Maximal freedom at minimum cost: linear large-scale structure in general modifications of gravity

Abstract: We present a turnkey solution, ready for implementation in numerical codes, for the study of linear structure formation in general scalar-tensor models involving a single universally coupled scalar field. We show that the totality of cosmological information on the gravitational sector can be compressed - without any redundancy - into five independent and arbitrary functions of time only and one constant. These describe physical properties of the universe: the observable background expansion history, fractional matter density today, and four functions of time describing the properties of the dark energy. We show that two of those dark-energy property functions control the existence of anisotropic stress, the other two - dark-energy clustering, both of which are can be scale-dependent. All these properties can in principle be measured, but no information on the underlying theory of acceleration beyond this can be obtained. We present a translation between popular models of late-time acceleration (e.g. perfect fluids, f (R), kinetic gravity braiding, galileons), as well as the effective field theory framework, and our formulation. In this way, implementing this formulation numerically would give a single tool which could consistently test the majority of models of late-time acceleration heretofore proposed.

• The paper introduces a compressed function framework that uses four key α functions to simplify analysis of scalar-tensor modifications of gravity.
• It reduces complex cosmological models into a unified formulation, enabling efficient numerical implementation in codes like CAMB or CLASS.
• The approach links background expansion with linear perturbations, advancing our capability to test dark energy theories and cosmic acceleration.

Analyzing Scalar-Tensor Models Using a Compressed Function Framework

The paper "Maximal freedom at minimum cost: linear large-scale structure in general modifications of gravity," authored by Emilio Bellini and Ignacy Sawicki, introduces a framework aimed at efficiently analyzing the linear structure formation in scalar-tensor theories. These models are pivotal in understanding late-time cosmic acceleration, providing alternatives to the conventional cosmological constant. Scalar-tensor models often involve a single universally coupled scalar field interacting with the gravitational sector, influencing the cosmos's expansion and structure formation.

Core Contribution

The authors present a formulation whereby the cosmological effects of these models are encapsulated by five arbitrary functions of time and one constant, reducing computational complexity without losing generality. This framework simplifies exploring various scalar-tensor theories, ranging from perfect fluids to more complex models like $f(R)$ gravity and kinetic gravity braiding. By translating diverse models into this new formulation, Bellini and Sawicki offer a unified tool for numerically testing and comparing a wide range of dark energy theories.

Formulating the Problem

The paper starts by acknowledging the breadth of theoretical frameworks proposed following the discovery of cosmic acceleration, many of which rely on modifications to general relativity or introduce new scalar degrees of freedom. Traditional analysis methods require substantial computational resources, primarily due to the need to explore model-specific parameters and dynamics deeply embedded in the equations of motion. By framing the problem in terms of background metric evolution and linear perturbations encoded in time-dependent functions, the authors sidestep these complexities.

Mathematical Framework

The authors propose to represent the effects of scalar-tensor theories through four dimensionalless functions, $\alpha_K$, $\alpha_B$, $\alpha_M$, and $\alpha_T$, each encapsulating different physical properties of dark energy and its coupling to matter:

• $\alpha_K$ captures kinetic terms associated with scalar field perturbations typical of perfect-fluid models.
• $\alpha_B$ reflects the mixing of scalar and metric kinetic terms, instrumental in characterizing dark energy clustering.
• $\alpha_M$ denotes the rate of change of the effective Planck mass, influencing anisotropic stress.
• $\alpha_T$ indicates deviations in the speed of gravitational waves from light's speed.

These functions, alongside the observable background expansion history and today's fractional matter density, constitute a complete description of structure formation within these models.

Implications and Applications

By translating various gravitational models into this compact formulation, the framework offers analytical insights and lays the groundwork for robust numerical implementations in cosmological codes such as CAMB or CLASS. This approach facilitates a consistent evaluation across models, enhancing our understanding of different dark energy scenarios. Furthermore, it establishes a methodology to explore the relationship between the observable expansion history and perturbation dynamics, thereby probing the plausibility of alternative gravitational theories.

Future Prospects

The versatility of this approach invites future extensions to multi-field scenarios or non-universal couplings, suggesting a path towards even richer model explorations. Given the ongoing advancements in observational cosmology, frameworks like this can play a significant role in testing wide-ranging hypotheses and eventually constraining the landscape of viable theories.

The work of Bellini and Sawicki provides a streamlined avenue for structured theoretical exploration, holding potential for impactful discoveries in the field of modified gravity and the nature of cosmic acceleration. It represents a step towards a more unified understanding of how modifications to gravity might underpin the observed acceleration of the universe.