REVIEW
On the non-Gaussian correlation of the primordial curvature perturbation with vector fields
Cosmology and Nongalactic Astrophysics · Submitted 12 Oct 2012 · 68 references
Pith's verdict: A careful in-in computation of the full ζ–B–B shape, with a genuine technical correction to the prior literature; the soft spots are minor and the dropped total derivative is standard practice, not a load-bearing flaw.
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Referee report
The paper
The paper claims
The paper computes how the primordial curvature perturbation cross-correlates with two powers of a vector field when the vector's conformal invariance is broken by a direct coupling to the inflaton, finding the signal peaks in flattened-tri
The reading
What carries the argument
The machinery is the in-in (Schwinger-Keldysh) formalism applied to a third-order interaction Hamiltonian between the curvature perturbation and the gauge field. The load-bearing step is the reduction of that Hamiltonian: starting from $S^{}_{BB}$ and using the conservation equation $S^{}_{BB}$, the authors rewrite the interaction so that the physical part is proportional only to the derivative of the coupling, $S^{}_{BB}$, while the term proportional to $S^{}_{BB}$ itself becomes a total time derivative and is dropped. The remaining integrals $S^{}_{BB}$ and $S^{}_{BB}$ are evaluated with Hankel-function mode functions; the logarithm $S^{}_{BB}$ appears only in $S^{}_{BB}$, cannot cancel, and dominates in the flattened limit, which is what produces the large $b^{}_{NL}$. The squeezed limit is obtained from the small-argument behaviour of the Hankel functions, where the integral reduces to $S^{}_{BB}$ for integer $n$.
Extended reading notes
Core claim
The central claim is that the cross-correlation of the curvature perturbation with two powers of a $U(1)$ vector field, in a model where the coupling $S^{}_{BB}$ is broken by a coupling $S^{}_{BB}$ to the inflaton, has a computable shape whose maximum sits at the flattened configuration $k_1 = 2k_2 = 2k_3$. There the integral that carries the correlation is dominated by a term $S^{}_{BB}$, giving $S^{}_{BB}$ and, for the case $n=2$ with $S^{}_{BB}$, a magnetic non-linearity parameter $|b^{}_{NL}|$ of order $5760$ for $S^{}_{BB}$. The authors stress that this is far above the order-unity level one might have guessed. The second claim is a check: taking $k_1$, the same calculation reduces to $S^{}_{BB}$ with $S^{}_{BB}$, matching the magnetic consistency relation of the authors' earlier work, and for a scale-invariant magnetic spectrum this gives $b^{}_{NL} = -4$. The paper also reports a disagreement with an earlier treatment of the same correlation, attributing it to a polarization sum and to an interaction Hamiltonian whose leading term is a total derivative.
Load-bearing premise
The argument requires that the total time derivative $S^{}_{BB}$ dropped from the interaction Hamiltonian can be removed by a field redefinition that only affects higher orders in perturbation theory; no proof of that order counting is given, and if the redefinition feeds back at the same order the shape function and the $b^{}_{NL}$ values change.
Editorial extensions
If this is right
- If the vector field is the electromagnetic field, a magnetic non-linearity parameter of order $10^3$ in flattened configurations is the specific non-Gaussian signature a search for inflation-generated magnetic fields should target.
- The squeezed-limit agreement means the earlier magnetic consistency relation $b^{}_{NL}^{local} = n_B - 4$ survives a full dynamical calculation, so it can be used as a consistency check on any future model of this type.
- Because the result follows from the same coupling that generates the magnetic field, a measurement of the cross-correlation and a measurement of the magnetic power spectrum are not independent: their ratio is fixed by the coupling's time dependence.
- The calculation generalizes directly to the case where the curvature perturbation is replaced by an isocurvature field or a curvaton, so the same shape function applies to mixed inflaton-curvaton scenarios without a strong-coupling problem.
Reading between the lines
- The logarithmic enhancement is a super-horizon effect, so its size depends on how long inflation lasted after the relevant modes exited the horizon; a short window of inflation would shrink the $10^3$ number toward order unity and make the flattened shape much harder to distinguish from ordinary local non-Gaussianity.
- Since the signal is a cross-correlation rather than an auto-correlation, it is odd under interchange of the two magnetic legs in a way the curvature auto-bispectrum is not, which suggests looking for it in oriented statistics such as Faraday-rotation maps cross-correlated with large-scale structure rather than in temperature bispectra alone.
