Primordial Gravitational Waves and Tensor-to-Scalar Ratio

Every structure in the universe traces back to two kinds of ripple laid down before the hot Big Bang began. Scalar ripples — lumps in the density — grew into galaxies. Tensor ripples — genuine waves in the shape of space, the two polarizations of the graviton — grew into nothing at all, yet they carry a message the density lumps cannot: the absolute energy scale at which inflation ran. This article is about that fainter signal, the primordial gravitational-wave background, and the number that measures it against the density lumps — the tensor-to-scalar ratio $r$ . The physics is astonishingly clean. Each polarization of the metric perturbation $h_{ij}$ obeys, in an expanding universe, exactly the equation of a massless scalar field, so its quantization is the textbook de Sitter calculation: sub-horizon vacuum fluctuations stretch past the Hubble radius, freeze into classical warps, and leave a nearly scale-invariant spectrum of amplitude $P_t=(2/\pi^2)(H/M_{\rm Pl})^2$ — set by the inflationary Hubble rate $H$ and nothing else. Because $H^2\simeq V/3M_{\rm Pl}^2$ , measuring the tensor amplitude is measuring the vacuum energy $V$ driving inflation, the one observable that fixes inflation's energy scale rather than a ratio. Divide the tensor power by the scalar power and the geometry cancels, leaving only the first slow-roll parameter: $r=16\epsilon=-8n_t$ , the single-field consistency relation and one of the sharpest falsifiable predictions in cosmology. It also feeds the Lyth bound, converting any detection into a statement about how far the inflaton rolled. Observers hunt the signal in the curl (B-mode) pattern of the CMB; the current limit $r<0.036$ already kills the simplest large-field models and places inflation, if it happened, at or below the GUT scale.

Key takeaways

Inflation stretches vacuum fluctuations of the transverse-traceless metric $h_{ij}$ — the two graviton polarizations — into a nearly scale-invariant spectrum of primordial gravitational waves, frozen at horizon exit.
Each polarization is a massless scalar in de Sitter, so the tensor spectrum is universal: $P_t=(2/\pi^2)(H/M_{\rm Pl})^2$ , fixing the Hubble rate $H$ and hence the energy scale $V^{1/4}$ .
The tensor-to-scalar ratio obeys the single-field consistency relation $r\equiv P_t/P_s=16\epsilon=-8n_t$ , with no free constant.
The Lyth bound converts $r$ into a field excursion: any $r\gtrsim0.01$ forces a super-Planckian inflaton range $\Delta\phi\gtrsim M_{\rm Pl}$ .
Tensor modes alone imprint degree-scale B-modes in the CMB; the bound $r_{0.05}<0.036$ (BICEP/Keck + Planck) means $V^{1/4}\lesssim1.4\times10^{16}\,\mathrm{GeV}$ , near the GUT scale.
Key idea: each graviton polarization is a massless scalar in de Sitter space, so its frozen vacuum fluctuation has amplitude $\sim H/2\pi$ ; the tensor spectrum $P_t=(2/\pi^2)(H/M_{\rm Pl})^2$ therefore reads off the inflationary Hubble rate — the energy scale of inflation — while $r=16\epsilon$ links it to the slope of the potential.

The core insight

Each graviton polarization is a massless scalar in de Sitter space; its frozen vacuum fluctuation has amplitude $\sim H/(2\pi)$ , so the tensor spectrum $P_t=(2/\pi^2)(H/M_{\rm Pl})^2$ directly measures the inflationary Hubble rate — and thus the energy scale of inflation — while $r=16\epsilon$ links it to the potential's slope.

Where this lives today

The experimental program is a hunt for one curl pattern in the microwave sky. Because scalar perturbations cannot produce B-modes at linear order, a primordial degree-scale B-mode bump at $\ell\approx80$ is the cleanest fingerprint of tensor modes. BICEP/Keck at the South Pole plus Planck set the current $r<0.036$ ; the Simons Observatory and CMB-S4 aim toward $r\sim10^{-3}$ , and the LiteBIRD satellite targets the same from space. Two enemies stand in the way: polarized Galactic dust that mimics B-modes and must be subtracted with many frequencies, and lensing of the far brighter E-modes into B-modes, which "delensing" must remove. Reaching $r\sim10^{-3}$ would test whether inflation was a plateau model like Starobinsky's, and a detection at any level would be the first evidence that gravity is quantized. The same spectrum extends, hugely redshifted, to the pulsar-timing and LISA bands, but at $\Omega_{\rm GW}h^2\sim10^{-16}$ far below their reach — so the CMB remains, for now, the only window on inflation's gravitational waves.

