Fuzzy Dark Matter at Cosmic Dawn: New 21-cm Constraints
PPrepared for submission to JCAP
Fuzzy Dark Matter at CosmicDawn: New 21-cm Constraints
Olof Nebrin, a Raghunath Ghara, a Garrelt Mellema a a Department of Astronomy and Oskar Klein Centre,Stockholm University, AlbaNova, SE-106 91 Stockholm, SwedenE-mail: [email protected], [email protected],[email protected]
Abstract.
Potential small-scale discrepancies in the picture of galaxy formation painted bythe Λ CDM paradigm have led to considerations of modified dark matter models. One suchdark matter model that has recently attracted much attention is fuzzy dark matter (FDM).In FDM models, the dark matter is envisaged to be an ultra-light scalar field with a particlemass m FDM ∼ − eV. This yields astronomically large de Broglie wavelengths which cansuppress small-scale structure formation and give rise to the observed kpc-sized density coresin dwarf galaxies. We investigate the evolution of the 21-cm signal during Cosmic Dawnand the Epoch of Reionization (EoR) in Λ FDM cosmologies using analytical models. Thedelay in source formation and the absence of small halos in Λ FDM significantly postpone the Ly α coupling, heating, as well as the reionization of the neutral hydrogen of the intergalacticmedium. As a result, the absorption feature in the evolution of the global 21-cm signal hasa significantly smaller full width at half maximum ( ∆ z (cid:46) ), than Λ CDM ( ∆ z (cid:39) ). Thisalone rules out m FDM < × − eV as a result of the σ lower limit ∆ z (cid:38) from EDGESHigh-Band. As a result, Λ FDM is not a viable solution to the potential small-scale problemsfacing Λ CDM. Finally, we show that any detection of the 21-cm signal at redshifts z > byinterferometers such as the SKA can also exclude Λ FDM models.
Keywords: reionization, first stars, X-rays, dark matter theory, axions, power spectrum a r X i v : . [ a s t r o - ph . C O ] A p r ontents ii Bubble Size Distribution 133.5 21-cm Signal 14
What is the nature of dark matter? When and how did the first stars and galaxies form inthe Universe? When and how did the Universe reionize? These are some of the most pressingquestions for modern-day cosmology. The cosmological backdrop provided by the Λ CDMparadigm renders it possible to begin to address them. The development and broad-brushsuccess of the inflationary Λ CDM paradigm has offered a new synopsis of the evolution ofthe Universe over large swathes of time and many decades in scale. The main aspects ofthis paradigm include the inflationary predictions of a flat Universe perturbed by a Gaussianand nearly scale-invariant spectrum of primordial density fluctuations [e.g. 1–3], a recentaccelerating expansion of the Universe driven by something resembling a cosmological constant[4–6], and the hierarchical assembly of large-scale structure expected in a cold dark matter(CDM) Universe [7].If Λ CDM accurately describes dark matter on sub-megaparsec (Mpc) scales, the rootmean square (RMS) amplitude of linear density fluctuations grows as smaller and smallerscales are surveyed until the free-streaming length is reached. The larger amplitudes on smallscales lead to a hierarchical picture of structure formation in which low-mass dark matterhalos form earlier and in greater abundance than high-mass halos.The fate of the baryons in these low-mass halos at high redshifts will be determined byhow rapidly the gas can cool and condense [7, 8]. The very first Population III (PopIII) starsin the Λ CDM model are expected to form in so-called “mini-halos” with masses ∼ M (cid:12) – 1 –t redshifts z ∼ − in which cooling by molecular hydrogen (H ) is effective [e.g. 9, 10].Soon thereafter, this avenue for star formation can be shut down as a result of Lyman-Wernerradiation produced by these first stars which can dissociate the H [11]. Without H -drivencooling, star formation is limited to the earliest “atomic-cooling” halos with virial temperaturesin excess of ∼ K , the limit below which Ly α cooling becomes ineffective [e.g. 12–16]. Thesehalos have characteristic masses ∼ M (cid:12) , typically collapse at redshifts z ∼ − , and areexpected to host the first galaxies. The formation of these halos marks the so-called CosmicDawn when the first galactic sources began to have an impact on the state of the intergalacticmedium (IGM). This eventually culminates in the Epoch of Reionization (EoR) where theIGM evolves into an ionized state in the presence of strong ionizing UV-sources, completingthe process around a redshift z ∼ [17].The progress and timing of events just outlined is intimately connected to the natureof dark matter. If the power spectrum of density fluctuations is suppressed on comovingmass-scales M (cid:46) M (cid:12) , this could substantially reduce the number of PopIII stars formingin minihalos. Stronger suppression of the power spectrum on mass scales M (cid:46) few × M (cid:12) could further delay the EoR and the heating of the IGM — effects that could be observation-ally probed.Such probes have become increasingly urgent in view of the reported small-scale problemswith galaxy formation in Λ CDM. These are the missing satellites problem [e.g. 18–20], theToo-Big-To-Fail Problem [e.g. 21–23], the Core-Cusp problem [e.g. 24–27] and the PlaneSatellites Problem [e.g. 28]. For a recent review mainly relevant for the first three, see [29].An up-to-date review of the Plane Satellites Problem can be found in [30]. Whether thesechallenges to Λ CDM are mainly indicative of unaccounted baryonic feedback mechanisms,new exotic dark matter physics on small scales, or a mixture of the two, remains debated. Ontop of the small-scale challenges to Λ CDM, Weakly Interactive Massive Particles (WIMPs)— long heralded as one of the most promising CDM candidates — have continued to eludethe most recent and sensitive experiments [e.g. 31].These concerns with Λ CDM models motivate a serious look at alternative dark mattermodels that could mitigate the small-scale challenges faced by Λ CDM, and at the same timebe unconstrained by recent experimental searches. One such dark matter model, fuzzy darkmatter (FDM), sometimes called wave dark matter or ψ DM [see e.g. 32–37], has gained anincreasing amount of attention as of late [for a recent review, see 38]. In FDM models, it ispostulated that most of the dark matter has its origin in a new scalar field, the quanta ofwhich (the FDM particles) have exceedingly small masses on the order of m FDM ∼ − eV.The small particle mass entails astronomically large de Broglie wavelengths which can simul-taneously suppress small-scale structure formation and yield kpc-sized cores in dark matterhalos. This renders possible the solution of the Missing Satellites Problem, the Core-CuspProblem, and perhaps the related Too-Big-To-Fail Problem [39], in terms of m FDM — theonly free parameter.The “all-in-one” solution to the small-scale problems facing Λ CDM provided by Λ FDMgives it an advantage over warm dark matter (WDM) models. WDM is, unlike FDM, ther-mally produced in the early Universe, with non-negligible free-streaming that can suppressessmall-scale structure formation [e.g. 40]. However, WDM models suffer from a
Catch 22 prob-lem when it comes to producing cores in halo density profiles: A kpc-sized core in a dwarfgalaxy translates into a suppression of the power spectrum that would prohibit the formationof the dwarf galaxy in the first place [41]. Thus, WDM cannot simultaneously solve the Miss-ing Satellites Problem and the Cusp-Core Problem. This trap is avoided in FDM because for– 2 – given suppression of the power spectrum, FDM generates significantly more extended coresthan WDM [42].For FDM particle masses m FDM ∼ − eV, the halo mass function is suppressedfor halo masses below ∼ M (cid:12) . This eliminates the bulk of halos expected to heat upthe IGM and reionize the Universe in the Λ CDM paradigm. The implied stark differencein the demographics of the first galaxies could potentially be tested by future observationsof the Cosmic Dawn and the EoR. The most direct probe of the state of the IGM duringthose eras is the redshifted 21-cm signal, originating from the hyperfine transition of neutralhydrogen. In principle it has the potential to constrain various details of these epochs suchas the exact timing of the reionization, properties of sources responsible for reionizing theIGM, the presence of X-ray sources, the efficiency of various feedback mechanisms, etc. Asthis signal crucially depend on the population and nature of the sources, it opens a windowto both constrain and distinguish between dark matter models. This is what we explore inthis article.A range of existing radio interferometers such as the Low Frequency Array (LOFAR) [43, 44], the Precision Array for Probing the Epoch of Reionization (PAPER) [45], and theMurchison Widefield Array (MWA) [46, 47] have dedicated substantial resources and effortsto detect fluctuations in the 21-cm signal from the EoR. In parallel, several ongoing experi-ments such as EDGES [48], SARAS [49], BigHorns [50], SciHi [51] and LEDA [52] strive todetect the global (i.e. sky-averaged) 21-cm signal from the EoR and Cosmic Dawn. Recently,[53] presented the first claimed detection of the global 21-cm signal using the EDGES low-bandexperiment. However, this detection has not been independently confirmed. Furthermore,[54] argues that better foreground modelling can remove the