Our knowledge of inflation, or whatever process generated the initial departures from homogeneity, comes almost entirely through our inference of the power spectrum of these initial departures. The CMB has been a powerful probe, but there are two ways in which LSST can improve upon the CMB: by using cosmic shear to extend to smaller scales and by using the power spectrum of the galaxies themselves to improve the determination of the primordial power spectrum on very large scales.
It is difficult to use the CMB to probe the power spectrum on angular scales much smaller l ~1000, because below this scale photon diffusion smooths the temperature anisotropies on the last-scattering surface. In contrast, there is plenty of fluctuation power in cosmic shear out to much higher l values. In addition, at fixed l value cosmic shear power spectra are sensitive to the matter power spectrum on smaller scales than is the case for the CMB, since the three-dimensional fluctuations are being projected into angular fluctuations over a smaller distance. Any large departures from the expected power-law behavior of the primordial power spectrum occurring at scales within a decade of those constrained well by the CMB, should be detectable by LSST.
If one assumes that departures from a power-law occur only at the small levels expected from the simplest models of inflation, the impact of cosmic shear may not be as dramatic as the detection of a large departure from scale-invariance outlined above. Nevertheless, cosmic shear can add significant statistical weight to the data, improving the determination of spectral index nS and dnS/dlnk each by a factor of two over what can be done with Planck alone (Song and Knox 2004). This may be the crucial factor of two improvement in precision necessary to detect a non-zero dnS/dlnk at the expected level for the simplest models.
Galaxy power spectrum
Results from the WMAP have revealed a CMB temperature power spectrum that is remarkably well-fit by a simple 5 or 6-parameter model. The success of this simple model is somewhat qualified by irregularities on large scales. There is evidence for departures from statistical isotropy, an anomalously low quadrupole amplitude, an anomalous absence of correlations on angular scales greater than 60 degrees, non-Gaussianity, unexplained correlations with the ecliptic, and sharp features in the temperature power spectrum. While these irregularities may eventually be revealed to be due to improper modeling of the instrument or astrophysical foregrounds, they certainly have served to call attention to the possibility of interesting departures from the standard paradigm on the largest observable scales.
Very large-volume spectroscopic or photo-z redshift surveys can actually probe these very large scales very well. In fact, the galaxy power spectra not only provide a complementary look at structure on very large scales, but they can also determine the power spectrum on a given scale even more precisely than with CMB measurements. The reason is that in determining a power spectrum one is fundamentally limited by the number of modes on a given scale, and there are more such modes with a three-dimensional survey than with a two-dimensional survey.
Determining the power spectrum of the fluctuations on very large scales is of great interest for several reasons. First and foremost, the power spectrum on these scales can be cleanly used to infer the power spectrum of primordial fluctuations, and the primordial fluctuation power spectrum on any scale is of interest. The primordial power spectrum is one of our only handles on the mechanism that led to the fluctuations that are responsible for the diversity of structures we see in the Universe today, including ourselves.
In the context of inflation, probing larger scales means probing inflation at an earlier epoch. Larger scales are larger because they exited the fixed-size horizon earlier and thus had more time to expand. Reaching back to these earlier epochs may provide us with valuable clues about inflation.
On smaller scales use of galaxies to infer the matter power spectrum are tricky since the relationship between the contrast in the number density of galaxies and the contrast in the mass density is complicated by baryonic effects. However, these complications simplify greatly on large scales.
Estimates of the matter power spectrum P(k) on very large scales are robust to errors in modeling of galaxy-dark matter biasing as well as contamination from photometric redshift errors, and redshift distortions.
This estimation of the dark matter power spectrum from LSST galaxies has been explored recently (Zhan et al. 2005). They have shown that LSST determination of the large-scale galaxy power spectrum can be used to test a hypothesized primordial power spectrum used to explain certain irregular features in the WMAP large-scale temperature power spectrum.