- The authors' disagreement with the earlier treatment traces back to whether the leading interaction term is a total derivative; that is a checkable point, and resolving it would tell us whether the polarization-sum discrepancy is a bookkeeping issue or a genuine difference in the predicted shape.
Original abstract
We compute the three-point cross-correlation function of the primordial curvature perturbation generated during inflation with two powers of a vector field in a model where conformal invariance is broken by a direct coupling of the vector field with the inflaton. If the vector field is identified with the electromagnetic field, this correlation would be a non-Gaussian signature of primordial magnetic fields generated during inflation. We find that the signal is maximized for the flattened configuration where the wave number of the curvature perturbation is twice that of the vector field and in this limit, the magnetic non-linear parameter becomes as large as |b_{NL}| ~ 10^3. In the squeezed limit where the wave number of the curvature perturbation vanishes, our results agree with the magnetic consistency relation derived in arXiv:1207.4187.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We compute the three-point cross-correlation function of the primordial curvature perturbation generated during inflation with two powers of a vector field in a model where conformal invariance is broken by a direct coupling of the vector field with the inflaton. If the vector field is identified with the electromagnetic field, this correlation would be a non-Gaussian signature of primordial magnetic fields generated during inflation. We find that the signal is maximized for the flattened configuration where the wave number of the curvature perturbation is twice that of the vector field and in this limit, the magnetic non-linear parameter becomes as large as |b_{NL}| ~ 10^3. In the squeezed limit where the wave number of the curvature perturbation vanishes, our results agree with the magnetic consistency relation derived in arXiv:1207.4187.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We compute the three-point cross-correlation function of the primordial curvature perturbation generated during inflation with two powers of a vector field in a model where conformal invariance is broken by a direct coupling of the vector field with the inflaton. | contribution 0.98 | 0.95 | 0.09 | asserted 0.99 |
| s1 | If the vector field is identified with the electromagnetic field, this correlation would be a non-Gaussian signature of primordial magnetic fields generated during inflation. | interpretation 0.99 | 0.11 | 0.08 | hedged possibility 0.93 |
| s2 | We find that the signal is maximized for the flattened configuration where the wave number of the curvature perturbation is twice that of the vector field and in this limit, the magnetic non-linear parameter becomes as large as |b_{NL}| ~ 10^3. | result 1.00 | 0.96 | 0.64 | asserted 1.00 |
| s3 | In the squeezed limit where the wave number of the curvature perturbation vanishes, our results agree with the magnetic consistency relation derived in arXiv:1207.4187. | result 0.97 | 0.76 | 0.08 | asserted 0.99 |
What was examined, and what was not
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References
How the literature uses it
35 arXiv papers cite this in 95 sentences
Papers citing it · 57 sentences without a confident label
Contests it 1 paper
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We recently found that there is actually a typo in [ref] which is also there in the results of [ref] .].
Inflationary cross-correlations of a non-minimal spectator and their soft limits · Cross-correlation with curvature perturbation and the consistency relation · contest 0.93
Builds on it 2 papers
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Following the approach of our previous papers [ref] for the case of a cross-correlation of the comoving curvature perturbation with primordial gauge fields, we can similarly define convenient and useful non-linearity parameters to characterise the cross-correlations of the inflationary tensor perturbation with primordial gauge fields.
On the primordial correlation of gravitons with gauge fields · The magnetic and electric non-linearity parameters · 0.90
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While such three-point functions have already been studied for the case of non-helical magnetic fields [ref] , we believe it is of utmost interest to study the non-Gaussianities produced due to the helical fields as well.
Enhancing the cross-correlations between magnetic fields and scalar perturbations through parity violation · Introduction · 0.81
Uses its method or tool 3 papers
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Unlike the equisided configuration, the four-point correlation in the counter collinear limit can be easily derived for a generic coupling by repeated use of the equation of motion of the gauge fields, similar to the squeezed limit evaluation of the cross-correlation bispectra [ref] .
Inflationary trispectrum of gauge fields from scalar and tensor exchanges · Counter collinear limit · 0.80
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Using leading order contribution of the interaction Hamiltonian, the cross-correlation function at the end of inflation, i.e., $η_end$ can simply be written as [ref] [equation] Now, for simplicity, we define a new quantity $G_ℛBB(_̨1, _̨2, _̨3)$ as [equation] Then, by using the expression [ref] along with the interaction Hamiltonian [ref], the definition of the mode functions [ref] and [ref], and the Wick's theorem, $G_ℛBB(_̨1, _̨2, _̨3)$ can be expressed as [equation] where six $𝒢$'s are defined as [equation]
Inflationary magnetogenesis: solving the strong coupling and its non-Gaussian signatures · Cross-correlations between magnetic fields and curvature perturbation · 0.91
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Such integrals can be evaluated using the same method as previously outlined in [ref] .
