  The Peters formula, which tells how the coalescence time of a binary system
emitting gravitational radiation is determined by the initial size and shape of
the elliptic orbit, is often used in estimating the merger rate of primordial
black holes and the gravitational wave background from the mergers. Valid as it
is in some interesting scenarios, such as the analysis of the LIGO-Virgo
events, the Peters formula fails to describe the coalescence time if the
orbital period of the binary exceeds the value given by the formula. This could
underestimate the event rate of mergers that occur before the cosmic time
$t\sim 10^{13}\ \text{s}$. As a result, the energy density spectrum of the
gravitational wave background could develop a peak, which is from mergers
occurring at either $t\sim 10^{13}\ \text{s}$ (for black holes with mass
$M\gtrsim 10^8 M_\odot$) or $t\sim 10^{26}(M/M_\odot)^{-5/3}\ \text{s}$ (for
$10^5 M_\odot \lesssim M\lesssim 10^8 M_\odot$). This can be used to constrain
the fraction of dark matter in primordial black holes (denoted by $f$) if
potential probes (such as SKA and U-DECIGO) do not discover such a background,
with the result $f\lesssim 10^{-6}\text{-}10^{-4}$ for the mass range
$10\text{-} 10^9M_\odot$. We then consider the effect of mass accretion onto
primordial black holes at redshift $z\sim 10$, and find that the merger rate
could drop significantly at low redshifts. The spectrum of the gravitational
wave background thus gets suppressed at the high-frequency end. This feature
might be captured by future detectors such as ET and CE for initial mass $M=
\mathcal{O}(10\text{-}100) M_\odot$ with $f\gtrsim 10^{-4}$.
