Primordial Perturbations and the Power Spectrum

Every galaxy, every cluster, every warm and cold speck on the cosmic microwave background is the descendant of a fluctuation that was present when the observable universe was smaller than a proton. The hot Big Bang, built on the FLRW metric and the Friedmann equations of the preceding articles, is silent about where those seeds came from: a perfectly homogeneous universe stays homogeneous forever. Inflation supplies the missing ingredient. A transient epoch of nearly de Sitter expansion shrinks the comoving Hubble radius $(aH)^{-1}$ , so a single causal patch inflates to engulf everything we can see, and — the miracle — the quantum vacuum fluctuations of the field driving the expansion are stretched to cosmological wavelengths and frozen in as classical density variations. This article is the synthesis where quantum field theory, general relativity, and observational cosmology meet in a single number. The central object is the gauge-invariant comoving curvature perturbation $\mathcal{R}$ ; its dynamics collapse, in conformal time, onto a canonically normalized Mukhanov-Sasaki variable obeying a harmonic-oscillator equation with a time-dependent mass. Quantize it in the Bunch-Davies vacuum, follow each Fourier mode from sub-horizon out through horizon crossing where it freezes, and the two-point function hands you the power spectrum. The answer is almost, but not exactly, scale invariant, the small departure set by the slow-roll parameters that measure how gently $H$ drifts. The same mechanism applied to the two polarizations of the metric yields a background of primordial gravitational waves, whose amplitude relative to the scalars — the ratio $r$ — weighs the energy scale of inflation directly. These are not soft predictions: Planck fixes the tilt to percent precision, and the hunt for the tensor signal is now the sharpest probe of physics at $10^{16}$ GeV.

Key takeaways

Inflation stretches quantum vacuum fluctuations of the inflaton until their physical wavelength exceeds the Hubble radius, where they freeze into classical, super-horizon curvature perturbations.
The comoving curvature perturbation $\mathcal{R}$ is conserved on super-horizon scales for adiabatic modes, so the prediction is insensitive to the unknown physics of reheating.
Single-field slow-roll inflation predicts a nearly scale-invariant scalar spectrum with tilt $n_s-1=-6\epsilon+2\eta$ , slightly red ( $n_s<1$ ).
Tensor modes (primordial gravitational waves) come free from the metric with amplitude set by $H$ ; the tensor-to-scalar ratio is $r=16\epsilon$ , obeying the consistency relation $r=-8n_t$ .
Planck measures $n_s=0.9649\pm0.0042$ and $A_s\approx2.1\times10^{-9}$ at $k_*=0.05\ \mathrm{Mpc}^{-1}$ , with $r<0.036$ — favouring slow roll over exact scale invariance at high significance.
Key idea: a quantum field in a quasi-de Sitter background fluctuates by $H/2\pi$ at horizon crossing; converting to curvature via $\mathcal{R}=-H\,\delta\phi/\dot\phi$ and freezing it super-horizon gives $\mathcal{P}_{\mathcal{R}}=\frac{1}{8\pi^2}\frac{H^2}{\epsilon\,M_{\rm Pl}^2}$ evaluated at exit, whose mild scale dependence is the tilt.

The core insight

A quantum field in a quasi-de Sitter background has fluctuations of order $H/2\pi$ at horizon crossing; converting these to curvature perturbations via $\mathcal{R}=-H\,\delta\phi/\dot\phi$ and using conservation of $\mathcal{R}$ on super-horizon scales yields a power spectrum $\mathcal{P}_{\mathcal{R}}=\frac{1}{8\pi^2}\frac{H^2}{\epsilon\,M_{\rm Pl}^2}$ evaluated at horizon exit, whose mild scale dependence is the spectral tilt.

Where this lives today

The primordial power spectrum is the working input of every precision cosmology pipeline. The frozen curvature perturbations re-enter the horizon and drive the acoustic oscillations whose snapshot at recombination is the CMB angular power spectrum $C_\ell$ — a transfer-function-weighted projection of $\mathcal{P}_{\mathcal{R}}(k)$ . Fitting Planck gives $A_s=(2.10\pm0.03)\times10^{-9}$ and $n_s=0.9649\pm0.0042$ , an $8\sigma$ detection that the spectrum is tilted, exactly as slow roll demands. The tensor sector is the live frontier: BICEP/Keck and Planck jointly bound $r<0.036$ , and next-generation experiments — the Simons Observatory, CMB-S4, and LiteBIRD — target $\sigma(r)\sim10^{-3}$ , deep enough to detect the $r\approx3\times10^{-3}$ of plateau models. On the low-redshift side, DESI, Euclid, and Rubin/LSST measure the matter power spectrum $P(k)\propto k^{n_s}T^2(k)$ , cross-checking the CMB tilt with different systematics. The open problems are sharp: does $n_s-1$ run with scale; are the fluctuations exactly Gaussian; and will a primordial gravitational-wave background ever be seen, giving the first direct evidence for the quantum nature of gravity?