signal all together.In anticipation of a confirmed detection of the 21-cm signal, a wide range of theoreti-cal approaches using analytical [e.g., 55, 56], semi-numerical [57–60], and numerical [61–67]methods have been considered for modelling this signal, so as to understand the impact ofvarious astrophysical and cosmological processes on this signal.Reionization in a Universe in which the dark matter is FDM has been studied in termsof the luminosity functions and the production rate of ionizing photons by [68] and [69]. Thelatter showed that reionization by a redshift z (cid:39) , and with a Thomson optical depth valueconsistent with the Planck results [70], is possible in Λ FDM provided m (cid:38) . but thatconstraints from the observed UV luminosity functions for z = 4 – 10 imply a lower limit of m ≥ . . These authors did not consider the 21-cm signal. [71] did study reionization 21-cmpower spectra in FDM cosmologies and derived a lower limit of m (cid:38) . for reionizationto reach 50 percent by z = 8 and also showed that the resulting 21-cm power is 2 – 10 higherthan for Λ CDM on observable scales. They did however not consider the X-ray heating and Ly α coupling effects on the 21-cm signal. [72] investigated the effect of FDM models on theglobal 21-cm signal from the Cosmic Dawn and place a lower limit of m ≥ for couplingthe spin temperature to the gas temperature at z = 20 , which would be required to explainthe timing of the claimed EDGES low-band detection.Unlike FDM, the effect of WDM on the 21-cm signal from the Cosmic Dawn and Epochof reionization has been investigated by many previous studies such as [73] and [74]. Thesestudies find that WDM delays the emergence of the 21-cm signal and accelerates the impactof X-ray heating. Recently, [75] have used the EDGES low band results to constrain the http://eor.berkeley.edu/ – 3 –article mass of WDM. This study constrains the mass of the WDM particles to be > keVif the star formation rate at z = 18 is dominated by atomic cooling.In this article we provide a detailed investigation of the effect of the FDM models onthe 21-cm signal in terms of the global signal, the power spectrum as well as bubble sizedistributions. In this we take into account the effects of Ly α coupling, X-ray heating, aswell as photo-ionization. We use a fast analytical framework for this investigation. We alsoexplore the possibilities for ongoing and future 21-cm experiments to constrain the FDMparticle mass, m .The paper is structured as follows. In Section 2, we briefly describe the FDM modelused in this study as well as their effects on the halo abundance in the Universe. The basicframework of the 21-cm signal from the Cosmic Dawn is presented in Section 3, and we presentour results in Section 4. We summarize our findings in Section 5. Throughout the paper, weadopt the following cosmological parameters: Ω m = 0 . , Ω B = 0 . , Ω Λ = 0 . , h = 0 . , σ = 0 . , and n s = 0 . [70]. Because the 21-cm signal is strongly governed by the nature of the ionizing and X-ray sources,it is important to have a plausible picture of how galaxy formation would proceed in Λ FDM. Inthis section the most important aspects of FDM, especially for galaxy formation, are discussed,some of which will be used in formulating the scenarios to be simulated. In Section 2.1 weuse natural units, wherein (cid:126) = c = 1 , for convenience, but not elsewhere in this paper. As discussed by [38, 76], the Lagrangian for the most well-motivated model of FDM — atleast one new ultra-light axion-like scalar field φ that can appear in the context of stringtheory — reads L = 12 ∂ µ φ∂ µ φ − m FDM F (cid:20) − cos (cid:18) φF (cid:19)(cid:21) , (2.1)where F is a constant predicted to lie somewhere in the range between ∼ GeV (theGUT scale) and ∼ GeV (the Planck scale). Assuming a Friedmann-Lemaître-Robertson-Walker background, the FDM scalar field is governed by the following Klein-Gordon equation: ¨ φ + 3 H ˙ φ + m FDM F sin (cid:18) φF (cid:19) = 0 , (2.2)where H = ˙ a/a is the Hubble parameter. The corresponding energy density of the scalar fieldis ρ FDM = 12 ˙ φ + m FDM F (cid:20) − cos (cid:18) φF (cid:19)(cid:21) . (2.3)When H ∼ t − (cid:28) m FDM , the scalar field starts to oscillate with a decaying amplitude. Inthis regime the approximate WKB solution of the Klein-Gordon equation is of the form [76] φ (cid:39) φ (cid:16) a osc a (cid:17) / cos( m FDM t ) , (2.4)where H ( a osc ) ∼ t − osc ∼ m FDM . From Eq. (2.3) we therefore see that the energy densityfrom then on scales as ∼ a − . The time at which this CDM-like behaviour turns on is– 4 – osc ∼ (10 − m eV ) − (cid:39) . m − yr, where m ≡ m FDM / − eV. For the particle massrange considered in this paper, this is safely in the radiation-dominated era. In summary, thelarge-scale evolution of the Universe in Λ FDM is indistinguishable from Λ CDM.The predicted present-day abundance of FDM is also entirely specified by φ , F , and m [77, 78]: Ω FDM (cid:39) . m / (cid:18) φ GeV (cid:19) ln / (cid:20) e − ( φ /πF ) (cid:21) . (2.5)As long as πF is greater than H I , the inflationary Hubble scale, then the initial field valueof the scalar field, φ , has a uniform distribution on the interval [ − πF, πF ] [e.g. 76, 79, 80].The condition πF > H I will indeed be satisfied for FDM models of interest, because weneed H I (cid:46) × m − / GeV in order to evade observational constraints on isocurvaturefluctuations [78] — far smaller than
F > GeV . We therefore expect that φ ∼ F ,showing that F ∼ GeV (in the range favoured by particle physics) and m ∼ (theparticle mass motivated by astrophysics) can explain the observed dark matter abundance( Ω DM = 0 . for our adopted cosmological parameters). Because of this, and in order tomaximize the potential of Λ FDM in solving small-scale problems facing Λ CDM, we assumethroughout the rest of the paper that Ω FDM = Ω DM . Modifications to the unfolding of events in Λ CDM make their first appearance in small-scalestructure formation. In the non-relativistic limit and on sub-horizon scales, the amplitude ofa density perturbation δ k with comoving wavenumber k is governed by [e.g. 33, 76, 78] ¨ δ k + 2 H ˙ δ k = (cid:32) H Ω m a − (cid:126) | k | a m FDM (cid:33) δ k , (2.6)where H = 100 h km s − Mpc − is the Hubble constant. Growth is only possible if the righthand side is positive, which defines a Jeans wavenumber k J,FDM , k J,FDM = (cid:0) H Ω m m FDM a (cid:1) / (cid:126) − (cid:39) m / (cid:18) Ω m h . (cid:19) / (cid:18) z (cid:19) − / Mpc − . (2.7)The main implications of this Jeans scale are twofold: • There can be no growth of density perturbations and therefore no halo formation onmass scales below M J,FDM = 4 π ( π/k J,FDM ) ρ m , , where ρ m , ≡ ρ m ( z = 0) is the present-day mean matter density, given by M J,FDM (cid:39) . × m − / (cid:18) Ω m h . (cid:19) / (cid:18) z (cid:19) / M (cid:12) . (2.8) • In Λ CDM, density perturbations can only grow significantly in the matter-dominatedera. Thus, structure formation will be delayed in Λ FDM relative to Λ CDM below mass Here we have used the expression for Ω FDM derived by [78], but including the logarithmic correction termfound by [77] that becomes important when | φ | /πF (cid:39) due to anharmonic effects. – 5 –cales on the order of M J,FDM evaluated at the redshift of matter-radiation equality, z eq : M Jeq,FDM (cid:39) . × m − / (cid:18) Ω m h . (cid:19) / (cid:18) z eq (cid:19) / M (cid:12) . (2.9)Qualitatively, we therefore expect the halo mass function (HMF) to peak around M Jeq,FDM ,and below that drop abruptly so that no halos with masses around M J,FDM are present.The HMF used in this study needs to capture both of these features in order to accuratelymodel the predicted 21-cm signal. We model the HMF as follows. For the FDM linear powerspectrum P FDM ( k ) , we use the fit from [32], P FDM ( k ) = T F ( k ) P CDM ( k ) , (2.10)with T F ( k ) (cid:39) cos( x )1 + x , (2.11)in which x ≡ . m / kk Jeq,FDM . (2.12)We see that there is a significant loss of power relative to P CDM ( k ) for k (cid:38) k Jeq,FDM asexpected. With this power spectrum, it is possible to naively compute the HMF using thePress-Schechter or Sheth-Tormen HMF [81, 82], or running a CDM-like N-body simulation[e.g. 35, 37, 83]. However, these approaches will not induce a sharp cut-off to the HMFnear M J,FDM since CDM-like N-body simulations and semi-analytical halo mass functions bythemselves do not incorporate the relevant pressure-like effect that gives rise to the Jeansscale. [84] followed the evolution of the HMF in Λ FDM in N-body simulations that bothincorporated and ignored the pressure-like effect, showing an expected sharp decline in thelow-end HMF when the effect was included.We model the HMF in a semi-analytic fashion following [42], [68], [85], and [36]. Theseauthors modelled the sharp cut-off to the HMF by including a mass dependent barrier δ crit ( M ) for halo formation. This simulates the scale-dependent growth implied by the solution of Eq.