Detection of gravitational lensing in the cosmic microwave background

Gravitational lensing of the cosmic microwave background (CMB), a long-standing prediction of the standard cosmological model, is ultimately expected to be an important source of cosmological information, but first detection has not been achieved to date. We report a $3.4\sigma$ detection, by applying quadratic estimator techniques to all sky maps from the Wilkinson microwave anisotropy probe (WMAP) satellite, and correlating the result with radio galaxy counts from the NRAO VLA sky survey (NVSS). We present our methodology including a detailed discussion of potential contaminants. Our error estimates include systematic uncertainties from density gradients in NVSS, beam effects in WMAP, galactic microwave foregrounds, resolved and unresolved CMB point sources, and the thermal Sunyaev-Zel’dovich effect.

Figure 1

Left panel: CMB signal power spectrum, and three-year WMAP noise power spectrum. Right panel: Fiducial NVSS signal power spectrum, and NVSS shot noise ($\overline{G}=159000\mathrm{gal}/\mathrm{sr}$).

Figure 2

Left panel: Auto power spectrum $C_{\ell }^{\varphi \varphi }$ of the CMB lensing potential, and reconstruction noise power spectrum $N_{\ell }^{\varphi \varphi }$ [Eq. (4)] at three-year WMAP noise levels. Right panel: Cross power spectrum $C_{\ell }^{\varphi g}$ between the CMB lensing potential and NVSS galaxy counts, and the effective noise power spectrum $[N_{\ell }^{\varphi \varphi }{N}_{\ell }^{gg}/2{]}^{1/2}$ for detecting the cross correlation. The “boost” in signal-to-noise between the two cases is sufficient to obtain a several-sigma detection of CMB lensing.

Figure 3

Mean contribution to the squared total detection significance ${\sigma }^2$ per NVSS galaxy multipole ${\ell }_3$ (left panel), and per unit increase in maximum CMB multipole ${\ell }_{\max }^{\mathrm{CMB}}=\max ({\ell }_1,{\ell }_2)$ (right panel). Most of the statistical weight comes from galaxy multipoles near $\ell \sim 50$, and CMB multipoles near $\ell \sim 400$.

Figure 4

$\mathrm{\text{Simulation}}+\mathrm{\text{analysis}}$ pipeline used in this paper; the stages (1)–(7) are described in detail in Sec. 4.

Figure 5

Detection of CMB lensing via the cross power spectrum $C_{\ell }^{\varphi g}$ between the reconstructed potential and galaxy counts. The three $1\sigma$ error bars on each bandpower represent different Monte Carlo methods: WMAP simulations vs NVSS simulations (left/black); WMAP data vs NVSS simulations (middle/blue); and WMAP simulations vs NVSS data (right/red). These error bars represent statistical errors only; the result with systematic errors included will be shown in Fig. 19.

Figure 6

CMB lensing detection obtained by analyzing Q-band (left/black error bar in each triple), V-band (middle/blue), and W-band (right/red) data from WMAP separately, showing consistency of the result between CMB frequencies.

Figure 7

Result of the curl null test ($C_{\ell }^{\psi g}=0$). As in Fig. 5, the three error bars on each bandpower represent different Monte Carlo methods: WMAP simulations vs NVSS simulations (left/black); WMAP data vs NVSS simulations (middle/blue); and WMAP simulations vs NVSS data (right/red).

Figure 8

Maximum likelihood NVSS galaxy power spectrum, calculated without (top panel) and with (bottom panel) marginalization of $m=0$ modes in equatorial coordinates. In the bottom panel, fiducial spectra are shown (both for $b_g=1.7$) from the model for $dN/dz$ by 73 (dotted line) and our fit in Eq. (21) (dashed line).

Figure 9

NVSS galaxy overdensity field in equatorial coordinates, low-pass filtered to multipoles $\ell \le 10$, showing visible azimuthal striping.

Figure 10

Change $\Delta C_{\ell }^{gg}$ in maximum likelihood galaxy power spectrum, when NVSS is analyzed with $m=0$ marginalization vs no marginalization (top panel) or $m=0$, 1 marginalization vs $m=0$ marginalization (bottom panel) in equatorial coordinates. The error bars represent the rms shift obtained when Monte Carlo simulations are analyzed in the same way.