Inflationary cross-correlations of a non-minimal spectator and their soft limits · Evaluation of integrals in the squeezed limit · 1.00
Uses its data or values 1 paper
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It was calculated in [ref] that $b_NL∼5760$ in this shape, and the dominating contribution to the cross-correlation function comes from A ≈p̂·r̂ k_1^3/(p r)^3/2 3(n_B-4)log(-(k_1+p+r) τ_I) , D ≈-k_1^3/(p r)^3/2 3(n_B-4)log(-(k_1+p+r) τ_I) , J=0, as discussed in appendix [ref].
Constraints on Gauge Field Production during Inflation · Folded shape · 0.93
Compares with it 5 papers
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While [ref] apparently boils down to the same two-point factorizable form as obtained for the BD case in [ref] , this reduction is actually quite non-trivial as no assumption has been made so far as to the details of the initial vacua, which have instead been kept as generic as possible.
Primordial magnetic non-Gaussianity with generic vacua and detection prospects in CMB spectral distortions · Squeezed limit · 0.95
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This result bears a striking resemblance with the squeezed limit correlator of the curvature perturbation with gauge fields as found in [ref] , which is not a mere coincidence[One may notice a sign difference in the non-linearity parameter in eq.
Inflationary cross-correlations of a non-minimal spectator and their soft limits · Cross-correlation with curvature perturbation and the consistency relation · 0.99
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This result is related with the consistency relation obtained in [ref] for the cross correlator in the $f(ϕ)F^2$ model.
de Sitter symmetries and inflationary correlators in parity violating scalar-vector models · Conclusions and final remarks · 0.93
Cites it as background 17 papers
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Such cross correlations was recently studied in [ref] , and in [ref] it was suggested that such cross correlations could be parametrized in a model independent way in terms of a magnetic non-linearity parameter, $b_NL$ of the form $< ζ B·B> ∝b_NL P_ζ P_B$ analogous to the definition of $f_NL$, where here $P_ζ$ and $P_B$ are the power spectra of the curvature perturbation and the magnetic field respectively.
Constraints on Gauge Field Production during Inflation · Introduction · 0.88
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Therefore, in the squeezed limit, the three-point function can be completely expressed in terms of the two-point function through a relation referred to as the consistency condition (see Refs. [ref] in the context of three-point functions involving scalar and tensor perturbations, and Refs. [ref] for cross-correlations between the scalar perturbation and the magnetic fields).
Cross-correlations between scalar perturbations and magnetic fields in bouncing universes · Introduction · 0.93
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This has been extensively studied in the literature [ref] .
Inflationary magnetogenesis: solving the strong coupling and its non-Gaussian signatures · Introduction · 0.93
What it cites · 68 sentences without a confident label
Contests 2 papers
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In eq.(46) of [ref] the interaction Hamiltonian is proportional to $λ$, while in our eq.([ref]), we showed that the physical part of the interaction Hamiltonian is proportional only to the derivative of the coupling, $λ̇$, and the term proportional to $λ$ is a total derivative, such that the cross-correlation vanishes to leading order in the limit of a constant coupling.
Non-Gaussian features of primordial magnetic fields in power-law inflation · The squeezed limit · contest 0.81
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Note that the polarization factor multiplying the integrals is slightly different than in equation (41) of [ref] , where it appears that part of the polarization tensor product was missed.
