Cosmological inflation: slow-roll scalar fields and the origin of structure

Intuition Beginner

The universe is uncannily flat. Measure how much the geometry of space deviates from perfectly flat and you find it is flat to better than one part in a thousand. In an ordinary hot Big Bang that flatness is unstable: any tiny curvature grows with time. For the universe to be this flat today, it must have begun flat to one part in $10^{60}$, a coincidence so extreme it begs for an explanation. This is the flatness problem.

Look at the cosmic microwave background, the afterglow of the early universe. It has almost exactly the same temperature in every direction, to one part in $100000$. Yet opposite sides of the sky were so far apart when that light was set free that no signal could ever have crossed between them. There is no reason they should agree. This is the horizon problem. Both puzzles dissolve if the very early universe went through a moment of extraordinarily rapid, accelerated expansion called inflation.

During inflation a patch of space doubled in size, then doubled again, dozens of times in a tiny fraction of a second. The engine was a field called the inflaton, filling all of space and slowly rolling down its energy landscape like a ball on a wide, nearly flat plateau. That slow roll made the expansion accelerate. Inflation stretched any curvature flat and brought distant regions into contact. And the same field carried tiny quantum jitters, which inflation stretched to cosmic size and froze into place as the seeds of every galaxy.

Visual Beginner

Picture a graph of the inflaton's energy $V$ as a function of the field value $\varphi$. The curve is a wide, gently sloping plateau on top, then drops steeply at one end. A ball placed on the plateau rolls down very slowly at first: while it creeps, the universe inflates. When the slope steepens the ball picks up speed, slow roll ends, and the field's stored energy converts into a hot bath of particles in a process called reheating.

The same story told in horizons: during inflation the distance light could travel in one expansion time, the Hubble radius, stayed nearly fixed while space itself ballooned, so regions once in contact were swept out of contact. Reversing that, any quantum ripple of the field smaller than the Hubble radius was stretched past it and frozen, becoming a permanent density seed.

Worked example Beginner

After $N$ doublings of the scale factor, the size of a region has grown by a factor $e^N$, where $e\approx 2.718$. Inflation needs roughly $N=60$ of these e-folds to solve the flatness and horizon problems. The total growth factor is $e^{60}$.

Since $e^{60}\approx 10^{26}$, a region the size of a proton, about $10^{-15}$ metres across, is stretched to roughly $10^{11}$ metres, a hundred million kilometres, in that brief instant. A patch far smaller than an atom becomes larger than the entire observable universe. The number $60$ comes from requiring the patch that contains today's visible cosmos to have been inflated beyond the smallest meaningful length, the Planck length, $10^{-35}$ metres.

What this tells us: the power of inflation is exponential. Sixty quiet e-folds turn subatomic patches into cosmic ones, which is exactly how a single process both flattens the geometry and lets every part of the sky share a common thermal history.

Check your understanding Beginner

Formal definition Intermediate+

This unit deepens the inflation sketch of 13.08.02 into a dynamical theory: a scalar field drives an epoch of accelerated expansion, and its quantum fluctuations generate the primordial perturbations. The FLRW background and Friedmann equations of 13.08.01 are taken as given.

An inflaton is a real scalar field $\varphi (t,\mathbf{x})$ minimally coupled to gravity with potential $V(\varphi )$. The action is ^{[Mukhanov 2005]}

$$S=\int d^{4}x\sqrt{-g}\left[\frac{M_{\mathrm{Pl}}^2}{2}R-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -V(\varphi )\right],$$

with reduced Planck mass $M_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}\approx 2.435\times 10^{18}$ GeV. For a homogeneous configuration $\varphi (t)$ the stress-energy is that of a perfect fluid,

$${\rho }_{\varphi }=\frac{1}{2}{\dot{\varphi }}^2+V(\varphi ),p_{\varphi }=\frac{1}{2}{\dot{\varphi }}^2-V(\varphi ),$$

so the equation of state $w_{\varphi }=p_{\varphi }/{\rho }_{\varphi }$ tends to $-1$ whenever the kinetic energy $\frac{1}{2}{\dot{\varphi }}^2$ is negligible against $V$. Feeding ${\rho }_{\varphi }$ into the Friedmann equations of 13.08.01 yields the coupled system

$$H^2=\frac{1}{3M_{\mathrm{Pl}}^2}\left(\frac{1}{2}{\dot{\varphi }}^2+V\right),\ddot{\varphi }+3H\dot{\varphi }+V_{,\varphi }=0,$$

the second equation being the Klein-Gordon equation in the FLRW background; the friction term $3H\dot{\varphi }$ drains kinetic energy into the expansion ^{[Baumann 2009]}.

