2. Semi-analytic Halos

We begin by looking at the semi-analytic parametrisations of ULDM and CDM halos. The well-known NFW profile of CDM (Navarro et al. Reference Navarro, Frenk and White1996; Maccio’, Dutton, & Bosch Reference Maccio’, Dutton and Bosch2008) is given by

(1) \begin{equation} \rho_\mathrm{NFW}(r)=\frac{\rho_0}{\frac{r}{R_s}\left(1+\frac{r}{R_s}\right)^2} \, .\end{equation}

At small radii the profile is proportional to $1/r$ , while at large radii it goes as $1/r^3$ . The parameters $\rho_0$ and $R_s$ vary from halo to halo; $\rho_0$ can be interpreted as a characteristic density, while $R_s$ is the scale radius and determines the distance from the centre at which the transition between the ‘small r’ and ‘large r’ limits occurs.

The NFW halo is assumed to be radially symmetric and requires truncation at a finite radius in order to prevent the integrated mass diverging as $r\rightarrow \infty$ . The truncation is typically set by the virial radius, which is itself determined approximately via the spherical top-hat collapse model describing the evolution of a uniform spherical overdensity in a smooth expanding background (White Reference White2001; Suto et al. Reference Suto, Kitayama, Osato, Sasaki and Suto2016; Herrera, Waga, & Jorás Reference Herrera, Waga and Jorás2017). Gravitational collapse of the overdensity halts when virial equilibrium is reached. In this scenario, the corresponding virial radius is the radius at which the mean internal density is $\Delta_c \rho_\mathrm{crit}(t)$ . Here $\rho_\mathrm{crit}(t)$ is the critical density of the universe at time t. The factor $\Delta_c$ is of order $10^2$ and while different conventions exist, we make the common choice $\Delta_c = 200$ (Richings et al. Reference Richings2018) in what follows.

Once the virial radius is specified as the outer limit of the halo, Equation (1) completely determines the density profile for given $\rho_0$ and $R_s$ . For any given virial mass, there is a range of corresponding NFW density profiles, with the distributions of $\rho_0$ and $R_s$ emerging from the mass–concentration–redshift relation seen in N-body simulations and observations (Ludlow et al. Reference Ludlow, Navarro, Angulo, Boylan-Kolchin, Springel, Frenk and White2014; Ragagnin et al. Reference Ragagnin, Dolag, Moscardini, Biviano and D’Onofrio2018).

Whereas CDM halos can be described by NFW distributions, a different approach must be taken in the case of ULDM. ULDM dynamics is governed by the Schrödinger–Poisson system of coupled differential equations. In a static background, they take the dimensionless form

(2) \begin{align} &i\dot{\psi} = -\frac{1}{2}\nabla^2\psi+\Phi\psi,\end{align}

(3) \begin{align} &\nabla^2\Phi = 4\pi \vert \psi\vert^2,\end{align}

where $\psi$ is the ULDM wavefunction, $\Phi$ is the Newtonian potential, and the density $\rho \propto |\psi|^2$ . The solitonic ground state profile cannot be written down analytically, but given a numerically computed spherically symmetric profile $\psi$ for $\psi(0)=1$ , the full family of solutions is then given by

(4) \begin{equation} \psi'(x) = \gamma\psi(\sqrt{\gamma}x),\end{equation}

where $\gamma$ is a scaling parameter and the dimensionless mass of the soliton is proportional to $\sqrt{\gamma}$ , while the dimensionless radius is proportional to $1/\sqrt{\gamma}$ . The dimensionless density $\vert\psi\vert^2$ and dimensionless radius x can be transformed into dimensionful quantities by

(5) \begin{align} \rho &= \mathcal{M}\mathcal{L}^{-3}\vert\psi\vert^2, \end{align}

(6) \begin{align} r &= \mathcal{L}x, \end{align}

where

(7) \begin{equation} \mathcal{L}=\left(\frac{8\pi\hbar^2}{3 m^2H_0^2\Omega_{m_0}}\right)^{\frac{1}{4}}\approx121\left(\frac{10^{-23}\,{eV}}{m}\right)^{\frac{1}{2}}{kpc},\end{equation}

and

(8) \begin{align} \mathcal{M}&=\frac{1}{G}\left(\frac{8\pi}{3 H_0^2\Omega_{m_0}}\right)^{-\frac{1}{4}}\left(\frac{\hbar}{m}\right)^{\frac{3}{2}}\nonumber\\[5pt] &\approx 7\times 10^7\left(\frac{10^{-23}{eV}}{m}\right)^{\frac{3}{2}}{M}_{\odot}.\end{align}

