---
title: "What the Tilt, Tensors and Squeezed Triangles Measure"
description: "Derive the spectral tilt and tensor ratio, test two potentials against Planck and PDG limits, and see why single-field squeezed non-Gaussianity is tiny."
canonical_url: "https://sandboxphysics.com/stories/inflation-perturbations/what-tilt-tensors-and-squeezed-triangles-measure/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/inflation-perturbations/what-tilt-tensors-and-squeezed-triangles-measure/"
last_modified: "2026-11-12"
---

[From quantum ripples to cosmic texture](https://sandboxphysics.com/stories/inflation-perturbations/)

# What the Tilt, Tensors and Squeezed Triangles Measure

Sandbox Physics · 2026-11-12

Three numbers carry most of what observations say about single-field inflation: the tilt $n_s$, the tensor-to-scalar ratio $r$ and the squeezed-limit three-point amplitude. This chapter derives each one, compares the Lab’s numerical values with their slow-roll estimates and states which published limits they face.

## Before you begin

The mode-equation chapter. You should be comfortable with logarithmic derivatives and Fourier modes.

## The tilt measures how fast the freezing conditions change

Define $n_s-1=d\ln\mathcal{P}_{\mathcal{R}}/d\ln k$. Each $k$ freezes when $k=aH$, and $d\ln k=(1-\epsilon)\,dN$ there, so the tilt measures how quickly $H^2/\epsilon$ changes per e-fold. To leading order in the potential slow-roll parameters this gives the expression below.

Worked example on the plateau at $\phi_*=5.3213$: $\epsilon_V=2.30\times10^{-4}$ and $\eta_V=V_{,\phi\phi}/V=-0.0173$, so $n_s\approx1-0.0360=0.9640$. The numerical spectrum, differenced between $0.049$ and $0.051\,\mathrm{Mpc}^{-1}$, gives $0.96490$. Leading order is good to about $10^{-3}$ here.

$$
n_s-1\equiv\frac{d\ln\mathcal{P}_{\mathcal{R}}}{d\ln k}\simeq2\eta_V-6\epsilon_V,\qquad \epsilon_V=\frac12\left(\frac{V_{,\phi}}{V}\right)^{2},\quad \eta_V=\frac{V_{,\phi\phi}}{V}
$$

Leading-order slow-roll tilt, evaluated where the mode exits.

Reading: Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025) · Original background and mode calculations with an independent Python reference

## Two potentials, two predictions

For the $R^2$ plateau, Planck 2018 X equation 48 quotes the lowest-order results $n_s-1\simeq-2/N$ and $r\simeq12/N^2$. At $N_*=55$ these give $0.9636$ and $0.00397$; the full calculation gives $0.9649$ and $0.00355$. The formulas are lowest order in $1/N$, and our $N$ counts to $\epsilon=1$.

For $\tfrac12m^2\phi^2$, $\epsilon_V=2/\phi^2$ and inflation ends near $\phi^2=2$, so $\phi_*^2\simeq4N+2$. Then $n_s-1\simeq-2/(N+\tfrac12)$ and $r\simeq8/(N+\tfrac12)$: $0.9640$ and $0.144$ at $N=55$. The numerical values are $0.9634$ and $0.1440$.

Both match Planck’s tilt. They differ by a factor of forty in $r$, which is what separates them.

Reading: Planck Collaboration, Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211v2 · Original background and mode calculations with an independent Python reference

## Tensors: the ratio that excludes the steep slope

Gravitational waves freeze like the curvature ripple but with friction $3-\epsilon$. At leading order $r=\mathcal{P}_t/\mathcal{P}_{\mathcal{R}}\simeq16\epsilon$, so $r$ measures the slope of the potential, and through $H$ the energy scale of inflation.

PDG 2025 quotes $r<0.036$ at $95\%$ confidence from BICEP, Keck Array and Planck data. The quadratic model gives $0.124$ to $0.179$ for $N_*$ between $64$ and $44$, three to five times too large, as the right-hand panel shows. The plateau sits well below the bound.

