Primordial Perturbations

4.1 Introduction: Quantum Seeds of Cosmic Structure

One of the most profound achievements of inflationary cosmology is the explanation of the origin of all structure in the universe. During inflation, quantum fluctuations of the inflaton field are stretched by the exponential expansion from sub-Planckian wavelengths to superhorizon scales. Once outside the Hubble radius, these fluctuations “freeze” and become classical perturbations in the spacetime metric — the primordial seeds that later grow under gravitational instability into galaxies, clusters, and the large-scale structure we observe today.

The cosmic microwave background (CMB) anisotropies, first detected by COBE in 1992 and mapped with exquisite precision by WMAP and Planck, are a direct photograph of these primordial perturbations at the time of last scattering ($z \approx 1100$). The observed power spectrum is nearly scale-invariant, Gaussian, and adiabatic — exactly the predictions of the simplest inflationary models.

This chapter derives the complete theory of primordial perturbations from first principles. We will:

• Quantize scalar field fluctuations during inflation and derive the Mukhanov-Sasaki equation
• Solve for the mode functions with Bunch-Davies vacuum initial conditions
• Compute the scalar power spectrum $\mathcal{P}_\mathcal{R}(k)$ and its amplitude
• Derive the spectral index $n_s$ and its running $\alpha_s$
• Quantize tensor perturbations and derive the tensor-to-scalar ratio $r$ and consistency relation
• Analyze non-Gaussianity and the $f_\text{NL}$ parameter

4.2 Quantization of Scalar Field Fluctuations

Perturbing the Inflaton

We split the inflaton field into a homogeneous background and a perturbation:

where $\phi_0(t)$ satisfies the background Klein-Gordon equation $\ddot{\phi}_0 + 3H\dot{\phi}_0 + V'(\phi_0) = 0$, and $\delta\phi$ is treated as a quantum field on the quasi-de Sitter background.

Fourier Decomposition

We expand the perturbation in Fourier modes:

Substituting into the perturbed Klein-Gordon equation on an FRW background (and working in spatially flat gauge), each Fourier mode satisfies:

This is the equation for a damped harmonic oscillator. The $3H\dot{\delta\phi}_k$ term represents Hubble friction from the expanding background. The $k^2/a^2$ term is the physical momentum squared — it redshifts as the universe expands. We have neglected the mass term $V''(\phi_0)\delta\phi_k$ since during slow-roll $|V''| \ll H^2$(this is the condition $|\eta_V| \ll 1$).

The Mukhanov Variable

To bring the equation into canonical form for quantization, we define the Mukhanov variable:

and switch from cosmic time $t$ to conformal time $\tau$, defined by $d\tau = dt/a$, so that primes denote $d/d\tau$. We also define the pump field:

More precisely, the gauge-invariant Mukhanov variable is $v = a\delta\phi + z\mathcal{R}$where $\mathcal{R}$ is the comoving curvature perturbation, but in spatially flat gauge ($\mathcal{R} = 0$ on the slice) this reduces to $v = a\delta\phi$.

Deriving the Mukhanov-Sasaki Equation

Starting from the mode equation for $\delta\phi_k$ and substituting $\delta\phi_k = v_k/a$, we compute:

Substituting into the mode equation and simplifying, the Hubble friction term combines with the time derivatives to produce a remarkably clean result:

This has the form of a simple harmonic oscillator with a time-dependent effective frequency$\omega_k^2(\tau) = k^2 - z''/z$. The term $z''/z$ encodes the effect of the expanding background on the field fluctuations.

The Pump Term in de Sitter

In exact de Sitter space ($H = \text{const}$, $\dot{\phi}_0 = \text{const}$), the scale factor is $a(\tau) = -1/(H\tau)$ where $\tau \in (-\infty, 0)$. Then $z = a\dot{\phi}_0/H \propto a$ and:

In slow-roll inflation, corrections to this are of order $\epsilon$ and $\eta$:$z''/z \approx (2 + 6\epsilon - 2\eta)/\tau^2$, but to leading order we use the de Sitter result. The Mukhanov-Sasaki equation becomes:

4.3 The Scalar Power Spectrum

Bunch-Davies Vacuum

To solve the Mukhanov-Sasaki equation, we need initial conditions. Deep inside the Hubble radius ($k \gg aH$, equivalently $|k\tau| \gg 1$), the mode wavelength is much smaller than the curvature scale, and the mode should behave as a positive-frequency plane wave in Minkowski space. This defines the Bunch-Davies vacuum:

This is the standard normalization from canonical quantization: promoting $v$ to an operator$\hat{v} = v_k \hat{a}_k + v_k^* \hat{a}_k^\dagger$, the Wronskian condition$v_k v_k^{*\prime} - v_k^* v_k' = -i$ fixes the normalization factor $1/\sqrt{2k}$.

