  Recently, the Hydrogen Epoch of Reionization Array (HERA) collaboration has
produced the experiment's first upper limits on the power spectrum of 21-cm
fluctuations at z~8 and 10. Here, we use several independent theoretical models
to infer constraints on the intergalactic medium (IGM) and galaxies during the
epoch of reionization (EoR) from these limits. We find that the IGM must have
been heated above the adiabatic cooling threshold by z~8, independent of
uncertainties about the IGM ionization state and the nature of the radio
background. Combining HERA limits with galaxy and EoR observations constrains
the spin temperature of the z~8 neutral IGM to 27 K < T_S < 630 K (2.3 K < T_S
< 640 K) at 68% (95%) confidence. They therefore also place a lower bound on
X-ray heating, a previously unconstrained aspects of early galaxies. For
example, if the CMB dominates the z~8 radio background, the new HERA limits
imply that the first galaxies produced X-rays more efficiently than local ones
(with soft band X-ray luminosities per star formation rate constrained to
L_X/SFR = { 10^40.2, 10^41.9 } erg/s/(M_sun/yr) at 68% confidence), consistent
with expectations of X-ray binaries in low-metallicity environments. The z~10
limits require even earlier heating if dark-matter interactions (e.g., through
millicharges) cool down the hydrogen gas. Using a model in which an extra radio
background is produced by galaxies, we rule out (at 95% confidence) the
combination of high radio and low X-ray luminosities of L_{r,\nu}/SFR > 3.9 x
10^24 W/Hz/(M_sun/yr) and L_X/SFR<10^40 erg/s/(M_sun/yr). The new HERA upper
limits neither support nor disfavor a cosmological interpretation of the recent
EDGES detection. The analysis framework described here provides a foundation
for the interpretation of future HERA results.
