Super-sample signal

When extracting cosmological information from power spectrum measurements, we must consider the impact of super-sample density fluctuations whose wavelengths are larger than the survey scale. These modes contribute to the mean density fluctuation ${\delta }_{\mathrm{b}}$ in the survey and change the power spectrum in the same way as a change in the cosmological background. They can be simply included in cosmological parameter estimation and forecasts by treating ${\delta }_{\mathrm{b}}$ as an additional cosmological parameter enabling efficient exploration of its impact. To test this approach, we consider here an idealized measurement of the matter power spectrum itself in the $\mathrm{\Lambda }\mathrm{CDM}$ cosmology though our techniques can readily be extended to more observationally relevant statistics or other parameter spaces. Using subvolumes of large-volume $N$-body simulations for power spectra measured with respect to either the global or local mean density, we verify that the minimum variance estimator of ${\delta }_{\mathrm{b}}$ is both unbiased and has the predicted variance. Parameter degeneracies arise since the response of the matter power spectrum to ${\delta }_{\mathrm{b}}$ and cosmological parameters share similar properties in changing the growth of structure and dilating the scale of features especially in the local case. For matter power spectrum measurements, these degeneracies can lead in certain cases to substantial error degradation and motivates future studies of specific cosmological observables such as galaxy clustering and weak lensing statistics with these techniques.

Figure 1

Power spectrum response to the mean density fluctuation in a survey region, calibrated with the separate universe simulations through a change in background parameters (see Ref. [14]). Solid curve represents the response of the power spectrum referenced to the true or global mean density, while the dashed curve to the mean survey or local density. The two are related by an additive constant 2, which changes the shape of the response.

Figure 2

Estimated (${\widehat{\delta }}_{\mathrm{b}}$) vs true (${\delta }_{\mathrm{b}}$) mean density fluctuations for the set of 3584 simulation subvolumes. The means ($\times$, $+$, slightly shifted for clarity) of the estimators using both globally and locally referenced power spectra show no trace of a bias for bins with sufficient statistics that the standard deviation (errorbars) can be estimated. This standard deviation is significantly smaller than $\pm {\sigma }_{\mathrm{b}}$ (gray band) for the global case and comparable for the local case.

Figure 3

Distribution or scatter of the density estimator ${\widehat{\delta }}_{\mathrm{b}}$ around the true value (histogram) versus the prediction from Eq. (16) with a Gaussian distribution (curve). In both the global and local cases, the predictions are within 7% of the simulation results in the standard deviation justifying their use in parameter forecasts below.

Figure 4

Standard deviation of the ${\delta }_{\mathrm{b}}$ estimator, ${\sigma }_{{\delta }_{\mathrm{b}}}$, in the local and global cases as a function of the maximum $k$ bin compared with the rms density fluctuation ${\sigma }_{\mathrm{b}}$. In the local case the standard deviation approaches the rms in the nonlinear regime and remains nearly constant to $2h{\mathrm{Mpc}}^{-1}$. In the global case, it drops below the rms in the nonlinear regime and continues to improve beyond $1h{\mathrm{Mpc}}^{-1}$.

Figure 5

Power spectrum response to cosmological parameters $\{\ln A_{\mathrm{s}},n_{\mathrm{s}},h\}$ as calibrated from simulations. Similarities between these responses and Fig. 1 cause degeneracies which degrade parameter errors once uncertainties in the super-sample mode ${\delta }_{\mathrm{b}}$ are marginalized.

Figure 6

Growth component of the power spectrum response to ${\delta }_{\mathrm{b}}$ and $h$ compared with the response to the initial amplitude of power $\ln A_{\mathrm{s}}$. The two growth responses are nearly identical whereas the $\ln A_{\mathrm{s}}$ one differs in the nonlinear regime. The difference can be attributed to a change in halo scale radii between models with the same linear power spectrum.

Figure 7

Dilation component of the power spectrum response to ${\delta }_{\mathrm{b}}$ and $h$ compared with that derived from the slope of the power spectrum $T^{\mathrm{d}}$. All three responses are nearly identical with the $T^{\mathrm{d}}$ exhibiting more noise and resolution dependence from numerical differencing.

Figure 8

Errors on cosmological parameters as a function of the maximum $k$ bin for ${\delta }_{\mathrm{b}}$ fixed (dot-dashed), marginalized in the local case (dashed) and in the global case (solid). Left panel: single cosmological parameter estimation with the remaining two held fixed. Right panel: joint cosmological parameter estimation.

Figure 9

Error contours (68%) and distributions for joint ${\delta }_{\mathrm{b}}$ and cosmological parameter estimation with $k_{\max }=2h{\mathrm{Mpc}}^{-1}$ with (inner, black) and without (outer, blue) an external prior from ${\sigma }_{\mathrm{b}}$. Left panel: local case, with dashed lines representing the degeneracy direction from Eq. (32). Right panel: global case.

Figure 10

Power spectrum responses in most degenerate directions in the local and global cases, Eqs. (32) and (33), respectively. In the local case, the direction effectively nulls the response in the nonlinear regime creating a near perfect degeneracy. In the global case, no direction nulls the response and the degeneracy is much weaker. Responses are normalized to represent $1\sigma$ deviations in degenerate parameter combination.

Figure 11

Power spectrum response to $h$ from finite differences of simulations with $\delta h=\pm 0.02$ and scales fixed in $h^{-1}\mathrm{Mpc}$. Thin gray lines show differences of 64 pairs of realizations, with solid black lines representing their means with standard errors. A single calibration from one pair of simulations is highlighted in blue to illustrate that stochasticity around the mean are highly correlated across nonlinear $k$. In the nonlinear regime where most information is located, correction from the second derivative in dotted line is much smaller compared to the response itself, demonstrating that linear response serves as a good approximation for the power spectrum variation.