Inflationary Models

5.1 The Landscape of Inflationary Models

The inflationary paradigm posits a period of quasi-exponential expansion in the very early universe, driven by the potential energy of a scalar field — the inflaton. While the general framework of slow-roll inflation makes robust predictions (a nearly scale-invariant, nearly Gaussian spectrum of adiabatic perturbations), the specific predictions for the spectral index $n_s$ and the tensor-to-scalar ratio $r$ depend critically on the choice of inflaton potential $V(\phi)$.

The observational constraints from the Planck satellite (2018) and BICEP/Keck Array place tight bounds:

These constraints already rule out several simple models and strongly favor plateau-like potentials. The key observables are related to the slow-roll parameters via:

where the potential slow-roll parameters are:

In this chapter, we derive the predictions of five major classes of inflationary models from first principles, compute their $n_s$ and $r$ values, and compare them against the data. We will see a remarkable convergence: many successful models share the universal prediction$n_s \approx 1 - 2/N$, differing only in $r$.

5.2 Chaotic Inflation: $m^2\phi^2$

The simplest inflationary model is chaotic inflation with a massive free field, proposed by Andrei Linde in 1983. The potential is:

Step 1: Compute the slow-roll parameters. We need the first and second derivatives of the potential:

Substituting into the slow-roll parameter definitions:

Note the special feature of $m^2\phi^2$: we have $\epsilon_V = \eta_V$ exactly.

Step 2: Find the end of inflation. Inflation ends when$\epsilon_V = 1$:

Step 3: Compute the number of e-folds. The e-fold integral is:

Step 4: Invert for $\phi_*(N)$. Solving for $\phi_*$:

Step 5: Derive $n_s$ and $r$. Substituting$\phi_*^2 \approx 4NM_P^2$ into the slow-roll parameters:

Therefore:

The $\phi^4$ Case (Also Ruled Out)

For completeness, consider $V(\phi) = \lambda\phi^4/4$. We have $V' = \lambda\phi^3$ and $V'' = 3\lambda\phi^2$. Then:

The end of inflation: $\epsilon_V = 1$ gives $\phi_\text{end}^2 = 8M_P^2$. The e-fold integral yields $N = (\phi_*^2 - 8M_P^2)/(8M_P^2)$, so $\phi_*^2 \approx 8NM_P^2$ for large $N$. Thus:

For $N = 60$: $n_s = 0.950$, $r = 0.267$ — both $n_s$ and $r$ are far outside the Planck contours. The $\phi^4$ model was already under pressure from WMAP and is now decisively ruled out.

5.3 Starobinsky $R^2$ Inflation

The Starobinsky model (1980) was historically the first inflationary model, proposed even before Guth's 1981 paper. It modifies the Einstein-Hilbert action by adding a quadratic curvature term:

where $M$ is a mass scale to be fixed by the amplitude of perturbations ($M \sim 10^{13}$ GeV).

Step 1: Conformal Transformation to Einstein Frame

Any $f(R)$ theory can be recast as standard Einstein gravity plus a scalar field via a conformal transformation. Define:

Perform the conformal (Weyl) rescaling of the metric:

In the Einstein frame (with tildes), the action becomes:

Step 2: Identify the Scalar Field

The canonical scalar field is related to $F$ by:

Inverting: $F = e^{\sqrt{2/3}\,\phi/M_P}$, so $R = 3M^2(F - 1) = 3M^2(e^{\sqrt{2/3}\,\phi/M_P} - 1)$.

Step 3: Derive the Plateau Potential

The Einstein-frame potential is given by the general $f(R)$ formula:

Substituting $f(R) = R + R^2/(6M^2)$:

Now express $R$ in terms of $F$: $R = 3M^2(F-1)$, so $R^2 = 9M^4(F-1)^2$:

Using $F = e^{\sqrt{2/3}\,\phi/M_P}$, we write $(F-1)/F = 1 - 1/F = 1 - e^{-\sqrt{2/3}\,\phi/M_P}$:

This is the celebrated Starobinsky plateau potential. For large$\phi$, the exponential is negligible and $V \to 3M^2 M_P^2/4$ = const — a flat plateau that naturally gives slow-roll inflation.