Exact de Sitter Solution

The equation $v_k'' + (k^2 - 2/\tau^2)v_k = 0$ is a Bessel equation. Its general solution involves spherical Hankel functions, but with the Bunch-Davies boundary condition, the unique solution is:

Verification: As $k\tau \to -\infty$, the factor $i/(k\tau) \to 0$ and we recover the Bunch-Davies initial condition. One can directly verify that this satisfies the equation by computing $v_k''$:

Superhorizon Limit

On superhorizon scales ($k\tau \to 0^-$), the $i/(k\tau)$ term dominates:

The power spectrum of $\delta\phi$ is computed from $|\delta\phi_k|^2 = |v_k/a|^2$. Using $a = -1/(H\tau)$:

The dimensionless power spectrum (power per logarithmic interval in $k$) is defined as:

This is the famous result: quantum fluctuations of a light scalar field in de Sitter space have an amplitude $H/(2\pi)$, independent of $k$ — an exactly scale-invariant spectrum.

From Field Fluctuations to Curvature Perturbations

The observable quantity is not $\delta\phi$ itself, but the comoving curvature perturbation$\mathcal{R}$, which is conserved on superhorizon scales. On uniform-density hypersurfaces, the relation is:

This follows from the $\delta N$ formalism: $\mathcal{R} = \delta N = (dN/d\phi)\delta\phi = -(H/\dot{\phi}_0)\delta\phi$, using $dN = Hdt$ and $d\phi = \dot{\phi}_0 dt$. Therefore:

Using the slow-roll relation $\dot{\phi}_0^2 = 2\epsilon H^2 M_P^2$ (from $\epsilon = -\dot{H}/H^2 = \dot{\phi}_0^2/(2H^2M_P^2)$):

All quantities are evaluated at the moment of horizon crossing $k = aH$. Since $H$and $\epsilon$ vary slowly during inflation, different $k$-modes cross the horizon at slightly different times and therefore see slightly different values of $H$ and $\epsilon$ — this is what gives the spectrum a slight tilt away from exact scale invariance.

Equivalently, using $3H^2M_P^2 \approx V$ in slow-roll:

Planck measures $\mathcal{P}_\mathcal{R}(k_*) = A_s \approx 2.1 \times 10^{-9}$at the pivot scale $k_* = 0.05\,\text{Mpc}^{-1}$. This fixes the energy scale of inflation:$V^{1/4} \sim 10^{16}\,\text{GeV} \times (r/0.01)^{1/4}$.

4.4 Spectral Index and Running

Parametrizing the Tilt

The scalar power spectrum is parametrized as a power law about a pivot scale $k_*$:

where $n_s = 1$ corresponds to exact scale invariance (Harrison-Zel'dovich spectrum). The spectral index is defined as:

Derivation of the Spectral Index

From $\mathcal{P}_\mathcal{R} = H^2/(8\pi^2 M_P^2 \epsilon)$, we take the logarithmic derivative:

The key step is converting $d/d\ln k$ to derivatives with respect to the inflaton field. At horizon crossing, $k = aH$, so:

to leading order in slow-roll (we drop $\epsilon$ corrections since $\epsilon \ll 1$). Now we use the slow-roll flow equations:

where we used $d\epsilon_V/dN = 2\epsilon_V(2\epsilon_V - \eta_V)$ from the slow-roll relations $\epsilon_V = (M_P^2/2)(V'/V)^2$ and $\eta_V = M_P^2 V''/V$. Combining:

Since $\epsilon_V > 0$ and typically $|\eta_V| < \epsilon_V$ in many models, inflation generically predicts a red-tilted spectrum ($n_s < 1$). Planck measures $n_s = 0.9649 \pm 0.0042$ (68% CL), a 8$\sigma$ detection of the tilt away from scale invariance — a spectacular confirmation of inflation.

Running of the Spectral Index

The running quantifies how $n_s$ itself varies with scale:

Taking $d/dN$ of $n_s - 1 = -6\epsilon_V + 2\eta_V$ and using the slow-roll flow equations ($d\epsilon_V/dN = 2\epsilon_V(2\epsilon_V - \eta_V)$ and$d\eta_V/dN = 2\epsilon_V\eta_V - \xi_V^2$), we obtain:

where the second-order slow-roll parameter is:

The running is second-order in slow-roll parameters, so $|\alpha_s| \sim O(\epsilon^2, \epsilon\eta) \sim 10^{-3}$in typical models. Planck constrains $\alpha_s = -0.0045 \pm 0.0067$, consistent with zero.

