Primordial Density Perturbations from Inflation

The prediction and its measurement are now the beating heart of observational cosmology. On the scalar side, galaxy surveys — DESI, Euclid, the Rubin Observatory's LSST — map the matter power spectrum $P(k)\propto k^{n_s}T^2(k)$ across billions of galaxies, independently pinning $n_s$ and testing scale invariance across scales the CMB cannot reach. On the tensor side, the whole game is the hunt for primordial B-mode polarization: BICEP/Keck at the South Pole has driven the bound to $r<0.036$, and the coming decade belongs to the Simons Observatory, LiteBIRD, and CMB-S4, which aim for $\sigma(r)\sim10^{-3}$ — enough to detect the gravitational-wave signal of plateau models like Starobinsky inflation or to falsify them. Searches for primordial non-Gaussianity ($f_{\rm NL}$) sharpen the single-field test, since standard slow roll predicts an essentially Gaussian sky ($f_{\rm NL}\sim\epsilon,\eta$) while multi-field models would leave a detectable imprint. Every one of these experiments is, at bottom, an attempt to read the shape of a potential $V(\phi)$ that operated at $10^{16}$ GeV a fraction of a nanosecond after the beginning.

Picture the inflaton as a still pond skinned everywhere with tiny quantum ripples of every wavelength — its vacuum fluctuations. Inflation is a violent, near-exponential stretching of space that grabs each ripple and drags it to ever longer wavelength. The Hubble radius $H^{-1}$ stays almost fixed in physical size while the ripple grows, so sooner or later the ripple is longer than $H^{-1}$: causal physics can no longer push it through even a single oscillation within a Hubble time, and it freezes at whatever amplitude it happened to have. Because $H$ barely changes during slow roll, ripples that freeze at different epochs all freeze at nearly the same amplitude — the seed of a scale-invariant spectrum. The first figure follows one comoving mode through this life story: it starts sub-horizon, exits the Hubble radius during inflation, waits frozen for many e-folds, and re-enters long after inflation ends.

Once frozen, the field fluctuation is not the end of the story — it must be recast as something physical and conserved. The trick is the "separate universe" picture: a patch where the inflaton lags slightly behind its neighbours simply inflates for a little longer, so neighbouring regions finish inflation at slightly different times. That time delay, $\delta N=-H\,\delta\phi/\dot\phi$, is exactly a perturbation in the local expansion — a comoving curvature perturbation $\mathcal{R}$. Because $\mathcal{R}$ is gauge-invariant and conserved on super-horizon scales, the messy, poorly-known physics of reheating cannot touch it while the mode waits outside the horizon. It is this conservation law that makes inflation predictive at all. The slight red tilt $n_s<1$ has a clean meaning visible in the second figure. As inflation proceeds the field rolls a touch faster and $H$ slowly decreases, so modes that exit later — smaller scales, larger $k$ — freeze at slightly smaller amplitude, while the largest scales (which exited earliest, when $H$ was highest) carry slightly more power. The measured spectrum is therefore not perfectly flat but gently sloped, and that slope is a direct fingerprint of a field that is rolling rather than sitting still.

The starting point is the background: a homogeneous inflaton $\phi(t)$ rolling on a potential $V(\phi)$, its energy density driving the expansion. The field obeys a damped equation of motion with Hubble friction $3H\dot\phi$, and $H$ is fixed by the Friedmann equation. Together these govern how fast the field rolls and how nearly de Sitter the background is.

where: $\phi$ — homogeneous inflaton field value; $V(\phi)$ — inflaton potential, $V'=\mathrm{d}V/\mathrm{d}\phi$; $H=\dot a/a$ — Hubble parameter; $M_{\rm Pl}$ — reduced Planck mass, $M_{\rm Pl}^2=(8\pi G)^{-1}$, $M_{\rm Pl}=2.435\times10^{18}$ GeV. Reading: inflation needs the potential term to dominate the kinetic term so that $H$ is large and nearly constant. Slow roll is the controlled approximation that makes the theory tractable and predictive. It is quantified by two dimensionless parameters built from the shape of the potential: $\epsilon$ measures its steepness and $\eta$ its curvature. Accelerated expansion requires $\epsilon\ll1$, and inflation ends when $\epsilon$ grows to order unity.

