Constraining warm dark matter mass with cosmic reionization and gravitational wave

We constrain the warm dark matter (WDM) particle mass with the observations of cosmic reionization and CMB optical depth. We suggest that the GWs from stellar mass black holes (BHs) could give a further constraint on WDM particle mass for future observations. The star formation rates (SFRs) of Population I/II (Pop I/II) and Population III (Pop III) stars are also derived. If the metallicity of the universe have been enriched beyond the critical value of $Z_{\rm crit}=10^{-3.5}Z_{\odot}$, the star formation shift from Pop III to Pop I/II stars. Our results show that the SFRs are quite dependent on the WDM particle mass, especially at high redshifts. Combing with the reionization history and CMB optical depth derived from the recent \emph{Planck} mission, we find that the current data requires the WDM particle mass in a narrow range of $1 < m_x < 3$ keV. Furthermore, we suggest that the stochastic gravitational wave background (SGWB) produced by stellar BHs could give a further constraint on the WDM particle mass for future observations. For $m_{x}=3$ keV with Salpeter (Chabrier) initial mass function (IMF), the SGWB from Pop I/II BHs has a peak amplitude of $\Omega_{\rm GW}\approx2.8\times 10^{-9}~(5.0\times 10^{-9})$ at $f= 316 {\rm Hz}$, while the GW radiation at $f<10$Hz is seriously suppressed. For $m_{\rm x}=1$ keV, the SGWB peak amplitude is the same as that of $m_{\rm x}=1$ keV, but a little lower at low frequencies. Therefore, it is hard to constrain the WDM particle mass by the SGWB from Pop I/II BHs. To assess the detectability of GW signal, we also calculate the signal to noise ratio (SNR), which are $\rm SNR=37.7~ (66.5)$ and $27~(47.7)$ for $m_{\rm x}=3$ keV and $m_{\rm x}=1$ keV for Einstein Telescope (ET) with Salpeter (Chabrier) IMF, respectively.