The Two-Measure Theory and an Overview of Some of its Manifestations

TL;DR

The paper surveys the Two-Measure Theory (TMT), a framework that augments gravity with a second volume form and a dynamical ratio $zeta=rac{dV_psilon}{dV_g}$, leading to pregeometry effects and spontaneous space-time orientation. Through toy models and a range of scale-invariant constructions, it shows how TMT can reproduce Einstein GR in tests while offering novel mechanisms for dark energy, quintessence without fine-tuning, and neutrino-driven DE, often via a dynamical CC term and $zeta$-dependent potentials that remain well-behaved in the Einstein frame. The approach yields zero or small effective CCs, plateau-like inflation, and a natural decoupling of extra scalar forces under normal matter densities, with intriguing links to fundamental questions about initial conditions for inflation and orientation of space-time. Collectively, TMT presents a versatile, mathematically rich avenue for addressing the cosmological constant problem, late-time acceleration, and inflationary dynamics within a unifying geometric framework.

Abstract

The Two-Measure theory (TMT) has been developing since 1998 and has yielded a number of highly interesting results, including those not realized in traditional field theory models. The most important advantage of TMT as an alternative theory is that, under the conditions under which all classical tests of general relativity are performed, TMT models are able to accurately reproduce Einstein's general relativity. Despite this, TMT is still often perceived as something too exotic to be relevant to reality. In fact, the fundamental idea underlying TMT seems undeniable: if we truly believe in the effectiveness of mathematics in studying nature, we must agree that there must be a correspondence between the fundamental laws of nature and the structure of the mathematical apparatus necessary to adequately describe them. It then turns out that there is no reason to ignore the volume measure existing on the differentiable manifold on which the theory of gravity and matter fields is built. This idea has far-reaching implications. The goals of this paper are: 1) to provide a clear mathematical and conceptual justification for TMT; 2) to collect in a single article some of the main results of TMT obtained over the past 25 years.

Figures (7)

• Figure 1: For the model with $\alpha =0.2$, $\delta =0.1$, $V_{1}=10M^{4}$ and $V_{2}=9.9b_{g}M^{4}$ and with the the initial conditions $\phi_{in}=M_{p}$, $\dot\phi_{in} =5.7M^2/\sqrt{b_g}$: crossing the phantom divide $w=-1$ and changing sign of the total volume measure $(\Upsilon +b_{\phi}\sqrt{-g})$ in the scalar field $\phi$ kinetic term (in the primordial action (\ref{['totaction k-ess']})) occur simultaneously. The figure is copied from ref.GK4, where $\Phi$ was used instead of $\Upsilon$.
• Figure 2: Total EoS $w=\bar{p}/\bar{\rho}$ vs $\log a$.
• Figure 3: Total energy density $\log\bar{\rho}$ vs $\log a$.
• Figure 4: Evolution of the averaged value $\bar{\zeta}$ that with very high accuracy coincides with the value of $\zeta$ in the maximal volume domains of low fermion density. The result of this numerical solution confirms the analytical estimates, the results of which are formulated in the paragraph after eq. (\ref{['Pf-sim_vac']}) under the name of the “self-locking” effect: as the total energy density decays from the value $\sim 10^{14}$ at the cold matter dominated epoch with $w\approx 0$ up to the value $\sim 10^{-2}$ at the DE dominated epoch with $w\approx -1$ (see fig.2), $\bar{\zeta}$ changes only from $\bar{\zeta}\approx 2.3$ to the fermion vacuum value $\bar{\zeta}=\zeta_0 =4$ defined by eq.(\ref{['zete-vacuum']}).
• Figure 5: $\Omega$ vs $\log a$ where fractions of clustered (dark) matter $\Omega_m$ (the black dash line) and effective DE $\Omega_{DE}$ (the blue solid line).