Always check the pivot. Planck 2018 X quotes $r_{0.002}<0.056$, at $k=0.002\,\mathrm{Mpc}^{-1}$. For the plateau our $r$ is $0.00318$ there and $0.00355$ at $0.05\,\mathrm{Mpc}^{-1}$. The Lab reports the value at $0.05\,\mathrm{Mpc}^{-1}$, because a large-scale step can distort $r_{0.002}$.

![Computed spectral tilt and tensor ratio against the number of e-folds after the pivot exits, for two potentials, with published bands](https://sandboxphysics.com/_astro/tilt.As5HpP3Z.svg)

Points are our numerical spectra for the $R^2$ plateau and $\tfrac12m^2\phi^2$, with $n_s$ from a finite difference at $k=0.05\,\mathrm{Mpc}^{-1}$. Bands are transcribed: Planck 2018 $n_s=0.9649\pm0.0042$; the ACT combination $0.975\pm0.003$, which PDG 2025 says is not yet a full joint analysis; and the PDG 2025 bound $r<0.036$ at $95\%$ confidence. [Open the full figure](https://sandboxphysics.com/_astro/tilt.As5HpP3Z.svg)

Reading: Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025) · Planck Collaboration, Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211v2

## Reheating moves the pivot along the potential

How many e-folds $N_*$ separate the pivot’s exit from the end of inflation depends on the expansion history afterwards, including reheating. PDG 2025 notes that the uncertainty in $n_s$ is comparable to the spread of many model predictions for $N_*$ between $50$ and $60$.

The left panel makes this concrete. On our grid the plateau runs from $n_s=0.9564$ at $N_*=44$ to $0.9697$ at $N_*=64$. It stays inside the Planck $1\sigma$ band for roughly $N_*$ from $49$ to $63$. The Lab’s shorter-reheating preset, $N_*=46$, gives $0.9583$.

So a tilt measurement constrains the potential and the post-inflation history together. That degeneracy returns in the reading guide.

Reading: Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025) · Original background and mode calculations with an independent Python reference

## Squeezed triangles and the consistency relation

The three-point function $\langle\zeta\zeta\zeta\rangle$ depends on a triangle of wavenumbers. In the squeezed limit $k_1\ll k_2\simeq k_3$, the long mode froze long before the short ones. To them it is a constant rescaling of coordinates, which shifts their power by the tilt.

That gives the first relation below, the argument of Maldacena’s equation 1.2. Creminelli and Zaldarriaga showed it holds for any single-field model. With the sign convention used by Planck and PDG, which amounts to $\zeta=\zeta_g+\tfrac35f_{\mathrm{NL}}\zeta_g^2$, the local template has squeezed limit $\tfrac{12}{5}f_{\mathrm{NL}}P(k_1)P(k_2)$. Matching the two gives $f_{\mathrm{NL}}=5(1-n_s)/12$.

For $n_s=0.9649$ this is $0.0146$. Planck 2018 IX measured $f_{\mathrm{NL}}^{\mathrm{local}}=-0.9\pm5.1$, so the single-field value is about $350$ times smaller than the error bar. A clear detection of a local signal near $1$ would therefore point beyond a single field.

$$
\lim_{k_1\to0}B_\zeta(k_1,k_2,k_3)=(1-n_s)\,P_\zeta(k_1)P_\zeta(k_2),\qquad f_{\mathrm{NL}}^{\mathrm{sq}}=\frac{5}{12}(1-n_s)
$$

Here $P_\zeta(k)=2\pi^2\mathcal{P}_{\mathcal{R}}(k)/k^3$ is the dimensional spectrum, and the tilt is that of the short mode.

