The SFR-M$_*$ Correlation Extends to Low Mass at High Redshift
  To achieve a fuller understanding of galaxy evolution, SED fitting can be
used to recover quantities beyond stellar masses (M$_*$) and star formation
rates (SFRs). We use Star Formation Histories (SFHs) reconstructed via the
Dense Basis method of Iyer \& Gawiser (2017) for a sample of $17,873$ galaxies
at $0.5<z<6$ in the CANDELS GOODS-S field to study the nature and evolution of
the SFR-M$_*$ correlation. The reconstructed SFHs represent trajectories in
SFR-M$_*$ space, enabling us to study galaxies at epochs earlier than observed
by propagating them backwards in time along these trajectories. We study the
SFR-M$_*$ correlation at $z=1,2,3,4,5,6$ using both direct fits to galaxies
observed at those epochs and SFR-M$_*$ trajectories of galaxies observed at
lower redshifts. The SFR-M$_*$ correlations obtained using the two approaches
are found to be consistent with each other through a KS test. Validation tests
using SFHs from semi-analytic models and cosmological hydrodynamical
simulations confirm the sensitivity of the method to changes in the slope,
normalization and shape of the SFR-M$_*$ correlation. This technique allows us
to further probe the low-mass regime of the correlation at high-z by $\sim 1$
dex and over an effective volume of $\sim 10\times$ larger than possible with
just direct fits. We find that the SFR-M$_*$ correlation is consistent with
being linear down to M$_*\sim 10^7 M_\odot$ at $z>4$. The evolution of the
correlation is well described by $\log SFR= (0.80\pm 0.029 - 0.017\pm
0.010\times t_{univ})\log M_*$ $- (6.487\pm 0.282-0.039\pm 0.008\times
t_{univ})$, where $t_{univ}$ is the age of the universe in Gyr.