(2.6). [85] found that δ crit ( M ) could be fitted in a redshift-independent manner, δ crit ( M ) (cid:39) G F ( M ) δ crit,CDM ,G F ( M ) = h F ( x ) exp( a x − a )+ [1 − h F ( x )] exp( a x − a ) ,x = M/M J,FDM , (2.13) h F ( x ) = 12 (cid:8) − tanh[ M J,FDM ( x − a )] (cid:9) ,M J,FDM = a × m − / (cid:18) Ω m h . (cid:19) / h − M (cid:12) . Here δ crit,CDM (cid:39) . is the mass-independent barrier for spherical collapse in the matter-dominated era of Λ CDM, and the fitted constants are, { a , a , a , a , a , a } = { . , . , . , . , . , . } . (2.14) When z (cid:28) (Ω m / Ω Λ ) / the barrier asymptotes to (cid:39) . [86]. But during the Cosmic Dawn (andeven at z (cid:39) ), adopting the Einstein-de Sitter value of δ crit,CDM (cid:39) . is a good approximation. – 6 –iven this expression for δ crit ( M ) , we use the Press-Schechter HMF [81] with the replacement δ crit,CDM → δ crit ( M ) , ∂n ( M, z ) ∂ ln M = (cid:114) π ρ m , M δ crit ( M ) σ ( M, z ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ln σ∂ ln M (cid:12)(cid:12)(cid:12)(cid:12) e − δ crit ( M ) / σ ( M,z ) . (2.15)Here σ ( M, z ) is the root-mean-square linear overdensity within a spherical region of comovingmass M , computed using the power spectrum in Eq. (2.10) and normalized so that σ ( R =8 h − Mpc , z = 0) = σ .The simple prescription δ crit,CDM → δ crit ( M ) was first used by [42] and [68]. The HMFin Eq. (2.15) is plotted at z = 6 for different m in the left-hand panel of Figure 1. It isseen that the HMF for FDM peaks at ∼ × h − M (cid:12) for m = 1 , fairly close to what weexpected from Eq. (2.9). A dramatic deviation from Λ CDM is evident for halo masses belowthis peak. For example, with m = 1 the number density of halos with masses ∼ M (cid:12) isfour dex below the predicted number density in Λ CDM.The fact that Eq. (2.15) captures the peak and the sharp decline in the HMF is animprovement over the fit provided by [35], who ran N-body simulations with the FDM powerspectrum from Eq. (2.10), but ignored the pressure-like effect on small scales. However, Eq.(2.15) is not self-consistent. This is because a proper derivation of the HMF for a mass-dependent barrier should make use of the excursion set formalism [87] and not the merereplacement δ crit,CDM → δ crit ( M ) .The excursion set problem for Λ FDM was numerically solved by [36], and comparedwith the approach we adopt here. These authors found that a self-consistent solution of theexcursion set problem will yield a slightly sharper cut-off in the HMF, resulting in even fewerlow-mass halos than predicted by Eq. (2.15). We neglect this for the following reasons: • [36] found that the difference between the excursion set solution and the adopted HMFhere is minimized at high redshifts relevant for the Cosmic Dawn. • In order to draw strong conclusions from the predicted evolution of the 21-cm signal,we take a conservative approach. A sharper cut-off in the HMF would only exacerbatethe deviation from Λ CDM, and therefore make our conclusions even stronger.To conclude this section we would like to point out that there exist ‘extreme’ versionsof FDM, having an initial field displacement | φ | /πF (cid:39) [88–90]. In this case, anharmoniceffects from the potential in Eq. 2.1 can, for a given m , lead to a far greater numberof low-mass halos than the ‘vanilla’ version of FDM described above. As the properties ofthe HMF determine the galaxy population in our models, as explained below, the presenceof large numbers of low-mass halos would lead to very different results for our calculations.Indeed, [90] finds that Λ FDM can evade Ly α constraints at least as easily as Λ CDM if δθ ≡ | φ | /F − π (cid:46) . , with a best fit somewhere in the range . (cid:46) δθ (cid:46) . for m = 1 . . However, the prior probability of the initial field value being this close to the topof the potential is at most P ( δθ < . . /π (cid:39) . , or in other words the ‘extreme’FDM model requires improbable initial conditions. Given this, we will only consider ‘vanilla’FDM in this paper. Since a small value of δθ only increases the abundance of low-mass halos hosting relatively few stars, itseems unlikely that anthropic arguments can escape this conclusion. – 7 – -5 -4 -3 -2 -1 z=6 dn / d l n M ( h / M p c ) - M (MO• h -1 ) Λ CDMm =1m =2m =4 10 -9 -8 -7 -6 -5 -4 -3 -2 -1
6 8 10 12 14 16 18 20 f c o ll ( z ) z 10 -9 -8 -7 -6 -5 -4 -3 -2 -1
1 10 100 f c o ll ( z ) m z=6z=10z=15 Figure 1 : Left-hand Panel:
The halo mass functions at redshift 6 for Λ CDM, and Λ FDMwith a selected few values of m considered in this work. Middle Panel:
Redshift evolutionof the collapse fractions corresponding to the same scenarios as in the left-hand panel.
Right-hand Panel:
Dependence of the collapsed fraction on the FDM particle mass, m at differentredshifts. The horizontal lines mark the collapse fractions corresponding to Λ CDM. Theminimum Virial temperature for star formation is fixed to T vir = 10 K . While T vir determinesthe minimum halo mass for star formation in Λ CDM, M min corresponding to Λ FDM modelsis estimated following Eq. 2.18.
The suppression of structure formation on small scales also implies that galaxy formationshould be significantly delayed in Λ FDM relative to Λ CDM. More specifically, since structureformation is only suppressed below ∼ M Jeq,FDM in Λ FDM, the hierarchical structure forma-tion above this scale implies that the first galaxies form in halos with masses ∼ M Jeq,FDM .For comparison, Λ CDM predict that the first proper galaxies formed in atomic cooling ha-los with characteristic masses of ∼ M (cid:12) set by efficient Ly α cooling [e.g. 14]. Thesewould have formed relatively early compared to the ∼ M Jeq,FDM halos in Λ FDM since M Jeq,FDM (cid:38) M (cid:12) for m (cid:46) .Furthermore, with no early minihalos with mass M ∼ M (cid:12) in which H cooling ispossible [see e.g. 9, 10], Λ FDM predicts that the sites for Pop III star formation would bethe first atomic cooling halos with masses ∼ M Jeq,FDM . The expectation of delayed galaxyformation and radically different host halos for Pop III star formation in Λ FDM was recentlyconfirmed in the hydrodynamical simulations of [37].The process of reionization as well as the thermal state of the IGM will be closely relatedto f coll , the fraction of baryons in collapsed structures wherein star formation can proceed.If star formation is possible in halos above a minimum mass M min , then the collapse fractioncan be computed by integrating the HMF, f coll ( z ) = 1¯ ρ m , (cid:90) ∞ M min d M ∂n ( M, z ) ∂ ln M . (2.16)In Λ FDM, both the inability of virialized gas to cool efficiently and the stability againstgravitational collapse below the Jeans scale compete to determine M min . As remarked earlier,minihalos where H cooling is possible can be completely neglected in Λ FDM. Thus, efficient– 8 –ooling is only possible in halos with virial temperatures above T vir (cid:39) K, determined bythe onset of rapid Ly α cooling. The halo mass corresponding to this limit is [e.g. 91] M Ly α (cid:39) . × (cid:18) T vir K (cid:19) / (cid:18) Ω m h . (cid:19) − / (cid:18) ∆ vir π (cid:19) − / × (cid:16) µ . (cid:17) − / (cid:18) z (cid:19) − / M (cid:12) , (2.17)where ∆ vir ≡ ρ vir / ¯ ρ m is the mean overdensity of a virialized halo, and µ is the mean molecularweight of the baryons at temperature T vir . Since we are focusing on the high-redshift matter-dominated Universe, we adopt ∆ vir = 18 π [92, 93]. The mean molecular weight is µ (cid:39) . for neutral primordial gas, and µ (cid:39) . for ionized primordial gas. Even though hydrogen incollisional ionization equilibrium at a temperature < × K is mostly neutral, we adopt µ = 0 . , which crudely takes into account photoionization of the accreting gas from the IGM.Given this, the minimum halo mass for star formation will simply be, M min = max ( M Ly α , M J,FDM ) . (2.18)Comparing Eq. (2.17) and Eq. (2.8), we see that M J,FDM > M Ly α for redshifts z > m / . However, both M J,FDM and M Ly α lie at masses below the peak in the HMF, andtherefore do not have much of an effect on the collapse fraction f coll ( z ) . The collapse fractionsfor Λ CDM and Λ FDM are plotted in the middle panel of Figure 1. The collapse fraction for Λ FDM, with m ∼ O (1) , remains significantly smaller than Λ CDM even towards the endof the EoR at z (cid:39) . Furthermore, the redshift gradient of f coll ( z ) is seen to be steeperfor Λ FDM. From these observations, a later and more rapid heating and reionization of theIGM is expected for Λ FDM. The right-hand panel of the Figure 1 show the dependence of thecollapsed fraction on m parameter at different redshifts. Although f coll decreases drasticallywith m , it approaches the Λ CDM values for large m values. Even for m = 15 we seethat the collapse fractions for Λ FDM and Λ CDM differ by a factor of ∼ at z = 15 . The analytical model used in this work follows previous works such as [94, 95]. The modelincorporates the effects of both UV and X-ray photons and tracks the evolution of the volumeaveraged ionization fractions x i and x e which correspond to the highly ionized H ii regions andlargely neutral gas in the IGM outside these regions. It also estimates the kinetic temperature( T K ) of the largely neutral medium outside the H ii regions, while T K is assumed to be ∼ K in the H ii regions. The basic formalism of the model is described as below.The rate of production of the UV photons per baryon can be expressed as, Λ i = ζ d f coll d t , (3.1)where ζ ≡ N ion f (cid:63) f esc is the ionization efficiency parameter. The ionization state of the IGMtherefore crucially depends on quantities such as the star formation efficiency ( f (cid:63) ), escapefraction of the UV photons ( f esc ), and mean number of ionizing photons produced per stellarbaryon ( N ion ). We describe how we set these three quantities in Sections 3.2 and 3.3.– 9 – z )5)4)3)2)1 l o g ̇ r ⋆ ( M ⋆ y r ) M p c ) ) Madau & Dickinson (2014)Bouwens et al. (2015)Oesch et al. (2013)Oesch et al. (2014)Coe et al. (2013)
