The Subhalo Mass Function and Ultralight Bosonic Dark Matter
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MIT-CTP/5173 The Subhalo Mass Function and Ultralight Bosonic Dark Matter Katelin Schutz Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 Warm dark matter has recently become increasingly constrained by observational inferences about the low-mass end of the subhalo mass function, which would be suppressed by dark matter free streaming in the early Universe. In this work, we point out that a constraint can be placed on ultralight bosonic dark matter (often referred to as \fuzzy dark matter") based on similar consid- erations. Recent limits on warm dark matter from strong gravitational lensing of quasars and from fluctuations in stellar streams separately translate to a lower limit of ∼ 2:1 × 10−21 eV on the mass of an ultralight boson comprising all dark matter. These limits are complementary to constraints on ultralight dark matter from the Lyman-α forest and are subject to a completely different set of assumptions and systematic uncertainties. Taken together, these probes strongly suggest that dark matter with a mass ∼ 10−22 eV is not a viable way to reconcile differences between cold dark matter simulations and observations of structure on small scales. I. INTRODUCTION Accordingly, the results can be interpreted in the context of any theory that predicts suppression of the SHMF. To date, dark matter (DM) has only been detected One such theory, often referred to as \fuzzy dark mat- via its gravitational influence on visible matter. Grav- ter" (FDM), posits that DM is an ultralight boson that itational probes may yet prove to be the optimal way is so low in mass that its de Broglie wavelength is man- to understand the properties of DM if it only interacts ifest on kiloparsec scales in galactic DM halos. For ∼ substantially with visible matter through gravitational the relevant range of speeds in halos like the Milky Way, interactions. DM candidates with unique clustering sig- v 10−3, FDM must have a mass near 10−22 eV to have ∼ natures would be especially ripe for exploration in this such a large de Broglie wavelength (often the convention manner. For instance, if DM decouples from the thermal is to write the mass of this boson in units of 10−22 eV, bath of the early Universe while relativistic, the growth of which we denote as m22 in this work). Ultralight bosons, cosmological structure will be affected. This \warm dark particularly axions, are compelling DM candidates in the matter" (WDM) thermal history has been extensively context of string theory and their discovery could shed studied to determine its effect on linear and non-linear light on physics at extremely high energies and address cosmology. In this scenario, lowering the WDM mass the lack of observed CP violation in quantum chromo- leads to a longer free-streaming length because WDM dynamics (see e.g. Refs. [8{12]). The large de Broglie moves too quickly to cluster for a longer period of time. wavelength of this DM candidate may reduce the abun- Thus, lowering the WDM mass leads to a predicted sup- dance and central density of dwarf galaxies and subha- pression of structure on progressively larger and larger los as compared to CDM, which has been invoked (for scales where the effects become more readily apparent in instance, by Refs. [13, 14]) as a potential explanation comparison to the predictions of cold DM (CDM). for the discrepancy between observations and CDM-only Limits on WDM have been set in the quasilinear simulations of these systems [15]. We collectively refer to regime using the Lyman-α forest as a tracer of the density these discrepancies in subhalo density and abundance as field at high redshifts. WDM masses mχ . 4{5 keV are small-scale structure issues. Note however that the dif- excluded by these measurements, with the exact limit de- ference between observations and CDM-only simulations pending on the modeling assumptions about the thermal is plausibly explained by baryonic physics and may not history of the intergalactic medium (IGM) [1{4]. Re- pose a challenge to the CDM paradigm (see for instance cently, limits on WDM have become even tighter based Ref. [16]). Also note that FDM may not be compati- on observational inferences about the low-mass end of ble with the observed properties of dwarf galaxy density the subhalo mass function (SHMF), which would be sup- profiles [17, 18]. Nevertheless, the possibility that DM is arXiv:2001.05503v2 [astro-ph.CO] 16 Jul 2020 pressed in a WDM cosmology. For instance, recent analy- an ultralight boson merits consideration and a number of ses of the strong gravitational lensing of quadruple-image ideas have been put forth to constrain the unique signa- quasars imply the existence of low-mass subhalos that tures of FDM using galactic-scale observations [19{32]. are abundant enough to exclude WDM masses below An analogy can be made between WDM and FDM mχ . 