The Magnetized UniverseTanmay Vachaspati

Cosmology Initiative

Theoretical Physics Colloquium, 20 May 2020

Observationsastrophysics; data

Early Universestandard model; extensions

EvolutionMHD, 𝜒MHD

Components

Observations

Galactic fields: primordial vs. dynamo (How to explain micro-Gauss fields in galaxies at z > 2?)

Blazars: lower bounds at the 10-16 Gauss level on Mpc scales.

Faraday rotation (quasars): upper bounds of about a nano Gauss on coherent (Gpc) magnetic fields.

CMB: temperature anisotropies, B-modes. Upper bounds at the nano Gauss level on coherent fields.

BBN: Upper bounds at the micro Gauss level on energy density of fields.

CMB: inhomogeneous recombination. (More later.)

[many groups, many decades]

Reviews by Kronberg; Widrow; Durrer & Neronov; Subramanian

BBN

Quasars; CMB; UHECR…

Blazars

10-6 G

10-9 G

10-15 G

B𝝽

kpc Mpc Gpc 𝝽

💫

Observations: summary plot

Can explain galactic magnetic fields with minimal dynamo amplification.

Several other constraints, e.g. structure of dwarf galaxies (Sanati et al, 2020).

Blazar CascadesGould & Schreder, 1967; Coppi & Aharonian, 1998; ..... Neronov & Semikoz, 2009

EBL

HESSCredit: Nina McCurdy and Joel R. Primack/UC-HiPACC

GeV

TeV

Spectrum: Depletion at TeV energies and excess at GeV energies.

Energy

Flux

TeVGeV

B reduces shower contribution to spectrum.

GeV halo

TeV interior

Morphology:

A Lower BoundNeronov & Vovk, 2010

Essey, Ando & Kusenko 2011(and several other groups since)

Fig. 1: A comparison of models of cascade emission from TeV blazars (thick solid black curves)with Fermi upper limits (grey curves) and HESS data (grey data points). Thin dashed curvesshow the primary (unabsorbed) source spectra. Dotted curves show the spectra of electromag-netic cascade initiated by pair production on EBL. Vertical lines with arrows show the energiesbelow which the cascade emission should be suppressed.

5

Hess data

primary (unabsorbed)

spectrum

FERMI bound

processed spectrum

missingGeV

photons

cascade

Effect of a magnetic field

CMB

EBL

HESSCredit: Nina McCurdy and Joel R. Primack/UC-HiPACC

GeV

TeV

GeV halo

TeV interior

Morphology:

GeV halo

TeV interiorB

~B = 0

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~B 6= 0

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Plasma instabilities?Broderick, Chang & Pfrommer, 2012

Missing GeV photons attributed to B > 10-16 Gauss

Stacked AnalysesAndo & Kusenko, 2010

Hints for cascade photons from (stacked) sources.

3

(a) (b)

(degree)(degree)

(counts/sr)(counts/sr)

(de

gre

e)

(de

gre

e)

FIG. 1. γ-ray counts maps of the stacked sources in the 1GeV-1.58GeV energy bin. The large circles show the outer edge ofthe detection region. (a) Counts map of the 24 stacked low-redshift HSP BL Lacs. (b) Smoothed counts difference be-tween the stacked BL Lacs and the center-normalized stackedFSRQs. Positive values indicate the BL Lacs’ counts is greaterthan the FSRQs’.

dN

/dΩ

(sr

)

-1

θ (degree)

θ (degree)θ (degree)

θ (degree)

FSRQs

Pulsars

BL Lacs

dN

/dΩ

(sr

)

-1

FIG. 2. Angular distribution of photon events around thestacked pulsars (black), the stacked FSRQs (red), and thestacked BL lacs (blue): vertical errors are the 95% confidenceintervals; horizontal errors show the size of angular bins.

understanding of the PSF is critical for this type of study.Pulsars with unresolved pulsar wind nebulae (PWN) canbe used as calibration sources since they are effectivepoint sources for Fermi-LAT [5, 7]; here we choose theCrab and Geminga pulsars as our calibration sources.No evidence for a PWN has been found associated withGeminga [16] making it a natural choice. The size of

the Crab PWN is about 0.05 [17], which is smaller thanFermi’s angular resolution, providing us with a secondgood source for verifying the PSF. To compare differentangular distribution profiles of different stacked sources,we calculate and remove the diffuse background for eachsource, sum the background-subtracted counts and thennormalize the profiles. We calculate the angular pro-files for stacked pulsars (Crab and Geminga), the 24 BLLacs, and the 26 FSRQs, as shown in Fig. 2. The errorsgive the 95% confidence intervals of getting the numberof counts in each angular bin. The angular profiles forstacked pulsars agree with the up-to-date PSFs in eachenergy range [10]. However, the normalized angular pro-files of stacked BL Lacs have a lower scaled counts perunit solid angle in the inner regions (small θ), provid-ing evidence for extended emission since the additionalcounts in the extended halo reduce the scaled counts atsmall angles after normalization. The deficit in counts atsmall θ (evidence for extended emission) is more signifi-cant at lower energy ranges, consistent with the expecta-tion that the angular extent of the halo is larger at lowerenergies, as indicated in Eq. 1. In contrast, the angularprofiles of the stacked FSRQs are indistinguishable fromour surrogate point-source data from pulsars, as shownin Fig. 2.

