Constraints on Cosmological Parameters from the Angular Power Spectrum of a Combined 2500 deg² SPT-SZ and Planck Gravitational Lensing Map

Gravitational lensing remaps CMB fluctuations in position space (Lewis & Challinor 2006):

$$\begin{eqnarray}&&{T}^{{\rm{L}}}(\hat{{\boldsymbol{n}}})={T}^{{\rm{U}}}(\hat{{\boldsymbol{n}}}+{\rm{\nabla }}\phi (\hat{{\boldsymbol{n}}})),\end{eqnarray} \tag{ 1 }$$

where $\phi (\hat{{\boldsymbol{n}}})$ is the projected gravitational lensing potential and superscripts L and U refer to the lensed and unlensed temperature fields respectively. To gain intuition, Equation (1) can be Taylor expanded as

$$\begin{eqnarray}&&{T}^{{\rm{L}}}(\hat{{\boldsymbol{n}}})={T}^{{\rm{U}}}(\hat{{\boldsymbol{n}}})+{\rm{\nabla }}{T}^{{\rm{U}}}\cdot {\rm{\nabla }}\phi (\hat{{\boldsymbol{n}}})+\ldots .\end{eqnarray} \tag{ 2 }$$

From the second term, it can be seen that the observed lensed temperature has a component that is the gradient of the unlensed field modulated by the lensing deflection ∇ϕ. If we transform to harmonic space, Equation (2) would have the second term on the right-hand side written as a weighted convolution of the temperature field and the lensing potential, where the harmonic transform for any particular mode for the lensed field could involve a sum over all of the modes of the unlensed field. Lensing thus introduces non-zero off-diagonal elements in the covariance of observed temperature fields in harmonic space (Okamoto & Hu 2003):

$$\begin{eqnarray}&&{\rm{\Delta }}\langle {T}_{{{\ell }}_{1}{m}_{1}}{T}_{{{\ell }}_{2}{m}_{2}}\rangle \\ \qquad =\,\displaystyle \sum _{{LM}}{(-1)}^{M}\left(\begin{array}{ccc}{{\ell }}_{1} & {{\ell }}_{2} & L\\ {m}_{1} & {m}_{2} & -M\end{array}\right){W}_{{{\ell }}_{1}{{\ell }}_{2}L}^{\phi }\,{\phi }_{{LM}},\end{eqnarray} \tag{ 3 }$$

where T_ℓm are the spherical harmonic expansion coefficients of the temperature fields and ϕ_LM the coefficients of the projected lensing potential. The weight

$$\begin{eqnarray}{W}_{{{\ell }}_{1}{{\ell }}_{2}L}^{\phi } & = & -\,\sqrt{\displaystyle \frac{(2{{\ell }}_{1}+1)(2{{\ell }}_{2}+1)(2L+1)}{4\pi }}\\ & & \times {C}_{{{\ell }}_{1}}^{{TT}}\left(\displaystyle \frac{1+{(-1)}^{{{\ell }}_{1}+{{\ell }}_{2}+L}}{2}\right)\left(\begin{array}{ccc}{{\ell }}_{1} & {{\ell }}_{2} & L\\ 1 & 0 & -1\end{array}\right)\\ & & \times \sqrt{L(L+1){{\ell }}_{1}({{\ell }}_{1}+1)}+({{\ell }}_{1}\leftrightarrow {{\ell }}_{2})\end{eqnarray} \tag{ 4 }$$

characterizes the mode coupling induced by lensing (i.e., the effect of the convolution in Equation (2)).

The lensing potential can be estimated from observed CMB maps by measuring the lensing-induced mode coupling of Equation (3) between pairs of modes in the observed temperature field (Zaldarriaga & Seljak 1999; Hu & Okamoto 2002). In general, it is best to use pairs in harmonic space that have good signal-to-noise for measuring lensing. For this purpose, it is useful to work with a filtered map: ${\bar{T}}_{{\ell }m}\equiv {F}_{{\ell }m}{T}_{{\ell }m}$, with the filter ${F}_{{\ell }m}\equiv {({C}_{{\ell }}+{N}_{{\ell }m})}^{-1}$ for a given CMB power spectrum C_ℓ and an anisotropic (m-dependent) noise power spectrum N_ℓm.

