The Starobinsky model predicts a primordial inflation period without the presence of an inflaton field. The modified version of this model predicts a simple time dependence for the Hubble parameter H(t), which decreases slowly between the Planck epoch and the end of the inflation, H(t) = M Pl -betaM²Pl t, where beta is a dimensionless constant to be adjusted from observations. We investigate an inflaton model which has the same time dependence for H(t). A reverse engineered inflaton potential for the time dependence of H is derived. Normalization of the derived inflaton potential is determined by the condition that the observed density fluctuations, <FONT FACE=Symbol>dr</FONT>/<FONT FACE=Symbol>r »</FONT> 10-5, are created at ~ 60 e-folds before the end of inflation. The derived potential indicates an energy (mass) scale, Mend ~ 10(13) GeV, at the end of inflation. Using the slow roll parameters, which are obtained from this potential, we calculate the spectral index for the scalar modes nS and the relative amplitude of the tensor to scalar modes r. A tensor contribution, r ~ 0.13, and an approximately Harrison-Zeldovich density perturbation spectrum, nS ~ 0.95, are predicted.

We investigate models that suggest that the vacuum energy could decay into cold dark matter (CDM) or into a homogeneous distribution of thermalized cosmic microwave background (CMB) photons. We show that the agreement of the density fluctuation spectra obtained from the CMB and galaxy distribution data puts strong limits on the rate of vacuum energy decay. A vacuum energy decaying into CDM increases the density of the CDM rho, diluting the CDM density fluctuations (<FONT FACE=Symbol>dr</FONT>/rho)². The temperature fluctuations of the CMB photons (deltaT/T)² are approximately proportional to (<FONT FACE=Symbol>dr</FONT>/rho)², at the recombination epoch. We define F as the predicted increase of (<FONT FACE=Symbol>dr</FONT>/rho)² (or (deltaT/T)²) at the recombination epoch. Since the present observed (<FONT FACE=Symbol>dr</FONT>/rho)² derived from the CMB and galaxy distribution data agree to ~ 10%, the maximum value for F is Fmax <FONT FACE=Symbol>@</FONT> 1.1. Our results indicate that the rate of decay of the vacuum energy into CDM or CMB photons is extremely small.