4.5 Tensor Power Spectrum and Consistency Relation

Gravitational Wave Perturbations

Besides scalar perturbations, inflation also generates tensor perturbations — primordial gravitational waves. The transverse-traceless part of the metric perturbation is:

where $h_{ij}$ is transverse ($\partial_i h_{ij} = 0$) and traceless ($h_{ii} = 0$). It has two independent polarizations, $h_+ $ and $h_\times$. The Einstein equations give, for each polarization $s = +, \times$:

Equivalence to Massless Scalar

Defining the canonical variable $u_s = (M_P/2) \cdot a \cdot h_s$ (the factor of $M_P/2$comes from the normalization of the gravitational action), the equation becomes:

This is identical to the Mukhanov-Sasaki equation but with $z''/z$ replaced by $a''/a$. In de Sitter space, $a''/a = 2/\tau^2 = z''/z$, so each tensor polarization satisfies the same equation as the scalar Mukhanov variable. The solution with Bunch-Davies initial conditions is identical, and the power spectrum of each polarization is:

Since there are two polarizations, the total tensor power spectrum is:

Tensor-to-Scalar Ratio

The tensor-to-scalar ratio is defined as:

Tensor Spectral Index and Consistency Relation

The tensor spectral index is defined analogously to $n_s$:

From $\mathcal{P}_t \propto H^2$, we get $n_t = 2(d\ln H/dN) = -2\epsilon_V$:

Combining with $r = 16\epsilon_V$, we obtain the consistency relation:

This is a model-independent prediction of single-field slow-roll inflation: the tensor tilt and the tensor-to-scalar ratio are not independent. If both $r$ and $n_t$could be measured, violation of $r = -8n_t$ would rule out all single-field slow-roll models. This is one of the most powerful tests of the inflationary paradigm. Multi-field models and non-standard kinetic terms can violate this relation.

4.6 Non-Gaussianity

The Bispectrum

Gaussian random fields are completely specified by their two-point function (power spectrum). Any departure from Gaussianity is encoded in higher-order correlation functions. The leading non-Gaussian signature is the three-point function, or bispectrum:

The delta function enforces that the three $k$-vectors form a closed triangle in Fourier space.

The $f_\text{NL}$ Parameter

Non-Gaussianity is conventionally parametrized by writing the curvature perturbation as a local expansion about a Gaussian field $\mathcal{R}_G$:

The dimensionless parameter $f_\text{NL}$ measures the ratio of the bispectrum to the square of the power spectrum:

Maldacena's Result for Single-Field Inflation

In a landmark calculation, Maldacena (2003) computed the full three-point function of $\mathcal{R}$in single-field slow-roll inflation by expanding the action to third order. The result is:

More precisely, the bispectrum has a specific momentum-dependent shape. For the squeezed limit ($k_1 \ll k_2 \approx k_3$), the consistency relation gives:

This tiny value means that single-field slow-roll inflation predicts undetectably small non-Gaussianity. A detection of large $f_\text{NL}$ would therefore rule out the simplest inflationary models.

Triangle Shapes

Different inflationary mechanisms produce different bispectrum shapes depending on which triangle configurations dominate:

All Planck constraints are consistent with $f_\text{NL} = 0$, as predicted by single-field slow-roll inflation. Future surveys (CMB-S4, large-scale structure) will push the sensitivity to$\sigma(f_\text{NL}^\text{local}) \sim 1$, approaching the single-field prediction.

4.7 Observational Constraints and Future Prospects

Planck Results

The Planck satellite (2018 final results) provides the most precise measurements of primordial perturbation parameters:

The $n_s$-$r$ Plane

The spectral index and tensor-to-scalar ratio together provide a powerful discriminant of inflationary models. In the $n_s$-$r$ plane:

• Monomial potentials ($V \propto \phi^p$):$n_s = 1 - (p+2)/(2N)$, $r = 4p/N$ — $\phi^2$ and $\phi^4$ are ruled out by Planck+BICEP
• Starobinsky / $R^2$:$n_s = 1 - 2/N$, $r = 12/N^2$ — excellent agreement with data for $N = 50{-}60$
• Natural inflation:$n_s$ and $r$ depend on $f/M_P$ — under tension for sub-Planckian $f$
• $\alpha$-attractors:$n_s = 1 - 2/N$, $r = 12\alpha/N^2$ — interpolate between Starobinsky ($\alpha = 1$) and monomial limits

Future Experiments

Several upcoming and proposed experiments will dramatically improve constraints:

4.8 Historical Development

The theory of primordial perturbations from inflation was developed in a remarkable burst of activity in the early 1980s, with independent and partly overlapping contributions:

4.9 Python Simulations

The following simulations visualize the primordial power spectra and the $n_s$-$r$plane for several inflationary models, compared with observational constraints.