Isotropic non-Gaussian $g_\mathrm{NL}$-like toy models that reproduce
  the cosmic microwave background anomalies
  Based on recent observations of the cosmic microwave background (CMB), claims
of statistical anomalies in the properties of the CMB fluctuations have been
made. Although the statistical significance of the anomalies remains only at
the $\sim2-3\sigma$ significance level, the fact that there are many different
anomalies, several of which support a possible deviation from statistical
isotropy, has motivated a search for models that provide a common mechanism to
generate them. The goal of this paper is to investigate whether these anomalies
could originate from non-Gaussian cosmological models, and to determine what
properties these models should have. We present a simple isotropic,
non-Gaussian class of toy models that can reproduce six of the most extensively
studied anomalies. We compare the presence of anomalies found in simulated maps
generated from the toy models and from a standard model with Gaussian
fluctuations. We show that the following anomalies, as found in the Planck
data, commonly occur in the toy model maps: (1) large-scale hemispherical
asymmetry (large-scale dipolar modulation), (2) small-scale hemispherical
asymmetry (alignment of the spatial distribution of CMB power over all scales
$\ell=[2,1500]$) , (3) a strongly non-Gaussian hot or cold spot, (4) a low
power spectrum amplitude for $\ell<30$, including specifically (5) a low
quadrupole and an unusual alignment between the quadrupole and the octopole,
and (6) parity asymmetry of the lowest multipoles. We note that this class of
toy model resembles models of primordial non-Gaussianity characterised by
strongly scale-dependent $g_{NL}$-like trispectra.