The physical picture

The engine of the whole story is one geometric fact: during inflation the comoving Hubble radius $(aH)^{-1}$ shrinks. A comoving scale that starts inside the horizon is overtaken as the horizon retreats past it, exits, and stays frozen; long after inflation ends the Hubble radius grows again and the scale re-enters to become observable structure. The first figure is that valley.

Figure

The comoving Hubble radius (aH)⁻¹ falls during inflation and rises afterward, carving a valley. A fixed comoving scale k⁻¹ crosses out of the horizon at exit (k = aH), stays frozen with ℛ = constant while super-horizon, and re-enters much later to seed a CMB-scale structure

Now zoom into a single mode. Picture each Fourier component as a harmonic oscillator whose frequency is the physical wavenumber $k/a$ . Deep inside the horizon, $k\gg aH$ , it oscillates as an ordinary vacuum mode; once $k\ll aH$ the Hubble friction overdamps it, the amplitude stops evolving, and the fluctuation is now a classical, super-horizon perturbation. The freeze-out amplitude is universal — a massless field in de Sitter picks up fluctuations of size $\delta\phi\sim H/2\pi$ per Hubble time, because $H$ is the only scale around. Since $H$ barely changes during slow roll, every mode freezes at nearly the same amplitude regardless of when it exits: that is the origin of near scale-invariance, and the tiny drift of $H$ and $\dot\phi$ is the whole of the tilt. The payoff is a prediction in the plane of the two observables — the scalar tilt $n_s$ and the ratio $r$ — where different inflaton potentials trace different points and the data draw a bullseye. The second figure is that plane, the chart a model-builder actually lives in.

Figure

The observational (n_s, r) plane. The shaded blue box is the Planck region n_s = 0.9649 ± 0.0042 with r < 0.036. The Starobinsky R² plateau (green, n_s ≈ 0.965, r ≈ 0.003) sits inside it; φ² chaotic inflation (red, r ≈ 0.13) is excluded from the top; exact scale invariance is the point n_s = 1, r = 0

Tensor modes need no field at all: gravitational waves are the spin-2 fluctuations of the metric itself, populated even in empty de Sitter with amplitude $\sim H/M_{\rm Pl}$ . Their power reads off $H$ directly, and through $H^2\simeq V/3M_{\rm Pl}^2$ the energy density $V$ — so a detection of primordial B-modes would measure physics no accelerator will ever reach.

The core equations

We work in the perturbed FLRW metric in conformal-Newtonian gauge, which splits it into two scalar Bardeen potentials $\Phi,\Psi$ and the transverse-traceless gravitational-wave field $h_{ij}$ .

\mathrm{d}s^2 = a^2(\tau)\left[-(1+2\Phi)\,\mathrm{d}\tau^2 + \left((1-2\Psi)\delta_{ij}+h_{ij}\right)\mathrm{d}x^i\mathrm{d}x^j\right] \qquad (1)

where: $a(\tau)$ — scale factor in conformal time $\tau$ , with $\mathrm{d}\tau=\mathrm{d}t/a$ ; $\Phi,\Psi$ — scalar Bardeen potentials, equal ( $\Phi=\Psi$ ) when anisotropic stress is negligible; $h_{ij}$ — transverse-traceless tensor perturbation (gravitational waves), $\partial^i h_{ij}=h^i{}_i=0$ . Reading: scalars source density perturbations, tensors are the two graviton polarizations, and the sectors decouple at linear order. The dynamics are governed by how slowly the expansion changes, captured by two dimensionless slow-roll parameters: the fractional change of $H$ per Hubble time, and the curvature of the potential.

