DGP Gravity and Crossover Behavior

Dvali–Gabadadze–Porrati gravity places an Einstein term both in a flat five-dimensional bulk and on its four-dimensional brane. The competition produces four-dimensional propagation at short distance and five-dimensional leakage at long distance. That linear crossover is not the whole theory: a brane-bending scalar becomes nonlinear around sources, and cosmology contains inequivalent embedding branches.

Required background. Gravitational Actions and Boundary Terms supplies the bulk-plus-brane variation and normal-orientation conventions; Gauge Theory and String-Theory Preparation supplies the brane and effective-mode language.

Helpful background. Einstein Equations supplies the bulk and brane equations, Formulations of General Relativity separates boundary data from evolution, and Exact Spacetimes supplies the Minkowski and FRW background checks.

Two kinetic terms and one crossover

Use $(-,+,+,+,+)$ signature, an infinite Minkowski bulk, and matter minimally coupled to the induced brane metric $q_{\mu\nu}$. With reduced Planck conventions, take

$$S=\frac{M_5^3}{2}\int d^5x\sqrt{-g_5}\,\mathcal R_5 +\frac{M_4^2}{2}\int_{y=0}d^4x\sqrt{-q}\,R[q] +S_m[q,\psi].$$

Boundary terms appropriate to the chosen one-sided or two-sided description are implicit; their orientation must be changed together with the junction convention. Define

$$r_c=\frac{M_4^2}{2M_5^3}.$$

The factor two corresponds to a brane with identical infinite bulk regions on both sides. Treating the brane as the boundary of one region moves that factor into the definition of $M_5^3$ or $r_c$. The original action and induced-gravity mechanism are given in Dvali, Gabadadze, and Porrati 2000, §§1–2, PDF.

A scalar analogue exposes the propagation cleanly. Fourier-transform parallel to the brane and impose decay away from it, $\widetilde G(p,y)\propto e^{-p|y|}$ for Euclidean $p=\sqrt{p_\mu p_\mu}$. Integrating the field equation through the brane yields

$$\widetilde G(p,0)=\frac{1}{M_4^2p^2+2M_5^3p} =\frac{1}{M_4^2}\frac{1}{p^2+p/r_c}.$$

The equality itself checks the normalization of $r_c$. For $p r_c\gg1$, the $p^2$ term wins and the static potential scales as $1/r$; for $p r_c\ll1$, the $p$ term wins and it scales as $1/r^2$. Dvali, Gabadadze, and Porrati derive the exact Green function and both asymptotic potentials in Dvali, Gabadadze, and Porrati 2000, §3, Eqs. (7)–(15), PDF. Since $[M_4^2/M_5^3]=L$, the crossover also passes a direct dimensional check.

The linear scalar and nonlinear recovery

For the graviton, the linear brane-to-brane propagator has the tensor structure of a massive four-dimensional graviton: a scalar polarization couples to the trace of the source. Therefore the formal limit $r_c\to\infty$ of the linearized solution is not by itself the GR limit. The propagator and its extra scalar are displayed in Dvali, Gabadadze, and Porrati 2000, §4, Eqs. (23)–(26), PDF.

Nonlinear brane bending changes the conclusion near a compact source. In a common decoupling normalization, the scalar $\pi$ obeys schematically

$$3\Box\pi+\frac{1}{\Lambda_3^3} \left[(\Box\pi)^2-(\nabla_\mu\nabla_\nu\pi)^2\right] =-\frac{T}{M_4}, \qquad \Lambda_3^3=\frac{M_4}{r_c^2}.$$

Numerical coefficients depend on the normalization of $π$ and the one- versus two-sided action, but the balance scale around a source of Schwarzschild radius $r_S$ does not:

$$r_V\sim\left(r_Sr_c^2\right)^{1/3}.$$

For $r\ll r_V$, the nonlinear derivative term dominates and suppresses the scalar response relative to the tensor field; for $r_V\ll r\ll r_c$, linear scalar–tensor behavior applies; only for $r\gg r_c$ does the five-dimensional falloff dominate. Luty, Porrati, and Rattazzi derive the strong-interaction scale and source-dependent radius in Luty, Porrati, and Rattazzi 2003, §1, Eqs. (1.1)–(1.4), and §§3–4, PDF. Screening is a classical background effect inside the EFT; it does not prove an ultraviolet completion above $\Lambda_3$.