STATISTICAL EVIDENCE FOR PAIR-HALO

EMISSION AND ESTIMATION OF THE IGMF

To model the normalized angular profiles g(θ), we use

g(θ; fhalo,Θ) = fhaloghalo(θ;Θ) + (1− fhalo)gpsf(θ), (2)

where fhalo is the fraction of the pair halo component,gpsf(θ) is the PSF and ghalo(θ;Θ) is a Gaussian functionof θ in the small angle approximation convolved withthe PSF. Then, the number of photon events in the i-thangular bin is estimated by

λi(fhalo,Θ,λb,i, N∗) = (N∗gi + µb)Ωi, (3)

where gi is the discrete value of the normalized angulardistribution g(θ) given by Eq. 2, N∗ is a normalizationfactor, µb is the assumed uniform background counts perunit solid angle, and Ωi is the solid angle of the i-th bin.For a given configuration of the angular bins, a set ofestimators λi is a function of fhalo, Θ, µb, and N∗.Maximum likelihood estimation is used for the model

fitting. The likelihood L is defined in the 4-dimensionalspace of the model parameters, x = (fhalo,Θ, µb, N∗), asthe joint probability for a number of Poisson processes ofgetting a set of observed γ-ray counts in all the n angularbins Ni(i = 1, 2, ..., n) with Nbg background counts inthe background bin:

L(x|Ni) =

(

n∏

i=1

P (Ni|λi)

)

× P (Nbg|µbΩbg), (4)

Chen, Buckley & Ferrer, 2015

3

(a) (b)

(degree)(degree)

(counts/sr)(counts/sr)

(de

gre

e)

(de

gre

e)

FIG. 1. γ-ray counts maps of the stacked sources in the 1GeV-1.58GeV energy bin. The large circles show the outer edge ofthe detection region. (a) Counts map of the 24 stacked low-redshift HSP BL Lacs. (b) Smoothed counts difference be-tween the stacked BL Lacs and the center-normalized stackedFSRQs. Positive values indicate the BL Lacs’ counts is greaterthan the FSRQs’.

dN

/dΩ

(sr

)

-1

θ (degree)

θ (degree)θ (degree)

θ (degree)

FSRQs

Pulsars

BL Lacs

dN

/dΩ

(sr

)

-1

FIG. 2. Angular distribution of photon events around thestacked pulsars (black), the stacked FSRQs (red), and thestacked BL lacs (blue): vertical errors are the 95% confidenceintervals; horizontal errors show the size of angular bins.

understanding of the PSF is critical for this type of study.Pulsars with unresolved pulsar wind nebulae (PWN) canbe used as calibration sources since they are effectivepoint sources for Fermi-LAT [5, 7]; here we choose theCrab and Geminga pulsars as our calibration sources.No evidence for a PWN has been found associated withGeminga [16] making it a natural choice. The size of

the Crab PWN is about 0.05 [17], which is smaller thanFermi’s angular resolution, providing us with a secondgood source for verifying the PSF. To compare differentangular distribution profiles of different stacked sources,we calculate and remove the diffuse background for eachsource, sum the background-subtracted counts and thennormalize the profiles. We calculate the angular pro-files for stacked pulsars (Crab and Geminga), the 24 BLLacs, and the 26 FSRQs, as shown in Fig. 2. The errorsgive the 95% confidence intervals of getting the numberof counts in each angular bin. The angular profiles forstacked pulsars agree with the up-to-date PSFs in eachenergy range [10]. However, the normalized angular pro-files of stacked BL Lacs have a lower scaled counts perunit solid angle in the inner regions (small θ), provid-ing evidence for extended emission since the additionalcounts in the extended halo reduce the scaled counts atsmall angles after normalization. The deficit in counts atsmall θ (evidence for extended emission) is more signifi-cant at lower energy ranges, consistent with the expecta-tion that the angular extent of the halo is larger at lowerenergies, as indicated in Eq. 1. In contrast, the angularprofiles of the stacked FSRQs are indistinguishable fromour surrogate point-source data from pulsars, as shownin Fig. 2.

STATISTICAL EVIDENCE FOR PAIR-HALO

EMISSION AND ESTIMATION OF THE IGMF

To model the normalized angular profiles g(θ), we use

g(θ; fhalo,Θ) = fhaloghalo(θ;Θ) + (1− fhalo)gpsf(θ), (2)

where fhalo is the fraction of the pair halo component,gpsf(θ) is the PSF and ghalo(θ;Θ) is a Gaussian functionof θ in the small angle approximation convolved withthe PSF. Then, the number of photon events in the i-thangular bin is estimated by

λi(fhalo,Θ,λb,i, N∗) = (N∗gi + µb)Ωi, (3)

where gi is the discrete value of the normalized angulardistribution g(θ) given by Eq. 2, N∗ is a normalizationfactor, µb is the assumed uniform background counts perunit solid angle, and Ωi is the solid angle of the i-th bin.For a given configuration of the angular bins, a set ofestimators λi is a function of fhalo, Θ, µb, and N∗.Maximum likelihood estimation is used for the model

fitting. The likelihood L is defined in the 4-dimensionalspace of the model parameters, x = (fhalo,Θ, µb, N∗), asthe joint probability for a number of Poisson processes ofgetting a set of observed γ-ray counts in all the n angularbins Ni(i = 1, 2, ..., n) with Nbg background counts inthe background bin:

L(x|Ni) =

(

n∏

i=1

P (Ni|λi)

)

× P (Nbg|µbΩbg), (4)

Halo detected at ~3.5 sigma.

