Constraints on scattering of keV–TeV dark matter with protons in the early Universe
We present the first cosmological constraint on dark matter scattering with protons in the early Universe for the entire range of dark matter masses between 1 keV and 1 TeV. This constraint is derived from the Planck measurements of the cosmic microwave background (CMB) temperature and polarization anisotropy, and the CMB lensing anisotropy. It improves upon previous CMB constraints by many orders of magnitude, where limits are available, and closes the gap in coverage for low-mass dark matter candidates. We focus on two canonical interaction scenarios: spin-independent and spin-dependent scattering with no velocity dependence. Our results exclude (with 95% confidence) spin-independent interactions with cross sections greater than cm for 1 keV, cm for 1 MeV, cm for 1 GeV, and cm for 1 TeV dark matter mass. Finally, we discuss the implications of this study for dark matter physics and future observations.
Introduction. One of the primary ways to investigate the fundamental nature of dark matter (DM) is to search for evidence of its non-gravitational interactions with the Standard Model of particle physics. None of the experimental or observational searches for DM interactions have yet made a confirmed detection. As a result, large portions of DM parameter space are excluded and new proposals are coming on the scene to broaden the search strategy and examine unexplored DM scenarios Alexander et al. (); Battaglieri et al. ().
A substantial effort to detect and characterize the properties of DM rests on the hypothesis that DM may be a weak-scale thermal relic particle (WIMP) with a mass in the range of tens of GeV to a few TeV Jungman et al. (1996). Virtually all traditional direct-detection searches are constructed and optimized to search for WIMPs from the local Galactic halo through their scattering on nuclei in underground targets Cushman et al. (). They have exquisite sensitivity: the tightest constraints to-date on spin-independent interactions exclude cross sections greater than cm for masses around 30 GeV Aprile et al. (2017); Frederico Pascoal da Silva (), while the next-generation experiments promise to push this bound further in the near future Cushman et al. (). These searches, however, are looking under the lamppost. Specifically, current nuclear-recoil–based measurements are effectively blind or background dominated below DM masses of about a GeV Cushman et al. (). Additionally, the extensive shielding inherent in underground experiments puts a “ceiling” on the interaction strength, above which the majority of particles would be stopped before reaching the detector Zaharijas and Farrar (2005); Emken and Kouvaris (2017); Shafi Mahdawi and Farrar (2017). Technological improvements Angloher et al. (2016); Agnese et al. () and analyses of electronic recoils are able to expand direct-detection sensitivities to somewhat lower DM masses Essig et al. (2012, 2017); however, the latter are only applicable to DM interactions with electrons, not protons. Entirely new experimental strategies are thus required to truly open up sub-GeV DM to broad, in-depth exploration that parallels dedicated WIMP searches Alexander et al. (); Battaglieri et al. ().
In addition to direct detection, there is a range of studies that constrain low-energy DM–baryon interactions in the local Universe, using results from balloon-borne experiments Erickcek et al. (2007), Galactic structure Wandelt et al. (2001), observations of galaxy clusters Hu and Lou (2008); Natarajan et al. (2002), cosmic rays Cyburt et al. (2002); Cappiello et al. (2017), and other astrophysical observations Starkman et al. (1990); Boehm and Schaeffer (2005); Mack et al. (2007); Kavanagh (). These studies explore various parts of the DM parameter space, but few focus specifically on sub-GeV particles.
Given the current null results, new DM models (e.g., hidden-sector DM Feng and Kumar (2008), asymmetric DM Kaplan et al. (2009), freeze-in DM Hall et al. (2010), SIMPs/ELDERs Hochberg et al. (2014, 2015); Kuflik et al. (2016)) have recently received much attention in theoretical and experimental communities. Many of these models comfortably accommodate DM particles with masses in the keV–GeV range. In this study, we produce the strongest cosmological constraint to date on DM interactions covering the entire DM mass range between 1 keV and 1 TeV, using Planck measurements of the cosmic microwave background (CMB) temperature and polarization anisotropy, and the CMB lensing anisotropy Adam et al. (2016); Aghanim et al. (2016).111Lyman- forest limits on warm DM exclude masses below a few keV Viel et al. (2013), and we thus focus only on masses above this limit.
