Scalar tachyonic instabilities in gravitational backgrounds: Existence and growth rate

It is well known that the Klein Gordon (KG) equation $\Box \Phi + m^2\Phi=0$ has tachyonic unstable modes on large scales ($k^2<\vert m \vert^2$) for $m^2<m_{cr}^2=0$ in a flat Minkowski spacetime with maximum growth rate $\Omega_{F}(m)= \vert m \vert$ achieved at $k=0$. We investigate these instabilities in a Reissner-Nordstr\"om-deSitter (RN-dS) background spacetime with mass $M$, charge $Q$, cosmological constant $\Lambda>0$ and multiple horizons. By solving the KG equation in the range between the event and cosmological horizons, using tortoise coordinates $r_*$, we identify the bound states of the emerging Schrodinger-like Regge-Wheeler equation corresponding to instabilities. We find that the critical value $m_{cr}$ such that for $m^2<m_{cr}^2$ bound states and instabilities appear, remains equal to the flat space value $m_{cr}=0$ for all values of background metric parameters despite the locally negative nature of the Regge-Wheeler potential for $m=0$. However, the growth rate $\Omega$ of tachyonic instabilities for $m^2<0$ gets significantly reduced compared to the flat case for all parameter values of the background metric ($\Omega(Q/M,M^2 \Lambda, mM)< \vert m \vert$). This increased lifetime of tachyonic instabilities is maximal in the case of a near extreme Schwarzschild-deSitter (SdS) black hole where $Q=0$ and the cosmological horizon is nearly equal to the event horizon ($\xi \equiv 9M^2 \Lambda \simeq 1$). The physical reason for this delay of instability growth appears to be the existence of a cosmological horizon that tends to narrow the negative range of the Regge-Wheeler potential in tortoise coordinates.