CDM, f ⋆ = 0.012m = 1, f ⋆ = 0.11m = 2, f ⋆ = 0.05CDM, f ⋆ = 0.012m = 1, f ⋆ = 0.11m = 2, f ⋆ = 0.05 z m = 4, f ⋆ = 0.031m = 16, f ⋆ = 0.015m = 4, f ⋆ = 0.031m = 16, f ⋆ = 0.015 Figure 2 : Predicted cosmic star formation history for the main scenarios considered in thiswork at redshifts between z = 5 . and z = 12 . , along with observational constraints. Thestar formation efficiency f (cid:63) is tuned individually for each dark matter scenario in order toroughly fit the observational constraints on the star formation history. In both panels we plotobservational constraints compiled by [97] from z = 5 . − . , as well as constraints at higherredshifts z ∼ − from [98], [99], [100], and [101]. The limits at redshifts z ∼ − shouldbe seen as lower limits to ˙ ρ (cid:63) as they only correspond to galaxies brighter than M UV ∼ − . Left-hand panel:
The evolution of the cosmic star formation rate in Λ CDM and Λ FDMwith m = 1 , . Right-hand panel:
Same as the left panel but for Λ FDM with m = 4 , .The ionization and thermal state of the mostly neutral medium beyond these highlyionized H ii regions crucially depend on the X-rays produced by these sources. The emissivityof the X-ray photons from the sources is assumed to follow the star formation rate density.We use two X-ray parameters, namely f X and α X which quantify the spectral distributionwhich is modelled as, (cid:15) X ( ν ) = L hν (cid:18) νν (cid:19) − α X − , (3.2)where L = 1 × f X erg s − Mpc − , hν = 1 keV .The 21-cm signal from the cosmic dawn depends also on the Ly α photon flux from thesources. Here we estimate the average Ly α photon flux using the formalism of [96]. Weconsider the stellar emission and the X-ray excitation of the neutral hydrogen to estimate theaverage Ly α background. The details of the source model used in this study is described inthe following section. Only a fraction, denoted f (cid:63) , of the baryons in an atomic-cooling halo will be able to formstars. Following previous work we assume for simplicity that this fraction is independent ofthe halo mass and redshift [e.g. 94, 102, 103]. It then follows that the cosmic star formation– 10 –ate per unit comoving volume is, d ρ (cid:63) d t = f (cid:63) Ω B Ω m ρ m , d f coll d t . (3.3)We can then tune f (cid:63) to roughly be consistent with observational constraints on the cosmicstar formation rate for redshifts z (cid:38) . In Figure 2 we compare the theoretically derived ˙ ρ (cid:63) from Eq. (3.3) with observational constraints on the cosmic star formation rate. For redshifts z = 5 . − . we use data compiled by [97]. At higher redshifts of z ∼ − we also showobservational constraints from [98], [99], [100], and [101]. These constraints at higher redshiftsshould be seen as lower limits to ˙ ρ (cid:63) as the UV luminosity function is only integrated down to M UV (cid:39) − . for the data points by [98], [99], and [100], whereas the constraint at z (cid:39) . by [101] integrates down to M UV (cid:39) − . .We choose f (cid:63) mainly so that the tighter constraints at z ∼ − are approximatelyrespected. Doing this, it is seen that Λ FDM with m = 1 is possibly in tension with currentlower limits to ˙ ρ (cid:63) at z ∼ − , whereas m = 2 is not. This is broadly consistent with the σ lower bounds m ≥ . and m ≥ . derived by a detailed modelling of the UV luminosityfunction for Λ FDM by [35] and [69] respectively. The values of f (cid:63) needed to be consistentwith data is seen to increase with m from f (cid:63) ( m = 1) (cid:39) . to f (cid:63) ( m = 16) (cid:39) . .For Λ CDM, which corresponds to the limit m → ∞ , we find f (cid:63) ( m → ∞ ) (cid:39) . . Theresulting f (cid:63) - m relation can be fitted with the following expression. f (cid:63), fit ( m ) = 0 .
012 + 0 .
098 exp[ − . m − / ] . (3.4)This fit agrees exactly with our chosen values for f (cid:63) ( m = 1) and f (cid:63) ( m → ∞ ) , and inbetween only deviates by less than . . The fit f (cid:63), fit ( m ) will only be used in our study ofthe parameter space in Section 4.2 where we need a continuous relationship between f (cid:63) and m . The fact that we need a higher f (cid:63) for lower m simply reflects the fact that Λ FDMproduces fewer low-mass halos than Λ CDM which means that in order to not violate con-straints on ˙ ρ (cid:63) the average fraction of baryons ending up in stars needs to be larger. Beyondthis empirical constraint, such a f (cid:63) - m relationship could also plausibly arise naturally. Ob-servations indicate that f (cid:63) is a non-linear function of the halo mass — growing with halomass for M (cid:46) M (cid:12) [e.g. 104, 105]. Since lower values of m suppresses the formation oflow-mass halos, the halo mass-weighted average of f (cid:63) would decrease with m .Figure 2 also shows the chosen values of f (cid:63) for our four FDM models which will bedescribe latter in section 4.1. We also list them in Table 1 . For the parameter study weadopt the fit in Eq. (3.4) which approximately reproduces the f (cid:63) - m relation in the mainscenarios. The ionization and thermal state of the IGM during the Cosmic Dawn and the EoR cruciallydepend on the properties of the radiating sources present during these epochs. However, thereis a huge uncertainty regarding these. For the 21-cm signal, the most important photons areUV, X-rays and Ly α photons which are drivers of ionization, heating and Ly α couplingrespectively. • Ionizing photons : The relative contributions from the PopIII and PopII stars to thetotal ionizing photon budget is not well understood. We fix N ion , PopII = 4000 and– 11 – ion , PopIII = 30000 for the PopII and PopIII source model respectively. The escapefraction of ionizing photons f esc is taken to be 0.1. For a given fraction f PopII of starsthat are PopII, the total ionization budget will then be N ion = f PopII N ion , PopII + (1 − f PopII ) N ion , PopIII . (3.5)The parameters N ion and f PopII are determined as follows:1. First we estimate the approximate value of N ion needed to reionize the Universeby z = 6 for a given value of m .2. Next, if N ion , PopIII ≥ N ion ≥ N ion , PopII using Eq. 3.5 we determine the correspond-ing value of f PopII . If (cid:54) f PopII (cid:54) , we adopt this value.3. If N ion > N ion , PopIII , our models cannot reionize the Universe by z = 6 . We do notconsider these cases here.4. If N ion < N ion , PopII , the Universe can easily be reionized by z = 6 from PopII starsalone, so we set f PopII = 1 . • X-rays : Similar to the ionizing sources of reionization, the X-ray sources are alsouncertain. Mini-quasars, supernova remnants, X-ray binaries and the hot interstellarmedium in starburst galaxies are some of the possible X-ray sources during this epoch.We choose f X = 1 and α X = 0 . as our fiducial X-ray spectrum which corresponds to amini-quasar type X-ray source. In the parameter study in Sect. 4.2 we consider a widerange of values for f X and α X . • Ly α photons : We follow [106] to model the Ly α emission from the sources. We assumea power law spectrum (cid:15) s ( ν ) ∝ ν − α s − between Ly α and Ly β and between Ly β and theLyman limit, where the power law indices can differ. The spectral index α s between Ly α and Ly β is taken to be 0.14 and 1.29 for PopII and PopIII stars, respectively.For PopII stars, the spectrum is normalized such that the number of Ly α photons perbaryon in the range Ly α - Ly β is 6520 and we adjust the spectral index in the range Ly α -Lyman limit so that the total number of photons per baryon for this wavelengthregime is 9690. These numbers are 2670 and 4800 respectively for PopIII stars. Notethat we consider Ly α photon contributions from both PopII and PopIII stars in modelswith < f PopII < .Note that the use of a redshift-independent fraction of PopII stars (i.e. f PopII = constant )is a consequence of assuming a redshift-independent escape fraction f esc and star formationefficiency f (cid:63) . In reality, the fraction of PopII and PopIII stars would evolve with time ashalos and the IGM become enriched with metals. The exact conditions needed to transitionfrom massive PopIII star formation to the formation of relatively low-mass PopII stars arestill debated. Ignoring the presence of dust, CII or OI mediated cooling can probably inducefragmentation to low-mass PopII stars for [C / H] (cid:38) − . or [O / H] (cid:38) − . respectively [107].However, dust cooling could push the critical metallicity for low-mass PopII star formationdown to − (cid:46) log Z/Z (cid:12) (cid:46) − [e.g. 108, 109], but it may also be the case that the dust isevacuated before this can happen [110]. These considerations imply that a realistic modelling– 12 –f the evolution of the PopII fraction in the context of the analytical models used here wouldbe fraught with uncertainties. However, f PopII = constant may be a good approximationfor Λ FDM given the suppression of small-scale structure formation. In particular, low-masshalos in Λ FDM would have few, if any, progenitors of lower masses, indicating a monolithiccollapse. With few progenitors containing enriched gas, the transition from PopIII to PopIIstar formation would be prolonged, and so f PopII would remain almost