The very same interactions sought locally by direct detection and other experiments also take place in the early Universe (in the first 400 000 years after the Big Bang) and can be tested with cosmological observations. If baryons scatter with DM particles in the primordial plasma prior to recombination, the heat transferred to the DM fluid can cool the photons, producing spectral distortions in the CMB; this effect was previously used to constrain DM masses below a few hundred keV from the null-detection of distortions in FIRAS data Ali-Haïmoud et al. (2015) (see also Figure 1). Furthermore, due to a drag force between the DM and photon–baryon fluids, small-scale matter fluctuations are suppressed, altering the shape of the CMB power spectra and of the matter power spectrum. This effect too was explored in previous studies Chen et al. (2002); Sigurdson et al. (2004); Dvorkin et al. (2014) and was most recently used to place constraints for DM much heavier than a GeV Dvorkin et al. (2014) (see again Figure 1).
We expand upon this previous work in several important ways. First, we probe DM with masses down to 1 keV, thereby closing the gap in mass coverage of previous cosmological studies. Additionally, only scattering with free protons in the early Universe was previously considered for the leading cosmological constraints on heavy DM Dvorkin et al. (2014); we also account for DM scattering with helium nuclei, which significantly improves constraints in that mass regime. Finally, we use the latest Planck 2015 data release Adam et al. (2016), and for the first time include CMB polarization and lensing measurements to search for evidence of DM–proton interactions. With these improvements in our analysis, the limits we obtain are a factor of 30 stronger than the best previous CMB limits of Ref. Dvorkin et al. (2014) for heavy DM.222When we make the same simplifying assumptions as Ref. Dvorkin et al. (2014), we restore consistency with their results. In addition, our constraints are several orders of magnitude stronger than those of Refs. Chen et al. (2002); Ali-Haïmoud et al. (2015) for lower DM masses.333The constraint of Ref. Chen et al. (2002) is not dominated by the CMB measurements, but rather by a reconstruction of the linear matter power spectrum from the 2dF galaxy survey Peacock et al. (2001), which may strongly depend on the choice of galaxy bias model.
Dark matter–proton scattering. We concentrate on two DM–proton interaction scenarios: spin-independent and spin-dependent elastic scattering, with no dependence on relative particle velocity. These simple interactions are the most widely considered and easily arise at leading order from high-energy theories (the literature on this subject is vast, and we refer the reader to an early review for reference Jungman et al. (1996)). In a companion paper Boddy and Gluscevic (), we expand this study to constrain DM–proton interactions in the broader context of non-relativistic effective field theory Fan et al. (2010); Fitzpatrick et al. (2013); Anand et al. (2014) and address a wide range of momentum- and velocity-dependent interactions.
In order to compute CMB power spectra in presence of the interactions, we modify the code CLASS Blas et al. (2011) to solve the following Boltzmann equations (in synchronous gauge) Ma and Bertschinger (1995)
for the evolution of DM and baryon density fluctuations, and , and velocity divergences, and , respectively. In the above expressions, is the wave number of a given Fourier mode; is the scale factor; is the trace of the scalar metric perturbation Ma and Bertschinger (1995); and are the speeds of sound in the two fluids Ma and Bertschinger (1995); and and are their respective energy densities. The overdot notation represents a derivative with respect to conformal time. The subscript pertains to photons, where represents the usual Compton scattering term Ma and Bertschinger (1995).
