As predicted, observations of CMB anisotropies have provided tight constraints on cosmological parameters as well as definitive discrimination against competing models of cosmological structure formation. The CMB has been a phenomenal success as a cosmological probe. In this section we elucidate the foundations of this success and argue that the same foundations are in place for cosmic shear.
The cosmic microwave background has been a very fruitful target of observation because of
We consider each of these items in turn.
Clean relation between theory and observables. The accuracy of theoretical predictions is due to the fact that on large angular scales (scales greater than a few arcminutes) the temperature anisotropies are chiefly due to very small (1 part in 100,000) departures from homogeneity. The low amplitude of these inhomogeneities makes linear perturbation theory an excellent approximation. Any second-order contributions to the angular power spectrum are much smaller than the uncertainty with which they can be measured.
Rich features. The accuracy of the predictions would not be worth much if the only thing predicted were a featureless power law power spectrum. Fortunately, the spectrum is rich with bumps and wiggles and the detailed properties of these features depend on many cosmological parameters. From the CMB power spectrum one can simultaneously determine the mean curvature, the baryon density, the dark matter density, the power spectrum of the primordial density perturbations, the distance to the epoch of recombination, and more. The richness in features translates into a richness in the parameters that can be constrained
Measurability. Finally, the same reason that CMB anisotropies are easy to understand theoretically, the smallness of the departures from isotropy, makes them difficult to measure. So we are fortunate that it is not too difficult using modern detectors, as has been so clearly proven over the last decade and a half.
Despite the success of the CMB as a cosmological probe it cannot do everything. The CMB anisotropies do not have much sensitivity to events at redshifts much less than that of recombination (z ~ 1100). This means that the CMB cannot be used as a precision probe of the dark energy, which only becomes a signficant contributor to the total energy density at z < 2.
Fortunately, tomographic cosmic shear on large angular scales (once again angles greater than a few arcminutes) is an observation that is sensitive to events at lower redshift and shares all three positive attributes of the CMB as listed above.
Clean relation between theory and observables. Just like with the CMB, on sufficiently large scales linear perturbation theory is an excellent approximation for predicting the statistical properties of cosmic shear. However, because cosmic shear is sensitive to density perturbations at lower redshifts than the CMB, and these density perturbations have been growing with time, linear perturbation theory breaks down at larger angular scales for cosmic shear than it does for the CMB, as can be seen in the figure. But, with cosmic shear the breakdown of linear perturbation theory does not mean a breakdown in calculability. On scales greater than a few arc minutes numerical simulations can be used to calculate the shear power spectra with very high accuracy. The key to this ability is the dominant role played by the dark matter, and the simplicity of its purely gravitational interactions.
Rich features. Although each shear power spectrum on its own is much poorer in features than the CMB power spectrum, there are many of these shear power spectra. Their relative shapes and amplitudes are highly informative about the primordial power spectrum and the subsequent development of structure at lower redshifts. Also, the non-linear (but calculable) evolution of the density field means there is significant (and non-redundant) information in not just the two-point correlation function (power spectrum in Fourier space) but also the three-point function (bispectrum in Fourier space).
Measurability. The 0.001 - 0.01 shear of background galaxies induced by foreground mass structure is actually easier to measure than the 10ppm variations in the CMB temperature. These shear measurements are made more reliable by having shear calibrators - foreground stars -- densely sprinkled over the sky (10,000 per square degree). The technical challenges took two decades to solve in each case: low noise receivers for CMB, and large CCD detectors covering significant fields of view for the weak lens shear. Analysis software algorithm development also played a critical role in both CMB and cosmic shear.
Cosmic shear multiple probes are poised to take the next definitive step in unveiling the physical nature of the dark energy. LSST is uniquely capable of this mission, with first light in 2012.
