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)
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
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
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)
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
Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
RB & Jedamzik 2004
turbulent decay: numerical simulations
Evolution of Magnetic Fields
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
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)
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
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
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
Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
Non-Helical Inverse Cascade?
Reppin et al., in prep.
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.
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?
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
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
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
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
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
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
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)
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
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
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
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
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”
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”
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”
Robi Banerjee, Cosmological Magnetic Fields, Nordita Stockholm, June. 24th, 2015
“First Stars”: subsequent evolution
Seifried, RB, Schleicher, 2014
evolution during PI-SN explosion
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
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
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