Robles et al. (Reference Robles, Bullock and Boylan-Kolchin2019) gives a piecewise parameterisation of the generic ULDM profile

(9) \begin{equation} \rho(r)= \begin{cases} \rho_{\mathrm{sol}}(r), & 0\leq r \leq r_{\alpha} \\[5pt] \rho_\mathrm{NFW}(r), & r_{\alpha}\leq r \leq r_{\mathrm{vir}}, \end{cases}\end{equation}

where $\rho_{\mathrm{sol}}(r)$ is the appropriately scaled density profile of the ground-state soliton solution. The contribution from the solitonic core and the overall virial mass is predicted to obey a scaling relationship (Schive et al. Reference Schive, Liao, Woo, Wong, Chiueh, Broadhurst and Hwang2014; Chavanis Reference Chavanis2019) which sets the central density, $\rho_c$ , of a ULDM halo with virial mass, ${M_{\rm vir}}$ . This yields an expression relating the core size to the velocity dispersion, and finally to the halo virial mass.Footnote b

This core–halo mass relation can also be understood simply by matching the virial velocities of the core and the wider halo (see Appendix A for details). At $z=0$ , the relationship is found to be Schive et al. (Reference Schive, Liao, Woo, Wong, Chiueh, Broadhurst and Hwang2014)

(11) \begin{equation} \rho_c = 2.94\times10^6 {M}_{\odot}{kpc}^{-3}\left(\frac{{M_{\rm vir}}}{10^9 \mathrm{M_{\odot}}}\right)^{4/3}{m_{22}}^{2}\end{equation}

and

(12) \begin{equation} r_c = 1.6 {kpc}\left(\frac{{M_{\rm vir}}}{10^9 \mathrm{M_{\odot}}}\right)^{-1/3}\frac{1}{{m_{22}}},\end{equation}

where $r_c$ is the radius at which the density is half of the central value, and ${m_{22}}$ is given by ${m_{22}} \equiv m / 10^{-22} {eV}$ , where m is the ULDM particle mass.

While the piecewise semi-analytic ULDM profile is a useful tool, one should be mindful of its limitations. For example, while a number of studies have attempted to establish ‘universal’ properties of ULDM halos, many of these analyses generated ULDM halos through the mergers of smaller compact objects (Schwabe et al. Reference Schwabe, Niemeyer and Engels2016; Mocz et al. Reference Mocz, Vogelsberger, Robles, Zavala, Boylan-Kolchin, Fialkov and Hernquist2017). This method of halo assembly is not representative of a realistic structure formation process; however, it has the advantage of avoiding the computational difficulty of undertaking large-scale ULDM cosmological simulations. For this reason, there is currently limited information from which to draw robust conclusions about the properties of astrophysical ULDM halos. In particular, universal application of the core–halo mass relation cannot be fully justified until more work is done to understand the characteristic timescales associated with the formation of quantum pressure-supported cores in scenarios including condensation from a fluctuating background, gravitational collapse in an expanding background, and mergers of objects with and without stable central cores. Moreover, it is difficult to accurately predict the effect that baryonic feedback will have on the formation of solitonic cores in halos of different masses, which could be significant at small radii in the present context.

Halo substructure is likewise missing from the semi-analytic model presented above. In simulations of soliton mergers, the resulting halos have turbulent outer regions, with fluctuations on scales comparable to the core size, as illustrated in Figure 1. In addition to the fluctuations inherent in a large ULDM halo, smaller halos are likely to orbit or interact with larger halos. This substructure is not captured by the semi-analytic model described above, and predictions for tracer velocity profiles may thus not match those of realistic astrophysical objects. Furthermore, temporal fluctuations in the core density are also missing from the semi-analytic model. Realistic halo cores are not exact soliton solutions of the Schrödinger–Poisson equation, they interact non-trivially with the fluctuating NFW-like outer halo, and their central densities can vary with time by as much as a factor of two (Veltmaat et al. Reference Veltmaat, Niemeyer and Schwabe2018).

Figure 1. Illustration of the scale of the fluctuations present in the incoherent outer halo for a merger of eight randomly located solitons. The contour plot represents the ( $\text{log}_{10}$ scaled) local density across a slice through the centre of the final halo. In this plot, distance is not log-scaled, and we see that the spatial size of the fluctuations is of the same order of magnitude as the solitonic core itself.

