Hydrogen atom in Palatini theories of gravity

We study the effects that the gravitational interaction of $f(R)$ theories of gravity in Palatini formalism has on the stationary states of the hydrogen atom. We show that the role of gravity in this system is very important for Lagrangians $f(R)$ with terms that grow at low curvatures, which have been proposed to explain the accelerated expansion rate of the universe. In fact, new gravitationally induced terms in the atomic Hamiltonian generate a strong backreaction that is incompatible with the very existence of bound states. In fact, in the $1/R$ model, hydrogen disintegrates in less than two hours. The universe that we observe is, therefore, incompatible with that kind of gravitational interaction. Lagrangians with high curvature corrections do not lead to such instabilities.

Figure 1

Contribution of the different potentials in the ground state. The solid line, which represents the sum of all the potentials, tends to the constant value $-0.134m_0$ (or $-5048$ in the units of the plot).

Figure 2

The different contributions in this plot are $V_e\sim -1$, $V_m\sim -5\times {10}^3$, and $V_{\Omega }\sim \pm {10}^{19}$. The $y$-axis is measured in units of $13.6\mathrm{eV}$; the $x$-axis in units of $a_0$.