The Primordial Power Spectrum

Every galaxy, every cluster, every filament of the cosmic web, and every hot and cold speck on the microwave sky traces back to a fluctuation already present when the observable Universe was a fraction of a second old. The primordial power spectrum is the statistical blueprint of those seeds: how much fluctuation power sits at each spatial scale before gravity acts. Because the seeds are, to excellent accuracy, a Gaussian random field, this single two-point function is a nearly complete description of the initial conditions — fix it, and cosmic structure follows by known physics. The variable it describes is not the density contrast but the comoving curvature perturbation $\mathcal{R}$ , and that choice is the heart of the subject. On scales larger than the Hubble radius $\mathcal{R}$ is conserved for adiabatic modes, so the amplitude stamped on a mode at horizon exit survives untouched through reheating, nucleosynthesis, and equality until the mode falls back inside the horizon billions of years later. The spectrum is therefore a clean bridge from physics at grand-unified energies to the maps made by telescopes. Empirically the spectrum is a gentle power law fixed by two numbers: an amplitude $A_s$ and a tilt $n_s$ . A perfectly scale-invariant spectrum — guessed by Harrison, Zeldovich, and Peebles before any theory demanded it — has $n_s=1$ ; inflation instead predicts a slight red tilt, and the measured $n_s\approx0.965$ sits more than eight standard deviations below unity. That small deficit is one of the most direct windows we have onto physics beyond any accelerator, and it is the universal input every downstream article — transfer function, matter power spectrum, CMB anisotropies — takes as given.

Key takeaways

The primordial power spectrum $\mathcal{P}_\mathcal{R}(k)$ is the variance per logarithmic wavenumber of the comoving curvature perturbation $\mathcal{R}$ , laid down at the end of inflation before any processing.
It is well fit by a power law, $\mathcal{P}_\mathcal{R}(k)=A_s\,(k/k_*)^{n_s-1}$ , with amplitude $A_s\approx2.1\times10^{-9}$ and tilt $n_s\approx0.965$ at the pivot $k_*=0.05\,\mathrm{Mpc}^{-1}$ .
A scale-invariant Harrison–Zeldovich spectrum has $n_s=1$ ; the measured $n_s<1$ is a $>8\sigma$ red tilt confirming that inflation dynamically ends.
$\mathcal{R}$ is conserved on super-horizon scales, so the inflationary amplitude directly seeds the CMB and matter spectra through the transfer function.
Slow-roll inflation predicts $n_s-1=-6\epsilon+2\eta$ and a tensor-to-scalar ratio $r=16\epsilon$ , tying the tilt to the inflaton potential's shape.
Key idea: inflation stretches the quantum vacuum fluctuations of a light scalar to super-horizon scales, where they freeze into a conserved, nearly scale-invariant spectrum of curvature perturbations $\mathcal{P}_\mathcal{R}(k)=A_s(k/k_*)^{n_s-1}$ — the universal initial condition for all cosmic structure.

The core insight

Inflation stretches the quantum vacuum fluctuations of a light scalar field to super-horizon scales, where they freeze into a conserved, nearly scale-invariant spectrum of curvature perturbations $\mathcal{P}_\mathcal{R}(k)=A_s(k/k_*)^{n_s-1}$ — the universal initial condition for all cosmic structure.

Where this lives today

The primordial spectrum is a live measurement at the centre of cosmology: its two numbers are among the six base parameters of $\Lambda$ CDM, fixed by Planck 2018 to $\ln(10^{10}A_s)=3.044\pm0.014$ and $n_s=0.9649\pm0.0042$ at $k_*=0.05\,\mathrm{Mpc}^{-1}$ . The amplitude, propagated forward, predicts a clustering strength $\sigma_8\approx0.81$ — and the disagreement between that CMB-based value and the lower $\sigma_8$ from weak-lensing is the " $S_8$ tension," one of the sharpest cracks in the standard model. On the inflationary side, the bound $r<0.036$ against $n_s\approx0.965$ in the $(n_s,r)$ plane favours plateau potentials such as Starobinsky $R^2$ while excluding the simplest monomial models. The frontier is twofold: next-generation CMB experiments (Simons Observatory, CMB-S4, LiteBIRD) chase $r$ toward $10^{-3}$ to test $n_t=-r/8$ , while galaxy surveys (DESI, Euclid, Rubin) push $\mathcal{P}_\mathcal{R}(k)$ to smaller scales, hunting a running tilt or features that would reveal the inflaton potential.