Figure 2. Density profiles as a function of radius (normalised to the virial radius) for ULDM and NFW halos of masses $10^{11}{M}_{\odot}$ (top) and $10^{12}{M}_{\odot}$ (bottom). The left panel represents the results for ${m_{22}} = 0.8$ , while the right panel corresponds to ${m_{22}}=2.5$ . The transition radius is fixed at $r_{\alpha} = 3.5*r_c$ . The blue shaded region represents the ULDM profile with ${M}_c = {M}_{\mathrm{cp}} \pm 50 \% {M}_{\text{cp}}$ , while the solid blue line represents the ULDM profile when the theoretical core–halo mass relation is taken to be exact. The red shaded region represents the range of NFW profiles for a halo of the same virial mass with a 2 $\sigma$ variation around the median (solid red line).

Taken together, these limitations suggest that the core–halo mass relation of the semi-analytic model should not be interpreted as an inviolable rule, but as a statement about the averaged characteristics of a statistical distribution. To estimate the variance corresponding to this distribution, we can consider a range of possible central densities for a given virial mass (somewhat analogous to the scatter in NFW concentration parameters Maccio’ et al. Reference Maccio’, Dutton and Bosch2008). The results of Schive et al. (Reference Schive, Liao, Woo, Wong, Chiueh, Broadhurst and Hwang2014) indicate that a scatter in the core mass ${M_c}$ of up to $\pm 50\%$ is possible for a given virial mass. Unfortunately, the small sample size and limited halo mass range ( ${M_{\rm vir}} \approx 10^8$ – $10^{11} {M}_{\odot}$ ) found in Schive et al. (Reference Schive, Liao, Woo, Wong, Chiueh, Broadhurst and Hwang2014) preclude a detailed analysis of the statistical properties of realistic astrophysical halos, but future simulations (especially those including baryonic feedback) should lead to improved predictions for this distribution.

To partially account for statistical variance in halo properties, one can allow for variation in the radius at which the solitonic profile of the ULDM halo transitions into an NFW profile. This is acknowledged in Robles et al. (Reference Robles, Bullock and Boylan-Kolchin2019) and is captured by the parameter $\alpha$ : the transition radius, $r_{\alpha}$ , is given by $r_{\alpha} = \alpha r_c$ , with $3 \leq \alpha \leq 4$ . For a given theoretical halo, an adjustment to the transition radius should be accompanied by changes in the parameters of the outer NFW halo, so as to maintain the core–halo mass ratio.

Thus, by taking the central soliton density and transition radius as variable parameters, we can create a range of plausible ULDM halo profiles for a given halo by using the virial mass to predict $\rho_c$ , and assuming variation of $\pm 50\% $ around this central value. Given specific values for the central density and transition parameter $\alpha$ , the solitonic piece of the ULDM profile is then completely specified, and its mass can be calculated. The remainder of the virial mass must be accounted for by the NFW tail of the profile. By matching the densities of the NFW tail to the inner soliton at the transition radius, the values of $R_s$ and $\rho_0$ for the NFW tail are obtained.

3. ULDM and CDM halos and astrophysical data

We now compare the radial profiles of ULDM halos to NFW halos using the semi-analytic profiles described above, focusing on masses in the range $10^{11}$ and $10^{12} {M}_{\odot}$ , which may show an apparent worsening of the core-cusp problem (Robles et al. Reference Robles, Bullock and Boylan-Kolchin2019). Figure 2 compares such halos; the shaded blue region represents the ULDM halos for which the core–halo mass relation has a scatter ${M}_c = {M}_{\mathrm{cp}} \pm 50 \% $ range, where ${M}_{\mathrm{cp}}$ is the theoretical prediction for the core mass. Note that because higher central densities correspond to narrower soliton profiles, the shaded region possesses ‘crossover points’ near the transition from the solitonic to NFW profile, appearing somewhat skewed from the median line. Were we to vary more parameters in the model (such as transition radius and axion mass), we would see a broadening of the shaded region, such that the median line would be fully encompassed by the shaded region. Because we are here focusing primarily on the core mass (and therefore central density), we illustrate only the changes in density profile attributable to this, hence the restricted range of profiles shown as the shaded blue region.

