Cosmology constraints from the weak lensing peak counts and the power spectrum in CFHTLenS data

Lensing peaks have been proposed as a useful statistic, containing cosmological information from non-Gaussianities that is inaccessible from traditional two-point statistics such as the power spectrum or two-point correlation functions. Here we examine constraints on cosmological parameters from weak lensing peak counts, using the publicly available data from the $154{\mathrm{deg}}^2$ CFHTLenS survey. We utilize a new suite of ray-tracing N-body simulations on a grid of 91 cosmological models covering broad ranges of the three parameters ${\mathrm{\Omega }}_m$, ${\sigma }_8$, and $w$, and replicating the Galaxy sky positions, redshifts, and shape noise in the CFHTLenS observations. We then build an emulator that interpolates the power spectrum and the peak counts to an accuracy of $\le 5\%$, and compute the likelihood in the three-dimensional parameter space (${\mathrm{\Omega }}_m$, ${\sigma }_8$, $w$) from both observables. We find that constraints from peak counts are comparable to those from the power spectrum, and somewhat tighter when different smoothing scales are combined. Neither observable can constrain $w$ without external data. When the power spectrum and peak counts are combined, the area of the error “banana” in the (${\mathrm{\Omega }}_m$, ${\sigma }_8$) plane reduces by a factor of $\approx 2$, compared to using the power spectrum alone. For a flat $\mathrm{\Lambda }$ cold dark matter model, combining both statistics, we obtain the constraint ${\sigma }_8({\mathrm{\Omega }}_m/0.27{)}^{0.63}={0.85}_{-0.03}^{+0.03}$.

Figure 1

Convergence maps for the four CFHTLenS fields. They are divided into 13 subfields of $12{\mathrm{deg}}^2$ in size to match our simulation configuration. Scattered white dots are masks. White lines mark the edges of our simulated maps. Three subfields are collages of the six rectangular patches in ${\mathrm{W}}_1,{\mathrm{W}}_2,{\mathrm{W}}_4$. Patches in black and white are not used in our simulation.

Figure 2

Visual representation of the cosmological parameters in the 91 models used in our simulations, and listed in Table 1.

Figure 3

Examples of the interpolated power spectrum (upper panel) with 0.5 arc min smoothing and peak counts (lower panel) with 1.0 arc min smoothing and 25 $\kappa$ bins, using the two different interpolation techniques RBF and GP. The solid curves show the true quantities for the given cosmological model (no. 49 in Table 1), and the dashed and dotted curves show the interpolations based on the other 90 models.

Figure 4

Interpolated power for $\ell =3,000$ (7.2 arc min) as a function of ${\mathrm{\Omega }}_m$, $w$, and ${\sigma }_8$ using simulations (upper panel) and fitting formula from Ref. [60] (lower panel). The third parameter for each plane is at a fixed value ($[{\mathrm{\Omega }}_m,w,{\sigma }_8]=[0.26,-1.0,0.8]$).

Figure 5

Comparison of the CFHTLenS (top panel) and simulated (bottom panel) power spectrum measured using 75% of the fields (solid line) which pass the PSF residual test and all fields (dashed line). The error is measured from our simulations.

Figure 6

Sixty-eight percent error contours from the power spectrum measured using only the 75% of the CFHTLenS fields that pass the PSF residual test (solid curves) and for all fields (dashed curves). Constraints are shown with (thick curves) and without (thin curves) imposing an upper limit $\ell <7,000$.

Figure 7

Interpolated number counts for typical low ($<1{\sigma }_{\kappa }$, top panel), medium ($1–3{\sigma }_{\kappa }$, middle panel), and high ($>3{\sigma }_{\kappa }$, bottom panel) peaks, where ${\sigma }_{\kappa }=0.03$ is the standard deviation of $\kappa$ measured in our simulations. As in Fig. 4, the third parameter in each panel is at a fixed value ($[{\mathrm{\Omega }}_m,w,{\sigma }_8]=[0.26,-1.0,0.8]$).

Figure 8

Comparison of the CFHTLenS (upper panel) and simulated (lower panel) peak counts measured using 75% of the fields (solid curves) which pass the PSF residual test and all fields (dashed curves). The error is measured from our simulations.

Figure 9

Sixty-eight percent error contours from peak counts using smoothing scales of 1.0 (dotted curve), 1.8 (thick dashed curve), 3.5 (thin solid curve), and 5.3 (thin dashed curve) arc min, as well as from peak counts with 1.0 and 1.8 arc min smoothing scales in combination (thick solid curve).

Figure 10

Constraints on the parameter ${\mathrm{\Sigma }}_8={\sigma }_8({\mathrm{\Omega }}_m/0.27{)}^{\alpha }$, using the power spectrum (dashed line), peak counts (thin solid line), and their combination (thick solid line). The power spectrum is computed using the 75% of the fields that pass the PSF residual test, and restricted to $\ell <7,000$. Peak counts are computed using all fields, and include measurements of peaks on two smoothing scales (1.0 and 1.8 arc min).

Figure 11

Sixty-eight percent (dark color) and 95% (light color) error contours from the power spectrum (dashed curves), peak counts (thin solid curves), and their combination (thick solid curves). The shaded region in the bottom panel is the 68% error contour for the combination. The power spectrum is computed using the 75% of the fields that pass the PSF residual test, and restricted to $\ell <7,000$. Peak counts are computed using all fields, and include measurements of peaks on two smoothing scales (1.0 and 1.8 arc min).

Figure 12

Correlation coefficients of the total covariance. Bins 1–40 are for the power spectrum, bins 41–65 are for peak counts with 1.0 arc min smoothing scale, and bins 66–90 are for peak counts with 1.8 arc min smoothing scale.

Figure 13

Sixty-eight percent (dark color) and 95% (light color) power $\text{spectrum}+\text{peak}$ counts error contours with (thin curves) and without (thick curves) the cross-covariance. The power spectrum is computed using the 75% of the fields that pass the PSF residual test, and restricted to $\ell <7,000$. Peak counts are computed using all fields, and include measurements of peaks on two smoothing scales (1.0 and 1.8 arc min).

Figure 14

Fits to the CFHTLenS (thick solid curves) power spectrum (upper panel) and peak counts (lower panel). The peak counts on 1.0 and 1.8 arc min scales are concatenated on the x axis. The best fits (thin solid curves) and two other models (dash-dotted and dashed curves) randomly selected from within the 68% error bananas are shown for reference.