The physical picture

Think of inflation as a cosmic microscope run backwards: it seizes the quantum jitters of a scalar field and stretches their wavelengths exponentially, so what began as sub-atomic quivering ends up spanning the observable Universe. A comoving mode starts deep inside the Hubble radius, where quantum fluctuations are active and the field behaves like an ordinary oscillator. As space inflates, the mode's fixed comoving wavelength is dragged outside the shrinking comoving Hubble radius $(aH)^{-1}$ ; at that instant of horizon exit its amplitude is "photographed" and frozen. Because $H$ is very nearly constant during inflation, every mode is photographed under almost identical conditions, so all scales inherit almost the same amplitude — the origin of near scale invariance. The first figure tells the whole story: the comoving Hubble radius forms a valley, and a mode of fixed size threads out of it and, much later, back in.

Figure

Comoving Hubble radius (aH)⁻¹ shrinks during inflation then grows afterward; a mode of fixed comoving wavelength 1/k exits the horizon when k = aH, stays frozen for roughly 60 e-folds, and re-enters much later — R is conserved throughout

The small departure from flatness is physical, and it is where the information lives. Inflation cannot last forever — the field must roll down its potential and end — so $H$ decreases slowly. A mode exiting later is a smaller comoving scale (larger $k$ ) that samples a slightly smaller $H$ and receives less power: a mild redward tilt, $n_s<1$ . Measuring $n_s\neq1$ thus measures that inflation is a genuine dynamical episode with a beginning and an end, not eternal de Sitter. Conservation of $\mathcal{R}$ makes this predictive: between horizon exit and re-entry the Universe passes through reheating, nucleosynthesis, and equality, and none of that mess disturbs the frozen amplitude. The second figure shows the surviving spectrum as measured.

Figure

The measured primordial spectrum P_R(k) = A_s (k/k) raised to (n_s − 1), with A_s = 2.1×10⁻⁹ at pivot k = 0.05 Mpc⁻¹ and n_s = 0.9649 — a gentle red tilt sitting above the scale-invariant n_s = 1 line at large scales

The core equations

Everything begins with the two-point function. Homogeneity forces the correlation of two Fourier modes to be diagonal, and isotropy makes the coefficient — the power spectrum — depend only on the wavevector magnitude.

\langle \mathcal{R}(\mathbf{k})\,\mathcal{R}^*(\mathbf{k}')\rangle = (2\pi)^3\,\delta_D^{(3)}(\mathbf{k}-\mathbf{k}')\,P_\mathcal{R}(k) \qquad (1)

where: $\mathcal{R}(\mathbf{k})$ — Fourier mode of the comoving curvature perturbation; $P_\mathcal{R}(k)$ — curvature power spectrum, dimensions of volume ( $\mathrm{Mpc}^3$ ); $\delta_D^{(3)}$ — three-dimensional Dirac delta enforcing homogeneity; $k=|\mathbf{k}|$ — comoving wavenumber. The delta makes different modes independent; a Gaussian field is then fully specified by $P_\mathcal{R}(k)$ alone. For intuition it is better to work with a dimensionless quantity measuring the variance each decade of scale contributes; the phase-space factor $k^3/2\pi^2$ does that.

\mathcal{P}_\mathcal{R}(k) \equiv \frac{k^3}{2\pi^2}\,P_\mathcal{R}(k),\qquad \langle\mathcal{R}^2\rangle=\int \mathcal{P}_\mathcal{R}(k)\,\mathrm{d}\ln k \qquad (2)

where: $\mathcal{P}_\mathcal{R}(k)$ — dimensionless variance per logarithmic interval in $k$ ; $k^3/2\pi^2$ — the factor turning a spectral density into a log-interval variance. A flat $\mathcal{P}_\mathcal{R}$ means equal power per octave — exactly what "scale-invariant" means. Measured across four decades of scale, $\mathcal{P}_\mathcal{R}(k)$ is a nearly featureless power law — an amplitude and a slope about a chosen pivot.

\mathcal{P}_\mathcal{R}(k) = A_s\left(\frac{k}{k_*}\right)^{n_s-1} \qquad (3)

where: $A_s$ — scalar amplitude at the pivot, $A_s\approx2.1\times10^{-9}$ ; $n_s$ — scalar spectral index or tilt, $n_s\approx0.965$ ; $k_*$ — pivot scale, conventionally $0.05\,\mathrm{Mpc}^{-1}$ . The exponent $n_s-1$ is the constant log slope; $n_s=1$ is exact scale invariance, $n_s<1$ a red tilt. Inflation predicts this amplitude from first principles: quantizing a light scalar in quasi-de Sitter (Derivation 1) gives a value set by the Hubble rate and the first slow-roll parameter at horizon exit.