The terms proportional to encapsulate the new interaction physics; is the coefficient for the rate of momentum exchange between the DM and baryon fluids, found by averaging the momentum-transfer cross section over the velocity distributions of particles in the early Universe Sigurdson et al. (2004); Dvorkin et al. (2014). Previous work considered DM scattering with only free protons Dvorkin et al. (2014); here, we include scattering with protons inside helium nuclei, and thus need a more general expression for to account for the nuclear structure of helium.
We start by summarizing the results for scattering with free protons. In this case, both the spin-independent (SI) and spin-dependent (SD) cross sections are the same as the corresponding momentum-transfer cross sections,
where = is the spin of the DM, is the mass of the proton, is the mass of the DM particle, and is the reduced mass of the DM–proton system. The coupling coefficients and set the strength of the spin-independent and spin-dependent interactions, respectively. We insert the weak-scale mass 246 GeV, as an overall normalization.444The choice of the normalization scale does not impact our constraints on the cross sections.
Moving on to helium, we first note that it has zero spin and thus cannot have spin-dependent interactions. For the spin-independent interaction, there is no inherent velocity dependence; however, the nuclear form factor is a function of the momentum transferred in the scattering process555The momentum transfer is given by =, where is the scattering angle in the center-of-mass frame, is the relative velocity between the DM and helium particles, and is the reduced mass of the DM–helium system. Catena and Schwabe (2015). Thus, the associated momentum-transfer cross section has a velocity-dependent part multiplying the following numerical factor
which depends on the strength of the interaction, quantified by , and is the reduced mass of the DM–helium system. When we average the full momentum-transfer cross sections over the velocity distributions for DM and baryons, we obtain
where , is the helium mass fraction, and and are the temperatures of the baryon and DM fluids. The internal spin degrees of freedom666The DM and baryonic spin degrees of freedom were omitted in similar expressions derived in Refs. Chen et al. (2002); Sigurdson et al. (2004); Dvorkin et al. (2014). are =, =, and =. In the nuclear shell model, the length parameter for helium is fm Fitzpatrick et al. (2013). For spin-independent scattering, the total rate coefficient is ; for spin-dependent scattering, the total rate coefficient is . Note that the velocity dependence of the cross section in the case of helium translates to the additional temperature-dependent term in the last line of the above expressions.
Since we are interested in light DM, we cannot neglect terms with in the above equations (as was done in Ref. Dvorkin et al. (2014) for heavy DM). We thus track the DM temperature evolution given by777At early times, when the interactions affect the evolution of density modes accessible to cosmological observables, baryons are in thermal contact with photons, and the backreaction on the baryon temperature is a subdominant effect; we thus ignore it. Sigurdson et al. (2004); Dvorkin et al. (2014)
The heat-exchange coefficients control when the DM and baryon fluids thermally decouple, and they are given by
Data analysis and results.
We use the CMB power spectra and likelihoods from the Planck 2015 data release, as available through the clik/plik distribution Aghanim et al. (2016); Adam et al. (2016). We analyze temperature, polarization, and lensing to jointly constrain the six standard CDM parameters: the Hubble parameter , baryon density , DM density , reionization optical depth , the amplitude of the scalar perturbations , and the scalar spectral index . We also include the coupling coefficient as an additional free parameter (with a wide flat prior probability distribution). We use the code MontePython Audren et al. (2013) with the PyMultinest Buchner et al. (2014) implementation of nested likelihood sampling Feroz and Hobson (2008); Feroz et al. (2009, ).888For the case of no DM–proton interactions (vanishing coupling coefficients), we recover CDM parameter values and constraints consistent with Planck published results Adam et al. (2016) (to within ). We repeat the fitting procedure for a range of 8 fixed DM mass values between 1 keV and 1 TeV for spin-independent and for spin-dependent interactions.