Combining data on the anisotropy of the CMB from WMAP and Planck with weak lensing data from LSST will break cosmological parameter degeneracies and yield tighter constraints on nature of dark energy. Combining a multiband hemisphere-scale WL survey with the expected Planck CMB data could constrain both the dark energy equation of state and its time derivative 60 or more times more accurately than the Planck data alone, and measure the neutrino mass to +/-0.02 eV (Song & Knox 2003). Combining the two measurements will pin down w to of order 3% and its time derivative wa to 0.05, as well as tightly constrain the dark matter power spectrum. To deepen our understanding of how these surveys are constraining dark energy, we have examined how they constrain the function w(z), rather than its simple parameterization by w0 and wa. We proceed by binning w(z) in redshift bins and then identifying the eigenmodes and eigenvalues of the binned w(z) error covariance matrix as was done for supernovae by Huterer & Starkman (2003).
We particularly emphasize the eigenmode/eigenvalue results shown in Figure 1. We see a striking difference in the modes for 2000 SNe vs. the modes for LSST WL: those for LSST stretch out to higher z. The reason for this is that lensing is less sensitive to the growth factor at the lower redshifts where the source density in a given redshift bin is small and the lensing window (for sources at higher z) is also small. Thus the supernovae are better at detecting changes in w(z) at lower z and LSST shear tends to be better at detecting changes at higher redshift. LSST and 2000 SNe also have strikingly different eigenvalue spectra. The error on the amplitude of the best determined mode is quite similar for each ~0.03. But the 2000 SNe spectrum is much steeper. LSST has seven modes with errors smaller than unity, whereas 2000 SNe has four.
Figure 1 Eigenvalues and first three eigenmodes of the w(z) error covariance matrix for LSST WL+Planck and 2000 SNe+Planck. The best-determined mode for each dataset has a standard deviation of about 0.03. This error rises quite slowly with increasing eigenmode number for the lensing data, reaching one only by the 7th mode. The eigenmode shape differences indicate that lensing is better at probing higher z while supernovae have their chief advantage at lower z.
These constraints combine those from shear power-spectrum tomography, CMB data, and limited use of the cosmography method. Cluster counting or non-Gaussian statistics methods, with Planck CMB priors, are expected to yield similar precision and are subject to very different systematic errors. Inclusion of SN Ia information would further tighten the constraints. The errors in Figure 1 are for a simultaneous fit to a full set of cosmological parameters, and include marginalization over all parameters except curvature (measured via CMB anisotropy data). Even if the CMB constraints are not included, Ωm and other parameters are tightly constrained. This over-determination allows for consistency checks between multiple WL, CMB, and SNIa methods.
3-D mass tomography uses the lens induced shear of background galaxies vs distances to these source galaxies (via photometric redshifts) to estimate the lens redshift (Wittman, et al. 2003). LSST will apply these techniques to the whole visible sky. Weak lens shear data provides a precision measure of cosmology complementary to SNe. The twin requirements of LSST are (1) control of systematic shear errors, and (2) sufficient optical throughput to cover the entire visible sky multiple times per month in muliple filter bands. We know that we can achieve this because of weak lensing surveys already underway. For example, the Deep Lens Survey (DLS) when completed in 2005 will cover up to 24 square degrees in four wavelength bands. While the DLS imaging goes nearly as faint as that proposed for the LSST, the DLS uses old technology wide-field 4-meter telescopes which do not have active optics. The shear systematics introduced by the telescope optics are a factor of ten larger than that achieved by new technology telescopes which use active feedback to the optics. Nevertheless the DLS data already reach a shear systematics floor within a factor of 5-10 of that required for LSST. New technology telescopes in good sites are also routinely delivering superior image quality. In its stack of 200 deep images per sky patch per band, LSST should easily deliver 50-60 resolved source galaxies per square arcminute while covering 20,000 square degrees. Based on recent data from new-technology 8-m telescopes and operations simulations using actually records from potential LSST sites, the LSST will cover 20,000 square degrees of sky with a source density of 60 per square arcminute, and a shear error floor of 0.0001.
The LSST imaging goal is a sky noise limited stack of at least 300 images per filter band for every patch in at least 20,000 square degrees. Several billion source galaxies will be detected in this imaging data. The resulting goal for the shear-redshift database is shear data limited purely by the shot noise of 50-60 randomly oriented source galaxies per square arcminute. The LSST telescope, camera, and survey are designed explicitly to meet these challenges.
