Weyl current, scale-invariant inflation, and Planck scale generation

Scalar fields, ${\varphi }_i$, can be coupled nonminimally to curvature and satisfy the general criteria: (i) the theory has no mass input parameters, including $M_P=0$; (ii) the ${\varphi }_i$ have arbitrary values and gradients, but undergo a general expansion and relaxation to constant values that satisfy a nontrivial constraint, $K({\varphi }_i)=\text{constant}$; (iii) this constraint breaks scale symmetry spontaneously, and the Planck mass is dynamically generated; (iv) there can be adequate inflation associated with slow roll in a scale-invariant potential subject to the constraint; (v) the final vacuum can have a small to vanishing cosmological constant; (vi) large hierarchies in vacuum expectation values can naturally form; (vii) there is a harmless dilaton which naturally eludes the usual constraints on massless scalars. These models are governed by a global Weyl scale symmetry and its conserved current, $K_{\mu }$. At the quantum level the Weyl scale symmetry can be maintained by an invariant specification of renormalized quantities.

Figure 1

Plot of the Hubble parameter, $H$, $\varphi$, $\chi$ and the ratio of the two components of the effective Planck mass, $M_{\varphi }^2$ and $M_{\chi }^2$, as a function of $a$; we have normalized the x axis to the scale factor at the end of inflation, $a_e$. The chosen parameters are ${\alpha }_1=-1.25\times {10}^{-2}$; ${\alpha }_2=-6.19$; $\delta =0$; $\lambda ={10}^{-24}$; $\xi ={10}^{-9}$; $\varphi (-60)=1$ and $\chi (-60)=0.5$ in Planck units; initial velocities are set to zero.

Figure 2

The $K$ ellipse in $(\varphi ,\chi )$. The potential flat direction lies along the $\varphi$ axis for the potential $\zeta {\chi }^4$. Initial values of fields and their velocities rapidly redshift to constant $K$ and then slow roll on the ellipse toward the fixed point. The ellipse is mapped into $x+y=1$; initial value of $x=x(0)$ slow rolls to the final fixed point $x_0$ (presently $x_0\approx 1$).

Figure 3

We expand the right-handed quadrant, $(\varphi ,\chi )>0$ where the classical ellipse has rightmost endpoint at $\chi =1$. The “quantum ellipse” turns back toward the origin due to the quantum running of ${\alpha }_1$ where $\varphi$ tracks $\chi$ as $\varphi \propto {\chi }^{{\gamma }_{\alpha }/(2+{\gamma }_1)}$. The potential flat direction is indicated as the nearly horizontal line.