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

where: $H$ — Hubble rate during inflation, at horizon exit $k=aH$ ; $M_{\rm Pl}=(8\pi G)^{-1/2}$ — reduced Planck mass; $\epsilon=-\dot H/H^2$ — first slow-roll parameter. Reading this for the observed $A_s$ pins $H/M_{\rm Pl}$ during inflation to about $10^{-5}\sqrt{\epsilon}$ . Because $H$ and $\epsilon$ drift slowly, the slope is a combination of slow-roll parameters, which also fix the primordial gravitational-wave amplitude.

n_s-1 = -6\epsilon + 2\eta,\qquad r = 16\epsilon \qquad (5)

where: $\eta=M_{\rm Pl}^2 V''/V$ — second slow-roll parameter; $r$ — tensor-to-scalar ratio, gravitational-wave to curvature power; $V,V''$ — the inflaton potential and its curvature. A red tilt is generic since $\epsilon>0$ always; a detection of $r$ would measure $\epsilon$ and the inflationary energy scale. The tensor sector carries its own spectrum and tilt, tied together by single-field slow-roll with no free parameters.

\Delta_T^2(k)=A_t\left(\frac{k}{k_*}\right)^{n_t},\qquad n_t=-\frac{r}{8} \qquad (6)

where: $\Delta_T^2$ — primordial tensor (gravitational-wave) power spectrum; $A_t=r\,A_s$ — tensor amplitude; $n_t$ — tensor tilt. The relation $n_t=-r/8$ has no free parameters — the sharpest single test of single-field slow-roll. Finally, the late-time observables are built from this input: the matter power spectrum and the CMB angular spectrum are weighted integrals of $\mathcal{P}_\mathcal{R}(k)$ against a processing kernel.

P(k,z) = \frac{2\pi^2}{k^3}\,\mathcal{P}_\mathcal{R}(k)\left(\frac{2}{5}\frac{k^2}{\Omega_m H_0^2}\right)^2 T^2(k)\,D^2(z),\qquad C_\ell \propto \int \frac{\mathrm{d}k}{k}\,\mathcal{P}_\mathcal{R}(k)\,\Delta_\ell^2(k) \qquad (7)

where: $P(k,z)$ — matter power spectrum at redshift $z$ ; $T(k)$ — transfer function encoding sub-horizon growth suppression; $D(z)$ — linear growth factor; $\Delta_\ell(k)$ — radiation transfer function projecting a wavenumber onto multipole $\ell\approx kD_A$ ; $\Omega_m,H_0$ — matter density parameter and Hubble constant. Everything downstream inherits the $k^{n_s}$ tilt; the rest is known causal physics.

Derivation AFrom the quantum vacuum to a classical curvature spectrum

The goal is to show where the amplitude $A_s=H^2/(8\pi^2 M_{\rm Pl}^2\epsilon)$ comes from — quantizing a light scalar in a nearly de Sitter background and following a single mode from deep inside the horizon, where it behaves like a Minkowski oscillator, out to super-horizon scales where it freezes. We track the Mukhanov–Sasaki variable, impose the Bunch–Davies vacuum, take the super-horizon limit, and convert the field fluctuation into a curvature fluctuation, so a quantum expectation value becomes the classical initial condition for structure.

Step 1. Work with the canonically normalized variable $v_k=a\,\delta\phi_k$ , which obeys $v_k''+(k^2-a''/a)\,v_k=0$ , primes being conformal-time derivatives. Why: this is a harmonic oscillator with time-dependent effective mass $-a''/a$ , so quantized-oscillator machinery applies directly, with $a''/a$ carrying the gravitational effect of expansion.

Step 2. In exact de Sitter $a=-1/(H\tau)$ with $\tau\in(-\infty,0)$ , so $a''/a=2/\tau^2$ and the equation is exactly solvable. Why: constant $H$ is the leading slow-roll approximation, and the simple $2/\tau^2$ mass term is what allows a closed-form mode function.

Step 3. Select the Bunch–Davies solution $v_k=\dfrac{e^{-ik\tau}}{\sqrt{2k}}\left(1-\dfrac{i}{k\tau}\right)$ , fixed by requiring that deep inside the horizon ( $k\tau\to-\infty$ ) it reduce to the Minkowski mode $e^{-ik\tau}/\sqrt{2k}$ . Why: on scales far smaller than the horizon spacetime looks flat, so the vacuum must match the Minkowski vacuum — the boundary condition that makes the answer unique.