The Schrödinger–Poisson soliton scaling relations show that the ${M}_c = {M}_{\mathrm{cp}} \pm\,50 \% $ mass range corresponds to a range of $ \gamma_p /4 \leq \gamma \leq 9\gamma_p/4$ , where $\gamma_p$ is the theoretical prediction of the square root of the dimensionless central density. Consequently, there is a large variation in the central density and thus widely varying predictions for the ULDM profiles. We fix $\alpha = 3.5$ (in the middle of the predicted range) which does not affect the central density as the core lies well within the solitonic region. Changing the value of $\alpha$ will, however, affect the predicted velocity profiles for each halo. We do not attempt to fit this parameter to data in this section; the previously discussed limitations of the semi-analytic models employed here suggest that this would be unlikely to be a meaningful exercise. The blue ULDM profiles are compared to the red shaded regions of Figure 2, showing the $2\sigma$ variation about the theoretical prediction for the concentration parameter of the NFW halo with the same virial mass (Maccio’ et al. Reference Maccio’, Dutton and Bosch2008).

Figure 3. Velocity distributions for galaxies with maximum velocities in the range $125 \leq v < 175{km\,s}^{-1}$ in the SPARC database. Data at innermost radii are limited for these galaxies, making it difficult to determine the overall characteristics of the profiles. The SPARC data are plotted alongside theoretical NFW (shaded blue) and ULDM (shaded red) profiles, assuming a virial mass of $10^{12} \mathrm{M}_{\odot}$ , ${m_{22}} = 2.5$ , and $\pm 50 \%$ scatter in the ULDM core–halo mass relation and $\pm2\sigma$ scatter in NFW concentration. Galaxies in the legend are ordered from highest maximum velocity (top) to lowest (bottom).

Following Robles et al. (Reference Robles, Bullock and Boylan-Kolchin2019), we plot to a minimum radius of $r/r_{\mathrm{vir}} = 10^{-4}$ and for the same choices of ${m_{22}}$ . For any ${M}_{\mathrm{vir}}$ , the NFW halo density will inevitably exceed that of the ULDM halo at very small radii, though the threshold for this transition may be arbitrarily small, and not observationally relevant. However, we note that the apparent worsening of the core-cusp discrepancy does depend on the choice of inner radial cut-off.

From Figure 2, we see that for halo masses of $10^{11}{M}_{\odot}$ there is a wide range of ${M_c}$ for which the ULDM profile is ‘less cuspy’ than its NFW counterpart. For a halo mass of $10^{12}{M}_{\odot}$ and a ULDM particle mass ${m_{22}}=0.8,$ the range of plausible ULDM profiles likewise includes those which are ‘less cuspy’ than the corresponding NFW profile. At higher particle mass ( ${m_{22}}=2.5$ ) for $10^{12}{M}_{\odot}$ halos, the NFW profiles tend to be less peaked than corresponding ULDM profiles at radial distances in the range $10^{-4}\leq r/r_{\mathrm{vir}} \leq 1$ .

To assess the suitability of these semi-analytic profiles, we compare to observations drawn from the SPARC database. Because observations yield the (line of sight) velocity distributions of tracer stars as a function of galactocentric radius rather than the halo density itself, we must first transform our theoretical density profiles into velocity profiles. In so doing, we acknowledge that the effects of non-circular motion and kinematic irregularities constitute a non-trivial source of random error in observed velocities, which should be kept in mind especially when working with limited data sets.

We convert density profiles to velocity distributions (Sofue, Honma, & Omodaka Reference Sofue, Honma and Omodaka2009) via

(13) \begin{equation} V(r)^2 = \frac{4\pi G}{r}\int_0^r \rho(r')r'^2 dr',\end{equation}

where

(14) \begin{equation} V^2 = V_{\mathrm{disk}}^2 + V_{\mathrm{bulge}}^2 + V_{\mathrm{gas}}^2 + V_{\mathrm{halo}}^2.\end{equation}

The SPARC database contains photometric data for 175 galaxies and rotation curves from ${H}_{{I}}$ / ${H}_{\alpha}$ studies. The disc and bulge velocities in the SPARC database are given for $\Upsilon = 1 {M}_{\odot}/{L}_{\odot}$ at $3.6{\mu m}$ . However, the greatest source of uncertainty in mass modelling is the assumed stellar mass-to-light ratio, $\Upsilon_\star$ (Lelli et al. Reference Lelli, McGaugh and Schombert2016). As in Robles et al. (Reference Robles, Bullock and Boylan-Kolchin2019), we assume a constant value of $\Upsilon_\star = 0.2 {M}_{\odot}/{L}_{\odot}$ at $3.6{\mu m}$ , likewise noting that this constitutes a non-trivial source of uncertainty. Moreover, there is significant uncertainty in the SPARC data itself. Error bars are omitted in the following graphs for ease of viewing; however, they are discussed in Appendix C.

