Starobinsky Late-Time f(R)
Starobinsky's 2007 'disappearing cosmological constant in f(R) gravity' framework: a separate parameterization of late-time geometric modified gravity that produces accelerating expansion without a true vacuum energy.
Placeholder for a 3D visualisation of Geometric Modified Gravity. The interactive scene will land in Phase 3. Geometric modified gravity replaces the Einstein-Hilbert Lagrangian, linear in the Ricci scalar R, with an arbitrary function f(R) or a related higher-curvature extension. The framework was developed in inflationary cosmology (Starobinsky 1980 R-squared inflation) and revived for late-time cosmic acceleration in the mid-2000s (Hu-Sawicki 2007, Starobinsky 2007). The unifying claim is that observed dark energy may be a manifestation of modified gravitational geometry rather than a separate energy component. Modern f(R) gravity comes with sharp constraints: solar-system tests, structure-growth measurements, and the chameleon mechanism for screening. The framework is editorially distinct from MOND because the mechanism is geometric (modifying the curvature-action relation) rather than a phenomenological acceleration scale.
§1 · The claim, in one sentence
Alexei Starobinsky proposed in 2007 a separate f(R) construction for late-time cosmic acceleration: *Disappearing cosmological constant in f(R) gravity*, JETP Lett. 86, 157 (1,381 INSPIRE citations). The framework parameterizes the deviation from Einstein-Hilbert differently from Hu-Sawicki but addresses the same phenomenology: producing late-time acceleration without invoking a true cosmological constant. The two constructions are complementary entries in the geometric-modified-gravity literature.
§2 · Why it might be true
Starobinsky's late-time f(R) form takes a specific functional shape that smoothly interpolates between Einstein-Hilbert at high curvature (solar system) and an effective de Sitter de Sitter at low curvature (cosmological), similar in spirit to Hu-Sawicki but with a different parameterization. The framework is often cited alongside Hu-Sawicki as the canonical pair of viable late-time f(R) constructions.
Starobinsky's 1980 R-squared model (covered in the sibling Starobinsky R-squared Inflation variant) demonstrated that higher-curvature gravity could produce the cosmic inflation phenomenology. The 2007 paper extended this insight to late-time physics: if higher-curvature gravity worked for the early-universe acceleration, perhaps a different higher-curvature construction could explain the late-time acceleration without a true cosmological constant. The two papers are bookends of a unified geometric-modified-gravity research program.
The 2007 framework has produced extensive technical follow-up: weak-lensing predictions, structure-growth modifications, comparison with N-body simulations. Like Hu-Sawicki, the current observational constraints place the deviation parameters in small ranges, making the construction difficult to distinguish from ΛCDM at current precision. Next-generation surveys (DESI, Euclid, LSST) should sharpen the distinction.
The family stance
The accelerating expansion of the universe and the structure of strong-gravity regimes may be explained by replacing Einstein's R Lagrangian with a more general function f(R). The standard dark-energy interpretation (a true [[cosmological constant]] or scalar dark-energy field) becomes unnecessary if the gravitational action itself is the source of the acceleration.
§2.5 · Evidence
- The Starobinsky 2007 construction is mathematically well-defined and produces a phenomenologically viable late-time f(R) model
- The framework is part of a unified geometric-modified-gravity research program (Starobinsky's own 1980 R-squared inflation and this 2007 late-time work) that bridges early-universe and late-universe physics
- Independent technical work (Cembranos, Cruz-Dombriz, others) has confirmed the basic structure-growth predictions; the framework remains in active use
- The 1,381 INSPIRE citations testify to the proposal's significant theoretical impact
§3 · What you'd need to test it
- Late-time cosmic acceleration emerges from the modified gravitational action without a true cosmological constant; the effective late-time behavior mimics ΛCDM at the background level
- Structure growth differs from ΛCDM at low redshifts; the deviations are calculable and constrained by weak-lensing and redshift-space-distortion measurements
- Solar-system tests are passed via a chameleon-like screening mechanism similar to that of Hu-Sawicki
- Specific differences from Hu-Sawicki in the detailed parameterization can in principle be distinguished by next-generation cosmological surveys, though current data does not favor one over the other strongly
§4 · Where it breaks
- Current data does not strongly distinguish Starobinsky late-time f(R) from Hu-Sawicki or from ΛCDM; the choice between these constructions awaits next-generation surveys
- Like Hu-Sawicki, the framework requires fine-tuning the deviation parameters to specific ranges; the deeper physics origin of these values is not addressed
- The construction shares the general f(R) limitations: structure growth tensions, σ8 sensitivity, the need for chameleon screening
- The framework does not directly address the cosmological-constant problem in the sense of explaining why the vacuum energy is small; it relabels the problem rather than solving it
Go deeper
Starobinsky's 2007 framework has the action f(R) form that produces a 'disappearing cosmological constant' at large R, smoothly interpolating to a de Sitter effective Lagrangian at small R. The construction is technically distinct from Hu-Sawicki in the choice of mathematical parameterization but produces phenomenologically similar late-time cosmology.
The Sotiriou-Faraoni 2010 review (arXiv:0805.1726) covers both Starobinsky's late-time and the Hu-Sawicki forms as the canonical viable late-time f(R) models, with detailed discussion of the screening mechanism and structure-growth implications. The review is the standard entry point for understanding the relationship between the different f(R) parameterizations.
Cross-references: the sibling Hu-Sawicki f(R) variant covers the most-cited alternative construction. The sibling Starobinsky R-squared Inflation variant covers Starobinsky's 1980 early-universe work that established the higher-curvature gravity research program. The Ch.5 MOND family covers a phenomenologically-different modified-gravity approach. The Standard Cosmological Model family in Ch.5 covers ΛCDM, which f(R) gravity competes against at the late-time level.
Variants in this family
▸§5 · Who built it, and when(1 source, 1 established)
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