Inflationary cosmology is now considered as the standard theory to naturally provide the desired initial conditions for the subsequent standard hot big bang cosmology. Not only making the observable universe homogeneous and isotropic, inflation also generates quantum fluctuations which are believed to have become the seeds of all the observed large-scale structures in the universe.
We review first the basic of inflation and see how inflation can solve the problem of the initial conditions necessary for successfully reproducing the observed universe according to the hot big bang theory. Primordial perturbations generated from quantum fluctuations during inflation are discussed as the origin of inhomogeneities of the present universe. The power spectrum of these perturbations opens an window to the early universe where the energy scale is so high that we cannot investigate with current particle accelerator, and we summarise an useful formalism to calculate the power spectrum in a systematic manner.
These perturbations experience gravitational interaction when inflation ends and they reenter the horizon. The matter and radiation perturbations induced by the primordial ones behave very differently, and we find that they are intimately related to each other. While matter inhomogeneities continue to increase throughout the history of the universe by gravitational instability, radiation perturbations hardly grow because of large pressure. These perturbations are observable as the power spectra, and we study how to calculate them to compare with observations.