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Acoustic signatures in the Cosmic Microwave Background bispectrum from primordial magnetic fields
Cosmology and Nongalactic Astrophysics · Submitted 15 Jun 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find that, for the string inspired primordial magnetic fields characterized by index $n_B=-2.9$ and mean-field amplitude $B_{\lam}=9{\rm nG}$, the angular bispectrum is dominated by two primordial magnetic shapes.
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Using the full radiation transfer function, we numerically calculate the CMB angular bispectrum seeded by the compensated magnetic scalar density mode. We find that, for the string inspired primordial magnetic fields characterized by index $n_B=-2.9$ and mean-field amplitude $B_{\lam}=9{\rm nG}$, the angular bispectrum is dominated by two primordial magnetic shapes. The first magnetic shape looks similar to the one from local-type primordial curvature perturbations, so both the amplitude and profile of the Komatsu-Spergel estimator (reduced bispectrum) seeded by this shape are almost the same as those of the primary CMB anisotropies. However, for different parameter sets ($l_1,l_2$), this "local-type" reduced bispectrum oscillates around different asymptotic values in the high-$l_3$ regime because of the effect of the Lorentz force, which is exerted by the primordial magnetic fields on the charged baryons. This feature is different from the standard case where all modes approach to zero asymptotically in the high-$l$ limit. On the other hand, the second magnetic shape appears only in the primordial magnetic field model. The amplitude of the Komatsu-Spergel estimator sourced by the second shape diverges in the low-$l$ regime because of the negative slope of shape. In the high-$l$ regime, this amplitude is approximately equal to that of the first estimator, but with a reversal phase.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Using the full radiation transfer function, we numerically calculate the CMB angular bispectrum seeded by the compensated magnetic scalar density mode. | method 0.93 | 0.96 | 0.05 | asserted 0.99 |
| s1 | We find that, for the string inspired primordial magnetic fields characterized by index $n_B=-2.9$ and mean-field amplitude $B_{\lam}=9{\rm nG}$, the angular bispectrum is dominated by two primordial magnetic shapes. | result 1.00 | 0.96 | 0.43 | asserted 1.00 |
| s2 | The first magnetic shape looks similar to the one from local-type primordial curvature perturbations, so both the amplitude and profile of the Komatsu-Spergel estimator (reduced bispectrum) seeded by this shape are almost the same as those of the primary CMB anisotropies. | result 0.97 | 0.87 | 0.19 | asserted 1.00 |
| s3 | However, for different parameter sets ($l_1,l_2$), this "local-type" reduced bispectrum oscillates around different asymptotic values in the high-$l_3$ regime because of the effect of the Lorentz force, which is exerted by the primordial magnetic fields on the charged baryons. | result 0.84 | 0.83 | 0.13 | asserted 1.00 |
| s4 | This feature is different from the standard case where all modes approach to zero asymptotically in the high-$l$ limit. | result 0.55 | 0.51 | 0.09 | asserted 1.00 |
| s5 | On the other hand, the second magnetic shape appears only in the primordial magnetic field model. | result 1.00 | 0.79 | 0.12 | asserted 1.00 |
| s6 | The amplitude of the Komatsu-Spergel estimator sourced by the second shape diverges in the low-$l$ regime because of the negative slope of shape. | result 1.00 | 0.89 | 0.09 | asserted 1.00 |
| s7 | In the high-$l$ regime, this amplitude is approximately equal to that of the first estimator, but with a reversal phase. | result 1.00 | 0.91 | 0.11 | asserted 0.98 |
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Original abstract
Using the full radiation transfer function, we numerically calculate the CMB angular bispectrum seeded by the compensated magnetic scalar density mode. We find that, for the string inspired primordial magnetic fields characterized by index $n_B=-2.9$ and mean-field amplitude $B_{\lam}=9{\rm nG}$, the angular bispectrum is dominated by two primordial magnetic shapes. The first magnetic shape looks similar to the one from local-type primordial curvature perturbations, so both the amplitude and profile of the Komatsu-Spergel estimator (reduced bispectrum) seeded by this shape are almost the same as those of the primary CMB anisotropies. However, for different parameter sets ($l_1,l_2$), this "local-type" reduced bispectrum oscillates around different asymptotic values in the high-$l_3$ regime because of the effect of the Lorentz force, which is exerted by the primordial magnetic fields on the charged baryons. This feature is different from the standard case where all modes approach to zero asymptotically in the high-$l$ limit. On the other hand, the second magnetic shape appears only in the primordial magnetic field model. The amplitude of the Komatsu-Spergel estimator sourced by the second shape diverges in the low-$l$ regime because of the negative slope of shape. In the high-$l$ regime, this amplitude is approximately equal to that of the first estimator, but with a reversal phase.
