comparing the densities of magnetic fields and radiation during
nucleosynthesis. In practice, considering that nuclear binding energies are of
the order of some MeV, which correspond (in thermal-statistical equilibrium)
to absolute temperatures of the order $T_{\text{NS}}\sim
1~{}\text{MeV}/(k_{B}=8.61\cdot 10^{-11}~{}\text{MeV~{}K}^{-1})\sim 10^{10}$ K
($k_{B}$ is the Boltzmann constant), we can estimate that nucleosynthesis
within the standard cosmological model takes place at redshift:222In the above
estimation, we have assumed that the cosmic radiation is found in
thermodynamic equilibrium, so that it can be approximated by the black-body
radiation model. In particular, the radiation density has to be proportional
to the fourth power of the cosmic fluid’s absolute temperature $T$, in
accordance with the Stefan-Boltzmann law,
$\rho_{\text{rad}}=\sigma_{SB}T^{4}\,,$ (4.3) where $\sigma_{SB}=5.670\times
10^{-8}~{}\text{W}~{}\text{m}^{-2}~{}\text{K}^{-4}$ represents the Stefan-
Boltzmann constant. Note that the combination of (the familiar)
$\rho_{\text{rad}}\propto a^{-4}$ and eq. (4.3) leads to the familiar relation
$T\propto a^{-1}$, for the cosmic temperature evolution.

$1+z_{\text{NS}}=\frac{T_{\text{NS}}}{T_{p}}\sim 10^{9}\,,\hskip
42.67912pt\text{which means that}\hskip 42.67912pta_{\text{NS}}\sim
10^{-9}a_{p}\,.$ (4.4)

Therefore, the ratio of magnetic over radiation density at nucleosynthesis
reads:

$\left(\frac{\rho_{B}}{\rho_{\text{rad}}}\right)_{\text{NS}}=\left(\frac{\rho_{B}}{\rho_{\text{rad}}}\right)_{p}\left(\frac{a_{\text{NS}}}{a_{p}}\right)^{6\lambda}=10^{-18}\cdot\left(10^{-9}\right)^{6\lambda}=10^{-54\lambda-18}\,,$
(4.5)

in accordance with (4.1a) and (4.4). A rough requirement for the magnetic
fields not to affect the present abundance of nuclei, is that the magnetic
density during nucleosynthesis would be of smaller order of magnitude than the
radiation density. In other words, the aforementioned requirement translates,
via eq. (4.5), into the condition:
$\boldsymbol{\lambda}\boldsymbol{\geq}\boldsymbol{-1/3}$.

####  4.2 Introductory remarks on magnetised Bianchi I cosmology

Bianchi cosmologies (see [7]-[13] for a list principal works) have
traditionally and thoroughly been studied due to their physically interesting
anisotropic features. In particular, the well known dipole anisotropy in the
cosmic microwave background along with the observation of large-scale
(intergalactic) magnetic fields, have both motivated the study of
(homogeneous) anisotropic cosmological models.
The simplest of the aforementioned models is the Bianchi I, whose isometry
group is produced by three commuting generators, Killing vector fields. Its
first exact general solution in vacuum space was found by Kasner [14] (some
variations include [15]-[16]) while, in the presence of dust, by Heckmann and
Schucking [17]. The latter behaves like the Kasner solution at the beginning
of cosmic evolution while it approaches an isotropic Friedmann regime at an
advanced stage of expansion. The Heckmann-Schucking solution has been
generalised to other kinds of isotropic perfect fluids [18]-[20] and to the
case of non vanishing cosmological constant [21, 22]. Finally, solutions in
the presence of homogeneous anisotropic fields333It is worth noting that exact
solutions with highly anisotropic geometry exist even in empty space or in
spaces filled with isotropic matter., in particular magnetic fields, have also
been explored, basically during the 60s [19], [23]-[27], but more recently as
well [28]. Although the magnetised Bianchi I is not envisaged a realistic
model of the universe, its study can shed new light on processes taking place
in very early cosmic stages. Its essential advantage over the FRW model is
that it allows for the natural incorporation of the magnetic field. Besides,
it is known that in the FRW model, magnetic fields are approximately
incorporated in the form of perturbations.
Of the various Bianchi models only I, II, III, $\text{VI}_{-1}$ and
$\text{VII}_{0}$ are known to be natural hosts of pure large-scale magnetic
fields [29]. As far as we have searched, all past works on magnetised Bianchi
I cosmology consider a diagonal spacetime metric. In fact, as we remark within
the main text, such a metric restricts by construction the orientation of the
magnetic field along one of the three independent spatial directions.
Moreover, most of the early relevant works (e.g. [19], [25], [27] or the more
recent [30]) derive a magnetic evolution formula based on the assumption that
the magnetic field is an eigenvector of the shear tensor. The assumption in
question translates practically into envisaging the magnetic field as the sole
source of spatial anisotropy. A common stronger (more restrictive and rarer)
assumption encountered in the modern literature, consists of neglecting the
shear contribution to magnetic evolution. The latter leads to the well known
inverse square law of magnetic variation (e.g. see [31] and [32]).
We begin the chapter’s main part with determining the magnetic evolution
formula and writing down explicitly Einstein equations for the Bianchi I
metric spacetime, filled with a magnetic field. We then derive the exact full
solution of Einstein equations for the model. Subsequently, we move to the
innovative part of this work, which involves the presentation of qualitative
solutions, describing in detail the small and large-scale cosmic limit.
Finally, we end up the chapter with a section introducing and revealing some
evolution features of the non-diagonal magnetised Bianchi I model.

