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| R^2 inflation | |
|---|---|
| Name | R^2 inflation |
| Field | Cosmology |
| Introduced | 1980 |
| Proponents | Alexei Starobinsky, Andrei Linde, Alan Guth |
| Related | Inflation (cosmology), Cosmic microwave background, General relativity |
R^2 inflation is a cosmological inflationary scenario in which a quadratic curvature term is added to the gravitational action, producing a phase of accelerated expansion in the early universe. Developed as an alternative to scalar-field slow-roll proposals, the model yields specific predictions for the spectrum of primordial perturbations and links to reheating, quantum gravity, and observational probes. Its realization and tests involve work by theorists and collaborations across observational projects.
R^2 inflation originated in a modification of General relativity introduced by Alexei Starobinsky in 1980, proposing that adding an R^2 term to the Einstein–Hilbert action can drive exponential expansion. The model sits alongside proposals by Alan Guth and Andrei Linde in the broader field of Inflation (cosmology), and it has been compared against data from missions such as COBE, WMAP, and Planck Collaboration. Connections have been drawn to high-energy frameworks including Quantum field theory, String theory, and approaches associated with the Effective field theory of gravity.
The theoretical underpinning emerges from semiclassical corrections to Einstein field equations motivated by vacuum polarization and anomaly-driven effects studied in contexts like De Sitter space and early analyses by Nikolai Nikolaevich Bogolyubov and colleagues. The R^2 term can be viewed as a leading higher-derivative correction analogous to curvature-squared operators appearing in the low-energy expansion of Supergravity and Heterotic string theory. Relations to scalar-tensor representations invoke transformations familiar from work on the Brans–Dicke theory and conformal mappings used in studies by Robert Wald and Paul Dirac. The model also connects to renormalization-group considerations explored by Kenneth Wilson and quantum correction analyses by Gerard 't Hooft.
Starobinsky formulated the action as the Einstein–Hilbert action supplemented by a term proportional to the square of the Ricci scalar. The action can be transformed via a Legendre or conformal transformation into an equivalent scalar-tensor form with a single scalar degree of freedom—often called the scalaron—whose potential was derived explicitly by Alexei Starobinsky and later analyzed in the context of slow-roll criteria by Andrei Linde and Viatcheslav Mukhanov. The scalaron potential has a characteristic plateau shape similar to potentials considered by Paul Steinhardt and Juan Maldacena in their respective inflationary studies. Variants and parametrizations connect to work on f(R) gravity explored by Thomas Sotiriou and Valerio Faraoni.
Dynamics in the Einstein frame produce slow-roll inflation with a graceful exit, predicting a scalar spectral index and tensor-to-scalar ratio consistent with observations. Detailed calculations of primordial perturbations employ techniques developed by Mukhanov–Sasaki formalism and perturbation theory refined by Viatcheslav Mukhanov and H. J. Heitmann. Predicted values for the spectral tilt and low tensor amplitude have been contrasted with data analyses by the Planck Collaboration, the BICEP2 collaboration, and the South Pole Telescope. The model's predictions often match constraints discussed in reviews by David Spergel and Max Tegmark.
Reheating after R^2-driven inflation proceeds via scalaron oscillations and decay; computations build on particle production frameworks introduced by Leonard Parker and Yakov Zel'dovich. Scalaron decay channels into standard-model fields have been modeled with techniques from Quantum field theory in curved spacetime and perturbative decay analyses influenced by Steven Weinberg and John Preskill. Estimates for reheating temperature depend on couplings studied in the context of Grand Unified Theory scenarios and phenomenological work by Gordon Kane and John Ellis that link to baryogenesis proposals associated with Andrei Sakharov conditions.
Empirical tests leverage anisotropy measurements from Planck Collaboration, WMAP, and COBE, polarization searches by BICEP2 collaboration and Keck Array, and large-scale structure surveys like Sloan Digital Sky Survey and Dark Energy Survey. Constraints on the tensor-to-scalar ratio, spectral index, and non-Gaussianity have favored models with low tensors such as the Starobinsky formulation, informing statements by review authors including Daniel Baumann and Luca Amendola. Future probes like LiteBIRD, CMB-S4, and surveys by Euclid (spacecraft) and Vera C. Rubin Observatory aim to sharpen discriminants between R^2 inflation and alternatives proposed by Andrei Linde and A. A. Starobinsky's contemporaries.
Extensions embed the R^2 term within broader f(R) constructions, supergravity completions, and string-inspired setups explored by Renata Kallosh, Andrei Linde, and Eugene Cremmer. Hybrid schemes combine plateau potentials with monodromy ideas from Silverstein–Westphal constructions and multifield interactions studied by Christoph Burgess and Serguei Odintsov. Connections to effective-operator approaches and ultraviolet completions have been pursued in the literature by John Donoghue, Antony Zee, and researchers working on asymptotically safe gravity such as Martin Reuter. These efforts continue to relate the Starobinsky paradigm to signatures targeted by collaborations like Planck Collaboration and experiments in primordial gravitational waves.