What Is the Halo Mass Function in a Fuzzy Dark Matter Cosmology?

MNRAS 000,1–6 (2020) Preprint 5 November 2020 Compiled using MNRAS LATEX style ﬁle v3.0

What is the Halo Mass Function in a Fuzzy Dark Matter Cosmology?

Mihir Kulkarni,1? Jeremiah P. Ostriker,1† 1Columbia University, Department of Astronomy, New York, NY 10025, U.S.A.

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT Fuzzy dark matter (FDM) or wave dark matter is an alternative theory designed to solve the small-scale problems faced by the standard cold dark matter proposal for the primary material component of the universe. It is made up of ultra-light axions having mass ∼ 10−22 eV that typically have de Broglie wavelength of several kpc, alleviating some of the apparent small-scale discrepancies faced by the standard ΛCDM paradigm. In this paper, we calculate the halo mass function for the fuzzy dark matter using a sharp-k window function and compare it with one calculated using 10 −22 numerical simulations, ﬁnding the peak mass at roughly 10 M for a particle mass of 2 × 10 eV. We also constrain the mass of FDM particle to be ≈> 2 × 10−22 eV using the observations of high-redshift (z ∼ 10) lensed galaxies from CLASH survey. Key words: cosmology: dark matter – cosmology: theory – cosmology: early universe – galaxies: high-redshift

1 INTRODUCTION formation of galaxies or their destruction by mergers and tidal stripping. The standard model of cosmology (ΛCDM) includes dark en- The other set of solutions focus on the nature of dark mat- ergy in the form of a cosmological constant and ‘cold’ dark ter, such as warm dark matter (WDM) and fuzzy dark matter matter. This model has been immensely successful at ex- (FDM). These models suppress the small-scale structure of plaining the large-scale structure of the universe, the statis- dark matter that results in a cut-off at the lower end of halo tics of the cosmic microwave background, and cluster abun- mass function. Warm dark matter, which is made up of less dances (Bennett et al. 2013). However, recent observations massive (∼ keV) particles, remains relativistic for a longer have pointed out drawbacks of ΛCDM at small scales. A se- time than CDM. Its thermal velocity wipes out perturba- rious concern is the ‘missing satellite problem’ (Klypin et al. tions at small scales (free streaming). It has been pointed out 1999). The number of satellite galaxies predicted for a Milky- that warm dark matter undergoes a ‘Catch-22’ problem sat- way mass galaxy is greater than what we observe by an order isfying constraints simultaneously from Lyman-α forests and of magnitude. This issue is sharpened by the ‘too big to fail’ having large enough cores in dwarf galaxies (Macciòet al. problem of galaxy formation that claims some of the pre- 2012). Macciòet al.(2012) argue that having large enough dicted satellites are so massive that it is impossible for them cores (∼ 1 kpc) in dwarf galaxies requires WDM mass to be to not have any stars (Boylan-Kolchin et al. 2011). ΛCDM around 0.1 keV, which prevents the formation of the dwarf also predicts a cusp in the center of the density profile of galaxies in the first place. dark matter halos (Navarro et al. 1997), whereas recent ob- Fuzzy dark matter which is made up of ultra-light axions servations of dwarf galaxies suggest a flat core (Burkert 1995; with masses ∼ 10−22 eV is another theory of dark matter to Goerdt et al. 2006), although, it is important to note that this solve the small-scale problems. See Hui et al.(2017) for a

arXiv:2011.02116v1 [astro-ph.CO] 4 Nov 2020 issue is not yet settled. In addition, the predicted dynamical detailed review. Their extremely low mass makes their de friction faced by globular clusters in dwarf spheroidal galaxies Broglie wavelengths typically of the order of kpc. This results is so high that the globular clusters should have spiraled and in a cut-off at small-scales in the power spectrum. The fuzzy merged to the center, well before we observe them (Tremaine dark matter has finite quantum pressure and hence a non-zero 1976). effective sound speed given as: There are typically two types of suggested solutions to 2 2 2 2 these problems. The first attributes these inconsistencies to cs,e f f ≈ k /4a ma, (1) baryonic physics which is not yet very well understood. For where k is the comoving wave number, a is the scale factor example, the density profile of halos can be flattened to form and ma is the mass of axion. Thus the growth of overdensity a core when supernovae and black hole feedback redistributes is governed by the equation matter in the galaxy (Navarro et al. 1996). The missing satel-  2 2  lite problem could be a result of baryonic physics halting the  cs,e f f k  δ¨ + 2Hδ˙ +  − 4πGρ δ = 0. (2) k k  a2  k