A formally optimal estimator (at first order) which maximizes signal to noise in the estimated lensing potential (Hu & Okamoto 2002) is

$$\begin{eqnarray}&&{\bar{\phi }}_{{LM}}\\ \quad =\,\displaystyle \frac{{(-1)}^{M}}{2}\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{{{\ell }}_{1},{m}_{1}}{{{\ell }}_{2},{m}_{2}}}\left(\begin{array}{ccc}{{\ell }}_{1} & {{\ell }}_{2} & L\\ {m}_{1} & {m}_{2} & -M\end{array}\right){W}_{{{\ell }}_{1}{{\ell }}_{2}L}^{\phi }\,{\overline{T}}_{{{\ell }}_{1}{m}_{1}}{\overline{T}}_{{{\ell }}_{2}{m}_{2}}.\end{eqnarray} \tag{ 5 }$$

We use Equation (5) as our ϕ estimator for this analysis. There are other choices (e.g., Namikawa et al. 2013) for how to weight the mode pairs that sacrifice some signal-to-noise but reduce foreground contamination. Lensing reconstruction is done with the quicklens code.³⁷

The relationship between the filtered estimate of the lensing potential resulting from Equation (5) and the true potential can be written as

$$\begin{eqnarray}&&{\bar{\phi }}_{{LM}}\equiv {{ \mathcal R }}_{{LM}}^{\phi }{\phi }_{{LM}},\end{eqnarray} \tag{ 6 }$$

defining a response function ${{ \mathcal R }}_{{LM}}$ that in general depends on both L and M. As outlined in O17, this response function has been calibrated using simulations. We estimate it by measuring the cross-spectrum of simulated lensing potential outputs with the input lensing potential maps and normalizing by the autospectrum of the inputs.

The true amplitude of mode coupling in the CMB temperature field induced by lensing is sensitive to the true (unknown) temperature power spectrum, as can be seen in Equations (3) and (4). What is measured in the data is some amount of mode coupling; to turn this into an estimate of the amplitude of the lensing potential, an assumption is made about the typical amplitudes of the modes being coupled. The response function thus depends on the assumed cosmological parameters. To explore this cosmological dependence, we use an isotropic approximation to the full anisotropic response function and its dependence on cosmology. In the case where both the signal and noise are isotropic (i.e., the CMB signal and noise only depend statistically on ℓ and not m), the response function can be written as

$$\begin{eqnarray}&&{{ \mathcal R }}_{L}^{\phi }=\displaystyle \frac{1}{2L+1}\displaystyle \sum _{{{\ell }}_{1},{{\ell }}_{2}}{W}_{{{\ell }}_{1}{{\ell }}_{2}L}^{\phi ,t}{W}_{{{\ell }}_{1}{{\ell }}_{2}L}^{\phi ,f}{F}_{{{\ell }}_{1}}{F}_{{{\ell }}_{2}},\end{eqnarray} \tag{ 7 }$$

where we have indicated extra superscripts on the weight functions for either the true amount of mode coupling (t) or the assumed amount for our fiducial cosmology (f). The filters F_ℓ are calculated for the fiducial cosmology. We use Equation (7) and its dependence on cosmology to determine the cosmology-dependent corrections to the simulation-based response function.

The survey mask, point source mask, and spatially varying noise all violate statistical stationarity in the data, and consequently they introduce mode coupling that can bias the lensing reconstruction. The result is that the lensing reconstruction has a non-zero mean signal—even in the absence of true lensing signal—that depends on the survey geometry, mask, and noise properties. This mean field ${\bar{\phi }}_{{LM}}^{\mathrm{MF}}$ is calculated using simulations and removed.

After removing the mean field and correcting for the response function, the final estimate of the lensing potential is

$$\begin{eqnarray}&&{\hat{\phi }}_{{LM}}=\displaystyle \frac{{\bar{\phi }}_{{LM}}-{\bar{\phi }}_{{LM}}^{\mathrm{MF}}}{{{ \mathcal R }}_{{LM}}^{\phi }}.\end{eqnarray} \tag{ 8 }$$

To estimate the angular power spectrum of the CMB lensing map obtained in the previous section, we multiply the estimate $\hat{\phi }$ by the survey mask (including point source and galaxy cluster masking) and use PolSpice³⁸ (Szapudi et al. 2001; Chon et al. 2004) to compute the spectrum of the masked map.