Recombination with BJedamzik and Abel, 1108.2517 Jedamzik and Saveliev, 1804.06115Jedamzik and Pogosian, 2004.09487

Magnetic fields at recombination induce inhomogeneities in the baryon density on scales below the photon mean free path.

Recombination rate is proportional to ⟨ne2⟩ and is larger than ⟨ne⟩2.

Therefore magnetic fields induce earlier recombination and a reduced sound horizon r∗ at recombination.

This would shift the CMB spectral peak positions which are at 𝑙p ∝rls/r∗ unless last scattering surface is closer in, i.e. H0 is larger.

Relieving the H0 tension with primordial magnetic fields

64 66 68 70 72 74 76 78H0

0.725

0.750

0.775

0.800

0.825

0.850

0.875

0.900

S8

DES

Planck §CDM

Planck+H3 M1

Planck+H3 M2

The baryon clumping parameter detected at ~4s

Corresponds to magnetic field strengths of ~0.05 nano-Gauss

Also solves the mild tension in S8 between Planck and DESK. Jedamzik and L. Pogosian, arXiv:2004.09487

Resolving the H0 tension with B(slide provided by Levon Pogosian)

Quasars; CMB; UHECR…

Observations + spectra

Can explain galactic magnetic fields with minimal dynamo amplification.

BBN

Blazars

10-6 G

10-9 G

10-15 G

B𝝽

kpc Mpc Gpc 𝝽

💫

baryonclustering

Origin of cosmological magnetic fields

Several ideas: inflation; electroweak phase transition;QCD epoch; turbulence at recombination; astrophysics.

All scenarios except inflationary magnetogenesis are “causal”.

Description of stochastic, isotropic magnetic fields:

hbi(k)bj (k0)i =EM (k)

4k2(ij kikj) + iijlkl

HM (k)

8k2

(2)6(3)(k k0)

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Power spectrum Helicity spectrum~B · ~r ~B; circular polarization;

Zd3x ~A · ~B

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Monin & Yaglom

Magnetic fields from the electroweak epoch

Simplifying assumptions: zero temperature, only bosonic sector, classical.(Self-generated plasma.)

TVBaym, Bödekar, McLerran

Diaz-Gil, Garcia-Bellido, Garcia-Perez, Gonzalez-ArroyoFerrer, Zhang & TV

Exact form of the potential will not be crucial for us since late time behavior is most interesting.

The classical dynamics, as theHiggs acquires a VEV, produces

significant magnetic fields.