Step 4. Take the super-horizon limit $k\tau\to0$ , where $|\delta\phi_k|^2=|v_k|^2/a^2\to H^2/2k^3$ , so the dimensionless field spectrum is $\mathcal{P}_{\delta\phi}=(k^3/2\pi^2)|\delta\phi_k|^2=(H/2\pi)^2$ . Why: once far outside the horizon a mode's amplitude stops evolving — it is "frozen" — and the Gibbons–Hawking result $(H/2\pi)^2$ is the de Sitter analogue of thermal fluctuations at temperature $H/2\pi$ .

Step 5. Convert to the conserved curvature perturbation via $\mathcal{R}=-(H/\dot\phi)\,\delta\phi$ , and use $\dot\phi^2=2\epsilon H^2 M_{\rm Pl}^2$ to get $\mathcal{P}_\mathcal{R}=H^2/(8\pi^2 M_{\rm Pl}^2\epsilon)$ at $k=aH$ . Why: $\mathcal{R}$ , not $\delta\phi$ , survives to late times, and the $1/\epsilon$ enhancement is why a sub-Planckian $H$ still yields $A_s\sim10^{-9}$ . *Reading the observed $A_s=2.1\times10^{-9}$ back through Eq. (4) gives $H/M_{\rm Pl}\approx10^{-5}\sqrt\epsilon$ , the standard inflationary energy scale.*

Go deeper · Derivation B · 1 minThe tilt from slow-roll: why the spectrum is red

Derivation BThe tilt from slow-roll: why the spectrum is red

The goal is to derive the headline result $n_s-1=-6\epsilon+2\eta$ directly from the slow-roll amplitude of Eq. (4), showing that the tilt is just the rate at which $H$ and $\epsilon$ drift, translated into a slope in logarithmic wavenumber. Because the amplitude is evaluated at horizon crossing $k=aH$ , a change in $\ln k$ is a change in the time a mode exits, so the slope of the spectrum reads out the slow evolution of the background — whose sign makes the spectrum red.

Step 1. Take the logarithm of Eq. (4), keeping only scale-dependent pieces: $\ln\mathcal{P}_\mathcal{R}=2\ln H-\ln\epsilon+\text{const}$ . Why: the tilt is a logarithmic derivative, so working with $\ln\mathcal{P}_\mathcal{R}$ isolates the two things that vary — $H$ and $\epsilon$ .

Step 2. Convert $\mathrm{d}/\mathrm{d}\ln k$ into a time derivative. At horizon crossing $\ln k=\ln a+\ln H$ , and since $\ln H$ barely moves, $\mathrm{d}\ln k\approx\mathrm{d}\ln a=H\,\mathrm{d}t$ , so $\mathrm{d}/\mathrm{d}\ln k\approx H^{-1}\,\mathrm{d}/\mathrm{d}t$ . Why: "later exit" and "smaller scale" are the same statement, so scale-dependence of the spectrum is time-dependence of the background along the horizon-crossing condition.

Step 3. Evaluate the two rates from the slow-roll definitions: $\mathrm{d}\ln H/\mathrm{d}\ln k=\dot H/H^2=-\epsilon$ and $\mathrm{d}\ln\epsilon/\mathrm{d}\ln k=2(2\epsilon-\eta)$ . Why: these are the evolution equations of slow-roll — $\epsilon$ sets how fast $H$ falls, the second how fast $\epsilon$ grows toward the end of inflation.

Step 4. Combine: $n_s-1=2(-\epsilon)-2(2\epsilon-\eta)=-6\epsilon+2\eta$ . Why: because $\epsilon>0$ always and typically dominates, the generic prediction is $n_s<1$ — a red tilt is the fingerprint of an $H$ that is slowly decreasing because inflation must end. *As a check, a chaotic $m^2\phi^2$ model at $N=60$ e-folds gives $\epsilon=\eta=1/120$ , hence $n_s=0.967$ , consistent with the sign and magnitude of the measured tilt.*

Go deeper · Derivation C · 1 minThe Sachs-Wolfe plateau: scale invariance flattens the low-multipole CMB

Derivation CThe Sachs-Wolfe plateau: scale invariance flattens the low-multipole CMB

The goal is to prove the most visible consequence of near scale invariance: that a scale-invariant spectrum produces a flat plateau in $\ell(\ell+1)C_\ell$ at low multipoles, the feature dominating the largest-angle CMB map. We start from the Sachs–Wolfe relation between temperature and curvature at last scattering, project onto multipoles with a spherical Bessel transform, and use a standard Bessel identity to collapse the integral into a closed form that lays bare why the plateau is flat and what its height measures.

