evolution and constraints of primordial magnetic...
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Evolution and Constraintsof
Primordial Magnetic Fields
Robi BanerjeeUniversity of Hamburg
Collaborators: Jacques Wagstaff (UHH), Dominik Schleicher (Göttingen), Johannes Reppin (UHH), Günter Sigl (UHH), S. Sur (Bangalore), Jennifer Schober (Nordita), C. Federrath (Canberra, Australia), Karsten Jedamzik (Montpellier)
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Neronov & Vovk, Science 2010
• Observations:• Galactic fields ~ 10 µG
(e.g. Beck 1999)
• Cluster fields ~ µG (e.g. Bonafede et al. 2010)
• Upper limits:• BBN ~ 10-7 G
(Grasso & Rubinstein 2001)
• CMBR ~ 10-9 G• Reionization ~ 10-9 G
(Schleicher et al. 2008)
• Lower limits ~ 10-15 G (?) (FERMI obs. e.g. Neronov & Vovk 2010; Taveccio et al. 2010)
Cosmological Magnetic Fields
possible scenarios for PMF:• inflation• causal processes (e.g. PT) ⟹ λc < lH(T) + blue spectra
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Subsequent evolution
• dilution by cosmic expansion:
B ∝ a-2
assumption: flux freezing (no dynamic damping/ amplification)
• but: damping/amplification is important (e.g. Kunze’s talk, Jedamzik et al. 98, Subramanian & Barrow 98, RB & Jedamzik 2003/04, Schleicher et al. 2010, Sur et al. 2010, Jedamzik & Sigl 2011, Seifried et al. 2014)
Evolution of Magnetic Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Evolution described by • incompressible MHD: v, vA ≪ cs
vA < cs if B < 5×10−5 G at recombination
dissipation: Reynolds number:
Evolution of Magnetic Fields
MHD eq. can used in the exp. Universe(Axel’s Talk)
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Re ≫ 1 ⟹ decay via MHD turbulence
k
Ek
cascade direction
1/Ldiss1/LInt
quasi-stationary transfer of energy in k-space⟹ Kolmogorov Turbulence
Evolution of Magnetic Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
RB & Jedamzik 2004
turbulent decay: numerical simulations
Evolution of Magnetic Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
turbulent decay laws
• Initial spectrum on large scales (l > L):
• with: τl = l/vl = l/√El = t:
energy decay:
increase of coherence length:
Evolution of Magnetic Fields
⟹ n-dependent decay law
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
• decay law:
• growths of coherence length:
Evolution of Magnetic Fieldsturbulent decay laws
⟹ for causally generated fields (n > 5, Durrer & Caprini 2003)
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Helical Fields
• Helicity (measures complexity of the field):
is conserved (no resistivity)
• maximal helical field: H ∼ B2 L ≈ E L
energy decay:
inverse cascade:
Helicity
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
• decay law:
• growths of coherence length: inverse cascade
• Fields with maximum helicity:
Evolution of Magnetic Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Evolution of small scale random magnetic fields
no initial helicity with max. initial helicity
Evolution of Magnetic Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Non-Helical Inverse Cascade?
Reppin et al., in prep.
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Non-Helical Inverse Cascade?
ν = 1×10−4
ν = 5×10−4
Reppin et al., in prep.
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Non-Helical Inverse Cascade?
Reppin et al., in prep.
⟹ viscosity/Re dependent inverse cascade?
⟹ applicable to the Early Universe?
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Viscous regime (Re < 1):
for lmfp ≪ L for lmfp ≫ L
⟹
⟹ overdamped modes for t < τvisc (Jedamzik et al. 1998, RB & Jedamzik 2004)
⟹ B(t) ≈ const
η
Evolution of Magnetic Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
apply to cosmic evolution⟹ evolution equation:
turbulent regime (Re ≫1):
viscous regime (Re < 1):
for lmfp ≪ L for lmfp ≫ L
Evolution of Primordial Magnetic Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
combine with cosmic evolution
assume magneto-genesis at EW-PT (Tgen = 100 GeV)
RB & Jedamzik 2004
h = hmax; n = 3h = 10-3 hmax; n = 3h = 0; n = 3h = 0; n = 3
Gcoherence length field strength
time →
Evolution of Primordial Magnetic Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
EGMF ?
