Primordial Scalar and Tensor Spectra

Primordial spectra compress the late-time two-point statistics of inflationary perturbations into dimensionless functions of comoving wavenumber. In the canonical single-field slow-roll limit, the scalar amplitude measures $H^2/\epsilon_H$ while the tensor amplitude measures $H^2$. Their ratio therefore isolates the first slow-roll parameter at leading order. Every part of that statement—the field content, vacuum, freeze-out, and slow-roll order—is an assumption that must accompany the formula.

Required background. The Einstein Equation supplies Einstein Equations for the tensor normalization and background constraint. Classical Field Modes and Canonical Variables supplies Flat-Spacetime Quantum Field Theory for Fourier correlators and canonical vacuum normalization.

Helpful background. Relativistic Matter and Compact Objects supplies stress–energy interpretation, while Relativistic Cosmology supplies the FLRW and perturbation conventions. Quantizing Primordial Perturbations derives the mode functions used below.

Dimensionless two-point spectra

Use the Fourier convention

$$\mathcal R(\mathbf x)= \int\frac{d^3k}{(2\pi)^3}\mathcal R_{\mathbf k} e^{i\mathbf k\cdot\mathbf x}.$$

Statistical homogeneity and isotropy define the dimensionless scalar spectrum by

$$\left\langle \mathcal R_{\mathbf k}\mathcal R_{\mathbf k'} \right\rangle =(2\pi)^3\delta^{(3)}(\mathbf k+\mathbf k') \frac{2\pi^2}{k^3}\mathcal P_{\mathcal R}(k).$$

Both $\mathcal R$ and $\mathcal P_{\mathcal R}$ are dimensionless. For the scalar field, the real-space variance formally satisfies

$$\langle\mathcal R(\mathbf x)^2\rangle =\int_0^\infty d\ln k\,\mathcal P_{\mathcal R}(k).$$

Thus $\mathcal P_{\mathcal R}$ measures variance per logarithmic interval of wavenumber. A constant spectrum contributes equally in each decade; it does not give a finite unsmoothed variance over an infinite range of scales. Practical variances specify infrared and ultraviolet limits or a smoothing window. For the direct tensor decomposition $h_{ij}=\sum_\lambda e_{ij}^{\lambda}h_\lambda$ and $e_{ij}^{\lambda}e^{\lambda'ij}=2\delta_{\lambda\lambda'}$ normalization of the previous page, define one spectrum per polarization in the same way. The standard tensor spectrum is the spectrum of the contracted field,

$$\mathcal P_t(k) \equiv2\sum_\lambda\mathcal P_{h_\lambda}(k) =2\left[\mathcal P_{h_+}(k)+\mathcal P_{h_\times}(k)\right].$$

The leading factor two is $e^\lambda_{ij}e^{\lambda ij}$; displaying it avoids confusing the component sum with the contracted tensor power used in $r$ and in gravitational-wave energy density.

Leading scalar and tensor amplitudes

Let $M=(8\pi G)^{-1/2}$ and $\epsilon_1\equiv-\dot H/H^2$. For a canonical scalar in the adiabatic Bunch–Davies state, with slowly varying background quantities and conserved long-wavelength curvature,

$$\boxed{ \mathcal P_{\mathcal R}(k) =\left. \frac{H^2}{8\pi^2M^2\epsilon_1} \right\rvert_{k=aH} =\left. \frac{GH^2}{\pi\epsilon_1} \right\rvert_{k=aH} }.$$

For either physical tensor polarization, $h_\lambda=\sqrt2u_\lambda/(aM)$. Contracting the metric perturbation and summing the two polarizations then gives

$$\boxed{ \mathcal P_t(k) =\left. \frac{2H^2}{\pi^2M^2} \right\rvert_{k=aH} =\left. \frac{16GH^2}{\pi} \right\rvert_{k=aH} }.$$

The tensor-to-scalar ratio at the same comoving pivot is consequently

$$\boxed{ r(k)\equiv \frac{\mathcal P_t(k)}{\mathcal P_{\mathcal R}(k)} =16\epsilon_1 }$$

at leading slow-roll order. The dimensions check directly: $H^2/M^2$ and $\epsilon_1$ are dimensionless in units with $\hbar=c=1$. The scalar expression is singular in the exact de Sitter limit because a single-field comoving clock disappears as $\epsilon_1\to0$; the tensor expression remains regular.

The tensor factor can also be checked one polarization at a time. The long-wavelength canonical mode has $\lvert u_{\lambda,k}\rvert^2=1/(2k^3\eta^2)$, while $h_\lambda=\sqrt2u_\lambda/(aM)$ and $a\simeq-1/(H\eta)$. Hence each polarization contributes $\mathcal P_{h_\lambda}=\frac{H^2}{2\pi^2M^2}$. The two polarizations and the norm-two tensor contraction give $\mathcal P_t=2\sum_\lambda\mathcal P_{h_\lambda} =2H^2/(\pi^2M^2)$, as in the box. Omitting the contraction factor while retaining the standard symbol $\mathcal P_t$ produces the familiar factor-of-two error.

The direct norm-two plus/cross package is matched explicitly by Caprini and Figueroa 2018, §5.1.2, printed p. 64 and pp. 67–68, eqs. (167)–(171) and (185)–(191) (Open PDF): their canonical field is $aMh_\lambda/\sqrt2$, and their $\mathcal P_h$ is the contracted spectrum used here as $\mathcal P_t$. The final contracted total provides an independent check in Baumann 2012, §§12.2.5–13, printed pp. 58–62, eqs. (197)–(233) (Open PDF), but Baumann’s displayed canonical component and one-polarization spectrum use a different normalization. A separate derivation, with a nonreduced Planck mass, appears in Ellis, Vennin, and Wands 2025/2026, §§23.3.2–23.3.3, printed pp. 7–10, eqs. (23.30)–(23.50) (Official PDF). A source’s component covariance, polarization norm, and tensor-spectrum definition must be translated together; agreement of the final symbol alone does not fix the factor of two.

