On scale invariance, the Milne universe and induced gravity

M. D. Pollock

doi:10.1140/epjp/i2019-12930-5

EPJ

On scale invariance, the Milne universe and induced gravity

M. D. Pollock^*

V.A. Steklov Mathematical Institute, Russian Academy of Sciences, Ulitsa Gubkina 8, 119991, Moscow, Russia

Received: 25 July 2019
Accepted: 8 August 2019
Published online: 9 December 2019

Abstract

We discuss the notion of global scale invariance, as realized in the theory of induced gravity $\mathcal{L}=\sqrt{-g} [-\frac{1}{2} \epsilon \phi^{2} R+ \frac{1}{2} (\nabla\phi)^2 -V(\phi) + \mathcal{R}^2] + \mathcal{L}_{\rm m}$ , whereby the Newton gravitational constant $G_{\rm N}$ is generated from the expectation value of a scalar field $\phi$ via the dimensional coupling parameter $\epsilon$ . Assuming the Friedmann space-time ${\rm d}s^2={\rm d} t^2-a^2(t){\rm d} { x}^2$ , where the radius function is of the form $a= (t/t_0)^{2/3\gamma}$ , and a power-law field $\phi=\phi_0 t^n$ and matter energy-density source $\rho_{{\rm m}} =\rho_{ot}^{m}$ , which imply that $m=-2 \gamma_{{\rm m}}/\gamma$ , $\gamma_{{\rm m}}$ being the matter adiabatic index, conditions are derived for the positivity of $\epsilon$ . In the special case of the Milne universe, defined by $\gamma = 2/3$ , which is also the string equation of state, the line element is invariant under global coordinate transformations $t\rightarrow t'=\zeta t$ . There is then no creation of particle-anti-particle pairs, nor, analogously in the superstring theory, of string-anti-string pairs. These results follow from the correspondence principle, the respective classical wave states propagating without amplification or attenuation. Some aspects of scale and modular invariance of the superstring are also described.

12 Internat. Congress of the Balkan Physical Union
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EPJ

On scale invariance, the Milne universe and induced gravity

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