  Evidence for almost spatial flatness of the Universe has been provided from
several observational probes, including the Cosmic Microwave Background (CMB)
and Baryon Acoustic Oscillations (BAO) from galaxy clustering data. However,
other than inflation, and in this case only in the limit of infinite time,
there is no strong a priori motivation for a spatially flat Universe. Using the
renormalization group (RG) technique in curved spacetime, we present in this
work a theoretical motivation for spatial flatness. Starting from a general
spacetime, the first step of the RG, coarse-graining, gives a
Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric with a set of parameters.
Then, we study the rescaling properties of the curvature parameter, and find
that zero spatial curvature of the FLRW metric is singled out as the unique
scale-free, non-singular background for cosmological perturbations.