? E-mail: [email protected] The solution to Eq.2 describes the linear growth of perturba- † E-mail: [email protected] tions with sound speed from Eq.1. For fuzzy dark matter, the

2 2 2 perturbations are growing if (cs,e f f k /a < 4πGρ). The scale at 2 CALCULATIONS OF HALO MASS FUNCTION which two terms are equal is called the Jeans scale, 2.1 Summary of previous calculations

4 2 1/4 2 1/4 kJ = (16πGρa m ) = (16πGρ0am ) . (3) Extended Press-Schechter formalism is widely used for calculating halo mass function for dark matter using the linear power spectrum (Press & Schechter 1974; Bond et al. 1991; Sheth & Tormen 2002). !1/4 Ω h2 m 1/2 Variance of perturbation amplitudes in real space k = 66.5a1/4 a , (4) J 0.12 10−22eV smoothed over a scale R is defined as follows: Z ∞ k2dk S (R) = σ2(R) = P(k)W2(k). (7) as given by Marsh(2016). A corresponding Jeans mass can 2 R 0 2π be defined as: A spherical top hat window function is typically used, defined !3 4π π as M = ρ ∝ a−3/4m−3/2. (5) J 3 0 k 3(sin(kR) − kR cos(kR)) J W (kR) = . (8) TH (kR)3 7 The Jeans mass at z = 0 is ∼ 2 × 10 M . Perturbations We calculate the trajectory δS for a point in space by start- corresponding to scales slightly larger than the Jeans scale ing with a large sphere of radius R and decreasing the radius, at z = 0 are growing in time, however their amplitudes are and calculating the smoothed density contrast for each R. highly suppressed, as they are smaller than the Jeans scale at Use of a sharp-k window function makes increments in δS some earlier time. An important scale that determines the rel- independent of previous steps, as δ(k) are independent Gaus- ative suppression of amplitudes is the Jeans scale at matter- sian processes for different k. This use of the sharp-k window 1/2 −1 radiation equality, kJeq = 9m22 Mpc . The power spectrum for function makes this problem analytically solvable. The Press- fuzzy dark matter is calculated using a redshift independent Schechter (PS) ansatz (Press & Schechter 1974) equates mass transfer function from Hu et al.(2000); element fraction with δS > δc with the mass fraction at time that resides in halos of mass > M. Bond et al.(1991) re- cos x3 P k, z T 2 k P k, z , T k , (6) moved cloud-in-cloud inconsistency in PS ansatz by equating FDM( ) = F ( ) CDM( ) F ( ) = 8 1 + x the fraction of trajectories with first upcrossing δS = δc at 2 S > S 1 = σ (M) with the fraction that resides in halos of 1/18 where x = 1.61m22 k/kJeq. mass M < M1 (extended Press Schechter formalism). The transfer function in Eq.6 is for matter-radiation equal- Following a few steps, we obtain the halo mass function, ity, as it depends on the Jeans scale at that epoch. If we take which is the comoving number density of halos per logarith- the initial power spectrum at matter-radiation equality and mic mass bin, given as evolve it numerically using Eq.2 to z = 0, the shape of the dn ρ d ln σ power spectrum remains largely unchanged because the scales = − 0 f (σ) . (9) smaller than the Jeans scale at matter-radiation equality will d ln M M d ln M still be highly suppressed, even if they start growing at a later epoch. This confirms the redshift independence of the r " !p# ! 2a σ2 δ aδ2 transfer function. f (σ) = A 1 + c exp c (10) ST π aδ2 σ 2σ2 In this paper, we calculate the halo mass function using c an extended Press-Schechter formalism (Press & Schechter is the fitting function for Sheth-Tormen mass function, with 1974; Bond et al. 1991). We first summarize the previous A = 0.3222, a = 0.707 and p = 0.3. The critical density δc = calculations of the halo mass function by Marsh & Silk(2014); 1.686 is the density contrast at collapse in linear theory for a Bozek et al.(2015); Du et al.(2017) that use a spherical top spherical collapse model. hat window function and a mass dependent critical density. The halo mass function calculated using this method We then point out inconsistencies in their methods and argue matches well with the numerical simulations for CDM. But why a sharp-k window function works better. We compare if we follow the same procedure for the fuzzy dark matter, our calculations with the halo mass function calculated using we do not obtain a cut-off in the halo mass function as ex- collision-less numerical simulations based on the FDM initial pected. Figure1 shows the halo mass functions for CDM and power spectrum (Schive et al. 2016). FDM calculated using a top-hat window function and a con- Early galaxy formation can be used to constrain the prop- stant critical density. This produces halos of masses smaller erties of FDM, since the short wavelength cutoff in FDM than the Jeans mass for FDM, which is unphysical. The main greatly delays galaxy formation at high redshift. Thus, we reason for this is that the top hat window collects the high- also use observations of high redshift (z ∼ 10) lensed galax- est contribution from very large scales (small-k modes). For 2 3 2 ies from CLASH survey to constrain the mass of fuzzy dark CDM, the dimensionless power spectrum ∆k = k P(k)/(2π ) matter, following a procedure similar to one used by Pacucci increases for increasing k and hence the spherical top hat win- et al.(2013) to constrain warm dark matter. dow reflects contributions from the scales of interest. FDM We use cosmological parameters consistent with WMAP9 has a cut-off in the power spectrum and hence, while calcu- data (Hinshaw et al. 2013) (Ωm0 = 0.284, ΩΛ = 0.716, h = lating σ(R) for small R, the major contribution still comes 0.696) so as to be able to compare our results with Schive from the large scales in the power spectrum, not reflecting et al.(2016). power spectrum amplitudes at corresponding scales.