Step 4: Slow-Roll Parameters on the Plateau

Define $x = e^{-\sqrt{2/3}\,\phi/M_P}$ for convenience (during inflation, $\phi$ is large and $x \ll 1$). Then:

More directly, we compute $V'/V$:

For $\eta_V$, we compute $V''/V$:

Step 5: Relate to e-folds and Derive $n_s$, $r$

The number of e-folds (for $x \ll 1$) is:

Substituting into the slow-roll parameters:

Therefore:

5.4 Natural Inflation

Natural inflation (Freese, Frieman & Olinto, 1990) uses a pseudo-Nambu-Goldstone boson as the inflaton, providing a natural mechanism for the flatness of the potential. The potential arises from explicit breaking of a shift symmetry:

where $\Lambda$ is the energy scale of the potential and $f$ is the axion decay constant (the symmetry-breaking scale).

Step 1: Slow-Roll Parameters

Computing the derivatives:

The slow-roll parameters become:

where we used $\sin^2\theta/(1+\cos\theta)^2 = (1-\cos\theta)/(1+\cos\theta)$ (the identity $\sin^2\theta = (1-\cos\theta)(1+\cos\theta)$).

Step 2: Predictions as Functions of $N$ and $f$

The e-fold integral can be evaluated analytically:

In the limit $f \gg M_P$ (large decay constant), the potential near its minimum becomes approximately quadratic: $V \approx \Lambda^4\phi^2/(2f^2)$, recovering chaotic inflation. In the opposite limit$f \lesssim M_P$, the potential is too steep and inflation cannot produce enough e-folds.

The spectral index and tensor-to-scalar ratio are:

In the large-$f$ limit, expanding to leading order in $M_P^2/f^2$:

recovering the $m^2\phi^2$ predictions, as expected.

The $f > M_P$ Requirement and Weak Gravity Conjecture

For natural inflation to produce $N \geq 50$ e-folds with $n_s$ consistent with Planck, one needs:

This super-Planckian decay constant is problematic from the perspective of quantum gravity. The weak gravity conjecture (Arkani-Hamed, Motl, Nicolis & Vafa, 2007) suggests that in any consistent theory of quantum gravity, the axion decay constant should satisfy $f \lesssim M_P$. If correct, this would rule out natural inflation in its simplest form.

Various mechanisms have been proposed to circumvent this tension, including alignment of multiple axions (Kim, Nilles & Peloso, 2005) and monodromy (Silverstein & Westphal, 2008), though each introduces additional model-building complications. Current Planck+BICEP data place natural inflation under significant observational pressure even for large $f$.

5.5 $\alpha$-Attractors

The $\alpha$-attractor models (Kallosh & Linde, 2013; Kallosh, Linde & Roest, 2013) represent a broad universality class of inflationary potentials motivated by supergravity and the geometry of the Kähler manifold. They come in two families:

T-models:

E-models:

The parameter $\alpha$ controls the curvature of the Kähler manifold, $n$ determines the power, and $V_0$ sets the energy scale. For $n = 1$, the E-model with $\alpha = 1$ reproduces exactly the Starobinsky potential.

Universal Predictions in the Large-Field Limit

The remarkable property of $\alpha$-attractors is that for any value of $n$, the predictions for $n_s$ and $r$ converge to universal values in the limit of large field excursion (large $N$). To see this, consider the T-model with general $n$. Define$y = \tanh(\phi/(\sqrt{6\alpha}\,M_P))$. For large $\phi$, $y \to 1$ and we write$y = 1 - \delta$ with $\delta \ll 1$.

The slow-roll parameter $\epsilon_V$ is:

The e-fold calculation gives:

Substituting back:

Note that $n$ has completely cancelled! Similarly, $\eta_V \approx -1/N$. Therefore:

This is the attractor behavior: the spectral index is universal ($n$-independent), while $r$ depends only on $\alpha$. The Starobinsky model corresponds to $\alpha = 1$ ($r = 12/N^2$).