\epsilon \equiv -\frac{\dot H}{H^2} = \frac{M_{\rm Pl}^2}{2}\left(\frac{V'}{V}\right)^2,\qquad \eta \equiv M_{\rm Pl}^2\,\frac{V''}{V} \qquad (2)

where: $H=\dot a/a$ — Hubble rate, dot is a cosmic-time derivative; $V(\phi)$ — inflaton potential, prime denotes $\mathrm{d}/\mathrm{d}\phi$ ; $M_{\rm Pl}$ — reduced Planck mass, $M_{\rm Pl}^2=(8\pi G)^{-1}$ , so $M_{\rm Pl}=2.435\times10^{18}\ \mathrm{GeV}$ ; $\epsilon,\eta\ll1$ during inflation. Reading: accelerated expansion requires $\epsilon<1$ , and inflation ends when $\epsilon\to1$ . The physical seed of structure is not the gauge-dependent $\delta\phi$ but the comoving curvature perturbation $\mathcal{R}$ — a gauge-invariant combination of metric and field measuring the local time-lag of inflation's end.

\mathcal{R} = \Psi + \frac{H}{\dot\phi}\,\delta\phi \;=\; -\frac{H}{\dot\phi}\,\delta\phi\Big|_{\rm flat\ gauge} \qquad (3)

where: $\mathcal{R}$ — gauge-invariant curvature perturbation on comoving slices; $\delta\phi$ — inflaton fluctuation; $\dot\phi$ — background field velocity. Reading: $\mathcal{R}$ is conserved once $k\ll aH$ ; a region where the field lagged inflates longer and ends up overdense. Substituting the perturbed Einstein equations into the second-order action collapses the coupled field-metric system onto a single canonically normalized variable obeying a flat-space oscillator equation with a time-varying mass — the Mukhanov-Sasaki equation.

v_k'' + \left(k^2 - \frac{z''}{z}\right)v_k = 0,\qquad v\equiv z\,\mathcal{R},\quad z\equiv a\sqrt{2\epsilon}\,M_{\rm Pl} \qquad (4)

where: $v$ — canonically normalized Mukhanov-Sasaki variable; $k$ — comoving wavenumber; prime is $\mathrm{d}/\mathrm{d}\tau$ ; $z''/z$ — effective time-dependent mass, $\to 2/\tau^2$ in de Sitter. Reading: for $k\gg aH$ the mass is negligible and $v$ oscillates; for $k\ll aH$ it dominates and drives the growing (freezing) solution. Imposing the Bunch-Davies condition — every mode starts in the flat-space vacuum deep inside the horizon — selects a unique exact de Sitter solution.

\frac{z''}{z}=\frac{2}{\tau^2}\;\Rightarrow\; v_k(\tau)=\frac{e^{-ik\tau}}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right) \qquad (5)

where: $\tau$ — conformal time, $\tau=-1/(aH)<0$ in de Sitter, $\tau\to0^-$ at late times; $e^{-ik\tau}/\sqrt{2k}$ — positive-frequency Minkowski vacuum, recovered as $k\tau\to-\infty$ . Reading: the $(1-i/k\tau)$ factor is the imprint of curved space, making $|v_k|$ grow as the mode exits. Squaring the frozen amplitude as the dimensionless variance per logarithmic interval in $k$ gives the scalar power spectrum, the theory's headline result.

\mathcal{P}_{\mathcal{R}}(k)=\frac{k^3}{2\pi^2}\,|\mathcal{R}_k|^2=\left.\frac{1}{8\pi^2}\frac{H^2}{\epsilon\,M_{\rm Pl}^2}\right|_{k=aH}=\left.\frac{1}{24\pi^2}\frac{V}{\epsilon\,M_{\rm Pl}^4}\right|_{k=aH} \qquad (6)

where: $\mathcal{P}_{\mathcal{R}}$ — dimensionless curvature power spectrum; $|_{k=aH}$ — evaluated at horizon crossing of the mode $k$ ; the second form uses $H^2\simeq V/3M_{\rm Pl}^2$ . Reading: the measured normalization is $A_s\equiv\mathcal{P}_{\mathcal{R}}(k_*)\approx2.1\times10^{-9}$ at the pivot $k_*=0.05\ \mathrm{Mpc}^{-1}$ . The same de Sitter computation applied to the two metric polarizations gives the tensor spectrum, and the ratio of the two is the single most coveted number in the field.