The resulting power spectrum is a biased estimate of the true lensing power spectrum. Known sources of bias include a straightforward noise bias, ${N}_{L}^{(0)}$, that comes from taking an autospectrum of data with noise in it (where "noise" here includes the Gaussian part of the CMB temperature field and any other sky signal), and a bias that arises from ambiguity in exactly which lensing modes are being measured in the power spectrum, ${N}_{L}^{(1)}$ (Kesden et al. 2003). The superscript denotes the order of the lensing power spectrum involved: ${N}_{L}^{(0)}$ is independent of the true lensing power and only depends on the instrument noise and sky power, while ${N}_{L}^{(1)}$ has a linear dependence on the lensing power. As detailed in O17, we calculate these biases using simulations and subtract them from the measured power spectrum

$$\begin{eqnarray}&&{\hat{C}}_{L}^{\phi \phi }={C}_{L}^{\hat{\phi }\hat{\phi }}-{N}_{L}^{(0)}-{N}_{L}^{(1)}.\end{eqnarray} \tag{ 9 }$$

We use a realization-dependent ${N}_{L}^{(0)}$ estimate that takes into account the power in the particular realization but does not depend on the assumed cosmology (Namikawa et al. 2013).

The ${N}_{L}^{(1)}$ bias depends linearly on the lensing power and will therefore depend on cosmological parameters. In the flat-sky limit (Kesden et al. 2003; Das et al. 2011b; Planck Collaboration 2014b) and assuming isotropic noise and filtering, the bias is

$$\begin{eqnarray}{N}_{L}^{(1)} & = & \displaystyle \frac{1}{{{ \mathcal R }}_{L}^{\phi }{{ \mathcal R }}_{L}^{\phi }}\displaystyle \int \displaystyle \frac{{d}^{2}{{\boldsymbol{\ell }}}_{1}}{{(2\pi )}^{2}}\displaystyle \int \displaystyle \frac{{d}^{2}{{\boldsymbol{\ell }}}_{3}}{{(2\pi )}^{2}}\\ & & \times \,{F}_{{{\ell }}_{1}}\,{F}_{{{\ell }}_{2}}\,{F}_{{{\ell }}_{3}}\,{F}_{{{\ell }}_{4}}\,{W}^{\phi ,f}({{\boldsymbol{\ell }}}_{1},{{\boldsymbol{\ell }}}_{2}){W}^{\phi ,f}({{\boldsymbol{\ell }}}_{3},{{\boldsymbol{\ell }}}_{4})\\ & & \times [{C}_{| {{\boldsymbol{\ell }}}_{1}-{{\boldsymbol{\ell }}}_{3}| }^{\phi \phi }{W}^{\phi ,t}(-{{\boldsymbol{\ell }}}_{1},{{\boldsymbol{\ell }}}_{3})\,{W}^{\phi ,t}(-{{\boldsymbol{\ell }}}_{2},{{\boldsymbol{\ell }}}_{4})\\ & & +\,\,{C}_{| {{\boldsymbol{\ell }}}_{1}-{{\boldsymbol{\ell }}}_{4}| }^{\phi \phi }{W}^{\phi ,t}(-{{\boldsymbol{\ell }}}_{1},{{\boldsymbol{\ell }}}_{4})\,{W}^{\phi ,t}(-{{\boldsymbol{\ell }}}_{2},{{\boldsymbol{\ell }}}_{3})],\end{eqnarray} \tag{ 10 }$$

where the weight ${W}^{\phi }({{\boldsymbol{\ell }}}_{1},{{\boldsymbol{\ell }}}_{2})$ is the flat-sky version of Equation (4).

There is a dependence on both the true CMB power (just as for ${{ \mathcal R }}_{L}^{\phi }$) and the lensing power. To explore this cosmological dependence (below), we will use Equation (10) to determine the cosmology-dependent corrections to the ${N}_{L}^{(1)}$ that is derived from simulations.

The next-order ${N}_{L}^{(2)}$ bias is largely removed by using the lensed theory temperature power spectrum rather than the unlensed spectrum when constructing the lensing estimator (Hanson et al. 2011). There are other biases, such as the ${N}_{L}^{(3/2)}$ bias (Böhm et al. 2016), which are small at the precision of the current work, and will be neglected.