Inflation is the regime of accelerated expansion, equivalently a shrinking comoving Hubble radius:

$$\frac{d}{dt}(aH{)}^{-1}<0⟺\ddot{a}>0⟺{\epsilon }_H<1,{\epsilon }_H\equiv -\frac{\dot{H}}{H^2}.$$

The Hubble-flow slow-roll parameters are

$${\epsilon }_H\equiv -\frac{\dot{H}}{H^2},{\eta }_H\equiv -\frac{\ddot{\varphi }}{H\dot{\varphi }},$$

and slow roll means ${\epsilon }_H\ll 1$ and $|{\eta }_H|\ll 1$, so $\ddot{\varphi }$ is negligible and the field tracks the slow-roll attractor

$$3H\dot{\varphi }\approx -V_{,\varphi },H^2\approx \frac{V}{3M_{\mathrm{Pl}}^2}.$$

Equivalent potential slow-roll parameters depend only on the shape of $V$ ^{[Liddle & Lyth 2000]}:

$${\epsilon }_V\equiv \frac{M_{\mathrm{Pl}}^2}{2}{\left(\frac{V_{,\varphi }}{V}\right)}^2,{\eta }_V\equiv M_{\mathrm{Pl}}^2\frac{V_{,\varphi \varphi }}{V},$$

with ${\epsilon }_H\approx {\epsilon }_V$ and ${\eta }_H\approx {\eta }_V-{\epsilon }_V$ to leading order. The number of e-folds before inflation ends at ${\epsilon }_H=1$ is

$$N(\varphi )={\int }_{t_{\mathrm{start}}}^{t_{\mathrm{end}}}Hdt=\frac{1}{M_{\mathrm{Pl}}^2}{\int }_{\varphi }^{{\varphi }_{\mathrm{end}}}\frac{V}{V_{,\varphi }}d\varphi ,$$

and viable inflation requires $N\gtrsim 60$ for the present horizon.

The comoving curvature perturbation $\mathcal{R}$ measures the spatial curvature on slices of uniform density; on super-horizon scales it is conserved and is related to a field fluctuation by $\mathcal{R}=H\delta \varphi /\dot{\varphi }$. A Fourier mode $k$ exits the horizon when $k=aH$; during inflation the shrinking $(aH{)}^{-1}$ carries each mode from sub- to super-horizon, where it freezes. Reheating is the post-inflationary conversion of the inflaton's coherent oscillation energy into relativistic particles at temperature $T_{\mathrm{reh}}\sim ({\Gamma }_{\varphi }{M}_{\mathrm{Pl}}{)}^{1/2}$, with ${\Gamma }_{\varphi }$ the inflaton decay width ^{[Mukhanov 2005]}.

The observable predictions are the scalar power spectrum $P_{\mathcal{R}}(k)=A_s(k/k_{\ast }{)}^{n_s-1}$, the tensor-to-scalar ratio $r=P_t/P_{\mathcal{R}}$, and the single-field consistency relation $r=-8n_t$, all derived in the next section.

Key derivation Intermediate+

The slow-roll mechanism and the attractor equation

In slow roll the acceleration $\ddot{\varphi }$ is a small correction to the friction $3H\dot{\varphi }$. Dropping $\ddot{\varphi }$ in the Klein-Gordon equation and the kinetic term $\frac{1}{2}{\dot{\varphi }}^2$ in Friedmann gives the attractor equations

$$3H\dot{\varphi }\approx -V_{,\varphi },3M_{\mathrm{Pl}}^2{H}^2\approx V.$$

Solving the first for $\dot{\varphi }$ and using $H^2\approx V/(3M_{\mathrm{Pl}}^2)$ gives the field velocity on the attractor,

$$\dot{\varphi }\approx -\frac{V_{,\varphi }}{3H}=-\frac{M_{\mathrm{Pl}}{V}_{,\varphi }}{\sqrt{3V}}.$$

