CMB spectral distortions from the decay of causally generated magnetic fields

Jacques M. Wagstaff and Robi Banerjee

Phys. Rev. D 92, 123004 – Published 14 December 2015

Abstract

We improve previous calculations of the cosmic microwave background spectral distortions due to the decay of primordial magnetic fields. We focus our studies on causally generated magnetic fields at the electroweak and QCD phase transitions. We also consider the decay of helical magnetic fields. We show that the decay of nonhelical magnetic fields generated at either the electroweak or QCD scale produce $μ$ - and $y$ -type distortions below $10^{- 8}$ which are probably not detectable by a future PIXIE-like experiment. We show that magnetic fields generated at the electroweak scale must have a helicity fraction $f_{*} > 1 0^{- 4}$ in order to produce detectable $μ$ -type distortions. Hence, a positive detection coming from the decay of magnetic fields would rule out nonhelical primordial magnetic fields and provide a lower bound on the magnetic helicity.

Received 28 July 2015

DOI:https://doi.org/10.1103/PhysRevD.92.123004

Authors & Affiliations

Jacques M. Wagstaff^1,2,* and Robi Banerjee^1,2

¹Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany
²Nordita KTH Royal Institute of Technology and Stockholm University Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

^*jwagstaff@hs.uni-hamburg.de

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Vol. 92, Iss. 12 — 15 December 2015

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Images

Figure 1
In plots (a) and (b), we show the evolution of magnetic energy from the time of magnetogenesis (at the EW and QCD scales respectively) to recombination. In plots (a) and (b), the (solid, blue) lines from top to bottom correspond to initial helicity fractions $f_{*} = {10^{- 3}, 10^{- 4}, 10^{- 6}, < 10^{- 14}}$ and $f_{*} = {10^{- 1}, 10^{- 4}, 10^{- 6}, < 10^{- 14}}$ respectively. The maximally helical case $f_{*} = 1$ (solid, red) is also shown. We also show (dashed, gray lines) the evolution of nonhelical magnetic fields with an inverse transfer of energy [see the discussion below Eq. (5)]. In all the above cases we consider $ϵ \equiv {\tilde{ρ}}_{B, *} / ρ_{γ, 0} \approx 1$ , which corresponds to an initial field strength of ${\tilde{B}}_{λ, *} ≃ 3 \times 10^{- 6} G$ , with $ϵ < 1$ the evolution history also changes.
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Figure 2
EW scale: In plot (a), we show the $μ$ -type distortion generated due to the decay of magnetic energy initially generated at the EW scale. Here, we plot the spectral distortion $μ$ vs $ϵ$ , where $ϵ \equiv {\tilde{ρ}}_{B, *} / ρ_{γ, 0} \approx 1$ corresponds to an initial field strength ${\tilde{B}}_{λ, *} ≃ 3 \times 10^{- 6} G$ . The (solid, blue) lines from top to bottom, in both plots, correspond to initial helicity fractions $f_{*} = {1 0^{- 3}, 1 0^{- 4}, 1 0^{- 6}, < 1 0^{- 14}}$ . The maximally helical case $f_{*} = 1$ (solid, red) is also shown. In plot (b), we show the final field strength $B_{0}$ and coherence length $λ_{B}$ that would be observed today, i.e. after MHD turbulent decay; see Eq. (22). We also show the approximate constraint on magnetic fields from CMB observations, $B_{0} ≲ 1 0^{- 9} G$ ; see Ref. [48] and references therein. The results for nonhelical magnetic fields with an inverse transfer of energy [see the discussion below Eq. (5)] are also shown (dashed, gray lines).
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Figure 3
QCD scale: In plot (a), we show the $μ$ -type distortion generated due to the decay of magnetic energy initially generated at the QCD scale. Here, we plot the spectral distortion $μ$ vs $ϵ$ , where $ϵ \equiv {\tilde{ρ}}_{B, *} / ρ_{γ, 0} \approx 1$ corresponds to an initial field strength ${\tilde{B}}_{λ, *} ≃ 3 \times 10^{- 6} G$ . The (solid, blue) lines from top to bottom, in both plots, correspond to initial helicity fractions $f_{*} = {1 0^{- 1}, 1 0^{- 4}, 1 0^{- 6}, < 1 0^{- 14}}$ . The maximally helical case $f_{*} = 1$ (solid, red) is also shown. In plot (b), we show the final field strength $B_{0}$ and coherence length $λ_{B}$ that would be observed today, i.e. after MHD turbulent decay; see Eq. (22). We also show the approximate constraint on magnetic fields from CMB observations, $B_{0} ≲ 1 0^{- 9} G$ ; see Ref. [48] and references therein. The results for nonhelical magnetic fields with an inverse transfer of energy [see the discussion below Eq. (5)] are also shown (dashed, gray lines). Here, we comment on the seemingly strange behavior of the plot for $f_{*} = 1 0^{- 1}$ in the QCD case (also applicable to the nonhelical inverse transfer case, gray dashed line). As $ϵ$ decreases, the onset of the viscous freezing regime begins earlier. For a certain value of $ϵ$ , the start of the viscous freezing regime coincides with the start of the $μ$ -era, and hence there is practically no magnetic energy decay in this time, and hence $μ$ becomes very small. As $ϵ$ decrease further, the start of the viscous damping stage occurs within the $μ$ -era, and a large $μ$ can once again be generated.
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Figure 4
In plots (a) and (b), we show the $y$ -type distortion produced before decoupling due to the decay of magnetic energy initially generated at the EW and QCD scales respectively. Here, we plot the spectral distortion $y$ vs $ϵ$ , where $ϵ \equiv {\tilde{ρ}}_{B, *} / ρ_{γ, 0} \approx 1$ corresponds to an initial field strength of ${\tilde{B}}_{λ, *} ≃ 3 \times 10^{- 6} G$ . In plots (a) and (b), the (solid, blue) lines from top to bottom correspond to initial helicity fractions $f_{*} = {1 0^{- 3}, 1 0^{- 4}, 1 0^{- 6}, < 1 0^{- 14}}$ and $f_{*} = {1 0^{- 1}, 1 0^{- 4}, 1 0^{- 6}, < 1 0^{- 14}}$ respectively. The maximally helical case $f_{*} = 1$ (solid, red) is also shown. The final field strength $B_{0}$ and coherence length $λ_{B}$ that would be observed today, i.e. after MHD turbulent decay, see Eq. (22), are the same as in plots (b) of Figs. 2 and 3. We also show (dashed, gray lines) the results from nonhelical magnetic fields with an inverse transfer of energy [see the discussion below Eq. (5)].
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