The primordial seeds of inhomogeneity, whose imprint is seen at last scattering in the CMB anisotropies, may be generated by quantum fluctuations during inflation in the very early universe. These seeds subsequently evolve from linear to nonlinear fluctuations via gravitational instability, and produce the large-scale matter distribution that is observed at lower redshifts. The previous chapter dealt with the CMB anisotropies. In this chapter we provide brief overviews of the primordial fluctuations from inflation, and then of the evolution of large-scale structure, as described via the power spectrum of matter. A key probe of the total matter (dark and baryonic) and its distribution is weak gravitational lensing by the large-scale structure of light from distant sources. We develop the theoretical framework for gravitational lensing and briefly describe how this is applied in cosmology. The following chapter will draw on this chapter and its predecessors to show how current observations constrain and describe the standard model of cosmology. We start with a summary of the statistical description of perturbations.
Correlation functions and power spectra
Perturbations on an FLRW background are treated as random variables in space at each time instant, and observations determine the statistical properties of these random distributions. (See Durrer (2008) for a more complete discussion.) A perturbative variable A(x) at some fixed time is associated with an ensemble of random functions, each with a probability assigned to it. We define the 2-point correlation function 〈A(x)A(x′)〉 as the average over the ensemble (incorporating the probability distribution).