(t = 0,x) = 0<latexit sha1_base64="ZHVafpFiawSXCKkE65hOOwPabrk=">AAACHXicbVDLSsNAFJ3UV62vqLhyM1iEClISEXRTKLpxWcE+oAllZjpph04yYWYiltBvEbf6He7ErfgZ/oGTNgvbeuDC4Zx7OZeDY86Udpxvq7Cyura+UdwsbW3v7O7Z+wctJRJJaJMILmQHI0U5i2hTM81pJ5YUhZjTNh7dZn77kUrFRPSgxzH1QzSIWMAI0kbq2UdeY8hgRdecc5h6OIBPk7Oa07PLTtWZAi4TNydlkKPRs3+8viBJSCNNOFKq6zqx9lMkNSOcTkpeomiMyAgNaNfQCIVU+en0/Qk8NUofBkKaiTScqn8vUhQqNQ6x2QyRHqpFLxP/87qJDq79lEVxomlEZkFBwqEWMOsC9pmkRPOxIYhIZn6FZIgkIto0NpeCQ3MNseD9ABEKsyjTkbvYyDJpXVRdp+reX5brN3lbRXAMTkAFuOAK1MEdaIAmICAFL+AVvFnP1rv1YX3OVgtWfnMI5mB9/QKw9aAO</latexit><latexit sha1_base64="ZHVafpFiawSXCKkE65hOOwPabrk=">AAACHXicbVDLSsNAFJ3UV62vqLhyM1iEClISEXRTKLpxWcE+oAllZjpph04yYWYiltBvEbf6He7ErfgZ/oGTNgvbeuDC4Zx7OZeDY86Udpxvq7Cyura+UdwsbW3v7O7Z+wctJRJJaJMILmQHI0U5i2hTM81pJ5YUhZjTNh7dZn77kUrFRPSgxzH1QzSIWMAI0kbq2UdeY8hgRdecc5h6OIBPk7Oa07PLTtWZAi4TNydlkKPRs3+8viBJSCNNOFKq6zqx9lMkNSOcTkpeomiMyAgNaNfQCIVU+en0/Qk8NUofBkKaiTScqn8vUhQqNQ6x2QyRHqpFLxP/87qJDq79lEVxomlEZkFBwqEWMOsC9pmkRPOxIYhIZn6FZIgkIto0NpeCQ3MNseD9ABEKsyjTkbvYyDJpXVRdp+reX5brN3lbRXAMTkAFuOAK1MEdaIAmICAFL+AVvFnP1rv1YX3OVgtWfnMI5mB9/QKw9aAO</latexit><latexit sha1_base64="ZHVafpFiawSXCKkE65hOOwPabrk=">AAACHXicbVDLSsNAFJ3UV62vqLhyM1iEClISEXRTKLpxWcE+oAllZjpph04yYWYiltBvEbf6He7ErfgZ/oGTNgvbeuDC4Zx7OZeDY86Udpxvq7Cyura+UdwsbW3v7O7Z+wctJRJJaJMILmQHI0U5i2hTM81pJ5YUhZjTNh7dZn77kUrFRPSgxzH1QzSIWMAI0kbq2UdeY8hgRdecc5h6OIBPk7Oa07PLTtWZAi4TNydlkKPRs3+8viBJSCNNOFKq6zqx9lMkNSOcTkpeomiMyAgNaNfQCIVU+en0/Qk8NUofBkKaiTScqn8vUhQqNQ6x2QyRHqpFLxP/87qJDq79lEVxomlEZkFBwqEWMOsC9pmkRPOxIYhIZn6FZIgkIto0NpeCQ3MNseD9ABEKsyjTkbvYyDJpXVRdp+reX5brN3lbRXAMTkAFuOAK1MEdaIAmICAFL+AVvFnP1rv1YX3OVgtWfnMI5mB9/QKw9aAO</latexit><latexit sha1_base64="ZHVafpFiawSXCKkE65hOOwPabrk=">AAACHXicbVDLSsNAFJ3UV62vqLhyM1iEClISEXRTKLpxWcE+oAllZjpph04yYWYiltBvEbf6He7ErfgZ/oGTNgvbeuDC4Zx7OZeDY86Udpxvq7Cyura+UdwsbW3v7O7Z+wctJRJJaJMILmQHI0U5i2hTM81pJ5YUhZjTNh7dZn77kUrFRPSgxzH1QzSIWMAI0kbq2UdeY8hgRdecc5h6OIBPk7Oa07PLTtWZAi4TNydlkKPRs3+8viBJSCNNOFKq6zqx9lMkNSOcTkpeomiMyAgNaNfQCIVU+en0/Qk8NUofBkKaiTScqn8vUhQqNQ6x2QyRHqpFLxP/87qJDq79lEVxomlEZkFBwqEWMOsC9pmkRPOxIYhIZn6FZIgkIto0NpeCQ3MNseD9ABEKsyjTkbvYyDJpXVRdp+reX5brN3lbRXAMTkAFuOAK1MEdaIAmICAFL+AVvFnP1rv1YX3OVgtWfnMI5mB9/QKw9aAO</latexit>

