Quintessence Dark Energy from non-perturbative Higgs-Yang-Mills mass gap

Anish Ghoshal; Marco Frasca; Massimiliano Rinaldi

REVIEW 5 major objections 4 minor 70 references

Quintessence Dark Energy from non-perturbative Higgs-Yang-Mills mass gap

T0 review · 5 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A Higgs–Yang–Mills field can drive cosmic acceleration

desk verdict The paper contains new explicit expressions for w and Ωde from a Higgs-YM dark sector, but the late-time solution used to get the density drives φ to the wrong vacuum and the central result is an artifact. read the letter →

arxiv 2505.11644 v2 pith:WZC4PDNV submitted 2025-05-16 astro-ph.CO gr-qchep-phhep-th

Marco Frasca , Anish Ghoshal , Massimiliano Rinaldi This is my paper · ORCID

classification astro-ph.COgr-qchep-phhep-th MSC 83F0581T1381T16 PACS 98.80.-k11.15.-q

keywords darkenergyquintessenceYang-Mills-HiggstheorySU(2)gaugefieldJacobiellipticfunctionsnon-perturbativemethodsmassgapcosmictriad

topics Dark Energy

open problems Dark Energy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that dark energy need not be a cosmological constant: the classical SU(2) Yang–Mills–Higgs action, minimally coupled to gravity and taken in its non-perturbative regime, can produce the observed late-time acceleration of the Universe. The claim is that the Higgs field, interacting with a strongly coupled gauge field, never settles exactly at the bottom of its potential; its asymptotic relaxation behaves as a slowly varying effective cosmological constant. The model yields an equation-of-state parameter that tends to $w=-1$ with a small time-dependent correction, and a dark-energy density parameter set by an integration constant (a phase $\theta$) rather than by fine-tuned physical couplings. If correct, this replaces the cosmological constant with a dynamical mechanism built from Standard-Model-like ingredients.

What carries the argument

The argument combines three ingredients. First, the cosmic-triad reduction: imposing isotropy and homogeneity forces the SU(2) gauge field into a single time-dependent component $A^a_i=f(t)\delta^a_i$ and the Higgs doublet into one real scalar $\phi(t)$, so the system reduces to two coupled Klein–Gordon equations plus the Friedmann equations. Second, the mapping theorem: in Lorenz gauge the SU(2) Yang–Mills equations reduce to a single scalar $\phi^4$ equation, so the exact Jacobi elliptic solutions ($\mathrm{dn}$, $\mathrm{sn}$, $\mathrm{cn}$) carry over to the gauge sector and generate the classical mass gap $m_0$. Third, a multiple-time-scale expansion ordered by couplings: the Hubble time is the slowest scale, the Higgs self-coupling an intermediate scale, and the strong gauge coupling the fastest scale; this hierarchy lets the fast gauge oscillations average out while the scalar field drives the slow late-time dynamics.

What would settle it

A lattice computation of the SU(2)-Higgs system in an FLRW background that found the classical mass gap and the $\mathrm{dn}$/$\mathrm{sn}$ solutions are strongly corrected by quantum fluctuations would break the derivation; observationally, high-precision $w(z)$ data from supernovae and baryon acoustic oscillations at $z<2$ showing no damped time variation around $w=-1$ at the predicted amplitude would falsify the dark-energy claim.

Watch

Extended reading notes

Core claim

The central discovery is that the coupled Einstein–Higgs–Yang-Mills system in the cosmic-triad SU(2) ansatz admits closed-form non-perturbative background solutions in terms of Jacobi elliptic functions, and that these solutions account for the present-day dark-energy density without fine-tuning physical constants. In the strong-coupling regime the scalar field is never exactly at its vacuum expectation value: it relaxes asymptotically to $\phi_0$ while the gauge field acquires a classical mass gap and decouples from gravity at large times. The effective dark energy is the residual, slowly decaying potential energy of the scalar, whose equation of state asymptotes to $-1$ and whose present density parameter $\Omega_{\mathrm{de}}$ is fixed by the phase $\theta$ of the solution rather than by the model couplings. This is presented as a dynamical alternative to a cosmological constant that uses only fields that exist in the Standard Model.

