Observational Signatures of Inflation

7.1 Introduction: Testing Inflation

Inflation is not merely an elegant theoretical framework — it makes precise, falsifiable predictions about the statistical properties of cosmic perturbations. The key observational signatures of inflation include:

• ●A nearly scale-invariant spectrum of primordial density perturbations with spectral index $n_s \approx 0.96$, slightly red-tilted
• ●Gaussian statistics of the perturbations to leading order, with small non-Gaussianity parameterized by $f_\text{NL}$
• ●A stochastic background of gravitational waves (tensor modes) with amplitude characterized by the tensor-to-scalar ratio $r$
• ●Adiabatic initial conditions for all matter species
• ●Spatial flatness: $|\Omega_K| \ll 1$

These predictions are imprinted on the Cosmic Microwave Background (CMB) — the relic radiation from $z \approx 1100$ — and on the large-scale structure (LSS) of the universe. In this chapter, we derive the precise relationships between inflationary parameters and observables, and compare with the latest data from Planck, BICEP/Keck, and other experiments.

7.2 CMB Angular Power Spectrum from the Primordial Spectrum

Starting Point: The Primordial Power Spectrum

Inflation predicts a nearly scale-invariant primordial curvature power spectrum. The standard parameterization around a pivot scale $k_* = 0.05\;\text{Mpc}^{-1}$ is:

where $A_s \approx 2.1 \times 10^{-9}$ is the scalar amplitude and $n_s \approx 0.965$ is the scalar spectral index. The dimensionless power spectrum relates to the Fourier-space two-point function as:

Transfer to the CMB: Angular Power Spectrum

The CMB temperature anisotropy field $\frac{\delta T}{T}(\hat{n})$ on the sky is expanded in spherical harmonics:

The angular power spectrum $C_\ell$ is defined by $\langle a_{\ell m} a^*_{\ell' m'} \rangle = C_\ell \delta_{\ell\ell'} \delta_{mm'}$, and relates to the primordial spectrum through:

Here $\Delta_\ell(k)$ is the radiation transfer function, which encodes all the physics between the primordial epoch and the last scattering surface. It is computed by solving the coupled Boltzmann-Einstein equations for photons, baryons, dark matter, and neutrinos. The transfer function captures three key physical effects:

• ●Sachs-Wolfe effect (large scales, $\ell \lesssim 30$)
• ●Acoustic oscillations (intermediate scales, $30 \lesssim \ell \lesssim 2000$)
• ●Silk damping (small scales, $\ell \gtrsim 1500$)

The Sachs-Wolfe Effect

On scales larger than the horizon at recombination, perturbations are frozen and the temperature anisotropy is determined by the gravitational redshift of photons climbing out of potential wells. The ordinary Sachs-Wolfe effect gives:

The factor of $-1/5$ arises from the combination of the intrinsic temperature perturbation at the last scattering surface ($\delta T/T = \Phi/3$ for adiabatic perturbations) and the gravitational redshift ($\delta T/T = -\Phi$), combined with the matter-radiation relation $\Phi = -3\mathcal{R}/5$ during matter domination. With the Sachs-Wolfe transfer function$\Delta_\ell(k) = \frac{1}{5} j_\ell(k\eta_0)$ where $j_\ell$ is a spherical Bessel function and$\eta_0$ is the conformal time today, we obtain:

For a scale-invariant spectrum $\mathcal{P}_\mathcal{R}(k) = A_s$ (i.e., $n_s = 1$), using the identity $\int_0^\infty \frac{dx}{x} j_\ell^2(x) = \frac{1}{2\ell(\ell+1)}$, we derive the celebrated Sachs-Wolfe plateau:

This means that $\ell(\ell+1)C_\ell$ is approximately constant at low $\ell$, forming the flat plateau seen in CMB power spectrum plots. The slight red tilt $n_s < 1$makes the plateau gently slope downward with increasing $\ell$.

