Introduction

Maxwell's Equations in an Expanding Universe

• Gauss's law: \[ \nabla \cdot \mathbf{E} = 4\pi \rho_e \]
• Ampère's law: \[ \nabla \times \mathbf{B} = \frac{1}{a} \left( 4\pi \mathbf{J} + \frac{\partial \mathbf{E}}{\partial t} \right) \]
• Faraday's law: \[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
• No magnetic monopoles: \[ \nabla \cdot \mathbf{B} = 0 \]

Generation of Primordial Magnetic Fields

1. Magnetogenesis During Inflation: During inflation, quantum fluctuations of the electromagnetic field can be stretched to cosmological scales. However, since the electromagnetic field is conformally invariant, no magnetic field is generated in the standard model. To overcome this, models of inflationary magnetogenesis break conformal invariance, often by coupling the electromagnetic field to a scalar field (such as the inflaton) via the interaction term: \[ S_{\text{EM}} = - \frac{1}{4} \int d^4 x \sqrt{-g} \, I(\phi) F_{\mu \nu} F^{\mu \nu} \] Where:
   • \( I(\phi) \) is a coupling function dependent on the inflaton field \( \phi \).
   • \( F_{\mu \nu} \) is the electromagnetic field tensor.
   By choosing a time-varying coupling \( I(\phi) \), the equation of motion for the electromagnetic field becomes: \[ \frac{d}{dt} \left( a^2 I(\phi) \mathbf{E} \right) = -\frac{\nabla \times \mathbf{B}}{a^2} \] This breaking of conformal symmetry leads to amplification of magnetic fields during inflation, with the strength depending on the coupling function \( I(\phi) \) and the inflationary dynamics.
2. Magnetogenesis from Cosmological Phase Transitions: Phase transitions in the early universe, such as the electroweak (EW) or quantum chromodynamics (QCD) phase transitions, can also generate magnetic fields. These transitions often involve the formation of topological defects (like cosmic strings or domain walls), which can seed magnetic fields.

   During a phase transition, magnetic helicity and turbulence can play crucial roles in amplifying fields. The evolution of the magnetic field in a turbulent plasma is governed by the magnetohydrodynamics (MHD) equations:

   \[ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left( \mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B} \right) \] Where:
   • \( \mathbf{v} \) is the velocity field of the plasma,
   • \( \eta \) is the magnetic diffusivity.
   In the absence of significant diffusion (\( \eta \rightarrow 0 \)), the magnetic field lines are "frozen" into the plasma, and magnetic energy can be amplified by plasma motions. The strength of the generated fields depends on the temperature and energy scale of the phase transition.
3. Magnetogenesis Through Baryogenesis: Magnetic fields can also arise during baryogenesis, particularly in scenarios where baryon asymmetry is generated through chiral processes. In such models, the chiral anomaly introduces an additional source of magnetic field amplification.

   The anomaly equation for the chiral current \( J_5^\mu \) is:

   \[ \partial_\mu J_5^\mu = \frac{e^2}{16\pi^2} F_{\mu \nu} \tilde{F}^{\mu \nu} \] Where \( \tilde{F}^{\mu \nu} \) is the dual of the electromagnetic field tensor. This anomaly can induce a current \( \mathbf{J}_5 \), which modifies the MHD equations to include a chiral magnetic term: \[ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left( \mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B} + \mathbf{J}_5 \right) \] This can amplify magnetic fields in the presence of a non-zero chiral asymmetry. Such processes can generate fields during baryogenesis, typically at energy scales close to the electroweak scale.

Evolution of Magnetic Fields

Magnetic Field Scaling:

Magnetic Field Constraints

• \( A_B \) is the amplitude of the magnetic field power spectrum,
• \( n \) is the spectral index.