Step 1. On super-horizon scales at last scattering the temperature anisotropy is a fixed fraction of the curvature, $\Delta T/T(\hat n)=\tfrac15\,\mathcal{R}(\mathbf{x}_{\rm ls})$ , in the matter-dominated adiabatic limit. Why: the largest angular scales were still outside the horizon at recombination, so no causal processing has occurred and the temperature inherits the frozen curvature, the $1/5$ being gravitational redshift net of the intrinsic term.

Step 2. Expand in spherical harmonics and take the variance: $C_\ell=\dfrac{2}{\pi}\displaystyle\int_0^\infty \mathrm{d}k\,k^2\,\left(\tfrac15\right)^2 P_\mathcal{R}(k)\,j_\ell^2(kD_{\rm ls})$ , with $D_{\rm ls}$ the distance to last scattering. Why: a plane wave of wavenumber $k$ projects onto the sphere as $j_\ell(kD_{\rm ls})$ , the transfer mapping a Fourier mode onto a multipole.

Step 3. Replace $P_\mathcal{R}=2\pi^2\mathcal{P}_\mathcal{R}/k^3$ to write the integral over $\mathrm{d}k/k$ : $C_\ell=\dfrac{4\pi}{25}\displaystyle\int\frac{\mathrm{d}k}{k}\,\mathcal{P}_\mathcal{R}(k)\,j_\ell^2(kD_{\rm ls})$ . Why: in the dimensionless spectrum the scale-invariant case is trivial — $\mathcal{P}_\mathcal{R}$ pulls out as a constant.

Step 4. For $\mathcal{P}_\mathcal{R}=A_s=$ const use the identity $\displaystyle\int_0^\infty \frac{\mathrm{d}x}{x}\,j_\ell^2(x)=\frac{1}{2\ell(\ell+1)}$ to obtain $\ell(\ell+1)C_\ell=\dfrac{2\pi A_s}{25}=\text{const}$ . Why: the $\ell(\ell+1)$ from the Bessel integral cancels the conventional $\ell(\ell+1)$ prefactor — which is exactly why cosmologists plot $\ell(\ell+1)C_\ell/2\pi$ , and why a scale-invariant input makes it flat. *The steps above yield the Sachs–Wolfe result $C_\ell^{\rm SW}=2\pi A_s/[25\,\ell(\ell+1)]$ ; a red tilt $n_s<1$ makes the plateau rise gently toward low $\ell$ , a slope that itself constrains $n_s$ .*

Limiting cases

The single power law is an approximation; the useful limits show what it becomes at the edges.

Case	Result	Meaning
Exact scale invariance (Harrison–Zeldovich)	$n_s=1,\ \ \mathcal{P}_\mathcal{R}(k)=A_s=\text{const}$	Equal power in every logarithmic scale; every mode re-enters the horizon with the same amplitude. Now excluded by data at $>8\sigma$ .
Slow-roll (observed) limit	$n_s-1=-6\epsilon+2\eta,\ \ \|n_s-1\|\ll1$	A small red tilt is generic for a slowly rolling field; this is the regime the real Universe occupies, $n_s\approx0.965$ .
Running of the tilt	$\mathcal{P}_\mathcal{R}=A_s(k/k_)^{n_s-1+\frac12\alpha_s\ln(k/k_)}$	Second-order curvature $\alpha_s\equiv\mathrm{d}n_s/\mathrm{d}\ln k$ ; consistent with zero, $\alpha_s=-0.004\pm0.007$ (Planck).
Blue tilt	$n_s>1$	More power on small scales; arises in some bouncing and hybrid models, and is disfavoured for the scalar sector.

Common trapthe misconception that a scale-invariant spectrum makes the matter power spectrum flat

It is tempting to read " $n_s=1$ , scale-invariant" as " $P(k)$ is flat," but that conflates two different spectra. Scale invariance means the dimensionless curvature spectrum $\mathcal{P}_\mathcal{R}(k)$ is constant in $\ln k$ . The matter power spectrum is a different object: $P(k)\propto k^{n_s}T^2(k)$ , so even for $n_s=1$ it rises as $k^1$ on large scales, turns over near matter–radiation equality, and falls as roughly $k^{n_s-4}\ln^2 k$ on small scales because the transfer function $T(k)$ suppresses modes that entered the horizon during radiation domination. Keep them separate: $\mathcal{P}_\mathcal{R}$ is the flat, dimensionless input; $P(k)$ is the processed, peaked output with units of volume, and its turnover is the most information-rich feature of galaxy surveys.