where: $\epsilon$ — first slow-roll parameter, also $\epsilon=-\dot H/H^2$; $\eta$ — second slow-roll parameter, the curvature of $V$; $V',V''$ — first and second derivatives of $V$ with respect to $\phi$. Reading: both must be small for a prolonged, nearly de Sitter phase; their smallness later fixes the smallness of the tilt. Perturbations are handled by promoting a single gauge-invariant combination of the field and metric to a canonically normalized variable, the Mukhanov variable $v$. In conformal time each Fourier mode obeys a wave equation with a time-dependent effective mass $z''/z$ — a harmonic oscillator whose "spring" softens as the mode leaves the horizon.

where: $v_k$ — Mukhanov variable, $v=z\mathcal{R}$; $'$ — derivative with respect to conformal time $\tau$, $\mathrm{d}\tau=\mathrm{d}t/a$; $k$ — comoving wavenumber; $z''/z\approx2/\tau^2$ in de Sitter. Reading: when $k^2\gg z''/z$ the mode oscillates; when $k^2\ll z''/z$ it is driven and freezes — horizon crossing is where the two terms balance. The quantum state is fixed by demanding that, deep inside the horizon where curvature is irrelevant, each mode sits in the ordinary flat-space vacuum. This Bunch–Davies condition selects one exact solution of equation (3) in de Sitter.

where: $\tau$ — conformal time, $\tau=-1/(aH)$ in de Sitter, running from $-\infty$ to $0^-$; the prefactor $e^{-ik\tau}/\sqrt{2k}$ is the flat-space positive-frequency form recovered as $\tau\to-\infty$. Reading: the $-i/(k\tau)$ term grows as $\tau\to0^-$, and it is this term that carries the frozen super-horizon amplitude. Squaring the super-horizon mode function and converting to curvature gives the central observable: the dimensionless power spectrum of $\mathcal{R}$, evaluated at horizon exit where all slowly varying quantities are read off.

where: $\mathcal{P}_\mathcal{R}$ — dimensionless curvature power spectrum; $\mathcal{R}_k=v_k/z$ — curvature mode; the subscript $k=aH$ — evaluated at horizon exit. Reading: the amplitude is set by the ratio of the Hubble scale to the Planck scale, enhanced by $1/\epsilon$ because a slowly rolling field converts field fluctuations into large curvature. Because $H$ and $\epsilon$ drift slightly between the exit of one scale and the next, the spectrum is not exactly flat. Its logarithmic slope defines the scalar spectral index, expanded to leading order in slow roll.

where: $n_s$ — scalar spectral index; $n_s=1$ is exact scale invariance (Harrison–Zel'dovich); $-6\epsilon+2\eta$ — the leading slow-roll tilt. Reading: a red tilt ($n_s<1$) needs $3\epsilon>\eta$, generic for the flat, concave potentials that data favour. Tensor perturbations — gravitational waves — are generated by the same de Sitter amplification, but couple only through the Planck mass, so their spectrum lacks the $1/\epsilon$ enhancement. Their ratio to the scalars isolates $\epsilon$ alone.

where: $\mathcal{P}_t$ — tensor power spectrum summed over the two polarizations; $r$ — tensor-to-scalar ratio. Reading: $r$ measures $\epsilon$ directly, and through $\mathcal{P}_t\propto H^2$ it fixes the energy scale of inflation — a detection would be a direct measurement of the inflationary Hubble rate. Finally, the whole prediction is compressed into a two-parameter fit that observations constrain: an amplitude and a tilt about a chosen pivot scale.

where: $A_s$ — scalar amplitude at the pivot; $k_*$ — pivot comoving wavenumber; $n_s$ — measured spectral index (Planck 2018). Reading: these three numbers, plus the bound $r<0.036$, are the entire observational contact between inflation and the sky — everything else is derived.