We find no evidence for DM–proton scattering in the data, and thus derive 95 confidence-level upper limits on and as a function of DM mass. We then convert these results into upper limits on the corresponding interaction cross sections; the resulting exclusion curves are shown and compared to previous results999The results of Ref. Dvorkin et al. (2014) are only valid for . The slope of their constraint starts to deviate noticeably from our exact calculation at 50 GeV. in Figure 1. For the spin-independent interaction, we exclude cross sections greater than cm for 1 keV, cm for 1 MeV, cm for 1 GeV, and cm for 1 TeV DM particle mass. To illustrate the effect of scattering, Figure 2 shows the percent difference in the CMB temperature power spectrum between the CDM model and a model with spin-independent DM–proton scattering.
Most of the constraining power in this analysis comes from the temperature measurements; lensing and polarization contribute to the limits at the level of .101010It has been noted that Planck high-multipole polarization may have systematic issues Aghanim et al. (2016); Adam et al. (2016); however, excluding 30 polarization degrades our reported constraints by only less than . On the other hand, while the inclusion of scattering on helium makes only a modest contribution for sub-GeV DM masses, it improves the limits by as much as a factor of 6 at high masses (in Figure 1, compare the spin-independent limit and spin-dependent limit; helium contributes only to the former). This is a consequence of the mass dependence of the momentum-transfer rate between DM and baryons. With helium included, the maximal momentum-transfer rate occurs at a higher mass (by a factor of a few, as compared to the proton-only case). Given the rapid loss of sensitivity with increasing mass (see Figure 1), this shift implies modest improvements in constraining power at masses around a GeV, but substantial improvements in the high-mass regime.
Finally, the scaling of the cross-section constraint with DM mass depends on two quantities that enter all relevant evolution equations: and . For heavy DM, both rates scale as , as does the resulting exclusion curve shown in Figure 1; thus, our result can be directly extended to higher masses by appropriately scaling our reported limit at 1 TeV. In the low-mass limit, the mass scaling of the rates is different [see Eqs. (4) and (6)], and the slope of the exclusion curve is a non-trivial combination of the two effects.
Conclusions. We analyze Planck measurements of temperature, polarization, and lensing anisotropy to perform the first cosmological search for dark matter–proton scattering in the early Universe in the full range of dark matter masses between 1 keV and 1 TeV. We find no evidence of such interactions and thus report an upper bound on the corresponding cross sections, shown in Figure 1. This analysis improves upon previous leading cosmological limits by one or more orders of magnitude, for masses where they were available.
We directly constrain cross sections for dark matter scattering with protons—the same quantities probed by direct detection and other experiments that operate at low energies, but extend to a regime in parameter space that is inaccessible to current underground experiments. Additionally, upper limits coming from all experimental probes seeking to detect dark matter in the Galactic halo are sensitive to the assumptions about the astrophysical properties of dark matter particles (their local velocity distribution and energy density, in particular). The limits we report directly address cosmological dark matter in the early Universe and thus sidestep these important caveats of the local low-energy probes. Therefore, our result provides highly complementary information on dark matter interaction physics, and paves the road for a broad approach to the dark matter problem.
The effect of dark matter interactions is progressively more prominent at smaller angular scales (see Figure 2), making it a prime target of investigation for a number of existing and upcoming low-noise, high-resolution, ground-based CMB experiments, such as the Atacama Cosmology Telescope (ACT) Louis et al. (2017), the South Pole Telescope (SPT) Henning et al. (), the Simons Observatory111111https://simonsobservatory.org, and the CMB Stage-4 experiment Abazajian et al. (). We expect a substantial improvement in the sensitivity of our analysis with data from ground-based CMB measurements in the near future.
Acknowledgments. VG gratefully acknowledges the support of the Eric Schmidt fellowship at the Institute for Advanced Study. The authors thank Yacine Ali-Haïmoud, John Beacom, Joanna Dunkley, Ely Kovetz, and Samuel McDermott for comments on the manuscript. We also thank Marc Kamionkowski, Jason Kumar, Zack Li, Joel Meyers, Vivian Poulin, and David Spergel for useful discussions.
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