We estimate uncertainties on the lensing power spectrum by averaging over N_s = 198 simulations:

$$\begin{eqnarray}&&{({\rm{\Delta }}{\hat{C}}_{L}^{\phi \phi })}^{2}=\displaystyle \frac{1}{{N}_{s}-1}\displaystyle \sum _{i=1}^{{N}_{s}}{({\hat{C}}_{L;i}^{\phi \phi }-\langle {\hat{C}}_{L}^{\phi \phi }{\rangle }_{{N}_{s}})}^{2}.\end{eqnarray} \tag{ 11 }$$

This procedure could be used to generate a full covariance matrix, but for this analysis we assume that uncertainties are uncorrelated between bins. This is expected for the relatively large bins that we use and the realization-dependent removal of the ${N}_{L}^{(0)}$ bias that strongly reduces the off-diagonal elements of the covariance matrix (Schmittfull et al. 2013). From simulations, we measured the correlation between bins to be no more than 5%.

The choice of cosmological model affects the computation of the estimated lensing bandpowers through the calculation of the response function and through the calculation of the ${N}_{L}^{(1)}$ bias term. These effects need to be included in the likelihood analysis.

The response function ${{ \mathcal R }}_{{LM}}^{\phi }$ and ${N}_{L}^{(1)}$ correction for the fiducial cosmology are obtained using simulations and calculated using two-dimensional, anisotropic weighting. To calculate the cosmological corrections to these terms, we use isotropic approximations to both the response function and the ${N}_{L}^{(1)}$ bias (see Equations (7) and (10)). Within the range of allowed parameters, the cosmological corrections are relatively small, and we expect the error on these corrections from using the isotropic approximation to be negligible.

At a given point in parameter space, we apply the cosmology-dependent response function and ${N}_{L}^{(1)}$ corrections to the theory spectrum (Planck Collaboration 2016b):

$$\begin{eqnarray}&&{C}_{L}^{\phi \phi ,\mathrm{th}}=\displaystyle \frac{{({{ \mathcal R }}_{L}^{\phi })}^{2}| {}_{{\rm{\Theta }}}}{{({{ \mathcal R }}_{L}^{\phi })}^{2}| {}_{f}}{C}_{L}^{\phi \phi ,\mathrm{th}}| {}_{{\rm{\Theta }}}+{N}_{L}^{(1)}| {}_{{\rm{\Theta }}}-{N}_{L}^{(1)}| {}_{f}.\end{eqnarray} \tag{ 13 }$$

To obtain these corrections, we use a linear approximation, Taylor-expanding around the response function or ${N}_{L}^{(1)}$ bias calculated for the fiducial cosmology. For a temperature or lensing power spectrum that differs by Δ from the fiducial spectrum, we obtain:

$$\begin{eqnarray}&&{\rm{\Delta }}{({{ \mathcal R }}_{L}^{\phi })}^{2}| {}_{{\rm{\Theta }}}\simeq {M}_{L,{\ell }^{\prime} }^{(R)}| {}_{f}\times {\rm{\Delta }}{C}_{{\ell }^{\prime} }^{{TT}}| {}_{{\rm{\Theta }}},\end{eqnarray} \tag{ 14 }$$

where ${M}_{L,{\ell }^{\prime} }^{(R)}\equiv \tfrac{{C}_{L}^{\phi \phi }}{{({{ \mathcal R }}_{L}^{\phi })}^{2}}\tfrac{\partial {({{ \mathcal R }}_{L}^{\phi })}^{2}}{\partial {C}_{{\ell }^{\prime} }^{{TT}}}$, and

$$\begin{eqnarray}&&{\rm{\Delta }}{N}_{L}^{(1)}| {}_{{\rm{\Theta }}}\simeq {M}_{L,L^{\prime} }^{(1)}| {}_{\mathrm{fid}}\times {\rm{\Delta }}{C}_{L^{\prime} }^{\phi \phi }| {}_{{\rm{\Theta }}},\end{eqnarray} \tag{ 15 }$$

where ${M}_{L,L^{\prime} }^{(1)}\equiv \tfrac{\partial {N}_{L}^{(1)}}{\partial {C}_{L^{\prime} }^{\phi \phi }}$. The matrices M were calculated using binned versions of the temperature and lensing power spectra. In principle there is also a dependence on the temperature power spectrum in the ${N}_{L}^{(1)}$ correction, but that term was found to be negligible.

An alternative way to parameterize the amplitude of the matter power spectrum is σ₈, the rms mass fluctuation today in 8 h⁻¹ Mpc spheres assuming linear theory. This parameter is convenient for comparisons with results from galaxy surveys.