####  4.3 Bianchi-I diagonal metric model with spatially homogeneous magnetic
field

Let us consider the simplest anisotropically expanding cosmological model,
namely the so-called Bianchi I, which has Euclidean spatial sections and is
known to allow for the existence of large-scale magnetic fields [33].

#####  4.3.1 Metric, energy-momentum tensor and magnetic field

The model’s diagonal metric reads:

$ds^{2}=dt^{2}-A^{2}(t)dx^{2}-B^{2}(t)dy^{2}-C^{2}(t)dz^{2}\,,$ (4.6)

where the average scale-factor is $a=\sqrt[3]{ABC}$. In covariant terms, the
only non-vanishing quantities in Bianchi I cosmologies are the relativistic
energy density and pressure, the anisotropic stress tensor, the volume scalar,
the shear and the electric Weyl tensor (i.e. $\rho$, $P$, $\pi_{ab}$,
$\Theta$, $\sigma_{ab}$ and $E_{ab}$ respectively) [16]. All the remaining
terms are zero by construction, namely
$\omega_{a}=0=\dot{u}_{a}=q_{a}=H_{ab}=\mathcal{R}_{ab}$ (with
$\mathcal{R}_{ab}=0$ implying Euclidean spatial sections). It is worth noting
that because of their non-zero anisotropic stress tensor ($\pi_{ab}\neq 0$)
Bianchi I models can generally host viscous fluids such as the electromagnetic
ones, however under the restriction of zero momentum density ($q_{a}=0$). In
case of an electromagnetic fluid the aforementioned limitation translates into
a zero Poynting vector, $q^{\text{(em)}}_{a}=\epsilon_{abc}E^{b}B^{c}=0$,
which means that on considering large-scale magnetic fields, the associated
electric components of the Maxwell field have to vanish. This means that the
Bianchi I cosmologies satisfy the MHD approximation by construction.
We recall that the Lagrangian and the energy-momentum tensor of the
electromagnetic field are (in the Heaviside-Lorentz system):

$L_{\rm(em)}=-\frac{1}{4}\,F_{ik}\,F^{ik}\hskip 14.22636pt\text{and}\hskip
14.22636ptT_{\
k}^{i}=-F^{il}\,F_{kl}+\frac{1}{4}\,\delta_{k}^{i}\,F_{lm}\,F^{lm}\,,$ (4.7)

where $F_{ab}$ denotes the Faraday tensor. In particular, we consider that our
model’s spacetime is filled with a homogeneous magnetic field along the
direction $z$. Therefore, the only non-vanishing component of the
electromagnetic field tensor is $F_{12}$. The sourceless set of Maxwell
equations, $\partial_{i}F_{jk}=0$ (in Riemannian framework), implies then that
$\partial_{0}F_{12}=0$, namely $F_{12}$ is constant. For the diagonal metric
(4.6), the only non-vanishing component of the fully contravariant
electromagnetic field tensor is thus given by:

$F^{12}=g^{11}\,g^{22}\,F_{12}\sim A^{-2}\,B^{-2}\,.$ (4.8)

The above expression means that all the contributions to the mixed components
of the energy-momentum tensor in eq. (4.7b) are proportional to
$A^{-2}\,B^{-2}$. On choosing a convenient parametrization, the individual
energy-momentum components can be written as:

$T_{\ 0}^{0}=-T_{\ 1}^{1}=-T_{\ 2}^{2}=T^{3}_{\
3}=\frac{\mathcal{B}_{0}^{2}}{A^{2}B^{2}}\,,$ (4.9)

where $\mathcal{B}_{0}^{2}$ is a positive constant characterizing the
intensity of the magnetic field. It is straightforward to check that the trace
$T$ of the energy-momentum tensor (4.9) vanishes, as it should.

