Equivalence Principle Implications of Modified Gravity Models

TL;DR

The paper investigates how extended bodies move under modified gravity theories that employ screening mechanisms. It shows that chameleon screening can produce order-one violations of the equivalence principle by renormalizing the object's scalar charge, while Vainshtein screening does not, reducing EP violations to post-Newtonian order. Through complementary Einstein-frame and Jordan-frame analyses, it derives how the center-of-mass motion depends on background fields and screening, introducing the thin-shell parameter $\epsilon$ and the relation $Q=\epsilon\alpha M$. It then outlines concrete observational tests—especially internal motions in voids and differential HI/stellar kinematics—that can distinguish chameleon effects from standard GR, with implications for $f(R)$ and DGP-like theories.

Abstract

Theories that attempt to explain the observed cosmic acceleration by modifying general relativity all introduce a new scalar degree of freedom that is active on large scales, but is screened on small scales to match experiments. We show that if such screening occurrs via the chameleon mechanism such as in f(R), it is possible to have order one violation of the equivalence principle, despite the absence of explicit violation in the microscopic action. Namely, extended objects such as galaxies or constituents thereof do not all fall at the same rate. The chameleon mechanism can screen the scalar charge for large objects but not for small ones (large/small is defined by the gravitational potential and controlled by the scalar coupling). This leads to order one fluctuations in the inertial to gravitational mass ratio. In Jordan frame, it is no longer true that all objects move on geodesics. In contrast, if the scalar screening occurrs via strong coupling, such as in the DGP braneworld model, equivalence principle violation occurrs at a much reduced level. We propose several observational tests of the chameleon mechanism: 1. small galaxies should fall faster than large galaxies, even when dynamical friction is negligible; 2. voids defined by small galaxies would be larger compared to standard expectations; 3. stars and diffuse gas in small galaxies should have different velocities, even on the same orbits; 4. lensing and dynamical mass estimates should agree for large galaxies but disagree for small ones. We discuss possible pitfalls in some of these tests. The cleanest is the third one where mass estimate from HI rotational velocity could exceed that from stars by 30 % or more. To avoid blanket screening of all objects, the most promising place to look is in voids.

Figures (4)

• Figure 1: A scalar potential for the chameleon mechanism. The effective potential felt by $\varphi$ in the presence of sources is the sum of the self-interaction potential $V(\varphi)$ and the scalar-matter coupling $(\alpha \, 8 \pi G ) \tilde{\rho} \, \varphi$. This can give $\varphi$ a large mass at locations where the matter density $\tilde{\rho}$ is large.
• Figure 2: Our mathematical trick.
• Figure 3: Schematic experimental bounds in the chameleon parameter space. The shaded region is already excluded, by demanding that the Milky way be screened. The dashed line is the improvement we propose. Further improvement is possible using Lyman-alpha clouds. For $\alpha \ll 1$ the chameleon is very weakly coupled to matter to begin with, and clearly this relaxes the bound on the other parameter. Curvature invariant theories such as $f(R)$ have $\alpha = 1/\sqrt{6}$, though this value is not protected by symmetry. The generic expectation is $\alpha \sim O(1)$.
• Figure 4: A schematic illustration of observational tests. $\Phi_{\rm self}$ and $\Phi_{\rm env}$ represent the gravitational potentials of an object and its environment. They can be thought of as $\sim (v/c)^2$, where $v$ is the internal velocity. The value $\varphi_*/(2\alpha)$ delineates screening or lack thereof---objects/environments with a potential deeper than $\varphi_*/(2\alpha)$ are screened (shaded), and those with a shallower potential are unscreened (unshaded). Current constraints tell us $\varphi_*/(2\alpha)$ has to be less than $\sim 10^{-6}$. There are many comparison tests one can make. For instance, an unscreened diffuse gas cloud residing in a dwarf galaxy (A), versus a screened star residing in the same galaxy (B)---A falls faster than B. This situation can be replicated on a larger scale e.g. a dwarf galaxy in the fields/voids (A) versus a massive galaxy in the same fields/voids (B). Another example: a dwarf galaxy out in the fields/voids (A), versus a dwarf galaxy residing in a group or cluster (D)---D is blanket screened by its environment and would exhibit no equivalence principle violations in its internal motions of gas clouds and stars, while A would have observable violations. On the other hand, a massive galaxy would have no such (internal) equivalence principle violations whether it be in the fields/voids (B) or in a group/cluster (C).