One-Parameter Deformation of Starobinsky Model

• The paper shows that introducing a deformation parameter n in the Starobinsky model adjusts the auxiliary scalar coupling, recovering various successful inflationary models.
• It details how the modified kinetic sector affects the UV cutoff, with n=1 restoring the Planck scale, while higher values yield lower cutoffs during inflation.
• The analysis connects nearly universal slow-roll predictions with small higher-order corrections, ensuring consistent alignment with current CMB observations.

A one-parameter deformation of the Starobinsky model refers to a controlled modification of the original R + R² inflationary action through the introduction of a single continuous parameter that changes the functional structure of the model, thereby modifying both its ultraviolet structure and its cosmological predictions. This mechanism subsumes many phenomenologically successful inflationary models as descendants of the Starobinsky model, distinguished by their kinetic sector, UV cut-off, and minor variations in slow-roll predictions, and, in special cases, alters the effective field theory's domain of validity.

1. Mathematical Formulation and Deformation Parameterization

The original Starobinsky model is based on a gravitational action of the form

$$S_{\mathrm{S}} = \frac{1}{2} \int d^4x\ \sqrt{-g}\ \left( M_P^2 R + \frac{1}{6M^2} R^2 \right)$$

where $M_P$ is the reduced Planck mass and $M$ is set by CMB normalization. By introducing an auxiliary scalar and transforming to the Einstein frame, the theory becomes equivalent to canonical gravity plus a scalar field $\phi$ with the potential

$$V_S(\phi) = \frac{3}{4} M_P^2 M^2 \left(1 - e^{-\sqrt{2/3}\,\phi/M_P}\right)^2$$

which supports slow-roll inflation on a plateau at large field values.

A generic one-parameter deformation is introduced via a modification of the function $f(\phi)$ that replaces the auxiliary coupling in the Jordan frame. In universal attractor-type models, the deformation is characterized by an exponent $n$ in

$f(\phi) \sim \frac{\phi^n}{M_P^{n-2}}$

with the Jordan frame potential $V_J(\phi) \sim f(\phi)^2$. The parameter $n$ serves as the deformation parameter; $n=1$ recovers non-minimally coupled quadratic chaotic inflation and $n=2$ corresponds to Higgs inflation.

This class of action generalizes as: $$S = \int d^4x\, \sqrt{-g}\, \left[\frac{M_P^2}{2} R + f(\phi)\, R - V_J(\phi) \right]$$ During the inflationary regime, the kinetic term is negligible, $\phi$ becomes auxiliary and can be integrated out, yielding an effective $R + R^2$ action with corrections set by $n$.

2. Role of the Deformation in Inflationary Dynamics

For large $\phi$, all models with $V_J(\phi) \sim f(\phi)^2$ and $f(\phi) \sim \phi^n$ collapse to an effective $R + R^2$ theory whose Einstein-frame potential asymptotes to the Starobinsky form. The inflationary predictions—scalar spectral index $n_s$ and tensor-to-scalar ratio $r$—are nearly universal at leading order: $$n_s \approx 1 - \frac{2}{N}, \quad r \approx \frac{12}{N^2}$$ where $N$ is the number of e-folds. The differences between descendants and the Starobinsky model appear in higher-order corrections ($10^{-3}$–$10^{-5}$) to $n_s$ and $r$, determined by the precise kinetic structure in the Jordan frame. These differences are observationally insignificant with current data, but may be relevant for future high-precision measurements.

3. Deformation and UV Cutoff: Effective Field Theory Implications

The kinetic term structure imposed by the deformation parameter $n$ controls the UV cutoff scale $A$ of the effective field theory. For universal attractor models,

• If $n < 7/3$ (including the Higgs inflation case $n=2$), the UV cutoff is lower than the inflationary energy, $A \lesssim V^{1/4}$.
• If $n=1$ (non-minimally coupled quadratic chaotic inflation), the UV cutoff is restored to the Planck scale, $A = M_P$.

This distinction is rooted in whether the inflaton is an auxiliary field (as in Starobinsky, implying no kinetic term and high cutoff) or a true propagating field (as in general attractor models, lowering the cutoff). As a result, only specific deformations (notably $n=1$) provide a radiatively stable inflationary sector valid up to Planckian scales, crucial for the reliability of any inflationary effective theory.

4. Special Case: Non-Minimally Coupled Quadratic Chaotic Inflation

The non-minimally coupled quadratic chaotic inflation model has the action

$$S = \int d^4x\, \sqrt{-g}\, \left[\left(\frac{M_P^2}{2} + \xi\phi^2\right) R - \frac{1}{2}(\partial\phi)^2 - \frac{1}{2} m^2 \phi^2\right]$$

During inflation, integrating out $\phi$ leads to an effective

$$S_{\mathrm{eff}} = \frac{1}{2} \int d^4x\, \sqrt{-g}\, \left[ M_P^2 R + \frac{1}{6M_{\mathrm{eff}}^2} R^2 \right]$$

with $M_{\mathrm{eff}}^2/(12M_P^2) = g(\xi,m)$. This mapping ensures matching slow-roll observables with the Starobinsky predictions, while the cutoff $A = M_P$ ensures theoretical consistency. Unlike the minimally coupled quadratic model (ruled out by the large predicted $r$), this non-minimal version (with $n=1$) supports inflation in agreement with Planck data and is valid up to the Planck scale.

5. Comparison of Descendant Models and UV Consistency

The following table summarizes the main features for different values of the deformation parameter $n$:

Model	$n$ value	UV Cutoff $A$	Inflationary Observables
Starobinsky	—	$M_P$	$n_s \simeq 1-2/N$, $r\simeq12/N^2$
Higgs inflation	2	$A < V^{1/4}$	As Starobinsky (small corrections)
Universal attractor (gen)	$<7/3$	$A < V^{1/4}$	As Starobinsky (small corrections)
Non-min. quadratic chaos	1	$M_P$	As Starobinsky

For $n < 7/3$ (Higgs inflation, generic attractors), the UV cutoff can be below the energy scale of inflation, potentially invalidating the effective theory during observable inflationary evolution. For $n=1$, both the inflationary predictions and the UV consistency of the theory are maintained.

6. Data Compatibility and Theoretical Selectivity

Planck (and subsequent) observations support a scalar tilt $n_s \sim 0.96$ and a low tensor-to-scalar ratio $r \lesssim 0.01$. All models reducing to the Starobinsky dynamics for large-field inflation—regardless of the detailed value of $n$—fit these observables up to negligible corrections for slow-roll parameters, as slow-roll dynamics erases dependence on nonleading kinetic terms. However, only certain one-parameter deformations (specifically, those with UV-safe cutoff, i.e., $n=1$) possess the theoretical consistency required of a robust inflationary setup, in particular regarding the validity of the effective field theory throughout the inflationary regime.

7. Implications and Theoretical Significance

The idea that a one-parameter deformation (encoded by $n$) governs both inflationary observables and the theory's ultraviolet cutoff links cosmological predictions to underlying high-energy theory structure. While a broad class of universal attractor models and their variants can fit CMB observations, theoretical viability requires attention to the UV cutoff set by the kinetic sector. In this framework, only specific deformations (notably, the non-minimally coupled quadratic chaotic inflation with $n=1$) remain viable: they reproduce Starobinsky-like inflation, comply with Planckian cutoff requirements, and avoid inconsistency with the anticipated energy scales during the early universe.

This synthesis clarifies the internal structure of the Starobinsky model and its deformations and demonstrates how a one-parameter deformation, controlled by the exponent $n$, maps out the space of inflationary models that both match observation and maintain theoretical consistency (Kehagias et al., 2013).