A Model for Cosmic Magnetic Field Amplification: Effects of Pressure Anisotropy Perturbations

Abstract

Magnetic field amplification in the early universe is a long-standing problem that has been extensively studied through theoretical and numerical approaches, focusing on turbulent dynamos and the growth of collisionless plasma instabilities. In the post-recombination era, pressure anisotropy drives plasma instabilities, and magnetic field amplification through these instabilities can be faster than that driven by turbulent dynamos. By considering the balance between isotropization through magnetic field amplification and anisotropy generation by external sources such as turbulence and anisotropic cosmic-ray scattering, it is reasonable to assume that the system evolves around an equilibrium anisotropy value. To improve the theoretical modeling of magnetic field amplification in such systems, this study specifically examines pressure anisotropy perturbations near the equilibrium anisotropy value, which may destabilize the system. By analyzing the effects of pressure anisotropy perturbations and their damping rates on the time evolution of cosmic magnetic fields, we highlight the importance of these perturbations in driving plasma instabilities and boosting cosmic magnetic field amplification.

1. Introduction

It has been shown that the large-scale structure of the universe could be magnetized, and the presence of magnetic fields in clusters of galaxies is particularly suggested by observations [1,2,3,4,5,6,7]. Magnetic fields play a significant role in particle acceleration and transport in the intergalactic and intracluster media [8,9,10,11,12]. Additionally, they have a significant impact on the anisotropies of the cosmic microwave background [13,14,15] as well as on the physics of Big Bang nucleosynthesis [16]. However, the origin and/or amplification of cosmic magnetic fields are not yet fully understood, and these remain long-standing problems.

Theoretically, weak seed magnetic fields can be generated in the early universe through several mechanisms, each of which plays a crucial role in the subsequent evolution of cosmic magnetic fields. These mechanisms include: (1) large-scale magnetohydrodynamic (MHD) processes, such as the Biermann battery effect, which generates magnetic fields from the gradients of pressure and density [17,18,19,20]; (2) currents induced by the electronic scattering of anisotropic radiation fields, a process that can create localized magnetic fields by transferring momentum from the radiation to charged particles [21,22]; (3) the evolution of turbulent fields, where the amplification of magnetic fields occurs through the dynamo action within the turbulent plasma, which could be particularly significant during the post-recombination era [23,24]; and (4) Weibel instabilities excited by shocks produced during structure formation, which can generate magnetic fields through the interaction of relativistic particles with the surrounding plasma [12,25,26]. These processes contribute to the generation of cosmic magnetic fields in the early universe. It has been shown that cosmic magnetic fields undergo damping from before the epoch of neutrino decoupling through to recombination, limiting their growth during this period [27]. This suggests that while the early universe experienced significant damping of magnetic fields, the amplification processes became more efficient after recombination, leading to the formation of the large-scale magnetic fields observed in galaxy clusters today.

Magnetic field amplification through turbulent dynamo has been extensively studied using both analytical and numerical methods [23,24,28,29,30,31,32]. The turbulent dynamo is a physical process in which the kinetic energy of turbulent motions within a conducting fluid or plasma is converted into magnetic energy. Turbulence stretches and folds the magnetic field lines, a process that leads to the amplification of magnetic field strength over time. This mechanism plays a crucial role in generating and enhancing magnetic fields in astrophysical environments, particularly in regions like galaxy clusters and the intergalactic medium. Here, and throughout this paper, we assume that intergalactic plasma is a highly conducting medium in the sense that its resistivity ($\eta$) is much smaller than its viscosity ($\mu$), i.e., the Prandtl number $Pm\equiv \mu /\eta \gg 1$. In this case, the small-scale dynamo theory predicts exponential growth of magnetic energy up to equipartition at the viscous scale. Specifically, magnetic field amplification through small-scale dynamo is enhanced during the radiation-dominated epoch due to turbulence generated by first-order phase transitions or primordial density perturbations [32]. In principle, magnetic field amplification through small-scale dynamo is efficient; however, it has been shown that pressure anisotropies generated by turbulence can drive plasma instabilities in locally unstable regions. These instabilities enhance effective collisionality, which could affect the exponential growth through turbulent dynamo [33]. In this regard, another mechanism may also contribute to the amplification of cosmic magnetic fields alongside small-scale dynamo.

As Grasso and Rubinstein [34] have highlighted, several processes, such as those occurring during cosmic inflation, phase transitions, and vorticity-driven processes, have been proposed as key contributors to the generation of primordial magnetic fields. In particular, magnetogenesis during inflation is often associated with the amplification of seed magnetic fields due to the rapid expansion of the universe, with quantum fluctuations in the electromagnetic field being stretched to cosmological scales. Similarly, phase transitions such as those occurring during the electroweak or QCD transitions can produce magnetic fields through the breaking of symmetries and the subsequent generation of currents. Vorticity-driven processes related to the motion of fluids or plasmas in the early universe can also lead to the generation of magnetic fields, as rotating fluids can create local currents that contribute to magnetic field generation. Within this broader framework, it has been highlighted that pressure-anisotropy-driven instabilities can be seen as an additional mechanism that could amplify or alter magnetic fields [35,36,37,38]. These instabilities, which arise from the anisotropic distribution of plasma particles, could act on already-existing magnetic fields, possibly leading to nonlinear amplification. The role of pressure anisotropy in magnetogenesis should, therefore, be considered alongside other mechanisms, such as vorticity-driven processes, to provide a more complete picture of the processes responsible for the generation and evolution of cosmic magnetic fields.

To enhance the understanding of magnetogenesis mechanisms, this study specifically focuses on the amplification of cosmic magnetic fields through pressure-anisotropy-driven plasma instabilities. In previous work, the factor describing pressure anisotropy is often assumed to be constant, which is justified by the fact that any process amplifying the pressure anisotropy is precisely counterbalanced by processes leading to isotropization [35]. Even in such a balanced system, small perturbations around an equilibrium anisotropy factor could be generated by turbulence. In this work, we additionally consider perturbations to the pressure anisotropy around a constant value. These perturbations could lead to a marginally unstable system, driving plasma instabilities in very weakly magnetized plasmas. Such effects could modify the evolution of the magnetic field, including the saturation timescales of magnetic field amplification.

2. Statement of the Problem

In this section, we introduce the problem setup based on pressure anisotropy and the perturbations that modify it. Depending on the characteristic frequencies for isotropizing pressure anisotropy, we distinguish between two different regimes for examining magnetic field amplification.

