RE-CAPTURING COSMIC INFORMATION

Gravitational lensing has emerged as a powerful tool to probe the distribution of matter in the universe (Bartelmann & Schneider 2001). Observations of the ellipticities of background galaxies can be transformed into estimates of the convergence field $\kappa (\vec{\theta })$. Along a given line of sight $\vec{\theta }$, the convergence measures a weighted integral of the total mass density field. Thus, by carefully studying κ as a function of position on the sky, we can learn about the underlying density field directly, without relying on the traditional assumption that every galaxy corresponds to an overdense region.

By measuring the convergence to sources at multiple background redshifts, cosmologists can infer not only the density field as a function of two-dimensional (2D) position (Kaiser 1992; Blandford et al. 1991; Jain & Seljak 1997; Jain et al. 2000), but also the evolution of this density field with time (Hu 1999). This information will be particularly valuable as a tool to study both dark matter and dark energy, which affect the growth of structure in the universe (Huterer 2002; Hu 2002). A number of wide-area surveys have been planned with the goal of mapping out the cosmic convergence field, and ultimately measuring properties of the dark energy (Aldering 2005; Miyazaki et al. 2006; Tyson 2006; Abbott et al. 2005; Refregier & the DUNE Collaboration 2009).

This goal appears attainable, as it is reminiscent of another cosmological success story: measurement of anisotropies in the cosmic microwave background (CMB; Hu & Dodelson 2002). In both cases, the values of the measured quantities—temperature in the case of the CMB and convergence from lensing—at any particular spot on the sky are not important. Rather, it is the statistics of the field that carries all the important information. The two-point function of the temperature of the CMB, the power spectrum of the anisotropies, is sensitive to a number of cosmological parameters, and some of these have now been measured to percent level accuracy (Komatsu et al. 2011). Similarly, the power spectrum of the convergence depends on cosmological parameters, and one can hope to extract information about these parameters from lensing surveys (Hu & Tegmark 1999; Abazajian & Dodelson 2003; Refregier et al. 2004; Hoekstra & Jain 2008).

However, the convergence field differs in an important way from the anisotropy maps. CMB anisotropies provide a snapshot of the universe when it was very young, and hence all deviations from homogeneity are very small (temperature differences in the maps are of order several parts in a hundred thousand). The physics describing these perturbations is linear. Further, the perturbations were drawn from a Gaussian distribution, so the two-point function captures all of the information in the field. On the other hand, the cosmic density field today is nonlinear and non-Gaussian, increasingly so on smaller scales, so some of the information initially stored in the two-point function when the fields were linear is no longer present.

Before quantifying this notion that information has left the two-point function, it is worthwhile to review some approaches to this problem. Takada & Jain (2004) pointed out that including information from both the two- and three-point functions significantly reduces the errors on cosmological parameters. This makes intuitive sense: the nonlinear process of gravitational instability transforms the initially Gaussian field into one with appreciable non-Gaussianity, one hallmark of which is a non-zero skewness. The goal of measuring both sets of functions may work, but it suffers from the drawback of requiring non-trivial covariance matrices (which involve the challenge of computing five- and six-point functions; Takada & Jain 2009).

A series of papers devoted to the three-dimensional (3D) density field $\delta (\vec{x})\equiv (\rho (\vec{x})-\bar{\rho })/\bar{\rho }$ (Rimes & Hamilton 2005, 2006; Neyrinck & Szapudi 2007; Neyrinck et al. 2006, 2009) have noted that information in the power spectrum of δ saturates at high wavenumbers k (or small length scales). That is, the power spectrum at high k is highly correlated, apparently due to the coupling of modes induced by nonlinear gravitational clustering. The most recent of these papers offered a useful proposal (Neyrinck et al. 2009) for re-capturing information about the 3D density field by pointing out that ln(1 + δ) has properties similar to the initial, linear density field. Its probability distribution is close to a Gaussian, the broadband shape is close to that of the linear power spectrum, and finally, the information content is close to the Gaussian case. Practically, this transform may be of limited utility because the 3D density field is typically estimated by using galaxies as tracers, and it is unlikely that the log transform of the galaxy density will be a useful tracer of the linear matter density field. However, we now show that the log transform can be applied to the 2D lensing convergence field to de-correlate modes and obtain information from higher order correlations back in the two-point function.

Using simulations, we study the statistics of a new field:

$$\begin{equation} \kappa _{\rm ln}(\vec{\theta }) \equiv {\kappa _0}\ln \left[ 1 + \frac{\kappa (\vec{\theta })}{\kappa _0}\right], \end{equation} \tag{ 1 }$$

where κ₀ is a constant with a value slightly larger than the absolute value of the minimum value of κ in the survey—this keeps the argument of the logarithm positive. In the limit of small κ, κ_ln reduces to the standard convergence, but the log alters it in very high or low density regimes. The parameter κ₀ tunes the degree of the alteration: the smaller κ₀, the more we alter the field.⁸ The log-mapping described above is motivated by our goal to de-correlate the Fourier modes of the convergence field. Although the mapping is local on the sky, it is nonlinear, so in Fourier space it has the potential to undo some of the correlations introduced by nonlinear clustering.