\mathcal{P}_t(k)=\frac{2}{\pi^2}\frac{H^2}{M_{\rm Pl}^2}\bigg|_{k=aH},\qquad r\equiv\frac{\mathcal{P}_t}{\mathcal{P}_{\mathcal{R}}}=16\epsilon \qquad (7)

where: $\mathcal{P}_t$ — power summed over both graviton polarizations; $r$ — tensor-to-scalar ratio. Reading: the scalar carries an extra $1/\epsilon$ enhancement (from dividing by the small $\dot\phi$ ), so $r$ is small; measuring it fixes $\epsilon$ and the inflationary energy scale. Finally, because $H$ and $\epsilon$ drift slightly over the modes that exit at different times, the spectra tilt. The logarithmic derivatives define the spectral indices.

n_s-1\equiv\frac{\mathrm{d}\ln\mathcal{P}_{\mathcal{R}}}{\mathrm{d}\ln k}=-6\epsilon+2\eta,\qquad n_t\equiv\frac{\mathrm{d}\ln\mathcal{P}_t}{\mathrm{d}\ln k}=-2\epsilon \qquad (8)

where: $n_s$ — scalar spectral index, $n_s=1$ is exact scale invariance; $n_t$ — tensor index, always negative in standard inflation. Reading: eliminating $\epsilon$ between $r=16\epsilon$ and $n_t=-2\epsilon$ gives the consistency relation $r=-8n_t$ .

Derivation AThe Mukhanov-Sasaki equation from the quadratic action

The goal is to show why the coupled system of a scalar field and the metric it perturbs — apparently a mess of constrained, gauge-redundant degrees of freedom — reduces to a single free field obeying a simple oscillator equation. We start from the second-order action for $\mathcal{R}$ , absorb the background into one function $z(\tau)$ , rescale to a canonical variable, and read off the Fourier-space equation of motion — so all of inflationary phenomenology becomes the physics of one oscillator.

Step 1. Expanding the Einstein-Hilbert plus scalar-field action to second order in perturbations, after using the constraint equations to eliminate the non-dynamical metric pieces, yields $S^{(2)}=\tfrac12\int \mathrm{d}\tau\,\mathrm{d}^3x\,z^2\big[(\mathcal{R}')^2-(\partial_i\mathcal{R})^2\big]$ with $z^2=2a^2\epsilon M_{\rm Pl}^2$ . Why: one function $z(\tau)$ packages the whole background, so the perturbation dynamics decouple from the expansion history.

Step 2. Define the canonically normalized field $v\equiv z\mathcal{R}$ . The kinetic term becomes standard and the action turns into $S^{(2)}=\tfrac12\int \mathrm{d}\tau\,\mathrm{d}^3x\big[(v')^2-(\partial_i v)^2+\tfrac{z''}{z}v^2\big]$ , i.e. a free field with a time-dependent mass $-z''/z$ . Why: canonical normalization lets us quantize $v$ with ordinary flat-space commutators; the price is the tachyonic-looking mass that drives the growth.

Step 3. Varying gives the Euler-Lagrange equation, and Fourier transforming in space produces $v_k''+(k^2-z''/z)v_k=0$ . In de Sitter $z\propto a\propto -1/\tau$ , so $z''/z=2/\tau^2$ and the equation is exactly solvable. Why: a harmonic oscillator whose frequency $\omega_k^2=k^2-z''/z$ is real for $k\gg aH$ (oscillation) and imaginary for $k\ll aH$ (freezing) — the mathematical face of horizon exit. *In de Sitter the equation reduces to the exactly solvable form $v_k''+(k^2-2/\tau^2)v_k=0$ .*

Go deeper · Derivation B · 1 minFrom the quantum vacuum to the scalar power spectrum

Derivation BFrom the quantum vacuum to the scalar power spectrum

The goal is to turn the classical mode equation into a prediction for a measurable variance, by treating $v$ as a quantum field, choosing its vacuum, evolving each mode through horizon crossing, and reading off the frozen amplitude of $\mathcal{R}=v/z$ . The answer depends only on $H$ and $\epsilon$ at the instant the mode crosses the horizon — all later evolution drops out, which is what makes inflation predictive despite our ignorance of reheating.

Step 1. Promote $v$ to an operator, $\hat v=\int\!\frac{\mathrm{d}^3k}{(2\pi)^3}\big[v_k(\tau)\,\hat a_{\mathbf k}e^{i\mathbf k\cdot\mathbf x}+\mathrm{h.c.}\big]$ , with $[\hat a_{\mathbf k},\hat a^\dagger_{\mathbf k'}]=(2\pi)^3\delta^3(\mathbf k-\mathbf k')$ . Why: the mode function $v_k$ carries the dynamics, the operators the quantum statistics — the vacuum two-point function is simply $|v_k|^2$ .

Step 2. Fix the state by the Bunch-Davies condition $v_k\to e^{-ik\tau}/\sqrt{2k}$ as $k\tau\to-\infty$ . Why: deep inside the horizon spacetime curvature is irrelevant and the mode cannot know it lives in an expanding universe, so the unique adiabatic choice is the flat-space vacuum.