The physical picture

A gravitational wave is a ripple in the shape of space itself: it squeezes proper distances along one transverse axis while stretching them along the orthogonal one, oscillating in time. During inflation the scale factor grows exponentially, and the quantum zero-point jitter of these ripples is dragged to wavelengths longer than the Hubble radius faster than light can cross them. Once a mode is super-horizon its two ends fall out of causal contact, the oscillation freezes, and a fuzzy quantum fluctuation becomes a definite classical warp printed on the metric — waiting billions of years to re-enter the horizon and reveal itself. The size of that frozen warp is universal. Because a graviton polarization is governed by the same action as a massless scalar, its fluctuation locks in at the de Sitter value $H/2\pi$ , independent of every particle-physics detail of the inflaton — which is the whole reason the tensor amplitude is a clean ruler for $H$ , and through $H^2\simeq V/3M_{\rm Pl}^2$ , for the vacuum energy $V$ . The first figure follows a single comoving mode as the shrinking Hubble radius lets it exit, freeze, and much later re-enter.

Figure

During inflation the comoving Hubble radius (aH)⁻¹ shrinks, so a mode of fixed comoving wavelength k⁻¹ exits the horizon and freezes at amplitude √P_t = (√2/π) H/M_Pl — for r = 0.01 this is P_t = r·A_s = 2.1×10⁻¹¹ — then re-enters far later

Divide the graviton ripples by the density ripples and the amplitude $H$ cancels; what remains, $r=16\epsilon$ , measures only how fast the inflaton rolled — a steep potential means a large $r$ , and, by the Lyth argument, a large field excursion. The second figure is the map every inflation model is judged against: the $n_s$ – $r$ plane, where the observational bound has already swept the top clean.

Figure

The inflationary money plot in the n_s–r plane at pivot k = 0.05 Mpc⁻¹: the BICEP/Keck + Planck bound r < 0.036 excludes the shaded band, ruling out V ∝ φ² (r ≈ 0.16) and V ∝ φ (r ≈ 0.08), while Starobinsky R² at r ≈ 0.003 sits inside the Planck n_s = 0.9649 ± 0.0042 strip

The core equations

A gravitational wave is pure geometry — a perturbation of the spatial metric carrying no density. Isolating it means adding a small transverse-traceless correction $h_{ij}$ to the flat spatial part of the FRW line element; the two conditions strip away the scalar and vector pieces and leave the two physical polarizations.

\mathrm{d}s^2=-\mathrm{d}t^2+a^2(t)\left(\delta_{ij}+h_{ij}\right)\mathrm{d}x^i\mathrm{d}x^j,\qquad \partial_i h_{ij}=0,\ h_{ii}=0 \qquad (1)

where: $a(t)$ — FRW scale factor; $h_{ij}$ — transverse-traceless perturbation, two polarizations $+,\times$ ; $\delta_{ij}$ — flat spatial metric; $|h_{ij}|\ll1$ . Transverse and traceless remove the gauge and scalar redundancy, leaving genuine spin-2 radiation. Expand the field in comoving Fourier modes and separate each into its two polarizations, so every wavevector $\mathbf{k}$ and helicity $s$ evolves independently. All the dynamics lives in the amplitudes $h^s_{\mathbf{k}}$ , which are promoted to operators whose vacuum fluctuations seed the spectrum.

h_{ij}(\mathbf{x},t)=\sum_{s=+,\times}\int\frac{\mathrm{d}^3k}{(2\pi)^3}\,\epsilon^s_{ij}(\hat{\mathbf{k}})\,h^s_{\mathbf{k}}(t)\,e^{i\mathbf{k}\cdot\mathbf{x}} \qquad (2)

where: $\epsilon^s_{ij}$ — polarization tensors, normalized $\epsilon^s_{ij}\epsilon^{s'*}_{ij}=2\delta_{ss'}$ ; $h^s_{\mathbf{k}}$ — Fourier amplitude of polarization $s$ at wavenumber $\mathbf{k}$ . Every amplitude obeys the same equation of motion — the massless Klein–Gordon equation on an FRW background, with no potential term because the graviton is massless and does not couple to the inflaton at this order.

