Despite being an integral over time, this expression gives the correct distance that would be measured by a hypothetical tape measure at fixed time t, i.e. the "proper distance" (as defined below) after accounting for the time-dependent comoving speed of light via the inverse scale factor term 1 / a ( t ′ ) in the integrand. By "comoving speed of light", we mean the velocity of light through comoving coordinates [ c / a ( t ′ ) ] which is time-dependent even though locally, at any point along the null geodesic of the light particles, an observer in an inertial frame always measures the speed of light as c in accordance with special relativity. For a derivation see "Appendix A: Standard general relativistic definitions of expansion and horizons" from Davis & Lineweaver 2004. In particular, see eqs. 16–22 in the referenced 2004 paper [note: in that paper the scale factor R ( t ′ ) is defined as a quantity with the dimension of distance while the radial coordinate χ χ is dimensionless.]