Inflation and Primordial Fluctuations

Inflation links four pieces of physics that are often taught separately: accelerated expansion shrinks the comoving Hubble radius, a slowly evolving field supplies a concrete background, canonical quantum modes acquire long-wavelength correlations, and reheating maps those modes into the later thermal universe. This chapter develops that chain with fixed signs and normalizations, then tests which parts are distinctive once noncanonical, multifield, bouncing, emergent, and modified-initial-state scenarios are allowed.

Helpful background. Einstein Equations introduce the gravitational constraints, while relativistic matter explains stress–energy and energy conditions. Classical Field Modes and Canonical Variables prepares the oscillator normalization and quantum-state discussion. The cosmology overview locates the background and perturbation chapters used here.

Enter the inflationary calculation

The six pages follow the logical dependency of the calculation.

1. Inflationary Kinematics and the Comoving Horizon proves that accelerated expansion is equivalent to a shrinking comoving Hubble radius, solves constant-$\epsilon_H$ expansion, and separates Hubble crossing from particle and event horizons.
2. Scalar-Field Inflation and Slow Roll derives the exact scalar–gravity system, obtains potential slow-roll conditions, and checks the e-fold approximation against an exact numerical trajectory through inflation end.
3. Quantizing Primordial Perturbations derives the scalar and tensor canonical oscillators, fixes their Wronskians and state assumptions, and explains what frozen, classical-looking correlations do and do not mean.
4. Primordial Scalar and Tensor Spectra fixes the two-point normalization, amplitudes, tilts, running, pivot dependence, and the approximation boundary of the canonical consistency relation.
5. Reheating and the Expansion History separates decay from thermalization and propagates matter-like, radiation-like, stiff, perturbative, resonant, and effective-$w$ histories into the $N_*$–$k$ map.
6. Alternatives and Tests of the Inflationary Paradigm compares complete primordial formulations through stability, matching, scalar and tensor spectra, non-Gaussian shapes, isocurvature, and relics under one late-time likelihood.

For the canonical calculation, read pages 1–4 in order and then page 5. For a model-comparison route, establish the convention package below, read pages 4 and 5, and use page 6 to avoid treating one shared scalar tilt as a unique diagnostic.

One convention package

Cosmic time is $t$, conformal time is $\eta$, and

$$H=\frac{\dot a}{a}, \qquad \mathcal H=\frac{a'}{a}=aH.$$

Uppercase $K$ denotes FLRW spatial curvature; lowercase $k$ is the magnitude of a comoving Fourier wavevector in the $e^{i\mathbf k\cdot\mathbf x}$ convention. The reduced Planck mass is an abbreviation,

$$M\equiv(8\pi G)^{-1/2},$$

while equations retain explicit $G$ when that makes the gravitational normalization clearer.

The e-fold variable $N=\ln a$ increases forward in time. By contrast,

$$N_*\equiv\ln\left(\frac{a_{\rm end}}{a_*}\right)$$

counts from a mode’s crossing time to the end of inflation. Confusing these orientations reverses slow-roll and reheating derivatives.

The curvature sign is inherited from the cosmological-perturbation chapter:

$$\mathcal R=-\psi+\mathcal H\frac{\delta q}{\rho+p}.$$

For a canonical inflaton this is the negative of the curvature variable in some standard inflation sources. With $z=a\dot\phi/H$, this chapter therefore uses

$$v=-z\mathcal R, \qquad z^2=2a^2M^2\epsilon_1.$$

Power spectra are unaffected by that overall sign. Tensor polarization tensors obey $e_{ij}^{\lambda}e^{\lambda'ij}=2\delta_{\lambda\lambda'}$, the metric uses the direct decomposition $h_{ij}=\sum_\lambda e_{ij}^{\lambda}h_\lambda$, and $u_\lambda=aMh_\lambda/\sqrt2$ is canonical. The standard tensor spectrum is the contraction-normalized quantity

$$\mathcal P_t=2\sum_\lambda\mathcal P_{h_\lambda},$$

not the unweighted sum of the two component spectra.

The equation network

The first Hubble-flow parameter controls the background:

$$\epsilon_1\equiv-\frac{\dot H}{H^2}, \qquad \frac{\ddot a}{a}=H^2(1-\epsilon_1),$$ $$\frac{d}{dN}\ln(aH)^{-1}=-(1-\epsilon_1).$$

Thus inflation is $\epsilon_1\lt 1$. A canonical homogeneous scalar realizes it through

$$H^2=\frac{8\pi G}{3} \left(\frac12\dot\phi^2+V\right), \qquad \ddot\phi+3H\dot\phi+V_{,\phi}=0.$$

The next flow parameter, $\epsilon_2=d\ln\epsilon_1/dN$ for $\epsilon_1\gt0$, measures how rapidly the first changes. It enters the scalar tilt because the curvature amplitude depends on both $H$ and the speed of the background field.