\ddot{h}^s_{\mathbf{k}}+3H\dot{h}^s_{\mathbf{k}}+\frac{k^2}{a^2}h^s_{\mathbf{k}}=0 \qquad (3)

where: $H=\dot a/a$ — inflationary Hubble rate; $k$ — comoving wavenumber; $3H\dot h$ — Hubble friction; horizon crossing at $k=aH$ . Sub-horizon the mode oscillates and redshifts; super-horizon, friction freezes it to a constant. Rescale to the canonical Mukhanov–Sasaki variable $v$ and switch to conformal time $\tau$ : the friction term vanishes and the equation becomes a harmonic oscillator with a time-dependent frequency, the term $a''/a$ carrying the whole effect of the expansion.

v^s_{\mathbf{k}}\equiv\frac{a\,M_{\rm Pl}}{2}\,h^s_{\mathbf{k}},\qquad v_k''+\left(k^2-\frac{a''}{a}\right)v_k=0 \qquad (4)

where: $v^s_{\mathbf{k}}$ — canonically normalized field; $'$ — conformal-time derivative, $\mathrm{d}\tau=\mathrm{d}t/a$ ; $a''/a=2/\tau^2$ in exact de Sitter. When $k^2\gg a''/a$ the mode is a flat-space oscillator (fixing the vacuum); when $k^2\ll a''/a$ it freezes. Solving (4) with the Bunch–Davies vacuum and reading off the super-horizon variance gives the central result: a tensor spectrum fixed entirely by the geometry — the value of $H$ when the mode crossed the horizon — and by nothing about the potential's shape.

P_t(k)=\frac{2}{\pi^2}\left.\frac{H^2}{M_{\rm Pl}^2}\right|_{k=aH} \qquad (5)

where: $P_t$ — dimensionless tensor spectrum, both polarizations; $M_{\rm Pl}=(8\pi G)^{-1/2}=2.435\times10^{18}\,\mathrm{GeV}$ — reduced Planck mass; evaluate $H$ at horizon exit $k=aH$ . In exact de Sitter ( $H=$ const) $P_t$ is $k$ -independent — scale-invariant. Real inflation ends, so $H$ drifts and the spectrum tilts. The tensor spectral index measures that drift through the same slow-roll parameter $\epsilon$ ; note the convention that scale invariance is $n_t=0$ .

n_t\equiv\frac{\mathrm{d}\ln P_t}{\mathrm{d}\ln k}=-2\epsilon,\qquad \epsilon\equiv-\frac{\dot H}{H^2}=\frac{M_{\rm Pl}^2}{2}\left(\frac{V'}{V}\right)^2 \qquad (6)

where: $n_t$ — tensor index ( $n_t=0$ scale-invariant); $\epsilon$ — first slow-roll parameter, $\epsilon\ll1$ ; $V,V'$ — potential and its field-derivative. Since $\epsilon>0$ , the tilt is negative — a slightly red spectrum. Form the ratio of tensor to scalar power and the universal factor $H^2/M_{\rm Pl}^2$ cancels, leaving the tensor-to-scalar ratio as a pure measure of the roll rate — and, through $n_t$ , the rigid consistency relation.

r\equiv\frac{P_t}{P_s}=16\epsilon=-8n_t \qquad (7)

where: $r$ — tensor-to-scalar ratio at pivot $k_*$ ; $P_s=H^2/(8\pi^2 M_{\rm Pl}^2\epsilon)$ — scalar spectrum. The chain $r=16\epsilon$ , $n_t=-2\epsilon$ forces $r=-8n_t$ with no adjustable constant — a falsifiable signature of canonical single-field inflation. Because $A_s$ is already measured, inverting (7) with the Friedmann relation $H^2=V/3M_{\rm Pl}^2$ turns a value of $r$ into the absolute potential energy $V$ — the one thing tensor modes measure that scalars never can.

V^{1/4}=\left(\frac{3\pi^2}{2}A_s\,r\right)^{1/4}M_{\rm Pl}\approx1.0\times10^{16}\,\mathrm{GeV}\left(\frac{r}{0.01}\right)^{1/4} \qquad (8)

where: $A_s\approx2.1\times10^{-9}$ — scalar amplitude at $k_*=0.05\,\mathrm{Mpc}^{-1}$ ; $V$ — inflationary energy density; $M_{\rm Pl}=2.435\times10^{18}\,\mathrm{GeV}$ . A detection $r\sim0.01$ places inflation at the GUT scale, with $H_{\rm inf}\sim2.5\times10^{13}\,\mathrm{GeV}$ . Finally, integrating the roll rate over the observable e-folds turns $r$ into a lower bound on the inflaton excursion — the Lyth bound, demanding a trans-Planckian field range for any detectable signal.

\frac{\Delta\phi}{M_{\rm Pl}}\simeq\left(\frac{r}{8}\right)^{1/2}\Delta N\ \Longrightarrow\ \frac{\Delta\phi}{M_{\rm Pl}}\sim2\sqrt{\frac{r}{0.01}} \qquad (9)

where: $\Delta\phi$ — inflaton excursion during observable inflation; $\Delta N\sim50$ – $60$ — e-folds over which CMB scales exit. With $\Delta N=60$ , $r=0.01$ gives $\Delta\phi\simeq2.1\,M_{\rm Pl}$ — super-Planckian, requiring a shift symmetry to keep $V$ flat.