Load-bearing premise

The whole argument turns on the mapping theorem that Lorenz-gauge SU(2) Yang-Mills reduces to a single scalar $\phi^4$ equation, and on the assumption that the resulting classical Jacobi-elliptic solutions capture the true non-perturbative regime of the quantum theory.

Editorial extensions

If this is right

Dark energy becomes dynamical: the equation-of-state parameter approaches $-1$ asymptotically with small time-dependent corrections, so the model is in principle distinguishable from a pure cosmological constant.
The gauge-field contribution to the energy density freezes out as the scale factor grows, leaving the scalar sector as the dominant source of current acceleration.
The dark-energy density parameter in the asymptotic limit depends on a single integration phase $\theta$, which the authors argue removes the need to fine-tune physical couplings.
The exact elliptic-function method replaces numerical dynamical-system scans, which are highly sensitive to initial conditions, with closed-form background solutions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A decisive observational test would be tomographic $w(z)$ data: the model predicts small damped oscillations around $w=-1$ at late times, whereas $\Lambda$CDM predicts exactly $-1$; current data are not yet precise enough to see them.
If the mapping theorem survives contact with quantum corrections, the same mechanism could be embedded in the full electroweak sector, making the dark-energy scale a phase-transition remnant rather than an input parameter; the paper does not perform this embedding.
One could extend the multi-scale expansion to include anisotropic or inhomogeneous perturbations and ask whether the cosmic-triad solution is stable, a question the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know. First, the paper does contain a genuinely new piece: explicit closed-form expressions for w_DE and Ω_de (eqs. 31, 45, 48) for an SU(2) Higgs-YM dark sector, worked out with a multiple-scale plus Jacobi elliptic function approach. Second, the central mechanism does not survive contact with the equations it writes. The approximate scalar solution used to derive the late-time density, eq. (24), has φ→0 as τ→∞. But the actual attractor of eq. (13) is φ=±φ0, where V=0. The stress-test note is right: φ=0 is an unstable fixed point of the damped double-well oscillator, and the gauge field also decays (eq. 42), so there is nothing holding the scalar field off the vacuum. The paper's own asymptotic solution, eq. (27), says the same. One cannot use eq. (24) to evaluate Ω_de at the present epoch.

There are also two more local problems, both load-bearing. Eq. (35) redefines f(t)=e^{-3Ht/2}\tilde f, but substituting into (11) gives \ddot{\tilde f}-2H\dot{\tilde f}+(3H²/4)\tilde f, not \ddot{\tilde f}-(9/4)H²\tilde f. The missing derivative term changes the solution. And the passage from (45) to (48) is not a limit: as ε→0, (45) tends to 9 λφ0⁴/(108 M_p²H²), not to (2 dn(θ)²-3)², while eq. (49) (from the φ→φ0 branch) tends to zero. Three different "late-time" Ωde's, only one of which matches data.

Finally, the abstract promises agreement without fine-tuning, but Fig. 3 says the opposite: θ must be "properly tuned," and the normalization λφ0⁴/(108 M_p²H²) ≈ 3×10^53 means θ has to be tuned to about one part in 10^53 to land Ωde near 0.7. That is exactly the fine-tuning the paper claims to avoid.

What good is here? The multiple-scale method and the use of Jacobi functions are applied transparently, the paper is honest that distances and growth rates have not been checked, and the appendices make the technique reproducible. The mapping theorem and mass-gap framework are imported from earlier work and not re-validated here, but the flaw is not the import; it's the internal inconsistency between eqs. (24), (27), (48) and (49).

Who is this for? Someone working with the authors' non-perturbative techniques might find the time-dependent w expression instructive as an exercise. As a dark-energy explanation, it fails.

My recommendation: don't send this to referees in its current form; it needs to resolve the attractor problem and re-derive the late-time density from the field actually relaxing to ±φ0. If the authors can do that and fit Ωde without tuning θ to 10^-53, it becomes a different paper worth another look.