In Figure 3, constraints from lensing experiments, both CMB lensing (Planck Collaboration 2016b; Sherwin et al. 2017) and cosmic shear (Hildebrandt et al. 2017; Joudaki et al. 2017; Troxel et al. 2017), are shown in the σ₈ − Ω_m plane, compared with expectations from the primary CMB fluctuations as measured by Planck. There have been hints of mild tension between Planck CMB power spectrum constraints and probes of low-redshift structure. The CMB lensing constraints are all highly consistent with each other, and it can be seen that the constraints from this paper (the SPT + Planck CMB lensing data) overlap with both the low-redshift probes and the primary CMB estimates, although the primary CMB data are substantially more precise. In making the CMB-lensing-only constraints, as was done in Planck Collaboration (2016b) and Sherwin et al. (2017), the corrections to the response function were held at the best-fit cosmology corresponding to the Planck TT and lowP likelihoods⁴¹ (Planck Collaboration 2016a). The close agreement between SPT + Planck and Planck is not simply from the combined SPT + Planck data set including data from Planck. The SPT + Planck data are based on only ∼2500 deg², and are mainly driven by the SPT data.

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Joint constraints on Ω_m and σ₈ obtained by combining the CMB lensing data with the primary CMB measurements from Planck are shown in Figure 4. In general, the CMB lensing data (either the full-sky Planck or 2500 deg² SPT + Planck) prefer lower values of σ₈, as could be expected from Figure 3. A commonly used parameter for lensing constraints is ${\sigma }_{8}{{\rm{\Omega }}}_{{\rm{m}}}^{0.25}$. For SPT + Planck we find ${\sigma }_{8}{{\rm{\Omega }}}_{{\rm{m}}}^{0.25}$ = 0.598 ± 0.024, in excellent agreement with both the value found using the Planck full-sky lensing reconstruction, 0.591 ± 0.021 (Planck Collaboration 2016b), and the estimate by ACT_pol of 0.643 ± 0.054 (Sherwin et al. 2017).

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CMB lensing data are most sensitive to overall shifts in the amplitude of matter fluctuations. This amplitude can be expressed as the rms deflection angle $\langle {d}^{2}{\rangle }^{1/2}$. For SPT + Planck, we use the samples from the MCMC chains for ${\rm{\Lambda }}\mathrm{CDM}$ to determine that this rms deflection angle is 2.27 ± 0.16 arcmin (68%), in good agreement with the extremely precise measurement of the full-sky Planck survey of 2.46 ± 0.06.

Gravitational lensing of the CMB leads to a small amount of smearing of the acoustic oscillations in the primary fluctuations, an effect that has been well measured (Das et al. 2011a; Keisler et al. 2011). The primary CMB fluctuations as observed by Planck show weak evidence for a slightly elevated amount of lensing-like smearing of the acoustic peaks, although the lensing power directly measured by Planck shows no such excess (Planck Collaboration 2016b). Using the same effect in SPT temperature (Story et al. 2013) and polarization (Henning et al. 2017) power spectra, there was no evidence for such an excess of peak smoothing, with a modest (∼1σ) preference for less lensing than expected.

The expected amount of lensing depends on the somewhat uncertain cosmological parameters. To explore this, we marginalize over cosmological parameters and use new parameters to artificially scale the amount of lensing: ${A}_{L}$ scales the lensing power spectrum in both the lens reconstruction power and in the smearing of the acoustic peaks, and ${A}^{\phi \phi }$ scales only the amplitude of the CMB lensing reconstruction power spectrum. This parameterization ensures that the ${\rm{\Lambda }}\mathrm{CDM}$ parameters that control the predicted degree of lensing (such as ${{\rm{\Omega }}}_{b}{h}^{2}$ and σ₈) are determined without considering the measured amount of peak smearing or mode coupling, and that these measurements are reflected entirely in ${A}_{L}$ and ${A}^{\phi \phi }$.

As these parameters are defined, the ${\rm{\Lambda }}\mathrm{CDM}$ prediction for the reconstructed lensing power spectrum gets multiplied by both ${A}_{L}$ and ${A}^{\phi \phi }$. Therefore, the combination ${A}^{\phi \phi }$ × ${A}_{L}$ represents the amplitude for the lensing power relative to the ${\rm{\Lambda }}\mathrm{CDM}$ prediction when the cosmological parameter fits are not sensitive to the observed amount of peak smearing.

