Cosmological Collider: Strong Mixing

2606.18248v1(2).pdf

PDF

This paper lacks details of calculation. Fill-in details as much as you can, so that students with grad-school level math & physics can understand it smoothly. Generate .md and .pdf output for download & read

Done. I generated expanded calculation notes based on the uploaded paper .

Files:

The notes include detailed derivations of the canonical normalization, equations of motion, Bogoliubov system, fourth-order reduction, exact solution basis, initial-condition matching, and power-spectrum limits. I also included a short technical note on the massless-limit logarithm convention.

[原始存档缺少此条消息 / This message was not captured in the archive]

Verdict

They are consistent. In fact, after translating notation, the 2017 paper’s numerical mode equations are the same linear system solved analytically in the 2026 paper. The main difference is notation:

\text{2017 mixing parameter: }\hat\mu\equiv \mu_{\rm 2017}/H, \qquad \text{2017 bare isocurvaton mass: }m ,

whereas the 2026 paper uses

\lambda = \text{dimensionless mixing}, \qquad M \equiv \mu_{\rm 2026}=\text{entropy mass}.

The dictionary is

\boxed{\lambda=-\hat\mu}

up to a sign convention that drops out of the power spectrum, and

\boxed{\frac{M^2}{H^2}=\hat\mu^2+\frac{m^2}{H^2}} .

With this identification, the old “non-perturbative” region

\left(\frac{\mu_{\rm 2017}}{H}\right)^2+\left(\frac{m}{H}\right)^2>\frac94

is exactly the 2026 paper’s heavy-field regime

\frac{M^2}{H^2}>\frac94 .

The 2017 paper explicitly studies the model with kinetic mixing $\mu \dot\pi S$ , focuses on $(\mu/H)^2+(m/H)^2>9/4$ , and solves the coupled de Sitter mode equations numerically in that regime. arXiv The 2026 paper presents exact linear solutions for the same type of coupled curvature-isocurvature system for arbitrary coupling and entropy mass, reducing the Bogoliubov system to a fourth-order equation and solving it in hypergeometric functions.

1. Direct equation-level comparison

The 2017 paper’s de Sitter mode equations are, in their notation $\eta=k\tau$ ,

\pi_k''-\frac{2}{\eta}\pi_k'+\pi_k -\frac{\mu_{\rm 2017}}{H} \left(\frac{s_k'}{\eta}-\frac{3s_k}{\eta^2}\right)=0,

s_k''-\frac{2}{\eta}s_k' +\left(1+\frac{m^2}{H^2\eta^2}\right)s_k +\frac{\mu_{\rm 2017}}{H}\frac{\pi_k'}{\eta}=0 .

These are their Eqs. (4.4) and (4.5). arXiv

The 2026 paper writes the system in terms of $z=k/(aH)=-\eta$ and

D_z\zeta_c=\zeta_c'+\frac{\lambda}{z}\sigma .

Expanding its Eqs. (8)–(10), one gets

\zeta_c''-\frac{2}{z}\zeta_c'+\zeta_c +\lambda\left(\frac{\sigma'}{z}-\frac{3\sigma}{z^2}\right)=0,

\sigma''-\frac{2}{z}\sigma' +\left[ 1+\frac{1}{z^2}\left(\frac{M^2}{H^2}-\lambda^2\right) \right]\sigma -\lambda\frac{\zeta_c'}{z}=0 .

Now set

z=-\eta,\qquad \zeta_c\leftrightarrow \pi,\qquad \sigma\leftrightarrow s,\qquad \lambda=-\frac{\mu_{\rm 2017}}{H}, \qquad \frac{M^2}{H^2}-\lambda^2=\frac{m^2}{H^2}.

Then the equations coincide. The last relation is just

M^2=m^2+\mu_{\rm 2017}^2 .

This also explains why the 2017 paper’s superhorizon index depends on $m^2+\mu_{\rm 2017}^2$ : their Eq. (4.20) gives

\alpha_\pm=\frac32\pm\sqrt{\frac94-\frac{m^2+\mu_{\rm 2017}^2}{H^2}},

which is precisely the 2026 paper’s Hankel index $\nu=\sqrt{9/4-M^2/H^2}$ . arXiv

2. Initial conditions are also consistent

There is a mild-looking difference in presentation, but not in physics.