constant. This isconsistent with the hydrodynamical simulations of [37], who found that PopIII star formationpersists down to z ∼ for m = 1 . H ii Bubble Size DistributionTo correctly model the fluctuations in the 21-cm signal towards the end of the EoR, we needto have an appropriate model of the size distribution of H ii regions around ionizing sources.We model the H ii bubble size distribution (BSD) mainly following the formalism of [111]and [95] with some modifications for Λ FDM. In the formalism of [111] we consider a regionof mass M with linear overdensity δ M , and RMS linear overdensity σ ( M ) ≡ σ ( M, z = 0) .The IGM within this region of mass M will be ionized at a redshift z if there are a sufficientnumber of ionizing photons at that time and in that region. Mathematically this criterioncan be written as ζ f coll [ z, δ M , σ ( M ) ] ≥ . (3.6)If this is fulfilled, there is at least one ionizing photon per hydrogen atom in our region of mass M . The collapse fraction in this expression can be derived using the extended Press-Schechterformalism [87]. In Λ CDM, the barrier for halo formation δ crit,CDM is constant and thus thereis an analytical solution for the collapsed fraction which can be expressed as f coll [ z, δ M , σ ( M ) ] = erfc δ crit,CDM ( z ) − δ M (cid:113) (cid:2) σ − σ ( M ) (cid:3) . (3.7)In this notation, all the redshift dependence from the growth of linear density perturbationshas been absorbed into the barrier: δ crit,CDM ( z ) = δ crit,CDM /D ( z ) where D ( z ) is the growthfactor of linear density perturbations. Because of this, the growth factor is not incorporatedinto σ min = σ ( M min ) even though M min , given by Eq. (2.18), is redshift dependent.Let δ M = δ x ( M, z ) be defined such that ζ f coll [ z, δ x ( M, z ) , σ ( M ) ] = 1 . (3.8)The physical interpretation of the barrier δ x ( M, z ) is that regions with linear overdensity δ M ≥ δ x ( M, z ) will be ionized. For Λ CDM we can use Eq. (3.7) to invert Eq. (3.6) and obtainan analytical expression for δ x ( M, z ) . However, for Λ FDM the barrier for halo formation isnot constant and Eq. (3.7) can therefore not be used to find δ x ( M, z ) . However, an excellentapproximate solution for δ x ( M, z ) can be found as follows: • We can always find some value of the RMS linear overdensity σ min , (1) such that:erfc (cid:34) δ crit,CDM ( z ) √ σ min , (1) (cid:35) = f coll ( z ) , (3.9)– 13 –here f coll ( z ) is the global collapse fraction in Λ FDM given by Eq. (2.16). We canreadily invert Eq. (3.9) to find, σ min , (1) = δ crit,CDM ( z ) √ erf − [1 − f coll ( z )] . (3.10)In the limit when m → ∞ we get σ min , (1) → σ min . Thus, σ min , (1) yields 1 st -ordercorrections to the erfc-formula for the conditional collapse fraction in Λ FDM: f (1)coll [ z, δ M , σ ( M ) ] = erfc δ crit,CDM ( z ) − δ M (cid:114) (cid:104) σ , (1) − σ ( M ) (cid:105) . (3.11) • Next, we can use Eq. (3.11) and Eq. (3.6) to derive δ (1)x ( M, z ) , the 1 st -order approxi-mation to δ x ( M, z ) : δ (1)x ( M, z ) = δ crit,CDM ( z ) − √ K ( ζ ) (cid:113) σ , (1) − σ ( M ) , (3.12)where K ( ζ ) = erf − (1 − ζ − ) .Using the approximate barrier for H ii bubble formation in Eq. (3.12), the H ii bubblemass distribution becomes: ∂n b ( M, z ) ∂ ln M = (cid:114) π ρ m , M | T ( M, z ) | σ ( M ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ln σ∂ ln M (cid:12)(cid:12)(cid:12)(cid:12) e − δ (1)x ( M,z ) / σ ( M ) , (3.13)where, T ( M, z ) = (cid:88) n =0 ( − S ) n n ! ∂ n δ (1)x ( M, z ) ∂S n , S ≡ σ ( M ) . (3.14)The function T ( M, z ) is a fit that takes into account the non-linearity of the H ii bubbleformation barrier in Eq. (3.12). If Eq. (3.12) is expanded linearly in S = σ ( M ) , werecover the exact same bubble mass distribution formula as in [111]. We do not performsuch an expansion because the non-linearity in the bubble formation barrier is not negligiblein Λ FDM. The fitting function T ( M, z ) was found by [82], and yields an approximate massfunction when the barrier is slightly non-linear in S . [112] confirmed that the H ii bubblemass function computed using T ( M, z ) and the form of the barrier function in Eq. (3.12)yielded a good approximation to the bubble mass function derived numerically in the extendedPress-Schechter formalism. The differential brightness temperature of the 21-cm signal at a region with coordinate x canbe written as, δT b ( x , z ) = 27 x HI ( x , z )[1 + δ B ( x , z )] (cid:18) Ω B h . (cid:19) × (cid:18) . m h z (cid:19) / (cid:18) − T γ T S (cid:19) mK , (3.15)where x HI , δ B , T γ = 2 . × (1 + z ) K and T S denote the neutral fraction, density contrast,the CMBR temperature and the spin temperature of hydrogen gas at position x at redshift– 14 – . Note that the above expression of δT b does not include the contribution from the peculiarvelocities of the gas in the IGM. As the effect of peculiar velocities is not very significant inpresence of the spin temperature fluctuations [67, 113], we will ignore their contribution tothe brightness temperature fluctuations in this study.The fluctuations in the differential brightness temperature can, to first order, be linearlyexpanded as [see e.g. 94] δT b = β B δ B + β x δ x + β α δ α + β T δ T , (3.16)where δ B is the baryon density fluctuation, δ x is the fluctuation in the neutral fraction, δ α the fluctuation in x α , and δ T the fluctuation in the kinetic temperature T K . The expressionof the β coefficients can be found in [94]. The spherically averaged power spectrum of δT b can then be written as P δT b ( k ) = δT b P δδ ( k )( β (cid:48) + 2 β (cid:48) / / , (3.17)where P δδ ( k ) and δT b denote the power spectrum of the density field and the mean brightnesstemperature respectively. The quantity β (cid:48) can be expressed as β (cid:48) = β B − β x ¯ x e g e / (1 + ¯ x e ) + β T g T + β α W α , (3.18)where we assume δ T = g T ( k, z ) δ, δ α = W α ( k, z ) δ and δ e = g e δ with δ e = (1 − /x e ) δ x . Wewill present our results in terms of the dimensionless power spectrum ∆ ( k ) = k P δT b ( k ) / π .We note that the linear approximation for the 21-cm power spectrum is a relatively crudeone as the cross-terms and non-linear terms can contribute substantially, see [114] and [115]. The main motive of this paper is to study the impact of different FDM models on the cos-mological 21-cm signal from the EoR and CD in terms of different quantities such as thepower spectrum of the brightness temperature, global signal, bubble statistics, etc. whichcan provide useful insights on these epochs. To gain intuition we first study in Sect. 4.1 a setof four FDM models which differ in their value of m and compare them to a CDM model.For this limited set of models we can present both the global signal evolution and the powerspectra in detail. After that we present in Sect. 4.2 a parameter study where we vary both m and the X-ray parameters f X and α X . In our four Λ FDM models we want to explore the impact of different values for m . Wechoose m =1, 2, 4 and 16. This parameter determines the halo mass function. We thencalculate the value of f ∗ which reproduces the star formation rate density between redshift6 and 7 from [97]. Finally we choose a value of N ion which ensures that reionization finishesby z = 6 . The values of these parameters are listed in Table 1. Note that the value of N ion automatically implies a value for f PopII . The Thomson scattering optical depth for all modelsare consistent with the Planck range τ = 0 . ± . from [116]. We denote S to beour fiducial model for Λ FDM scenarios. For all the models we use f esc = 0 . , f X = 1 and α X = 0 . . The models are also consistent, to within σ , with the tighter constraint on τ from Planck 2018: τ =0 . ± . [ σ ; 117]. – 15 – cenario DM model m f (cid:63) N ion f PopII τ ( − ) < δT b > min (mK) z δT b , min ∆ z ∆ max S0 Λ CDM – 0.012 12173 0.68 6.70 -109.4 14.5 67.8 12.4S1 Λ FDM 1.0 0.110 16841 0.51 4.65 -78.5 8.5 198.3 7.9S2 Λ FDM 2.0 0.050 18235 0.45 4.88 -79.4 9.0 147.3 8.3S3 Λ FDM 4.0 0.031 17152 0.49 5.13 -85.8 9.6 125.0 8.8S4 Λ FDM 16.0 0.015 17253 0.50 5.70 -93.0 11.0 90.0 9.9
Table 1 : The table shows the choice of parameters for the main scenarios considered in thiswork, as well as the main results. We fixed f esc = 0 . for all the main scenarios, but explorevariations in Section 4.2. The value of N ion corresponding to a model is estimated such thatreionization ends at z ∼ . N ion sets the fraction of population II stars ( f PopII ) consideredin each model. We consider S be our fiducial Λ FDM model. We have used f X = 1 and α X = 0 . for these models. See the body of the text for details. a ∆ is evaluated at k = 0 . h Mpc − .We also include one Λ CDM scenario for which we used the same procedure to set theparameters choices. It uses a minimum halo mass for star formation M min = M Ly α , seeEq. 2.17. This will make the Λ CDM results very different from the Λ FDM results justbecause of the difference in halo population. If the minimum mass of halos that can formstars would be much higher in the CDM case, the results could start to resemble the FDMones. These kinds of degeneracies are hard to break, see for example the study of [73] for thecase of WDM. Our study does not attempt to identify unique features for FDM which woulddistinguish it from all possible CDM models but rather includes a CDM model for referenceand shows what can be expected from FDM models.
