Chaotic Inflation With Time-Variable Space Dimensions

Abstract

Assuming the space dimension is not constant but decreases during the expansion of the Universe, we study chaotic inflation with the potential m 2 φ 2 /2. Our investigations are based on a model Universe with variable space dimensions. We write down field equations in the slow-roll approximation, and define slow-roll parameters by assuming the number of space dimensions decreases continuously as the Universe expands. The dynamical character of the space dimension shifts the initial and final value of the inflaton field to larger values. We obtain an upper limit for the space dimension at the Planck length. This result is in agreement with previous works for the effective time variation of the Newtonian gravitational constant in a model Universe with variable space dimensions.

Dynamics of chaotic inflation with variable space dimension

“…The reader can find the answers of these questions in detail in Refs. [2,3]. It is worth mentioning that for generalization of Friedmann equations from three-space to variable space dimension, we cannot define a metric approach.…”

“…As shown in [2], for different values of C, corresponding to the space dimension at the Planck length D P = 4, 10, 25 and +∞, we calculated the e-folding number N ∼ 69, 98, 116 and 132, respectively. Using the number of e-foldings, we have also obtained the size of the Universe at the onset of inflation, a i , see Table 1.…”

“…[2]. As discussed in [2], constraints on the dimensionality of the space are derived from requiring that the radius of the space at the onset of inflation, a i , be larger than the Planck length, and than the minimum size of the model, which is δ. Particularly, for D P = 25 and +∞, we have a i < δ < l P , and for D P = 10, δ < a i < l P .…”

“…It should be noticed that the values of Table 2 are given for D P = 3, 4, 10 and not for D P = 25 or so. Since for D P ≥ 25 we have a i < δ < l P , we rule out D P ≥ 25 [2].…”

“…Interest in a speculative model in which the number of space dimension decreases continuously as the Universe expands has increased during the past few years [1][2][3][4][5].…”