Quartic Galileon Models

• Quartic Galileon models are scalar-tensor theories featuring unique higher-derivative terms that yield second-order equations of motion and avoid instabilities.
• They enable self-accelerating cosmologies and employ the Vainshtein mechanism to screen fifth forces in high-density regions, aligning with solar-system constraints.
• These models impact cosmic structure formation and primordial non-Gaussianity, while offering a framework for ghost-free effective field theories and even supersymmetric extensions.

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 $\pi \to \pi + c + b_\mu x^\mu$ (Rham et al., 2012, Andrews et al., 2013, Hinterbichler et al., 2015).

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 (Fasiello, 2013). 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 (Deffayet et al., 2017).

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 (Barreira et al., 2014, Barreira et al., 2013, Barreira et al., 2014).

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 (Aoki et al., 2018).

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 $r^{-2/3}$ (Andrews et al., 2013, Rham et al., 2012).

The screening is parametrically efficient: the scalar-mediated force inside $r_{V}$ is suppressed as $(r/r_{V})^2$ relative to Newtonian gravity (Bolis et al., 2018). 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 (Andrews et al., 2013). 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 (Barreira et al., 2014, Li et al., 2013).

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 (Renk et al., 2017, Barreira et al., 2014). 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 (Renk et al., 2017, Barreira et al., 2014). 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 (Barreira et al., 2014).

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 (Barreira et al., 2013, Barreira et al., 2014).

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 (Li et al., 2013, Barreira et al., 2014).

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 (Fasiello, 2013). 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 (Elvang et al., 2017). 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 (Hinterbichler et al., 2015).

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 (Gerhardinger et al., 8 Feb 2024, Rham et al., 8 Feb 2024, Rham et al., 2012).

• "Trispectrum from Co-dimension 2(n) Galileons" (Fasiello, 2013)
• "Spherical collapse in Galileon gravity: fifth force solutions, halo mass function and halo bias" (Barreira et al., 2013)
• "Simulating the quartic Galileon gravity model on adaptively refined meshes" (Li et al., 2013)
• "Classifying Galileon $p$-form theories" (Deffayet et al., 2017)
• "Scalar Radiation with a Quartic Galileon" (Rham et al., 8 Feb 2024)
• "Galileon and generalized Galileon with projective invariance in a metric-affine formalism" (Aoki et al., 2018)
• "Simulating a numerical UV Completion of Quartic Galileons" (Gerhardinger et al., 8 Feb 2024)
• "Galileon Gravity in Light of ISW, CMB, BAO and $H_0$ data" (Renk et al., 2017)
• "The observational status of Galileon gravity after Planck" (Barreira et al., 2014)
• "A Hidden Symmetry of the Galileon" (Hinterbichler et al., 2015)
• "Galileon forces in the Solar System" (Andrews et al., 2013)
• "Generalized Galileon Scenario Inspires Chaotic Inflation" (López et al., 2019)
• "Halo model and halo properties in Galileon gravity cosmologies" (Barreira et al., 2014)
• "The Parameterized Post-Newtonian-Vainshteinian formalism for the Galileon field" (Bolis et al., 2018)
• "On the Covariant Galileon and a consistent self-accelerating Universe" (Germani, 2012)
• "On the Supersymmetrization of Galileon Theories in Four Dimensions" (Elvang et al., 2017)
• "Galileon Radiation from Binary Systems" (Rham et al., 2012)