Step 3. The exact de Sitter solution matching this is $v_k(\tau)=\frac{e^{-ik\tau}}{\sqrt{2k}}\big(1-\frac{i}{k\tau}\big)$ , whose modulus in the super-horizon limit $k\tau\to0^-$ becomes $|v_k|^2\to \frac{1}{2k^3\tau^2}$ . Why: the amplitude no longer oscillates but grows and locks — a frozen classical number, the seed we observe.

Step 4. Convert to curvature with $\mathcal{R}=v/z$ and $z^2=2a^2\epsilon M_{\rm Pl}^2$ : $|\mathcal{R}_k|^2=\frac{1}{2k^3\tau^2}\cdot\frac{1}{2a^2\epsilon M_{\rm Pl}^2}$ , and using $\tau=-1/(aH)$ the combination $a^2\tau^2=1/H^2$ cancels the scale factor. Why: the explicit $a$ -dependence vanishes — the technical statement that $\mathcal{R}$ has frozen.

Step 5. Form the dimensionless spectrum $\mathcal{P}_{\mathcal{R}}=\frac{k^3}{2\pi^2}|\mathcal{R}_k|^2=\frac{1}{8\pi^2}\frac{H^2}{\epsilon M_{\rm Pl}^2}$ , evaluated at $k=aH$ . Why: evaluating $H$ and $\epsilon$ at horizon crossing captures the leading scale dependence — the near-flat spectrum with its small tilt. *As a numerical check, for $H\sim10^{13}$ GeV and $\epsilon\sim0.01$ this gives $\mathcal{P}_{\mathcal{R}}\sim\mathcal{O}(10^{-9})$ , consistent with the observed amplitude.*

Go deeper · Derivation C · 1 minThe tensor spectrum and the single-field consistency relation

Derivation CThe tensor spectrum and the single-field consistency relation

The goal is to obtain the gravitational-wave spectrum and the relation $r=-8n_t$ from essentially the scalar computation done twice — once per polarization — and to see why the tensor amplitude is not enhanced by $1/\epsilon$ . The consistency relation is the cleanest test of the single-field hypothesis: a measurement of both $r$ and $n_t$ that violated it would rule out the simplest inflation in one stroke.

Step 1. Expanding $\sqrt{-g}R$ to second order in the transverse-traceless perturbation gives $S^{(2)}_t=\frac{M_{\rm Pl}^2}{8}\int \mathrm{d}\tau\,\mathrm{d}^3x\,a^2\big[(h_{ij}')^2-(\partial_k h_{ij})^2\big]$ , and $h_{ij}$ carries two independent polarizations $h_+,h_\times$ . Why: gravity alone contains propagating wave degrees of freedom, so the tensor spectrum is a pure prediction of general relativity in de Sitter.

Step 2. Each polarization $h_s$ maps onto a canonical field $v^{(s)}=\frac{aM_{\rm Pl}}{\sqrt2}h_s$ obeying the Mukhanov-Sasaki equation with $z_t=aM_{\rm Pl}/\sqrt2$ — a massless scalar, with no factor of $\epsilon$ . Why: the tensor " $z$ " lacks the $\sqrt{2\epsilon}$ of the scalar case — the missing factor that makes $r$ small.

Step 3. The per-polarization spectrum is therefore $\mathcal{P}_h^{(s)}=\frac{1}{\pi^2}\frac{H^2}{M_{\rm Pl}^2}$ , and summing the two gives $\mathcal{P}_t=\frac{2}{\pi^2}\frac{H^2}{M_{\rm Pl}^2}$ . Why: the tensor amplitude depends on $H$ alone, so $\mathcal{P}_t$ directly measures the inflationary energy scale.

Step 4. Forming the ratio, the common $H^2/M_{\rm Pl}^2$ cancels and the scalar's $1/\epsilon$ survives: $r=\mathcal{P}_t/\mathcal{P}_{\mathcal{R}}=16\epsilon$ , and with $n_t=-2\epsilon$ this gives $r=-8n_t$ . Why: two observables are fixed by the one parameter $\epsilon$ , so they are not independent — the falsifiable signature of single-field slow roll.

