Scale invariance, horizons, and inflation

Abstract

ABSTRACTMaxwell equations and the equations of general relativity are scale invariant in empty space. The presence of charge or currents in electromagnetism or the presence of matter in cosmology are preventing scale invariance. The question arises on how much matter within the horizon is necessary to kill scale invariance. The scale-invariant field equation, first written by Dirac in 1973 and then revisited by Canuto et al. in 1977, provides the starting point to address this question. The resulting cosmological models show that, as soon as matter is present, the effects of scale invariance rapidly decline from ϱ = 0 to ϱc, and are forbidden for densities above ϱc. The absence of scale invariance in this case is consistent with considerations about causal connection. Below ϱc, scale invariance appears as an open possibility, which also depends on the occurrence of inflation in the scale-invariant context. In the present approach, we identify the scalar field of the empty space in the scale-invariant vacuum context to the scalar field φ in the energy density $\varrho = \frac{1}{2} \dot{\varphi }^2 + V(\varphi)$ of the vacuum at inflation. This leads to some constraints on the potential. This identification also solves the so-called ‘cosmological constant problem’. In the framework of scale invariance, an inflation with a large number of e-foldings is also predicted. We conclude that scale invariance for models with densities below ϱc is an open possibility; the final answer may come from high redshift observations, where differences from the ΛCDM models appear.