Observation of magnetic fields

Magnetic Fields in Galaxies

1. Synchrotron Radiation: Magnetic fields cause relativistic charged particles (such as electrons) to spiral along magnetic field lines, emitting synchrotron radiation. This radiation is observed in the radio spectrum and is a direct indicator of magnetic fields. In galaxies, magnetic fields have typical strengths of about \( 1 - 10 \, \mu G \).
   • Milky Way: In our galaxy, the magnetic field is structured with both a regular (coherent) and a random (turbulent) component. The regular field follows the spiral arms, while the random component is associated with turbulent processes in the interstellar medium (ISM).
   • External Galaxies: Spiral galaxies like M51 (the Whirlpool Galaxy) show magnetic fields aligned with the spiral arms, detected through synchrotron radiation. This suggests that magnetic fields are involved in galactic structure formation and evolution.
2. Faraday Rotation: When polarized light from distant radio sources passes through a magnetized plasma, the plane of polarization rotates. This effect, known as Faraday rotation, is proportional to the magnetic field strength and the electron density along the line of sight. The rotation measure (RM) is given by: \[ RM \propto \int n_e \mathbf{B} \cdot d\mathbf{l} \] Where:
   • \( n_e \) is the electron density,
   • \( \mathbf{B} \) is the magnetic field along the line of sight,
   • \( d\mathbf{l} \) is the differential element along the line of sight.
   Faraday rotation measurements provide information about both the strength and direction of magnetic fields in galaxies and the intergalactic medium.

Magnetic Fields in Galaxy Clusters

1. Radio Halos and Relics: Galaxy clusters exhibit diffuse radio emission known as radio halos and radio relics, believed to be caused by synchrotron radiation from relativistic electrons interacting with cluster magnetic fields. These large-scale structures can be as large as \( 1 \, \text{Mpc} \) across and indicate the presence of magnetic fields over vast regions.
   • Radio Halos: They are centrally located and are thought to arise from turbulence in the ICM, which reaccelerates relativistic particles. The magnetic fields inferred from radio halos are typically \( 1 - 5 \, \mu G \).
   • Radio Relics: These are found at the periphery of clusters and are associated with shock waves caused by cluster mergers. Magnetic fields in radio relics are estimated to be \( 0.1 - 1 \, \mu G \).
2. X-Ray Observations of Galaxy Clusters: Galaxy clusters emit X-rays from hot, ionized gas in the ICM. The presence of magnetic fields can influence the thermal pressure and structure of the gas. Observations of the Sunyaev-Zel’dovich (SZ) effect in clusters (interaction between CMB photons and hot gas) can also provide constraints on the magnetic field strength in these environments.

Magnetic Fields in the Intergalactic Medium (IGM)

• Upper Limits on IGM Fields: Using techniques like blazar observations, upper limits have been placed on magnetic fields in the IGM. Blazars are active galactic nuclei that emit high-energy gamma rays. If these gamma rays interact with the extragalactic background light, they produce electron-positron pairs, which subsequently emit secondary gamma rays through inverse Compton scattering. The presence of an IGM magnetic field can deflect the electron-positron pairs, suppressing the secondary gamma ray emission. By comparing the observed gamma ray flux from blazars, constraints on IGM magnetic fields have been derived. These observations have placed upper limits on the magnetic field strength in the IGM, typically around: \[ B_{\text{IGM}} \lesssim 10^{-16} \, \text{G} \] This suggests that, if primordial magnetic fields were generated, they have been significantly diluted by the expansion of the universe or are very weak on large scales.
• Faraday Rotation in the IGM: Faraday rotation measurements through the IGM have also provided constraints on the presence of magnetic fields. However, since the electron density in the IGM is low, the Faraday rotation signal is weak, and stronger constraints are challenging to obtain.

Magnetic Fields in the Cosmic Microwave Background (CMB)

• CMB Polarization: Primordial magnetic fields can affect the polarization of the CMB through Faraday rotation of the CMB photons as they travel through a magnetized plasma. This rotation would modify the angular power spectrum of the CMB’s E-mode and B-mode polarization, potentially providing evidence of magnetic fields at early times.
• Scalar and Vector Modes: Magnetic fields generate both scalar and vector perturbations in the early universe, leading to anisotropies in the CMB. These perturbations can affect the temperature and polarization power spectra, allowing for constraints on the strength and structure of primordial magnetic fields.

Large-Scale Structure and Magnetic Fields

• Cosmic Filaments: Magnetic fields are believed to exist in the filaments connecting galaxy clusters. These fields can be detected through synchrotron emission or Faraday rotation.
• Structure Formation: Magnetic fields may also play a role in the collapse of gas into galaxies and galaxy clusters. During the gravitational collapse, magnetic fields can be amplified through magnetic dynamo mechanisms.

1. Magnetic fields in galaxies: Strong and well-ordered fields are seen in galaxies, with evidence for both primordial and dynamo amplification processes.
2. Magnetic fields in galaxy clusters: Diffuse radio halos and relics in clusters indicate large-scale magnetic fields, possibly enhanced by cluster mergers and turbulence.
3. IGM and CMB constraints: Upper limits on magnetic fields in the intergalactic medium and the CMB place stringent constraints on the strength of primordial magnetic fields, providing insight into the possible mechanisms of magnetogenesis.