Limiting cases

The framework is anchored by several physical regimes and consistency limits:

Case	Result	Meaning
Exact de Sitter ( $\epsilon\to0$ )	$n_s\to1,\quad r\to0$	perfectly scale-invariant Harrison-Zel'dovich spectrum; excluded by the measured $n_s<1$
Sub-horizon ( $k\gg aH$ )	$v_k\to \frac{1}{\sqrt{2k}}e^{-ik\tau}$	modes are Minkowski-vacuum oscillators; this fixes the Bunch-Davies initial condition
Super-horizon ( $k\ll aH$ )	$\mathcal{R}_k\to\text{const}$	adiabatic curvature freezes; decaying mode dies, growing mode is conserved
Consistency relation	$r=-8\,n_t$	single-field slow roll ties tensor amplitude to tensor tilt — a key falsifiable test
Lyth bound	$\frac{\Delta\phi}{M_{\rm Pl}}\gtrsim\left(\frac{r}{0.01}\right)^{1/2}$	detectable tensors demand trans-Planckian field excursions, sharply constraining models

Common trapEvaluate the spectrum at horizon exit, not at the end of inflation

The single most common error is to compute $\mathcal{P}_{\mathcal{R}}$ using $H$ and $\epsilon$ at one fixed time — the end of inflation, say — which erases the tilt. Every comoving mode $k$ exits the horizon at its own moment ( $k=aH$ ), when $H$ and $\epsilon$ have their own slightly different values, and it is precisely this spread that produces $n_s\neq1$ ; freeze the spectrum mode-by-mode at each exit. A second, deeper trap: $\mathcal{R}$ is conserved super-horizon only for adiabatic perturbations. In multi-field models, entropy (isocurvature) perturbations keep sourcing $\mathcal{R}$ after exit, so the naive single-field formula and the relation $r=-8n_t$ can both fail — which is why measuring them is worthwhile.

Worked examples

1 · The energy scale of inflation from a tensor detection (fully worked)

Suppose a future experiment detects primordial B-modes at $r=0.01$ , with the scalar amplitude fixed at $A_s=2.1\times10^{-9}$ . Find the slow-roll parameter $\epsilon$ , the inflationary Hubble rate $H$ , and the energy scale $V^{1/4}$ . Take $M_{\rm Pl}=2.435\times10^{18}$ GeV.

Step 1. From $r=16\epsilon$ , $\epsilon=r/16=0.01/16=\mathbf{6.25\times10^{-4}}$ — the inflaton rolling extremely slowly, the smallness that makes the scalar spectrum $1/\epsilon\approx1600$ times the tensor one.

Step 2. The tensor spectrum is $\mathcal{P}_t=rA_s=2.1\times10^{-11}$ . Setting this equal to $\frac{2}{\pi^2}H^2/M_{\rm Pl}^2$ gives $H^2=\frac{\pi^2}{2}\mathcal{P}_t M_{\rm Pl}^2=6.1\times10^{26}\ \mathrm{GeV}^2$ , so $H=\mathbf{2.5\times10^{13}\ GeV}$ (the same $H$ the scalar formula returns with $\epsilon=6.25\times10^{-4}$ ).

Step 3. The energy scale follows from $V\simeq3M_{\rm Pl}^2H^2\approx1.1\times10^{64}\ \mathrm{GeV}^4$ , so $V^{1/4}=\mathbf{1.0\times10^{16}\ GeV}$ — within a factor of a few of the grand-unification scale, the deepest energy probe in physics. Scaling as $V^{1/4}\propto r^{1/4}$ , the current bound $r<0.036$ gives $H\lesssim4.7\times10^{13}$ GeV and $V^{1/4}\lesssim1.4\times10^{16}$ GeV. ε = 6.25×10⁻⁴, H = 2.5×10¹³ GeV, V¹ᐟ⁴ = 1.0×10¹⁶ GeV.

2 · Discriminating models in the (n_s, r) plane

Compare two classic potentials at $N=60$ e-folds before the end of inflation: the plateau Starobinsky $R^2$ model, which predicts $n_s\simeq1-2/N$ and $r\simeq12/N^2$ , and monomial $\phi^2$ chaotic inflation, which predicts $n_s\simeq1-2/N$ and $r\simeq8/N$ . Which survives the data $n_s=0.9649\pm0.0042$ , $r<0.036$ ?

Step 1. Starobinsky at $N=60$ : $n_s=1-2/60=\mathbf{0.9667}$ and $r=12/3600=\mathbf{0.0033}$ . Both land inside the Planck box; at $N=55$ the tilt shifts to $n_s=0.964$ , still within $1\sigma$ of the central value.