Tilts and slow-roll order

Define the Hubble-flow hierarchy

$$\epsilon_{n+1}\equiv \frac{d\ln\lvert\epsilon_n\rvert}{dN}, \qquad n\ge1.$$

At Hubble crossing,

$$d\ln k=d\ln(aH)=(1-\epsilon_1)dN, \qquad \frac{d}{d\ln k} =\frac{1}{1-\epsilon_1}\frac{d}{dN}.$$

Define

$$n_s-1\equiv\frac{d\ln\mathcal P_{\mathcal R}}{d\ln k}, \qquad n_t\equiv\frac{d\ln\mathcal P_t}{d\ln k}.$$

Differentiating the leading amplitudes and retaining first slow-roll order gives

$$n_s-1=-2\epsilon_1-\epsilon_2, \qquad n_t=-2\epsilon_1,$$

and hence the leading single-field consistency relation

$$r=-8n_t.$$

In potential variables these become

$$n_s-1\simeq-6\epsilon_V+2\eta_V, \qquad r\simeq16\epsilon_V.$$

The scalar running begins one order higher:

$$\alpha_s\equiv\frac{dn_s}{d\ln k} =-2\epsilon_1\epsilon_2-\epsilon_2\epsilon_3 +\mathcal O(\epsilon^3).$$

Next-order amplitudes depend on the precise evaluation convention and include terms from the slow variation during crossing. One must use a consistent order for amplitudes, tilts, and the relation between Hubble and potential parameters; simply appending the first-order relation $r=16\epsilon_1$ to a second-order scalar spectrum is not a second-order prediction. Baumann gives the potential translation and leading consistency relation in Baumann 2012, §13.2, printed pp. 62–63, eqs. (234)–(239) (Open PDF).

Pivot scales and local expansions

Observational parameters are quoted at a declared pivot $k_*$:

$$\ln\mathcal P_{\mathcal R}(k) =\ln A_s+(n_s-1)L+\frac12\alpha_sL^2+\cdots, \qquad L\equiv\ln\left(\frac{k}{k_*}\right).$$

Here $k$ and $k_*$ must use the same units, conventionally inverse megaparsecs. A change of pivot changes the correlated numerical parameters, not the underlying spectrum. Moreover, this series is local in $\ln k$; a finite truncation cannot faithfully represent a sharp feature or a long extrapolation to vastly different scales. Such a model requires direct mode evolution rather than a high-order polynomial fitted around one pivot.

Where the canonical relation changes

If the scalar perturbation propagates with sound speed $c_s$, its leading amplitude becomes

$$\mathcal P_{\mathcal R} \simeq\left. \frac{H^2}{8\pi^2M^2\epsilon_1c_s} \right\rvert_{c_sk=aH}.$$

For slowly varying $c_s$ with no later scalar transfer, $r\simeq16\epsilon_1c_s$, up to corrections because scalar and tensor modes of the same $k$ cross their respective freeze-out scales at different times. In multifield inflation, entropy modes can source curvature after crossing while leaving the leading tensor spectrum unchanged; then even the single-field equality becomes model-dependent, and in a common canonical setup one finds $r\le16\epsilon_{1*}$. Excited initial states, nonattractor evolution, modified gravity, and tensor propagation effects introduce further changes.

The effective-sound-speed and multifield qualifications are reviewed in Ellis, Vennin, and Wands 2025/2026, §§23.7.2–23.7.3, printed pp. 24–26, eqs. (23.87)–(23.99) (Official PDF). A bound on $r$ is therefore not, without a model declaration, a bound on one universal inflationary slow-roll parameter.

Exercises

Suppose the scalar spectrum is truncated at constant running. Derive the parameters at a new pivot $k_*'=e^\Delta k_*$. Then use the quadratic slow-roll example at $N_*=60$ as a normalization check.

Solution

Writing $L'=\ln(k/k_*')=L-\Delta$ and matching powers of $L'$ gives

$$\ln A_s'=\ln A_s+(n_s-1)\Delta +\frac12\alpha_s\Delta^2,$$ $$n_s'-1=(n_s-1)+\alpha_s\Delta, \qquad \alpha_s'=\alpha_s.$$

For $V=m^2\phi^2/2$, the result derived in Scalar-Field Inflation and Slow Roll is $\epsilon_V=1/(2N_*+1)$. At $N_*=60$,

$$\epsilon_V=\frac1{121}=0.00826446, \qquad n_s=1-4\epsilon_V=0.966942,$$ $$r=16\epsilon_V=0.132231, \qquad r+8n_t=0.$$

This is an algebraic consistency check of the declared approximation, not an observational endorsement of the quadratic potential.

References

• Baumann, D., “TASI Lectures on Inflation,” arXiv:0907.5424v2 (2012), Open PDF.
• Caprini, C., and D. G. Figueroa, “Cosmological Backgrounds of Gravitational Waves,” Classical and Quantum Gravity 35, 163001 (2018), DOI, Open PDF.
• Ellis, J., V. Vennin, and D. Wands, “Inflation,” in F. Takahashi et al. (Particle Data Group), Review of Particle Physics, International Journal of Modern Physics A 41, 2630011 (2026), revised August 2025, Official PDF.