2.1. Pressure Anisotropy

Throughout the paper, we define the pressure anisotropy factor with respect to the magnetic field direction as follows:

$$∆\equiv {\displaystyle \frac{p_{\perp }-p_{\parallel }}{p_{\perp }}}.$$

(1)

To consider the time evolution of pressure anisotropy, we apply the time derivative to Equation (1):

$${\displaystyle \frac{d∆}{dt}}={\displaystyle \frac{1}{p_{\perp }}}\left[\left(1-∆\right){\displaystyle \frac{d{p}_{\perp }}{dt}}-{\displaystyle \frac{d{p}_{\parallel }}{dt}}\right].$$

(2)

Here, the first term on the right-hand side of Equation (2) accounts for the time evolution of magnetic field intensity. Considering the conservation of the first adiabatic invariant $\mu =m_i{v}_{\perp }^2/2B$, the emergence of pressure anisotropy is a natural consequence of changes in magnetic field strength. By summing up the first adiabatic invariants of all particles following a Maxwellian distribution with thermal velocity $v_{\mathrm{t}\mathrm{h}}$, we obtain $p_{\perp }/B=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. Including the pressure isotropization due to collisions, parameterized by the Braginskii collision frequency ${\nu }_{ii}$, Equation (2) can be rewritten as follows:

$${\displaystyle \frac{d∆}{dt}}\approx \left(1-∆\right){\displaystyle \frac{1}{B}}{\displaystyle \frac{dB}{dt}}-{\displaystyle \frac{1}{p_{\perp }}}{\displaystyle \frac{d{p}_{\parallel }}{dt}}-{\nu }_{ii}\left|∆\right|.$$

(3)

The term $p_{\perp }^{-1}\left(d{p}_{\parallel }/dt\right)$ describes the interchange of parallel and perpendicular pressures, mediated by magnetic field fluctuations, which represents the net magnetic scattering of the particle distributions. The relevant scattering frequency is defined as follows:

$${\nu }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}~{\displaystyle \frac{1}{p_{\perp }}}{\displaystyle \frac{d{p}_{\parallel }}{dt}}{\displaystyle \frac{1}{∆}}.$$

(4)

Based on the characteristic frequencies for isotropization of pressure anisotropy, this work considers two different regimes, summarized as follows:

• Collision-dominated regime, for ${\nu }_{ii}\gg {\nu }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}$;
• Intermediate regime, for ${\nu }_{ii}~{\nu }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}$;
• Scattering-dominated regime, for ${\nu }_{ii}\ll {\nu }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}$.

As the magnetic field evolves in the system, the relative significance of ${\nu }_{ii}$ and ${\nu }_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{t}}$ also changes. Initially, the pressure anisotropy is primarily isotropized by collisions. As the magnetic fields increase, scattering processes mediated by plasma instabilities become more significant. In this context, the paper describes the evolution of three regimes, as outlined below.

2.2. Magnetic Field Amplification in the Collision-Dominated Regime and Perturbations Enhancing Pressure Anisotropy

The analytic approach with a constant $∆$ in regions involving pressure, anisotropy-driven instabilities can be conducted by assuming that any process amplifying the pressure anisotropy, such as turbulence or anisotropic cosmic-ray scattering, is precisely counterbalanced by processes leading to isotropization. Under this assumption, the evolution of the system can be formulated using the equilibrium anisotropy value $∆\left(t\right)\approx {∆}_0$ [35]. We additionally consider perturbations to the pressure anisotropy around ${∆}_0$ that marginally destabilize the system near equilibrium ($\delta ∆\ll {∆}_0$). Depending on the damping rate (${\Gamma }_d$) of the oscillation frequency (${\omega }_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}}$) of the perturbations, $∆\left(t\right)$ can be described as follows:

$$∆\left(t\right)\approx {∆}_0+\delta ∆\exp \left(-{\Gamma }_{d}t\right)\cos \left({\omega }_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}}t\right).$$

(5)

Assuming that oscillation is negligible compared to the damping (i.e., ${\Gamma }_d\gg {\omega }_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}}$), $∆\left(t\right)$ is simplified as follows:

$$∆\left(t\right)\approx {∆}_0+\delta ∆\exp \left(-{\Gamma }_{d}t\right).$$

(6)

The generation of $\delta ∆$ through turbulence is further examined in the regime of collision-dominated isotropization, which could be relevant to the early stages of the evolution of a cosmological seed field after the recombination era, especially given that $\beta =2v_{\mathrm{t}\mathrm{h}}^2/v_{\mathrm{A}}^2\gg 1$ and ${\nu }_{ii}\gg {\Omega }_i$. Ignoring the isotropization governed by magnetic scattering, the equation describing the time evolution of the anisotropy factor can be reduced as follows:

$${\displaystyle \frac{d∆}{dt}}\approx \left(1-∆\right){\displaystyle \frac{1}{B}}{\displaystyle \frac{dB}{dt}}-{\nu }_{ii}\left|∆\right|.$$

(7)

In this regime, the magnetic field is primarily amplified by the dynamics of the plasma (i.e., the turbulence dynamo in the kinetic phase) rather than by instabilities driven by the pressure anisotropy.

Throughout this paper, we assume the intergalactic plasma to be a highly conductive medium (i.e., the resistivity is much smaller than the viscosity). In such a system, the motion of the fluid elements is coupled to the magnetic field (i.e., magnetic field lines are “frozen” into the fluid). Applying the “frozen-in” condition to the plasma, we describe the magnetic field amplification through turbulent dynamos in the collision-dominated regime [37]:

$${\displaystyle \frac{1}{B}}{\displaystyle \frac{dB}{dt}}\approx \widehat{\mathit{b}}\widehat{\mathit{b}}:\nabla u-\nabla ·\mathit{u}~{\displaystyle \frac{\delta v_L}{L}}{\mathrm{R}\mathrm{e}}^{1/2}={\left({\displaystyle \frac{\delta v_L}{v_{\mathrm{t}\mathrm{h}}}}\right)}^{3/2}{\left({\displaystyle \frac{{\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}}{L}}\right)}^{1/2}{\nu }_{ii},$$

(8)

where ${\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}=v_{\mathrm{t}\mathrm{h}}/{\nu }_{ii}$ is the mean free path with the thermal velocity $v_{\mathrm{t}\mathrm{h}}$, $\mathit{u}$ is the fluid velocity, $\widehat{\mathit{b}}=\mathit{B}/B_0$ is the normalized magnetic field with respect to the seed field intensity $B_0$, and $\delta v_L$ is the velocity fluctuations at the outer scale of turbulence $L$. The term $\widehat{\mathit{b}}\widehat{\mathit{b}}:\nabla u$ represents the projection of the gradient of the fluid velocity $\nabla u$ onto the magnetic field direction, represented by the unit vector $\widehat{\mathit{b}}$. The Reynolds number ($\mathrm{R}\mathrm{e}$) is defined as $\mathrm{R}\mathrm{e}={\delta v}_{L}L/\nu$, where $\nu$ is the fluid viscosity. The Kolmogorov scaling relation is employed to scale the turbulent velocity with respect to the viscous scale ${\delta v}_{l_v}\propto l_v^{1/3}$. In the case without the perturbation term, we obtain the value of ${∆}_0$ from Equation (8):

$$\left|{∆}_0\right|~{\left({\displaystyle \frac{v_{\mathrm{t}\mathrm{h}}}{\delta v_L}}\right)}^{3/2}{\left({\displaystyle \frac{L}{{\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}}}\right)}^{1/2}-1.$$

(9)

Equation (9) indicates that the turbulence during the early evolutionary phase generates pressure anisotropy by itself [35], as the rapid growth of the magnetic field is not counterbalanced by collision-mediated isotropization.