Cosmological embedding branches

For a spatially flat FRW brane in an empty Minkowski bulk, the two embeddings give

$$H^2-\varepsilon\frac{H}{r_c}=\frac{\rho}{3M_4^2}, \qquad \varepsilon=+1\ \text{or}\ -1.$$

At $Hr_c\gg1$, both branches approach the usual Friedmann relation. When $\rho\to0$, the $ε=+1$ branch admits $H\to r_c^{-1}$ and is called self-accelerating; the $ε=-1$ normal branch approaches $H\to0$ unless another stress-energy component drives acceleration. The sign is an embedding choice produced when the junction equation is square-rooted, not a gauge convention. Maartens and Koyama derive the action, crossover, and cosmological branch equation in Maartens and Koyama 2010, §9.1, Eqs. (414)–(417), PDF.

The self-accelerating de Sitter branch contains a ghost in its scalar gravitational sector, whereas the normal branch avoids that particular pathology but does not self-accelerate without additional energy. This is a theory-side distinction between backgrounds, not an observational verdict; the perturbative evidence and branch contrast are summarized with their regime in Maartens and Koyama 2010, §§9.1–9.2, PDF. Background expansion alone is also insufficient: cosmological perturbations depend on the five-dimensional bulk response and boundary conditions.

Independent checks and applicability

Three scales must not be conflated. The crossover $r_c$ compares the two quadratic kinetic terms. The Vainshtein radius $r_V$ depends on the source and marks failure of linearization. The length $\Lambda_3^{-1}$ corresponds to the vacuum strong-coupling scale; it is not automatically the ultraviolet cutoff or the local fluctuation scale on a screened background. Those require power counting about that background. A solar-system source may have $r_V$ far larger than its physical radius even though $r_V\ll r_c$.

The GR recovery check must therefore specify a route through scale space: keep $r_S$ fixed, work at $r\ll r_V$, and remain at energies below the EFT cutoff. Taking only $r/r_c\to0$ leaves the linear scalar unsuppressed. Conversely, taking $M_5\to0$ at fixed $M_4$ sends $r_c\to\infty$ but can also lower $\Lambda_3$, making the limit singular rather than uniformly predictive.

Mutable empirical constraints on $r_c$, cosmological growth, and lensing belong in Tests of Gravity. The question of ultraviolet completion belongs in Quantum Gravity. This page establishes the theory prediction and its regimes, not a likelihood comparison.

Common pitfalls

Using $r_c$ as the screening radius. $r_c$ is source independent; $r_V$ scales as $r_S^{1/3}$ and marks nonlinear scalar dynamics.

Calling the short-distance linear limit GR. The linear propagator retains an extra scalar polarization. GR is recovered only after the nonlinear solution is controlled inside $r_V$.

Erasing the branch sign. Self-accelerating and normal cosmologies are different embeddings with different perturbative health. They cannot be averaged into one prediction.

Exercises

Let $M_4^2=100$ and $M_5^3=2$ in consistent length units, and let a source have $r_S=10^{-6}$. Compute $r_c$ and $r_V$. Classify the dominant description at $r=10^{-3}$, $1$, and $10^3$.

Solution

The chosen two-sided normalization gives $r_c=M_4^2/(2M_5^3)=25$. The Vainshtein estimate is $r_V=(10^{-6}\times25^2)^{1/3}\approx0.0855$. Thus $r=10^{-3}$ lies in the nominal nonlinear screening region. At $r=1$, one has $r_V\ll r\ll r_c$, the nominal four-dimensional scalar–tensor region. At $r=10^3\gg r_c$, the bulk term dominates and the potential has five-dimensional falloff. These ratios classify classical balances only. Indeed $M_4=10$ gives $\Lambda_3=(10/625)^{1/3}\simeq0.252$ in the stated units, so the two smaller radii probe inverse lengths above the vacuum strong-coupling scale. Establishing predictive control there requires the background-dependent fluctuation scale, omitted operators, and a specified source radius. The exercise therefore does not certify a controlled solution at every listed radius.

References

• G. Dvali, G. Gabadadze, and M. Porrati, “4D Gravity on a Brane in 5D Minkowski Space,” Physics Letters B 485 (2000) 208–214, DOI, Open PDF.
• M. A. Luty, M. Porrati, and R. Rattazzi, “Strong Interactions and Stability in the DGP Model,” Journal of High Energy Physics 09 (2003) 029, DOI, Open PDF.
• R. Maartens and K. Koyama, “Brane-World Gravity,” Living Reviews in Relativity 13 (2010) 5, DOI, Open PDF.