MNRAS 000,1–6 (2020) Halo mass function for FDM 3

scale-independent critical density. On the other hand, FDM 106 CDM growth is scale-dependent and hence one should use a critical 104 FDM density which is higher for lower masses. Marsh & Silk(2014) and Bozek et al.(2015) use this mass 102 dependent critical density and fitting function f (σ) given by ] 3 Sheth & Tormen(2002). Du et al.(2017) use the same mass − 0 10 dependent critical density, but argue that the same fitting Mpc [ 2 function cannot be used. The fitting function f (σ) and the

M 10 − critical density (barrier) are related to each other in the Press- ln 4 10− Schechter formalism as: dn/d 6 10− Z S Z B(S ) f (S 0)dS 0 + P(δ, S )dδ = 1, (17) 8 10− 0 −∞ where S = σ2, B(S ) is the mass dependent barrier and P(δ, S ) 10 10− is the probability for a trajectory to lie between δ and δ+dδ for 107 109 1011 1013 1015 M[M ] variance S . Du et al.(2017) numerically calculate f (σ) for the mass dependent critical density. This changes few properties of the halo mass function. Both calculations by Marsh & Silk(2014); Bozek et al. Figure 1. Halo mass functions for CDM and FDM calculated with (2015) and Du et al.(2017) give sharp cut-offs in the halo the extended Press-Schechter formalism using a top hat window mass function and the cut-offs increase as we go to higher function and a constant critical density for z = 0. We can see that the halo mass function for FDM does not have a cut-off here as redshifts. The cut-off for Marsh & Silk(2014) and Bozek et al. 8 −22 expected. (2015) is ∼ 2 × 10 M for ma = 10 eV at z = 0. Whereas one calculated by Du et al.(2017) is about four times higher. For redshift 14, Marsh & Silk(2014) and Bozek et al.(2015) 9 This can be understood analytically as follows: obtain a cut-off at 2 × 10 M , whereas Du et al.(2017) have 9 ∞ 2 it to be 3 × 10 M . dσ2(R) Z k2dk dW (k) = P(k) R , (11) Du et al.(2017) calculate the cut-offs to be higher than dR 2π2 dR 0 those calculated by Marsh & Silk(2014) or Bozek et al. (2015). It is also worth noting that the cut-off mass changes ! 3 3 sin(x) W0 (x) = ((x2 −3) sin x+3x cos x) = − W (x) , (12) less strongly with redshift for Du et al.(2017). TH x4 x x TH

