  We establish a quantitative relationship between photometric and
spectroscopic detections of solar-like oscillations using ab initio,
three-dimensional (3D), hydrodynamical numerical simulations of stellar
atmospheres. We present a theoretical derivation as proof of concept for our
method. We perform realistic spectral line formation calculations to quantify
the ratio between luminosity and radial velocity amplitude for two case
studies: the Sun and the red giant $\epsilon$ Tau. Luminosity amplitudes are
computed based on the bolometric flux predicted by 3D simulations with
granulation background modelled the same way as asteroseismic observations.
Radial velocity amplitudes are determined from the wavelength shift of
synthesized spectral lines with methods closely resembling those used in BiSON
and SONG observations. Consequently, the theoretical luminosity to radial
velocity amplitude ratios are directly comparable with corresponding
observations. For the Sun, we predict theoretical ratios of 21.0 and 23.7
ppm/[m/s] from BiSON and SONG respectively, in good agreement with observations
19.1 and 21.6 ppm/[m/s]. For $\epsilon$ Tau, we predict K2 and SONG ratios of
48.4 ppm/[m/s], again in good agreement with observations 42.2 ppm/[m/s], and
much improved over the result from conventional empirical scaling relations
which gives 23.2 ppm/[m/s]. This study thus opens the path towards a
quantitative understanding of solar-like oscillations, via detailed modelling
of 3D stellar atmospheres.
