The Impact of Nonlinear Structure Formation on the Power Spectrum of
  Transverse Momentum Fluctuations and the Kinetic Sunyaev-Zel'dovich Effect
  Cosmological transverse momentum fields, whose directions are perpendicular
to Fourier wave vectors, induce temperature anisotropies in the cosmic
microwave background via the kinetic Sunyaev-Zeldovich (kSZ) effect. The
transverse momentum power spectrum contains the four-point function of density
and velocity fields, $\langle\delta\delta v v\rangle$. In the post-reionization
epoch, nonlinear effects dominate in the power spectrum. We use perturbation
theory and cosmological $N$-body simulations to calculate this nonlinearity. We
derive the next-to-leading order expression for the power spectrum with a
particular emphasis on the connected term that has been ignored in the
literature. While the contribution from the connected term on small scales
($k>0.1\,h\,\rm{Mpc}^{-1}$) is subdominant relative to the unconnected term, we
find that its contribution to the kSZ power spectrum at $\ell = 3000$ at $z<6$
can be as large as ten percent of the unconnected term, which would reduce the
allowed contribution from the reionization epoch ($z>6$) by twenty percent. The
power spectrum of transverse momentum on large scales is expected to scale as
$k^2$ as a consequence of momentum conservation. We show that both the leading
and the next-to-leading order terms satisfy this scaling. In particular, we
find that both of the unconnected and connected terms are necessary to
reproduce $k^2$.
