Constraints on dark matter scenarios from measurements of the galaxy luminosity function at high redshifts

We use state-of-the-art measurements of the galaxy luminosity function (LF) at $z=6$, 7, and 8 to derive constraints on warm dark matter (WDM), late-forming dark matter, and ultralight axion dark matter models alternative to the cold dark matter (CDM) paradigm. To this purpose, we have run a suite of high-resolution $N$-body simulations to accurately characterize the low-mass end of the halo mass function and derive dark matter (DM) model predictions of the high-z luminosity function. In order to convert halo masses into UV magnitudes, we introduce an empirical approach based on halo abundance matching, which allows us to model the LF in terms of the amplitude and scatter of the ensemble average star formation rate halo mass relation, $⟨\mathrm{SFR}({\mathrm{M}}_{\mathrm{h}},z)⟩$, of each DM model. We find that, independent of the DM scenario, the average SFR at fixed halo mass increases from $z=6$ to 8, while the scatter remains constant. At halo mass ${\mathrm{M}}_{\mathrm{h}}\gtrsim {10}^{12}{\mathrm{M}}_{\odot }{\mathrm{h}}^{-1}$, the average SFR as a function of halo mass follows a double power law trend that is common to all models, while differences occur at smaller masses. In particular, we find that models with a suppressed low-mass halo abundance exhibit higher SFR compared to the CDM results. Thus, different DM models predict a different faint-end slope of the LF which causes the goodness of fit to vary within each DM scenario for different model parameters. Using deviance statistics, we obtain a lower limit on the WDM thermal relic particle mass, $m_{\mathrm{WDM}}\gtrsim 1.5\mathrm{keV}$ at $2\sigma$. In the case of LFDM models, the phase transition redshift parameter is bounded to $z_t\gtrsim 8\times {10}^5$ at $2\sigma$. We find ultralight axion dark matter best-fit models with axion mass $m_a\gtrsim 1.6\times {10}^{-22}\mathrm{eV}$ to be well within $2\sigma$ of the deviance statistics. We remark that measurements at $z=6$ slightly favor a flattening of the LF at faint UV magnitudes. This tends to prefer some of the non-CDM models in our simulation suite, although not at a statistically significant level to distinguish them from CDM.

Figure 1

WDM best-fit models against LF data at $z=6$. Curves from bottom to top correspond to WDM-1 to WDM-5 respectively.

Figure 2

Top panel: Linear matter power spectra at $z=0$ for CDM (black solid line), WDM (blue lines), LFDM (red lines), and ULADM models (green lines). Bottom panel: Transfer function of the cutoff matched DM models.

Figure 3

The growth rate ratio, $\xi$, defined in Ref. [80] for the ULADM models for $z_{\text{start}}=100$ and $z_{\text{end}}=4$ as function of the mass scale $M(k)=4\pi (\pi /k{)}^3{\rho }_m/3$, where ${\rho }_m$ is the comoving matter density. The vertical dotted lines denote the lowest mass scales probed by HST observations of the faintest galaxies at $4\lesssim z\lesssim 8$. Unlike Ref. [80], our initial conditions use the correctly evolved initial conditions from axionCAMB; we use $\xi$ as a measure of the importance of scale-dependent growth to assess how well ULAs can be modeled with $N$-body simulations. The lightest ULA model we consider is well described by collisionless simulations at the percent level for halo masses of interest.

Figure 4

Halo mass function at $z=4$, 6, 7, and 8 (panels left to right) in mass bins of size $\mathrm{\Delta }{\mathrm{M}}_{\mathrm{h}}/{\mathrm{M}}_{\mathrm{h}}=0.20$ for WDM-2, LFDM-1, and ULADM-1 models (top to bottom panels respectively) before (blue empty squares) and after (red filled triangles) halo selection. In each panel, the CDM halo mass function is shown as empty circles. Error bars correspond to the Poisson errors in each mass bin.

Figure 5

Halo mass function of the reference CDM model simulation at $z=4$, 5, 6, 7, and 8 in mass bins of size $\mathrm{\Delta }{\mathrm{M}}_{\mathrm{h}}/{\mathrm{M}}_{\mathrm{h}}=0.20$. The dashed lines corresponds to Sheth-Tormen multiplicity function, Eq. (2), with best-fit parameters at the different redshifts given in Table 2.