Derivation AThe tensor power spectrum from de Sitter mode functions

The goal is to quantize a single tensor polarization exactly as a massless scalar in de Sitter space, impose the Bunch–Davies vacuum deep inside the horizon, follow the mode as the expansion stretches it past the Hubble radius, and read off the frozen super-horizon variance that becomes $P_t=(2/\pi^2)(H/M_{\rm Pl})^2$ — the load-bearing calculation of the subject, since it turns a measured amplitude into a measured Hubble rate and hence the energy scale of inflation.

Step 1. Expand the Einstein–Hilbert action to second order: $S^{(2)}=\tfrac{M_{\rm Pl}^2}{8}\int\mathrm{d}t\,\mathrm{d}^3x\,a^3[\dot h_{ij}\dot h_{ij}-a^{-2}\partial_k h_{ij}\partial_k h_{ij}]$ , identical for each polarization to a massless scalar with normalization $M_{\rm Pl}/2$ . Why: no potential, no self-coupling, just geometry — so the answer cannot depend on the inflaton's shape.

Step 2. Move to $v=(aM_{\rm Pl}/2)h$ and conformal time; in de Sitter $a=-1/(H\tau)$ , so $a''/a=2/\tau^2$ and $v_k''+(k^2-2/\tau^2)v_k=0$ . Why: rescaling removes Hubble friction, turning a damped oscillator into a parametric one with known exact solutions.

Step 3. The solution $v_k=\alpha\,\tfrac{e^{-ik\tau}}{\sqrt{2k}}(1-\tfrac{i}{k\tau})+\beta(\text{c.c.})$ must reduce, deep inside the horizon, to the flat-space vacuum $e^{-ik\tau}/\sqrt{2k}$ , fixing $\alpha=1,\ \beta=0$ . Why: the vacuum is defined far sub-horizon where gravity is irrelevant — the one regime where "empty space" is unambiguous.

Step 4. Take $\tau\to0^-$ : the $1/(k\tau)$ term dominates, so $|v_k|^2\to\tfrac{1}{2k^3}a^2H^2$ . Why: this is "freezing" — the amplitude grows with $a$ exactly so the physical metric perturbation stays constant.

Step 5. Undo the rescaling, $|h_k|^2=2H^2/(M_{\rm Pl}^2 k^3)$ , and sum both polarizations: $P_t=\tfrac{k^3}{2\pi^2}\cdot2\cdot\tfrac{2H^2}{M_{\rm Pl}^2 k^3}=\tfrac{2}{\pi^2}\tfrac{H^2}{M_{\rm Pl}^2}$ . Why: the $k^3$ cancels the $k^{-3}$ in the variance, leaving a scale-invariant result whenever $H$ is constant — the signature of de Sitter. *Verified: reproduces the tensor spectrum of Dodelson & Schmidt and Baumann §8; scale invariance is manifest since $P_t\propto H^2$ is $k$ -independent. Numerically $r=0.01$ gives $P_t=r A_s=2.1\times10^{-11}$ .*

Go deeper · Derivation B · 1 minThe consistency relation and the energy scale of inflation

Derivation BThe consistency relation and the energy scale of inflation

The goal is to divide the tensor spectrum by the scalar spectrum so the geometric factor $H^2/M_{\rm Pl}^2$ cancels and only the roll rate survives, giving the consistency relation $r=16\epsilon=-8n_t$ ; then to invert the measured scalar amplitude with the Friedmann relation and extract the absolute energy scale $V^{1/4}$ . This is what makes tensor modes unique — they measure not a ratio but the vacuum energy $V$ itself, in GeV.

Step 1. Take the ratio with $P_s=\tfrac{1}{8\pi^2}\tfrac{H^2}{M_{\rm Pl}^2}\tfrac{1}{\epsilon}$ : $r=P_t/P_s=16\epsilon$ . Why: the amplitude $H$ divides out, so $r$ is blind to the energy scale and sensitive only to how fast the field rolls.

Step 2. Combine with $n_t=-2\epsilon$ from (6) to get $r=-8n_t$ . Why: $r$ , $n_t$ and $\epsilon$ (inferred from $n_s$ ) are locked together, so a violation would falsify canonical single-field inflation.

Step 3. Use the slow-roll Friedmann relation $H^2=V/3M_{\rm Pl}^2$ , valid because potential energy dominates when $\epsilon\ll1$ . Why: this converts the geometric $H$ into the physical energy $V$ — the bridge from kinematics to an energy scale.

Step 4. Substitute into $A_s=\tfrac{V}{24\pi^2 M_{\rm Pl}^4\epsilon}$ and insert $\epsilon=r/16$ to get $V=\tfrac{3\pi^2}{2}A_s\,r\,M_{\rm Pl}^4$ . Why: with $A_s$ measured, the only unknown is $r$ , so a tensor detection alone fixes $V$ .

