Testing the warm dark matter paradigm with large-scale structures

We explore the impact of a $\Lambda \mathrm{WDM}$ cosmological scenario on the clustering properties large-scale structure in the Universe. We do this by extending the halo model. The new development is that we consider two components to the mass density: one arising from mass in collapsed haloes, and the second from a smooth component of uncollapsed mass. Assuming that the nonlinear clustering of dark matter haloes can be understood, then from conservation arguments one can precisely calculate the clustering properties of the smooth component and its cross correlation with haloes. We then explore how the three main ingredients of the halo calculations, the halo mass function, bias and density profiles are affected by warm dark matter (WDM). We show that, relative to cold dark matter (CDM), the halo mass function is suppressed by 50%, for masses $\sim 100$ times the free-streaming mass scale $M_{\mathrm{fs}}$. Consequently, the bias of low mass haloes can be boosted by as much as $\sim 20\%$ for 0.25 keV WDM particles. Core densities of haloes will also be suppressed relative to the CDM case. We also examine the impact of relic thermal velocities on the density profiles, and find that these effects are constrained to scales $r<1h^{-1}\mathrm{kpc}$, and hence of little importance for dark matter tests, owing to uncertainties in the baryonic physics. We use our modified halo model to calculate the nonlinear matter power spectrum, and find that there is significant small-scale power in the model. However, relative to the CDM case the power is suppressed. The amount of suppression depends on the mass of the WDM particle, but can be of order 10% at $k\sim 1h{\mathrm{Mpc}}^{-1}$ for particles of mass 0.25 keV. We then calculate the expected signal and noise that our set of $\Lambda \mathrm{WDM}$ models would give for a future weak lensing mission. We show that the models should in principle be separable at high significance. Finally, using the Fisher matrix formalism we forecast the limit on the WDM particle mass for a future full-sky weak lensing mission like Euclid or the Large Synoptic Survey Telescope. With Planck priors and using only multipoles $l<5000$, we find that a lower limit of 2.6 keV should be easily achievable.

Figure 1

Free-streaming mass scale ($M_{\mathrm{fs}}$) as a function of the mass of the WDM particle mass $m_{\mathrm{WDM}}$. Haloes that have masses $M>M_{\mathrm{fs}}$ may form hierarchically (hatched region). Haloes with $M<M_{\mathrm{fs}}$ (empty region) will have their peaks in the primordial density field erased and so may not form hierarchically. Haloes in this region may possibly form through fragmentation. The solid red band shows the $m_{\mathrm{WDM}}$ allowed by the Lyman alpha forest [27] [note that we have rescaled $m_{{\nu }_s}\to m_{\mathrm{WDM}}$ using Eq. (5)].

Figure 2

Linear mass power spectra as a function of wave number in the WDM and CDM scenarios. The solid black line shows results for CDM. From bottom to top, the dot-dashed lines represent the WDM power spectra for the cases where $m_{\mathrm{WDM}}\equiv m_X\in \{0.25,0.5,0.75,1.0,1.25\}\mathrm{keV}$. The lighter the WDM particle the more the power is damped on small scales.

Figure 3

Left panel: Halo mass function as a function of halo mass. Lines show the theoretical predictions from the ST model of Eq. (40). The uppermost (black) solid line denotes the result for CDM; from bottom to top, the dot-dash lines denote results for WDM for our standard set of particle masses ($m_{\mathrm{WDM}}\equiv m_X=\{0.25,0.50.751.0,1.25\}\mathrm{keV}$) with no cutoff mass applied; the solid colored lines denote the same, but with a cutoff mass scale as given in Eq. (49), and here we use ${\sigma }_{\log M}=0.5$. Right panel: Fractional difference between the WDM and CDM mass functions, as a function of halo mass scaled in units of the WDM free-streaming mass scale $M_{\mathrm{fs}}$. Line styles are the same as in left panel, but this time the mass of the WDM particle increases from top to bottom.

Figure 4

Left panel: Halo bias as a function of halo mass for various dark matter models. The lowermost solid (black) line denotes the results for CDM; from top to bottom, the dot-dashed line denotes the halo bias for our standard set of WDM particle masses ($m_{\mathrm{WDM}}\equiv m_X=\{0.25,0.5,0.75,1.0,1.25\}\mathrm{keV}$). The dashed lines denote the bias of the smooth component of mass as given by Eq. (35). Right panel: fractional difference between the bias in WDM and CDM models as a function of halo mass scaled in units of the free-streaming mass scale $M_{\mathrm{fs}}$. WDM particle mass increases from bottom to top.

