---
title: Quartic Galileon Models
url: https://www.emergentmind.com/topics/quartic-galileon-models
type: topic
---

# Quartic Galileon Models

Quartic Galileon models are a class of scalar-tensor field theories characterized by higher-derivative self-interactions that yield strictly second-order equations of motion and that exhibit robust non-renormalization and Vainshtein screening properties. These models are key elements in infrared modifications of gravity, notably within the covariant Galileon/Horndeski sector, and have been actively investigated in cosmological, astrophysical, and field-theoretic contexts. Quartic Galileon interactions are structurally unique in four spacetime dimensions and play a fundamental role in realizing consistent self-accelerating cosmologies, constructing ghost-free effective field theories, understanding non-Gaussian statistical properties of primordial fluctuations, and formulating mechanisms for screening fifth forces in strongly gravitating environments.

## 1. Quartic Galileon Lagrangian and Structural Uniqueness

The defining feature of quartic Galileon models is a Lagrangian term, built from scalar (or multi-field) degrees of freedom, that involves four field powers and derivatives organized so as to ensure the Euler–Lagrange equations remain second-order—thus avoiding Ostrogradsky instabilities. The canonical flat-space scalar action reads
\[
\mathcal{L}_4 = -\frac{1}{4\Lambda^6}\,(\partial\pi)^2 \left[\,(\Box\pi)^2 - (\partial_\mu\partial_\nu\pi)(\partial^\mu\partial^\nu\pi)\,\right]
\]
where $\Lambda$ is the strong-coupling scale. This term is invariant under the Galilean shift symmetry $\pi \to \pi + c + b_\mu x^\mu$ [1212.5212, 1305.2194, 1501.07600].

In multi-field generalizations, e.g., co-dimension-2(n) brane embeddings, SO(N) invariance ensures that only even-order Galileon interactions remain (quadratic and quartic) and cubic terms are forbidden [1303.5015]. This exclusivity is mirrored in the full group-theoretic classification of Galileon $p$-form theories, where only a unique scalar quartic as well as a 3-form quartic in $D=9$ or higher exists [1704.02980].

## 2. Covariant Extensions and Metric-Affine Realizations

Quartic Galileon terms admit covariant generalizations consistent with second-order dynamics. In curved spacetime, the quartic interaction becomes
\[
\mathcal{L}_4 = \frac{1}{M^6}(\nabla\varphi)^2\big[\,2(\Box\varphi)^2 - 2(\nabla_\mu\nabla_\nu\varphi)^2 - \frac{1}{2}R\,(\nabla\varphi)^2\,\big]
\]
where $M^3 \sim M_{\rm Pl} H_0^2$ is a cosmological scale [1406.0485, 1308.3699, 1401.1497].

Within the metric-affine formalism, quartic Galileon densities can be expressed via epsilon contractions of covariant derivatives and are uniquely selected by projective invariance—leading to actions of the form
\[
S_4= \int d^4x\,\sqrt{-g}\,\frac{c_4}{\Lambda^6}\,\epsilon^{\alpha\beta\gamma\delta}\,\epsilon^{\alpha'\beta'\gamma'}{}_\delta\,\nabla_\alpha\phi\,\nabla_{\alpha'}\phi\,\nabla_{\beta}\nabla_{\beta'}\phi\,\nabla_{\gamma}\nabla_{\gamma'}\phi
\]
which yields a Riemannian effective action in the quadratic DHOST class Ia, with novel non-minimal couplings to fermionic matter [1806.02589].

## 3. Vainshtein Screening and Solar-System Constraints

Quartic Galileon models realize the Vainshtein mechanism, suppressing fifth forces and screening scalar gravitational modifications near massive sources. The spherically symmetric background field equation is algebraic and, for a point mass $M$,
\[
x + \frac{2}{3\Lambda_3^3}x^2 + \frac{2}{\Lambda_4^6}x^3 = \frac{M}{12\pi M_{\rm Pl} r^3}\quad \text{with}\quad x = \frac{d\pi_0/dr}{r}
\]
Defining crossover Vainshtein radii $r_{*,3}, r_{*,4}$, the field profile transitions between linear, cubic, and quartic scaling regimes—deep in the quartic region, gradients fall as $r^{-2/3}$ [1305.2194, 1212.5212].

The screening is parametrically efficient: the scalar-mediated force inside $r_{V}$ is suppressed as $(r/r_{V})^2$ relative to Newtonian gravity [1810.02725]. Solar-system perihelion precession and laboratory fifth-force constraints currently require order-unity quartic couplings; weaker couplings would allow cubic terms to dominate and are tightly bounded [1305.2194]. However, residual time variation of $G_{\rm eff}$ due to the quartic term in high-density environments is problematic for lunar-laser-ranging and local gravity tests, unless extra covariant terms further suppress dynamics [1406.0485, 1308.3491].