Referee Report

5 major / 4 minor

Summary. The manuscript studies a dark SU(2) Yang–Mills–Higgs sector minimally coupled to a flat FLRW spacetime, using the cosmic-triad ansatz to reduce the gauge field to a single function f(t) and the Higgs doublet to a single real scalar φ(t). The authors devise a multiple-time-scale approximation whose solutions are expressed in terms of Jacobi elliptic functions, and they claim that the scalar field generates an effective cosmological constant with equation-of-state parameter w→−1 and a dark-energy density Ωde that agrees with observations without fine-tuning the physical constants of the model. The central quantitative result is Eq. (48), plotted in Fig. 3, which is said to match the observed Ωde for suitably chosen values of the phase θ. Appendices A and B review the mapping of SU(2) Yang–Mills to a scalar φ⁴ equation and compute a secular correction to the scalar amplitude.

Significance. If the central claim were correct, the paper would offer a dynamical dark-energy mechanism sourced by Standard-Model-like fields, replacing a bare cosmological constant with a non-perturbative mass-gap scale. The analytical machinery is nontrivial: closed-form elliptic-function solutions are exhibited and Figs. 1–2 provide a numerical check for the scalar approximation, which is commendable. However, the central claim is not currently supported. The approximate solution used to compute Ωde is inconsistent with the late-time attractor of the scalar equation, the advertised agreement with data is obtained by tuning an integration constant, the limiting step leading to the main Ωde formula is invalid as written, and the gauge-field equation of motion contains an algebraic error. These are load-bearing problems rather than presentation issues. As it stands, the manuscript does not deliver a viable cosmological model.

major comments (5)

[§III, Eqs. (13), (24)–(27); §V, Eq. (48)] The approximate solution used to compute Ωde decays to φ=0, but φ=0 is not the late-time attractor of Eq. (13). Equation (13) is a damped anharmonic oscillator whose stable fixed points are φ=±φ0 with V(φ0)=0; linearizing about φ=0 gives φ¨+3Hφ˙−λφ0²φ=0, which contains a growing mode. The paper itself states in Eqs. (25)–(27) that φ→φ0 plus exponentially damped oscillations and says that "the points ±φ0 are asymptotically stable due to the Hubble constant." The multiple-scale solution in Eq. (24), used in Eq. (45) and leading to Eq. (48), describes the unstable φ=0 branch rather than the physical late-time field. The gauge field does not rescue the result: Eq. (42) gives f0∼e^{−3τ/4}→0, so the coupling term g²f²φ/(4a²) in Eq. (12) vanishes at late times. If φ relaxes to ±φ0, the potential vanishes and the claimed dark-energy density disappears. This is an internal inconsistency in the central argument.
[§IV, Eq. (35)] The field redefinition f=e^{−3Ht/2}f̃ is applied incorrectly. Starting from Eq. (11), the transformation gives f¨+Hf˙ = e^{−3Ht/2}(f̃¨−2Hf̃˙+(3/4)H²f̃), not f̃¨−(9/4)H²f̃. Equation (35) drops the −2Hf̃˙ term and has the wrong sign and coefficient for the H² term. Consequently Eqs. (36)–(42), including the mass-gap solution and the late-time estimate f0∼e^{−3τ/4}, solve a different equation from the one derived from the action. The gauge-field part of the paper is therefore not established.
[§V, Eqs. (45)–(48), Fig. 3] The limiting step from Eq. (45) to Eq. (48) is not valid. In Eq. (45), as ε(τ)→0, the bracket [3−2ε²dn²(...)]² tends to 9 and the second line is O(ε²), so the expression tends to 9λφ0⁴/(108M_P²H²)=λφ0⁴/(12M_P²H²), which is independent of dn(θ,−1). Equation (48), by contrast, retains the factor [2dn(θ,−1)²−3]², which cannot be obtained from Eq. (45) in the stated limit. The derivation of the central dark-energy formula is therefore missing, and Fig. 3 is not a plot of the limiting form of Eq. (45).
[Abstract; §V, Fig. 3; §VI] The claim of agreement with cosmological data without fine-tuning is not supported by the paper's own analysis. Equation (48) depends on θ, and Fig. 3 shows agreement only "provided θ is properly tuned," as stated in Sec. V; Sec. VI repeats this admission. Since θ is a free integration constant of the approximate solution, choosing θ to match Ωde is parameter fitting, not a prediction from the model. The abstract's contrast with fine-tuning of physical constants does not address this circularity.
[Appendix A, Eqs. (65)–(66)] The reduction of SU(2) Yang–Mills to a single scalar φ⁴ equation in Lorenz gauge is imported from Refs. [27,28] and is neither proved nor tested in this manuscript. Since this mapping underlies the gauge-field solution and the mass-gap interpretation, the non-perturbative content of the model rests on an unexamined domain assumption. This is a correctness risk independent of the internal inconsistency discussed above; even if the mapping is granted, the scalar-sector problem in Sec. III remains.