Figure 11

Result of convolving a single noiseless CMB realization with the WMAP V1 beam, including beam asymmetry. We have shown the output map separated into contributions from different beam multipoles: $s=0$ (isotropic component, top left); $s=1$ (top right); $s=2$ (bottom left); and $s=3$ (bottom right). Each map has been scaled independently for visibility; the rms temperature in the $s=0,\dots ,3$ maps is 88, 0.4, 1.0, $0.04\mu \mathrm{K}$. The convolution with the $s>0$ multipoles is scan dependent and shows alignments with the ecliptic poles reflecting the WMAP scan strategy.

Figure 12

Foreground templates used in this paper, shown with Kp0 mask (Sec. 2) applied. Left panel: Dust template, based on 77 with frequency dependence given by Eq. (23). Right panel: Free-free template, based on 76, 78 with frequency dependence given by Eq. (24). The masked rms of the templates in V-band is $6.4\mu \mathrm{K}$ and $4.8\mu \mathrm{K}$, respectively.

Figure 13

Point source estimator ${\widehat{C}}_b^{sg}$ applied to the WMAP and NVSS data sets, showing no evidence for CMB point source power which is correlated to NVSS. The error bars were obtained from Monte Carlo $\mathrm{WMAP}+\mathrm{NVSS}$ simulations without point sources.

Figure 14

Histogrammed 1.4 GHz flux distribution in NVSS, with the fitting function in Eq. (35) shown for comparison.

Figure 15

Ensemble of simulations in the fiducial point source model [Eqs. (35, 36)] with varying point source amplitude $\Lambda$. For each realization, we show the observed point source level ${\widehat{C}}_b^{sg}$ in the band $b=({\ell }_{\min },{\ell }_{\max })=(20,40)$ and the change in the lensing estimator $\Delta {\widehat{C}}_b^{\varphi g}$ due to the point source contribution. The dotted vertical line shows the point source level in this band estimated from the real $\mathrm{WMAP}+\mathrm{NVSS}$ data; the smaller vertical error bar shows the mean and rms $\Delta {\widehat{C}}_b^{\varphi g}$ among simulations whose observed point source level matches the measured value.

Figure 16

Illustration of our procedure [Eq. (39)] for simulating correlations between the NVSS galaxy field (left) and source mask (right). For visibility, we have bandlimited the galaxy field $\delta g$ to $\ell \le 6$, and used 100 sources with masking radius 4° rather than the mask parameters of the data sets (Sec. 2).

Figure 17

The Compton-y profile for three different cluster masses at $z=1$ (thick lines) and $z=0.1$ (thin lines). The profiles have been normalized to 1 at $l=0$ to facilitate comparison. According to this panel, at high redshift it may be possible to approximate even rare and massive clusters as point sources on the scale where our lensing estimator gathers most of its information, $l\simeq 400$.

Figure 18

The thermal Sunyaev-Zel’dovich angular power spectrum contributions (in the Rayleigh-Jeans limit) from Poisson and clustering terms. On the scale of most interest for our lensing reconstruction, $l\simeq 400$, the SZ Poisson term dominates by an order of magnitude over the clustering part. The angular power spectra were calculated using the gas-pressure profile model by 92, 93.

Figure 19

Final result from Table , showing statistical errors alone (blue/inner error bars) and $\mathrm{\text{statistical}}+\mathrm{\text{systematic}}$ errors (red/outer).

Figure 20

Preconditioner chain for multigrid $(S+N{)}^{-1}$ filtering, using noise maps from the three-year WMAP data set. From left to right, each set of maps represents one conjugate gradient inversion problem, which is preconditioned by the “faster and cruder” approximation which appears next in the chain, obtained by either reducing resolution or the number of distinct beams retained in the problem.

Figure 21

Sequence of coarsifying and decoarsifying operations in an instance of the multigrid method with $N_{(1)}=3$, $N_{(2)}=2$, showing the W-cycle structure. Each solid circle represents one “forward” operation of the operator $X=(1+S^{1/2}{N}^{-1}{S}^{1/2})$ at the appropriate resolution.