Step 2. $\phi^2$ at $N=60$ : the same $n_s=0.9667$ , but $r=8/60=\mathbf{0.133}$ — nearly forty times the observational ceiling, so the predicted B-mode signal is simply not there.

Step 3. Verdict: the tilt alone cannot separate these models — both give $n_s\approx0.967$ — but $r$ is decisive. Starobinsky (green in the second figure) sits in the bullseye; $\phi^2$ (red) is excluded from the top. This is why $r$ , not the tilt, is the frontier: it is the axis on which surviving models still differ. As a check, the Sachs-Wolfe relation $\delta T/T\simeq\tfrac15\sqrt{A_s}\approx9\times10^{-6}$ recovers the one-part-in- $10^5$ CMB anisotropy COBE measured in 1992. Starobinsky survives (r ≈ 0.003); φ² is excluded (r ≈ 0.13 ≫ 0.036).

Physics of the portrait

The primordial power spectrum is where quantum mechanics wrote its signature across the sky — and the person who first read it off a de Sitter phase, before inflation had a name in the West, is Viatcheslav Mukhanov.

Origins · 1 minViatcheslav Mukhanov and how this was discovered

Viatcheslav Mukhanov

b. 1952 · inflationary cosmological perturbation theory

Viatcheslav Mukhanov, a Soviet-born physicist trained at the Moscow Physical-Technical Institute, did the decisive calculation as a graduate student. In 1981, with Gennady Chibisov at the Lebedev Institute, he studied quantum fluctuations in Alexei Starobinsky's $R^2$ model of a de Sitter stage and found they emerge with a nearly scale-invariant spectrum of density perturbations — the first prediction of the spectrum we now measure, preceding the same conclusion reached at the 1982 Nuffield workshop by Guth, Hawking, Starobinsky, and Bardeen-Steinhardt-Turner. Through the 1980s Mukhanov built the systematic gauge-invariant theory of cosmological perturbations — the variable now written $v$ , the equation now called Mukhanov-Sasaki (independently obtained by Misao Sasaki), and the conserved curvature $\mathcal{R}$ — collected in the landmark 1992 review with Feldman and Brandenberger. Drawing on Lifshitz's 1946 perturbation theory and Bardeen's 1980 gauge-invariant potentials, he turned a philosophical question about the origin of structure into a calculation with a definite, testable answer that every Planck constraint on $n_s$ now measures.

History

1946

— Lifshitz formulates relativistic cosmological perturbation theory

1970

— Harrison and Zel'dovich propose a scale-invariant primordial spectrum

1981

— Mukhanov and Chibisov compute perturbations from a de Sitter phase

1982

— the Nuffield workshop fixes the inflationary spectrum and its amplitude

1992

— COBE detects the 10⁻⁵ CMB temperature anisotropies

2018

— Planck pins n_s = 0.9649 and bounds r below 0.06

Going further

Texts. Baumann, TASI Lectures on Inflation (arXiv:0907.5424) · Mukhanov, Feldman & Brandenberger, Theory of Cosmological Perturbations, Phys. Rep. 215 (1992) 203 · Weinberg, Cosmology (2008), ch. 10 · Dodelson & Schmidt, Modern Cosmology, 2nd ed. (2020) · Liddle & Lyth, Cosmological Inflation and Large-Scale Structure (2000) · Carroll, Spacetime and Geometry (2004), ch. 8 · Mukhanov, Physical Foundations of Cosmology (2005). The key relations are $\mathcal{P}_{\mathcal{R}}=H^2/(8\pi^2\epsilon M_{\rm Pl}^2)$ , $\mathcal{P}_t=2H^2/(\pi^2 M_{\rm Pl}^2)$ , $r=16\epsilon$ , $n_s-1=-6\epsilon+2\eta$ , and $n_t=-2\epsilon$ . The observational anchors are $n_s=0.9649\pm0.0042$ , $A_s\approx2.1\times10^{-9}$ at $k_*=0.05\ \mathrm{Mpc}^{-1}$ , and $r<0.036$ . At $r=0.01$ the worked example gives $\epsilon=r/16$ , $H=2.5\times10^{13}$ GeV, and $V^{1/4}=1.0\times10^{16}$ GeV; the Starobinsky and $\phi^2$ predictions at $N=60$ are as computed above.

Prerequisites. The FLRW Metric · The Scale Factor and Hubble Parameter · The Friedmann Equations · The Cosmological Fluid and Equation of State · Comoving and Proper Coordinates. Leads to. Cosmological Redshift · The Radiation-Dominated Era · The Matter-Dominated Era.