|(t = 1,x)| = <latexit sha1_base64="NbbKCSsMkx6eud+K7QnvUckmrOM=">AAACKnicbVDLSgMxFM3U97vq0k2wCBWkzIigm4LoxmUFW4VOKZn0ThuayQzJHXEYu/djxK1+hztx6w/4B6aPhbUeCBzOvYdzc4JECoOu++EU5uYXFpeWV1bX1jc2t4rbOw0Tp5pDnccy1ncBMyCFgjoKlHCXaGBRIOE26F8O57f3oI2I1Q1mCbQi1lUiFJyhldrF/Ue/1hO0jFVfqBCzI5r7QUgfBof0kVapD8jaxZJbcUegs8SbkBKZoNYufvudmKcRKOSSGdP03ARbOdMouITBqp8aSBjvsy40LVUsAtPKR38Z0AOrdGgYa/sU0pH625GzyJgsCuxmxLBn/s6G4n+zZorhWSsXKkkRFB8HhamkGNNhMbQjNHCUmSWMa2FvpbzHNONo65tKCSLrpkEsOyHjQIdRtiPvbyOzpHFc8dyKd31SOr+YtLVM9sg+KROPnJJzckVqpE44eSIv5JW8Oc/Ou/PhfI5XC87Es0um4Hz9AGbtpaQ=</latexit><latexit sha1_base64="NbbKCSsMkx6eud+K7QnvUckmrOM=">AAACKnicbVDLSgMxFM3U97vq0k2wCBWkzIigm4LoxmUFW4VOKZn0ThuayQzJHXEYu/djxK1+hztx6w/4B6aPhbUeCBzOvYdzc4JECoOu++EU5uYXFpeWV1bX1jc2t4rbOw0Tp5pDnccy1ncBMyCFgjoKlHCXaGBRIOE26F8O57f3oI2I1Q1mCbQi1lUiFJyhldrF/Ue/1hO0jFVfqBCzI5r7QUgfBof0kVapD8jaxZJbcUegs8SbkBKZoNYufvudmKcRKOSSGdP03ARbOdMouITBqp8aSBjvsy40LVUsAtPKR38Z0AOrdGgYa/sU0pH625GzyJgsCuxmxLBn/s6G4n+zZorhWSsXKkkRFB8HhamkGNNhMbQjNHCUmSWMa2FvpbzHNONo65tKCSLrpkEsOyHjQIdRtiPvbyOzpHFc8dyKd31SOr+YtLVM9sg+KROPnJJzckVqpE44eSIv5JW8Oc/Ou/PhfI5XC87Es0um4Hz9AGbtpaQ=</latexit><latexit sha1_base64="NbbKCSsMkx6eud+K7QnvUckmrOM=">AAACKnicbVDLSgMxFM3U97vq0k2wCBWkzIigm4LoxmUFW4VOKZn0ThuayQzJHXEYu/djxK1+hztx6w/4B6aPhbUeCBzOvYdzc4JECoOu++EU5uYXFpeWV1bX1jc2t4rbOw0Tp5pDnccy1ncBMyCFgjoKlHCXaGBRIOE26F8O57f3oI2I1Q1mCbQi1lUiFJyhldrF/Ue/1hO0jFVfqBCzI5r7QUgfBof0kVapD8jaxZJbcUegs8SbkBKZoNYufvudmKcRKOSSGdP03ARbOdMouITBqp8aSBjvsy40LVUsAtPKR38Z0AOrdGgYa/sU0pH625GzyJgsCuxmxLBn/s6G4n+zZorhWSsXKkkRFB8HhamkGNNhMbQjNHCUmSWMa2FvpbzHNONo65tKCSLrpkEsOyHjQIdRtiPvbyOzpHFc8dyKd31SOr+YtLVM9sg+KROPnJJzckVqpE44eSIv5JW8Oc/Ou/PhfI5XC87Es0um4Hz9AGbtpaQ=</latexit><latexit sha1_base64="NbbKCSsMkx6eud+K7QnvUckmrOM=">AAACKnicbVDLSgMxFM3U97vq0k2wCBWkzIigm4LoxmUFW4VOKZn0ThuayQzJHXEYu/djxK1+hztx6w/4B6aPhbUeCBzOvYdzc4JECoOu++EU5uYXFpeWV1bX1jc2t4rbOw0Tp5pDnccy1ncBMyCFgjoKlHCXaGBRIOE26F8O57f3oI2I1Q1mCbQi1lUiFJyhldrF/Ue/1hO0jFVfqBCzI5r7QUgfBof0kVapD8jaxZJbcUegs8SbkBKZoNYufvudmKcRKOSSGdP03ARbOdMouITBqp8aSBjvsy40LVUsAtPKR38Z0AOrdGgYa/sU0pH625GzyJgsCuxmxLBn/s6G4n+zZorhWSsXKkkRFB8HhamkGNNhMbQjNHCUmSWMa2FvpbzHNONo65tKCSLrpkEsOyHjQIdRtiPvbyOzpHFc8dyKd31SOr+YtLVM9sg+KROPnJJzckVqpE44eSIv5JW8Oc/Ou/PhfI5XC87Es0um4Hz9AGbtpaQ=</latexit>

Φ

V

V () = (||2 2)2<latexit sha1_base64="Gv/yR5fNC0a9L+wS496wgtCQAp0=">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</latexit><latexit sha1_base64="Gv/yR5fNC0a9L+wS496wgtCQAp0=">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</latexit><latexit sha1_base64="Gv/yR5fNC0a9L+wS496wgtCQAp0=">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</latexit><latexit sha1_base64="Gv/yR5fNC0a9L+wS496wgtCQAp0=">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</latexit>

Triggering the EWPT

Energy with Φ=0 is the same as with:

C =p2 1

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Choose center of bubble to be in true vacuum:

|| = 2p2CemHr/

p2

1 + C2ep2mHr

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“trigger bubble” profile

Nucleate “trigger bubbles” uniformly in Higgs=0 phase at constant rate.

||2 = 21 + 2

2 + 23 + 2

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Ferrer, Zhang & TV

A perturbation is required to initiate the phase transition.Energy

I*÷¥÷¥

"

,

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:

-

IEtz fieldspace .

“Direction” of Higgs VEV chosen on 3-sphere:

Evolution @20 = DiDi 2(||2 2) @0 ln ||,

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@20Bi = @jBij + g0 Im[†Di],

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@20W

ai = @kW

aik g abcW b

kWcik + g Im[†aDi].

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γ is a damping coefficient for Higgs field oscillations.

We want to damp out radial oscillations but not the angular dynamicsbecause electric charge has to be conserved.

Numerical techniques: (1) lattice inspired, (2) numerically-relativity inspired.

Magnetic Field: Definition

Aµ = sin wnaW a

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Aµ =sin wnaW a

µ + cos wBµ

i2

g2sin w[(Dµ)

†D (D)†Dµ]