minor comments (4)

[Introduction] The text contains a typo: "Cosmic Mircowave Background" should read "Cosmic Microwave Background."
[§V, Eq. (45)] The expression for Ωde is split across Eq. (45) and a second numbered display (46) without a clear indication that they form a single equation; this formatting should be corrected for readability.
[§III, after Eq. (27)] The sentence "the points ±ϕ0 are asymptotically stable due to the Hubble constant" directly contradicts the behavior of Eq. (24), where φ→0; regardless of the substantive issue raised above, the presentation should acknowledge and resolve this apparent contradiction.
[§IV, Eq. (36)] The replacement t→gt is not carried out consistently in the notation: after rescaling, the manuscript writes f̃ as a function of t while the scalar field is evaluated at t/g, and the argument of the exponential is not displayed consistently. Clarifying the time variables would improve the derivation.

Circularity Check

2 steps flagged · score 6.0 of 10

The claimed agreement with cosmological Ωde data is obtained by tuning the free phase θ, and the central Yang-Mills-to-scalar reduction is imported from the first author's own mapping theorem.

fitted input called prediction [Section V, Eq. (48) and Fig. 3; also Sec. VI bullet list]
"This result grants the proper limit for Ωde at the present time provided θ is properly tuned. This avoids to tune physical constants that, due to their running, cannot be fine-tuned. … whose value can be set by tuning an integration constant, rather than physical constants."

The central cosmological output, Ωde, is given in Eq. (48) as (λφ0^4/(108 M_P^2 H^2)) [2 dn(θ,−1)^2 − 3]^2, with θ an arbitrary integration constant introduced in the elliptic solution Eq. (20)/(24). The paper explicitly states that 'infinite choices of θ exists granting agreement with data' and that the result holds 'provided θ is properly tuned.' Therefore the 'agreement with cosmological data for the dark energy density' claimed in the Abstract is not a model prediction; it is a one-parameter fit of θ to the target quantity. The fine-tuning is not eliminated, only moved from physical constants to the phase θ.
self citation load bearing [Appendix A, Eqs. (65)–(66)]
"To solve these equations, we use the mapping theorem proven in [27, 28] where the Yang-Mills potential can be written as A a μ (x) = η a μ φ(x), where η a μ are numerical coefficients with both color and Lorentz indexes and φ(x) satisfies the equation in the Lorentz gauge □φ + Ng 2 φ 3 = 0, for SU (2)."

The reduction of SU(2) Yang-Mills to a single φ^4 scalar equation is the load-bearing premise for every non-perturbative solution in the paper, including the mass-gap solution for the gauge field and the dark-energy density. The theorem is cited to Refs. [27,28], both authored by the first author of the present paper, and no proof or independent, machine-checkable verification is included here. Invoking this same-author result as an external mathematical fact makes the central derivation rest on a self-citation chain rather than on evidence established inside the paper or by independent work.

full rationale

The paper's own equations show that the headline 'agreement with cosmological data for the dark energy density' is secured by tuning the free integration constant θ: Fig. 3 and Section V state this explicitly, and Section VI repeats that the effective cosmological constant 'can be set by tuning an integration constant.' That is a fitted input renamed as a no-fine-tuning prediction, so the central cosmological claim is partially circular. The second circular element is the 'mapping theorem' of Appendix A, which is imported from the first author's prior papers [27,28] and is the only justification for replacing SU(2) Yang-Mills by a single φ^4 scalar; because the cited theorem is not independently substantiated in this manuscript, this is a load-bearing self-citation. The multi-scale algebra itself (Eqs. (19)–(24), (41)–(42)) is internally consistent given those premises, and the paper candidly notes in Section VI that 'further thorough and comprehensive analysis is required to verify if this is truly viable,' which is an acknowledged limitation rather than a circular step. There is also a potential internal inconsistency flagged in the reviewer context: the solution used for Ωde decays to φ=0, whereas the paper's own asymptotic analysis, Eqs. (25)–(27), asserts φ→φ0 with V(φ0)=0, which would make the late-time dark-energy density vanish; that is a correctness risk, not a circularity, but it further weakens the claim of a genuine prediction. Overall score 6: a central reported 'prediction' reduces by construction to a fitted parameter, while the self-cited mapping theorem adds partial circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The derivation rests on a chain of nontrivial assumptions from prior work, chiefly the mapping theorem and the treatment of classical solutions as the non-perturbative regime. The only free parameter tuned to data is the phase theta, which is fine-tuned to extreme precision. No machine-checked proofs or reproducible code are provided.