To connect this to ${\epsilon }_H$, differentiate $H^2={\rho }_{\varphi }/(3M_{\mathrm{Pl}}^2)$ and use the continuity equation ${\dot{\rho }}_{\varphi }=-3H{\dot{\varphi }}^2$ of 13.08.01:

$$2H\dot{H}=\frac{{\dot{\rho }}_{\varphi }}{3M_{\mathrm{Pl}}^2}=-\frac{H{\dot{\varphi }}^2}{M_{\mathrm{Pl}}^2}⟹\dot{H}=-\frac{{\dot{\varphi }}^2}{2M_{\mathrm{Pl}}^2},$$

so ${\epsilon }_H=-\dot{H}/H^2={\dot{\varphi }}^2/(2M_{\mathrm{Pl}}^2{H}^2)$. Substituting the attractor ${\dot{\varphi }}^2=M_{\mathrm{Pl}}^2{V}_{,\varphi }^2/(3V)$ and $H^2=V/(3M_{\mathrm{Pl}}^2)$,

$${\epsilon }_H=\frac{M_{\mathrm{Pl}}^2{V}_{,\varphi }^2/(3V)}{2M_{\mathrm{Pl}}^2\cdot V/(3M_{\mathrm{Pl}}^2)}=\frac{M_{\mathrm{Pl}}^2}{2}{\left(\frac{V_{,\varphi }}{V}\right)}^2={\epsilon }_V.$$

Thus ${\epsilon }_H={\epsilon }_V$ on the attractor, and inflation (${\epsilon }_H<1$) is the statement that the potential is flat enough that $V_{,\varphi }$ is small compared with $V/M_{\mathrm{Pl}}$. The attractor is also stable: an upward perturbation of $|\dot{\varphi }|$ raises ${\dot{\varphi }}^2$, raises ${\epsilon }_H$, raises the Hubble friction, and damps the perturbation back.

Flatness, horizons, and the spectrum of perturbations

The flatness and horizon problems are resolved kinematically by the shrinking comoving Hubble radius. The curvature density parameter is ${\Omega }_k=-k{c}^2/(a^2{H}^2)\propto (aH{)}^{-2}$, so when $(aH{)}^{-1}$ shrinks, ${\Omega }_k$ decays. During slow roll $a\propto e^{Ht}$ with $H$ nearly constant, hence $|{\Omega }_k|\propto e^{-2N}$, an exponential suppression. The horizon problem is resolved because the shrinking $(aH{)}^{-1}$ means comoving regions that are outside contact today were inside it before inflation stretched them apart.

The central dynamical output is the spectrum of primordial curvature perturbations. Quantising the field fluctuation $\delta \varphi$ on the Bunch-Davies vacuum (treated in the Master tier) gives each mode variance $(H/2\pi {)}^2$ at horizon exit $k=aH$. Since $\mathcal{R}=H\delta \varphi /\dot{\varphi }$ is conserved after exit,

$$P_{\mathcal{R}}^{1/2}(k)=\frac{H}{2\pi }\cdot \frac{H}{|\dot{\varphi }|}⟹P_{\mathcal{R}}(k)=\frac{H^4}{4{\pi }^2{\dot{\varphi }}^2}=\frac{H^2}{8{\pi }^2{M}_{\mathrm{Pl}}^2{\epsilon }_H}\text{(at }k=aH\text{)},$$

using ${\dot{\varphi }}^2=2M_{\mathrm{Pl}}^2{H}^2{\epsilon }_H$. Substituting $H^2\approx V/(3M_{\mathrm{Pl}}^2)$ and ${\epsilon }_H\approx {\epsilon }_V$ gives the scalar amplitude

$$A_s\equiv P_{\mathcal{R}}(k_{\ast })\approx \frac{V}{24{\pi }^2{M}_{\mathrm{Pl}}^4{\epsilon }_V}{|}_{k_{\ast }},$$

evaluated at a pivot scale $k_{\ast }=0.05$ Mpc${}^{-1}$. Planck measures $A_s=2.1\times 10^{-9}$, so the inflationary energy scale is $V^{1/4}\sim 10^{16}$ GeV ^{[Dodelson 2003]}. Because $H$ varies slowly, every mode acquires nearly the same amplification; differentiating $\ln P_{\mathcal{R}}$ with respect to $\ln k\approx dN$ at horizon exit gives the scalar tilt

