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.