To study the properties of κ_ln, we use a suite of numerical simulations: 100 convergence fields, each 5° × 5° (a total of 2500 deg²) with 2048² pixels (i.e., 0.15 arcmin per pixel) were generated using N-body simulations as described in Sato et al. (2009). All source galaxies are taken to be at redshift z_s = 1 for all the results shown below, though we have also checked other source redshifts.

A first glimpse into the advantages of the log transform can be seen from Figure 1 which shows the probability distribution function (PDF) of both κ and κ_ln, compared to the (linear) Gaussian PDF. The new field is much closer to Gaussian, a promising sign since the loss of information in κ is attributed to gravity transforming the initially Gaussian random field into one that is highly non-Gaussian.

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To evaluate the log transform quantitatively, we take the Fourier transform of the three different convergence fields (linear, κ, and κ_ln) in each of the simulations. The angular power spectrum is estimated from the Fourier transforms (denoted $\tilde{\kappa }(\vec{l})$) by summing over all modes with wavenumber $\vert \vec{l}\vert$ in a given bin l_bin.

Figure 2 shows these spectra. As expected, the power spectrum of the nonlinear κ field is much larger than the linear field on small scales (large l). This excess power on small scales is suppressed when κ_ln is used. Again the result is not surprising, as the high-density regions are smoothed out: κ_ln ≪ κ for large κ.

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Although the power spectrum of κ_ln is smaller than that of κ, it contains more cosmological information. To see this, consider a model with one free parameter, the amplitude of the observed, nonlinear power spectrum before and after the log transform. The projected fractional error on this parameter is the inverse of the signal-to-noise ratio (S/N) defined as

$$\begin{equation} \rm{S/N} (l_{\rm max}) \equiv \left[ \sum _{l,l^{\prime }<l_{\rm max}} C_l {\rm Cov}^{-1}(l,l^{\prime }) C_{l^{\prime }} \right]^{1/2}, \end{equation} \tag{ 2 }$$

where C_l is the power spectrum of multipole l before and after the transform, Cov is the covariance matrix describing correlations between the power spectra of multipoles l and l' (l, l' < l_max), and the summation runs over all the multipoles l and l' subject to l, l' < l_max (Sato et al. 2009; Takahashi et al. 2009). We measure the non-Gaussian covariance matrix from the dispersions of the 100 convergence fields before and after the transform. We follow Neyrinck et al. (2009) and call the square of the S/N the information content. Heuristically, then, "information" quantifies how accurately parameters will be determined. To compute the expected error on the chosen cosmological parameter (here the amplitude of the power spectrum; Lee & Pen 2008), one needs to know the covariance matrix of the spectra. If the field was Gaussian random, the covariance matrix would be diagonal. In the absence of shape noise,⁹ it would arise from sample variance and be equal to the spectrum squared divided by the number of independent modes in the bin. In that case, since the number of modes in a bin grows as l for log binning, the $\rm{(S/N)}^2$ would grow as l²_max.

Figure 3 shows the $\rm{(S/N)}^2$ as a function of l_max. The linear κ field is shown by the dotted gray line. The information obtained from the nonlinear κ field falls well below this ideal limit, as seen in the figure. This arises because the nonlinearities significantly affect the covariance matrix. Non-zero off-diagonal elements in the covariance matrix mean that many of the modes carry redundant information, so the total gain is significantly below the l²_max Gaussian limit. The log transform undoes a large portion of this damage. The left panel of Figure 3 shows that the information in κ_ln is well above that in κ and close to the Gaussian case. In other words, we measure the amplitude of the power spectrum with higher precision if we use the log-transformed field. We find a factor of ∼1.3 improvement in $\rm{(S/N)}^2$ at l_max ∼ 250, a factor of ∼2.6 at l_max ∼ 1000, a factor of 4 at l_max ∼ 2000, and a factor of 8 at l_max ∼ 5000.

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The restored information in the κ_ln field can be understood by examining the covariance matrix of the power spectra. Figure 4 shows two rows of the covariance matrix for the fields, with one of the wavenumbers fixed at l' = 253 and l' = 1049 in the two cases (upper and lower panel). The κ covariance matrix has large off-diagonal elements in adjacent bins—these carry redundant information and therefore do not add much to the $\rm{S/N}$. The transformed κ_ln, on the other hand, is much more nearly diagonal. A nearly diagonal covariance matrix implies another important advantage of the log transform: the approximation of a Gaussian covariance matrix for cosmological parameter estimation is more accurate for κ_ln.