Next, we take into account the perturbation term in Equation (7) to examine the perturbation produced by turbulence during the early evolutionary phase. Assuming that the time variation of the perturbation term can be neglected compared to the damping rate (i.e., we consider the damping-dominated regime, ${\Gamma }_d\gg {\omega }_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}}$), we obtain the following expression from Equation (7):

$${\displaystyle \frac{d∆}{dt}}\approx -{\Gamma }_d\delta ∆\exp \left(-{\Gamma }_{d}t\right),$$

(10)

$$\left|{\displaystyle \frac{\delta ∆}{{∆}_0}}\right|\exp \left(-{\Gamma }_{d}t\right)\approx \left|{\displaystyle \frac{\left(1+{\left|{∆}_0\right|}^{-1}\right)\left(\frac{1}{B}\frac{dB}{dt}-{\nu }_{ii}\right)}{{\Gamma }_d+{\nu }_{ii}-\frac{1}{B}\frac{dB}{dt}}}\right|={\displaystyle \frac{1}{\frac{{\Gamma }_d}{{\nu }_{ii}}+\frac{1}{1+{\left|{∆}_0\right|}^{-1}}}}.$$

(11)

Equation (11) shows that the time evolution of the perturbation term depends on the ratio between the damping and collision rates, as well as the anisotropy factor around the equilibrium state, ${∆}_0$. For systems with sufficiently small ${∆}_0$ satisfying $\beta \gg {\left|{∆}_0\right|}^{-1}\gg 1$, the perturbation term at the saturation timescale ${\tau }_c^{\left(turb\right)}$, which satisfies ${\tau }_c^{\left(turb\right)}\ll {\Gamma }_d^{-1}$, is given as follows:

$$\left|{\displaystyle \frac{\delta ∆}{{∆}_0}}\right|~{\left({\displaystyle \frac{{\Gamma }_d}{{\nu }_{ii}}}\right)}^{-1}~{\left({\displaystyle \frac{{\Gamma }_d}{{\Omega }_i}}\right)}^{-1}.$$

(12)

This is because the strength of the saturated magnetic field through turbulent dynamo is roughly ${\Omega }_i~{\nu }_{ii}$, which is relevant to the condition for a gyrotropic collisionless plasma (i.e., the Larmor radius of the ions ($r_{L,i}~v_{\mathrm{t}\mathrm{h}}/{\Omega }_i$) is smaller than the mean free path (${\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}$)). Figure 1 shows the calculated perturbation terms induced by the turbulent dynamo. It is evident that Equation (11) for the case with sufficiently small ${∆}_0$ provides an anti-correlation between the perturbation term and ${\Gamma }_d/{\nu }_{ii}$.

3. A Model for Amplifying the Cosmic Magnetic Field Mediated by Plasma Instabilities

In this section, we investigate the impact of perturbations on the evolution of the magnetic field in a regime where isotropization is affected by scattering. Once the magnetic field reaches a sufficient strength for the plasma to become gyrotropic, magnetic field amplification will be primarily driven by instabilities. The condition for a plasma to be gyrotropic is given by $B_{\mathrm{g}\mathrm{y}\mathrm{r}\mathrm{o}}\left(\mathrm{G}\right)>~{10}^{-20}n\left({\mathrm{c}\mathrm{m}}^{-3}\right){T\left(\mathrm{e}\mathrm{V}\right)}^{-1/2}$. This leads to the conclusion that for magnetic fields stronger than $~{10}^{-14}\mathrm{G}$, the condition $r_{L,i}<{\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}$ holds, which is relevant to the parameters of standard $\mathsf{\Lambda }$ CDM cosmology (with baryon density $n_{z1000}~{10}^6{\mathrm{c}\mathrm{m}}^{-3}$ and temperature $T_{z1000}~1\mathrm{e}\mathrm{V}$ at $z~1000$). For the gyrotropic collisionless plasma, we adopt a magnetic field amplification model based on plasma instabilities that enable fast interchange of internal, kinetic, and magnetic energies. The characteristics of pressure-anisotropy-driven instabilities are discussed in Section 3.1, while the model for magnetic field amplification is presented in Section 3.2 and Section 3.3. In particular, the effects of perturbations in pressure anisotropy on the magnetic field amplification process are examined. A discussion of the nonlinear saturation stage of magnetic field amplification is provided briefly in Section 3.4.

3.1. Instabilities Induced by Pressure Anisotropy

Pressure-anisotropy-driven instabilities have been explored as possible mechanisms for magnetic field amplification [35,36,37,38]. In this paper, we repeat the formulation of pressure-driven instabilities introduced in earlier studies, including the linear wave analysis [36], to ensure the paper remains self-contained. Additionally, we investigate the impact of perturbation terms on the growth of these instabilities.

For magnetized ions satisfying ${\Omega }_i\gg {\nu }_{ii}$, the following equations for the fluid velocity $\mathit{u}$ and magnetic field $\mathit{B}$ are valid, where $u=\left|\mathit{u}\right|$ and $B=\left|\mathit{B}\right|$ represent the moduli of the respective vectors:

$$\rho {\displaystyle \frac{d\mathit{u}}{dt}}=-\nabla \left(p_{\perp }+{\displaystyle \frac{B^2}{8\pi }}\right)+\nabla ·\left[\widehat{\mathit{b}}\widehat{\mathit{b}}\left(p_{\perp }-p_{\parallel }\right)\right]+{\displaystyle \frac{\mathit{B}·\nabla B}{4\pi }}\widehat{\mathit{b}},$$

(13)

$${\displaystyle \frac{d\mathit{B}}{dt}}=\mathit{B}·\nabla u-\mathit{B}\nabla ·\mathit{u}.$$

(14)

Equation (13) represents the equation of motion for the fluid velocity, where the term $\rho d\mathit{u}/dt$ accounts for the inertial term and the right-hand side includes forces due to pressure gradients, magnetic pressure, and the interaction between the magnetic field and the velocity gradient. Equation (14) governs the evolution of the magnetic field by accounting for the advection of the field by the plasma flow and the stretching of the magnetic field lines due to velocity gradients. Additionally, the kinetic equation for the magnetized ions, following an anisotropic ion distribution function $f(v_{\parallel },v_{\perp })$, is written as follows:

$$\begin{gathered} \left({\displaystyle \frac{df}{dt}}+v_{\parallel }\widehat{\mathit{b}}·\nabla f\right)+{\displaystyle \frac{1}{2}}{v}_{\perp }\left(\widehat{\mathit{b}}\widehat{\mathit{b}}:\nabla u-\nabla ·\mathit{u}+v_{\parallel }{\displaystyle \frac{\widehat{\mathit{b}}·\nabla B}{B}}\right){\displaystyle \frac{\partial f}{\partial v_{\perp }}} \\ -\left[\widehat{\mathit{b}}·\left({\displaystyle \frac{d\mathit{u}}{dt}}+v_{\parallel }\widehat{\mathit{b}}·\nabla u\right)+{\displaystyle \frac{1}{2}}{v}_{\perp }^2{\displaystyle \frac{\widehat{\mathit{b}}·\nabla B}{B}}\right]{\displaystyle \frac{\partial f}{\partial v_{\parallel }}}=C\left[f_0,f\right], \end{gathered}$$

(15)

$$C\left[f_0,f\right]={\displaystyle \frac{1}{2}}{\nu }_{ii}{\left({\displaystyle \frac{v_{th}}{v}}\right)}^3{\displaystyle \frac{\partial }{\partial \xi }}\left(1-{\xi }^2\right){\displaystyle \frac{\partial f}{\partial \xi }}.$$

(16)

here, $C\left[f_0,f\right]$ represents the collision operator with $\xi =v_{\parallel }/v$, and $v=\sqrt{v_{\parallel }^2+v_{\perp }^2}$, where $f_0$ is the zeroth-order Maxwellian distribution function. Equation (15) describes how the distribution function changes due to the interactions between the plasma velocity and magnetic field. The first term accounts for the convective transport of ions along the magnetic field direction and their movement in the velocity space. The second term describes the effect of the plasma velocity and the magnetic field gradient on the distribution function, incorporating the anisotropic velocity and field interactions. Particularly, the term $\widehat{\mathit{b}}\widehat{\mathit{b}}:\nabla u$ represents how the velocity of the plasma varies in the direction of the magnetic field. The third term addresses the magnetic field’s effect on the ion dynamics, including energy exchange between the field and the plasma.

