Fundamental Cosmology from Precision Spectroscopy: II. Synergies with
  supernovae
  In previous work [Amendola {\it et al.}, Phys. Rev. D86 (2012) 063515],
Principal Component Analysis based methods to constrain the dark energy
equation of state using Type Ia supernovae and other low redshift probes were
extended to spectroscopic tests of the stability fundamental couplings, which
can probe higher redshifts. Here we use them to quantify the gains in
sensitivity obtained by combining spectroscopic measurements expected from
ESPRESSO at the VLT and the high-resolution ultra-stable spectrograph for the
E-ELT (known as ELT-HIRES) with future supernova surveys. In addition to
simulated low and intermediate redshift supernova surveys, we assess the dark
energy impact of high-redshift supernovas detected by JWST and characterized by
the E-ELT or TMT. Our results show that a detailed characterization of the dark
energy properties beyond the acceleration phase (i.e., deep in the matter era)
is viable, and may reach as deep as redshift 4.