The 2017 paper diagonalizes the large- $|\eta|$ behavior and finds UV modes of the form

\pi_k^{(1,2)},\,s_k^{(1,2)} \propto e^{-i\eta}(-\eta)^{1\pm i\mu_{\rm 2017}/(2H)},\qquad s_k^{(1,2)}=\pm i\,\pi_k^{(1,2)} ,

with normalization fixed by the canonical commutators. arXiv

The 2026 paper instead expands in the decoupled Hankel-function basis and encodes the interaction in time-dependent Bogoliubov coefficients. In the large- $z_0$ matching, the integration constants carry phases

\exp\left[\pm \frac{i\lambda}{2}\ln(2z_0)\right],

and the leading coefficients are independent of the entropy mass, as expected in the UV. With $z_0=-\eta_0$ and $\lambda=-\mu_{\rm 2017}/H$ , this is the same logarithmic UV phase.

So the two papers choose different bases for the two positive-frequency solutions, but the bases are related by a constant unitary rotation. The summed power spectrum is invariant under that choice.

3. Power-spectrum comparison

The 2026 exact heavy-regime result is

\frac{\Delta_\zeta}{\Delta_0} = \frac{\left|\Gamma\!\left(\frac34+\frac{i\rho}{2}\right)\right|^4} { \left|\Gamma\!\left(\frac34+\frac{i}{2}(\lambda+\rho)\right)\right|^2 \left|\Gamma\!\left(\frac34+\frac{i}{2}(\lambda-\rho)\right)\right|^2 }, \qquad \rho=\sqrt{\frac{M^2}{H^2}-\frac94}.

Using the dictionary above,

\rho=\sqrt{\hat\mu^2+\frac{m^2}{H^2}-\frac94}, \qquad \lambda=\hat\mu

after dropping the irrelevant sign. Therefore the old numerical result should be reproduced by

\boxed{ \frac{P_\zeta}{P_\zeta^{(0)}} = \frac{\left|\Gamma\!\left(\frac34+\frac{i\rho}{2}\right)\right|^4} { \left|\Gamma\!\left(\frac34+\frac{i}{2}(\hat\mu+\rho)\right)\right|^2 \left|\Gamma\!\left(\frac34+\frac{i}{2}(\hat\mu-\rho)\right)\right|^2 } }.

This is the analytic completion of the 2017 numerical power-spectrum calculation. The 2026 paper gives this expression as its heavy-field result.

A useful numerical check is the point emphasized in the 2017 paper:

\hat\mu=1.5,\qquad m=0.5H .

Then

\rho=\sqrt{1.5^2+0.5^2-\frac94}=0.5 .

The 2026 formula gives

\frac{P_\zeta}{P_\zeta^{(0)}}\simeq 7.2305, \qquad \frac{\Delta P_\zeta}{P_\zeta^{(0)}}\simeq 6.2305 .

The 2017 paper reports the corresponding non-perturbative numerical value as $6.2$ , while the perturbative result would incorrectly give $310$ . arXiv This is a very direct consistency check.

4. Large-mixing EFT limit

The 2017 paper found that for large mixing,

P_\zeta(k)\simeq C\left(\frac{\mu_{\rm 2017}}{H}\right)^{1/2}P_\zeta^{(0)}(k), \qquad C=\frac{16\pi}{\Gamma^2(-1/4)}\simeq 2.09 .

This is their Eq. (4.13)–(4.14). arXiv The same result follows from the 2026 gamma-function formula in the limit

\hat\mu\to\infty,\qquad m/H=\text{fixed}, \qquad \rho=\sqrt{\hat\mu^2+m^2/H^2-9/4}\simeq \hat\mu .

Using $\Gamma(3/4)=(-1/4)\Gamma(-1/4)$ , the leading asymptotic form is

\frac{P_\zeta}{P_\zeta^{(0)}}\sim \frac{\pi}{\Gamma^2(3/4)}\sqrt{\hat\mu} = \frac{16\pi}{\Gamma^2(-1/4)}\sqrt{\hat\mu}.

So the large- $\mu$ EFT result of the 2017 paper is not merely qualitatively consistent; it is the large- $\lambda$ , $\rho\simeq\lambda$ limit of the exact 2026 expression.

There is, however, an important conceptual distinction:

Limit	Scaling	Physical meaning
$M/H\to\infty$ at fixed $\lambda$	$P_\zeta/P_0=1+\lambda^2/(2\rho^2)+\cdots$	heavy-field decoupling
$\hat\mu\to\infty$ at fixed old $m/H$ , so (M/H\simeq	\lambda	)

The 2026 paper’s fixed- $\lambda$ , large- $M$ expansion gives the standard decoupling result. The 2017 large- $\mu$ EFT is a different trajectory in parameter space: the mass and coupling grow together. This distinction is one of the most useful lessons from putting the two papers together.

