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# Impact of a primordial magnetic field on cosmic microwave background $$B$$ modes with weak lensing

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### Abstract

We discuss the manner in which the primordial magnetic field (PMF) suppresses the cosmic microwave background (CMB) $$B$$ mode due to the weak-lensing (WL) effect. The WL effect depends on the lensing potential (LP) caused by matter perturbations, the distribution of which at cosmological scales is given by the matter power spectrum (MPS). Therefore, the WL effect on the CMB $$B$$ mode is affected by the MPS. Considering the effect of the ensemble average energy density of the PMF, which we call "the background PMF," on the MPS, the amplitude of MPS is suppressed in the wave number range of $$k>0.01~h$$ Mpc$$^{-1}$$.The MPS affects the LP and the WL effect in the CMB $$B$$ mode; however, the PMF can damp this effect. Previous studies of the CMB $$B$$ mode with the PMF have only considered the vector and tensor modes. These modes boost the CMB $$B$$ mode in the multipole range of $$\ell > 1000$$, whereas the background PMF damps the CMB $$B$$ mode owing to the WL effect in the entire multipole range. The matter density in the Universe controls the WL effect. Therefore, when we constrain the PMF and the matter density parameters from cosmological observational data sets, including the CMB $$B$$ mode, we expect degeneracy between these parameters. The CMB $$B$$ mode also provides important information on the background gravitational waves, inflation theory, matter density fluctuations, and the structure formations at the cosmological scale through the cosmological parameter search. If we study these topics and correctly constrain the cosmological parameters from cosmological observations including the CMB $$B$$ mode, we need to correctly consider the background PMF.

### Most cited references32

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### BICEP2 / Keck Array VI: Improved Constraints On Cosmology and Foregrounds When Adding 95 GHz Data From Keck Array

,  ,   (2016)
We present results from an analysis of all data taken by the BICEP2 & Keck Array CMB polarization experiments up to and including the 2014 observing season. This includes the first Keck Array observations at 95 GHz. The maps reach a depth of 50 nK deg in Stokes $$Q$$ and $$U$$ in the 150 GHz band and 127 nK deg in the 95 GHz band. We take auto- and cross-spectra between these maps and publicly available maps from WMAP and Planck at frequencies from 23 GHz to 353 GHz. An excess over lensed-LCDM is detected at modest significance in the 95x150 $$BB$$ spectrum, and is consistent with the dust contribution expected from our previous work. No significant evidence for synchrotron emission is found in spectra such as 23x95, or for correlation between the dust and synchrotron sky patterns in spectra such as 23x353. We take the likelihood of all the spectra for a multi-component model including lensed-LCDM, dust, synchrotron and a possible contribution from inflationary gravitational waves (as parametrized by the tensor-to-scalar ratio $$r$$), using priors on the frequency spectral behaviors of dust and synchrotron emission from previous analyses of WMAP and Planck data in other regions of the sky. This analysis yields an upper limit $$r_{0.05}<0.09$$ at 95% confidence, which is robust to variations explored in analysis and priors. Combining these $$B$$-mode results with the (more model-dependent) constraints from Planck analysis of CMB temperature plus BAO and other data, yields a combined limit $$r_{0.05}<0.07$$ at 95% confidence. These are the strongest constraints to date on inflationary gravitational waves.
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### Tensor microwave anisotropies from a stochastic magnetic field

(2000)
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### Evidence for strong extragalactic magnetic fields from Fermi observations of TeV blazars

(2010)
Magnetic fields in galaxies are produced via the amplification of seed magnetic fields of unknown nature. The seed fields, which might exist in their initial form in the intergalactic medium, were never detected. We report a lower bound $$B\ge 3\times 10^{-16}$$~gauss on the strength of intergalactic magnetic fields, which stems from the nonobservation of GeV gamma-ray emission from electromagnetic cascade initiated by tera-electron volt gamma-ray in intergalactic medium. The bound improves as $$\lambda_B^{-1/2}$$ if magnetic field correlation length, $$\lambda_B$$, is much smaller than a megaparsec. This lower bound constrains models for the origin of cosmic magnetic fields.
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### Author and article information

###### Journal
24 May 2018
###### Article
10.1103/PhysRevD.97.103525
1805.09903