Ask the tutor: Want to work through how a specific inflaton potential $V(\phi)$ lands as a point in the $(n_s,r)$ plane, or how the frozen $\mathcal{R}$ becomes the acoustic peaks of the CMB?

Why is the curvature perturbation ℛ, and not δφ, the right variable?

Because $\delta\phi$ is gauge-dependent — its value changes if you shift your time slicing — while the physics of structure cannot. The combination $\mathcal{R}=\Psi+H\delta\phi/\dot\phi$ is gauge-invariant: the intrinsic spatial curvature of comoving slices, with a clean meaning as the local time-lag of the end of inflation, so a patch where the inflaton lagged inflates longer and ends up overdense. Crucially, $\mathcal{R}$ is conserved super-horizon for adiabatic modes, so its value at exit is exactly what re-enters to seed the CMB — which is why we can compute during inflation and compare to data, skipping everything in between.

What does "horizon exit" actually mean if space itself is expanding?

It means the comoving Hubble radius $(aH)^{-1}$ shrinks below the mode's fixed comoving wavelength $k^{-1}$ . The Hubble radius $H^{-1}$ is the distance over which causal processes act in one expansion time; during inflation $H$ is nearly constant while wavelengths grow like $a$ , so a fluctuation quickly becomes larger than any causal patch. Once super-horizon, the two ends of a wave can no longer communicate, the mode stops evolving, and its amplitude freezes. This is the Hubble horizon, not the much larger particle horizon of the Big Bang; the mechanism is that it retreats past scale after scale, freezing a fluctuation at each.

Where does the famous δφ ∼ H/2π come from?

It is the amplitude of vacuum fluctuations of a light field in de Sitter space. Since $H$ is the only scale there, a nearly massless field must fluctuate by an amount set by $H$ ; the coefficient $1/2\pi$ comes from evaluating $\sqrt{k^3/2\pi^2}\,|\delta\phi_k|$ at horizon crossing with the Bunch-Davies mode function. Intuitively, each Hubble time the expansion stretches sub-horizon jitters out past the horizon, depositing a fresh classical increment of size $H/2\pi$ — a random walk that is the microscopic origin of the spectrum, which through $\mathcal{R}=-H\delta\phi/\dot\phi$ becomes $\mathcal{P}_{\mathcal{R}}=H^2/(8\pi^2\epsilon M_{\rm Pl}^2)$ .

Why does the scalar spectrum carry a factor 1/ε but the tensor spectrum does not?

Tensor modes are fluctuations of the metric itself, normalized with $z_t=aM_{\rm Pl}/\sqrt2$ , so their spectrum depends on $H$ alone: $\mathcal{P}_t\propto H^2/M_{\rm Pl}^2$ . The scalar mode is a fluctuation of the inflaton, converted to curvature via $\mathcal{R}=-H\delta\phi/\dot\phi$ , and that division by the small velocity $\dot\phi$ amplifies it. Since $\dot\phi^2=2\epsilon H^2M_{\rm Pl}^2$ , the conversion inserts a factor $1/\epsilon$ , giving $\mathcal{P}_{\mathcal{R}}\propto H^2/(\epsilon M_{\rm Pl}^2)$ and hence $r=16\epsilon$ : slow roll (small $\epsilon$ ) makes the scalar spectrum much larger than the tensor one, so $r$ is small.

Is a superluminal issue involved when modes leave the horizon?

No causal law is broken. Horizon exit is not anything moving faster than light through space; it is the comoving Hubble radius shrinking because $a$ grows quasi-exponentially while $H$ stays nearly fixed. The proper distance between two comoving points does grow faster than $c$ once their separation exceeds $H^{-1}$ , but that is the expansion of space permitted by general relativity, not signal propagation. The fluctuation carries no information across the horizon at exit — it simply ceases to evolve as its two ends fall out of causal contact.

How would a measurement of r overturn or confirm the theory?

A detection of $r$ at the plateau-model level ( $r\sim10^{-3}$ ) would directly measure the inflationary energy scale, $V^{1/4}\sim10^{16}$ GeV, and — because tensors come from quantizing the metric — arguably be the first observation of a quantum-gravitational effect. Measuring the tensor tilt $n_t$ would then test the consistency relation $r=-8n_t$ : agreement favours single-field slow roll, a violation demands extra fields. Conversely, pushing the bound to $r\lesssim10^{-3}$ with no detection excludes every large-field model via the Lyth bound. Either way, $r$ is the discriminating axis of modern cosmology.