5. The $m=0$ issue is clarified

The 2017 paper stressed that perturbation theory diverges at $m=0$ , while the full non-perturbative numerical answer remains finite because the heavy mode has mass

\sqrt{m^2+\mu_{\rm 2017}^2},

which does not vanish as $m\to0$ for fixed nonzero mixing. arXiv

The 2026 notation makes this transparent. Old $m=0$ corresponds to

M^2=\mu_{\rm 2017}^2,

not to a truly massless entropy field unless $\mu_{\rm 2017}=0$ as well. Therefore one should not use the 2026 paper’s special massless formula for old $m=0$ at finite mixing. One should use the light/heavy formula with

M/H=|\hat\mu|.

This resolves a possible apparent contradiction: the 2026 paper’s true $M=0$ case has secular superhorizon growth, while the 2017 paper’s $m=0$ but $\mu_{\rm 2017}\neq0$ case is regulated by the mixing-induced entropy mass. The 2026 paper explicitly treats the true massless case separately and gives the logarithmically growing expression.

6. What new insight comes from combining them?

6.1 The 2017 numerical power-spectrum surface is now analytic

The 2017 paper had to solve the coupled mode equations numerically for the intermediate region, and concluded that much of parameter space is neither well described by small- $\mu/H$ perturbation theory nor by the large- $\mu/H$ EFT. arXiv The 2026 result turns that entire intermediate region into a closed gamma-function expression for the power spectrum.

In old variables:

\nu=\sqrt{\frac94-\hat\mu^2-\frac{m^2}{H^2}}

for the light side, and

\rho=\sqrt{\hat\mu^2+\frac{m^2}{H^2}-\frac94}

for the heavy side. Thus the transition at

\hat\mu^2+\frac{m^2}{H^2}=\frac94

is simply the analytic continuation $\nu\to i\rho$ . This is the clean analytic explanation of the oscillatory/non-oscillatory transition seen in the older numerical study.

6.2 The old “clock frequency” is the same analytic index

The 2017 paper observes that the non-decaying/massive component outside the horizon has logarithmic oscillations governed by the effective mass $\sqrt{m^2+\mu_{\rm 2017}^2}$ , and that these modes matter for non-Gaussianity. arXiv In the 2026 language, this is precisely the imaginary Hankel index

\nu=i\rho,\qquad \rho=\sqrt{\hat\mu^2+\frac{m^2}{H^2}-\frac94}.

So the “cosmological collider clock” frequency is not just the bare isocurvaton mass $m/H$ ; it is the entropy eigenmass $M/H=\sqrt{m^2+\mu_{\rm 2017}^2}/H$ , while the amplitude is controlled separately by the mixing $\lambda=\hat\mu$ .

6.3 The exact linear solutions can replace numerical mode functions in old bispectrum calculations

The 2017 paper computes equilateral and squeezed bispectra numerically using the non-perturbative mode functions, and it has to Wick rotate the mode equations for the imaginary-time integral. arXiv The 2026 paper supplies exact linear solutions in terms of confluent hypergeometric functions and differential hypergeometric operators.

That means the natural next step is to redo the 2017 bispectrum integrals with the 2026 analytic mode basis. This would separate the problem into:

\text{exact linear transfer} \quad \times \quad \text{chosen cubic interaction}.

The old paper supplies the cubic interactions, such as $s\,\partial\pi\partial\pi$ and $s^3$ , plus the phenomenological constraints; the new paper supplies the exact non-perturbative propagating basis.

6.4 The large- $\mu$ EFT validity threshold can be sharpened

The 2017 paper found empirically that the EFT is typically good only for $\mu_{\rm 2017}/H\gtrsim 10$ , especially for non-Gaussianities. arXiv The exact power-spectrum formula lets one quantify the approach to the EFT analytically.

For example:

(\hat\mu,m/H)=(10,2)

gives from the exact formula

P_\zeta/P_0\simeq 5.83,

whereas the asymptotic EFT formula gives

2.09\sqrt{10}\simeq 6.62.

So even at $\hat\mu=10$ , the leading EFT estimate is only moderately accurate for the power spectrum. This agrees with the old paper’s qualitative conclusion that $\mu/H\sim10$ is roughly the beginning, not the end, of EFT convergence.

6.5 The exact result exposes nonuniform limits

There are at least three distinct limits that should not be conflated:

\lambda\to0 \quad \text{at fixed } M,

M/H\to\infty \quad \text{at fixed } \lambda,

\lambda\to\infty,\quad M/H\simeq|\lambda| \quad \text{with old } m/H \text{ fixed}.