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References
How the literature uses it
15 arXiv papers cite this in 17 sentences
Papers citing it · 11 sentences without a confident label
Cites it as background 6 papers
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[ref] for those on higher-order correlation functions.
Signatures of anisotropic sources in the squeezed-limit bispectrum of the cosmic microwave background · Helical and non-helical magnetic fields · 0.96
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[ref] ), there are a lot of literature which have been analyzing the impacts of the PMF on the CMB bispectrum and seeking its reasonable constraints [ref] .
Optimal limits on primordial magnetic fields from CMB temperature bispectrum of passive modes · Introduction · 0.92
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The effects of a stochastic primordial magnetic field on the CMB are many: spectral distortions of the monopole [ref] , the generation of scalar, vector and tensor perturbations in the metric affecting both the temperature and the polarization spectra, see e.g. [ref] , the production of non-Gaussian signatures leading to a non-zero temperature bispectrum and trispectrum, see e.g. [ref] , Faraday rotation of CMB polarization, see e.g. [ref] .
Magnetic fields from inflation: The CMB temperature anisotropies · Introduction · 1.00
What it cites · 36 sentences without a confident label
Uses the method or tool of 9 papers
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By virtue of the above initial conditions, we formally integrate the set of evolution equations over the line of sight [ref] I_l = 4∫^t_Rdt e^-τ≤{≤[ka_k+316n_e_T(_2^)^-1 I_2]≤[13j_l(x)+^2x^2j_l(x)]. ≤.-≤(kaA_k-n_e_Tv_k)xj_l(x)-≤[13≤(kaZ_k-ΘA_k) -14n_e_TI_l] j_l(x)} ,where $τ=∫n_e_Tt$ is the optical depth, $x=kχ$ with $χ$ is the comoving radial distance along the line of sight and $j_l(x)$ are the spherical Bessel functions.
A Line of Sight Approach to Cosmic Microwave Background Anisotropies · CMB power spectrum · 1.00
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By virtue of the above initial conditions, we formally integrate the set of evolution equations over the line of sight [ref] I_l = 4∫^t_Rdt e^-τ≤{≤[ka_k+316n_e_T(_2^)^-1 I_2]≤[13j_l(x)+^2x^2j_l(x)]. ≤.-≤(kaA_k-n_e_Tv_k)xj_l(x)-≤[13≤(kaZ_k-ΘA_k) -14n_e_TI_l] j_l(x)} ,where $τ=∫n_e_Tt$ is the optical depth, $x=kχ$ with $χ$ is the comoving radial distance along the line of sight and $j_l(x)$ are the spherical Bessel functions.
CMB Anisotropies: Total Angular Momentum Method · CMB power spectrum · 1.00
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By virtue of the above initial conditions, we formally integrate the set of evolution equations over the line of sight [ref] I_l = 4∫^t_Rdt e^-τ≤{≤[ka_k+316n_e_T(_2^)^-1 I_2]≤[13j_l(x)+^2x^2j_l(x)]. ≤.-≤(kaA_k-n_e_Tv_k)xj_l(x)-≤[13≤(kaZ_k-ΘA_k) -14n_e_TI_l] j_l(x)} ,where $τ=∫n_e_Tt$ is the optical depth, $x=kχ$ with $χ$ is the comoving radial distance along the line of sight and $j_l(x)$ are the spherical Bessel functions.
