Cosmological perturbations in the theory of gravity with non-minimal derivative coupling. I. Modes of perturbations

Abstract: We consider perturbations in the isotropic and homogeneous cosmological model with the spatially flat Friedmann-Lemaitre-Robertson-Walker metric in the framework of the theory of gravity with non-minimal derivative coupling. The Lagrangian of the theory contains the coupling term $ηG^{μν}\nabla_μφ\nabla_νφ$ and represents the particular example of a general Horndeski Lagrangian, which results in second-order field equations. It is known that the non-minimal derivative coupling crucially changes scenarios of the Universe evolution on early times. In particular, the $η$-term is dominating on early times and leads to a primary quasi-de Sitter (inflationary) stage which needs no fine-tuned potential. On late times the influence of non-minimal derivative coupling on the Universe evolution completely disappears, and this naturally leads to the transition to the standard cosmological evolution (post-inflationary stage). We have derived a complete set of equations which describe an evolution of scalar, vector and tensor modes of perturbations. All modes are analyzed analytically in two asymptotic cases, and then we construct exact numerical solutions which describe an entire evolution of the modes. We show that all modes, including vector ones, are amplified in the quasi-de Sitter (inflationary) stage, and such the behavior is cardinally distinct from that in Friedmann cosmology.

• The paper demonstrates that non-minimal derivative coupling induces strong amplification of scalar, vector, and tensor perturbations during inflation.
• It derives analytic and numerical solutions for perturbation modes, revealing distinct growth exponents (5 for scalar, 4 for vector, and 3 for tensor) along with observable predictions.
• The work shows that vector perturbations, typically negligible in GR, can achieve observable amplitudes, offering a novel test for early universe dynamics.

Cosmological Perturbations in Gravity with Non-Minimal Derivative Coupling: Modes of Perturbations

Introduction

This paper investigates linear cosmological perturbations within the framework of gravity theories incorporating non-minimal derivative coupling, where the scalar field derivatives couple directly to the Einstein tensor in the action. This specific scenario constitutes a special subclass of the general Horndeski class of scalar-tensor theories. Such models are of fundamental interest given their ability to alter early universe dynamics substantially, in particular by providing a gravitationally-driven inflationary phase without the necessity of fine-tuned potentials for the scalar field. The analysis comprehensively derives and assesses the governing equations for scalar, vector, and tensor metric perturbations throughout both the inflationary (quasi-de Sitter) and post-inflationary phases.

Model Formulation

The action for the model is: $$S = \frac{1}{2}\int d^4x\sqrt{-g}\left[ \frac{1}{8\pi}R -(g^{\mu\nu}+ G^{\mu\nu}) \nabla_\mu\phi \nabla_\nu\phi \right]$$ where the non-minimal derivative coupling is realized by the $G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$ term, fundamentally modifying the dynamics of $\phi$ and the background geometry, especially at high curvature. The analysis specializes to spatially flat FLRW backgrounds with a homogeneous scalar field, and neglects ordinary matter in the early universe limit.

Background Cosmological Evolution

In the early universe, the derivative coupling dominates: the relevant terms in the Friedmann equation imply $H^2 = 1/9\eta$ ($\eta$ is the coupling parameter), driving quasi-de Sitter expansion even with a massless scalar. The model therefore realises "kinetic" inflation without a scalar potential. As the universe expands and $a(t)$ increases, the impact of the non-minimal coupling decays (since $\dot{\phi} \propto a^{-3}$), naturally effecting a graceful exit to a standard post-inflationary FLRW regime with conventional massless scalar behavior ($a(t) \propto t^{1/3}$).

Linear Cosmological Perturbations

The perturbative analysis is conducted in the conformal Newtonian (Poisson) gauge. The paper systematically derives the full set of coupled equations for scalar ($\Psi$, $\Phi$), vector ($G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$0), and tensor ($G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$1) metric perturbation modes, together with the scalar field perturbation.

