1. 宇宙磁场的起源
郭宗宽
中山大学宇宙学研讨班

2. 内容提纲
§宇宙磁场的探测
§大尺度磁场的观测限制
§天体物理过程对大尺度磁场的放大
§原初磁场的产生机制

3. § Large-scale magnetic fields
Evidences: Magnetic fields are ubiquitous in our Universe. They are detected in astronomical structures of different sizes, from stars up to galaxies, galaxy clusters and voids.
Methods: Faraday rotation measurements, synchrotron radio emission, the Zeeman splitting of radio spectral lines, …
𝐵 𝑔𝑎𝑙𝑎𝑥𝑦 ~ 𝐵 𝑐𝑙𝑢𝑠𝑡𝑒𝑟 ~ 10 −6 Gauss
𝐵 𝑣𝑜𝑖𝑑 ≥ 10 −16 Gauss

5. § Observational constraints
CMB (Planck Collaboration, arXiv: )
the energy momentum tensor
Faraday rotation
magnetically-induced bispectra
the breaking of statistical isotropy
BBN (B. Cheng et al, arXiv:astro-ph/ )
increasing the weak reaction rates
increasing the electron density
increasing the expansion rate of the universe
𝐵 1𝑀𝑝𝑐 <4.4× 10 −9 Gauss
𝐵 𝐵𝐵𝑁 < 10 −6 Gauss

6. Intergalactic (voids) magnetic fields
𝐵≥ 10 −16 Gauss
EW phase transition
QCD phase transition
Recombination epoch
Inflationary magnetogenesis

7. § Astrophysical processes
dynamo mechanism (Y.B. Zeldovich et al, 1980)
compression mechanism
tiny seed magnetic fields ≳ 10 −13 G
galactic dynamo
galactic magnetic fields ~1𝜇 G

8. § Cosmological phase transitions
EW phase transition [C.J. Hogan, Phys. Rev. Lett. 51 (1983) 1488]
QCD phase transition [J.M. Quashnock et al, ApJ 344 (1989) L49]
(1) strength of magnetic fields, 𝐵
(2) correlation length, 𝜆 𝐵 ~ 𝐻 −1

9. § Inflationary magnetogenesis
Models
problems
(1) strong coupling problem
(2) back reaction problem
(3) curvature perturbation problem
breaking of conformal invariance
amplified fluctuations

10. ℒ 𝐸𝑀 =− 1 4 𝐹 𝜇𝜈 𝐹 𝜇𝜈 −𝑏𝑅 𝐴 𝜇 𝐴 𝜇 −𝑐 𝑅 𝜇𝜈 𝐴 𝜇 𝐴 𝜈
𝑈(1)
ℒ 𝐸𝑀 =− 1 4 𝐹 𝜇𝜈 𝐹 𝜇𝜈 −𝑏𝑅 𝐹 𝜇𝜈 𝐹 𝜇𝜈 −𝑐 𝑅 𝜇𝜈 𝐹 𝜇𝜅 𝐹 𝜅 𝜈 −𝑑 𝑅 𝜇𝜈𝜆𝜅 𝐹 𝜇𝜈 𝐹 𝜆𝜅
~ 10 −40 G
ℒ 𝐸𝑀 =− 1 4 𝐹 𝜇𝜈 𝐹 𝜇𝜈 − 𝐷 𝜇 𝜙 ( 𝐷 𝜇 𝜙) ∗
ℒ 𝐸𝑀 =− 1 4 𝐹 𝜇𝜈 𝐹 𝜇𝜈 − 1 2 𝜕 𝜇 𝜃 𝜕 𝜇 𝜃+ 𝑔 𝑎 𝜃 𝐹 𝜇𝜈 𝐹 𝜇𝜈
Non-Gaussianity

11. ℒ 𝐸𝑀 =− 1 4 𝑒 𝛼𝜙 𝐹 𝜇𝜈 𝐹 𝜇𝜈

12. ℒ 𝐸𝑀 =− 1 4 𝐼 2 (𝜙) 𝐹 𝜇𝜈 𝐹 𝜇𝜈
❶ strong coupling problem, ❷ back reaction problem, ❸ curvature perturbation problem.
𝐴 𝜇 → 𝑔𝐴 𝜇
ℒ=− 1 4 𝐹 𝜇𝜈 𝐹 𝜇𝜈 +𝑖 𝜓 𝛾 𝜇 ( 𝜕 𝜇 +𝑖𝑔 𝐴 𝜇 )𝜓
ℒ=− 1 4 𝑔 2 𝐹 𝜇𝜈 𝐹 𝜇𝜈 +𝑖 𝜓 𝛾 𝜇 ( 𝜕 𝜇 +𝑖 𝐴 𝜇 )𝜓
(1) 𝐼>1 during inflation, 𝐼~1 at the end of inflation
(2) 𝜌 𝐸𝑀 < 𝐻 𝐼 2 during inflation
(3) 𝜁 𝐸𝑀 < 𝐴 𝑠

13. § A model of inflationary magnetogenesis
our action
the equation of motion
a spatially-flat FRW metric
𝑑 𝑠 2 = 𝑎 2 (𝜂)(−𝑑 𝜂 2 +𝑑 𝑥 2 )
Coulomb gauge: 𝐴 0 =0 and 𝜕 𝑖 𝐴 𝑖 =0
Fourier expansion

14. normalization condition
assuming
normalization condition
where 𝜒 2 = 𝑐 1 −4 12 𝑐 2 +3 𝑐 3 +2 𝑐 4 𝐻 𝐼 2
defining a new variable 𝑣 𝑘 =𝜒𝑓 𝐴 𝑘
𝜒 𝑣 𝑘 ′′ + 𝑘 2 − 𝑓′′ 𝑓 𝑣 𝑘 =0
assuming 𝑓 𝜂 = 𝑓 𝑒 𝑎 𝑎 𝑒 𝑛
❶ solution to the strong coupling problem,
requiring 𝑓 𝑒 ~1 and 𝑛<0

15. for short waves
for long waves
If −1/2<𝑛<0, the first term dominates. a strong blue spectrum
If 𝑛<−1/2, the second term dominates.
The power spectrum is
a scale invariant spectrum for 𝑛=−3
the entropy conservation,
for 𝐻 𝐼 ~ 10 −6 , 𝑎 0 / 𝑎 𝑒 ~ , 𝐵 𝜆 ~ 10 −6 𝐻 𝐼 ~ 10 −12 Gauss

16. The energy-momentum tensor is
The energy density is
where
The trace of i-j components is
where

17. The main contribution to the energy density and pressure comes from the power spectrum of the electric fields.
❷ solution to the back reaction problem, requiring 𝑄 1 = 𝑄 3 =0
the evolution of the curvature perturbation on super-Hubble scales
❸ The curvature perturbation problem is avoided if 𝑄 1 = 𝑄 3 =0.

18. 谢谢！