Weak lensing is a powerful probe of cosmological models, beautifully complementary to those that have given rise to the current standard model of cosmology. The statistics of shear and mass maps on large scales over a wide range in redshift holds much promise for fundamental cosmology. The underlying physics is extremely simple General Relativity: FRW Universe plus the GR deflection formula. One measures angles, dimensionless ellipticities, and redshifts. Mass structure formation proceeds through a linear regime to non-linear compact clumps at late times. The WL shear caused by gravitational light deflection from foreground mass concentrations, modified by dimensionless cosmic distance ratios -- both of which are sensitive to dark energy -- naturally fall into two regimes, suggesting two methods of data analysis and interpretation:

**Method 1:** Operate on large scales in the nearly linear regime. Predictions are as good as for CMB. One also needs to know the redshift distribution of sources, which is measurable via multi-color photometry.

**Method 2:** Operate in non-linear, non-Gaussian regime. This applies to shear correlations at small angle and to cluster counting. Predictions require N-body calculations, but to ~1% level are dark-matter dominated and hence purely gravitational and calculable.

**Hybrids:** Combine CMB and weak lens shear vs redshift data. Fully utilize cross correlations on all scales, over-constraining models of dark energy.

Weak gravitational lensing maps source galaxies to new positions on the sky, systematically distorting their images. The resulting shear *γ* of their images is related to the projected foreground mass contrast inside an angular radius θ: *γ _{t}* =<κ(<θ)> -κ(θ) where

If the galaxies can be separated into n multiple redshift bins, then we can create n shear maps. A most interesting statistical property of these maps are the two-point functions. These n(n+1)/2 unique shear power spectra can be written as projections of the matter power spectrum along the line of sight out to some redshift. Jointly, these correlations contain enough information to break degeneracies and determine cosmological parameters, including dark energy parameters.

Weak lensing has high information content. A big advantage for WL over CMB is *Tomography*. The rich cosmological harvest from CMB comes from a single redshift of 1100, but because dark energy creates a change in observables as a function of redshift, CMB data alone is relatively insensitive to dark energy model parameters. If we know distances to source galaxies, the mass distribution and cosmic geometry can be measured *as a function of redshift*. WL thus has sensitivity to evolution of dark energy. Weak lensing can be pursued from the ground because it is already known that the maximal effects of dark energy occur at modest redshift around 0.5, and galaxies at z=0.5 are well resolved in deep optical imaging. The gravitational light deflection by a foreground mass is given by a product of the mass (enclosed inside the impact radius) and a dimensionless ratio of distances. Both of these terms are affected by dark energy and other cosmological parameters (see Figure 1). Combining with CMB data brakes degeneracies and enables separate direct investigations of the growth of dark matter structure and multiple probes of the geometry from z~1 to present. Another advantage over CMB is that recent and/or small-scale fluctuations are *non-Gaussian*, so there is information present beyond the power spectrum. Research is in progress to see how much cosmological information can be extracted from higher-order behavior: N-point correlations, bispectrum, etc. For all these data analysis techniques a large number of source galaxies and coverage of a significant fraction of the sky are crucial.

The number of clusters of mass, and the distribution with redshift, is an exponentially sensitive probe of dark energy. Because of the exponential sensitivity to mass it is important to avoid mass proxies. Other techniques for surveying for clusters are baryon biased; this bias is complex and scale dependent (Majumdar & Mohr 2003). Cluster counting from lensing is much cleaner than other methods (x-rays, Sunyaev-Zeldovich) because mass is measured directly, no baryonic astrophysics enters. Non-Gaussian signatures give constraints on cosmological parameters that are complementary to those from power-spectrum, breaking degeneracies. There is yet another advantage over CMB: Cross-correlation of WL and visible matter can reveal the relation between dark matter and the visible Universe.

Weak lensing measurements on small scales are limited by the noise from the intrinsic ellipticities of the background galaxies. To average down this shot noise one wants deep images to maximize the number of resolved galaxies per unit area on the sky. On larger scales, one is limited by cosmic variance; we therefore need to maximize the sky coverage of the survey. A very deep survey of a significant fraction of the visible sky in enough wavelength bands to allow photometric redshifts, will enable multiple cosmological tests to be carried out. No other existing or planned observatory would be capable of such a survey.

The dark energy equation of state affects the comoving volume, the evolution of structure on all scales (cosmic shear), and the tail of the mass distribution (massive cluster counts versus redshift). Complementary analyses of the same shear data probe the underlying geometry and its evolution over cosmic time. 2-D cosmic shear can be extended to 3-D by using distances to the source galaxies based on their colors, so-called "photometric redshifts." An all sky deep shear survey with photometric redshifts captures all these cosmological effects and they may be measured and analyzed separately. Thus several complementary approaches will be used in analyzing the shear and color-redshift data:

The number density of clusters as a function of redshift ("cluster tomography"). Individual massive clusters provide shear signals large enough to be easily detectable in LSST mass maps over a redshift range 0.1 - 1, bracketing the redshift of maximal dark energy effects. The number of these clusters depends exponentially on the amplitude of density fluctuations and on the cosmological distance scale, which sets the volume-redshift relation. Both of these inputs depend on the details of dark energy (Haiman, Mohr, & Holder 2001; Tyson 2003). Hence, counts of mass clusters constrains the equation of state of dark energy.

With a sky coverage of 20,000 square degrees, the LSST is expected to find about 300,000 clusters in its mass maps; a cluster sample of this size offers 2-3% precision on the equation of state parameter w if a degeneracy between w and Ω_{m} is broken using other probes (Tyson et al 2003). Using shear data for source galaxies at different redshifts, the detected cluster redshift can be estimated in a baryon independent way, in addition to mapping the cluster mass on the plane of the sky. This 3-D technique has been developed recently (Wittman et al 2003), and LSST can apply it to the whole visible sky. A large area is needed to reach the required statistical precision.

There is information beyond tangential shear. The shear is a spin-2 field and consequently we can measure two independent ellipticity correlation functions. The lensing signal is caused by a scalar gravitational potential in the lens and therefore should be curl-free. We can project each correlation function into one that measures the divergence and one that measures the curl: E-B mode decomposition. While we cannot correct the E-mode data with the B-mode, the existence of a B-mode is a useful diagnostic of systematic shear error. The current generation of weak lens surveys suffer some degree of both of these systematics. The good news is that tests show that the new generation of optical telescopes provide sufficient PSF control to eliminate optics induced B-modes. By using color-redshifts of sources to eliminate pairs of sources nearby in redshift the B-mode contamination of shear-shear cospectra is reduced to insignificant levels in current surveys. As a check, the LSST lensing pipeline software will deliver B-mode diagnostics live.

Financial support for Rubin Observatory comes from the National Science Foundation (NSF) through Cooperative Agreement No. 1258333, the Department of Energy (DOE) Office of Science under Contract No. DE-AC02-76SF00515, and private funding raised by the LSST Corporation. The NSF-funded Rubin Observatory Project Office for construction was established as an operating center under management of the Association of Universities for Research in Astronomy (AURA). The DOE-funded effort to build the Rubin Observatory LSST Camera (LSSTCam) is managed by the SLAC National Accelerator Laboratory (SLAC).

The National Science Foundation (NSF) is an
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NSF and DOE will continue to support Rubin Observatory in its Operations phase. They will also provide support for scientific research with LSST data.

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