  According to the CMB observations, Mielczarek (\cite{Mielczarek}) evaluated
the reheating temperature, which could help to determine the history of the
Universe. In this paper, we recalculate the reheating temperature using the new
data from WMAP 7 observations. Based on that, we list the approximate solutions
of relic gravitational waves (RGWs) for various frequency bands. With the
combination of the quantum normalization of RGWs when they are produced and the
CMB observations, we obtain the relation between the tensor-to-scalar ratio $r$
and the inflation index $\beta$ for a given scalar spectral index $n_s$. As a
comparison, the diagram $r-\beta$ in the slow-roll inflation model is also
given. Thus, the observational limits of $r$ from CMB lead to the constraints
on the value of $\beta$. Then, we illustrate the energy density spectrum of
RGWs with the quantum normalization for different values of $r$ and the
corresponding $\beta$. For comparison, the energy density spectra of RGWs with
parameters based on slow-roll inflation are also discussed. We find that the
values of $n_s$ affect the spectra of RGWs sensitively in the very high
frequencies. Based on the current and planed gravitational wave detectors, we
discuss the detectabilities of RGWs.