Figure 6

Ratio of the best-fit mass functions at $z=4$, 6, 7, and 8 (left to right panels respectively) for WDM-2/LFDM-1 and WDM-2/ULADM-1 (top panels), WDM-3/LFDM-2 and WDM-3/ULADM-2 (central panels), and WDM-5/LFDM-3 and WDM-5/ULADM-3 (bottom panels). Ratios with respect to the LFDM are given by the red dotted lines, while those relative to the ULADM models are shown as blue solid lines. The short-dash-dot lines above and below each curve given the numerical statistical errors around the ratio of the best-fit functions.

Figure 7

Galaxy luminosity function at $z=4$ (red points) and $z=5$ (blue points) from Bouwens et al. [6] before (filled circles) and after (empty triangles) correction for dust extinction. The arrow indicates the direction of the shift of the uncorrected data due to the extinction correction.

Figure 8

SFR density function estimates at $z=4$ (red empty squares) and $z=5$ (blue filled squares) obtained using dust-corrected LF data from Bouwens et al. [6]. The dashed lines represent the Schechter function (Eq. (11) with best-fit coefficients (see Table 3).

Figure 9

${\mathrm{M}}_{\mathrm{h}}$-SFR relation at $z=4$ (short dashed line) and $z=5$ (long dashed line) for CDM (top left panel), WDM (top right panel), LFDM (bottom left panel), and ULADM (bottom right panel). The solid line in each panel represents the redshift-averaged relation ${\mathrm{SFR}}_{\mathrm{av}}({\mathrm{M}}_{\mathrm{h}})$ at fixed halo mass between $z=4$ and 5, which we use as a template to model the ensemble average relation at higher redshifts (see Sec. 3c).

Figure 10

Marginalized mean and $1\sigma$ error on ${\alpha }_{\mathrm{LF}}$ as inferred from the likelihood analysis of the Schechter fit to the LF measurements from B15 (red solid circles), A15 (green solid triangles), L16 (blue empty circles), and B16 (magenta empty squares) at $z=6$, 7, and 8. The points have been slightly displaced in redshift to facilitate readability of the figure.

Figure 11

Best-fit luminosity function for the various DM models against data at $z=6$ (top left panel), $z=7$ (top right panel), and $z=8$ (bottom panel) from B15 (red filled circles), B16 (magenta empty squares), A15 (green filled triangles), and L16 (blue empty circles). Within each panel, CDM is shown in the top left; WDM models are shown in the top right (lines from bottom to top correspond to WDM-1, WDM-2, WDM-3, WDM-4, and WDM-5 respectively); LFDM models are shown in the bottom left (LFDM-1, LFDM-2, and LFDM-3 lines from bottom to top); and ULADM modes are shown in the bottom right (ULADM-1, ULADM-2, and ULADM-3 lines from bottom to top).

Figure 12

Triangle plot of the one- and two-dimensional distributions of ${\log }_{10}{ϵ}_{\mathrm{SFR}}$ and ${\log }_{10}{\sigma }_{\mathrm{SFR}}$ for the CDM model at $z=6$. We find similar results using the data at higher redshifts and for the other DM models.

Figure 13

Average SFR vs halo mass, $⟨\mathrm{SFR}({\mathrm{M}}_{\mathrm{h}}),\mathrm{z}⟩$ at $z=6$ (black lines), $z=7$ (blue lines), and $z=8$ (red lines) for CDM (top left panel), WDM-3 (top right panel), LFDM-2 (bottom left panel), and ULADM-3 (bottom right panel). The solid lines correspond to the marginal mean value of ${ϵ}_{\mathrm{SFR}}^z$, while the dashed lines correspond to the 68% statistical uncertainties quoted in Table 6 with the hatched area in between covering the $\pm 1\sigma$ errors. In each panel, we also plot the marginal mean and $1\sigma$ error on ${\sigma }_{\mathrm{SFR}}$ at $z=6$, 7, and 8.

Figure 14

As in Fig. 11 for L16 data at $z=6$ (cyan filled circles).

Figure 15

Halo mass function from $N$-body halo catalogs of WDM-2 (left panel), LFDM-1 (center panel), and ULADM-1 (right panel) and best-fit functions at $z=4$, 5, 6, 7, and 8 (data points and curves from top to bottom).