  A central limit theorem is shown for moderately interacting particles in the
whole space. The interaction potential approximates singular attractive or
repulsive potentials of sub-Coulomb type. It is proved that the fluctuations
become asymptotically Gaussians in the limit of infinitely many particles. The
methodology is inspired by the classical work of Oelschl\"ager on fluctuations
for the porous-medium equation. The novelty in this work is that we can allow
for attractive potentials in the moderate regime and still obtain asymptotic
Gaussian fluctuations. The key element of the proof is the mean-square
convergence in expectation for smoothed empirical measures associated to
moderately interacting $N$-particle systems with rate $N^{-1/2-\varepsilon}$
for some $\varepsilon>0$. To allow for attractive potentials, the proof uses a
quantitative mean-field convergence in probability with any algebraic rate and
a law-of-large-numbers estimate as well as a systematic separation of the terms
to be estimated in a mean-field part and a law-of-large-numbers part.