5{6 keV [5,6]. Additionally, a population of low- cosmologies because FDM exhibits a similar suppression mass subhalos has recently been suggested as the origin of density perturbations below a characteristic scale: in a of fluctuations in the linear density of stellar streams. FDM universe, the de Broglie scale is effectively a Jeans When combined in a joint analysis with classical Milky scale below which perturbations cannot grow due to the Way satellite counts, the fluctuations in stellar streams uncertainty principle. The Lyman-α forest has already exclude WDM masses below mχ < 6:3 keV [7]. Aside been used to exclude m22 . 20, again with the exact from constraining this particular DM scenario, these re- number depending on modeling of the IGM [33{36]. As sults constitute measurements of the shape of the SHMF. is the case for WDM, inferences about the SHMF may 2 improve upon | and independently corroborate | the consequences of FDM. Lyman-α forest constraints on FDM. The details of how fluctuations are suppressed are not exactly the same for WDM and FDM, and therefore the predictions for the II. WARM DARK MATTER SUBHALO MASS SHMF will not necessarily have a one-to-one mapping FUNCTIONS between the two theories. However, in this work we show that a conservative limit on FDM of m22 21 can be set WDM suppresses the growth of structure for modes ∼ based on recent SHMF WDM constraints from stellar that are shorter than the DM free-streaming length. Typ- streams and from gravitational lensing. In particular, ically, WDM becomes non-relativistic in the radiation- FDM with m22 21 predicts a suppression of the SHMF dominated epoch, where the growth of perturbations in ∼ at low subhalo masses that is equal to or stronger than CDM would occur logarithmically in the scale factor a via the WDM scenarios that are excluded by stellar streams the M´esz´aroseffect [41]. After matter-radiation equality and quasar lensing. where perturbations grow linearly in a, WDM can still stream freely (although it is nonrelativistic, it is still mov- Taken at face value, the Lyman-α constraints on m22 . 20 already exclude the possibility that FDM is respon- ing too quickly to cluster) until the typical WDM speed is sible for solving any small-scale structure issues, which sufficiently Hubble damped. In this situation, the WDM would require m22 1. However, independent con- transfer function can be computed with a Boltzmann straints are particularly∼ helpful in this case due to un- code (for instance CAMB [42]) and is well described by certainties regarding the astrophysical modeling of the the fitting form [43{45] IGM. Some have called the strength of the Lyman-α con- WDM 1=2 straints into question (see for instance, Refs. [14, 37]), PL eff 2ν −5/ν TWDM CDM = 1 + (λfs k) ) (1) arguing that the key observable effect of FDM, i.e. the ≡ PL suppression of the flux power spectrum on small scales, could potentially be mimicked by spatial fluctuations in where PL is the linear matter power spectrum, ν = 1:12, the ionizing background radiation, large instantaneous and the effective free-streaming length is temperature inhomogeneities in the IGM due to patchy −1:11 0:11 0:22 eff mχ ΩWDM h reionization, or integrated IGM thermal histories that λfs = 0:07 Mpc, (2) lead to nontrivial gas pressure effects. Since the path to- 1 keV 0:25 0:7 ward solidifying and improving Lyman-α constraints is where ΩWDM is the fraction of the critical density of the bound to be full of nuances and theoretical challenges, Universe comprised of WDM and where h is defined so there is considerable strength in having a diversity of that the Hubble parameter is H0 = 100 h km/s/Mpc. A different probes disfavoring m22 1 with different sys- scale often introduced to parameterize the spectrum of tematics and underlying assumptions.∼ Unlike the case of WDM perturbations is the half-mode length scale, λhm, the Lyman-α forest, thermal properties of the IGM do which corresponds to the scale where the amplitude of not affect the bound on m22 from stellar streams or lens- the WDM transfer function is reduced by half compared ing, and these limits all derive from different astrophys- to CDM. From the transfer function, ical environments, redshifts, and clustering regimes (i.e. eff ν=5 −1=2ν eff quasilinear versus collapsed subhalos). Limits on FDM λhm = 2πλfs (2 1) 13:93λfs : (3) specifically from the Milky Way SHMF are also a very − ≈ direct way to test whether FDM can be responsible for A related quantity used to parameterize the SHMF is solving small-scale structure issues in the local Universe. the half-mode mass, which is the average mass contained It would be difficult to argue that FDM could explain the within a mode of spatial size λhm, fluctuations in the GD-1 stellar stream while still creat- 3 ing a \missing satellites problem" in the Milky Way halo.