$$n_s-1=\frac{d\ln P_{\mathcal{R}}}{d\ln k}{|}_{k=aH}=-6{\epsilon }_V+2{\eta }_V,$$

and quantising tensor modes gives the tensor spectrum $P_t=2H^2/({\pi }^2{M}_{\mathrm{Pl}}^2)$, hence the tensor-to-scalar ratio

$$r\equiv \frac{P_t}{P_{\mathcal{R}}}=16{\epsilon }_H\approx 16{\epsilon }_V.$$

The observed $n_s=0.9649\pm 0.0042$ (slightly below unity) and $r<0.06$ pin the inflationary potential to be flat, ${\epsilon }_V<0.004$.

Bridge. This slow-roll derivation builds toward the quantum-field-theoretic origin of the primordial perturbations developed in the Master tier, where the Mukhanov variable is quantised on the Bunch-Davies vacuum and the horizon-exit condition $k=aH$ appears again in the freezing that imprints the spectrum. The foundational reason inflation can serve both as the resolution of the flatness and horizon puzzles and as the source of all cosmic structure is that a single slowly rolling field $\varphi$ sets a nearly constant $H$ while its fluctuations generate $A_s$: this is exactly the unification that makes inflation predictive rather than decorative. Putting these together, the measured values $(A_s,n_s,r)$ become a consistency check on the shape of $V(\varphi )$ itself. The bridge is that the Friedmann background of 13.08.01 supplies the clock $H(t)$, and inflation reinterprets that $H$ as a field-driven quasi-de Sitter phase whose stretched fluctuations seed everything catalogued in 13.08.02.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib does not yet contain the inflationary physics this unit develops. The gravitational substrate (Lorentzian metrics, the Einstein-Hilbert action, curvature tensors, the Levi-Civita connection) is partially in place, and the ODE library covers existence and uniqueness for the Friedmann system at a formal level. What is missing is any structured encoding of the inflationary objects: a minimally coupled scalar field with a potential $V(\varphi )$ as a stress-energy source in the Friedmann equations, the Hubble-flow parameters ${\epsilon }_H$ and ${\eta }_H$, the potential slow-roll parameters ${\epsilon }_V$ and ${\eta }_V$ and their leading-order equality ${\epsilon }_H\approx {\epsilon }_V$, the slow-roll attractor equation $3H\dot{\varphi }=-V_{,\varphi }$ together with its stability, the e-fold count $N=(1/M_{\mathrm{Pl}}^2)\int (V/V_{,\varphi })d\varphi$, the quantisation of the comoving curvature perturbation $\mathcal{R}$ on the Bunch-Davies vacuum with mode equation and horizon-exit freeze, the resulting power spectra $P_{\mathcal{R}}$ and $P_t$, the consistency relation $r=-8n_t$, or the Lyth bound on the field excursion.

Each of these is an exact, machine-checkable statement once the physical scaffolding is defined: ${\epsilon }_H={\epsilon }_V$ on the attractor, $r=16{\epsilon }_V$, and the quadratic-chaotic predictions $n_s=1-2/N$, $r=8/N$ are all identities that admit Lean proofs on top of a modest inflation-physics layer. The named human reviewer attests intermediate-tier correctness; the lean_status: none field reflects the absence of such a layer in Mathlib today.

Advanced results Master

Quantum origin of the perturbations

The curvature perturbation $\mathcal{R}$ is not postulated but derived by quantising the inflaton fluctuation on the quasi-de Sitter background. Writing $\varphi (\mathbf{x},t)={\varphi }_0(t)+\delta \varphi (\mathbf{x},t)$ and using the Friedmann and Klein-Gordon equations to second order yields the Mukhanov-Sasaki equation for the canonically normalised variable $v=z\mathcal{R}$, with $z=a\dot{\varphi }/H$ ^{[Mukhanov & Chibisov 1981]}:

$$v_k^{\prime \prime }+\left(k^2-\frac{z^{\prime \prime }}{z}\right)v_k=0,$$

where primes denote derivatives with respect to conformal time $\eta$. Imposing the Bunch-Davies initial condition $v_k\to e^{-ik\eta }/\sqrt{2k}$ as $k|\eta |\to \infty$ (positive-frequency Minkowski vacuum deep inside the horizon), and evaluating the mode at horizon exit $k=aH$ where $z^{\prime \prime }/z\approx 2/{\eta }^2$, gives the frozen amplitude $|v_k{|}^2\to H^2/(2k^3)$ and hence $P_{\mathcal{R}}=H^2/(8{\pi }^2{M}_{\mathrm{Pl}}^2{\epsilon }_H)$ as derived above. Two features make this robust: the Bunch-Davies vacuum is the unique de-Sitter-invariant Hadamard state, and $\mathcal{R}$ is conserved on super-horizon scales, so the prediction is insensitive to the messy post-inflationary physics.