Figure 5

Density profiles of CDM and WDM haloes as a function of radius scaled in units of the characteristic scale radius $r_s$. Top left, top right, bottom left and bottom right panels show the results for haloes with masses: $M\in \{{10}^{15},{10}^{13},{10}^{11},{10}^9\}{h}^{-1}{M}_{\odot }$. In all panels: the uppermost solid (black) line denotes the CDM profile; from bottom to top the (red, green, blue, cyan, magenta) lines denote the WDM profiles for masses $m_{\mathrm{WDM}}\in \{0.25,0.5,0.75,1.0,1.25\}\mathrm{keV}$, respectively. The solid lines denote the case where there are no thermal relic velocities; the dot-dash lines denote the case where profiles have been smoothed with a Gaussian filter of the scale of the mean relic velocity speed $l_{v_{\mathrm{relic}}}\sim \overline{v}/H_0\sim 3.15v_0/H_0$. In each panel we also indicate the scale radius that the halo would have for the various WDM particle masses considered, where from top to bottom we show this for decreasing $m_{\mathrm{WDM}}$.

Figure 6

Comparison of the nonlinear matter power spectrum in the halo model for a particular WDM model ($m_{\mathrm{WDM}}\equiv m_x=0.25\mathrm{keV}$) and the CDM model, as a function of wave number. The solid upper (black) and lower (red) lines denote the total CDM and total WDM power spectra in the halo model. The dashed, dot-dash, dotted and triple dot-dashed lines denote, $P_{\mathrm{ss}}$, $P_{\mathrm{sh}}$, $P_{1\mathrm{h}}$, and $P_{2\mathrm{h}}$, respectively. The solid blue and red lines correspond to results computed with and without the relic velocities of the WDM particles—note that these lines are indistinguishable in the plot.

Figure 7

Comparison of the halo model predictions for the WDM and CDM matter power spectra as a function of wave number. Left and right panels: results as obtained from the Halo model and halofit, respectively. Top panels: absolute power. Solid and dashed lines denote the nonlinear and linear theory predictions, respectively. Uppermost black line denotes CDM; and and from bottom to top the (red, green, blue, cyan, magenta) lines denote WDM particle masses $m_{\mathrm{WDM}}\equiv m_X\in \{0.25,0.5,0.75,1.0,1.25\}\mathrm{keV}$. Bottom panels: ratio of WDM and CDM power spectra. Line styles unchanged.

Figure 8

Comparison of the weak-lensing convergence power spectrum in the CDM and WDM models, as a function of angular multipole for source galaxies at $z_s=1$. Left and right panels: predictions obtained using the halo model and halofit, respectively. Top panels: Absolute power. Dashed and solid lines show the predictions from linear theory and from the nonlinear halo model. Uppermost (black) lines represent results for CDM. From bottom to top, the (red, green, blue, cyan, magenta) lines denote results for the WDM model with particle masses $m_{\mathrm{WDM}}\equiv m_{\mathrm{WDM}}\in \{0.25,0.5,0.75,1.0,1.25\}\mathrm{keV}$. Bottom panel: Ratio of the WDM model predictions to the CDM predictions. Lines styles and colors are as above.

Figure 9

We plot 65% and 95% likelihood contours, marginalized over all other parameters, for the fiducial (Euclid/LSST-like) survey (green) as well as the fiducial survey combined with Planck (blue). Our underlying fiducial model is the best fit WMAP7 $\Lambda \mathrm{CDM}$ model.

Figure 10

Dependence of the halo model predictions for the WDM power spectra on the minimum mass scale $M_{\mathrm{cut}}$, as a function of wave number. Top panel: absolute power. Solid lines of increasing thickness denote the predictions with $M_{\mathrm{cut}}\in \{0.1,1.0,10.0\}{M}_{\mathrm{fs}}$, respectively. Black lines denote CDM; and colors {red, green, blue, cyan, magenta} denote WDM particle masses $m_{\mathrm{WDM}}\equiv m_X\in \{0.25,0.5,0.75,1.0,1.25\}\mathrm{keV}$. Bottom panels: ratio of WDM power spectra with varying cutoff mass scales to the fiducial WDM predictions with $M_{\mathrm{cut}}=M_{\mathrm{fs}}$.