Linearized Gravity in the Starobinsky Model: Perturbative Deviations from General Relativity

Abstract

In this work, we linearize the field equations of $f(R)$ gravity using the Starobinsky model, $R+R^2/(6m^2)$, and examine the modifications to General Relativity. We derive an equation for the trace, $T$, of the energy-momentum tensor, which we then decompose using an auxiliary field. This field satisfies the wave equation with $T$ as its source, while simultaneously acting as an effective source for the classical deviation, $\bar{h}$, governed by the Klein-Gordon equation. The fields were expressed in terms of Green's functions, whose symmetry properties facilitated the solution of the trace equation. Then ${\bar{h}}_{\mu \nu }$ was determined in terms of a modified or effective matter-energy distribution. From this, the effective energy density was obtained as the usual energy density $T_{00}$, plus a perturbative correction proportional to $m^{-2}$, involving the Laplacian of the integral of $T$, weighted by the retarded propagator of the Klein-Gordon equation. Finally, we numerically computed the perturbative term in a binary star system, evaluating it as a function of $m$ and spatial position near the stars. In all cases, the results illustrate how the gravitational influence of the stars diminishes with distance. Additionally, the perturbation decreases as $m$ increases, consistently recovering the relativistic limit. These results highlight the role of modified gravity corrections in the vicinity of compact objects.

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References in corpus (34)

• Observation of Gravitational Waves from a Binary Black Hole Merger
• f(R) Theories Of Gravity
• Modified Gravity and Cosmology
• f(R) theories
• Disappearing cosmological constant in f(R) gravity
• Modified f(R) gravity consistent with realistic cosmology: from matter dominated epoch to dark energy universe
• Conditions for the cosmological viability of f(R) dark energy models
• f(R) Gravity and scalar-tensor theory
• Black Holes in f(R) theories
• Modified theories of Gravity: Why, How and What?
• Polarizations of gravitational waves in $f(R)$ gravity
• Constraining early and interacting dark energy with gravitational wave standard sirens: the potential of the eLISA mission
• Is exponential gravity a viable description for the whole cosmological history?
• Neutron star mergers as a probe of modifications of general relativity with finite-range scalar forces
• Analytic extensions of Starobinsky model of inflation
• Gravitational Waves in $F(R)$ Gravity: Scalar Waves and the Chameleon Mechanism
• Gravitational Wave Polarizations in $f(R)$ Gravity and Scalar-Tensor Theory
• Gauge invariant formulation of gravitational waves in metric $f(R)$ gravity
• A Panorama of Viable $F(R)$ Gravity Dark Energy Models
• Early Dark Energy with Power-law F(R) Gravity
• Reinterpretation of the Starobinsky model
• Gravitational Waves in f(R) Gravity Power Law Model
• f(R) gravity constraints from gravitational waves
• Neutron star-black hole mergers in next generation gravitational-wave observatories
• Gravitational wave in f(R) gravity: possible signature of sub- and super-Chandrasekhar limiting mass white dwarfs
• New rotating AdS/dS black holes in $\mathrm{f}(\mathrm{R})$ gravity
• Polarization Modes of Gravitational Wave for Viable f(R) Models
• Higher Order Curvature Theories of Gravity Matched with Observations: a Bridge Between Dark Energy and Dark Matter Problems
• Dynamics of viable $f(R)$ dark energy models in the presence of Curvature-Matter interactions
• Bimetric Starobinsky model
• Spherically symmetric configurations in the quadratic $f(R)$ gravity
• Spherically symmetric and static solutions in $f(R)$ gravity coupled with EM fields
• Hypergeometric viable models in $f(R)$ gravity
• Viable f(R) gravity and future cosmological evolution