Reading: Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models (2003), astro-ph/0210603v5 · Creminelli & Zaldarriaga, Single field consistency relation for the 3-point function (2004), astro-ph/0407059v2 · Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025) · Planck Collaboration, Planck 2018 results. IX. Constraints on primordial non-Gaussianity, arXiv:1905.05697v1

## What the Lab’s squeezed readout is, and is not

The Lab evaluates $\tfrac{5}{12}(1-n_s(k))$ with the local tilt of the computed spectrum at each short-mode $k$. On the smooth plateau it stays below $0.02$. With the default step the local tilt swings, and the largest value reaches $-0.651$; the exaggerated sharp step reaches $3.93$.

This is a restricted bispectrum: only the squeezed limit, and only through the consistency relation. Triangles with comparable sides need the full cubic interaction and are not computed. It covers only this limit of the step’s three-point signal. Even so, it shows that a feature in the spectrum must also appear in squeezed triangles.

![Computed primordial spectrum with and without a small step, over a Planck power-law band, and the percentage ripple the step adds](https://sandboxphysics.com/_astro/spectrum.BCtcaVss.svg)

Our calculation on $110$ wavenumbers for the Lab’s default step ($c=2\times10^{-5}$, crossed $59$ e-folds before the end) and for the same plateau without it, both scaled so the smooth curve meets $A_s=e^{3.044}\times10^{-10}$ at $k=0.05\,\mathrm{Mpc}^{-1}$. The shaded band is the Planck 2018 power law with $n_s=0.9649\pm0.0042$, an ideal reference rather than a fit to these curves. [Open the full figure](https://sandboxphysics.com/_astro/spectrum.BCtcaVss.svg)

Reading: Creminelli & Zaldarriaga, Single field consistency relation for the 3-point function (2004), astro-ph/0407059v2 · Original background and mode calculations with an independent Python reference

## Exercises and explained answers

1. Use the leading formulas to predict $n_s$ and $r$ for $\tfrac12m^2\phi^2$ at $N=60$, then compare with the numerical $0.96644$ and $0.1322$.

   Read the explanation

   $n_s\simeq1-2/60.5=0.96694$ and $r\simeq8/60.5=0.1322$. The tensor ratio matches to four digits; the tilt is off by $5\times10^{-4}$, the size of the next-order terms.

2. If the ACT combination $n_s=0.975$ were right, what squeezed amplitude would single-field inflation predict?

   Read the explanation

   $f_{\mathrm{NL}}^{\mathrm{sq}}=5\times0.025/12\approx0.0104$. It is smaller than for Planck’s tilt and still far below any foreseeable error bar.

3. For the plateau at $N_*=46$, the formula $1-2/N$ gives $0.9565$, while the numerical tilt is $0.9583$. Which is more trustworthy, and why do they differ?

   Read the explanation

   The numerical value: it solves the mode equation without slow-roll truncation. The formula is lowest order in $1/N$, and the correction grows as $N$ falls, so the gap at $N=46$ is larger than at $55$.

4. A step raises the Lab’s largest squeezed amplitude to $-0.651$. Does that contradict Planck’s $-0.9\pm5.1$?

   Read the explanation

   No. The Lab’s number is the largest value at a single scale, while Planck’s local template assumes a scale-independent $f_{\mathrm{NL}}$ and averages over many triangles. A fair comparison projects the scale-dependent bispectrum onto that template first, a step beyond the Lab’s squeezed readout.

## Try the prediction

### [Link the squeezed chart to the local tilt](https://sandboxphysics.com/labs/inflation-mode-freezing/)

1. Choose Smooth plateau. The squeezed chart lies just above zero, inside the Planck band.
2. Choose A step rings the spectrum. Each place where the spectrum rises steeply with $k$ becomes a negative squeezed value, and each falling stretch a positive one.
3. Choose Steep slope, strong gravitational waves and read the tensor ratio: about $0.144$, well above $0.036$.