First, we will discuss the impact of different FDM models in terms of the global variables ofEoR such as the volume averaged ionization fraction ( Q ), kinetic temperature of the partiallyionized IGM and the mean brightness temperature. The redshift evolution of the volumeaveraged ionization fraction for these different models are shown in the left panel of Figure 3.We find the evolution of Q is more rapid for the FDM model compared to the CDM model.For example, Q evolves from 0.2 to 1 between the redshifts 8 and 6 for the FDM model S ,while this occurs between z = 11 and z = 6 for the CDM model S . This is expected asthe ionization history follows f coll which evolves rapidly for the FDM models as we have seenpreviously.The evolution of the average T K for the different Λ FDM models as well as the CDM modelare shown in the middle panel of Figure 3. One can easily notice that the heating processis severely delayed for the Λ FDM models compared to the Λ CDM model. For example, theaverage T K becomes larger than T γ at z ∼ for the Λ CDM model, while this only happensat z ∼ . for the S model. The delay in the heating process is due to the delay in formationof the collapsed objects in the FDM models. As the number density of low mass haloesdecrease with m in the FDM models, the heating of the IGM occurs later for FDM modelswith smaller m (e.g, compare models S and S ). In comparison to the CDM model, T K – 16 – Q ( z ) z S0S1S2S3S4 10
6 10 14 18 22 26 30 T ( K ) z T γ -120-100-80-60-40-20 0 20 6 10 14 18 22 26 30 δ T b ( m K ) z Figure 3 : Left-hand panel:
Redshift evolution of the volume averaged ionization fraction( Q ) for different FDM models S (red dashed), S (green dotted), S (blue dash-dotted), S (magenta dash-double dotted ) and CDM model S (black solid). Middle panel:
Redshiftevolution of the average gas temperature (thick) and spin temperature (thin) of the neutralIGM. The thin black dotted curve represent the redshift evolution of the CMB temperature.
Right-hand panel : Differential brightness temperature of the 21-cm signal as a function ofredshift for these models.increases faster in the FDM models after T K reaches its minimum value. This is because f coll increases faster in the FDM models compared to the CDM model.Similar to the evolution of Q and the average T K of the IGM, the Ly α coupling alsofollows the collapsed fraction. Thus, this process is also delayed for the FDM models comparedto the CDM model. For example, T S follows T K from z ∼ for the CDM model, while samehappens at z ∼ for the FDM model S . In addition, the strength of the Ly α coupling isweaker for the FDM models and has a positive correlation with the value of m .All these effects are also reflected in the evolution of the average δT b as shown in theright-hand panel of Figure 3. While the absorption signal for the CDM model becomes thestrongest at z ∼ , the 21-cm signal is negligible for all the FDM models at that redshift.On the other hand the CDM model shows an emission signal for z (cid:46) whereas the FDMmodels show a strong absorption signal in that regime. A lower value for f X in the CDMmodel would of course shift the absorption signal to lower redshifts. However, because the Ly α coupling is stronger in the CDM models, this absorption signal would be much strongerthan what we see for the FDM cases.Another way to delay the the X-ray heating process would be to lower the star formationefficiency f (cid:63) in the CDM models, which would also make the Ly α coupling weaker. However,in that case the absorption profile will be much wider compared to those from the FDMmodels. Thus, the growth rate of the average δT b can in principle break the dark matterdegeneracies [also see 73, for the equivalent case for WDM].The detection of a global 21-cm signal at redshifts z (cid:38) will rule out or place strongconstrains on the FDM models. Recently, [118] reported a detection of the absorption signalcentered at a redshift of z (cid:39) using low-band observations with EDGES. The depth of theabsorption signal is much stronger than can be explained by standard physics and appearsto require an unknown cooling agent operating at very high redshifts [119]. Combined with– 17 – Q - V dn / d l n R R (Mpc/h) Λ CDM (S0)x
HII =0.10.30.50.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 1 10R (Mpc/h)m =2 (S2) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 1 10R (Mpc/h)z=9 S0S1S2S3S4 Figure 4 : Bubble size distribution at different stages of reionization with ionization fraction0.1 (solid), 0.3 (dashed), 0.5 (dotted) and 0.7 (dash-dotted). The left-hand and middle panelscorrespond to the CDM model S and FDM model S (fiducial model) respectively. Thecorresponding redshifts for S are 11.8, 9.5, 8.3 and 7.3 respectively, while these are 8.2, 7.3,6.9 and 6.5 for S . The right-hand panel represents the BSDs for all five models consideredin this work at redshift 9.the spectral shape of the signal [120] as well as worries about the handling of the strongforeground signals [54], this raises considerable doubts about the reliability of the claimedresult and confirmation by an independent group is required to give it strong credence. Asit stands, however, this measurement disfavours our FDM models and would require FDMmasses of m ≥ [121] or even m ≥ [122].[123] presented constraints from EDGES High-Band observations on the global 21-cmsignal in the range < z < . Among these are constraints on the width of the absorptionprofile. For a depth between 50 and 100 mK, the EDGES observations disfavour Gaussianabsorption profiles which have a FWHM of less than ∆ z ∼ . FDM models S1–S4 in factshow absorption features which are narrower than this. This implies that these four modelsare ruled out by the EDGES High-Band observations. This conclusion is of course dependenton the assumption that these constraints are reliable. In Section 4.2 we will investigate awider range of FDM models using this constraint. The difference in source abundances between the FDM models and the CDM model lead todifferent bubble size distributions (BSDs). The BSDs for the CDM model S and FDM model S are shown in the left-hand and middle panels of Figure 4 respectively at different stagesof reionization with ionization fraction 0.1, 0.3, 0.5 and 0.7. The most probable size of theionised bubbles, as given by the peak of the BSDs, increases as reionization progresses. Asone can easily notice, the most probable sizes of H ii regions are larger in the FDM modelsthan in the CDM model when compared at the same stage of reionization, characterized hereby the volume averaged ionization fraction. This is due to the fact that the FDM model lacksthe low mass halos which produce small H ii regions. In fact the FDM BSD shows a clearsmallest size which corresponds to the H ii region size produced by the lowest mass halo.The right-hand panel of the figure present the BSDs for all 5 models considered in table1 at redshift 9. The BSDs from the FDM models are more peaked than the one from the– 18 – β X − δ T b z S0S1S2S3S4 −150−100−50 0 50 100 150 10 15 20 25 30 β α − δ T b z−150−100−50 0 50 100 150 10 15 20 25 30 β T − δ T b z Figure 5 : Redshift evolution of the fluctuations of the fundamental quantities for differentFDM and CDM models. The left to right panels represent fluctuations due to ionization, gastemperature and Ly α coupling respectively. We do not plot the fluctuation from the baryonicdensity β B δT b as these follows the β x δT b curves.CDM model and the lower m the more peaked they become. In principle, if these BSDscan be measured in 21-cm tomographic imaging data [124], these differences will help breakthe degeneracies between the two dark matter models [125, 126]. Before discussing the power spectrum of the expected 21-cm signal from our five models, wewill first study the individual fluctuation terms that contribute to the power spectrum (seeequation 3.16). As shown in Figure 5, all models show that the 21-cm signal is dominated bythe Ly α fluctuation initially, followed by T K and then x HI fluctuations. These fluctuationsare weaker and delayed for the FDM models compared to the CDM models and also becomeweaker for smaller m values.These effects are also visible in the evolution of the large scale ( k = 0 . h Mpc − ) powerspectrum of the 21-cm signal as shown in Figure 6. Note that the galaxy bias is higherfor the FDM models which produce higher values of power spectrum compared to the CDMmodels. From the Cosmic Dawn to the end of reionization, different peaks of the curves in theleft-hand panel of the figure correspond to Ly α coupling, heating and ionization fluctuationsrespectively. The effects of delayed Ly α coupling and X-ray heating in the FDM models areclearly visible by the shifts of the peaks. There is also a significant amount of overlap betweenthe contributions from these fluctuations which makes the detection of these individual peaksmore difficult for the FDM models.One can notice that the 21-cm signal is insignificant for the FDM models beyond redshift ∼ , while the CDM model predicts a strong signal due to the Ly α and heating fluctuations.Thus, detection of a fluctuation signal at redshift (cid:38)