Correlation of inflation-produced magnetic fields with scalar fluctuations · The correlation of curvature perturbations with the magnetic fields · contest 0.85
Uses the method or tool of 2 papers
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From the interaction Hamiltonian in ([ref]) and using the rules of [ref] , we obtain the cross-correlation of the curvature perturbation with the electromagnetic field <ζ(τ_I,k_1)A_i(τ_I,k_2)A_j(τ_I,k_3)> = - (2π)^3δ^(3)(k_1+k_2+k_3) × [(δ_il-k_2,ik_2,l/k_2^2)(δ_lj-k_3,l k_3,j/k_3^2) (ℐ_1+k_2·k_3 ℐ_2). -.(δ_il-k_2,i k_2,l/k_2^2)k_3,l(δ_jm-k_3,j k_3,m/k_3^2)k_2,m ℐ_2] with the integrals ℐ_1= 2 Im [ζ_k_1(τ_I)A_k_2(τ_I)A_k_3(τ_I)∫dττλ'(τ) ζ_k_1^*(τ)_τ(A_k_2^*(τ))_τ(A_k_3^*(τ))] and ℐ_2=2 Im [ζ_k_1(τ_I)A_k_2(τ_I)A_k_3(τ_I)∫dττλ'(τ) ζ_k_1^*(τ)A_k_2^*(τ)A_k_3^*(τ)] where the mode function $ζ_k(τ)$ is given in ([ref]) and the mode function $A_k(τ)$ is obtained from ([ref]) which is given by A_k(τ) = 1/√(λ_I)√(π)/2e^iπ(1+n)/2√(-τ)(τ/τ_I)^nH^(1)_1/2+n(-kτ) .
Cosmological diagrammatic rules · The correlation of curvature perturbations with the magnetic fields · 1.00
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In order to compute the higher-order correlation function during inflation, we adopt a very useful and powerful tool of the in-in formalism [ref] .
Non-Gaussian features of primordial fluctuations in single field inflationary models · The correlation of curvature perturbations with the magnetic fields · 0.99
Uses data or values from 3 papers
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The strength of magnetic fields in galaxies and galaxy clusters has been determined to be of the order of a micro Gauss [ref] , while a lower bound of $0.1$ femto Gauss on magnetic fields in the intergalactic medium has also been claimed [ref] .
Fermi/LAT observations of 1ES 0229+200: implications for extragalactic magnetic fields and background light · Introduction · 0.85
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The strength of magnetic fields in galaxies and galaxy clusters has been determined to be of the order of a micro Gauss [ref] , while a lower bound of $0.1$ femto Gauss on magnetic fields in the intergalactic medium has also been claimed [ref] .
EGMF Constraints from Simultaneous GeV-TeV Observations of Blazars · Introduction · 0.85
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The strength of magnetic fields in galaxies and galaxy clusters has been determined to be of the order of a micro Gauss [ref] , while a lower bound of $0.1$ femto Gauss on magnetic fields in the intergalactic medium has also been claimed [ref] .
Evidence for strong extragalactic magnetic fields from Fermi observations of TeV blazars · Introduction · 0.85
Compares with 11 papers
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This agrees with the squeezed limit result in equation (64) of [ref] , when using $λ̇ζ= -_ϕλH δϕ$ and inserting the specific form of the coupling $λ(ϕ) = exp(2ϕ/M)$ used there.
Correlation of inflation-produced magnetic fields with scalar fluctuations · The squeezed limit · 0.98
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Upon comparing ([ref]) with <ζ(τ_I,k_1)B(τ_I,k_2)·B(τ_I,k_3)> =b_NL^local(2π)^3δ^(3)(k_1+k_2+k_3)P_ζ(k_1) P_B(k_2) in the squeezed limit, we obtain the local type magnetic non-linearity parameter b_NL^local= -1/Hλ̇_I/λ_I , in agreement with the magnetic consistency relation [ref] .
Consistency relation for cosmic magnetic fields · The squeezed limit · 0.84
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Inserting directly the energy-momentum tensor of the gauge field T_μν = λF_μρF_ν^ρ-1/4g_μνλF_ρκF^ρκ , one finds an expression that agrees with equation (46) of [ref] and is of order $λζA A$.
Non-Gaussian features of primordial magnetic fields in power-law inflation · The interaction Hamiltonian · 0.93
Cites as background 5 papers
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In the case where the vector field is identified with the electromagnetic field, the strength of the cross-correlation is parametrized by the magnetic non-linearity parameter $b_NL$ introduced in [ref] .
Consistency relation for cosmic magnetic fields · Discussion and conclusions · 0.84
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If the vector field is identified with the electromagnetic field this is a well studied model of primordial magnetogenesis [ref] .
Origin of Galactic and Extragalactic Magnetic Fields · Introduction · 0.91
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While writing up this paper, a related paper appeared [ref] with a discussion of the effect on the CMB of the cross-correlation bispectra between the curvature perturbation and the vector modes.
CMB power spectra induced by primordial cross-bispectra between metric perturbations and vector fields · Note added · 0.97
Discussion
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