Evolution and Constraints of Primordial Magnetic...

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