14 will rule out the FDM models or providestrong constrains on the FDM models. The dotted thin curves correspond to 1- σ error on thepower spectrum due to thermal noise for 100 hours of observation time, 32 MHz bandwidthwith SKA1-low and for intervals d k = k/ . We follow [127, 128] for estimating the error dueto thermal noise for 512 antenna of recent SKA1-low configuration. SKA1-Low observationsat high redshifts should therefore be able to easily rule out or put strong constraints on FDMmodels. The recent SKA1-low antenna configurations is taken from http://astronomers.skatelescope.org/ – 19 – -1 ∆ δ T b ( m K ) z 10 -1 -1 ) S0S1S2S3S4SKA:100h Figure 6 : Left-hand panel : The redshift evolution of the power spectrum of δT b at scale k = 0 . h Mpc − for different models of FDM and CDM. The dotted curve corresponds to σ error on the power spectrum at scale k = 0 . h Mpc − from the system noise from 100 h ofobservation with SKA1-low. We choose a bandwidth of 32 MHz and scale intervals dk = k/ to estimate the thermal noise. Right-hand panel : The power spectrum of δT b at z = 9 asa function of scales for the selected FDM and CDM models. The dips in the power spectrumat scales ∼ . h Mpc − for S and S occur as β T is negative whereas the other β valuesare positive. Physically, this is due to the fact that the denser regions become hotter whicheventually changes the signal from absorption into emission. See [94] for more details. Thedotted curve corresponds to the same noise as shown in the left-hand panel at different scales.Below z ∼ the FDM models show a much stronger signal than the CDM model. Thefull power spectra at z = 9 are shown in the right-hand panel of Figure 6 and reveal thatthis is true over a wide range of k values. This implies that the detection of the 21-cm signalbelow z = 10 with telescopes such as LOFAR, MWA, HERA and SKA1-low will be easier ifthe background cosmology is driven by FDM. However, observations at multiple redshifts willbe necessary to distinguish between FDM and CDM models with inefficient heating and orstar formation. Also, as pointed out above, it will be challenging to distinguish CDM modelsin which star formation in low mass halos is suppressed from FDM models. In this sense it iseasier to rule out than to positively confirm FDM models. Above we showed the results for four distinct FDM scenarios in which we varied m but kept f X and α X constant. In this section we present a study in which we vary f X , α X and m over a wide range so as to investigate the impact on the global signal as well as the powerspectrum of the 21-cm signal. We choose f X = 1 , α X = 0 . and m = 2 as our fiducialvalues. We vary two parameters at a time while the third parameter is fixed to its fiducialvalue. We characterize the results using the minimum value of δT b as well as the maximumvalue of the large scale power spectrum, ∆ ( k = 0 . h Mpc − ) , together with the redshiftswhen these occur. These numbers provide a quantitative hint on the detectability of thesignal in global experiments as well as with the interferometers.The left-hand panels of Fig 7 present the minimum of the brightness temperature δT b , min ,while the panels in the middle column show the associated redshifts z ( δT b , min ) . We see that– 20 – α X log(f X ) δ T b, min (mK) −25 mK−100 mK −180−160−140−120−100−80−60−40−20A X )z( δ T b, min ) 8 8.5 9 9.5 10 10.5 11 11.5 12A X ) ∆ z 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3A m log(f X ) −180−160−140−120−100−80−60−40−20B X ) 8 9 10 11 12 13 14B X ) 1.5 2 2.5 3 3.5 4 4.5 5 5.5B m α X −140−120−100−80−60−40C α X α X ∆ z=2 ∆ z=4 Figure 7 : Left-hand panels:
The minimum of the average brightness temperature ( δT b , min )throughout the reionization history. Middle panels:
The corresponding redshift to δT b , min . Right-hand panels:
The redshift width corresponding to the FWHM of the absorptionprofile of the brightness temperature. We choose f X = 1 , α X = 0 . and m = 2 as ourfiducial parameter values. While we vary two parameters at a time, we fixed the thirdparameter to its fiducial value. δT b , min is more sensitive to the X-ray parameters f X and α X than to m . The absorptionsignal during the Cosmic Dawn becomes weaker for larger values of f X as this leads toincreased X-ray heating. Thus, the amplitude of the absorption trough (or the strongestabsorption signal) decreases when f X increases as shown in the panel A of the figure. Theheating also occurs earlier for larger values of f X , and thus, the position of the absorptiontrough shifts towards higher redshifts (see panel A of the figure). The right-hand columnof figure 7 represents the redshift width ( ∆ z , full width at half maximum FWHM) of theabsorption profile around z ( δT b , min ) . The value of ∆ z decreases with increasing f X and– 21 – α X log(f X ) ∆ (mK ) 10 100 1000P X )z( ∆ ) 6.5 7 7.5 8 8.5 9 9.5 10 10.5P m log(f X ) 10 100 1000Q X ) 6 7 8 9 10 11 12 13 14Q m α X
80 100 120 140 160 180 200 220 240 260R α X Figure 8 : The left-hand panels present the maximum amplitude of power spectrum ( ∆ )at scale k = 0 . h Mpc − throughout the entire reionization period, while the right-handpanels present the corresponding redshifts to ∆ max . The fiducial choice of our parameters are f X = 1 , α X = 0 . and m = 2 . While we vary two parameters in each panel, we fixed thethird parameter to its fiducial value.varies from 1.4 to 3.2 for f X ∼ to . respectively for the fiducial value of m .As the value of α X in increased, the balance between soft and hard X-ray photons shiftsmore and more to soft X-rays. As soft X-rays more efficiently heat the neutral/partiallyionised regions outside the H ii regions, heating become more efficient and occurs earlierwhen we increase α X . This results in a lower value for δT b , min at earlier redshifts as can beseen in panels A and A . The same conclusion can also derived from panels C and C whichshow the δT b , min and its corresponding redshift as a function of m and α X respectively.Panels B and B show that the value of δT b , min depends weakly on the parameter m within the explored range. However, δT b , min decreases slightly when increasing m as can– 22 –e seen from panels B and C . On the other hand, z ( δT b , min ) shifts to higher redshifts when m is increased. This effect we had already seen in Fig. 3. As the absorption signal becomesstronger for larger values of m , its FWHM increases. For example, for the fiducial values of f X and α X , ∆ z changes from 2 to 3.8 when m increases from 1 to 20.As pointed out above [123] constrained the FWHM for different values of δT b , min usingEDGES high-band data. Their study excluded models with ∆ z smaller than 2 and 4 for δT b , min deeper than -25 mK and -100 mK respectively using a Gaussian δT b profile. Theabsorption profiles from our model can be well approximated by a Gaussian profile (see right-hand panel of Figure 3).The thin and thick solid curves in the left-hand and right-hand panelsof figure 7 corresponds to the contours of -25 mK and -100 mK respectively, while the dottedand dashed curves in the right-hand panels of the figure represents ∆ z equal to 2 and 4respectively. For example, the parameter space between the curves corresponding to ∆ z = 4 and δT b , min =-25 mK in panel B is inconsistent with the results of [123]. One can see thatall the FDM models for the fiducial X-ray source in this study are in disagreement with theirresults.Based on the EDGES High-Band upper limits for ∆ z , our parameter study thereforeimplies that m (cid:38) but then the X-ray efficiency has to be either very high or very low.For nominal X-ray efficiencies we find that m > is needed. These constraints are morestringent than the ones previously derived on the basis of reionization histories [68, 69, 71],becoming comparable to constraints on m from the Ly α forest for nomimal X-ray efficiencies[78, 83, 129, 130]. This shows the power of the 21-cm signal to constrain DM models. Aspointed out above, the results from the EDGES Low-Band observations have raised concernsabout the reliability of the EDGES results. Even though the high-band results are of adifferent character than the low-band results, we should caution that the constraints on FDMobviously depend on the reliability of the claimed upper limits in [123].Interferometers such as LOFAR, MWA, HERA and the SKA will not be able to measurethe global signal but are sensitive to the 21-cm power spectrum. We therefore also presentresults for this quantity. We chose to focus on the maximum value of the power spectrum ∆ at a scale k = 0 . h Mpc − over the entire reionization history. All currently activeand future interferometers are capable of measuring this scale. The left-hand panels of Fig. 8show the maximum value of ∆ ( k = 0 . h Mpc − ) while the right-hand panels present theassociated redshifts z (∆ ) .Prediction of the maximum amplitude of the large-scale power spectrum and its positionin redshift is not straightforward since the redshift evolution of the large-scale power spec-trum shows three peaks, as described before. As astrophysical parameters are changed, thestrongest peak may change from one of these three to another, leading to a sudden change ofthe redshift of the strongest peak. As one can see from panel P , similar to δT b , min , ∆ isalso very sensitive to the X-ray parameters. For intermediate values of f X , ∆ decreasesslightly and appears towards higher redshift for higher values of f X . However, for large valuesof f X ∼ , the IGM is heated very early leading to a smaller ∆ . In these cases, ∆ corresponds to the ionization peak and thus appears in the redshift range 6–7. Also when f X is very low ( ∼ . ), the ionization peak is stronger than the heating peak, see panel P and P . Panels R and R show that for larger values of α X , ∆ increases slightly and shiftstowards higher redshifts. This can be understood as an increase of the ratio of the number ofsoft X-rays to hard X-rays enhances the inhomogeneity of the heating process. On the otherhand, ∆ decreases and appears later for higher values of m which is consistent with Fig– 23 –. LOFAR is capable of measuring the 21-cm power spectrum at