free parameters (2)

theta (phase/integration constant) = not specified; tuned to match Omega_de
The phase theta in the Jacobi elliptic solution is the only parameter adjusted to reproduce the observed dark-energy density. The prefactor in eq. (48) is ~3x10^53, so theta must be tuned to about one part in 10^27 near a zero of [2 dn^2 - 3]^2, which is an extreme fine-tuning of an integration constant.
mu (gauge-field integration constant) = not specified
The integration constant mu appears in the gauge-field solution (eqs. 39-42) but does not enter the reported Omega_de. Its value is not determined by the model.

assumptions (5)

domain assumption In Lorenz gauge, SU(2) Yang-Mills field equations reduce to a single scalar phi^4 equation via a mapping theorem (Appendix A, eqs. 65-66).
Imported from Refs. [27,28] by the first author; not proved in this paper and central to the gauge-field mass-gap solution.
domain assumption The exact classical solutions in terms of Jacobi elliptic functions describe the non-perturbative strongly coupled regime of the quantum field theory.
The paper uses classical solutions to derive the DE density and equation of state without addressing quantum corrections; this is the core premise of the Frasca approach, cited to Refs. [21,23-26].
domain assumption The time-scale hierarchy: H is the slowest, the scalar self-coupling sqrt(lambda) is intermediate, and the gauge coupling g is the fastest time scale.
Stated in Sec. IV; the multiple-scale expansion and the dropping of Hubble friction terms rely on this hierarchy.
domain assumption H is approximately constant on the fast oscillation time scales, so the multi-scale redefinition phi = e^{-3Ht/2} chi is valid.
Used throughout Secs. III and IV to derive eqs. (16) and (35).
domain assumption The scale factor for the scalar-field-dominated universe is given by eq. (47) from [60,61], and taking Omega_0 to zero yields the present-time Omega_de.
The limit is asserted without derivation and is inconsistent with eq. (45) at large tau, where Omega_de tends to 9 times the prefactor rather than the theta-dependent bracket of eq. (48).

invented entities (1)

Dark SU(2) Yang-Mills-Higgs sector
purpose: Serves as the source of dark energy via a non-perturbative mass gap and an effective quintessence field.
The paper introduces a new hidden sector not present in the Standard Model; no couplings to Standard Model fields are specified besides gravity, and no experimental signature or falsifiable prediction is provided.

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Pith. "Pith review of Quintessence Dark Energy from non-perturbative Higgs-Yang-Mills mass gap." pith.science (2026). https://pith.science/paper/WZC4PDNV

@misc{pith2026250511644,
  author       = {Pith},
  title        = {Pith review of: Quintessence Dark Energy from non-perturbative Higgs-Yang-Mills mass gap},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZC4PDNV}},
  note         = {Machine review of arXiv:2505.11644}
}

abstract

We discuss the equations that arise from a Higgs--Yang-Mills dark sector coupled to gravity on a flat Friedmann-Lemaitre-Robinson-Walker metric. We choose the simplest $SU(2)$ representation, which we show to be compatible with the Cosmological Principle. We devise a multiple time scale approach to solve the equations of motion through a hierarchy of the couplings, utilizing exact solutions in terms of Jacobi elliptic functions. This novel method implements the dynamical system approach used in the literature and can shed new light on the possibility that this model can describe dark energy.

