[Paper Review] Revisiting chameleon gravity - thin-shells and no-shells with appropriate boundary conditions

Takashi Tamaki, Shinji Tsujikawa|arXiv (Cornell University)|Aug 17, 2008

Structural Analysis and Optimization4 citations

TL;DR

This paper re-examines chameleon gravity by deriving analytic solutions for scalar field profiles in spherically symmetric bodies, distinguishing between thin-shell and no-shell regimes through proper boundary conditions. It shows that under $ m_A r_c \gg 1 $, both regimes yield identical effective couplings, and no-shell solutions can satisfy local gravity constraints even when thin-shell solutions fail, depending on the scalar potential shape.

ABSTRACT

We derive analytic solutions of a chameleon scalar field $ϕ$ that couples to a non-relativistic matter in the weak gravitational background of a spherically symmetric body, paying particular attention to a field mass $m_A$ inside of the body. The standard thin-shell field profile is recovered by taking the limit $m_A*r_c o \infty$, where $r_c$ is a radius of the body. We show the existence of "no-shell" solutions where the field is nearly frozen in the whole interior of the body, which does not necessarily correspond to the "zero-shell" limit of thin-shell solutions. In the no-shell case, under the condition $m_A*r_c \gg 1$, the effective coupling of $ϕ$ with matter takes the same asymptotic form as that in the thin-shell case. We study experimental bounds coming from the violation of equivalence principle as well as solar-system tests for a number of models including $f(R)$ gravity and find that the field is in either the thin-shell or the no-shell regime under such constraints, depending on the shape of scalar-field potentials. We also show that, for the consistency with local gravity constraints, the field at the center of the body needs to be extremely close to the value $ϕ_A$ at the extremum of an effective potential induced by the matter coupling.

Motivation & Objective

• To re-derive chameleon scalar field solutions in spherically symmetric bodies with correct boundary conditions at r=0, r₁, r_c, and ∞.
• To clarify the distinction between thin-shell and no-shell solutions, showing no-shell solutions are not simply the zero-shell limit of thin-shell solutions.
• To assess the viability of chameleon models under experimental constraints from equivalence principle violation and solar-system tests.
• To determine under what conditions the field remains near φ_A in the body's interior, even when the matter coupling term Qρ_A is not dominant.

Proposed method

• Derives analytic solutions for the chameleon scalar field φ in three regions: 0 < r < r₁, r₁ < r < r_c, and r > r_c, using the field equation with a potential V(φ) and matter coupling Q.
• Imposes physical boundary conditions at r=0 (regularity), r=r₁ (transition), r=r_c (continuity of φ and φ'), and r→∞ (asymptotic behavior).
• Analyzes the field mass m_A inside the body and its role in determining whether the field is frozen (no-shell) or evolves in a thin-shell region.
• Compares the effective coupling Q_eff = 3Qε_th in both thin-shell and no-shell regimes under the limit m_A r_c → ∞.
• Applies constraints from equivalence principle tests (e.g., lunar laser ranging) and solar-system experiments to model parameters.
• Evaluates two scalar potentials: V(φ) ∝ φ^{−n} and V(φ) ∝ (φ − φ₀)^{p}, assessing their consistency with local gravity.

Experimental results

Research questions

• RQ1 Under what conditions does the chameleon field remain nearly frozen throughout the interior of a body, forming a no-shell solution rather than a thin-shell?
• RQ2 How do the field profiles and effective couplings in the no-shell regime compare to those in the thin-shell regime when m_A r_c ≫ 1?
• RQ3 Can no-shell solutions satisfy local gravity constraints even when thin-shell solutions are ruled out by experiments?
• RQ4 What is the role of the field mass m_A inside the body in determining whether the field stays near the potential minimum φ_A?
• RQ5 How do different scalar field potentials (inverse power-law vs. power-law) affect the transition between thin-shell and no-shell behavior?

Key findings

• No-shell solutions exist where the field remains near φ_A throughout the body’s interior, and they are not the zero-shell limit of thin-shell solutions.
• In the limit m_A r_c ≫ 1, the effective coupling Q_eff ≃ 3Qε_th is identical in both thin-shell and no-shell regimes.
• For the inverse power-law potential V(φ) ∝ φ^{−n}, no-shell solutions are viable for n ≳ 4, while thin-shell solutions apply for n ≲ 4 under current experimental bounds.
• For the power-law potential V(φ) ∝ (φ − φ₀)^p, the field mass inside the body is extremely large (m_A r_c ≳ 10^39), forcing the solution into the thin-shell regime.
• The field at the center of the body must be extremely close to φ_A for consistency with local gravity, especially in the power-law potential case.
• No-shell solutions with nearly massless exterior fields violate experimental bounds unless m_A r_c is large, but even then, constraints from equivalence principle tests rule out large couplings Q = O(1) for n > 0.