The Lyth bound and the cost of detectable tensors

Combining $|d\varphi /dN|=M_{\mathrm{Pl}}\sqrt{2{\epsilon }_V}$ with $r=16{\epsilon }_V$ at horizon exit gives the Lyth bound ^{[Lyth 1997]}

$$\frac{\Delta \varphi }{M_{\mathrm{Pl}}}\gtrsim \sqrt{\frac{r}{8}}\Delta N.$$

For $N_{\ast }\approx 60$ the bound is $\Delta \varphi /M_{\mathrm{Pl}}\gtrsim 0.3(r/0.01{)}^{1/2}$. Detectable tensors, $r\gtrsim 10^{-2}$, force a super-Planckian excursion, which selects large-field models and exposes inflation to assumptions about UV completion (the effective field theory of $\varphi$ must remain controlled over a trans-Planckian range). Sub-Planckian excursion, $\Delta \varphi \ll M_{\mathrm{Pl}}$, predicts unobservably small $r$, as in Starobinsky and Higgs inflation.

Reheating and the thermal start

When slow roll ends at ${\epsilon }_H=1$, the field oscillates about the minimum of $V$ with frequency $m_{\varphi }=\sqrt{V_{,\varphi \varphi }}$, and the coherent oscillation energy ${\bar{\rho }}_{\varphi }\propto a^{-3}$ (for a quadratic minimum) redshifts like matter. Couplings of $\varphi$ to Standard Model fields through a decay width ${\Gamma }_{\varphi }$ transfer this energy to relativistic particles; the universe becomes radiation-dominated when ${\Gamma }_{\varphi }\sim H$, at reheating temperature

$$T_{\mathrm{reh}}\sim {\left(\frac{90}{{\pi }^2{g}_{\ast }}\right)}^{1/4}({\Gamma }_{\varphi }{M}_{\mathrm{Pl}}{)}^{1/2},$$

with $g_{\ast }$ the relativistic degrees of freedom. The reheating history shifts the mapping between $N_{\ast }$ and the physical pivot scale $k_{\ast }$, producing a model-dependent uncertainty of one to two e-folds in the predicted $n_s$ and $r$.

Model landscape

The flatness constraints ${\epsilon }_V\lesssim 4\times 10^{-3}$, ${\eta }_V\lesssim 0.03$ select a narrow family of potentials. Large-field models ($\Delta \varphi >M_{\mathrm{Pl}}$): quadratic $V=\frac{1}{2}{m}^2{\varphi }^2$ gives $(n_s,r)\approx (0.967,0.13)$, now excluded; quartic $V=\lambda {\varphi }^4$ gives $r\approx 0.2$, also excluded. Plateau models: Starobinsky $R^2$ inflation and Higgs inflation with non-minimal coupling $\xi$ give $(n_s,r)\approx (0.965,0.003)$, favoured by Planck. Small-field and hilltop models ($\Delta \varphi \ll M_{\mathrm{Pl}}$) predict $r\ll 10^{-3}$. The current bound $r<0.06$ (BICEP/Keck and Planck) eliminates the steep large-field region and concentrates viable models near the plateau class.

Synthesis. The quantum-to-classical passage of a single inflaton fluctuation builds toward the entire observed large-scale structure of the universe: the foundational reason a nearly scale-invariant, nearly Gaussian, adiabatic perturbation spectrum is generic is that de-Sitter-like expansion amplifies every comoving mode identically at horizon exit, and this is exactly what the CMB and galaxy surveys measure ($n_s\approx 0.965$, $A_s\approx 2.1\times 10^{-9}$). The tensor sector generalises the scalar story to spacetime itself, predicting a primordial gravitational-wave background at ratio $r=16{\epsilon }_V$ whose detection would fix the inflationary energy scale $V^{1/4}\sim 10^{16}$ GeV. The central insight of the Lyth bound is that this tensor signal costs a super-Planckian field excursion, so detectable $r$ selects large-field models and binds inflation to UV physics. Putting these together, the consistency relation $r=-8n_t$ ties two independent observables to a single parameter ${\epsilon }_V$, turning the inflationary spectrum into a falsifiable, quantitative theory of initial conditions. The bridge is that all of this rides on the Friedmann background clock of 13.08.01 and refines the inflation sketch of 13.08.02 into a precision, model-discriminating framework.