With ${A}_{L}$ fixed to unity the known preference in the Planck primary data for ${A}_{L}$ > 1 will instead drive a preference for models with higher intrinsic lensing amplitudes, leading to a preference for lower values of ${A}^{\phi \phi }$ when compared with lensing reconstruction measurements that are otherwise consistent with ${\rm{\Lambda }}\mathrm{CDM}$. When ${A}_{L}$ is free, the peak-smoothing preference for ${A}_{L}$ > 1 increases the predicted lensing reconstruction power and therefore causes a lower ${A}^{\phi \phi }$ for a given model compared with the lensing power spectrum measurement. The combination ${A}_{L}$ × ${A}^{\phi \phi }$ thus gives the amplitude of the lensing power spectrum compared to Planck-allowed ${\rm{\Lambda }}\mathrm{CDM}$ predictions when the peak smoothing effect is not reflected in the Planck constraints.

Posterior distributions for ${A}_{L}$, ${A}^{\phi \phi }$, and ${A}^{\phi \phi }$ × ${A}_{L}$ from chains using combinations of Planck primary CMB data, the Planck lensing power spectrum, and the lensing power spectrum in this work are shown in Figure 5. For models with ${A}_{L}$ = 1 the measured SPT + Planck lensing power spectrum is somewhat low, with ${A}^{\phi \phi }$ = 0.91 ± 0.06. The CMB lensing reconstruction power spectrum measurements show no evidence for an anomalous amount of lensing relative to the amount predicted from the best-fit ${\rm{\Lambda }}\mathrm{CDM}$ parameters determined in the primary CMB data when the peak smearing effect has been marginalized over. Using SPT + Planck data, we find ${A}^{\phi \phi }$ × ${A}_{L}$ = 1.01 ± 0.08 relative to the predicted level of lensing for ${\rm{\Lambda }}\mathrm{CDM}$ marginalized over ${A}_{L}$; using the full-sky Planck full-sky lensing reconstruction, the result is only slightly higher, ${A}^{\phi \phi }$ × ${A}_{L}$ = 1.05 ± 0.06. The peak smearing in the Planck primary CMB power spectra, meanwhile, indicates mild evidence for enhanced lensing, with ${A}_{L}$ = 1.22 ± 0.10.

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Inflationary models predict that the universe should be close to spatially flat, and the combination of observations of the primary CMB, supernovae Ia, baryon acoustic oscillations, and local Hubble constant measurements shows that spatial curvature is not large (e.g., Komatsu et al. 2011). Constraints from the primary CMB have a geometrical degeneracy that allows spatial curvature to be increased while the Hubble constant is adjusted to keep the angular diameter distance to last scattering fixed; as a result, CMB measurements have historically relied on Hubble constant priors or external measurements to constrain curvature. As a probe of the local universe, lensing partially lifts this degeneracy (Sherwin et al. 2011). Figure 6 demonstrates this degeneracy-breaking by adding 2500 deg² SPT + Planck or Planck full-sky lensing reconstruction information to the Planck primary CMB measurements. The constraint on spatial curvature from adding SPT + Planck lensing information to Planck primary CMB is ${{\rm{\Omega }}}_{k}=-{0.012}_{-0.023}^{+0.021}$ at 95% confidence.

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CMB lensing, as a measurement of the amplitude of clustering at intermediate redshifts, is a potentially powerful probe of neutrino masses (Smith et al. 2006; Abazajian et al. 2015). Neutrino oscillation experiments have precisely measured the differences in the squares of the masses between the neutrino eigenstates, but the absolute masses have not been measured. Laboratory limits constrain the mass of the electron neutrino, but the strongest constraints on absolute neutrino masses currently come from cosmology. Having a substantial amount of the energy density in the form of massive neutrinos leads to a suppression of structure on small scales in the matter power spectrum. The Planck primary CMB measurements limit the sum of the masses to be ${M}_{\nu }$ < 0.72 eV at 95% confidence (Planck Collaboration 2016a). This constraint is strongly driven by the measurement of lensing through the smearing of peaks in the CMB power spectra.

As was seen in Planck Collaboration (2016a), Figure 7 shows that adding information from the lensing reconstruction power spectrum reduces the ${M}_{\nu }$ posterior value at zero, but the lensing reconstruction data also rule out large values of ${M}_{\nu }$, with the combined result being a similar 95% upper limit. Using SPT + Planck, the upper limit on neutrino masses is ${M}_{\nu }$ < 0.70 eV at 95% confidence, compared to ${M}_{\nu }$ < 0.68 eV for adding Planck full-sky lensing reconstruction data.

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