The first gives perturbation theory in mixing. The second gives decoupling. The third gives the old large-mixing EFT with $P_\zeta/P_0\sim\sqrt{\lambda}$ . The 2026 paper unifies all three, while the 2017 paper gives the physical interpretation of the third as a modified-dispersion, nonrelativistic-like light mode.

Bottom line

The two papers are not in tension. The 2017 paper is essentially a numerical and phenomenological exploration of the same linear system in old QSFI variables, plus cubic interactions and observational estimates. The 2026 paper gives the exact analytic linear solution for that system.

The most useful combined statement is:

\boxed{ \text{2017 numerical non-perturbative QSFI power spectrum} = \text{2026 exact gamma-function result} }

after the parameter map

\boxed{ \lambda=-\mu_{\rm 2017}/H,\qquad M^2=m^2+\mu_{\rm 2017}^2 . }

The 2017 paper then tells us what to do next: use the exact 2026 mode functions to analytically control the non-Gaussianities, squeezed-limit oscillations, EFT matching errors, and observational constraints that the 2017 paper had to study numerically.

Put this as the last section of the above md and pdf files, for me to download

Updated files with the comparison section appended as the final section, using the uploaded 2606.18248 paper and the 2017 An–McAneny–Ridgway–Wise QSFI paper as the comparison target. The 2017 paper is Quasi Single Field Inflation in the non-perturbative regime, arXiv:1706.09971. arXiv

Now I'd like to calculate non-Gaussianitites based on the provided analytic solution. The calculation is similar to what was numerically done in https://arxiv.org/pdf/1706.09971, but in modern notations (in addition to 2606.18248, the modern notations can also be found, in for example https://arxiv.org/pdf/2604.07434), and more importantly using the analytic mode functions. This sould be a difficult calculation. Do the following things: (1) Append a section to the md and pdf files, set up the basis and conventions for the calculation, to the point of the integrals to be evaluated, for bispectrum and trispectrum. (2) Append another section to the md and pdf files, proceed as far as possible for the calculation. If it cannot be done, mark clearly that they are on-going attempts, and show explicitly the difficulties instead of hiding any difficulty -- it is used to assist human calculation, by what is possible and what failed, instead of showing appearent progress. Here, if bispectrum and trispectrum has similar difficulty, do them all. If the difficulty level is significantly different, do the bispectrum only. Don't mess things up too much to consider trispectrum. (3) Take the squeezed limit of the integral (probably from step (1) which is simpler from starting from step (2) -- evaluate where is the best place to take squeezed limit). Try find analytical result with the only approximation squeezed limit, i.e., k1 -> 0 while keeping other k's finite. Also check the analytical squeezed limit calculation against the numerical full calculation and plot them in figures. See if they are consistent. Append this part to the pdf and md as another section. Note that the Appendices are always the real last sections. By last sections, i actually mean before the appendices.

Updated files are ready. I appended the new non-Gaussianity sections before the appendices, so the appendices remain the final sections.

The added material uses the exact coupled-mode basis and Bogoliubov formulation from the uploaded 2606.18248 paper , the non-perturbative QSFI numerical setup of An--McAneny--Ridgway--Wise arXiv, and the modern cosmological-collider full-shape / coupled-mode-function notation from arXiv:2604.07434 arXiv.

Files:

The PDF includes the numerical squeezed-limit comparison plots. The ZIP is useful if you want the Markdown file with its referenced PNG figures available locally.

please also provide source code for figure generation for download

Done.

The ZIP contains:


generate_squeezed_benchmark_figures.py
requirements.txt
README.md
generated_figures/
  squeezed_benchmark_data.csv
  ng_squeezed_benchmark_mu15.png
  ng_squeezed_error_mu15.png

The script recomputes the plotted data by solving the coupled quadratic mode equations equivalent to the exact linear system in the uploaded 2606.18248 paper, then validates the numerical mode normalization against the exact gamma-function power spectrum before evaluating the squeezed-limit integrals.

Important technical note: the code uses the ODE form of the exact coupled linear system for stable numerical figure generation. It does not symbolically evaluate the hypergeometric differential-operator solution. That symbolic Mellin/hypergeometric evaluation remains one of the open calculation steps marked in the notes.

[原始存档缺少此条消息 / This message was not captured in the archive]

Updated and generated the requested outputs.

