Zel’dovich Λ and Weinberg’s relation: an explanation for the cosmological coincidences

Abstract

In 1937 Dirac proposed the large number hypothesis (LNH). The idea was to explain that these numbers were large because the Universe is old. A time variation of certain “constants” was assumed. So far, no experimental evidence has significantly supported this time variation. Here we present a simplified cosmological model. We propose a new cosmological system of units, including a cosmological Planck’s constant that “absorbs” the well known large number 10¹²⁰. With this new Planck’s constant no large numbers appear at the cosmological level. They appear at lower levels, e.g. at the quantum world. We note here that Zel’dovich formula, for the cosmological constant Λ, is equivalent to the Weinberg’s relation. The immediate conclusion is that the speed of light c must be proportional to the Hubble parameter H, and therefore decrease with time. We find that the gravitational radius of the Universe and its size are one and the same constant (Mach’s principle). The usual cosmological Ω’s parameters for mass, lambda and curvature turn out to be all constants of order one. The anthropic principle is not necessary in this theory. It is shown that a factor of 10⁶¹ converts in this theory a Planck fluctuation (a quantum black hole) into a cosmological quantum black hole: the Universe today. General relativity and quantum mechanics give the same local solution of an expanding Universe with the law a(t)≈const⋅t. This constant is just the speed of light today. Then the Hubble parameter is exactly H=a(t)′/a(t)=1/t.