Figures

Figures reproduced from arXiv: 2505.11644 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

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Works this paper leans on

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= 0. (12) In [56] it was suggested that these equations might give an alternative explanation to the dark energy problem since the potential does not vanish, although it can be arbitrarily small. This is evident from eq. (12): here, the third term prevents ϕ0 to be an exact solution of the equations of motion. It acts as a friction term that does not allo...

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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= 0. (12) In [56] it was suggested that these equations might give an alternative explanation to the dark energy problem since the potential does not vanish, although it can be arbitrarily small. This is evident from eq. (12): here, the third term prevents ϕ0 to be an exact solution of the equations of motion. It acts as a friction term that does not allo...

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Now, we assume that the dynamics is ruled by two different time scales: the scalar field time-scale tϕ = √ λϕ0t and the Hubble time-scale τ = 3Ht

= 0, (13) and H2 = 1 3M 2 P " ˙ϕ2 2 +V (ϕ) +ρm +ρr # , (14) ˙H = − 1 2M 2 P ˙ϕ2 +ρm + 4ρr 3 . Now, we assume that the dynamics is ruled by two different time scales: the scalar field time-scale tϕ = √ λϕ0t and the Hubble time-scale τ = 3Ht. We also define the new field ϕ(t) =e− 3 2Htχ(t). (15) As a result, the Klein-Gordon equation (13) can be written, ap...

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[3] [3]

Now, we can apply a multiple scale technique by assuming that χ = χ(tϕ,τ )

= 0, (16) where we assumed that the Hubble parameter changes very slowly compared to χ. Now, we can apply a multiple scale technique by assuming that χ = χ(tϕ,τ ). This entails a redefinition of the derivative as d dt = √ λϕ0 ∂ ∂tϕ + 3H ∂ ∂τ . (17) We make the perturbative expansion χ =χ0 + 1√ λ χ1 +O(λ−1), (18) to obtain the set of equations ϕ2 0 ∂2 ∂t2 ...

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(19) 7 This is also known as the averaging method [58]

= 0, ϕ2 0 ∂2 ∂t2 ϕ χ1 + (−ϕ2 0 + 3e−τχ2 0)χ1 + 6Hϕ0 ∂2 ∂tϕ∂τχ0 = 0, .... (19) 7 This is also known as the averaging method [58]. The leading order solution is given by χ0 =e τ 2 r 2 3ϕ0 dn 1√ 3tϕ +θ,−1 , (20) where dn is a Jacobi elliptic function (see section 22 of Ref. [59] for further details) 1 and θ an arbitrary integration constant. This function ne...

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3− 2ϵ2(t)dn2 θ + tϕ0 √ λ√ 3 ,−1 !#2 + 4λϕ4 0 108H2M 2ϵ2(t)× (45)

= 0, (34) together with the Friedmann equations (9). In the preceding section, we observed that the Hubble constant and the Higgs field run on different time scales, the latter much faster than the former. Now, due to the fact that the strong coupling is very large, towards the confining regime, this should represent the fastest time scale. Similarly as b...

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[6] [6]

When λϕ2 0≫H2, this reduces to Ωde =e−3Ht 20− 8 √ 6 3 H2 L, (50) where we have neglected the exponential e−6Ht with respect to the leading term

+ (49− 20 √ 6)e−3Ht)λϕ2 0 i , (49) after averaging on the fast time scale determined by the period T = 2π/ p 2λϕ2 0− 9H2/4. When λϕ2 0≫H2, this reduces to Ωde =e−3Ht 20− 8 √ 6 3 H2 L, (50) where we have neglected the exponential e−6Ht with respect to the leading term. 15 VI. DISCUSSION AND CONCLUSION To recall the main results, we derive closed analytical...

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(58) We see we have got massive non-linear waves extending to all the space-time. The 2-point function in the momenta space becomes [37]: G(1) c (k) = ∞X n=0 Bn k2−m2 n +iϵ (59) with Bn = (2n + 1)2 π2 2K2(i) e−(n+ 1 2)π 1 +e−(2n+1)π (60) and mn = (2n + 1) π 2K(i) λ 2 1 4 µ. (61) From this we can see that, in the limit µ→ 0 the free theory is recovered at ...

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Planck 2018 results. VI. Cosmological parameters,

N. Aghanim et al. [Planck Collaboration], “Planck 2018 results. VI. Cosmological parameters,”

work page 2018