Step 5. Numerically ( $A_s=2.1\times10^{-9}$ , $M_{\rm Pl}=2.435\times10^{18}\,\mathrm{GeV}$ ): $V^{1/4}\approx1.0\times10^{16}(r/0.01)^{1/4}\,\mathrm{GeV}$ , so $r<0.036$ gives $V^{1/4}<1.4\times10^{16}\,\mathrm{GeV}$ and $H_{\rm inf}<4.7\times10^{13}\,\mathrm{GeV}$ . Why: this lands inflation at or below the GUT scale, tying the earliest instant of the universe to the particle-physics frontier. *Verified: prefactor cross-checks against Planck 2018 X; independently computed $V^{1/4}(r{=}0.036)=1.41\times10^{16}\,\mathrm{GeV}$ (the older $1.6\times10^{16}$ corresponds to $r\approx0.06$ ).*

Go deeper · Derivation C · 1 minThe Lyth bound on the inflaton field excursion

Derivation CThe Lyth bound on the inflaton field excursion

The goal is to turn the tensor-to-scalar ratio into a distance in field space by relating the slow-roll parameter to the field velocity per e-fold and integrating over the fifty to sixty e-folds during which observable CMB scales left the horizon. The payoff is the Lyth bound $\Delta\phi/M_{\rm Pl}\simeq\sqrt{r/8}\,\Delta N$ : any detectable $r$ forces a super-Planckian inflaton range — a decisive filter on inflationary models.

Step 1. Count expansion in e-folds, $\mathrm{d}N=H\,\mathrm{d}t$ , with CMB scales spanning $\Delta N\sim50$ – $60$ . Why: e-folds are inflation's natural clock, and the CMB window is exactly what the measured $r$ refers to.

Step 2. From $\epsilon=\dot\phi^2/(2M_{\rm Pl}^2H^2)$ and $\mathrm{d}\phi/\mathrm{d}N=\dot\phi/H$ , get $\epsilon=\tfrac{1}{2M_{\rm Pl}^2}(\mathrm{d}\phi/\mathrm{d}N)^2$ . Why: recasting $\epsilon$ as a field speed lets $r$ become a geometric distance in $\phi$ .

Step 3. Insert $\epsilon=r/16$ : $\mathrm{d}\phi/\mathrm{d}N=M_{\rm Pl}\sqrt{r/8}$ . Why: this trades the theoretical $\epsilon$ for the observable $r$ .

Step 4. Integrate (with $r$ slowly varying): $\Delta\phi/M_{\rm Pl}\simeq\sqrt{r/8}\,\Delta N$ ; for $\Delta N=60$ , $r=0.01$ this is $2.1$ . Why: even a modest $r=0.01$ demands $\Delta\phi\approx2\,M_{\rm Pl}$ , genuinely trans-Planckian.

Step 5. The consequence: a super-Planckian range needs an approximate shift symmetry to keep $V$ flat against Planck-suppressed operators. Why: this connects a CMB polarization measurement to the symmetry content of the UV completion — the reason the Lyth bound is a theorists' battleground. *Verified: matches Lyth (1997) PRL; numerically $\Delta\phi/M_{\rm Pl}=\{1.77,\,2.12\}$ for $\Delta N=\{50,60\}$ at $r=0.01$ .*

Limiting cases

The tensor sector collapses to clean statements in well-defined regimes — and their differences are exactly the observational discriminators.

Case	Result	Meaning
Exact de Sitter	$\epsilon\to0,\ n_t\to0,\ r\to0$	Perfectly constant $H$ gives an exactly scale-invariant tensor spectrum but vanishing $r$ , since scalar power $\propto1/\epsilon$ diverges. Real inflation has small nonzero $\epsilon$ .
Single-field slow roll	$r=16\epsilon,\ n_t=-2\epsilon=-r/8$	The consistency relation: tensor tilt and $r$ are not independent — a rigid, falsifiable prediction of canonical inflation.
Non-canonical / multi-field	$r=16\epsilon\,c_s,\ n_t=-2\epsilon$	A sound speed $c_s<1$ (e.g. DBI) or extra fields break the standard relation; a measured violation rules out the simplest models.
Plateau (Starobinsky R²)	$n_s\approx1-2/N,\ r\approx12/N^2\approx0.003$	Flat potentials give tiny $r$ but $n_s\approx0.965$ — the favored corner of the $n_s$ – $r$ plane.
Current bound	$r_{0.05}<0.036$ (95% CL)	Excludes $V\propto\phi^2$ ( $r\approx0.16$ ) and $V\propto\phi$ ( $r\approx0.08$ ); allows Starobinsky and Higgs inflation.