Full proof set Master

Proposition 1 (potential and Hubble-flow slow-roll parameters agree on the attractor). On the slow-roll attractor $3H\dot{\varphi }=-V_{,\varphi }$, $H^2=V/(3M_{\mathrm{Pl}}^2)$, the Hubble-flow parameter ${\epsilon }_H=-\dot{H}/H^2$ equals the potential parameter ${\epsilon }_V=(M_{\mathrm{Pl}}^2/2)(V_{,\varphi }/V{)}^2$ to leading order.

Proof. Differentiating $H^2=(\frac{1}{2}{\dot{\varphi }}^2+V)/(3M_{\mathrm{Pl}}^2)$ and using the continuity equation ${\dot{\rho }}_{\varphi }=-3H{\dot{\varphi }}^2$ (equivalently ${\nabla }_{\mu }{T}^{\mu \nu }=0$ on the homogeneous field),

$$2H\dot{H}=\frac{{\dot{\rho }}_{\varphi }}{3M_{\mathrm{Pl}}^2}=-\frac{H{\dot{\varphi }}^2}{M_{\mathrm{Pl}}^2}⟹\dot{H}=-\frac{{\dot{\varphi }}^2}{2M_{\mathrm{Pl}}^2}⟹{\epsilon }_H=\frac{{\dot{\varphi }}^2}{2M_{\mathrm{Pl}}^2{H}^2}.$$

On the attractor $\dot{\varphi }=-V_{,\varphi }/(3H)$, so ${\dot{\varphi }}^2=V_{,\varphi }^2/(9H^2)=V_{,\varphi }^2\cdot 3M_{\mathrm{Pl}}^2/(9V)=M_{\mathrm{Pl}}^2{V}_{,\varphi }^2/(3V)$. Substituting,

$${\epsilon }_H=\frac{M_{\mathrm{Pl}}^2{V}_{,\varphi }^2/(3V)}{2M_{\mathrm{Pl}}^2\cdot V/(3M_{\mathrm{Pl}}^2)}=\frac{M_{\mathrm{Pl}}^2}{2}{\left(\frac{V_{,\varphi }}{V}\right)}^2={\epsilon }_V,$$

which is the claimed leading-order equality. $\square$

Proposition 2 (single-field consistency relation). In single-field slow-roll inflation, the tensor-to-scalar ratio and the tensor spectral index satisfy $r=-8n_t$.

Proof. The tensor spectrum evaluated at horizon exit $k=aH$ is $P_t=2H^2/({\pi }^2{M}_{\mathrm{Pl}}^2)$, with $M_{\mathrm{Pl}}$ constant, so $\ln P_t=2\ln H+\mathrm{const}$. The spectral index is

$$n_t=\frac{d\ln P_t}{d\ln k}{|}_{k=aH}.$$

During slow roll $H$ varies slowly, so $\ln k=\ln (aH)\approx \ln a$, and $d\ln k=Hdt=dN$. Thus

$$n_t=2\frac{d\ln H}{dN}=2\cdot \frac{\dot{H}/H}{H}=2\frac{\dot{H}}{H^2}=-2{\epsilon }_H.$$

Since $r=P_t/P_{\mathcal{R}}=16{\epsilon }_H$ and $n_t=-2{\epsilon }_H$, eliminating ${\epsilon }_H$ gives $n_t=-r/8$, equivalently $r=-8n_t$. $\square$

Proposition 3 (Lyth bound on the inflaton excursion). The total field excursion during single-field slow-roll inflation satisfies $\Delta \varphi /M_{\mathrm{Pl}}\ge \sqrt{r/8}\Delta N$ over the interval of $\Delta N$ e-folds ending at horizon exit.

