Why there is no Newtonian backreaction
  In the conventional framework for cosmological dynamics the scale factor
$a(t)$ is assumed to obey the `background' Friedmann equation for a perfectly
homogeneous universe while particles move according to equations of motions
driven by the gravity of the density fluctuations. It has recently been
suggested that the emergence of structure modifies the evolution of $a(t)$ via
Newtonian (or `kinematic') backreaction and that this may avoid the need for
dark energy. Here we point out that the conventional system of equations is
exact in Newtonian gravity and there is no approximation in the use of the
homogeneous universe equation for $a(t)$. The recently proposed modification of
Racz et al.\ (2017) does not reduce to Newtonian gravity in the limit of low
velocities. We discuss the relation of this to the `generalised Friedmann
equation' of Buchert and Ehlers. These are quite different things; their
formula describes individual regions and is obtained under the restrictive
assumption that the matter behaves like a pressure-free fluid whereas our
result is exact for collisionless dynamics and is an auxiliary relation
appearing in the structure equations. We use the symmetry of the general
velocity autocorrelation function to show how Buchert's $\cal Q$ tends very
rapidly to zero for large volume and that this does not simply arise `by
construction' through the adoption of periodic boundary conditions as has been
claimed. We conclude that, to the extent that Newtonian gravity accurately
describes the low-$z$ universe, there is no backreaction of structure on $a(t)$
and that the need for dark energy cannot be avoided in this way.