The entire behaviour of a comoving mode is dictated by its size relative to the comoving Hubble radius $1/(aH)$, and the observables emerge in specific limits of the slow-roll expansion:

1 · The energy scale of inflation from the observed amplitude (fully worked)

Given the scalar amplitude $A_s=2.1\times10^{-9}$ and the observational bound $r<0.036$, find the upper limits on the inflationary Hubble rate $H$ and on the energy scale $V^{1/4}$. Use $M_{\rm Pl}=2.435\times10^{18}$ GeV.

Step 1. Convert the tensor bound into a bound on $\epsilon$ using equation (7): $\epsilon=r/16<0.036/16=\mathbf{2.25\times10^{-3}}$. This is the largest slow-roll steepness still allowed by the non-detection of primordial B-modes.

Step 2. Solve equation (5), $A_s=H^2/(8\pi^2\epsilon M_{\rm Pl}^2)$, for the Hubble rate at the pivot: $H=M_{\rm Pl}\sqrt{8\pi^2\epsilon A_s}$. Inserting $\epsilon=2.25\times10^{-3}$ and $A_s=2.1\times10^{-9}$ gives $8\pi^2\epsilon A_s=78.96\times2.25\times10^{-3}\times2.1\times10^{-9}=3.73\times10^{-10}$, so $H=1.93\times10^{-5}\,M_{\rm Pl}=\mathbf{4.7\times10^{13}\ GeV}$.

Step 3. Convert to an energy density via $V\simeq3H^2M_{\rm Pl}^2$ (potential domination) and take the fourth root: $V=3\times3.73\times10^{-10}M_{\rm Pl}^4=1.12\times10^{-9}M_{\rm Pl}^4$, so $V^{1/4}=(1.12\times10^{-9})^{1/4}M_{\rm Pl}=5.78\times10^{-3}\times2.435\times10^{18}=\mathbf{1.4\times10^{16}\ GeV}$.

Result. The current tensor bound places inflation at $H<4.7\times10^{13}$ GeV and $V^{1/4}<1.4\times10^{16}$ GeV — squarely at the scale of grand unification, some twelve orders of magnitude above anything a collider can reach. This is why a B-mode detection would be a direct measurement of GUT-scale physics.

2 · Does the simplest model survive? Confronting $V=\tfrac12 m^2\phi^2$

The textbook chaotic-inflation potential is a pure mass term, $V=\tfrac12 m^2\phi^2$. Compute its predictions for $n_s$ and $r$ at $N=60$ e-folds before the end of inflation, and compare with Planck ($n_s=0.965$, $r<0.036$).

Step 1. Evaluate the slow-roll parameters. With $V'=m^2\phi$ and $V''=m^2$: $\epsilon=\tfrac12M_{\rm Pl}^2(V'/V)^2=\tfrac12M_{\rm Pl}^2(2/\phi)^2=2M_{\rm Pl}^2/\phi^2$, and $\eta=M_{\rm Pl}^2 V''/V=2M_{\rm Pl}^2/\phi^2$. So $\epsilon=\eta$ for this potential.

Step 2. Relate the field value to the number of e-folds via $N=\int V/(M_{\rm Pl}^2V')\,\mathrm{d}\phi=\phi^2/(4M_{\rm Pl}^2)$ (with $\phi_{\rm end}$ negligible), giving $\phi^2=4NM_{\rm Pl}^2$. Substituting, $\epsilon=2M_{\rm Pl}^2/(4NM_{\rm Pl}^2)=\mathbf{1/(2N)}$, and likewise $\eta=1/(2N)$.

Step 3. At $N=60$: $\epsilon=\eta=1/120=8.33\times10^{-3}$. Then $n_s-1=-6\epsilon+2\eta=-4\epsilon=-4/(2N)=-2/N=-0.0333$, so $n_s=\mathbf{0.967}$, and $r=16\epsilon=8/N=\mathbf{0.133}$.

