We analyse the implications for inflationary models of the cosmic microwave background ( cmb ) anisotropy measured by COBE .
Vacuum fluctuations during inflation generate an adiabatic density perturbation , and also gravitational waves .
The ratio of these two contributions to the cmb anisotropy is given for an arbitrary slow-roll inflaton potential .
Results from the IRAS/QDOT and POTENT galaxy surveys are used to normalise the spectrum of the density perturbation on the scale 20 h ^ { -1 } \hbox { Mpc } , so that the COBE measurement on the scale 10 ^ { 3 } h ^ { -1 } \hbox { Mpc } provides a lower bound on the spectral index n .
For ‘ power law ’ and ‘ extended ’ inflation , gravitational waves are significant and the bound is n > 0.84 at the 2 -sigma level .
For ‘ natural ’ inflation , gravitational waves are negligible and the constraint is weakened to n > 0.70 , at best marginally consistent with a recent proposal for explaining the excess clustering observed in the APM galaxy survey .
Many versions of extended inflation , including those based on the Brans–Dicke theory , are ruled out , because they require n \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ < $ } } 0.75 in order that bubbles formed at the end of inflation should not be observed now in the cmb .