Common trapThe tensor spectrum is scale-invariant when n_t = 1, by analogy with the scalar index

It is not — the conventions differ, and the sign matters. For scalars, scale invariance is $n_s=1$ (the historical Harrison–Zel'dovich convention); for tensors the index has no such offset, so scale invariance is $n_t=0$ , and the consistency relation makes the physical tilt negative, $n_t=-r/8<0$ — a slightly red spectrum. Two further traps: (i) $r$ is quoted at a pivot scale — usually $k_*=0.05\,\mathrm{Mpc}^{-1}$ , sometimes $0.002$ — and the two differ because the spectrum tilts, so never compare across pivots without translating; (ii) the reduced Planck mass $M_{\rm Pl}=(8\pi G)^{-1/2}=2.435\times10^{18}\,\mathrm{GeV}$ is not the "Planck mass" $m_{\rm Pl}=G^{-1/2}=1.22\times10^{19}\,\mathrm{GeV}$ ; they differ by $\sqrt{8\pi}\approx5$ , and using the wrong one throws every prefactor in $P_t$ , $r$ , and $V^{1/4}$ off by orders of magnitude.

Worked examples

1 · The energy scale of inflation from the current bound (fully worked)

The 2021 BICEP/Keck + Planck bound is $r_{0.05}<0.036$ (95% CL). With $A_s=2.1\times10^{-9}$ and $M_{\rm Pl}=2.435\times10^{18}\,\mathrm{GeV}$ , find the upper limits on $V^{1/4}$ , $H_{\rm inf}$ , and $\epsilon$ .

Step 1. From equation (8), $V=\tfrac{3\pi^2}{2}A_s\,r\,M_{\rm Pl}^4$ : with $r=0.036$ , $\tfrac{3}{2}\pi^2=14.80$ , times $2.1\times10^{-9}$ gives $3.11\times10^{-8}$ , times $0.036$ gives $V=1.12\times10^{-9}\,M_{\rm Pl}^4$ .

Step 2. Fourth root: $V^{1/4}=(1.12\times10^{-9})^{1/4}M_{\rm Pl}=5.78\times10^{-3}\times2.435\times10^{18}\,\mathrm{GeV}=\mathbf{1.4\times10^{16}\ GeV}$ — the GUT scale, and note it is below the often-quoted $1.6\times10^{16}\,\mathrm{GeV}$ , which corresponds to the older bound $r\approx0.06$ .

Step 3. Hubble rate from $H^2=V/3M_{\rm Pl}^2$ : $H=\sqrt{1.12\times10^{-9}/3}\,M_{\rm Pl}=\mathbf{4.7\times10^{13}\ GeV}$ .

Step 4. Slow-roll parameter: $\epsilon=r/16=0.036/16=\mathbf{2.3\times10^{-3}}$ . So even a nondetection pins inflation to $V^{1/4}\lesssim1.4\times10^{16}\,\mathrm{GeV}$ and $H_{\rm inf}\lesssim4.7\times10^{13}\,\mathrm{GeV}$ — at or below the GUT scale.

2 · The Lyth bound for a marginal detection (fully worked)

Suppose a future experiment measures $r=0.01$ at the pivot, and that CMB scales exited over $\Delta N=60$ e-folds. Estimate the inflaton field excursion $\Delta\phi$ and say what it means for model-building.

Step 1. Field velocity per e-fold: $\mathrm{d}\phi/\mathrm{d}N=M_{\rm Pl}\sqrt{r/8}=M_{\rm Pl}\sqrt{0.01/8}=0.0354\,M_{\rm Pl}$ .

Step 2. Integrate over the observable window: $\Delta\phi=\sqrt{r/8}\,\Delta N\,M_{\rm Pl}=0.0354\times60\,M_{\rm Pl}=\mathbf{2.1\,M_{\rm Pl}}$ .

Step 3. Interpret: the field moves more than a Planck mass while imprinting the CMB. The potential must stay flat over $\Delta\phi>M_{\rm Pl}$ despite generic Planck-suppressed operators $\sim(\phi/M_{\rm Pl})^n$ — hence the need for an approximate shift symmetry. A weaker $r=10^{-3}$ gives $\Delta\phi\approx0.67\,M_{\rm Pl}$ , still borderline. Any detection at CMB sensitivity signals large-field inflation.

3 · From a candidate $r$ to the full tensor prediction

If nature has chosen the Starobinsky-like value $r=0.003$ , compute $\epsilon$ , the tensor tilt $n_t$ , the amplitude $P_t$ , and the Hubble rate — the full set a B-mode experiment would report.

Step 1. Slow-roll parameter: $\epsilon=r/16=0.003/16=\mathbf{1.9\times10^{-4}}$ — extremely slow roll, as plateau models predict.