Under the assumption that the fluid velocity is much smaller than the thermal speed and varies over scales much longer than the mean free path, we can construct a perturbation theory to derive the dispersion relation for pressure-anisotropy-driven instabilities. Assuming Kolmogorov scaling, the velocity at the viscous scale ($u$) and the normalized wavenumber associated with the viscous scale relative to the mean free path ($ϵ=k_{\nu }{\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}$) are given as follows:

$$u~{\mathrm{R}\mathrm{e}}^{-1/4}{v}_{\mathrm{t}\mathrm{h}},$$

(17)

$$ϵ=k_{\nu }{\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}~{\mathrm{R}\mathrm{e}}^{-1/4}.$$

(18)

To perform the ordering of Equation $\left(15\right)$ with respect to $ϵ$, we assume the following relations:

$$k_{\parallel }{\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}~ϵ,$$

(19)

$${\displaystyle \frac{d}{dt}}~\nabla u~{ϵ}^2{\displaystyle \frac{v_{\mathrm{t}\mathrm{h}}}{{\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}}}.$$

(20)

The anisotropic ion distribution function, up to order ${ϵ}^2$, is then calculated as follows:

$$f\left(v_{\parallel },v_{\perp }\right)~{\displaystyle \frac{n_0}{{\left(2\pi v_{\mathrm{t}\mathrm{h}}^2\right)}^{3/2}}}\exp \left(-{\displaystyle \frac{v^2}{2v_{\mathrm{t}\mathrm{h}}^2}}\right)\left[1+{\displaystyle \frac{\widehat{\mathit{b}}\widehat{\mathit{b}}:\nabla u}{{\nu }_{ii}}}{\displaystyle \frac{v^3\left(v^2-3v_{\parallel }^2\right)}{3v_{\mathrm{t}\mathrm{h}}^5}}\right],$$

(21)

where $n_0$ is the plasma number density.

Note that the kinetic model in this study assumes small velocity fluctuations and long spatial scales relative to the mean free path, which are standard approximations in low-collision and weakly magnetized plasmas. However, these assumptions may be sensitive to variations in collisional frequency (${\nu }_{ii}$) and magnetic field dynamics ($\widehat{\mathit{b}}\widehat{\mathit{b}}:\nabla u$, $\nabla B$), especially under highly dynamic conditions such as those in the early universe. A sensitivity analysis reveals that decreasing ${\nu }_{ii}$ enhances the anisotropic ion distribution’s deviation from Maxwellian behavior, increasing susceptibility to pressure-anisotropy-driven instabilities. Conversely, increasing ${\nu }_{ii}$ suppresses these deviations, stabilizing the plasma. Furthermore, changes in magnetic field strength and spatial gradients ($\nabla B$) significantly modify the anisotropic distribution by amplifying the contribution of terms proportional to $\widehat{\mathit{b}}\widehat{\mathit{b}}:\nabla u$ in the distribution function. Strong magnetic fields or steep gradients exacerbate anisotropy, potentially triggering instabilities such as mirror or firehose instabilities.

We proceed with the stability analysis of the velocity and magnetic fields that vary on time scales larger than $~{\left(\nabla u\right)}^{-1}$ and spatial scales larger than $~k_{\nu }^{-1}$. By considering the linear perturbations $\mathit{\delta }\mathit{u}$ and $\mathit{\delta }\mathit{B}$, with frequencies $\omega \gg \nabla u$ and wavenumbers $k\gg k_{\nu }$, we linearize Equations (13) and (14), neglecting temporal and spatial derivatives of the unperturbed quantities. The resulting equations are as follows:

$$\begin{gathered} -i\omega \rho \mathit{\delta }\mathit{u}=-i\mathit{k}\delta \left(p_{\perp }+{\displaystyle \frac{B^2}{8\pi }}\right)+i{k}_{\parallel }\delta \widehat{\mathit{b}}\left(p_{\perp }-p_{\parallel }+{\displaystyle \frac{B^2}{4\pi }}\right) \\ +\widehat{\mathit{b}}\delta \left[\widehat{\mathit{b}}·\nabla \left(p_{\perp }-p_{\parallel }\right)-\left(p_{\perp }-p_{\parallel }+{\displaystyle \frac{B^2}{4\pi }}\right){\displaystyle \frac{\widehat{\mathit{b}}·\nabla B}{B}}\right], \end{gathered}$$

(22)

$$\delta \widehat{\mathit{b}}=-{\displaystyle \frac{k_{\parallel }}{\omega }}\mathit{\delta }{\mathit{u}}_{\perp },$$

(23)

$${\displaystyle \frac{\delta B}{B}}={\displaystyle \frac{1}{\omega }}{\mathit{k}}_{\perp }·\mathit{\delta }{\mathit{u}}_{\perp }.$$

(24)

The dispersion relation for modes with shear-Alfvén-wave polarization ($\mathit{\delta }\mathit{u}\propto {\mathit{k}}_{\perp }\times \widehat{\mathit{b}}$) is derived as follows:

$${\displaystyle \frac{{\omega }^2}{k_{\parallel }^2{v}_{\mathrm{t}\mathrm{h}}^2}}=∆+2{\beta }^{-1}.$$

(25)

In the limit, $\beta \gg {\left|∆\right|}^{-1}$, and for $∆<0$ (i.e., $p_{\parallel }>$ $p_{\perp }$), an instability appears with the growth rate ${\Gamma }_g\approx k_{\parallel }{v}_{\mathrm{t}\mathrm{h}}{\left|∆\right|}^{1/2}$, which is known as the firehose instability. The instability stabilizes when $\left|∆\right|~2{\beta }^{-1}$, with the maximum instability occurring around the ion-gyration scale (i.e., $k_{\parallel }{r}_{L,i}~1$) [37,39]:

$${\Gamma }_{g,max}~{\left(\left|∆\right|-2{\beta }^{-1}\right)}^{1/2}{\Omega }_i$$

(26)