-12 -10 -8 -6 -4
-14
-12
-10
-8
-6
-4
LogHlB êMpcL
LogHB
lêGL
non-helicalmaximally helical
EGMF - 10 -15lB -1ê 2
EGMF - 10 -18lB -1ê 21ê a
HQCD
1ê aH
EW
bê aH
EW
f* >5.6â10 -10
f* min>5.6â10 -14
recombinatio
n
Blmax
Ha LÊ
HbLÊ
HcLÊ
Hd LÊ
Ha LÊ
HbLÊ
HcLÊ
Hd LÊ
Wagstaff & RB, arXiv:1409.4223
• bubble size: β ≈ 10−2 lH
• EW bariogenesis ⟹ f* ≲ 10−24
(T.Vachaspati 2001)
⟹ EGMF cannot be explained by EW-PT magnetogenesis
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Brandenburg & Subramanian 2005
B-Field AmplificationSmall-scale dynamo (Batchelor 1950, Kazantsev 1968, see also Brandenburg & Subramanian 2005, Schober et al. 2012a,b)
• exponential growth of weak seed fields
• growth rate depends on Reynolds number (Pm ≫ 1):
Γ ∝ Re1/2
• mag. spectrum: Emag,k ∝ k3/2
• saturation at Emag ~ 0.1 Ekin
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Evolution of Primordial Magnetic FieldsSmall scale dynamo (SSD) during the radiation dominated era⟹ turbulence from primordial density perturbation
⟹ SSD efficient even during the ‘smooth’ Universe phase
⟹ B0 ~ 10−15 ε1/2 G @ λc ~ 0.1 pc
PT: phase transition (e.g. Kosowsky et al. 2002), PDP primordial density perturbations
Wagstaff et al., PRD 2014
Γ
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Small scale dynamo
Federrath et al., PRL, 2011
• SSD saturation level:
⟹ strong dependence on the Ma and turbulent mode (compressive vs. rotational)
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
(Jedamzik et al. 1998, Subramanian & Barrow 1998)
(Sethi & Subramanian 2005)
(Subramanian & Barrow 1998)• Magnetic Jeans mass:
• Ambipolar diffusion heating:
• Smallest scale:
Effects of Primordial Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Schleicher et al. (2009)
• Thermal / magnetic Jeans masses: Critical mass scale for gravity to overcome thermal / magnetic pressure note: MJB > MJT for B ≳ 0.1 nG • Both are significantly increased in the presence of strong magnetic fields.
Effects of Primordial Fields
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Modification of primordial star formation⟹ constraints from the optical depth + zreion ≤ 6
Constraints of Primordial Fields
Reionisation by Pop. III stars Reionisation by Pop. III & Pop. II starsSchleicher, RB & Klessen (2008)
used τ = 0.087 +/- 0.017 (WMAP 5: Komatsu et al. 2008)
PLANCK 2015:τ = 0.066 +/- 0.016
⟹ strong correlation of SFE (f*) and B0
⟹ for f* < 0.01 ⟹ B0 ≲ 1 nG
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
B ∝ ρ2/3
mean(B)
Dolag et al. 2005
B-fields during compression
•maximum growth by adiabatic compression: B ∝ ρ2/3
• small-scale dynamo works in cluster forming models (e.g. Dolag et al. 1999, 2000; Xu et al. 2009, 2010)
• depends on numerical resolution ⟹ Re(Ngrid)
B-Field Amplification
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
• turbulent infall motions (e.g. Abel et al. 2002, Greif et al. 2008)
• baryonic core modelled on a supercritical hydrostatic sphere:• Mbaryon = 1500 Msol• ρ0 = 5x10-20 g cm-3
• weak random field: B = 1nG, β = 1010
• transonic turbulence: vrms = 1.1 km sec-1
Federrath, Sur, Schleicher, RB, Klessen,2011
characteristic length: Jeans length:
Dynamo during “First Star Formation”
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
characteristic length: Jeans length:
• turbulent infall motions (e.g. Abel et al. 2002, Greif et al. 2008)
• baryonic core modelled on a supercritical hydrostatic sphere:• Mbaryon = 1500 Msol• ρ0 = 5x10-20 g cm-3
• weak random field: B = 1nG, β = 1010
• transonic turbulence: vrms = 1.1 km sec-1
Dynamo during “First Star Formation”
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
• growth rate depends on Rm, i.e. resolution:
Rm ∝ NJ4/3 (e.g. Haugen et al. 2004)
NJ: number of grid cells per local Jeans length; realization with adaptive mesh refinement (AMR)
• minimum resolution: ~ 30 grid cells per Jeans length
⟹ num. simulations cannot capture the full kinetic phase
Sur et al. 2011
Dynamo during “First Star Formation”
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
“First Stars”: subsequent evolution
Seifried, RB, Schleicher, 2014
evolution during PI-SN explosion
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
“First Stars”: subsequent evolutionevolution during PI-SN explosion
~ 4.4 Myr
⟹ strong SNe: efficient to magnetise the IGM ?
Seifried, RB, Schleicher, 2014
spectra autocorrelation
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
• Primordial Magnetic Fields undergo strong dynamic evolution (not only B ∝ a−2)
• damping and amplification in the turbulent regime• helical fields ⟹ inverse cascade, larger λc, slower decay
• non-helical inverse cascade: we have to work on it!
• EW-PT generated fields cannot explain the observed extragalactic fields
• turbulent dynamo: efficient amplification of weak fields even during the RD era
• strong fields (~ nG) change cosmic evolution (i.e. star formation history, reionisation)
Summary
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Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Helicity from first order PT
0.2 0.4 0.6 0.8 1.0
10-10
10-9
10-8
10-7
10-6
10-5
Bubble wall velocity vb
Helicity
fraction
f *minâl I,*
l EW
aN =0.01
0.03
0.1
0.3
1
3
deflagrations detonations
hybridsv b=c s
Wagstaff & RB, arXiv:1409.4223