The characteristics of the velocity profiles in the SPARC database vary widely from galaxy to galaxy; however, we qualitatively identify two subsets of galaxies; those with maximum tracer velocities $75 \leq v < 125{km\,s}^{-1}$ , and those for which $125 \leq v < 175{km\,s}^{-1}$ . The former group tends to exhibit a strong steepening in the radial velocity profile towards the inner halo, while the profiles for the latter group are comparatively flatFootnote c. We assume that higher asymptotic velocities correspond to a larger halo mass, and consider halo masses in the range $10^{11}-10^{12} {M}_{\odot}$ , expecting that masses at the top end of the range will give a better match to galaxies with higher asymptotic velocities.

Figure 4. Velocity distributions for galaxies with maximum velocities in the range $75 \leq v < 125{km\,s}^{-1}$ in the SPARC database. Data at outer radii are limited for these galaxies, making it difficult to determine the overall characteristics of the profiles. The SPARC data are plotted along with theoretical NFW (shaded blue) and ULDM (shaded red) profiles, assuming a virial mass of $5\times10^{11} \mathrm{M}_{\odot}$ , ${m_{22}} = 0.1$ , and $\pm 50 \%$ scatter in the ULDM core–halo mass relation and $\pm2\sigma$ scatter in NFW concentration. Galaxies in the legend are ordered from highest maximum velocity (top left) to lowest (bottom right).

In Figure 3, we see that galaxies with asymptotic velocities at the higher end of the range do not always exhibit a pronounced steepening of the velocity profile at small radii. Indeed in some cases there is simply no data at small radii. From this figure, we see that while a halo mass of $10^{12} \mathrm{M}_{\odot}$ with ${m_{22}} = 2.5$ provides a reasonable fit to the data at radii $>10 \mathrm{\,kpc}$ , it is difficult to judge the fit at small radii, where the ULDM and NFW profiles differ most strongly, due to a lack of data. Furthermore, while the data at higher radii seem to be relatively clustered, there are significant deviations within the limited data that exist at small radii. For example, the curves for NGC1090 and NGC6946 are widely disparate at small radii, but seem to converge at larger radii. Attempting to fit such data to a single set of model parameters would be of limited utility without a much more comprehensive data set from which statistical outliers could be properly identified. Furthermore, we note that there are substantial changes in theoretical ULDM velocity profiles under variation in the ULDM particle mass. The scale of these changes is exhibited in Appendix B. As such, we remark that analyses of the sort presented here would benefit greatly from tighter constraints on the ULDM particle mass.

By contrast, for galaxies with smaller maximum velocities ( $75 \leq v < 125{km\,s}^{-1}$ ), there are more data at smaller radii. For such galaxies, we see the steepening rotation curves characteristic of cored density profiles, as shown in Figure 4. In this case, choosing parameters such that the theoretical profiles overlap with the data at small radii is easy (in this case ${m_{22}} = 0.1$ , ${M}_{\mathrm{vir}}5\times10^{11} \mathrm{M}_{\odot}$ ); however, it is not clear whether this profile would fit data at larger radii were it available. Furthermore, while the choice ${m_{22}} = 0.1$ provides a reasonable fit to the data in this case, a ULDM particle mass ${m_{22}} = 0.1$ is in tension with constraints from the Lyman- $\alpha$ forest, as well as high-redshift UV luminosity function comparisonsFootnote d (Amendola and Barbieri Reference Amendola and Barbieri2006; Bozek et al. Reference Bozek, Marsh, Silk and Wyse2015; Armengaud et al. Reference Armengaud, Palanque-Delabrouille, Yãĺche, Marsh and Baur2017; Ni et al. Reference Ni, Wang, Feng and Di Matteo2019; Nebrin, Ghara, & Mellema Reference Nebrin, Ghara and Mellema2020). It must be acknowledged, however, that baryonic feedback is expected to have the greatest impact in the innermost regions of realistic halos. As such, agreement between our semi-analytic DM-only model and observational data at small radii should be interpreted cautiously, especially since this is also the region where assumptions regarding the stellar mass to light ratio have the greatest significance.