Measurement of Magnetic Fields

1. Faraday Rotation: Faraday Rotation is one of the most common methods for measuring magnetic fields in space. It occurs when polarized electromagnetic radiation (usually from radio sources such as quasars) passes through a magnetized plasma. The magnetic field causes the plane of polarization to rotate, and this rotation is proportional to the strength of the magnetic field along the line of sight, as well as the electron density. The Rotation Measure (RM) quantifies the amount of this rotation and is given by: \[ RM = 0.81 \int_0^L n_e(z) B_{\parallel}(z) \, dz \] Where:
   • \( RM \) is the rotation measure in units of \(\text{rad}/\text{m}^2\)
   • \( n_e(z) \) is the electron density in \(\text{cm}^3\)
   • \( B_{\parallel}(z) \) is the magnetic field component along the line of sight in \(\mu\) G
   • \( z \) is the position along the line of sight in pc (parsecs), and \( L \) is the distance to the source.
   The angle of polarization rotation \( \Delta \theta \) is related to the RM and the wavelength \( \lambda \) of the radiation by the equation: \[ \Delta \theta = RM \cdot \lambda^2 \] Thus, the rotation angle \( \Delta \theta \) depends on the wavelength of the radiation, and measuring \( \Delta \theta \) at different wavelengths allows astronomers to calculate the RM. Once the RM is known, the magnetic field strength along the line of sight can be determined, provided the electron density \( n_e \) is estimated from other observations (e.g., X-ray observations of the hot gas in galaxy clusters).

   Data Source: Radio telescopes like the Very Large Array (VLA), LOFAR, and ASKAP are used for Faraday Rotation studies. RM grids are constructed using observations of polarized radio emissions from extragalactic sources.

   Key points:

   • Faraday Rotation is sensitive to the magnetic field component along the line of sight \( B_{\parallel} \).
   • The RM integrates the product of the magnetic field and the electron density along the line of sight.
2. Synchrotron Radiation: Synchrotron radiation is produced when relativistic electrons (electrons moving at nearly the speed of light) spiral around magnetic field lines. The radiation is emitted in the radio spectrum and is polarized, making it a useful tool for measuring both the strength and structure of magnetic fields. The total synchrotron power emitted by a relativistic electron spiraling in a magnetic field is given by: \[ P_{\text{synch}} \propto N_e B_{\perp}^{1.5} \nu^{-0.5} \] Where:
   • \( P_{\text{synch}} \) is the synchrotron power.
   • \( N_e \) is the number density of relativistic electrons.
   • \( B_{\perp} \) is the magnetic field component perpendicular to the line of sight.
   • \( \nu \) is the frequency of the emitted radiation.
   The synchrotron power depends on both the electron density and the magnetic field strength. By observing the radio flux at different frequencies, one can estimate the magnetic field strength, assuming a model for the electron population.

   The synchrotron emission spectrum follows a power law:

   \[ I(\nu) \propto \nu^{-\alpha} \] Where \( \alpha \) is the spectral index, typically in the range \( 0.5 - 1 \) for cosmic sources. From the synchrotron spectrum, we can derive the spectral index, which gives insights into the energy distribution of relativistic electrons and the magnetic field strength.

   Polarized Synchrotron Emission:

   In addition to the total synchrotron power, synchrotron emission is partially polarized, and the degree of polarization can give information about the ordering of the magnetic field: \[ \Pi = \frac{p}{p+1} \] Where:
   • \( \Pi \) is the fractional polarization of the synchrotron radiation.
   • \( p = 2\alpha \) is the spectral index of the electron energy distribution.

   Highly polarized synchrotron emission indicates a well-ordered magnetic field, whereas less polarized emission suggests a more turbulent or random field.

   Data source:

   • Radio telescopes detect synchrotron emission in the radio frequency range (e.g., VLA, GMRT, SKA). Observations can also be made at high frequencies, such as millimeter wavelengths (e.g., with ALMA).
   • Fermi Gamma-ray Space Telescope can observe synchrotron radiation from high-energy electrons.

   Key points:

   • Synchrotron Radiation provides information about the magnetic field component perpendicular to the line of sight \( B_{\perp} \).
   • The intensity of the emission depends on both the magnetic field strength and the density of relativistic electrons.
3. Zeeman Effect: The Zeeman effect refers to the splitting of spectral lines in the presence of a magnetic field. This effect can be used to directly measure the strength of magnetic fields, particularly in molecular clouds and star-forming regions.

