  In the search for the mechanism of high-temperature superconductivity,
intense research has been focused on the evolution of the spin excitation
spectrum upon doping from the antiferromagnetic insulating to the
superconducting states of the cuprates. Because of technical limitations, the
experimental investigation of doped cuprates has been largely focused on
low-energy excitations in a small range of momentum space. Here we use resonant
inelastic x-ray scattering to show that a large family of superconductors,
encompassing underdoped YBa$_2$Cu$_4$O$_8$ and overdoped YBa$_2$Cu$_3$O$_{7}$,
exhibits damped spin excitations (paramagnons) with dispersions and spectral
weights closely similar to those of magnons in undoped cuprates. %The results
are in excellent agreement with the spin excitations obtained by exact
diagonalization of the $\bf t-J$ Hamiltonian on finite-sized clusters. The
comprehensive experimental description of this surprisingly simple spectrum
permits quantitative tests of magnetic Cooper pairing models. A numerical
solution of the Eliashberg equations for the magnetic spectrum of
YBa$_2$Cu$_3$O$_{7}$ reproduces its superconducting transition temperature
within a factor of two, a level of agreement comparable to Eliashberg theories
of conventional superconductors.