scales of k = 0 . h Mpc − for redshifts below 11. The results from Fig. 8 show that for a large part of the parameterspace explored here FDM models show appreciable power at redshifts accessible to LOFAR.Depending on the upper limits that LOFAR will be able to reach, these may lead to furtherconstraints on the mass of the FDM particle. In this study, we have examined the evolution of the redshifted 21-cm signal from CosmicDawn and EoR in fuzzy dark matter cosmologies. For this we use an analytic model whichincorporates the effects of Ly α coupling, X-ray heating and ionization to generate the ioniza-tion history and the expected 21-cm signal. As far as we know this is the first study for FDMincorporating all of these effects and considering the entire Cosmic Dawn and reionizationepoch. Here we summarize our main findings. • Compared to standard Λ CDM models, the severe reduction in the number of low-masshalos in Λ FDM models leads to a smaller number of the collapsed objects hostingionizing sources. As a consequence the rate of ionizing photons per baryon produced bythe sources must be high to ensure a reionization history which is both consistent withthe CMB observations such as by Planck and a completion of reionization by z = 6 . Wefind that the required rates imply a significant contribution from PopIII stars assumingstandard star formation efficiency and escape fraction values. The lower the value of m , the higher the required photon per baryon rate and the larger the contributionfrom PopIII stars. • Compared to standard Λ CDM models, there is also a considerable delay in the formationof collapsed objects which host ionizing sources. When requiring the end of reionizationto be at z = 6 , this has as a result that the mean ionization fraction of the Universeevolves more rapidly in FDM models, as do the Ly α coupling and X-ray heating. Thelower the value of m , the more rapid the evolution. Furthermore, whereas it is possible,although not necessary, to separate the three eras of Ly α coupling, X-ray heating andionization in Λ CDM models, there will always be considerable overlap between them inFDM models. This implies that the assumption of spin temperature saturation in suchmodels, as was for example used by [71], is not valid. • The redshift evolution of the globally averaged 21-cm signal is delayed in Λ FDM scenar-ios relative to Λ CDM. As a consequence the absorption profile is narrower and shallower.Narrower absorption profiles are easier to detect in global signal experiments. In fact,[123] used results from the EDGES High-Band experiment to constrain the width ofthe absorption profile. For the parameter range which we explored these constraintstranslate into a lower limit for the mass of the FDM particle of m (cid:38) but only foreither very low or very high X-ray efficiencies. For nominal values of the X-ray efficiency m would need to be higher than 20. • Another consequence of the delay in the redshift evolution of the globally averaged 21-cm signal is that the 21-cm signal is very weak above a redshift z (cid:39) for m ≤ .Therefore a corroborated detection of the signal for z > from ongoing global signal– 24 –etection experiments such as EDGES, SARAS, and LEDA will be able to put strongconstraints on the dark matter particle mass m in FDM scenarios. In fact, the claimeddetection of an absorption signal at z (cid:39) by the EDGES low band experiment [53]has already been shown to imply m ≥ [121]. • The bubble size distribution for FDM cases differs considerably from the CDM case.The relative lack of small H ii bubbles in the FDM models lead to a narrower bubblesize distribution. • The evolution of the large scale 21-cm power spectrum in the FDM models resemblesthat of the CDM models, but just as for the global signal the features shift to lowerredshifts and the evolution is more rapid. One consequence of the latter will be thatfor the FDM case the light cone effect [see e.g., 113] will have a stronger impact on the21-cm signal power spectra than for the typical CDM case. • As the maximum of the power spectrum, which generally correspond to the heatingpeak, occurs at lower redshifts in the FDM models, the detectability of the signal isexpected to be higher for both ongoing and future experiments such as LOFAR andSKA. The detection of a power spectrum signal at high redshifts ( z (cid:38) ) may rule outor put strong constrain on the FDM models.We want to reiterate that the FDM models may be degenerate with certain parameterchoices for CDM models. Specifically, a higher minimal virial temperature for halos to produceionizing photons can introduce features in the CDM models which will resemble those seenin the FDM models. Breaking these degeneracies is not easy and needs further investigation.However, since the conclusions are based on the HMF and the mapping of halo mass to photonproduction is complex, it will always be easier to rule out FDM models than to positivelyconfirm them.The FDM mass constraint derived from the EDGES High-Band results [123], m > ,has strong implications for Λ FDM as a viable model in solving small-scale challenges facing Λ CDM. Explaining observed constant-density cores in dwarf spheroidal galaxies in the contextof Λ FDM favour (at σ ) m = 1 . +0 . − . or m = 1 . +0 . − . , depending on the data set used[131]. This is inconsistent with the 21-cm constraints derived in this paper, showing that Λ FDM cannot solve the cusp-core problem. Similarly, Λ FDM with m > is ill-equipped tofully explain the dearth of dwarf galaxies. This is due to the fact that, as explained in Section2.2, the halo mass function is only significantly suppressed on mass scales below M Jeq,FDM (cid:39) . × ( m / − / M (cid:12) . For m > , this is below M Ly α (cid:39) . × (1 + z ) − / M (cid:12) — the T vir (cid:39) K lower limit for star formation set by efficient Ly α cooling and the reionization ofthe IGM — for redshifts at least up to z (cid:39) . . Thus, the observed scarcity of dwarf galaxiesin halos with masses < few × M (cid:12) in the local Universe would have to be indicativeof baryonic processes rather than exotic FDM effects. In summary, if the EDGES High-Band constraints are reliable, they entail the failure of Λ FDM in addressing the astrophysicalproblems that motivated it in the first place. This shows the potential of 21-cm observationsfor dark matter studies, but in view of the important implications for the viability of Λ FDMalso argues for independent confirmation of the results claimed by the EDGES team.We caution that our models have made use of the Press-Schechter (ST) HMF, ratherthan the more accurate Sheth-Tormen (ST) HMF [82]. This is done mainly for consistencywith our modelling of the H ii bubble distribution (see Section 4.1.2), the derivation of which– 25 –mplicitly assumes the Press-Schechter form to arrive at an expression for the H ii bubbleformation barrier. As our model only uses the total collapsed fraction, we are only sensitive todifferences between the two models in this quantity. The ST HMF has a weaker exponentialcut-off for high-mass halos (i.e. halo masses M for which δ crit ( M ) /σ ( M, z ) (cid:38) ) than thePS HMF. This results in a slightly larger collapse fraction at higher redshifts which couldease our constraints on m somewhat. However, this would probably be offset by a moreaccurate semi-analytical modelling of the HMF for Λ FDM as done by [36], who found thatsimply applying the ST HMF with δ crit,CDM → δ crit ( M ) leads to an underestimate (by a factor ∼ − judging from their Figure 3) of the halo mass below which there is a sharp cut-offto the HMF, as well as an overestimate of the value of ∂n/∂ ln M at the peak of the HMF.Correcting for these effects would again lower the collapse fraction and lead to constraintson m closer to what is found here using the PS HMF (if not more severe). Furthermore,we note that, while more accurate than the PS collapse fraction, the ST collapse fraction overestimate the actual collapse fraction derived from N-body simulations [see e.g. Figure 3in 132].As discussed in Section 2.2, there are ‘extreme’ versions of FDM which do not exhibitthe same severe lack of low-mass halos as the ‘vanilla’ FDM model studied in this paper.As a consequence, it is likely that these ‘extreme’ versions of FDM would avoid the 21-cm constraints derived here. However, as we argued in Section 2.2, these versions assumeimprobable initial conditions, making FDM far less attractive from the perspective of particlephysics.Although the exact value of the constraint from 21-cm observations may depend on thedetails of for example the star formation model, it is obvious that the severely delayed struc-ture formation combined with an end of reionization by z (cid:39) will always push the redshiftwidth of the absorption signal to values of ∆ z (cid:46) . In fact, we would expect interesting con-straints also for WDM models as they in this respect resemble FDM models. Future 21-cmobservations of either the global signal or the power spectrum can therefore be expected tolead to important constraints on the nature of dark matter. Acknowledgements
We would like to thank T. Roy Choudhury and Kanan K. Datta for useful discussions onthis work, Jonathan Pritchard and Yue Bin for input regarding the analytical modelling,and Sunny Vagnozzi and Luca Visinelli for helpful comments on axion-like dark matter. Weacknowledge the support from Swedish Research Council grant 2016-03581. We have also usedresources provided by the Swedish National Infrastructure for Computing (SNIC) (proposalnumber SNIC 2018/3-40) at PDC, Royal Institute of Technology, Stockholm.
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