## 4. Cosmological Dynamics and Observational Signatures

The quartic Galileon model, in the cosmological context, admits a tracker (self-accelerating) solution with $\dot\varphi/H = \xi M_{\rm Pl} H_0^2$, yielding late-time cosmic acceleration without an explicit cosmological constant [1707.02263, 1406.0485]. The background Friedmann equation and linear scalar field dynamics are analytically tractable:
\[
3 M_{\rm Pl}^2 H^2 = \rho_m + \rho_r + \rho_\varphi
\]
with
\[
\rho_\varphi = (c_2/2)\dot\varphi^2 + (6c_3/M^3) H \dot\varphi^3 + (45 c_4/M^6) H^2 \dot\varphi^4
\]
Parameter fits to CMB+BAO+ISW data favor $c_4\sim -0.0045$ and neutrino mass $\Sigma m_\nu \sim 0.5$ eV, and uniquely set $H_0$ consistent with local measurements, unlike $\Lambda$CDM [1707.02263, 1406.0485]. The quartic term impacts the low-$\ell$ power in CMB temperature (ISW effect) and lensing spectra, reducing the late-time deepening of the lensing potential and matching Planck data closely [1406.0485].

## 5. Structure Formation, Nonlinear Power Spectrum, and Halo Phenomenology

Nonlinear structure formation in quartic Galileon cosmologies is distinctively affected by the interplay between increased large-scale $G_{\rm eff}$ and residual screening at high density. Excursion-set and spherical collapse analysis yield a flat collapse barrier, leading to a significant overabundance of high-mass halos at $z=0$ (30–50% above $\Lambda$CDM), suppressed linear halo bias, and lower halo concentrations relative to GR [1308.3699, 1401.1497].

N-body simulations incorporating the full quartic Galileon equation reveal scale-dependent matter power spectrum modifications: large-scale modes are boosted, but clustering is suppressed ($\sim$20–30% at $k\sim1\,h/$Mpc) on small scales due to weakened gravity inside screened regions. The halo-model predictions, with calibrated Sheth–Tormen mass function and concentration–mass relations, account for these phenomena [1308.3491, 1401.1497].

## 6. Primordial Non-Gaussianity and Trispectrum

In inflationary scenarios, an SO(N)-invariant quartic Galileon leads to unique non-Gaussian signatures. Absence of a cubic operator yields a naturally small bispectrum ($|f_{\rm NL}| \lesssim 1$), while the trispectrum exhibits distinctive shapes: in equilateral and double-squeezed momentum configurations, contact-term contributions are finite and differ sharply from DBI or $P(X,\phi)$ models [1303.5015]. The bispectrum and trispectrum scale identically with the model's parameters (i.e., $\tau_{\rm NL}/f_{\rm NL} \sim \mathcal{O}(1)$), precluding any "large-trispectrum, small-bispectrum" separation possible in other scenarios.

## 7. UV Completion, Supersymmetry, and Theoretical Extensions

The quartic Galileon is the only member of the Galileon hierarchy admitting a non-trivial $\mathcal{N}=1$ supersymmetric extension in $D=4$ compatible with Galileon shift symmetry for the scalar and an ordinary shift for the fermion, uniqueness holding at up to six-field order [1712.09937]. The special Galileon theory, enjoying an enhanced quadratic shift symmetry, contains only even powers of the field and features an improved soft-momentum limit—amplitudes vanish as $\mathcal{O}(q^2)$ for soft external scalar momentum [1501.07600].

Numerical simulations of the classical initial-value problem for the quartic Galileon encounter ill-posedness at high frequencies due to loss of hyperbolicity in the principal symbol. Recent developments employ auxiliary field UV completions and low-pass filtering to restore hyperbolicity and maintain correct IR physics, enabling stable evolution even when quartic interactions dominate. These methods correctly reproduce the suppression of monopole and dipole radiation and quadrupole-dominated scalar emission in binary systems, confirming that Vainshtein screening is dynamically realized [2402.05897, 2402.05898, 1212.5212].

## References

- "Trispectrum from Co-dimension 2(n) Galileons" [1303.5015]
- "Spherical collapse in Galileon gravity: fifth force solutions, halo mass function and halo bias" [1308.3699]
- "Simulating the quartic Galileon gravity model on adaptively refined meshes" [1308.3491]
- "Classifying Galileon $p$-form theories" [1704.02980]
- "Scalar Radiation with a Quartic Galileon" [2402.05898]
- "Galileon and generalized Galileon with projective invariance in a metric-affine formalism" [1806.02589]
- "Simulating a numerical UV Completion of Quartic Galileons" [2402.05897]
- "Galileon Gravity in Light of ISW, CMB, BAO and $H_0$ data" [1707.02263]
- "The observational status of Galileon gravity after Planck" [1406.0485]
- "A Hidden Symmetry of the Galileon" [1501.07600]
- "Galileon forces in the Solar System" [1305.2194]
- "Generalized Galileon Scenario Inspires Chaotic Inflation" [1908.03155]
- "Halo model and halo properties in Galileon gravity cosmologies" [1401.1497]
- "The Parameterized Post-Newtonian-Vainshteinian formalism for the Galileon field" [1810.02725]
- "On the Covariant Galileon and a consistent self-accelerating Universe" [1207.6414]
- "On the Supersymmetrization of Galileon Theories in Four Dimensions" [1712.09937]
- "Galileon Radiation from Binary Systems" [1212.5212]

Source: https://www.emergentmind.com/topics/quartic-galileon-models