  The shapes of galaxy N-point correlation functions can be used as standard
rulers to constrain the distance-redshift relationship and thence the expansion
rate of the Universe. The cosmological density fields traced by late-time
galaxy formation are initially nearly Gaussian, and hence all the cosmological
information can be extracted from their 2-Point Correlation Function (2PCF) or
its Fourier-space analog the power spectrum. Subsequent nonlinear evolution
under gravity, as well as halo and then galaxy formation, generate higher-order
correlation functions. Since the mapping of the initial to the final density
field is, on large scales, invertible, it is often claimed that the information
content of the initial field's power spectrum is equal to that of all the
higher-order functions of the final, nonlinear field. This claim implies that
reconstruction of the initial density field from the nonlinear field renders
analysis of higher-order correlation functions of the latter superfluous. We
here show that this claim is false when the N-point functions are used as
standard rulers. Constraints available from joint analysis of the galaxy power
spectrum and bispectrum (Fourier-space analog of the 3-Point Correlation
Function) can, in some cases, exceed those offered by the initial power
spectrum even when the reconstruction is perfect. We provide a mathematical
justification for this claim and also demonstrate it using a large suite of
N-body simulations. In particular, we show that for the z = 0 real-space matter
field in the limit of vanishing shot noise, taking modes up to k_max = 0.2
h/Mpc, using the bispectrum alone offers a factor of two reduction in the
variance on the cosmic distance scale relative to that available from the power
spectrum.