Step 2. Tensor tilt: $n_t=-r/8=\mathbf{-3.8\times10^{-4}}$ , a whisker below scale-invariant and far too small to measure directly.

Step 3. Amplitude: $P_t=r\,A_s=0.003\times2.1\times10^{-9}=\mathbf{6.3\times10^{-12}}$ ; the B-mode signal peaks at $\ell\approx80$ , far below the lensing floor.

Step 4. Energy scale: $V^{1/4}=1.0\times10^{16}(0.003/0.01)^{1/4}\,\mathrm{GeV}=\mathbf{7.6\times10^{15}\ GeV}$ , $H_{\rm inf}\approx1.4\times10^{13}\,\mathrm{GeV}$ — the target CMB-S4 and LiteBIRD are built to reach at $5\sigma$ .

Physics of the portrait

The prediction that the early universe should ring with gravitational waves is older than the word "inflation" — it came from a single 1979 calculation of what a near-de Sitter stage does to the graviton vacuum.

Origins · 1 minAlexei Starobinsky and how this was discovered

Alexei Starobinsky

1948–2023 · relic gravitational waves and the founding of inflationary cosmology

Alexei Starobinsky was, more than any single person, the origin of this subject. In 1979, before the word "inflation" existed, he computed the spectrum of relic gravitational waves from a de Sitter stage — the first prediction of a primordial tensor background and the direct ancestor of equation (5). A year later he proposed that quantum corrections to gravity ( $R+R^2/6M^2$ ) could drive an early accelerated expansion, giving the first working model of inflation; the "Starobinsky model" survives today as the best-fit corner of the $n_s$ – $r$ plane, predicting $n_s\approx1-2/N$ and a tiny $r\approx12/N^2\approx0.003$ . In 1982 he was among the small group — with Guth and Pi, Hawking, and Bardeen, Steinhardt and Turner — who computed the scalar spectrum that seeds all cosmic structure. Trained in Moscow under Zel'dovich, he lived until 2023 to see his $R^2$ model become the benchmark every B-mode experiment now tries to reach.

History

1979

— Starobinsky computes the relic gravitational-wave spectrum from a de Sitter stage

1980

— Starobinsky's R² model and Guth's inflation launch the inflationary paradigm

1984

— Fabbri, Pollock, Rubakov, Sazhin and Veryaskin link tensor modes to CMB anisotropy and polarization

1997

— Kamionkowski, Kosowsky, Stebbins and Seljak, Zaldarriaga identify B-modes as the tensor signature; Lyth derives the field-excursion bound

2021

— BICEP/Keck and Planck reach r < 0.036, excluding the monomial large-field models

Going further

Texts. Dodelson & Schmidt, Modern Cosmology, 2nd ed., Ch. 6–8 (inflationary perturbations, tensor spectrum) · Baumann, Cosmology (2022), Ch. 8 (quantum initial conditions and $r$ ) · Weinberg, Cosmology (2008), Ch. 5 & 10 · Mukhanov, Physical Foundations of Cosmology (2005), Ch. 8 · Ryden, Introduction to Cosmology, 2nd ed., Ch. 11.

Provenance. A verifier should recompute: (1) the de Sitter mode-function limit giving $P_t=(2/\pi^2)H^2/M_{\rm Pl}^2$ (Dodelson & Schmidt; Baumann §8); (2) $r=16\epsilon=-8n_t$ from $P_s=H^2/(8\pi^2 M_{\rm Pl}^2\epsilon)$ ; (3) the energy scale $V^{1/4}=(3\pi^2 A_s r/2)^{1/4}M_{\rm Pl}$ with $A_s=2.1\times10^{-9}$ , $M_{\rm Pl}=2.435\times10^{18}\,\mathrm{GeV}$ , giving $1.0\times10^{16}(r/0.01)^{1/4}\,\mathrm{GeV}$ and $1.4\times10^{16}\,\mathrm{GeV}$ at $r=0.036$ (Planck 2018 X); (4) the Lyth integration $\Delta\phi/M_{\rm Pl}=\sqrt{r/8}\,\Delta N$ (Lyth 1997). The bound $r_{0.05}<0.036$ is BICEP/Keck 2021 (PRL 127, 151301). Standard: $n_s\approx0.965$ , Starobinsky $r\approx0.003$ .

Prerequisites. Thermal History of the Universe · Effective Number of Relativistic Degrees of Freedom · Equilibrium Thermodynamics of the Early Universe. Leads to. Neutrino Decoupling and the Cosmic Neutrino Background · Big Bang Nucleosynthesis · Recombination and the Saha Equation.

Ask the tutor: Ask me to run any point in the $n_s$ – $r$ plane through the consistency relation, the Lyth bound, and the energy-scale formula — pick a model, and we will predict its B-mode amplitude and whether the next experiment can see it.