5.6 Higgs Inflation

Higgs inflation (Bezrukov & Shaposhnikov, 2008) is the remarkable idea that the Standard Model Higgs boson itself can serve as the inflaton, provided it has a large non-minimal coupling to gravity. The Jordan-frame action is:

where $h$ is the Higgs field (in unitary gauge), $\xi$ is the non-minimal coupling ($\xi \sim 10^4$ for the correct amplitude of perturbations), $\lambda$ is the Higgs quartic coupling, and $v = 246$ GeV is the electroweak vacuum expectation value.

Step 1: Conformal Transformation

Define the conformal factor:

Perform the Weyl rescaling $\tilde{g}_{\mu\nu} = \Omega^2 g_{\mu\nu}$ to go to the Einstein frame:

Step 2: Canonical Scalar Field

After the conformal transformation, the kinetic term for $h$ is non-canonical. The canonically normalized field $\chi$ satisfies:

In the inflationary regime where $h \gg M_P/\sqrt{\xi}$ (so $\Omega^2 \approx \xi h^2/M_P^2 \gg 1$and $6\xi \gg 1$):

Integrating: $\chi = \sqrt{6}\,M_P\,\ln(h\sqrt{\xi}/M_P)$, or equivalently:

Step 3: The Einstein-Frame Plateau Potential

The Einstein-frame potential is:

For $h \gg M_P/\sqrt{\xi}$:

This is a constant — a plateau! More precisely, keeping the correction:

This has exactly the same functional form as the Starobinsky potential! Therefore, the predictions are identical:

The mass scale is fixed by the amplitude of scalar perturbations: $\lambda/\xi^2 \sim 10^{-10}$, which for $\lambda \sim 0.01$ (the Higgs self-coupling at the inflationary scale) gives$\xi \sim 10^4$.

Unitarity Concerns

The main theoretical concern with Higgs inflation is perturbative unitarity. With $\xi \sim 10^4$, the effective cutoff scale of the Jordan-frame theory in the vacuum ($h \sim v$) is:

This is below the inflationary Hubble scale $H_\text{inf} \sim 10^{13}$ GeV only marginally. The concern is that new physics below this scale could invalidate the inflationary calculation. However, Bezrukov and Shaposhnikov argued that the relevant cutoff during inflation is background-dependent and higher: $\Lambda \sim M_P/\sqrt{\xi} \sim 10^{16}$ GeV, which is safely above $H_\text{inf}$. The debate on this subtle point continues in the literature.

Despite these concerns, Higgs inflation remains attractive as a minimal model: it requires no new particles beyond the Standard Model, only a single new coupling constant $\xi$.

5.7 Model Comparison and Bayesian Selection

The $n_s$–$r$ Plane

The most informative way to compare models is in the $n_s$–$r$ plane. Each model traces a curve or region parameterized by the number of e-folds $N$ (and sometimes additional parameters like $f/M_P$ or $\alpha$). The Planck 2018 + BICEP/Keck 2021 data define confidence contours in this plane. Here is a summary:

Bayesian Model Selection with Planck Data

Bayesian model comparison computes the evidence (marginal likelihood) for each model:

The Bayes factor between models $M_i$ and $M_j$ is $B_{ij} = \mathcal{Z}_i/\mathcal{Z}_j$. Planck 2018 analyses using this framework consistently find:

• Strong preference for Starobinsky/$R^2$ and $\alpha$-attractor models with small $\alpha$
• Decisive evidence against $\phi^4$ and monomial models with $p \geq 2$
• Moderate tension for natural inflation even with $f = 10\,M_P$
• Higgs inflation is statistically indistinguishable from Starobinsky (identical predictions for $n_s$, $r$)

The key discriminant between surviving models is the tensor-to-scalar ratio $r$. Future CMB experiments (CMB-S4, LiteBIRD) aim for sensitivity $\sigma(r) \sim 10^{-3}$, which would definitively test Starobinsky-class models.

5.8 Historical Context

5.9 Python Simulation

The following simulation plots the $n_s$–$r$ plane with approximate Planck 2018 + BICEP/Keck confidence contours, overlaid with predictions from all five models discussed in this chapter. It also plots the potential $V(\phi)$ for each model.