Result. The predicted tilt $n_s=0.967$ is a beautiful match to Planck's $0.965$ — but $r=0.13$ overshoots the bound $r<0.036$ by nearly a factor of four. The simplest quadratic model is ruled out by the tensor data, not the tilt — a textbook case of how the $r$ axis, not the $n_s$ axis, killed the large-field monomials and pushed the field toward plateau potentials like Starobinsky's $R^2$ model.

3 · Number sense: the size of the seed fluctuation

If $H\sim10^{13}$ GeV during inflation, how large is the inflaton's field fluctuation, and how does it become a $10^{-5}$ perturbation? The de Sitter fluctuation is $\delta\phi\sim H/2\pi\sim10^{13}/(2\pi)\approx\mathbf{1.6\times10^{12}\ GeV}$ per logarithmic interval — set by the Gibbons–Hawking scale $H/2\pi$, the effective temperature of the de Sitter horizon. The curvature perturbation is then $\mathcal{R}\sim(H/2\pi)/(\dot\phi/H)=(H/2\pi)/\sqrt{2\epsilon}M_{\rm Pl}$, and with $\epsilon\sim10^{-2}$ and $H/M_{\rm Pl}\sim10^{-5}$ this lands at $\mathcal{R}\sim10^{-5}$ — exactly the amplitude of the temperature ripples COBE first measured. The $1/\sqrt\epsilon$ conversion is what turns a $10^{12}$ GeV field wobble into a part-in-$10^{5}$ dent in the metric.

The quantum theory of inflationary perturbations was not the work of a single person, but one name is attached to its central equation and to the first calculation of the spectrum that the CMB would later confirm.

Texts. Dodelson, Modern Cosmology, 2nd ed., ch. 6–7 (inflationary perturbations, the Mukhanov–Sasaki equation) · Baumann, TASI Lectures on Inflation, arXiv:0907.5424 (the modern pedagogical standard) · Mukhanov, Feldman & Brandenberger, Phys. Rep. 215 (1992) 203 (the technical foundation) · Weinberg, Cosmology (2008), ch. 10 · Ryden, Introduction to Cosmology, 2nd ed., ch. 11 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 30.

Provenance. The equations are standard single-field slow-roll cosmological perturbation theory in the Mukhanov–Sasaki formalism. A verifier should reproduce: (i) the de Sitter mode function (4) and the cancellation of $\tau^2$ factors giving $\mathcal{P}_\mathcal{R}=H^2/(8\pi^2\epsilon M_{\rm Pl}^2)$; (ii) the slow-roll identity $\mathrm{d}\ln\epsilon/\mathrm{d}N=4\epsilon-2\eta$ yielding $n_s-1=-6\epsilon+2\eta$; (iii) $r=16\epsilon$ and $r=-8n_t$; (iv) the worked-example numbers $H<4.7\times10^{13}$ GeV, $V^{1/4}<1.4\times10^{16}$ GeV for $r<0.036$, and $\{n_s,r\}=\{0.967,0.133\}$ for $V=\tfrac12 m^2\phi^2$ at $N=60$. Observational values ($A_s\approx2.1\times10^{-9}$, $n_s=0.9649\pm0.0042$, $r<0.036$, $k_*=0.05\,\mathrm{Mpc}^{-1}$) are Planck 2018 + BICEP/Keck 2021.

Prerequisites. Equilibrium Thermodynamics of the Early Universe · Thermal History of the Universe · Recombination and the Saha Equation. Leads to. Big Bang Nucleosynthesis · CMB Anisotropies and Acoustic Peaks · Large-Scale Structure and the Matter Power Spectrum.

Ask the tutor: Ask me to walk through the Mukhanov–Sasaki mode function step by step, or to plot how a specific potential $V(\phi)$ lands on the $n_s$–$r$ plane against the Planck contours.