Integral Solution for the Microwave Background Anisotropies in Non-fl at Universes · CMB power spectrum · 1.00
Uses data or values from 2 papers
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Then, we can express the reduced bispectrum ([ref]) in the following form b_l_1l_2l_3 = b^(1)_l_1l_2l_3+b^(2)_l_1l_2l_3+b^(3)_l_1l_2l_3 , b^(1)_l_1l_2l_3 = ∫_0^∞x^2dx N_1 {b^()_l_1(x)b^()̱_l_2(x)b^()_l_3(x) +(l_1,l_2,l_3) perm.} , N_1 = 3(8π)^3A^3n_B48π^2(n_B+3)(2n_B+3)^̊()^3 , b^(2)_l_1l_2l_3 = ∫_0^∞x^2dx N_2 {b^()_l_1(x)b^()̣_l_2(x)b^()_l_3(x) +(l_1,l_2,l_3) perm.} , N_2 = 3(8π)^3A^3n_B48π^2(3n_B+3)(2n_B+3)^̊()^3 , b^(3)_l_1l_2l_3 = ∫_0^∞x^2dx N_3 {b^()_l_1(x)b^()_l_2(x)b^()_l_3(x) +(l_1,l_2,l_3) perm.} , N_3 = 3(8π)^3A^3k_D^3n_B+348π^2(3n_B+3)^̊()^3 .The seven-year WMAP data [ref] give $P_∼2.441×10^-9$.
Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation · Analytic formulas · 1.00
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The current limitations on the primordial bispectra from WMAP-$7$yr data are $-10<f_NL^local<74$ and $-214<f_NL^equil<266$ at $95%$ CL [ref] , where $f_NL^local$ and $f_NL^equil$ are the non-linear parameters of the “squeezed" and “equilateral" momentum configurations, respectively.
Hunting for Primordial Non-Gaussianity in the Cosmic Microwave Background · Introduction · 0.98
Compares with 5 papers
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From Figure ([ref]) and ([ref]), we can see that in the Sachs-Wolfe (SW) regime ($l≤10$) our result presents a SW plateau, which is consistent with that in [ref] .
The cosmic microwave background temperature bispectrum from scalar perturbations induced by primordial magnetic fields · Numerical results · 0.98
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For the reduced bispectrum $b^(1)_l_1l_2l_3$ ([ref]), which is seeded by the “local-type” shape $f^(1)(k,q,p)$ ([ref]), both the profile and amplitude look similar to those of the primary CMB anisotropies [ref] , (see Figure [ref], [ref], [ref] and [ref]).
Acoustic Signatures in the Primary Microwave Background Bispectrum · Conclusion · 0.97
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For the reduced bispectrum $b^(1)_l_1l_2l_3$ ([ref]), which is seeded by the “local-type” shape $f^(1)(k,q,p)$ ([ref]), both the profile and amplitude look similar to those of the primary CMB anisotropies [ref] , (see Figure [ref], [ref], [ref] and [ref]).
The Pursuit of Non-Gaussian Fluctuations in the Cosmic Microwave Background · Conclusion · 0.97
Cites as support 2 papers
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However, as demonstrated in [ref] , one cannot neglect the matter contributions in the case of existence of PMFs due to the compensation mechanism between the radiation density perturbations and those of PMFs.
Evolution of Cosmological Perturbations in the Presence of Primordial Magnetic Fields · Compensated magnetic initial conditions · 0.98
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However, as demonstrated in [ref] , one cannot neglect the matter contributions in the case of existence of PMFs due to the compensation mechanism between the radiation density perturbations and those of PMFs.
Massive Neutrinos and Magnetic Fields in the Early Universe · Compensated magnetic initial conditions · 0.98
Cites as background 54 papers
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The contributions to the CMB angular power spectrum from the scalar perturbations induced by PMFs are investigated in [ref] , from the vector perturbations in [ref] and from tensor mode in [ref] , respectively.
Microwave Background Signatures of a Primordial Stochastic Magnetic Field · Introduction · 0.94
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The contributions to the CMB angular power spectrum from the scalar perturbations induced by PMFs are investigated in [ref] , from the vector perturbations in [ref] and from tensor mode in [ref] , respectively.
The Cosmic Microwave Background and Helical Magnetic Fields: the tensor mode · Introduction · 0.94
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The contributions to the CMB angular power spectrum from the scalar perturbations induced by PMFs are investigated in [ref] , from the vector perturbations in [ref] and from tensor mode in [ref] , respectively.
Magnetized CMB observables: a dedicated numerical approach · Introduction · 0.94
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