Emergent Planck mass and dark energy from affine gravity

Abstract

We introduce a novel model of affine gravity, which implements the no-scale scenario. Namely, the Planck mass and Hubble constant emerge dynamically in our model, through the mechanism of spontaneous breaking of scale invariance. This naturally gives rise to inflation, thus introducing a new inflationary mechanism. Moreover, the time direction and nondegenerate metric emerge dynamically as well, which allows considering the usual General Relativity as an effective theory. We show that our model is phenomenologically viable, both from the perspective of the direct tests of gravity and from the standpoint of cosmological evolution.

Appendix: Linearized limit

To verify the phenomenological validity of scalar–affine gravity, we show that the linearized limit of our model coincides with that of GR. We parameterize the fluctuations of the metric as

$$ g_{00}=\eta_{00}+h_{00},\qquad g_{ij}=a^2(\eta_{ij}+h_{ij}),\qquad g_{0i}=h_{0i},$$

(A.1)

where $h_{\mu\nu}$ are fluctuations of the metric. In the remainder of this appendix, the sum is taken with respect to the flat metric $\delta_{ij}$, and we do not distinguish between upper and lower indices. We use the $3+1$ decomposition

$$h_{00}= 2\Phi,$$

(A.2a)

$$h_{0i}=\partial_i Z+Z_i^T,$$

(A.2b)

$$h_{ij}=-2\Psi\delta_{ij}+2\partial_i\partial_j E+\partial_{(i}W_{j)}^T+h_{ij}^{TT},$$

(A.2c)

where, as usual,

$$ \partial_i Z_i^T=0,\qquad \partial_i W_i^T=0,\qquad \partial_i h_{ij}^{TT}=0,\qquad h_{ii}^{TT}=0,$$

(A.3)

and impose the gauge $h_{0i}=0$.

For our choice of the constant $c=-3$, $\varphi$ follows the dynamics of the determinant of the metric. Hence, their fluctuations are the same. However, we also need to obtain a formula governing the fluctuations of the covariant derivatives of $\varphi$. For the $v_\mu$ fluctuations, $v_\mu=v_\mu^{\mathrm{vac}}+u_\mu$, we then obtain the equation

$$ -(a^3\partial_0+3a^2\partial_0 a)\biggl(u_0+\frac{v}{2}(h+h_{00})\biggr)+a\partial_i u_j=0,$$

(A.4)

where $h=h_i^i$. The solution of this equation is

$$ u_0=-\frac{v}{2}(h-h_{00})-u,\qquad \dot u+3Hu+a^{-2}\partial_i u_i=0.$$

(A.5)

We now consider the equations of motion for the perturbations of the metric. As follows from Eq. (3.13), the terms quadratic in $h_{\mu\nu}$ are suppressed if

$$ h_{\cdot\cdot}\ll 1,\qquad v h_{\cdot\cdot} h_{\cdot\cdot}\ll\partial_{\cdot} h_{\cdot\cdot},$$

(A.6)

where dots stay for some indices. The first condition is standard. The second constraint appears due to the nonzero nonmetricity and its validity should be verified after obtaining the solution of the equations of motion.

We use Cadabra software [63], [64] for obtaining the equations of motion for the perturbed metric. The result is that the equations of motion in the spin-2 and spin-1 sectors are the same as those in GR on the de Sitter background. Hence, gravitational waves are the same in our model as in GR, and vector perturbations are stable.

In the spin-0 sector in the gauge $E=0$, the equations of motion are

$$\begin{alignedat}{3} &(00):&\quad &3H(2H+3v)\Phi+27Hv\Psi+3(u'+3Hu+a^{-2}\partial_i u_i)-{} \nonumber\\ &&&-6H\Psi'+2a^{-2}\triangle\Psi=0, \end{alignedat}$$

(A.7a)

$$\begin{alignedat}{3} &(0i): &\quad &\partial_i(H\Phi-\Psi')=0, \end{alignedat}$$

(A.7b)

$$\begin{alignedat}{3} &(ij): &\quad &\partial_i\partial_j(\Phi+\Psi)-\delta_{i j}\bigl(\triangle(\Phi+\Psi)+a^2(3H(2H+3v)\Phi+{} \nonumber\\ &&&+27Hv\Psi+2H\Phi'-6H\Psi'-2\Psi'')\bigr)=0, \end{alignedat}$$

(A.7c)

where $\triangle=\partial_i\partial_i$. After taking Eq. (A.5) into account, these equations become the same as in GR. Thus, scalar–affine gravity is fully equivalent to GR in the linearized limit. In particular, it follows from Eqs. (A.7c) that $\Psi+\Phi=0$. Hence, our model passes experimental constraints on the difference of these potentials [65]. In particular, by introducing a point-like particle of mass $M$ at the origin of the coordinates, we recover Newton’s law. Requirement (A.6) is then fulfilled if

$$ H\ll\frac{M_{\mathrm{Pl}}^2}{M},$$

(A.8)

which holds in all cases of physical interest.