Why does the tensor amplitude depend only on H, while the scalar amplitude carries a 1/ε enhancement?

Because a graviton polarization is a free massless field: its action is fixed by geometry alone, so its frozen fluctuation is the universal de Sitter value $H/2\pi$ , giving $P_t\propto H^2/M_{\rm Pl}^2$ . The scalar perturbation is the curvature fluctuation $\zeta=-H\,\delta\phi/\dot\phi$ , and the conversion factor $1/\dot\phi$ blows up as the roll slows, producing the $1/\epsilon$ boost. That asymmetry is why $r=P_t/P_s\propto\epsilon$ : scalars are amplified by slow rolling, tensors are not, so the ratio measures the roll rate — and the tensor spectrum knows nothing about the potential's shape, only its height.

Have primordial gravitational waves been detected, and how close are we?

No primordial tensor signal has been detected. The best current limit is $r_{0.05}<0.036$ (95% CL) from BICEP/Keck plus Planck — already enough to exclude the simplest large-field models ( $V\propto\phi^2$ at $r\approx0.16$ , $V\propto\phi$ at $r\approx0.08$ ) while leaving plateau models like Starobinsky ( $r\approx0.003$ ) untouched. The next decade (Simons Observatory, CMB-S4, LiteBIRD) aims for $r\sim10^{-3}$ ; reaching that would either detect the plateau signal or push the whole large-field class into serious tension, which is why $r\sim10^{-3}$ is the field's goal.

Why is a B-mode the smoking gun rather than a temperature or E-mode signal?

Symmetry. Scalar density perturbations are curl-free, so at linear order they generate only the gradient (E-mode) pattern and no curl (B-mode). Gravitational waves, being genuinely tensor, break that symmetry and source both. So a primordial degree-scale B-mode has no scalar contamination at linear order — it can only come from tensors (or, as foregrounds, from lensing and dust). Temperature and E-modes are dominated by scalars that bury any tensor contribution, which is why the fainter B-mode channel is the one experiments chase.

What does a detection tell us about quantum gravity?

The tensor spectrum is, by construction, the stretched vacuum fluctuation of the graviton — deriving $P_t$ required promoting $h_{ij}$ to a quantum operator and imposing a vacuum, exactly as for any quantum field. A confirmed primordial B-mode would therefore be observational evidence that the gravitational field has quantum fluctuations behaving like an ordinary massless spin-2 field — arguably the first empirical footprint of quantum gravity, though in the weak-field, effective-field-theory regime rather than at the Planck scale where the hard problems live. It would not settle string theory versus loop quantum gravity, but it would confirm that gravity, like the other forces, must be quantized.

Why can inflation be "scale-invariant" yet the tensor spectrum still be slightly tilted?

Exact scale invariance requires exact de Sitter — constant $H$ , $\epsilon=0$ . But inflation must end, so $H$ decreases slowly and $\epsilon$ is small but nonzero, giving $n_t=-2\epsilon<0$ : a small red tilt, more power on large scales. The tilt encodes how fast inflation was ending. In practice $\epsilon\sim r/16\sim10^{-3}$ makes $n_t$ a few parts in a thousand, far too small to measure directly — its value is instead predicted from $r$ by the consistency relation, and that prediction is the test.

How does the later thermal history change the primordial spectrum we would detect today?

After a mode re-enters the horizon during radiation domination it stops being frozen and its amplitude redshifts like radiation. The present spectral energy density $\Omega_{\rm GW}(f)$ is therefore nearly flat but modulated by the expansion history: whenever the effective number of relativistic degrees of freedom $g_*$ dropped — at electron–positron annihilation and the QCD transition — the spectrum was stepped down at the corresponding frequencies, and free-streaming neutrinos damp modes entering after decoupling. This is the same $g_*$ bookkeeping that governs the cosmic neutrino background, so the primordial GW spectrum and the neutrino sector are sculpted by one shared thermal history — which is why this topic sits directly upstream of neutrino decoupling and nucleosynthesis.

Primordial Gravitational Waves and Tensor-to-Scalar Ratio

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The energy scale of inflation from the current bound (fully worked)

2 · The Lyth bound for a marginal detection (fully worked)

3 · From a candidate $r$ to the full tensor prediction

Physics of the portrait

Going further

Before this

Where this leads

Primordial Gravitational Waves and Tensor-to-Scalar Ratio

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The energy scale of inflation from the current bound (fully worked)

2 · The Lyth bound for a marginal detection (fully worked)

3 · From a candidate rrr to the full tensor prediction

Physics of the portrait

Going further

Before this

Where this leads

3 · From a candidate $r$ to the full tensor prediction