Worked examples

1 · The Sachs-Wolfe plateau in microkelvin (fully worked)

Using the scale-invariant result from Derivation 3 and $A_s=2.1\times10^{-9}$ , estimate the temperature fluctuation of the low- $\ell$ CMB plateau, given the monopole $T=2.725\,\mathrm{K}$ .

Step 1. The plateau height in the flat-band convention is $\ell(\ell+1)C_\ell/2\pi=A_s/25=2.1\times10^{-9}/25=8.4\times10^{-11}$ — the variance in $(\Delta T/T)^2$ per logarithmic band.

Step 2. Convert to a temperature: $\Delta T_{\rm rms}=T\sqrt{A_s/25}=2.725\times10^6\,\mu\mathrm{K}\times\sqrt{8.4\times10^{-11}}$ . The square root is $9.17\times10^{-6}$ , so $\Delta T_{\rm rms}=\mathbf{25\ \mu K}$ .

Step 3. Compare to observation: the measured large-angle plateau is $\sqrt{\ell(\ell+1)C_\ell/2\pi}\,T\approx28$ – $30\,\mu\mathrm{K}$ . The pure Sachs–Wolfe estimate lands within 15 percent, the shortfall coming from the late-time integrated Sachs–Wolfe term. A single number, $A_s\approx2.1\times10^{-9}$ , predicts a $\sim25$ – $30\,\mu\mathrm{K}$ plateau — the amplitude of the whole microwave sky at large angles.

2 · The tilt: significance and the power lost on small scales

Planck reports $n_s=0.9649\pm0.0042$ . (a) How many standard deviations is this from scale invariance? (b) By what factor is the primordial power reduced from a near-horizon scale $k=10^{-3}\,\mathrm{Mpc}^{-1}$ to a galaxy scale $k=0.5\,\mathrm{Mpc}^{-1}$ ?

Step 1. Significance: the deviation is $1-n_s=0.0351$ against an uncertainty $0.0042$ , so $0.0351/0.0042=\mathbf{8.4\sigma}$ . Exact scale invariance is excluded at high confidence — a landmark, since $n_s=1$ had been the default guess for four decades.

Step 2. Power ratio: $\mathcal{P}_\mathcal{R}(0.5)/\mathcal{P}_\mathcal{R}(10^{-3})=(0.5/10^{-3})^{n_s-1}=500^{-0.0351}$ . Taking logs, $-0.0351\times\ln 500=-0.218$ , so the ratio is $e^{-0.218}=\mathbf{0.80}$ .

Step 3. Interpretation: over these 2.7 decades the primordial power drops by about 20 percent — small, but measured, and exactly the redward trend inflation predicts, since those small-scale modes exited later when $H$ had fallen. The tilt is an 8.4σ effect removing roughly a fifth of the power between horizon and galaxy scales.

3 · Slow-roll predictions for chaotic $m^2\phi^2$ inflation

The simplest chaotic model has $V=\tfrac12 m^2\phi^2$ . For scales exiting $N=60$ e-folds before the end of inflation, $\epsilon=\eta=1/(2N)$ . Compute $n_s$ and $r$ , and decide whether it survives.

Step 1. Slow-roll parameters at $N=60$ : $\epsilon=\eta=1/120=8.33\times10^{-3}$ , safely small, so slow-roll is self-consistent.

Step 2. Tilt from Eq. (5): $n_s-1=-6\epsilon+2\eta=-4\times8.33\times10^{-3}=-0.0333$ , so $n_s=\mathbf{0.967}$ — comfortably close to the measured $0.9649$ .

Step 3. Tensors from Eq. (5): $r=16\epsilon=16/120=\mathbf{0.133}$ , and as a check $n_t=-r/8=-0.0167=-2\epsilon$ , satisfying the consistency relation. Verdict: $\phi^2$ inflation predicts the right $n_s$ but $r=0.13$ , far above the bound $r<0.036$ — so the simplest chaotic model is ruled out, while its plateau-potential cousins with much smaller $r$ survive.

