keitaro takahashi (nagoya university, japan) 1. introduction 2. electromagnetic properties of the...
TRANSCRIPT
![Page 1: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/1.jpg)
Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation
Magnetic Fieldsfrom
Cosmological Perturbations
![Page 2: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/2.jpg)
1. Introduction
![Page 3: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/3.jpg)
primordial fluctuations
observation ・ CMB ・ galaxy distributiontheory ・ inflation (initial condition) ・ cosmological perturbation theory (linear)
![Page 4: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/4.jpg)
magnetogenesismagnetogenesis fromcosmological perturbationsbefore recombination Hogan (2000) Berezhiani & Dolgov (2004) Matarrese et al. (2005) Gopal & Sethi (2005) KT et al. (2005, 2006, 2007, 2008) Siegel & Fry (2006) Hollenstein et al. (2008) Maeda et al. (2009)
based on ・ cosmological perturbation theory (nonlinear) ・ observational facts ・ no physical assumption
KT et al., Science 311 (2006) 827
![Page 5: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/5.jpg)
basic idea
photons→ CMB
protons
Thomsonscattering
electrons
Coulombinteraction
baryon
Thomson scattering→ deviation in motion due to mass difference→ net electric charge density and electric current→ magnetic fields
![Page 6: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/6.jpg)
extensions to the conventional formalism
What do we need for magnetogenesis?
electric field ・ Conventionally, baryons ・ Separate treatment of p and e is necessary.rotational part (Roy’s talk) ・ No rotational part at the linear order ・ generated by nonlinear effect Linear order is sufficient for CMB but insufficient for B.
electric field and its rotation
Two extensions are needed for magnetogenesis.
![Page 7: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/7.jpg)
this talk
understanding the physics of magnetogenesisfrom cosmological perturbations ● electromagnetic properties of the early universe ・ solve Maxwell and Ohm ・ Newtonian ・ neglect anisotropic stress ● tight coupling approximation ・ express B by familiar quantities (δγ) ・ compare with another approach
![Page 8: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/8.jpg)
2. Electromagnetic properties of the early universe
KT, Ichiki & Sugiyama, PRD 77 (2008) 124028
![Page 9: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/9.jpg)
EOMs(Newtonian) EOMs for photons, protons & electrons,neglecting the anisotropic stress of photons.
Thomson
Coulomb Thomson
![Page 10: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/10.jpg)
simplifying EOMsrelative and center-of-mass quantities
cosmological perturbations up to 2nd order
→ Hall term and Lorentz force are higher order.
![Page 11: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/11.jpg)
rewriting EOMs
p-e relative motion, γ, γ-baryon relative motion
generalized Ohm’s law
![Page 12: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/12.jpg)
conventional CMB
Conventionally, we deal with photons and baryons.
![Page 13: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/13.jpg)
Solving Maxwell + Ohmbasic equations ・ EOM of photons ・ EOM of relative motion between photons and baryons ・ generalized Ohm’s law ・ Maxwell equationshow to solve ・ up to 2nd order in cosmological perturbations ・ regard Thomson term as an external source → Electric charge, current and EM fields are expressed as functions of Thomson term.
solve in this section
![Page 14: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/14.jpg)
basic equations
Thomson term (source) effective resistivity
EOM of Vpe↓
Ohm’s law
![Page 15: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/15.jpg)
first order in CP
B = 0 at the 1st order in CP.
First, take the divergence of the Ohm’s law.
![Page 16: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/16.jpg)
charge densitydivergence of the generalized Ohm’s law
damped oscillationwith a source
In cosmological timescale, plasma oscillation damps.The equilibrium is nonzero due to the source.
![Page 17: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/17.jpg)
solutions for the 1st order in CP
Electric charge,current and E fieldare expressed bythe Thomson term.
![Page 18: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/18.jpg)
second order in CP
B field joins at the second order.
![Page 19: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/19.jpg)
solutions up to second order in CP
zero at the 1st order
![Page 20: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/20.jpg)
interpretation
・ electric current term is not important in Ohm’s law → photon pressure balances with E field・ current → (displacement current) + (B field)・ E and charge vanish when Thomson term disappear. B and current do not because they are integral.
electronproton
E fieldphotonpressure
![Page 21: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/21.jpg)
summary of section 2
solving Maxwell + Ohm・ up to second order in CP・ Thomson term was treated as an external source → Electromagnetic quantities are expressed by the Thomson term.・ We need the Thomson term (photon-baryon relative velocity) to evaluate B.
![Page 22: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/22.jpg)
3. Magnetogenesis & tight coupling approximation
![Page 23: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/23.jpg)
magnetogenesis
B field and Thomson term
vector product ofdensity gradient of photonsand velocity difference
vorticitydifference
We solve for δv by tight coupling approximation.(Peebles & Yu, 1970; Kobayashi, Maartens, Shiromizu & KT, 2007)
![Page 24: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/24.jpg)
EOM for γ-b relative motion
Electric current has a negligible effect onthe dynamics of the γ-b relative motion.
tight coupling approximation (TCA)
Time derivativeis less important.
![Page 25: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/25.jpg)
TCA I
(I,1) → TCA I & CP 1, (I,2) → TCA I & CP 2
However, the source of B is zero at TCA I.
We must go on to TCA II.
![Page 26: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/26.jpg)
TCA II
This is not parallel to ∇ρ in general.
B is generated at TCA II.
![Page 27: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/27.jpg)
results
All EM quantitiesare expressed byfamiliar quantities.
Evaluation of B spectrum is now in progress,but the anisotropic stress must be included to be complete.
![Page 28: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/28.jpg)
another approach
Ichiki, KT et al. (2006)・ derive the source term (relativistic)
・ numerically calculate the spectrum contributed from (1st order) × (1st order), neglecting the vorticity (purely 2nd order)
![Page 29: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/29.jpg)
spectrum
-44
-40
-36
-32
-28
-24
-20
1Gpc 1Mpc 1kpc 1pcscale
com
ovin
g B
(lo
g B
(G
))
![Page 30: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/30.jpg)
toward the complete evaluation
numerical approach ・ numerically cancel TCA I ・ include vorticity (purely 2nd order) ・ confirm the validity of TCA
TCA approach ・ TCA I already canceled by hand ・ include anisotropic stress
These two approaches are necessaryto cross check and evaluate the spectrum.
![Page 31: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/31.jpg)
3 components
proton electric current
electron baryon photon - baryon
photon center of mass
timescales
BecauseHτ << 1,we will usetight couplingapproximation.
![Page 32: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/32.jpg)
If (scattering time) << (dynamical timescale)
tight coupling expansion is good. (Peebles & Yu, 1970)
tight coupling approximation (TCA)We need the velocity difference.
TCA I
TCA II
![Page 33: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/33.jpg)
EOM of photon-baryon relative motion
contribution from electric current
Electric current is negligible.Protons and electrons can be treated as one fluid.
![Page 34: Keitaro Takahashi (Nagoya University, Japan) 1. Introduction 2. Electromagnetic properties of the early universe 3. Magnetogenesis & tight coupling approximation](https://reader035.vdocuments.us/reader035/viewer/2022070306/5516e3ae550346f5558b463a/html5/thumbnails/34.jpg)
order estimation
deviation between photons and baryons
deviation between protons and electrons