Proof. On the slow-roll attractor, $d\varphi /dN=\dot{\varphi }/H$, so $|d\varphi /dN|=|\dot{\varphi }|/H=M_{\mathrm{Pl}}\sqrt{2{\epsilon }_H}\approx M_{\mathrm{Pl}}\sqrt{2{\epsilon }_V}$ using ${\epsilon }_H={\dot{\varphi }}^2/(2M_{\mathrm{Pl}}^2{H}^2)$ from Proposition 1. The excursion over the interval $\Delta N$ is

$$\Delta \varphi =\int \left|\frac{d\varphi }{dN}\right|dN=M_{\mathrm{Pl}}\int \sqrt{2{\epsilon }_V(N)}dN\ge M_{\mathrm{Pl}}\sqrt{2{\epsilon }_{V,\min }}\Delta N,$$

the inequality bounding the integrand by its minimum. During slow roll ${\epsilon }_V$ is monotonic increasing as $\varphi$ rolls toward ${\varphi }_{\mathrm{end}}$, so its minimum on the interval is its value at horizon exit, ${\epsilon }_{V,\ast }=r/16$. Substituting $\sqrt{2{\epsilon }_{V,\ast }}=\sqrt{r/8}$,

$$\frac{\Delta \varphi }{M_{\mathrm{Pl}}}\ge \sqrt{\frac{r}{8}}\Delta N.$$

For $\Delta N=N_{\ast }\approx 60$ and $r=0.01$ this gives $\Delta \varphi /M_{\mathrm{Pl}}\gtrsim 2.1$, forcing super-Planckian excursion whenever $r$ is detectable. $\square$

Proposition 4 (inflation exponentially drives the universe flat). During slow-roll inflation with $H$ approximately constant, the curvature density parameter evolves as $|{\Omega }_k(a)|=|{\Omega }_{k,\mathrm{start}}|e^{-2N(a)}$ where $N(a)=\ln (a/a_{\mathrm{start}})$, so $N\gtrsim 60$ e-folds suffice to suppress any initial curvature below present observational bounds.

Proof. By definition ${\Omega }_k=-k{c}^2/(a^2{H}^2)$, so $\ln |{\Omega }_k|=-2\ln a-2\ln H+\mathrm{const}$. Differentiating with respect to $\ln a$,

$$\frac{d\ln |{\Omega }_k|}{d\ln a}=-2-2\frac{d\ln H}{d\ln a}=-2+2{\epsilon }_H,$$

since $d\ln H/d\ln a=(\dot{H}/H)/(\dot{a}/a)=\dot{H}/H^2=-{\epsilon }_H$. During slow roll ${\epsilon }_H\ll 1$, so $d\ln |{\Omega }_k|/d\ln a\approx -2$, giving $\ln |{\Omega }_k(a)|=\ln |{\Omega }_{k,\mathrm{start}}|-2N(a)$ and hence ${\Omega }_k(a)={\Omega }_{k,\mathrm{start}}{e}^{-2N(a)}$. Combined with the post-inflationary dilution of roughly $10^{-25}$ from radiation and matter domination, reaching today's $|{\Omega }_k|\lesssim 10^{-2}$ from an initial value of order unity requires only $N\sim 50$ to $60$ e-folds of inflation. $\square$

Connections Master

• To the FLRW and Friedmann foundation 13.08.01. Every equation in this unit is the Friedmann system of 13.08.01 with a scalar-field stress-energy ${\rho }_{\varphi }=\frac{1}{2}{\dot{\varphi }}^2+V$ fed in. The Hubble parameter $H(t)$ that governs the slow-roll attractor, the e-fold count $N=\int Hdt$, and the curvature evolution ${\Omega }_k\propto (aH{)}^{-2}$ are all quantities defined in the background unit; inflation simply arranges for $H$ to be nearly constant and $(aH{)}^{-1}$ to shrink.

• To the cosmological overview 13.08.02. The sibling unit introduces inflation qualitatively and quotes the slow-roll predictions $n_s\approx 0.965$, $r<0.06$. This unit supplies the derivation: the attractor mechanism, the potential slow-roll parameters, the amplitude $A_s=V/(24{\pi }^2{M}_{\mathrm{Pl}}^4{\epsilon }_V)$, the consistency relation, the Lyth bound, and reheating. The perturbations seeded here are exactly those processed by the acoustic oscillator and gravitational instability analysed in 13.08.02.