Acoustic Peaks

On sub-horizon scales at recombination, the baryon-photon fluid undergoes acoustic oscillations driven by the competition between gravitational infall and radiation pressure. The photon temperature perturbation evolves as:

where $\Theta_0 = \delta T/(T)$ is the monopole perturbation,$R = 3\rho_b/(4\rho_\gamma)$ is the baryon loading, $c_s = 1/\sqrt{3(1+R)}$ is the sound speed, and $F(k,\eta)$ is a gravitational driving term. The solution gives oscillations:

where $r_s(\eta_*) = \int_0^{\eta_*} c_s\, d\eta$ is the sound horizon at recombination. Peaks in the CMB power spectrum occur when modes are at extrema of oscillation at the moment of recombination, giving the peak locations:

The first peak at $\ell \approx 220$ is slightly shifted from $302$ by the driving effect of gravity on the oscillation. Odd peaks (1st, 3rd, ...) correspond to compression (maximal infall) and are enhanced by baryons, while even peaks (2nd, 4th, ...) correspond to rarefaction and are suppressed by baryons. The ratio of odd to even peak heights directly measures $\Omega_b h^2$.

7.3 B-mode Polarization from Tensor Modes

E/B Decomposition of CMB Polarization

CMB polarization is generated by Thomson scattering of anisotropic radiation at the last scattering surface. The polarization field is described by the Stokes parameters $Q$and $U$, which form a spin-2 field on the sky. This field can be decomposed into gradient (E-mode) and curl (B-mode) components:

where the $Y_{\ell m}$ with spin weight $\pm 2$ are spin-weighted spherical harmonics. The E and B modes have distinct parity properties: E-modes are parity-even ($(-1)^\ell$) while B-modes are parity-odd ($(-1)^{\ell+1}$).

Scalar Perturbations Produce Only E-modes

Scalar (density) perturbations create a quadrupole anisotropy in the radiation field through velocity gradients at the last scattering surface. The key symmetry argument is: a scalar perturbation with wavevector $\mathbf{k}$ has azimuthal symmetry around$\mathbf{k}$, so the polarization pattern it produces must also have this symmetry. Under a parity transformation, the polarization pattern from a scalar mode is symmetric, which corresponds to a pure E-mode pattern:

Tensor Perturbations Produce Both E and B Modes

Gravitational waves (tensor perturbations) break the azimuthal symmetry. A gravitational wave has two polarization states ($+$ and $\times$) and the quadrupole it induces in the photon distribution has components of both parities. Specifically, for a tensor mode with wavevector along $\hat{z}$:

The $\times$ polarization generates a polarization pattern that is rotated by 45 degrees relative to the $+$ polarization, breaking the reflection symmetry and producing B-modes. The resulting angular power spectra are:

Since the tensor power spectrum is $\mathcal{P}_h(k) = r \cdot A_s (k/k_*)^{n_t}$with $n_t \approx -r/8$, the B-mode power spectrum scales as:

The B-mode Spectrum Structure

The primordial B-mode spectrum has two characteristic features:

• ●Reionization bump ($\ell \lesssim 10$): When the universe is reionized at $z \sim 8$, a new last scattering surface is created. Tensor modes that are super-horizon at reionization regenerate B-mode polarization, producing a bump at very low $\ell$.
• ●Recombination peak ($\ell \sim 80$): This peak corresponds to the horizon scale at recombination. Tensor modes that are entering the horizon at $z \approx 1100$ produce the maximum B-mode signal. The peak location is set by $\ell \sim \pi \eta_0/\eta_*$.

7.4 Non-Gaussianity and the Bispectrum

The Primordial Bispectrum

If the primordial perturbations were exactly Gaussian, all their statistical properties would be encoded in the two-point function (power spectrum). Deviations from Gaussianity are captured by higher-order correlators. The leading non-Gaussian signature is the three-point function, or bispectrum:

The delta function enforces the triangle condition (momentum conservation), so the bispectrum is a function of three wavenumbers forming a closed triangle. Different inflationary models predict different shapes for $B(k_1, k_2, k_3)$.

The f_NL Parameterization: Local Shape

The simplest parameterization of non-Gaussianity uses $f_\text{NL}$, defined through a local ansatz in real space:

where $\zeta_G$ is the Gaussian part. Taking the three-point function of this expression and keeping terms to first order in $f_\text{NL}$:

where $P_\zeta(k) = (2\pi^2/k^3)\mathcal{P}_\mathcal{R}(k)$. The local shape is maximized in the squeezed limit where $k_1 \ll k_2 \approx k_3$(one side of the triangle is much smaller than the other two). This configuration corresponds to a long-wavelength mode modulating the amplitude of short-wavelength fluctuations.