Physics of the portrait

The primordial power spectrum is where a bold guess made in the 1970s met a quantum mechanism supplied by inflation — and where both were confirmed by the microwave sky. The portrait below traces the concept from an aesthetic hunch to a measured function of scale.

Origins · 1 minprimordial power spectrum and how this was discovered

primordial power spectrum

the initial condition P_R(k) for all cosmic structure · inflationary cosmology and the CMB

The idea that the Universe began with a specific spectrum of fluctuations predates any theory that could explain it. In 1970 Edward Harrison, and independently Yakov Zeldovich in 1972 (with Jim Peebles arguing along parallel lines), proposed that the primordial fluctuations should be scale-invariant — the same amplitude on every scale as each mode crossed the horizon — because any other choice would overproduce black holes on small scales or excess inhomogeneity on large ones. This "Harrison–Zeldovich–Peebles" spectrum, $n_s=1$ , was a pure symmetry argument with no dynamical justification for a decade. The justification arrived with inflation: in 1981 Viatcheslav Mukhanov and Gennady Chibisov showed that the quantum vacuum fluctuations of the field driving a de Sitter phase are stretched to cosmological scales with an almost scale-invariant spectrum — the first calculation of $\mathcal{P}_\mathcal{R}(k)$ from microscopic physics. At the 1982 Nuffield workshop several groups — Guth and Pi, Hawking, Starobinsky, and Bardeen–Steinhardt–Turner — converged on the slow-roll amplitude and its small tilt: inflation must end, so $H$ drifts and the spectrum reddens. The prediction sat untested until COBE detected the anisotropies in 1992, and became precise when WMAP and Planck measured $n_s=0.9649$ , excluding $n_s=1$ at more than eight sigma. A number first written down on grounds of taste is now a measured window onto physics at $10^{16}$ GeV.

History

1970

— Harrison derives the scale-invariant fluctuation spectrum from consistency arguments

1981

— Mukhanov and Chibisov compute inflationary quantum fluctuations, finding a near-scale-invariant spectrum

1982

— the Nuffield workshop fixes the slow-roll amplitude and predicts a slight red tilt

1992

— COBE detects CMB anisotropies at the level scale invariance predicted

2018

— Planck measures n_s = 0.9649, excluding exact scale invariance at more than eight sigma

Going further

Texts. Dodelson, Modern Cosmology, 2nd ed., Ch. 6, 8 · Baumann, Cosmology (2022), Ch. 8 · Ryden, Introduction to Cosmology, 2nd ed., Ch. 11–12 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 4 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30 · Weinberg, Cosmology (2008), Ch. 10. The quantum-to-classical derivation uses the Bunch–Davies calculation and the $(H/2\pi)^2$ result, and the Sachs–Wolfe plateau uses the standard identity $\int \mathrm{d}x/x\, j_\ell^2(x)=1/[2\ell(\ell+1)]$ . Numerical values — $A_s\approx2.1\times10^{-9}$ , $\ln(10^{10}A_s)=3.044\pm0.014$ , $n_s=0.9649\pm0.0042$ , $\alpha_s=-0.004\pm0.007$ , $\sigma_8\approx0.81$ — are from Planck 2018, with $r<0.036$ from BICEP/Keck 2021. Key derived checks: the tilt significance is $8.4\sigma$ , the power ratio across 2.7 decades is $500^{-0.0351}=0.80$ , the Sachs–Wolfe plateau is $25\,\mu\mathrm{K}$ , and $\phi^2$ inflation at $N=60$ predicts $n_s=0.967$ , $r=0.133$ .

Builds on: Density Perturbations and the Density Contrast · Jeans Instability and Cosmological Perturbation Regimes · The ΛCDM Concordance Model. Leads to: Transfer Function and the Matter Power Spectrum · The CMB Anisotropy Power Spectrum · Linear Growth of Structure.

Ask the tutor: Ask me to walk through how the frozen amplitude $H^2/(8\pi^2 M_{\rm Pl}^2\epsilon)$ becomes the 25 μK plateau you see on the microwave sky, step by step.

Why use the curvature perturbation R instead of the density contrast?

Because $\mathcal{R}$ is conserved on super-horizon scales for adiabatic perturbations, while the density contrast $\delta$ is not. Between horizon exit and re-entry the Universe passes through reheating, nucleosynthesis, and equality; over that span $\delta$ grows, changes character, and is gauge-dependent on super-horizon scales, making it a hopeless bookkeeping variable for initial conditions. $\mathcal{R}$ simply sits frozen at its horizon-exit value, so the amplitude stamped by inflation is exactly the amplitude that seeds structure. That is why the "primordial" spectrum is always quoted for $\mathcal{R}$ (or the related $\zeta$ ): it alone carries early-universe information intact across the intervening physics.