• To the free Klein-Gordon quantum field 12.05.04. Quantising the inflaton fluctuation $\delta \varphi$ to obtain $P_{\mathcal{R}}$ is the curved-spacetime version of the canonical quantisation of the free scalar field treated in 12.05.04. The Mukhanov-Sasaki mode equation is a Klein-Gordon equation with a time-dependent effective mass $z^{\prime \prime }/z$ supplied by the FLRW background; the Bunch-Davies vacuum is the curved-space upgrade of the Minkowski positive-frequency vacuum.

• To the Einstein field equations 13.04.01. The Friedmann equations driving the inflaton are the $tt$ and spatial components of $G_{\mu \nu }+\Lambda g_{\mu \nu }=(8\pi G/c^4)T_{\mu \nu }$ with $T_{\mu \nu }$ that of a homogeneous scalar field, and the perturbation equation for $\mathcal{R}$ is the linearised Einstein equation about FLRW. Inflation adds no new gravitational dynamics beyond 13.04.01; it specialises that dynamics to a scalar-field source.

Historical & philosophical context Master

Inflation was first proposed by Alan Guth in 1980-1981 as a resolution of the horizon and flatness problems ^{[Guth 1981]}. Guth's original "old inflation" relied on a first-order phase transition through bubble nucleation and was quickly recognised to fail (the bubbles never percolate). Alexei Starobinsky had already constructed, in 1980, an inflationary solution driven by $R^2$ corrections to the Einstein-Hilbert action, motivated by singularity avoidance rather than by the horizon and flatness puzzles ^{[Starobinsky 1980]}; his model, after a conformal transformation, becomes the plateau potential that carries his name and remains the data-favoured template. Linde and, independently, Albrecht and Steinhardt proposed "new inflation" in 1982, in which the field slowly rolls down a plateau-shaped potential rather than tunnelling, giving smooth reheating ^{[Linde 1982]}.

The decisive second act was the realisation that inflation also generates structure. Mukhanov and Chibisov in 1981 showed that the quantum fluctuations of the inflaton, stretched by the expansion and frozen at horizon exit, produce a nearly scale-invariant spectrum of curvature perturbations ^{[Mukhanov & Chibisov 1981]}. This turned inflation from a graceful-exit scenario into a quantitative, falsifiable theory of initial conditions, whose predictions ($A_s\approx 2.1\times 10^{-9}$, $n_s\approx 0.965$, near-Gaussianity, near-adiabaticity) have been confirmed by COBE, WMAP, and Planck to percent precision. The tensor-to-scalar ratio $r$ and its relation to the field excursion through the Lyth bound ^{[Lyth 1997]} remain the key open observational target: a detection would fix the inflationary energy scale near $10^{16}$ GeV and probe physics near the Planck scale, while a continued upper limit concentrates viable models in the plateau class. The deepest open question is the status of the inflaton within a UV-complete theory, since most detectable-$r$ models require a controlled effective field theory over a trans-Planckian field range.

Bibliography Master

1. Guth, A. H. (1981). Inflationary universe: A possible solution to the horizon and flatness problems. Physical Review D 23, 347-356.

2. Starobinsky, A. A. (1980). A new type of isotropic cosmological models without singularity. Physics Letters B 91, 99-102.

3. Linde, A. D. (1982). A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Physics Letters B 108, 389-393.

4. Albrecht, A. & Steinhardt, P. J. (1982). Cosmology for grand unified theories with radiatively induced symmetry breaking. Physical Review Letters 48, 1220-1223.

5. Mukhanov, V. F. & Chibisov, G. V. (1981). Quantum fluctuations and a nonsingular universe. JETP Letters 33, 532-535.

6. Lyth, D. H. (1997). What would we learn by detecting a gravitational wave signal? Physical Review Letters 78, 1861-1863.

7. Mukhanov, V. (2005). Physical Foundations of Cosmology. Cambridge University Press.

8. Baumann, D. (2009). TASI lectures on inflation. arXiv:0907.5424; in Physics of the Large and the Small, TASI 2009, World Scientific.

9. Dodelson, S. (2003). Modern Cosmology. Academic Press.

10. Liddle, A. R. & Lyth, D. H. (2000). Cosmological Inflation and Large-Scale Structure. Cambridge University Press.