Main result: for fixed old QSFI bare mass $m/H=0.5$ , the heavy/light threshold is $\hat\mu_c=\sqrt{2}\simeq1.414$ . Below this threshold there is no logarithmic cosmological-collider oscillation; just a light-field squeezed power law. Just above threshold, the clock branch exists, but the frequency $\rho$ is small, so one period in $\ln(k_S/k_L)$ is enormous. The amplitude-level Boltzmann factor is $e^{-\pi\rho}$ , while a rate/number-density convention would use $e^{-2\pi\rho}$ . The computation uses the exact coupled linear system from 2606.18248, the old non-perturbative QSFI parameter map from 1706.09971, and the modern coupled-mode-function viewpoint used in recent full-shape cosmological-collider calculations. arXiv+1

Files:

The new section was inserted before the appendices, so the appendices remain the final sections.

2606_18248_small_mu_ng_webpage.html

[原始存档缺少此条消息 / This message was not captured in the archive]

Done. I continued the calculation proposed in PDF Sec. 19.1 and appended the new material before the appendices, so the appendices remain the final sections.

Main progress added:

Derived a local reconstruction formula for the physical entropy mode from the exact Bogoliubov coefficient $q_b^+$ .
Derived explicit gamma-function expressions for the soft squeezed-limit coefficients $C_\pm$ .
Checked those $C_\pm$ against direct numerical Frobenius fits.
Reduced the remaining hard coefficients $J_{\pm\nu}$ to exact non-oscillatory two-dimensional hypergeometric integrals, and also gave a one-fold Mellin-Barnes representation.
Marked clearly that the remaining unsolved step is the final generic contour summation of the Mellin-Barnes expression; I do not claim a generic gamma-only closed form for $J_{\pm\nu}$ .

This extends the exact linear-solution program of 2606.18248 , with the non-perturbative QSFI comparison target from An--McAneny--Ridgway--Wise and the modern full-shape cosmological-collider context from Kumar--Lu--Xianyu--Zhang. arXiv+1

Files:

2606_18248_calculation_details_continued_19_1.html

cc_phase_explanation(1).pdf

PDF

1. In 2109.14635, there's a technique for using filtering method to plot oscillations. Use plot similar to this attached figure in this paper to plot the squeezed limit oscillations. 2. Also add analysis to the squeezed limit result: (1) what's the phase angle? (for the phase angle, see the discussion in the attached pdf, but don't have a long discussion here, only see the attached pdf for background and get the result of the squeezed limit phase, using the same notation) (2) what's the exponential boltzmann factor? (3) Apart from the exponential Boltzmann factor, what's the polynomal mass power? For these ananysis, first write as accurate result as possible, and then expand in the large mu limit to get a clean and easy-to-understand result. Put them to a section: physical discussion of the squeezed limit. For output, only provide: (1) code, (2) pdf, and (3) sourcecode to produce the pdf (either md or tex, depending on how you produced the pdf). No need of html.

Updated outputs:

Summarize the valuable part of the chat history of this thread, into a downloadable markdown file to be used in a new thread. In the new thread we would like to study phase information with chemical potential, and averaging over cosmological collider signal with multiple fields with a mass distribution function or mass matrix. Summarize relevant information, especially useful equations and references.

Summarize the valuable part of the chat history of this thread (only some key equations following which the phase information in the squeezed limit can be derived), into a downloadable markdown file to be used in a new thread. In the new thread we would like to study phase information with chemical potential, and averaging over cosmological collider signal with multiple fields with a mass distribution function or mass matrix. Summarize relevant information, especially useful equations and references.

Created the focused Markdown summary:

It keeps the key equations for squeezed-limit phase extraction, chemical-potential generalization, and mass/matrix averaging. It draws on the uploaded phase note and the exact linear-solution paper , with references to the filtering method in arXiv:2109.14635, the non-perturbative QSFI benchmark arXiv:1706.09971, and the 2026 full-shape scalar-collider/chemical-potential study. arxiv.org+2arxiv.org+2

ChatGPT can make mistakes. Check important info.

Chat history

You said:

ChatGPT said:

ChatGPT said:

Verdict

1. Direct equation-level comparison

2. Initial conditions are also consistent

3. Power-spectrum comparison

4. Large-mixing EFT limit

5. The m=0m=0m=0 issue is clarified

6. What new insight comes from combining them?

6.1 The 2017 numerical power-spectrum surface is now analytic

6.2 The old “clock frequency” is the same analytic index

6.3 The exact linear solutions can replace numerical mode functions in old bispectrum calculations

6.4 The large-μ\muμ EFT validity threshold can be sharpened

6.5 The exact result exposes nonuniform limits

Bottom line

You said:

ChatGPT said:

You said:

ChatGPT said:

You said:

ChatGPT said:

ChatGPT said:

ChatGPT said:

You said:

ChatGPT said:

You said:

ChatGPT said:

You said:

ChatGPT said:

5. The $m=0$ issue is clarified

6.4 The large- $\mu$ EFT validity threshold can be sharpened