady-state, diffuse background of CHAMPs. That is, the CHAMPs ejected from galactic disks get smeared over the entire universe so that disk densities are essentially diluted by a factor . More rigorously, consider the transport equation for the total extragalactic spectrum expelled from a disk
[TABLE]
where is the IGM diffusion constant, which may depend on particle rigidity and position. Taking the initial spectrum to be a point source in space and time (on cosmological scales),141414The condition of the spectrum initially being localized in space and time relative to cosmological scales just simplifies the solution of (83) to be the Green’s function. Since we are considering intergalactic distances on the Mpc scale, and typical disk sizes are kpc size, localization in space is a good approximation. Likewise, an order one number of the CHAMPs ejected from disks occurs in a billion years or so, which is less (though not much less) than the age of the universe. the solution to (83) is
[TABLE]
where . is the number of CHAMPs evacuated from the disk.
If we take to be the position of our Milky Way, the total observed CHAMP spectrum is the superposition of all other extragalactic spectra
[TABLE]
where is the differential number of galaxies with halo mass a distance away. For now we assume that can be treated as continuous on the scale . By the Press-Schechter formalism, where Loeb and Furlanetto (2013), so that (85) becomes
[TABLE]
Note both time and the diffusion constant drop out from the integration over when going from the first to second line.
Integrating over the massive galaxies which dominantly eject CHAMPs relativistically, we find a steady-state, diffuse CHAMP differential momentum spectrum
[TABLE]
where is the fractional abundance of CHAMPs to dark matter, is the fraction of halo CHAMPs exposed to SNs, is the fraction of CHAMPs ejected from the disk as given by (24), and , with given by (82). The fraction is determined by whether or not CHAMPs collapse into the disks and the diffusion into the disks,
[TABLE]
where the diffusion current is given in Eq. (28). Eq. (86) shows the extragalactic CHAMP signal is subdominant in comparison with the galactic CHAMP signal discussed in the main text.
Finally, we investigate the validity of (86) for the slowest moving CHAMPs (with speeds near ), which may not be able to diffuse intergalactic distances within the age of the universe. The intergalactic magnetic field is not well known, though flux-freezing arguments suggest magnetic fields below a nanogauss permeate the IGM, with up to hundred nanogauss fields within large galactic clusters Adams et al. (1997); Aloisio and Berezinsky (2004). In contrast to the MW disk, it is unlikely that a turbulent (nanogauss) spectrum can form over intergalactic distances within the age of the universe Aloisio and Berezinsky (2004), so that the magnetic field lines connecting nearby galaxies are taut and the mean free path of CHAMPs is just the coherence length of the field , independent of magnetic field strength Adams et al. (1997). Thus the diffusion length of a CHAMP in the IGM is approximately
[TABLE]
Consequently, if the typical intergalactic magnetic field is non-turbulent, then expelled CHAMPS easily traverse intergalactic distances and (86) should be a good approximation to the diffuse CHAMP background.151515According to the Press-Schecter halo distribution function, the large spirals we consider are typically separated slightly farther apart than . However, our galactic neighbor, Andromeda, is atypically close (). In fact, the slow-moving CHAMP spectrum is likely dominated by Andromeda: Inserting (88) into (84), implies the low-speed CHAMPs escaping Andromeda produce a spectrum a few times greater than the diffuse background (86).
On the other hand, if the intergalactic magnetic field is actually turbulent, then the CHAMP mean free path is generally much less than . For a turbulent, Kolmogorov spectrum, the mean free path is Aloisio and Berezinsky (2004)
[TABLE]
and hence the slowest moving CHAMPs expelled from disks are unable to reach the Milky Way within the lifetime of the universe.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1Nomura and Tweedie (2005) Y. Nomura and B. Tweedie, Phys. Rev. D 72 , 015006 (2005) , ar Xiv:hep-ph/0504246 [hep-ph] . · doi ↗
- 2Harigaya and Nomura (2016) K. Harigaya and Y. Nomura, JHEP 03 , 091 (2016) , ar Xiv:1602.01092 [hep-ph] . · doi ↗
- 3De Luca et al. (2018) V. De Luca, A. Mitridate, M. Redi, J. Smirnov, and A. Strumia, Phys. Rev. D 97 , 115024 (2018) , ar Xiv:1801.01135 [hep-ph] . · doi ↗
- 4Dunsky et al. (2019) D. Dunsky, L. J. Hall, and K. Harigaya, ar Xiv preprint ar Xiv:1902.07726 (2019).
- 5Holdom (1986) B. Holdom, Phys. Lett. 166B , 196 (1986) . · doi ↗
- 6Goldberg and Hall (1986) H. Goldberg and L. J. Hall, Phys. Lett. B 174 , 151 (1986) . · doi ↗
- 7De Rujula et al. (1990) A. De Rujula, S. L. Glashow, and U. Sarid, Nucl. Phys. B 333 , 173 (1990) . · doi ↗
- 8Dimopoulos et al. (1990) S. Dimopoulos, D. Eichler, R. Esmailzadeh, and G. D. Starkman, Phys. Rev. D 41 , 2388 (1990) . · doi ↗