On the slow-roll branch,

$$3H\dot\phi\simeq-V_{,\phi}, \qquad H^2\simeq\frac{V}{3M^2},$$

and the approximate e-fold integral is

$$N_*\simeq\frac{1}{M^2} \int_{\phi_{\rm end}}^{\phi_*} \frac{V}{V_{,\phi}}\,d\phi.$$

After the scalar constraints are eliminated, every Fourier mode obeys

$$v_k''+\left(k^2-\frac{z''}{z}\right)v_k=0, \qquad v_kv_k^{*\prime}-v_k^*v_k'=i.$$

An adiabatic Bunch–Davies state begins as $v_k=e^{-ik\eta}/\sqrt{2k}$. On the canonical slow-roll attractor the long-wavelength solution freezes, giving at leading order

$$\mathcal P_{\mathcal R} =\frac{H_*^2}{8\pi^2M^2\epsilon_{1*}}, \qquad \mathcal P_t=\frac{2H_*^2}{\pi^2M^2},$$ $$n_s-1=-2\epsilon_{1*}-\epsilon_{2*}, \qquad n_t=-2\epsilon_{1*}, \qquad r=16\epsilon_{1*}=-8n_t.$$

Reheating then closes the scale map. Relative to a radiation-like history, a constant effective $w_{\rm re}\ne-1$ with fixed endpoint densities and degrees of freedom shifts the crossing assignment by

$$\Delta N_* =\frac{1-3w_{\rm re}}{12(1+w_{\rm re})} \ln\left(\frac{\rho_{\rm re}}{\rho_{\rm end}}\right).$$

These relations form one chain: reheating changes the field time associated with a measured $k$, which changes the background quantities inserted into the primordial spectrum.

Approximation boundaries

Each compact result has a different domain.

Result	Assumptions	Failure or extension
shrinking $(aH)^{-1}$	expanding background with $H\gt 0$ and $\ddot a\gt 0$	it is not a particle or event horizon and is undefined as a crossing variable at $H=0$
potential slow roll	canonical scalar on a reached attractor with small $\epsilon_V$ and $\eta_V$	overshoot, sharp features, nonattractor phases, turns, and noncanonical kinetics require exact evolution
Bunch–Davies mode	an adiabatic short-wavelength regime below the effective-theory cutoff	finite onset, excited states, and nonadiabatic evolution require another state prescription
frozen $\mathcal R$	regular adiabatic growing mode on an expanding attractor	entropy transfer or a decreasing pump field can evolve curvature outside the Hubble radius
$r=16\epsilon_1$	canonical single-field, leading slow roll, common pivot, no later scalar transfer	sound speed, multifield transfer, modified gravity, and next-order corrections alter the relation
constant-$w$ reheating map	one effective background equation of state and a declared thermalization endpoint	preheating, entropy production, or time-dependent mixtures require integration of the full history

The standard derivations and their normalization are developed in Baumann 2012, §§4–13, printed pp. 22–63 (Open PDF) and Ellis, Vennin, and Wands 2025/2026, §§23.1–23.6, printed pp. 1–22 (Official PDF).

Scope and onward routes

This chapter treats representative mechanisms, not a catalog of inflationary potentials or particle embeddings. Its first four pages present durable analytic physics. The reheating and alternatives pages give dated evidence through 2026-08-19 and state what would require reassessment rather than turning current null results into permanent verdicts.

Cosmological Perturbations and Gauge Invariance develops the general gauge-consistent linear system. Inflationary Fluctuations as Curved-Spacetime Fields develops the deeper curved-spacetime field theory. Thermal History and Relic Abundances develops entropy, thermalization, and relic calculations after inflation. The Cosmic Microwave Background carries primordial spectra through transfer functions, foregrounds, and likelihoods to observed anisotropies.

Alternative primordial backgrounds are compared here only far enough to show why stability, matching, full correlators, and common late-time inference are essential; Brandenberger and Peter 2017, §§II–VI, pp. 3–22 (Open PDF) surveys the contracting and bouncing mechanisms. Global singularity and completeness questions belong to Cosmological Horizons and Global Evolution.

Review the chapter

A canonical single-field stage has constant $\epsilon_1=0.01$, $\epsilon_2=0$, and lasts 50 e-folds after a selected mode is well inside the Hubble radius. Assume an adiabatic Bunch–Davies state and use leading slow-roll order. It is followed by a matter-like reheating phase with $\rho_{\rm re}/\rho_{\rm end}=10^{-12}$. Find the Hubble-radius suppression, the leading $(n_s,n_t,r)$, and the reheating shift in $N_*$. State one reason these results do not uniquely establish the inflationary paradigm.

Solution

Constant $\epsilon_1$ gives

$$\frac{(aH)^{-1}_f}{(aH)^{-1}_i} =e^{-(1-\epsilon_1)50} =e^{-49.5} =3.18\times10^{-22}.$$

At leading slow-roll order,

$$n_s=1-2\epsilon_1-\epsilon_2=0.98, \qquad n_t=-2\epsilon_1=-0.02,$$ $$r=16\epsilon_1=0.16=-8n_t.$$

For $w_{\rm re}=0$,

$$\Delta N_*=\frac1{12}\ln(10^{-12})=-2.302585.$$

Thus a fixed observed $k$ is assigned about $2.30$ fewer inflationary e-folds before the end than in the radiation-like reference history. In this constant-$\epsilon_1$ exercise the leading tilt and ratio do not vary with that reassignment; in a realistic potential they generally do.

The calculation remains model conditional. Matter contraction can also produce a scale-invariant scalar spectrum in an idealized linear treatment, while multifield or noncanonical inflation can violate the displayed consistency relation. Distinguishing scenarios requires tensor tilt, non-Gaussian and isocurvature information, transition stability, and a common late-time likelihood—not the scalar tilt alone.

References

• Baumann, D., “TASI Lectures on Inflation,” arXiv:0907.5424v2 (2012), Open PDF.
• Brandenberger, R., and P. Peter, “Bouncing Cosmologies: Progress and Problems,” Foundations of Physics 47, 797–850 (2017), DOI, Open PDF.
• Ellis, J., V. Vennin, and D. Wands, “Inflation,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), revised August 2025, Official PDF.