Fluctuation of the Hubble parameter

We study the Hubble parameter $H(z)$ in perturbed Friedmann universe and obtain an expression of the perturbed Hubble parameter $H(z,\mathbf{n})$. We derive the Hubble parameter power spectrum by using the initial spectrum during inflation and the Bardeen transfer function. We obtain a semianalytical expression in the case of cold dark matter universe. Similar with luminosity distance, the Hubble parameter spectrum is suggested to be a useful observational tool to determine some cosmological parameters. In addition, we show that the Hubble parameter power spectrum could be used to check whether the expansion is accelerated by the second order small scale fluctuation.