! 2.2 Shortcomings of previous calculations dW2 6W (kR) sin(kR) TH = TH − W (kR) , (13) dR R kR TH The argument for a mass dependent critical density is based on the fact that σ(R, z) ∝ D+(z) or inversely for δc for CDM which in the limit of R → 0 becomes ∝ −R. This tells us that and that this growth factor D+(z) should be replaced with the derivative of the window function has a tail stretching D+(k, z) for FDM. Let us take a closer look at this argument. dσ2(R) over small R. Plugging this into Eq. 11 gives that dR ∝ R For fuzzy dark matter, σ(R) becomes nearly constant for for small R. So, R smaller than the cut-oﬀ scale in the power spectrum. The shape of σ(R) is very weakly dependent on the shape of the dn ρ d ln σ 1 d ln σ 1 dW2 1 = − 0 f (σ) ∝ − ∝ − R TH ∝ , power spectrum for scales smaller than the cut-oﬀ scale, as d ln M M d ln M M d ln M M dR R the spherical top hat window draws contributions mainly (14) from large scales.

which means that the halo mass function diverges as M → 0 σ(R) ≈ σ(R0) for R < R0 (18) and does not give a cut-off as required. Hence, the redshift dependence of σ(R) can be given as Solutions by Marsh & Silk(2014); Bozek et al.(2015); Du  D (R, z) et al.(2017) involve using a mass dependent critical density  + σ(R, z ) for R > R ,  0 0 to suppress the halo mass function at lower masses. The new  D+(R, z0) σ(R, z) =  (19) critical density is defined as  D+(R0, z)  σ(R, z0) for R < R0. f dm cdm D+(R0, z0) δc (M, z) = G(k, z)δc (z), (15) where G(k, z) is given as D+(z) ∴ σ(R, z) = σ(R, z0). (20) D+(z0) δ (k, z)δ (k , z ) δ f dm(k, z)δ f dm(k , zh) G(k, z) = cdm cdm 0 h / 0 . (16) Here, R corresponds to the scale where the FDM power δ (k, z )δ (k , z) δ (k, z )δ (k , z) 0 cdm h cdm 0 f dm h f dm 0 spectrum starts to differ from CDM (approximately the Jeans The argument used for using this critical density is as fol- scale at the matter-radiation equality). The scales larger than lows: For CDM, we can take δc = 1.686 to be a constant and or equal to R0 grow the same as CDM, which concludes that take σ(R, z) ∝ D+(z), or we can make δc redshift dependent as σ(R) for FDM grows in a fashion similar to CDM with red- δc ∝ 1/D+(z) and σ(R) to be redshift independent. The growth shift. Hence, the critical density also cannot be dependent on rate for CDM is scale independent and hence we can use a the scale through the FDM growth rate.

MNRAS 000,1–6 (2020) 4 Kulkarni & Ostriker

It is also interesting to note that the redshift dependence in the halo mass functions calculated by Marsh & Silk(2014); CDM top-hat Du et al.(2017) comes mainly from the redshift dependence 8 FDM top-hat of the Jeans scale. The shape of the power spectrum and σ(R) FDM sharp-k does not change with the redshift, as the transfer function is redshift independent (Hu et al. 2000). However, the shape 6 of the critical density barrier changes with redshift, as it is )