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Another way of understanding the gain in information in the log field is to consider the Taylor expansion of the log transform κ_ln. For −1 < κ/κ₀ ⩽ 1, one sees that κ_ln contains the standard convergence field, but also a piece that scales as κ² (and higher orders). Considered perturbatively, then, the spectrum of κ_ln will depend not only on the two-point function of κ, C_l, but also on the three-point function, the bi-spectrum, as well as higher-point functions. Effectively, then this rather simple transform captures information in the power spectrum, bi-spectrum, tri-spectrum, etc., in a compact way. Of course, it does not contain all the information in these higher point functions, but the improvement seen in Figure 3 suggests that using κ_ln as a transform in future surveys may be a simple, powerful way to bundle much of this information into one simple spectrum. We have tested this by measuring the information contained in κ'_ln ≡ κ − κ²/(2κ₀) and found that, once we apply an appropriate cutoff¹⁰ on high κ values to make the polynomial expansion more sensible, the single extra term replicates most of the improvement observed in the log transform. Meanwhile, the cross-correlation of the κ and κ − κ²/κ₀ which involves only up to bi-spectrum, with an appropriate cutoff, replicates most of the improvement up to l ∼ 1000. This implies that the bi-spectrum is the dominant contributor to this improvement up to the scales.

There are several caveats to this analysis. So far, we have neglected noise, in particular shape noise due to the random orientations of galaxies on the sky. We have studied this issue for several survey parameters. Surveys with higher number density have lower shape noise and therefore the advantages of κ_ln approach those depicted in the left panel of Figure 3. For a galaxy number density of 30 arcmin^–2 at z_s = 1,¹¹ as expected for the planned Subaru Hyper SuprimeCam survey (Miyazaki et al. 2006), we find an improvement of 1.7 (2.4) in the information content for l_max ∼ 1000 (2000) (right panel in Figure 3). The gain is larger for more ambitious surveys like LSST or Euclid (Tyson 2006; Refregier et al. 2010) and smaller for shallower surveys like the Dark Energy Survey (Abbott et al. 2005). Further details will be presented in our future paper.

Second, although κ_ln has some of the advantages of the linear κ field, it does not actually recover the initial field phase by phase since the cross-correlation between the initial and final fields, when tested for the density fields, does not improve by this transformation. Third, our analysis (and our definition of information) revolved around only one parameter, the amplitude of the power spectrum. Its shape and evolution certainly contain additional cosmological information, as discussed by Takada & Jain (2009). We therefore need to test that the improvement in $\rm{(S/N)}^2$ in the amplitude translates to the improved precision on the parameters such as σ₈, matter density, and dark energy parameters. We plan to conduct a full Fisher matrix analysis using N-body simulations in our future paper to investigate this issue.

Finally, we have assumed that the convergence field, reconstructed from the shear, will be available over the entire survey area—in practice such a reconstruction adds additional noise. We are in the process of studying these issues, but they are not expected to affect our main point: that the log transform κ_ln recovers cosmological information.

We have found that the log transform of Equation (1) alters the nonlinear convergence field to one that mimics the properties of a Gaussian field. It returns a PDF that is close to a Gaussian—analogous to the findings of Neyrinck et al. (2009) for the 3D density field. The signal to noise (i.e., precision) of the measurement of the amplitude of the power spectrum is greatly improved over a wide range of angular scales, 200 ≲ l ≲ 10⁴. Even in the presence of shape noise, this improvement holds, to a greater or lesser extent depending on the galaxy number density. The improvement arises from the effect on the covariance matrix: the off-diagonal elements of the covariance matrix are substantially reduced for the log transform. We find that the bi-spectrum that is embedded in the log transform is the dominant contributor to this improvement. Therefore, the log transform appears to bundle much of the information from higher order statistics into the power spectrum.

Upcoming imaging surveys will collect data on the shapes of galaxies at an unprecedented rate, with an eye toward understanding the physics that drives the acceleration of the universe. It is imperative that we use algorithms to analyze these data that extract as much of the cosmological information as possible: the log transform κ_ln is a step in this direction.

We thank David Johnston, Nick Gnedin, Mark Neyrinck, Alex Szalay, and Ryuich Takahashi for useful comments.

This work is supported in part by World Premier International Research Center Initiative (WPI Initiative), by Grant-in-Aid for Scientific Research on Priority Areas No. 467 "Probing the Dark Energy through an Extremely Wide and Deep Survey with Subaru Telescope," MEXT, Japan, by the JSPS Research Fellows, by the DOE at Fermilab through grant DE-FG02-95ER40896, and by NSF grants AST-0908072 and AST-0607667.