Does a scale-invariant spectrum mean the matter power spectrum is flat?

No, and this is the most common trap. Scale invariance ( $n_s=1$ ) means the dimensionless curvature spectrum $\mathcal{P}_\mathcal{R}(k)$ is flat in $\ln k$ . The matter power spectrum is a different object, $P(k)\propto k^{n_s}T^2(k)D^2(z)$ : even with $n_s=1$ it rises as $k^1$ on large scales, peaks near the horizon at matter–radiation equality, and falls off as roughly $k^{-3}\ln^2 k$ on small scales because the transfer function suppresses modes that entered during radiation domination. The observed $P(k)$ has a pronounced turnover; the flatness lives only in the primordial curvature spectrum before processing.

What exactly is the pivot scale, and why 0.05 Mpc⁻¹?

The pivot $k_*$ is the reference wavenumber at which $A_s$ is quoted, since a power law needs an anchor. The choice $0.05\,\mathrm{Mpc}^{-1}$ is a convention (Planck's; WMAP used $0.002\,\mathrm{Mpc}^{-1}$ ) sitting near the middle of the scales the CMB constrains, which minimizes the correlation between the fitted $A_s$ and $n_s$ . Because the spectrum is a power law, changing the pivot simply rescales $A_s$ by $(k_*'/k_*)^{n_s-1}$ while leaving $n_s$ unchanged — no physics depends on it, but always check which pivot a paper used before comparing amplitudes.

How does inflation produce a red tilt rather than a blue one?

The amplitude of each mode is set by the Hubble rate $H$ at horizon exit, through $\mathcal{P}_\mathcal{R}\propto H^2/\epsilon$ . Since inflation must end, the field rolls down its potential and $H$ decreases slowly. Large-scale modes (small $k$ ) exit earlier, when $H$ was larger, and carry slightly more power; small-scale modes exit later and carry slightly less — a red tilt, $n_s<1$ . A blue tilt would require $H$ to increase during the relevant window, which does not happen in standard single-field slow-roll, hence the generic (now confirmed) prediction of a slightly red spectrum.

What is the tensor-to-scalar ratio r, and why does everyone chase it?

Inflation stretches tensor fluctuations — primordial gravitational waves — as well as scalar ones, and $r$ is the ratio of tensor to scalar power at the pivot. It matters for two reasons. In single-field slow-roll $r=16\epsilon$ , so a detection measures the first slow-roll parameter and, through Eq. (4), the energy scale of inflation: $r\sim0.01$ means $H\sim10^{13}$ – $10^{14}$ GeV. And tensors imprint a distinctive curl ("B-mode") pattern in CMB polarization that nothing else in $\Lambda$ CDM produces, a nearly background-free signature. The bound $r<0.036$ already rules out the simplest models; reaching $r\sim10^{-3}$ is the central goal of the next CMB generation.

Is the running of the tilt detected, and would it matter?

Not yet. The running $\alpha_s=\mathrm{d}n_s/\mathrm{d}\ln k$ is the second-order curvature of the spectrum, and Planck measures $\alpha_s=-0.004\pm0.007$ — consistent with zero. This is what single-field slow-roll predicts, since a running is second order in the slow-roll parameters, of order $(n_s-1)^2\sim10^{-3}$ . A robust detection of a large running, or of sharp features breaking the smooth power law, would be a major result: it would point to structure in the inflaton potential or physics beyond the simplest single-field picture — which is why galaxy surveys and future CMB spectral-distortion probes work to extend the lever arm in $k$ .

The Primordial Power Spectrum

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The Sachs-Wolfe plateau in microkelvin (fully worked)

2 · The tilt: significance and the power lost on small scales

3 · Slow-roll predictions for chaotic $m^2\phi^2$ inflation

Physics of the portrait

Going further

Before this

Where this leads

The Primordial Power Spectrum

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The Sachs-Wolfe plateau in microkelvin (fully worked)

2 · The tilt: significance and the power lost on small scales

3 · Slow-roll predictions for chaotic m2ϕ2m^2\phi^2m2ϕ2 inflation

Physics of the portrait

Going further

Before this

Where this leads

3 · Slow-roll predictions for chaotic $m^2\phi^2$ inflation