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na = †a

2<latexit sha1_base64="O1+vtYd9iuPFxWRJUgLw+bIf578=">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</latexit><latexit sha1_base64="O1+vtYd9iuPFxWRJUgLw+bIf578=">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</latexit><latexit sha1_base64="O1+vtYd9iuPFxWRJUgLw+bIf578=">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</latexit><latexit sha1_base64="O1+vtYd9iuPFxWRJUgLw+bIf578=">AAACNnicbVDLSgMxFM34flt16SZYBDeWGRF0I4huXFawVei05U7mThtMMkOSEcrQv/BjxK3+hBt34lb/wEzbha8DgcM593JuTpQJbqzvv3hT0zOzc/MLi0vLK6tr65WNzaZJc82wwVKR6psIDAqusGG5FXiTaQQZCbyObs9L//oOteGpurKDDNsSeoonnIF1UrdSUx2gJ3SfhokGVoT1Pu+EMfRoaHhPgjNLaViEaKFzMOxWqn7NH4H+JcGEVMkE9W7lM4xTlktUlgkwphX4mW0XoC1nAodLYW4wA3YLPWw5qkCiaRejfw3prlNimqTaPWXpSP2+UYA0ZiAjNynB9s1vrxT/81q5TY7bBVdZblGxcVCSC2pTWpZEY66RWTFwBJjm7lbK+uAKsq7KHymRdNs0SkWcAENaRrmOgt+N/CXNg1rg14LLw+rp2aStBbJNdsgeCcgROSUXpE4ahJF78kieyLP34L16b977eHTKm+xskR/wPr4A92WrHQ==</latexit>

Ambiguities (|Φ|2 vs. η2) diminish at late times.

Two bubbles

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Many bubbles

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Magnetic Field

~6% conversion of vacuum energy into magnetic field energy.

Most of the magnetic field energy is produced while the Higgs oscillates around the true vacuum, after bubbles have percolated.

Magnetic Field: Spectral Features

Coherence scale = bubble percolation scale?

Peak at small k.

Zhang, Ferrer, TV

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3k2 |

B(k)

|2 / V

k/m

mt=105

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

k2 |B(

k)|2 /

V

k/m

mt=145

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

k2 |B(

k)|2 /

V

k/m

mt=185

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

k2 |B(

k)|2 /

V

k/m

mt=265

Figure 12: The same as in Fig. 11 but for the magnetic component: hk2| ~B(k)|2i/V.

bution being a factor of 5-10 larger than the one coming from seed fields. Its profile

is very well described by:

1

Vh| ~Erad(k)|2i =

2wE

e(wEµE) 1(5.3)

1

Vh| ~Brad(k)|2i =

2k

e(wBµB) 1,

with wE(B) =q

k2 + m2E(B) and parameters given in Table 2. As illustrated in figures

11 and 12, this distribution fits very well the high momentum part of the spectrum

but fails in reproducing the low momentum peak. Eq. (5.3) represents free massive

thermal radiation with non zero chemical potential at temperatures slightly rising

with time, which we interpret as an e↵ect induced by the plasma of the W -fields.

Similar information can be extracted from the distribution of local values of the

norm of the transverse electric and magnetic fields. For free photons this should

follow a Maxwellian distribution (see Appendix C):

P (B) =

r2

3

hB2i

3/2

B2 e 3B

2

2hB2i , . (5.4)

– 26 –

Diaz-Gil, Garcia-Bellido, Garcia Perez, Gonzalez-ArroyoarXiv:0805.4159

Evolution of causal cosmological magnetic fields

Causal magnetic fields at production have a blue power spectrum, i.e. more energy at small length scales.

k_in k_diss ln(k)

ln(E_M(k)) t

Power at large length scales (small k) remains small.

Brandenburg & Kahniashvili & TevzadzeBanerjee & Jedamzik & Sigl

Olesen(other groups/collaborators)

Evolution of helical magnetic fields

k_in k_diss ln(k)

ln(E_M(k)) t

Power at large length scales (small k) grows with time.

Brandenburg & Kahniashvili & TevzadzeBanerjee & Jedamzik & Sigl(other groups/collaborators)

“inverse cascade”

Helicity is crucial for the survival of causal fields.

Spectrum: numerical results

EM (k) =k

2|HM (k)| =

(E0(k/kd)4, 0 k kd

0, kd < kmax. hel.

kin kdiss

7

FIG. 2: (color online). EM (ξ) (solid) and EK(ξ) (dashed) forthe helical (thin, red) and non-helical (thick, blue) cases.

FIG. 3: (color online). Evolution of EM (k, ξ) (solid) andEK(k, ξ) (dashed) versus k for η = 5, 10, 20, 50, and 100for the non-helical run. Thick lines are for η = 10. The reddash-dotted lines give the k2 and k4 scalings for comparison.All spectra are normalized by

∫EM (k, 0) dk/k0.

lence); (ii) other approaches are given in Refs. [27, 31].In particular, Ref. [27] refers to a more general case withmagnetic helicity evolving as HM (η) ∝ η−2s. Also, theratio between kinetic and magnetic energy densities hasin some studies assumed to be around 1; Ref. [31] assumesthat the magnetic field evolves toward a force-free regimewith constant magnetic helicity and with a constant ratiobetween magnetic and kinetic energy densities. All mod-els [28, 30–33] show that the scaling laws are independentof the initial magnetic field spectrum. In fact, there aretwo main behaviors described: (i) Refs. [30, 32, 33] claimEM (η) ∝ η−2/3 and ξM (η) ∝ η2/3; (ii) Refs. [27, 31]claim ξM (η) ∝ η1/2; with EM (η) ∝ η−1/2. In both scal-ing laws ξMEM ∼ const. The main difference betweenthese two scaling laws consists in choosing the turbulencemodel. Refs. [27, 31] assume a force-free developmentof the MHD turbulence decay, while Ref. [33], see theirEq. (4), assumes a linear dependence between vorticity

