Chaotic Inflation with Variable Space Dimensions

Assuming the space dimension is not constant but decreases during the expansion of the Universe, we study chaotic inflation with the potential $m^2 \phi^2/2$. We write down field equations in the slow-roll approximation and define slow -roll parameters by assuming the space dimension to be a dynamical parameter. The dynamical character of the space dimension shifts the initial and final value of the inflaton field to larger values, producing delayed chaotic inflation. We obtain an upper limit for the space dimension at the Planck length. This result is in agreement with previous works on the effective time variation of the Newtonian gravitational constant in a model Universe with variable space dimensions. We present some cosmological consequences and calculate observable quantities including the spectral indices, their scale-dependence, and the mass of the inflaton field.Comment: 4 pages, 2 figures, to appear in the proceedings of the CAPP2000, eds. Ruth Durrer, Juan Garcia-Bellido, and Mikhail Shaposhnikov, Verbier (Switzerland), July 17-28, 200

General Relativistic Relation Between Density Contrast and Peculiar Velocity

Concepts like peculiar velocity, gravitational force, and power spectrum and their interrelationships are of utmost importance in the theories of structure formation. The observational implementation of these concepts is usually based on the Newtonian hydrodynamic equations, but used up to scales where general relativistic effects come in. Using a perturbation of FRW metric in harmonic gauge, we show that the relativistic effects reduce to light cone effects including the expansion of the universe. Within the Newtonian gravitation, the linear perturbation theory of large scale structure formation predicts the peculiar velocity field to be directly proportional to gravitational force due to the matter distribution. The corresponding relation between peculiar velocity field and density contrast has been given by Peebles. Using the general relativistic perturbation we have developed, this familiar relation is modified by doing the calculation on the light cone in contrast to the usual procedure of taking a space-like slice defined at a definite time. The velocity and density--spectrum are compared to the familiar Newtonian expressions. In particular, the relativistic $\beta$--value obtained is reduced and leads to an increased bias factor or a decreased expected amount of the dark matter in a cluster.Comment: 14 pages, 1 postscript figur