  String scenarios generically predict that we live in a so called axiverse:
the Universe with about a hundred of light axion species which are decoupled
from the Standard Model particles. However, the axions can couple to the
inflaton which leads to their production after inflation. Then, these axions
remain in the expanding Universe contributing to the dark radiation component,
which is severely bounded from present cosmological data. We place a general
constraint on the axion production rate and apply it to several variants of
reasonable inflaton-to-axion couplings. The limit merely constrains the number
of ultralight axions and the relative strength of inflaton-to-axion coupling.
It is valid in both large and small field inflationary models irrespectively of
the axion energy scales and masses. Thus, the limit is complementary to those
associated with the Universe overclosure and axion isocurvature fluctuations.
In particular, a hundred of axions is forbidden if inflaton universally couples
to all the fields at reheating. In the case of gravitational sector being
responsible for the reheating of the Universe (which is a natural option in all
inflationary models with modified gravity), the axion production can be
efficient. We find that in the Starobinsky $R^2$-inflation even a single axion
(e.g. the standard QCD-axion) is in tension with the Planck data, making the
model inconsistent with the axiverse. The general conclusion is that an
inflation with inefficient reheating mechanism and low reheating temperature
may be in tension with the presence of light scalars.