To analyze the stability of perturbations with other polarizations, it is necessary to compute the pressure perturbations $\delta p_{\perp }$ and $\delta p_{\parallel }$. To obtain these, we first linearize the kinetic equation for magnetized ions around the distribution function in Equation (21) and calculate $\delta p_{\perp }$ and $\delta p_{\parallel }$ using the perturbed distribution function. The perturbations in the distribution function are assumed to have frequency $\omega \gg \nabla u$ and wavenumber $k\gg {\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p}}^{-1}$. We then linearize Equation (13) with the perturbations $\mathit{\delta }\mathit{u}$, $\mathit{\delta }\mathit{B}$,$\delta p_{\perp }$, and $\delta p_{\parallel }$, while neglecting the terms of the order ${\left(\omega /k_{\parallel }{v}_{\mathrm{t}\mathrm{h}}\right)}^3$ and higher, as high-frequency perturbations are subject to strong collisionless damping and cannot become unstable with small anisotropies. Under these assumptions, we obtain the following equation:

$$\begin{gathered} {\omega }^2\mathit{\delta }\mathit{u}=k_{\parallel }^2{v}_{\mathrm{t}\mathrm{h}}^2\left(∆+2{\beta }^{-1}\right)\mathit{\delta }{\mathit{u}}_{\perp }-\widehat{\mathit{b}}{\displaystyle \frac{{\omega }^2}{k_{\parallel }}}{\mathit{k}}_{\perp }·\mathit{\delta }{\mathit{u}}_{\perp } \\ +{\mathit{k}}_{\perp }\left({\mathit{k}}_{\perp }·\mathit{\delta }{\mathit{u}}_{\perp }\right)\left[2\left(-∆+{\beta }^{-1}\right)v_{\mathrm{t}\mathrm{h}}^2-i\omega \sqrt{2\pi }{\displaystyle \frac{v_{\mathrm{t}\mathrm{h}}}{\left|k_{\parallel }\right|}}+2{\displaystyle \frac{{\omega }^2}{k_{\parallel }^2}}\right]. \end{gathered}$$

(27)

The first term on the right-hand side represents modes with shear-Alfvén-wave polarization. To analyze the characteristics of $\delta u_{\parallel }$, we take the dot product using both sides of Equation (27) with $\widehat{b}$. This gives the following:

$$\delta u_{\parallel }\propto -{\mathit{k}}_{\perp }·{\displaystyle \frac{\mathit{\delta }{\mathit{u}}_{\perp }}{k_{\parallel }}}.$$

(28)

Perturbations with $\delta u_{\parallel }\ne 0$ are incompressible and slow-wave-polarized. The dispersion relation for these perturbations, with the condition $\mathit{\delta }{\mathit{u}}_{\perp }\propto {\mathit{k}}_{\perp }$, is as follows:

$$\left(1-{\displaystyle \frac{2k_{\perp }^2}{k_{\parallel }^2}}\right){\omega }^2+i\omega \sqrt{2\pi }{\displaystyle \frac{k_{\perp }^2{v}_{\mathrm{t}\mathrm{h}}}{\left|k_{\parallel }\right|}}-\left[\left(k_{\parallel }^2-2k_{\perp }^2\right)∆+2k^2{\beta }^{-1}\right]v_{\mathrm{t}\mathrm{h}}^2=0.$$

(29)

For parallel propagating modes where $k_{\perp }\ll k_{\parallel }$, the second term is negligible. For $∆<0$, the growth rate is ${\Gamma }_g\approx k_{\parallel }{v}_{\mathrm{t}\mathrm{h}}{\left|∆\right|}^{1/2}$. For modes propagating at an oblique angle to the magnetic field (i.e., $k_{\perp }$ is not small), the second term in Equation (29) dominates over the first term. In this case, the growth rate is as follows:

$${\displaystyle \frac{{\Gamma }_g}{k_{\parallel }{v}_{\mathrm{t}\mathrm{h}}}}\approx {\left({\displaystyle \frac{2}{\pi }}\right)}^{{\displaystyle \frac{1}{2}}}\left[∆\left(1-{\displaystyle \frac{k_{\parallel }^2}{2k_{\perp }^2}}\right)-{\beta }^{-1}\left(1+{\displaystyle \frac{k_{\parallel }^2}{k_{\perp }^2}}\right)\right].$$

(30)

Slow-wave-polarized firehose instability can be triggered when $k_{\parallel }>\sqrt{2}{k}_{\perp }$ for $∆<0$, while mirror instability can develop if $k_{\perp }>k_{\parallel }/\sqrt{2}$ for $∆>0$. In both cases, the growth rate is approximately ${\Gamma }_g~k_{\parallel }{v}_{\mathrm{t}\mathrm{h}}\left|∆\right|$. For the regime where $\left|∆\right|\ll 1$, the growth of oblique modes occurs much more slowly than the growth of parallel modes or shear-Alfvén-wave polarized modes.

The dispersion relations derived above do not account for second-order perturbation terms, such as $\delta p_{\perp }\mathit{\delta }{\mathit{u}}_{\perp }$, $\delta p_{\parallel }\mathit{\delta }{\mathit{u}}_{\perp }$, $\delta p_{\perp }\mathit{\delta }{\mathit{u}}_{\parallel }$, and $\delta p_{\parallel }\mathit{\delta }{\mathit{u}}_{\parallel }$, which include the effects of perturbations in the pressure anisotropy factor ($\delta ∆$). These terms account for more complex interactions between the plasma pressure anisotropy and the plasma velocity fields, which allow for a more accurate description of the system’s response to pressure anisotropy, especially near the critical stability condition, where plasma instabilities are triggered. In this context, we specifically investigate the impact of pressure anisotropy perturbations on the saturation timescale of magnetic field amplification due to plasma instabilities, denoted by ${\tau }_c^{\left(inst\right)}=1/{\Gamma }_c^{\left(inst\right)}$. Near the stability condition for the system where $\left|∆\right|~2{\beta }^{-1}$, the instability driven by pressure anisotropy perturbations may influence the system. In the case of rapid damping (${\Gamma }_d\gg {\Gamma }_c^{\left(inst\right)}$), the effects of pressure anisotropy are minimal, and the system tends toward equilibrium. In contrast, when ${\Gamma }_d\ll {\Gamma }_c^{\left(inst\right)}$, the perturbation can trigger plasma instabilities in a marginally unstable system, depending on the anisotropy factor $\left|\delta ∆\right|$.

Considering the perturbations $\delta p_{\perp }\mathit{\delta }{\mathit{u}}_{\perp }$ and $\delta p_{\parallel }\mathit{\delta }{\mathit{u}}_{\perp }$, the dispersion relation for the modes with shear-Alfvén-wave polarization is modified to include the pressure anisotropy perturbation factor ($\delta ∆$) as follows:

$$-i\omega \rho \mathit{\delta }{\mathit{u}}_{\perp }=-i{\displaystyle \frac{k_{\parallel }^2}{\omega }}\mathit{\delta }{\mathit{u}}_{\perp }\left(p_{\perp }+\delta p_{\perp }-p_{\parallel }-\delta p_{\parallel }+{\displaystyle \frac{B^2}{4\pi }}\right),$$

(31)

$${\displaystyle \frac{{\omega }^2}{k_{\parallel }^2{v}_{\mathrm{t}\mathrm{h}}^2}}=∆+\delta ∆\exp \left(-{\displaystyle \frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}}\right)+2{\beta }^{-1}.$$

(32)

In the limit, $\left|∆\right|~2{\beta }^{-1}$, and if $∆<0$ (i.e., $p_{\parallel }>$ $p_{\perp }$), the growth rate of the instability is given as follows:

$${\displaystyle \frac{{\tilde{\Gamma }}_{g,c}}{k_{\parallel }{v}_{\mathrm{t}\mathrm{h}}}}\approx {\left|\delta ∆\exp \left(-{\displaystyle \frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}}\right)\right|}^{{\displaystyle \frac{1}{2}}}.$$

(33)

Figure 2 illustrates the growth rate due to the perturbed anisotropy factor at $t~{\tau }_c^{\left(inst\right)}$, with $\left|∆\right|={10}^{-2}$ and $\left|\delta ∆/∆\right|={10}^{-1}$. This demonstrates how the perturbed anisotropy affects the growth of plasma instabilities. In the regime where ${\Gamma }_d<{\Gamma }_c^{\left(inst\right)}$, the growth rate depends weakly on the value of ${\Gamma }_d$, whereas in the regime ${\Gamma }_d>{\Gamma }_c^{\left(inst\right)}$, the growth rate rapidly decreases as ${\Gamma }_d$ increases. The growth of plasma instabilities and its dependence on the damping of the perturbed anisotropy can also accelerate or decelerate the growth of the magnetic field through plasma instabilities, which is discussed in the following sections.