based on which scales are growing and which are not. Hence, R ( they end up predicting a higher number of low mass halos at σ 4 smaller redshifts, even though the shape of the power spectrum does not change with redshift. In recent years, there have been many simulations that 2 solve the Schrödinger-Poisson equations for accurately evolv- ing FDM (Schive et al. 2014a,b; Mocz et al. 2017; Li et al. 2019). Although they accurately evolve fuzzy dark matter 0 1 0 1 2 and reproduce the soliton profiles in the halos, they do not 10− 10 10 10 have a sufficient number of halos to calculate the halo mass R[Mpc] function because of their small box sizes. A number of works (Schive et al. 2016; Sarkar et al. 2016; Zhang et al. 2018; Nori et al. 2019) have run collision-less N-body simulations with Figure 2. σ(R) using top hat window function for CDM and FDM the initial power spectrum of FDM. These numerical simula- as well as σ(R) for FDM using sharp-k window function. tions predict many lower mass halos too. Schive et al.(2016) mark and delete those ‘spurious’ halos and calculate the halo mass function for ‘genuine’ halos. Hence, we use the halo mass 0 function estimated in Schive et al.(2016) to compare with our 10 results. 1 10− Du et al.(2017) return their halo mass function for m22 = 1, 2 whereas Schive et al.(2016) return their results for m22 = 10− −

22 ]

0.8, 1.6, 3.2 (m22 = ma/10 eV). We compare the results for 3 − 3 m22 = 0.8 with results from Du et al.(2017). Du et al.(2017) 10− 9 Mpc calculate the cut-oﬀ mass for redshift 4 to be ∼ 1.5 × 10 M . [ 10 4 M − This cut-oﬀ will be slightly higher for m22 = 0.8. Schive et al.

8 ln (2016) have halos as small as 4 × 10 M which is clearly 5 10 m22 = 0.5 smaller than the predicted cut-oﬀ from Du et al.(2017). − dn/d m22 = 1 6 10− m22 = 2 m22 = 3 2.3 HMF using a sharp-k window function 7 10− m22 = 4 Instead of using the spherical top hat function and a mass CDM 8 10− dependent critical density, we use a sharp-k window function 107 108 109 1010 1011 1012 1013 and a constant critical density in this work. This approach M[M ] has been previously used for warm dark matter (Benson et al. 2013; Schneider et al. 2013).

 Figure 3. Halo mass functions for m22 of 0.5, 1, 2, 3, 4 for FDM and 1 for k ≤ k0 for CDM at z = 0. WR(k) =  (21) 0 for k > k0 A drawback of the sharp-k window is that the enclosed mass is not clearly defined in it. It has contributions from This halo mass function does not give a sharp cut-off like all scales in real space, making it difficult to assign a mass Marsh & Silk(2014); Bozek et al.(2015); Du et al.(2017). M for a given k0. A similar problem is faced by a Gaussian Figure3 shows the halo mass function for different FDM window function, but it can be assigned a mass based on its mass. The cut-off does not depend on redshift as strongly as integration in real space. The integration of the sharp-k win- previous works. Figure4 shows FDM halo mass functions for dow function in real space diverges (Maggiore & Riotto 2010). different redshifts for m22 = 2. Previous works (Benson et al. 2013; Schneider et al. 2013) us- The calculated halo mass function is shown in figure5 along ing a sharp-k window with warm dark matter use k0 = α/R with the halo mass function calculated by Schive et al.(2016) keeping a free parameter to be fit by the numerical simula- for m22 = 3.2 and m22 = 1.6 at z = 4. Schive et al.(2016) use tions. Following Benson et al.(2013), we use α = 2.5. We also collision-less dark matter simulation with an initial power need to rescale the critical density, as σ(R) with sharp-k win- spectrum for fuzzy dark matter. dow is higher than that of top hat window for higher masses We note that the analytical halo mass function we have (see Figure2). We rescale δc by multiplying it with 1.195. derived predicts a lower number of halos compared to the This is also to ensure that the halo mass function matches numerical simulations at low masses as shown in Figure5. with CDM at higher masses. There could be two possible reasons for this:

MNRAS 000,1–6 (2020) Halo mass function for FDM 5

For this exercise however, we will use the analytical halo 100 CDM mass function that we have calculated to constrain the mass FDM m22 = 2 of FDM. 2 10− ] 3 − 4 10− 3 LIMITS ON THE FDM MASS Mpc [ The fuzzy dark matter paradigm has only one parameter . M ma −18 ln 6 The FDM particle is lighter than 10 eV and massive than 10− z = 0 10−33 eV (Marsh 2016). The mass can further be constrained dn/d z = 2 by various astrophysical processes including the following. z = 4 8 Schive et al.(2016); Bozek et al.(2015) use galaxy UV lu- 10− z = 6 z = 8 minosity function and Schive et al.(2016) constrain it to be −22 −23 z = 10 > 1.2 × 10 eV. Sarkar et al.(2016) found ma > 10 eV using 10 10− damped Lyman-α observations and simulations. Calabrese & 8 9 10 11 12 13 10 10 10 10 10 10 Spergel(2016) have used the cored density profile from Schive M[M ] et al.(2014b) and observations of newly discovered ultra faint −22 dSphs and estimated ma to be 3.7 − 5.6 × 10 eV. Amorisco −22 & Loeb(2018) constrain ma > 1.5 × 10 eV based on the dy- Figure 4. Halo mass functions calculated using the sharp-k window namics of stellar streams in the Milky Way. Observations of function for various redshifts for ma = 2e − 22eV. The halo mass high-resolution Lyman-α spectra were used to constrain the 10 function is smoothed for masses lower than 5 × 10 M axion mass to be more than 10−21 eV (Armengaud et al. 2017; . Irˇsiˇcet al. 2017; Kobayashi et al. 2017), although further work needs to be done to understand the effects of baryonic physics better. In particular, late reionization can leave a patchy dis- 100 tribution of neutral hydrogen that can be falsely interpreted as due to gravitational fluctuations. In this work, we constrain the mass of axions using observa- z = 4 tions of high-redshift lensed galaxies following a method used 1 10− by Pacucci et al.(2013) to constrain the warm dark matter. ] 3

− Cluster Lensing And Supernova survey with Hubble (CLASH) is a survey using the Hubble telescope. Zheng et al. Mpc [ 10 2 (2012) report the observation of a galaxy at redshift 9.6 with M − a magniﬁcation of 15 (MACS 1149-JD). Coe et al.(2013) re- ln port the observation of another galaxy at redshift ∼ 10.8 with

dn/d CDM a magniﬁcation of 8 (MACS0647-JD). Coe et al.(2013) esti- 3 10− m22 = 3.2, Schive et al. 2016 mate intrinsic (unlensed) magnitude of MACS0647-JD to be m22 = 3.2, analytic 28.2 in F160W band and calculate the rest frame UV lumi- m = 1.6, Schive et al. 2016 22 nosity to be L ∼ 2.8 × 1028erg s−1Hz−1. They conclude that m = 1.6, analytic UV 22 8 9 4 the stellar mass of galaxy is most likely 10 −10 M and expect 10− 10 108 109 1010 1011 1012 the dark halo mass to be ∼ 10 M . We do not use the mass M[M ] of the galaxy for constraining the axion mass. The magniﬁca-

tion factor can be used to calculate the eﬀective volume of the galaxies. Existence of galaxies at z ∼ 10 with their eﬀective

Figure 5. Halo mass functions for m22 = 1.6 and m22 = 3.2 at redshift volume can be used to constrain the fuzzy dark matter, even 4 from analytic calculations using a sharp-k window function, and without explicit knowledge about masses of the galaxies. from simulations by Schive et al.(2016). 3 Object ID µ Ve f f (Mpc ) MACS1149-JD 15 ∼ 700 MACS0647-JD 8 ∼ 2000 • In their simulation, Schive et al.(2016) find many low mass halos. They flag “spurious” halos and remove them from Z ∞ dn their calculations. It is possible that a sufficient number of n(m , z) = (M, z, m ) d ln M (22) a d ln M a halos have not been removed. A full non-linear simulation 0 solving the Schrödinger-Poisson equations is required to make We calculate the halo mass function for the fuzzy dark an accurate comparison. matter at redshift 10. We integrate the halo mass function to • Fuzzy dark matter power spectrum falls much more get the integrated number density of halos as a function of rapidly as compared to the warm dark matter. Hence, halo axion mass. Since this includes halos of all masses, the number formation from non-linear effects such as fragmentation of density calculated from observations cannot be higher than larger structures could be significant, and this is not consid- this. ered in this calculation. This could increase the number of If we conservatively use the effective volume for two 3 −3 halos at low masses. This needs further investigation. galaxies to be 2000 Mpc , that gives ntot = 2/2000 Mpc =

MNRAS 000,1–6 (2020) 6 Kulkarni & Ostriker

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