FIG. 4: (color online). Evolution of EM (k, ξ) (solid) andEK(k, ξ) (dashed) versus k for η = 5, 10, 20, 50, and 100 forthe helical run. Thick lines are for η = 10. The red dash-dotted lines give the k2 and k4 scalings for comparison. Allspectra are normalized by

∫EM (k, 0) dk/k0.

and Lorentz force. Our new numerical results supportthe former scenario (i). Fig. 3 shows the evolution of ki-netic and magnetic spectral energies. As we can see, thek2 and k4 laws are established at large scales for kineticand magnetic spectral energies, respectively. We can alsosee the slight increase of power at large scales, even in thecase of non-helical fields.We have performed a study of the large-scale decay

of a maximally helical magnetic field under conditionssimilar to those in the non-helical case, and for differentmagnetic Prandtl numbers as well as different values ofmagnetic resistivity. We have recovered the EM (k) ∝ k4

spectral shape and the scaling laws as ξM ∝ η2/3 andEM ∝ η−2/3, so that nξ = 2/3, nE = −1/3 [46]; see also[29] for more general discussion; see Figs. 1 and 2. Againthese scaling laws are valid when the correlation lengthis greater than the damping scale, so that dissipationdoes not play an important role. Similarly to the non-helical run, we find for k $ k0 and early times thatEM (k, ξ) ∝ k4 and EK(k, ξ) ∝ k2; see Figure 4. Inthis case there is a strong inverse cascade with a strongincrease of spectral energies with time for k $ k0. Wecan also see the constant magnetic power while the peakis moving toward large scales (inverse cascade). Thiscorresponds to the constant helicity case.

IV. IMPLICATION FOR COSMOLOGICALEVOLUTION OF MAGNETIC FIELD

Decay of cosmological MHD turbulence occurs to-gether with the cooling of the Universe and the increase ofmagnetic correlation length. Therewith, the correlationlength increases only up to the point when the Universereaches the temperature T = 1 eV [35]. The initial valuesof correlation length and magnetic field strength, ξ0 and

Kahniashvili, Tevzadze, Brandenburg & Neronov, 2013

Magnetic field becomes“maximally helical”

(only dominant circular polarization survives).

Observation of helicityElyiv, Nerolnov & Semikoz

Tashiro & TVDuplessis & Long & TV

Batista, Saveliev, Sigl & TVBroderick et al

Tiede et alFitousi et al

…

lower energy

Morphology:

Parity-odd statistic:

(improved statistic in Duplessis & TV)

Q(R;E1 < E2 < E3) = hnE1 nE2 · nE3iR

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Figure 11. (Top Left) A set of simulated observed photons from the halo formed by a blazar’s jetwith half-opening angle of 5. The surrounding random magnetic field was created with parametersdescribed in Eq. (6.2) and has the form given by Eq. (6.1). (Bottom Left) The PP locations ofthe photons responsible for the halo. (Right) The result of applying the Q-statistic to the observedphotons on the right.

6 The Q statistic Applied to Stochastic Magnetic Fields

The result of Fig. 11 is noisy and can be misleading as we are dealing with random magneticfields. Indeed, these fields can sometimes create halos whose Q-statistics suggest the wronghelicity. It is therefore important to average over many realizations of the magnetic field andthe jet orientation. Each realization will simulate a blazar with a jet of half-opening anglejet = 5 and having Earth in its LoS. The jet is also constrained to generate a halo withat least 3 events in order for the statistics to be applied; this condition is easily satisfied ifEarth is in the jet’s LoS. Jets pointing further away from the LoS might still yield observablephotons but we would not be able to identify these blazars and so we don’t simulate thosecases.

We will consider magnetic fields of the form,

B(x) =1

2N2 + 2

X

k2Kb(k, fH, Brms)e

ik·x (6.1)

with the set K consisting of 2N2+2 vectors which have magnitude kmag and whose directionsare approximatively uniformly spread over the unit sphere. Half of the Fourier coecientsb(k, fH, Brms) are drawn from their respective distribution as outlined in Appendix A, whilethe other half are set by the requirement b(k, fH, Brms) = b

(k, fH, Brms), necessary forobtaining a real value for the magnetic field. The value of 1 fH 1 controls the

– 14 –

Figure 6. Example of how a blazar with a jet will only shine and activate a small region of the PPsurface (left) and the resultant halo (right). The magnetic field is given in Eq. (3.4) with B0 = 1014G, = 250 Mpc.

Figure 7. Monte Carlo simulation with stochastic PDs using the magnetic field of Eq. (3.4) andwith the same setup as in Fig. 6 but with B0 reduced to 5 1015

G. Compared to the right panelof Fig. 6, we see that the high energy gamma rays (blue and green points) are more clustered and sothe halo size is smaller at fixed energy.

The magnetic field strength directly a↵ects the amount of bending of the lepton trajec-tories since the gyroradius RL / 1/B0. Therefore a weak magnetic field will require that theinitial TeV gamma ray is already propagating nearly towards Earth. Thus reducing B0 willshrink the size of the halo at any given gamma ray energy, although lower energy gammarays may now enter the field of view. This can be seen in Fig. 7 which was created usingB0 = 5 1015 G and = 250 Mpc. The plot looks almost identical to Fig. 6, which wascreated using B0 = 1014 G, except that the extent of the halo in the x and y directions, forphotons of the same energy, has shrunk by a factor of 2.