Considering the perturbations $\delta p_{\perp }\mathit{\delta }{\mathit{u}}_{\parallel }$ and $\delta p_{\parallel }\mathit{\delta }{\mathit{u}}_{\parallel }$, the dispersion relation for the parallel modes with slow-wave-polarization around the saturation timescale ($\left|∆\right|~2{\beta }^{-1}$) becomes the following:

$$\left(1-{\displaystyle \frac{2k_{\perp }^2}{k_{\parallel }^2}}\right){\omega }^2-\left[\left(k_{\parallel }^2-2k_{\perp }^2\right)\left(∆+\delta ∆\exp \left(-{\displaystyle \frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}}\right)\right)+2k^2{\beta }^{-1}\right]v_{\mathrm{t}\mathrm{h}}^2\approx 0.$$

(34)

The growth rate for the instability is as follows:

$${\displaystyle \frac{{\tilde{\Gamma }}_{g,c}}{k_{\parallel }{v}_{\mathrm{t}\mathrm{h}}}}\approx {\left|\delta ∆\exp \left(-{\displaystyle \frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}}\right)\right|}^{{\displaystyle \frac{1}{2}}}.$$

(35)

For the oblique modes with slow-wave-polarization around the saturation timescale ($\left|∆\right|~2{\beta }^{-1}$), the dispersion relation becomes:

$$i\omega \sqrt{2\pi }{\displaystyle \frac{k_{\perp }^2{v}_{\mathrm{t}\mathrm{h}}}{\left|k_{\parallel }\right|}}-\left[\left(k_{\parallel }^2-2k_{\perp }^2\right)\left(∆+\delta ∆\exp \left(-{\displaystyle \frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}}\right)\right)+2k^2{\beta }^{-1}\right]v_{\mathrm{t}\mathrm{h}}^2\approx 0.$$

(36)

The growth rate is then as follows:

$${\displaystyle \frac{{\tilde{\Gamma }}_{g,c}}{k_{\parallel }{v}_{\mathrm{t}\mathrm{h}}}}\approx {\left({\displaystyle \frac{2}{\pi }}\right)}^{{\displaystyle \frac{1}{2}}}\left[\delta ∆\exp \left(-{\displaystyle \frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}}\right)\left(1-{\displaystyle \frac{k_{\parallel }^2}{2k_{\perp }^2}}\right)-3{\beta }^{-1}\right].$$

(37)

Assuming $\left|\delta ∆\right|\ll {\beta }^{-1}$, the mirror modes are stable regardless of the amplitude of $\delta ∆$. Firehose modes could grow for ${\Gamma }_d\ll {\Gamma }_c^{\left(inst\right)}$, and the minimum condition for unstable modes is as follows:

$${\displaystyle \frac{k_{\parallel }}{k_{\perp }}}>{\left[2\left({\displaystyle \frac{3{\beta }^{-1}}{\left|\delta ∆\right|}}\exp \left({\displaystyle \frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}}\right)-1\right)\right]}^{{\displaystyle \frac{1}{2}}}.$$

(38)

Additionally, the maximum condition for $k_{\parallel }/k_{\perp }$ depends on whether the contribution of $k_{\perp }$ is negligible. This condition implies that the second term in Equation (29) must be larger than the first term:

$$\left(1-{\displaystyle \frac{2k_{\perp }^2}{k_{\parallel }^2}}\right){\omega }^2\le i\omega \sqrt{2\pi }{\displaystyle \frac{k_{\perp }^2{v}_{\mathrm{t}\mathrm{h}}}{\left|k_{\parallel }\right|}}.$$

(39)

The growth rate is then constrained by the following:

$${\displaystyle \frac{{\tilde{\Gamma }}_{g,c}}{k_{\parallel }{v}_{\mathrm{t}\mathrm{h}}}}\le {\left(2\pi \right)}^{{\displaystyle \frac{1}{2}}}{\displaystyle \frac{k_{\perp }^2}{k_{\parallel }^2-2k_{\perp }^2}}.$$

(40)

This inequality provides the upper limit for $k_{\parallel }/k_{\perp }$, which is given by the following:

$${\displaystyle \frac{k_{\parallel }}{k_{\perp }}}\le {\left\{1+{\displaystyle \frac{3{\beta }^{-1}}{\left|\delta ∆\right|}}+{\left[{\left(1+{\displaystyle \frac{3{\beta }^{-1}}{\left|\delta ∆\right|}}\right)}^2-{\displaystyle \frac{12{\beta }^{-1}-2\pi }{\left|\delta ∆\right|}}\right]}^{{\displaystyle \frac{1}{2}}}\right\}}^{{\displaystyle \frac{1}{2}}}.$$

(41)

Note that the conditions for $k_{\parallel }/k_{\perp }$, described in Equations (38) and (41), represent the obliquity of the oblique mode with respect to the background magnetic field. Figure 3 shows the possible range of $k_{\parallel }/k_{\perp }$ with $\left|\delta ∆/∆\right|=0.1–0.3$ as a function of ${\Gamma }_d/{\Gamma }_c^{\left(inst\right)}$. This analysis demonstrates how the properties of oblique modes depend on ${\Gamma }_d$ and ${\Gamma }_c^{\left(inst\right)}$. The minimum condition is saturated below ${\Gamma }_d/{\Gamma }_c^{\left(inst\right)}~{10}^{-1}$. The properties of waves can be modified with respect to $\left|\delta ∆/∆\right|$, where the parallel mode is favored (i.e., $k_{\parallel }/k_{\perp }$ becomes larger) as $\left|\delta ∆/∆\right|$ decreases.

3.2. Amplification of the Cosmic Magnetic Field in the Intermediate Regime

Based on the maximum growth rate of the shear-Alfvén-wave polarized firehose instability, ${\Gamma }_{g,max}~{\left(\left|∆\right|-2{\beta }^{-1}\right)}^{1/2}{\Omega }_i$, the saturated magnetic fluctuation is estimated as follows:

$${\left.{\displaystyle \frac{{\delta B}^2}{B^2}}\right|}_{sat}~{\left(\left|∆\right|-2{\beta }^{-1}\right)}^{{\displaystyle \frac{1}{2}}}.$$

(42)

We further assume that the effect of the saturated instabilities is to scatter particles at a rate given as follows:

$${\nu }_{scatt}~{\left.{\displaystyle \frac{{\delta B}^2}{B^2}}\right|}_{sat}{\Gamma }_{g,max}~{\left(\left|∆\right|-2{\beta }^{-1}\right)}^{{\displaystyle \frac{3}{2}}}{\Omega }_i.$$

(43)

