  Broadband secondary-eclipse measurements of hot Jupiters have indicated the
existence of atmospheric thermal inversions, but their presence is difficult to
determine from broadband measurements because of degeneracies between molecular
abundances and temperature structure. We apply high-resolution (R = 100 000)
infrared spectroscopy to probe the temperature-pressure profile of HD 209458 b.
This bright, transiting hot-Jupiter has long been considered the gold standard
for a hot Jupiter with an inversion layer, but this has been challenged in
recent publications. We observed the thermal dayside emission of HD 209458 b
with CRIRES / VLT during three nights, targeting the carbon monoxide band at
2.3 microns. Thermal inversions give rise to emission features, which means
that detecting emission lines in the planetary spectrum, as opposed to
absorption lines, would be direct evidence of a region in which the temperature
increases with altitude.
  We do not detect any significant absorption or emission of CO in the dayside
spectrum of HD 209458 b, although cross-correlation with template spectra
either with CO absorption lines or with weak emission at the core of the lines
show a low-significance correlation signal with a signal-to-noise ratio of 3 -
3.5. Models with strong CO emission lines show a weak anti-correlation with
similar or lower significance levels. Furthermore, we found no evidence of
absorption or emission from H2O at these wavelengths.
  The non-detection of CO in the dayside spectrum of HD 209458 b is interesting
in light of a previous CO detection in the transmission spectrum. That there is
no signal indicates that HD 209458 b either has a nearly isothermal atmosphere
or that the signal is heavily muted. Assuming a clear atmosphere, we can rule
out a full-disc dayside inversion layer in the pressure range 1 bar to 1 mbar.