Scalar Modes

• Post-inflationary behavior: After inflation, the perturbation equations reduce to those of standard massless scalar cosmology. The Bardeen potentials coincide ($G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$2, and their evolution is governed by Bessel-type equations. In the long-wavelength limit ($G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$3), modes freeze-out and remain constant. In the short-wavelength limit ($G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$4), oscillatory decay transpires, with amplitude $G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$5.
• Inflationary regime: During the quasi-de Sitter (kinetic inflation) stage, the coupling terms dominate. The scalar perturbation equations yield solutions where both Bardeen potentials are amplified, scaling as $G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$6 for generic initial conditions. The amplification factor for a mode exiting the inflationary phase is $G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$7, where $G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$8 and $G^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi$9 denote initial and final conformal times for inflation. This is an anomalously steep growth compared to standard slow-roll inflation, and it occurs even in the absence of a fine-tuned potential.

Tensor Modes

• Post-inflationary regime: Tensor perturbations satisfy a damped wave equation; solutions are Bessel functions. Long-wavelength gravitational waves remain constant, while short-wavelength modes oscillate and decay.
• Inflationary regime: For the inflationary phase, the tensor equation again admits an instability: $\phi$0 generically for growing combinations. Thus, primordial tensor fluctuations are strongly amplified, with growth determined by $\phi$1.

Vector Modes

• Post-inflationary regime: Vector perturbations decay rapidly, $\phi$2, as in standard GR, supporting the conventional argument that primordial vector modes are dynamically negligible in standard cosmology.
• Inflationary regime: In sharp contrast, during kinetic inflation, vector perturbations experience exponential enhancement: $\phi$3. The net amplification factor for a generic mode is $\phi$4 across inflation.

The robust numerical analysis confirms the analytic asymptotics and tracks the mode evolution across the full dynamical range.

Distinctive Physical Consequences

The principal finding is that all three classes of linear perturbations (scalar, vector, tensor) undergo significant amplification during the inflationary era in non-minimal derivative coupling theories, including vector perturbations, which do not grow in GR-based inflation. The amplification exponents differ: $\phi$5 for scalar, $\phi$6 for vector, $\phi$7 for tensor modes. The final amplitude ratios at the end of inflation are:

• Tensor-to-scalar ratio $\phi$8
• Vector-to-scalar ratio $\phi$9

With the numerical evaluation for typical model parameters, $H^2 = 1/9\eta$0, compatible with current CMB bounds (e.g., $H^2 = 1/9\eta$1). The estimate $H^2 = 1/9\eta$2 implies that vector perturbations, typically neglected, could play a nontrivial role in the post-inflationary universe if their post-inflationary decay—and subsequent reheating/matter fields—do not suppress their observable imprints.

Theoretical and Observational Implications

From a theoretical standpoint, this work rigorously demonstrates that non-minimal derivative coupling in the inflationary sector can produce an inflationary phase, smoothly connected to the standard cosmological expansion, without resorting to potential-driven slow-roll dynamics. The characteristic amplification of vector modes is a distinctive prediction, challenging the prevailing assumption that vector perturbations are always negligible in early universe cosmology.

The strong mode amplification may also have consequences for the linear regime's breakdown, especially for scalars—affecting predictions for power spectra, non-Gaussianity, and primordial gravitational waves. The parameter space yielding CMB-compatible amplitude (for $H^2 = 1/9\eta$3), and constraints on $H^2 = 1/9\eta$4, provide a direct empirical handle on this class of gravitational theories.

Conclusion

This paper establishes a detailed, analytic and numerical framework for the evolution of cosmological perturbations in gravity theories with non-minimal derivative coupling. The findings highlight sharply different perturbative dynamics compared to both GR and potential-driven single field inflation. Notably, vector perturbations undergo strong growth during kinetic inflation—contradicting standard lore—and can reach potentially observable amplitudes. These results motivate extensions, including power spectrum calculation and matter-coupled late-time evolution, and provide a foundation for future empirical tests of non-minimal derivative gravity as a candidate for early universe inflationary dynamics.

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Reference:

"Cosmological perturbations in the theory of gravity with non-minimal derivative coupling. I. Modes of perturbations" (2605.13732)