In the intermediate regime, the equation for the evolution of $∆$ is written as follows:

$${\displaystyle \frac{d∆}{dt}}\approx \left(1-∆\right){\displaystyle \frac{1}{B}}{\displaystyle \frac{dB}{dt}}-{\nu }_{ii}\left|∆\right|+{\left(\left|∆\right|-2{\beta }^{-1}\right)}^{{\displaystyle \frac{3}{2}}}{\Omega }_i.$$

(44)

Using the time derivative of $∆$ described in Equation (10), the time variation of the magnetic field is obtained from Equation (44):

$$\begin{gathered} {\displaystyle \frac{dB}{dt}}\approx -{\Omega }_{i,0}{\displaystyle \frac{1}{1+\left|∆\right|}}{\displaystyle \frac{{\nu }_{ii}}{{\Omega }_i}}{\displaystyle \frac{B}{B_0}}+{\Omega }_{i,0}{\displaystyle \frac{{\left(\left|∆\right|-2{\beta }^{-1}\right)}^{\frac{3}{2}}}{1+\left|∆\right|}}{\left({\displaystyle \frac{B}{B_0}}\right)}^2 \\ +{\Omega }_{i,0}{\displaystyle \frac{2\exp \left(-{\Gamma }_{d}t-\frac{1}{2}\frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}\right){\left|\delta ∆\right|}^{\frac{3}{2}}}{1+\left|∆\right|}}{\left({\displaystyle \frac{B}{B_0}}\right)}^2. \end{gathered}$$

(45)

Here, ${\Omega }_{i,0}$ represents the cyclotron frequency for the seed magnetic field $B_0$. The first term represents the collisional damping term, while the second and third terms are responsible for the magnetic field growth through pressure anisotropy effects. Particularly, the third term describes the magnetic field amplification due to the anisotropy perturbation alone. Figure 4 shows that the magnetic field growth is suppressed at higher collisional rates.

In the very early stages of magnetic field amplification through plasma instabilities, collisional effects are significant. However, However, as the magnetic field strengthens (i.e., ${\Omega }_i$ increases), the collisional effects become negligible (i.e., ${\nu }_{ii}/{\Omega }_i\ll 1$). In this context, magnetic field growth through plasma instabilities is primarily driven by the scattering-dominated regime.

3.3. Amplification of the Cosmic Magnetic Field in the Scattering-Dominated Regime

In the scattering-dominated regime, the equation for the evolution of $∆$ is written as follows without the collisional term:

$${\displaystyle \frac{d∆}{dt}}\approx \left(1-∆\right){\displaystyle \frac{1}{B}}{\displaystyle \frac{dB}{dt}}+{\left(\left|∆\right|-2{\beta }^{-1}\right)}^{{\displaystyle \frac{3}{2}}}{\Omega }_i.$$

(46)

The time variation of the magnetic field is obtained from Equation (46):

$${\displaystyle \frac{dB}{dt}}\approx {\Omega }_{i,0}{\displaystyle \frac{{\left(\left|∆\right|-2{\beta }^{-1}\right)}^{\frac{3}{2}}}{1+\left|∆\right|}}{\left({\displaystyle \frac{B}{B_0}}\right)}^2+{\Omega }_{i,0}{\displaystyle \frac{2\exp \left(-{\Gamma }_{d}t-\frac{1}{2}\frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}\right){\left|\delta ∆\right|}^{\frac{3}{2}}}{1+\left|∆\right|}}{\left({\displaystyle \frac{B}{B_0}}\right)}^2,$$

(47)

In the limit $\left|\delta ∆\right|\to 0$, the result is consistent with that obtained for a constant $∆={∆}_0$ [35].

It is possible to obtain an analytical solution for the exponential growth of the magnetic field before saturation (i.e., $t\ll 1/{\Gamma }_d\ll 1/{\Gamma }_c^{\left(inst\right)}$ and $\left|∆\right|\ll {\beta }^{-1}$). In this limit, the system reduces to a first-order nonlinear ordinary differential equation with an analytical solution:

$$B\left(t\right)~B_0{\left[1-\left({\displaystyle \frac{{\left|∆\right|}^{\frac{3}{2}}+2\exp \left(-\frac{1}{2}\frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}\right){\left|\delta ∆\right|}^{\frac{3}{2}}}{1+\left|∆\right|}}\right){\Omega }_{i,0}t\right]}^{-1}.$$

(48)

From the quasi-stability condition, given by $\beta ~2{\left|∆\right|}^{-1}$, the saturation timescale for $\left|∆\right|\ll 1$ is obtained as follows:

$${\tau }_c^{\left(inst\right)}~{\left({\left|∆\right|}^{{\displaystyle \frac{3}{2}}}+2\exp \left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\Gamma }_d}{{\Gamma }_c^{\left(inst\right)}}}\right){\left|\delta ∆\right|}^{{\displaystyle \frac{3}{2}}}\right)}^{-1}{\Omega }_{i,0}^{-1}.$$

(49)

Equations (48) and (49) indicate that the time evolution of the magnetic field is influenced by the characteristic damping rate and the amplitude of anisotropy perturbations. We examine these effects on magnetic field evolution, as shown in Figure 5. While the contribution of anisotropy perturbations to amplifying the magnetic field strength is negligible, they significantly modify the saturation timescale. When considering a strong damping rate (i.e., ${\Gamma }_d>{\Gamma }_c^{\left(inst\right)}$) or very weak perturbations (i.e., $\left|\delta ∆/{∆}_0\right|<~{10}^{-3}$), the solution converges to the result without anisotropy perturbation. Figure 6 illustrates the saturation timescales for varying damping rates, ranging from weak (${\Gamma }_d\ll {\Gamma }_c^{\left(inst\right)}$) to strong (${\Gamma }_d\gg {\Gamma }_c^{\left(inst\right)}$) damping. For instance, in the weak damping regime, the saturation timescale can vary by up to $~10\mathrm{\%}$ when $\left|\delta ∆/{∆}_0\right|~{10}^{-1}$. This highlights that perturbations driving pressure anisotropy around an equilibrium value can substantially affect the evolution of cosmic magnetic fields, although the field amplitude remains independent of such effects.

3.4. Saturation of Plasma Instabilities Driven by Pressure Anisotropy

In this section, we discuss the saturation of plasma instability in turbulent media. Assuming the outer scale of the turbulence, $L$, and the rms velocity, $U$, at this scale, the effective Reynolds number (${\mathrm{R}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f}}$) is defined as follows:

$${\mathrm{R}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f}}~{\displaystyle \frac{UL}{{\mu }_{\parallel ,\mathrm{e}\mathrm{f}\mathrm{f}}}}~{\displaystyle \frac{UL}{v_{th}^2}}{\nu }_{\mathrm{e}\mathrm{f}\mathrm{f}}.$$

(50)

Here, ${\nu }_{\mathrm{e}\mathrm{f}\mathrm{f}}\equiv {\nu }_{ii}+{\nu }_{scatt}$ is the effective collision rate, and ${\mu }_{\parallel ,\mathrm{e}\mathrm{f}\mathrm{f}}~v_{th}{\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p},\mathrm{e}\mathrm{f}\mathrm{f}}$ is the effective viscosity of the turbulent system, with the effective mean free path of the particles ${\lambda }_{\mathrm{m}\mathrm{f}\mathrm{p},\mathrm{e}\mathrm{f}\mathrm{f}}~v_{th}/{\nu }_{\mathrm{e}\mathrm{f}\mathrm{f}}$. ${\mathrm{R}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ takes into account the combined effects of turbulent motions and collisional damping in the plasma, and it plays a crucial role in determining how efficiently magnetic fields can grow through turbulence and plasma instabilities. In the saturation stage ($d∆/dt~0$), Equation (3) is rewritten as follows:

$${\nu }_{\mathrm{e}\mathrm{f}\mathrm{f}}∆~{\displaystyle \frac{1}{B}}{\displaystyle \frac{dB}{dt}}~{\mu }_{\parallel ,\mathrm{e}\mathrm{f}\mathrm{f}}\widehat{\mathit{b}}\widehat{\mathit{b}}:\nabla u.$$