If one does not track the photon’s energy, the e↵ect of a change in B0 is not easily

– 10 –

"particle production surface”

Observation of helicity

°150

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050

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20-40

5 10 15 20 25 30°100

0

100

200

300

40030-40

North + South

R

Q(R

)£

106

Tashiro, Chen, Ferrer & TV…

[Asplund, Brandenberger, Guölaugur]

Q is non-zero at about 3 sigma but results may suffer from galactic foreground contamination (even though galaxy is masked out).

Gives B~10-14 G on ~10 Mpc. Technique can be used to determine the magnetic field spectrum.

The Full Spectrumhbi(k)bj (k0)i =

EM (k)

4k2pij + iijlkl

HM (k)

8k2

(2)6(3)(k k0)

TV, 2016

k_in k_diss ln(k)

ln(EM(k))

k_10

~k4

Dissipation mainly to magnetosonic modes: ld0 1 pc 1 kpc

Banerjee & Jedamzik, 2003

maximally helical|HM | = 2EM/k

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Estimates

E0 (3 1010 G)21 kpc

ld0

2

Energy density:

H0 3 1020 G2 kpc

1 kpc

ld0

2 1017 cm3

1 kpc

ld0

Helicity density:

B0 =E00

108

1 kpc

ld0

2

.

Magnetic to photon energy density:

Sets target parameters for any origin mechanism.E.g. electroweak can produce sufficient energy density but sufficient

helicity will require new physics.

Galactic Magnetic Field

Compression of IGMF can directly give a relatively large (micro Gauss), random (parsec scale), galactic magnetic field*.

Banerjee & JedamzikTV

During galaxy formation 3 1010 G on 1 kpc scales gets compressed

by (g/c)2/3 105 to 3 105 G on 10 pc scales.

Observations + spectra

Can explain galactic magnetic fields with minimal dynamo amplification.

Quasars; CMB; UHECR…

BBN

Blazars

10-6 G

10-9 G

10-15 G

B𝝽

kpc Mpc Gpc 𝝽

💫

“acausal”

“causal”

baryonclustering

helicity

Origin of helicity?

Large observed helicity needs explanation. More CP violation; helicity amplification.

Baryogenesis?

Every B = H = h #nb

Cornwall, 1997TV, 2001

Copi et al, 2008Chu, Dent, TV 2011

Kharzeev, Shuryak, Zahed 2019

Helicity density to baryon number density ratio: Bb h

nb 102

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Bb|obs 2 10241 kpc

ld

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sphaleron=twisted monopole-antimonopole

Example: chiral effectsIn a magnetized chiral plasma

JB =e2

22(µL µR)B

Vilenkin, 1980…

Cosmological-Chiral-MHD:

@Bc = rc (vc Bc) +1

cr2

cBc +eµ2

c

42crc (rc vc) +

e2µc

22crc Bc

chiral-vorticity chiral-magnetic term

Chiral anomaly equation:dµc

d= c↵@hc Fµc

anomaly term Higgs (mass) term

Joyce & Shaposhnikov, 1997Boyarsky, Fröhlich & Ruchaiskiy, 2012

Tashiro, TV & Vilenkin, 2012Rogachevskii, Ruchaiskiy, Boyarsky,

Fröhlich, Kleeorin, Brandenburg, Schober 2017Schober et al 2018

Kharzeev, 2013

advection diffusion

compare to Ohm’s law

conformal time

k

B-B+

Solution to the chiral-MHD equations

Joyce & Shaposhnikov, 1997Tashiro, TV & Vilenkin, 2012

|B±(, k)| = |B±0 | exp(K2

p/) exp(k Kp)

2/

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Exponential helicity growth:

but saturates

h ApKpe

2K2p/

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h µ(i)

↵

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Decompose magnetic field in left- and right- circular polarization modes.

Chiral-Electroweak Phase Transition?

So far we have been discussing chiral-MHD but we are interested inchiral effects at the electroweak phase transition. Our numerical analysis of

magnetic fields generated at the EWPT ignored fermions.

How can we include fermions in our simulations of the EWPT?Could the medium be chiral during the EWPT?What is a good description of the chiral electroweak plasma?Can there be undiscovered large CP violation?Can inclusion of SM chiral fermions suppress/amplify one

handedness of the magnetic field?

More questions…

Can axions play the role of a chiral asymmetry?Leptogenesis. (Baryogenesis from magnetic fields.)

µc ! a

Long, Sabancilar, TV

Fujita & KamadaSchober, Fujita & Dürrer

Conclusions• Several observations and lines of reasoning point to maximally

helical IGMF of ~10-14 G on ~10 Mpc scales and sub-nano-Gauss fields at kpc scales. (But still in need of a “WMAP moment”.)

• Electroweak phase transition can produce magnetic fields with sufficient energy density but large helicity does not seem to fit within the standard model.

• “Large helicity puzzle” requires new particle physics that may be tied to the cosmic baryon asymmetry, e.g. large CP violation, chiral magnetic effect, kinetic helicity, ???.