   The energy shift due to the Zeeman effect is proportional to the magnetic field strength:

   \[ \Delta E = g \mu_B B \] Where:
   • \( \Delta E \) is the energy splitting of the spectral lines.
   • \( g \) is the Landé g-factor.
   • \( \mu_B \) is the Bohr magneton.
   • \( B \) is the magnetic field strength.
   In terms of frequency shift, the splitting of a spectral line is given by: \[ \Delta \nu = \frac{g \mu_B B}{h} \]

   Where \( h \) is Planck’s constant. By measuring the frequency shift \( \Delta \nu \) in the spectral line, the magnetic field strength \( B \) can be directly determined.

   The Zeeman effect is particularly useful for measuring line-of-sight magnetic fields in dense regions, such as molecular clouds where strong magnetic fields are present.

   Data source: Radio and submillimeter telescopes like Arecibo, Effelsberg, and ALMA, which observe specific spectral lines like the 21-cm line of neutral hydrogen or OH lines in molecular clouds.

   Key points:

   • Zeeman Effect is sensitive to the magnetic field along the line of sight.
   • It provides a direct measurement of the magnetic field strength in regions where the effect is strong enough to be detected.
4. CMB Polarization (Faraday Rotation in the CMB): Magnetic fields can also be probed through their effects on the Cosmic Microwave Background (CMB) polarization. As CMB photons pass through magnetized regions, their polarization plane can be rotated due to Faraday rotation, similar to the effect on radio waves. The Faraday rotation angle for CMB photons is given by: \[ \Delta \theta_{\text{CMB}} = \frac{e^3}{2 \pi m_e^2} \int n_e(z) B_{\parallel}(z) \frac{\lambda^2}{a^2(z)} dz \] Where:
   • \( e \) is the electron charge.
   • \( m_e \) is the electron mass.
   • \( a(z) \) is the scale factor at redshift \( z \).
   This equation shows that the rotation angle depends on the magnetic field strength along the line of sight, the electron density, and the wavelength of the CMB photons. Measuring this rotation requires extremely sensitive polarization measurements of the CMB, which can provide constraints on the strength of primordial magnetic fields.

   Data Source: CMB experiments such as Planck and BICEP2 aim to detect B-mode polarization, which could be related to primordial magnetic fields.

5. Polarized Dust Emission: In the interstellar medium (ISM), elongated dust grains tend to align with magnetic field lines. As these grains emit thermal radiation, the radiation is polarized in a direction aligned with the magnetic field. By mapping polarized dust emission, we can infer the orientation of magnetic fields in star-forming regions, galaxies, and the Milky Way.

   Data Source:

   • Planck satellite measured polarized dust emission across the sky and provided an all-sky map of the magnetic field in the Milky Way.
   • Ground-based submillimeter observatories (e.g., the Submillimeter Array) also contribute to the study of polarized dust emission.

6. Magnetic Fields from Large-Scale Structure and Turbulence: In galaxy clusters and filaments, magnetic fields are measured through their interactions with turbulent plasma. Magnetohydrodynamics (MHD) equations govern the behavior of such magnetized plasmas: \[ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B}) \] Where:
   • \( \mathbf{v} \) is the velocity of the plasma.
   • \( \eta \) is the magnetic diffusivity.
   In the limit of low diffusivity (\( \eta \approx 0 \)), magnetic field lines are "frozen" into the plasma, and turbulent motions can amplify the magnetic field. Observing turbulence in galaxy clusters and filaments can provide insights into how magnetic fields evolve and amplify on large scales.

Summary of Magnetic Field Measurement Methods:

References

• How the universe got its magnetic field
• Baryon-Dark matter interaction in presence of magnetic fields in light of EDGES signal, Jitesh R Bhatt, Pravin Kumar Natwariya, Aleka C. Nayak, Arun Kumar Pandey [arXiv:1905.13486 [astro-ph.CO] (2019)], Eur. Phys. J. C 80, 334 (2020)
• Implications of baryon-dark matter interaction on IGM temperature and tSZ effect with magnetic field, Arun Kumar Pandey, Sunil Malik, T. R. Seshadri, arXiv:2006.07901 [astro-ph.CO] (2020), Mon.Not.Roy.Astron.Soc. 500 (2020)
• Thermal SZ effect in a magnetized IGM dominated by interacting DM decay/annihilation during dark ages, Arun Kumar Pandey, Sunil Malik (2022) [arXiv:2204.08088]