  Dilepton invariant-mass spectra for heavy-ion collisions at SIS 18 and
BEVALAC energies are calculated using a coarse-grained time evolution from the
Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model. The
coarse-graining of the microscopic simulations enables to calculate thermal
dilepton emission rates by application of in-medium spectral functions from
equilibrium quantum-field theoretical calculations. The results show that
extremely high baryon chemical potentials dominate the evolution of the created
hot and dense fireball. Consequently, a significant modification of the $\rho$
spectral shape becomes visible in the dilepton invariant-mass spectrum,
resulting in an enhancement in the low-mass region $M_{ee} = 200$ to 600
MeV/$c^{2}$. This enhancement, mainly caused by baryonic effects on the $\rho$
spectral shape, can fully describe the experimentally observed excess above the
hadronic cocktail contributions in Ar+KCl ($E_{\mathrm{lab}}=1.76$ $A$GeV)
reactions as measured by the HADES collaboration and also gives a good
explanation of the older DLS Ca+Ca ($E_{\mathrm{lab}}=1.04$ $A$GeV) data. For
the larger Au+Au ($E_{\mathrm{lab}}=1.23$ $A$GeV) system, we predict an even
stronger excess from our calculations. A systematic comparison of the results
for different system sizes from C+C to Au+Au shows that the thermal dilepton
yield increases stronger ($\propto A^{4/3}$) than the hadronic background
contributions, which scale with $A$, due to its sensitivity on the time
evolution of the reaction. We stress that the findings of the present work are
consistent with our previous coarse-graining results for the NA60 measurements
at top SPS energy. We argue that it is possible to describe the dilepton
results from SIS 18 up to SPS energies by considering the modifications of the
$\rho$ spectral function inside a hot and dense medium within the same model.