[Previous: Deriving the Mode Equation Behind a Computed Spectrum](https://sandboxphysics.com/stories/inflation-perturbations/slow-roll-and-the-mode-equation/) [Next: Recompute a Primordial Spectrum and Test the Solver](https://sandboxphysics.com/stories/inflation-perturbations/recompute-a-primordial-spectrum/)

## Sources and calculation scope

1. [Planck Collaboration, Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211v2](https://arxiv.org/abs/1807.06211v2)

   Full v2 PDF (2 August 2019) read for Sections 4.2 and 7.1–7.3: equations 47–48, the step template in equations 57–62, Table 11 priors, Table 12 step results and Figure 28. The headline spectral index and amplitude quoted in this course come from its parameter tables.

2. [Planck Collaboration, Planck 2018 results. IX. Constraints on primordial non-Gaussianity, arXiv:1905.05697v1](https://arxiv.org/abs/1905.05697v1)

   Full v1 PDF read for Sections 2.1–2.2 (equations 1–3, written for the Bardeen potential rather than the curvature perturbation), Table 5 final SMICA T+E values after lensing-ISW subtraction, and Table 10 feature-model peak statistics with the look-elsewhere correction.

3. [Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025)](https://pdg.lbl.gov/2025/reviews/rpp2025-rev-inflation.pdf)

   Full review read. Used: Section 23.3.4 equations 23.55–23.57 and its sentence on the ACT combination, Section 23.6 on reheating and the number of e-folds after the pivot exits, and Section 23.7.3.2 equations 23.101–23.103 for the sign convention of the local non-Gaussian amplitude and its single-field squeezed-limit value.

4. [Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models (2003), astro-ph/0210603v5](https://arxiv.org/abs/astro-ph/0210603v5)

   Full v5 read. Equation 1.2 gives the squeezed-limit argument. In that paper the tilt symbol means the deviation from scale invariance, which modern papers write as the spectral index minus one.

5. [Creminelli & Zaldarriaga, Single field consistency relation for the 3-point function (2004), astro-ph/0407059v2](https://arxiv.org/abs/astro-ph/0407059v2)

   Full v2 read. Equation 1 states the consistency relation for any single-field model. Equation 5 defines the local non-Gaussian amplitude with the opposite sign to Planck and PDG; we use the PDG sign and do not use or rewrite equation 6.

6. [Original background and mode calculations with an independent Python reference](https://sandboxphysics.com/stories/inflation-perturbations/materials/inflation-spectrum-recompute-v1.zip)

   Our own calculations: the Lab integrator and a separate NumPy/SciPy reference agree to better than three parts in ten thousand on all five presets. The ZIP holds the Python reference, closed-form benchmarks, five inputs and expected outputs. No Engine code and no paper text is included.

## Keep reading

From quantum ripples to cosmic texture · Reading path

### Horizon exit, the mode equation and spectral constraints

1. [Article **How a Quantum Ripple Becomes Cosmic Structure**](https://sandboxphysics.com/stories/inflation-perturbations/how-a-quantum-ripple-becomes-cosmic-structure/)
2. [Article **Why a Stretched Ripple Stops Changing After Horizon Exit**](https://sandboxphysics.com/stories/inflation-perturbations/why-a-stretched-ripple-stops-changing/)
3. [Article **Deriving the Mode Equation Behind a Computed Spectrum**](https://sandboxphysics.com/stories/inflation-perturbations/slow-roll-and-the-mode-equation/)
4. [You are here **What the Tilt, Tensors and Squeezed Triangles Measure**](https://sandboxphysics.com/stories/inflation-perturbations/what-tilt-tensors-and-squeezed-triangles-measure/)
5. [Article **Recompute a Primordial Spectrum and Test the Solver**](https://sandboxphysics.com/stories/inflation-perturbations/recompute-a-primordial-spectrum/)
6. [Article **Reading Planck’s Searches for Inflation Features and Non-Gaussianity**](https://sandboxphysics.com/stories/inflation-perturbations/read-feature-and-non-gaussianity-searches/)