  We deduce the normal state angle-resolved single-particle self-energy
$\Sigma(\theta, \omega)$ and the Eliashberg function (i.e., the product of the
fluctuation spectrum and its coupling to fermions) $\alpha^2 F(\theta,\omega)$
for the high temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ from
the ultra high resolution laser angle-resolved photoemission spectroscopy
(ARPES). The self-energy $\Sigma(\theta, \omega)$ at energy $\omega$ along
several cuts normal to the Fermi surface at the tilt angles $\theta$ with
respect to the nodal direction in a slightly underdoped
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ were extracted by fitting the ARPES momentum
distribution curves. Then, using the extracted self-energy as the experimental
input, the $\alpha^2 F(\theta,\omega)$ is deduced by inverting the Eliashberg
equation employing the adaptive maximum entropy method. Our principal new
result is that the Eliashberg function $\alpha^2F(\theta,\omega)$ collapse for
all $\theta$ onto a single function of $\omega$ up to the upper cut-off energy
despite the $\theta$ dependence of the self-energy. The in-plane momentum
anisotropy is therefore predominantly due to the anisotropic band dispersion
effects. The obtained Eliashberg function has a small peak at
$\omega\approx0.05$ eV and flattens out above 0.1 eV up to the angle-dependent
cut-off. It takes the intrinsic cut-off of about 0.4 eV or the energy of the
bottom of the band with respect to the Fermi energy in the direction $\theta$,
whichever is lower. The angle independence of the $\alpha^2 F(\theta,\omega)$
is consistent only with the fluctuation spectra which have the short
correlation length on the scale the lattice constant. This implies among others
that the antiferromagnetic fluctuations may not be underlying physics of the
deduced fluctuation spectrum.