Equilateral Shape

Models with higher-derivative operators in the inflaton Lagrangian (such as DBI inflation or k-inflation with $\mathcal{L} = P(X, \phi)$ where $X = -(\partial\phi)^2/2$) generate non-Gaussianity that peaks in the equilateral configuration$k_1 \approx k_2 \approx k_3$. The equilateral template is:

For DBI inflation with sound speed $c_s$, the non-Gaussianity amplitude is$f_\text{NL}^\text{equil} \sim -1/c_s^2$, so reduced sound speed gives large non-Gaussianity. Planck bounds $f_\text{NL}^\text{equil} = -26 \pm 47$ imply $c_s \gtrsim 0.02$.

The Maldacena Consistency Relation

For single-field slow-roll inflation, Maldacena (2003) derived an exact result for the bispectrum in the squeezed limit. The key insight is that a long-wavelength mode $\zeta_{k_L}$ with $k_L \to 0$ is equivalent to a local rescaling of the scale factor, which simply shifts the power spectrum. This leads to:

Comparing with the local template, this gives the celebrated Maldacena consistency relation:

Since $n_s \approx 0.965$, this gives $f_\text{NL}^\text{local} \approx 0.015$, far below current observational sensitivity ($\sigma(f_\text{NL}) \sim 5$ from Planck). This is a fundamental prediction: any detection of large local non-Gaussianity would rule out all single-field models of inflation. Conversely, multi-field models, curvaton scenarios, and models with non-standard initial states can produce$f_\text{NL}^\text{local} \gg 1$.

7.5 The Lyth Bound

The Lyth bound (1997) establishes a fundamental connection between the tensor-to-scalar ratio $r$ and the total field excursion of the inflaton during the observable epoch of inflation. This has profound implications for the UV sensitivity of inflationary model building.

Derivation

We start from the relationship between $r$ and the slow-roll parameter $\epsilon$:

Solving for the field velocity:

where $N$ is the number of e-folds ($dN = Hdt$). The total field excursion during the observable $\Delta N \approx 50$–$60$ e-folds of inflation is:

If $r$ varies slowly over the observable range (which is the case for most models), we can approximate $r(N) \approx r_*$ (the value at the pivot scale), giving:

where in the last step we used $\Delta N \approx 56$ and simplified numerically:$56\sqrt{r/8} = 56/(2\sqrt{2})\sqrt{r} \approx 19.8\sqrt{r}$. For $r = 0.01$:$\Delta\phi/M_P \approx 19.8 \times 0.1 \approx 2.0$.

Implications

This has deep implications for the interplay between inflation and quantum gravity:

• ●Large-field models ($r \gtrsim 0.01$): Require $\Delta\phi > M_P$. The inflaton potential must be controlled over super-Planckian distances in field space. This is challenging because quantum gravity corrections are expected to become important, requiring special symmetry structures (such as shift symmetries in natural inflation or the geometric structure in Starobinsky/$R^2$ inflation).
• ●Small-field models ($r \ll 0.01$): Have $\Delta\phi \ll M_P$ and are less sensitive to UV physics. Hilltop models, inflection-point models, and many string-inspired constructions fall in this category.

7.6 Current Constraints and Future Experiments

Planck 2018 Results

The Planck satellite (ESA, 2009–2013) provided the most precise measurement of the CMB temperature and E-mode polarization power spectra. The key inflationary parameters from Planck 2018 (TT,TE,EE+lowE+lensing) are:

BICEP/Keck Constraints

The BICEP/Keck Array experiment at the South Pole directly targets B-mode polarization. The BICEP/Keck 2021 analysis (using data through 2018, BK18) combined with Planck and BAO gives:

This already rules out several models: monomial potentials $V \propto \phi^n$ with$n \geq 2$ are excluded, as is natural inflation with $f \lesssim 7 M_P$. The Starobinsky/$R^2$ model with $r \approx 12/N^2 \approx 0.004$ remains well within bounds.

Future Experiments

7.7 Historical Context

The observational confirmation of inflation's predictions has unfolded over three decades of increasingly precise CMB measurements:

7.8 Python Simulations

The following simulations illustrate the key observational signatures. All computations use numpy only (no scipy).