(51)

Using $\left|\nabla u\right|~\left(U/L\right){\mathrm{R}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{1/2},$ we obtain the saturated $\left|∆\right|$ as follows:

$$\left|∆\right|~{\left({\displaystyle \frac{U}{v_{th}}}\right)}^2{\mathrm{R}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{-{\displaystyle \frac{1}{2}}}.$$

(52)

Considering that the instability is suppressed when the magnetic field is sufficiently amplified (i.e., $\left|∆\right|~2{\beta }^{-1}$), we obtain the critical effective Reynolds number (${\mathrm{R}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{c}}$) for the magnetic field amplification through plasma instabilities:

$${\mathrm{R}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{c}}~{\displaystyle \frac{{\beta }^2}{4}}{\left({\displaystyle \frac{U}{v_{th}}}\right)}^4.$$

(53)

This indicates that the magnetic field has grown enough to suppress further instability-driven growth when ${\mathrm{R}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f}}<{\mathrm{R}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{c}}$.

While this paper focuses on the exponential growth of the magnetic field before the saturation stage, as the magnetic field grows, nonlinear effects become increasingly important, particularly the saturation of the amplification process. Wagstaff et al. [32], for instance, emphasize the critical role of nonlinear feedback mechanisms that arise as the magnetic field interacts with the plasma. As the magnetic field strength increases, it induces back-reactions on the plasma, altering the dynamics of the turbulence and affecting plasma transport properties, such as viscosity and conductivity. These changes can significantly influence the efficiency of the dynamo process and ultimately lead to the saturation of magnetic field growth. In particular, as the field strength increases, the magnetic back-reaction acts to dampen further growth, leading to a state where the energy injected into the system by turbulence is balanced by the magnetic field energy. This nonlinear feedback loop causes the magnetic field to approach an equilibrium, limiting its growth. The timescale for saturation is dependent on various factors, including the plasma’s resistivity, the intensity of turbulence, and the rate of energy transfer between the kinetic and magnetic components of the system. Moreover, the feedback between magnetic field growth and plasma properties can lead to modifications in the plasma’s viscosity and conductivity, further influencing the saturation process. As the magnetic field amplifies, it can change the turbulence dynamics, leading to a decrease in the effective viscosity and a potential reduction in the efficiency of magnetic field amplification. These effects are particularly significant in the context of the early universe, where pressure anisotropies and instabilities could accelerate or slow down the saturation process, depending on the specific plasma conditions at the time.

4. Summary and Discussion

In an unstable system driven by pressure anisotropy, the possible scenario for amplifying the cosmic magnetic field is summarized as follows: (1) pressure anisotropies could naturally arise in the intergalactic medium during the early universe, at timescales comparable to the eddy turnover time at viscous dissipation scales (i.e., the post-recombination era); (2) these pressure anisotropies drive instabilities, leading to the exponential amplification of the magnetic field via the isotropization of the pressure anisotropy. In such a system, this study aims to examine the effects of pressure anisotropy perturbations on the evolution of cosmic magnetic fields.

The magnetic field amplification in three different regimes is summarized as follows. In the collision-dominated regime, the amplitude of pressure anisotropy can be calculated based on the evolution of the magnetic field through turbulent dynamo processes. Specifically, the amplitude of anisotropy perturbations around the equilibrium anisotropy value can be determined by the ratio of the perturbation damping rate to the collision frequency. It is possible to expect the presence of pressure anisotropy perturbations near the equilibrium value even after the magnetic field evolution has saturated. In the intermediate region, we examine the influence of collisional effects on magnetic field growth through plasma instabilities during the very early stages of magnetic field amplification. In the scattering-dominated regime, we examine the time evolution of the magnetic field driven by pressure anisotropy instabilities, as well as the effects of pressure anisotropy perturbations on the saturation of the magnetic field evolution. The significance of these effects is measured in terms of the amplitude and damping rate of the anisotropy perturbations. While the strength of the saturated magnetic field is not significantly affected by small anisotropy perturbations near the equilibrium value (denoted as ${∆}_0$), the saturation timescale of magnetic field amplification through instabilities is notably influenced by the pressure anisotropy perturbations. For perturbations with amplitude $\left|\delta ∆/{∆}_0\right|>~{10}^{-3}$ in the weak damping regime (${\Gamma }_d\ll {\Gamma }_c^{\left(inst\right)}$), the system reaches saturation faster compared to a system without anisotropy perturbation. However, the effect of perturbations becomes negligible in the strong damping regime (${\Gamma }_d\gg {\Gamma }_c^{\left(inst\right)}$).

Before concluding, we further discuss the following issues: (1) Scaling of magnetic fields in an expanding universe; (2) Observational implications of pressure-anisotropy-driven instabilities.

The scaling relation for magnetic fields in an expanding universe is derived under the assumption of conformal invariance, which assumes that the plasma behaves ideally with no resistivity or anisotropic effects. This scaling is typically valid in the early universe, when ideal magnetohydrodynamics (MHD) conditions hold, and magnetic field strength decreases as the universe expands. However, deviations from this ideal scaling law may occur under non-ideal MHD conditions [34]. Specifically, in the presence of finite resistivity or anisotropy-driven effects such as pressure anisotropy, these ideal scaling relations may no longer apply. Resistive effects introduce dissipation into the magnetic field evolution, leading to a slower decay rate or possible amplification of magnetic fields in certain regions. Similarly, anisotropy-driven instabilities can lead to nonlinear evolution of the magnetic field structure, altering its scaling behavior. Therefore, a more comprehensive treatment of magnetic field evolution should account for these non-ideal effects, especially in later stages of the universe’s expansion or in regions where plasma anisotropy is significant.

Magnetogenesis scenarios driven by such instabilities can have significant effects on observational signatures, particularly in the cosmic microwave background (CMB) anisotropies and the intergalactic magnetic field [40]. These effects could manifest as subtle distortions in CMB anisotropies, where the pressure anisotropy in the early universe could affect the polarization patterns and the temperature fluctuations of the CMB. Such distortions could serve as indirect evidence for anisotropic instabilities in the early universe, providing a testable prediction of this model. Additionally, gamma-ray halos produced by high-energy particles interacting with cosmic microwave background photons could be used to probe the nature of pressure-anisotropy instabilities. Specifically, energetic particles accelerated by anisotropic instabilities may generate gamma-ray emission, which could be detected through high-resolution gamma-ray telescopes. Furthermore, the presence of a large-scale intergalactic magnetic field, potentially shaped by pressure-anisotropy-driven processes, could leave its imprint on the polarization and intensity of gamma rays from distant astrophysical sources. Therefore, building the theoretical framework for magnetic field amplification through plasma instabilities would enhance the understanding of these observational signatures.