#####  4.3.2 Einstein equations and evolution formulae

We are going to work with Einstein equations in the mixed tensor form:

$G^{i}_{\ j}\equiv R^{i}_{\ j}-\frac{1}{2}\,R\,\delta^{i}_{\ j}=T^{i}_{\
j}\,.$ (4.10)

Adopting the following parametrization for the scale factors:

$\begin{split}A(t)&=a(t)\,e^{\alpha(t)+\beta(t)}\,,\\\
B(t)&=a(t)\,e^{\alpha(t)-\beta(t)}\,,\\\
C(t)&=a(t)\,e^{-2\alpha(t)}\,,\end{split}$ (4.11)

where $a^{3}=ABC$ and the parameters $\alpha(t)$, $\beta(t)$ determine the
model’s anisotropy; the Ricci components read then:

$\displaystyle R_{\ 0}^{0}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-3\frac{\ddot{a}}{a}-6\dot{\alpha}^{2}-2\dot{\beta}^{2}\,,$
(4.12) $\displaystyle R_{\ 1}^{1}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{\ddot{a}}{a}-2\frac{\dot{a}^{2}}{a^{2}}-\ddot{\alpha}-3\dot{\alpha}\frac{\dot{a}}{a}-\ddot{\beta}-3\dot{\beta}\frac{\dot{a}}{a}\,,$
(4.13) $\displaystyle R_{\ 2}^{2}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{\ddot{a}}{a}-2\frac{\dot{a}^{2}}{a^{2}}-\ddot{\alpha}-3\dot{\alpha}\frac{\dot{a}}{a}+\ddot{\beta}+3\dot{\beta}\frac{\dot{a}}{a}\,,$
(4.14) $\displaystyle R_{\ 3}^{3}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{\ddot{a}}{a}-2\frac{\dot{a}^{2}}{a^{2}}+2\ddot{\alpha}+6\dot{\alpha}\frac{\dot{a}}{a}\,,$
(4.15)

whilst Einstein equations reduce therefore to:

$G_{\ 0}^{0}=-G^{1}_{\ 1}=-G^{2}_{\ 2}=G_{\
3}^{3}=\frac{B_{0}^{2}}{a^{4}}\,e^{-4\alpha}\,,\hskip
17.07164pt\text{where}\hskip
17.07164ptR=-6\frac{\ddot{a}}{a}-6\frac{\dot{a}^{2}}{a^{2}}-6\dot{\alpha}^{2}-2\dot{\beta}^{2}$
(4.16)

is the Ricci scalar. Note that the latter must vanish (implying
$\dot{\alpha}^{2}=\dot{\beta}^{2}/3$) since $T=0$.
Taking the difference of the mixed $11$ and $22$–components of Einstein’s
equations (4.16) with the expressions (4.13) and (4.14) for the Ricci tensor,
we obtain:

$R_{\ 1}^{1}-R_{\
2}^{2}=-2\ddot{\beta}-6\,\dot{\beta}\frac{\dot{a}}{a}=0\,,\hskip
17.07164pt\text{accepting the solution}\hskip
17.07164pt\dot{\beta}=\frac{\beta_{0}}{a^{3}}\,,$ (4.17)

just like in the case of Kasner and Heckmann-Schucking solutions [21]. Note
that it is due to our simplifying assumption: $B^{a}\parallel z^{a}$, that the
magnetic field does not enter the temporal evolution equation for $\beta$
parameter. Likewise, combining eqs (4.13), (4.14), and (4.15) yields to:

$R_{\ 1}^{1}+R_{\ 2}^{2}-2R_{\
3}^{3}=-6\ddot{\alpha}-18\,\dot{\alpha}\frac{\dot{a}}{a}=-\frac{4B_{0}^{2}}{a^{4}}\,e^{-4\alpha}\hskip
17.07164pt\text{or}\hskip
17.07164pt\ddot{\alpha}+3\dot{\alpha}\frac{\dot{a}}{a}=\frac{2\,B_{0}^{2}}{3\,a^{4}}\,e^{-4\alpha}\,.$
(4.18)

We observe that the magnetic field presence, introducing a preferred spatial
direction, directly affects the evolution of the metric (anisotropic)
parameter $\alpha$. Moreover, the combination:

$R_{\ 1}^{1}+R_{\ 2}^{2}+2R_{\
3}^{3}=-4\frac{\ddot{a}}{a}-8\frac{\dot{a}^{2}}{a^{2}}+6\dot{\alpha}\frac{\dot{a}}{a}+2\ddot{\alpha}=0\hskip
17.07164pt\text{leads to}\hskip
17.07164pt\ddot{\alpha}+3\dot{\alpha}\frac{\dot{a}}{a}=2\frac{\ddot{a}}{a}+4\frac{\dot{a}^{2}}{a^{2}}\,,$
(4.19)

a linear differential equation in terms of the anisotropy factor $\alpha$. By
multiplying both sides with the spatial volume, $V=a^{3}$, the above equation
recasts into:

$\frac{d}{dt}\left(\dot{\alpha}a^{3}\right)=\frac{d}{dt}\left(2\dot{a}\,a^{2}\right)\hskip
17.07164pt\text{which upon integration yields}\hskip
17.07164pt\dot{\alpha}=\frac{2\dot{a}}{a}+\frac{\alpha_{0}}{a^{3}}\,,$ (4.20)

where $\alpha_{0}$ is a constant. Also, making use of eqs. (4.18) and (4.19),
we arrive at:

$\frac{1}{a^{3}}\,\frac{d^{2}a^{3}}{dt^{2}}=\frac{\mathcal{B}_{0}^{2}}{a^{4}}\,e^{-4\alpha}\,.$
(4.21)

According to the above, we can determine $\alpha(t)$ for a given scale factor
$a(t)$. Note that the second time derivative of the spatial volume, $V=a^{3}$,
must always be positive. Substituting eqs. (4.12), (4.16b), (4.17), (4.20) and
(4.21) into the $(00)$–component of Einstein eqs. (4.16), we obtain:

$\frac{1}{a^{3}}\,\frac{d^{2}a^{3}}{dt^{2}}+9\frac{\dot{a}^{2}}{a^{2}}+12\frac{\alpha_{0}}{a^{3}}\frac{\dot{a}}{a}+3\,\frac{\alpha_{0}^{2}}{a^{6}}+\frac{\beta_{0}^{2}}{a^{6}}=0\hskip
17.07164pt\text{or}\hskip
17.07164ptV\,\ddot{V}+\dot{V}^{2}+4\alpha_{0}\dot{V}+3\alpha_{0}^{2}+\beta_{0}^{2}=0\,,$
(4.22)

which consists of a differential equation in terms of the spatial volume.
Remarkably, this equation is integrable. Before proceeding to its solution
however, we will present its detailed qualitative analysis in the first place.
To begin with, let us note that not all solutions of eq. (4.22) solve the
complete system of Einstein and Maxwell equations. In fact, eq. (4.22) gives:

$\ddot{V}=-\frac{1}{V}\left(\dot{V}^{2}+4\alpha_{0}\,\dot{V}+3\alpha_{0}^{2}+\beta_{0}^{2}\right)\,,$
(4.23)

which has to be positive according to (4.21). Since $V$ should always be
nonnegative, the positivity of $\ddot{V}$ implies that $\alpha_{0}\dot{V}<0$,
with $\alpha_{0}\neq 0$, and

$-2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}\,\leq\,\dot{V}\leq-2\alpha_{0}+\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}\,,\hskip
11.38109pt\text{where}\hskip 11.38109pt\alpha_{0}^{2}\geq\beta_{0}^{2}\,.$
(4.24)

According to the above condition, the model’s rate of contraction/expansion is
constrained between two limiting values.

#####  4.3.3 Qualitative analysis of the model’s small and large-scale limit

In the following we present a detailed, qualitative description of the model’s
behaviour at its small and large-scale limit. Our analysis is based on the
consideration of approximate solutions obeying the above presented evolution
constraints.

###### Contracting universe: small scale limit

Let us first consider $\alpha_{0}>0$, implying a contracting universe with
$\dot{V}<0$. One can start at a certain moment in time with a positive value
of $V$ and a negative value of $\dot{V}$, satisfying the inequality (4.24).
Since $\ddot{V}>0$, the time derivative of $V$ grows, remaining negative, and
the absolute value of $\dot{V}$ decreases, always obeying the constraint:

$\bigl{|}\dot{V}\bigr{|}\,\geq\,2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}\equiv
W_{1}>0\,.$ (4.25)

The universe will therefore reach the singularity characterised by $V=0$ in a
finite period of time. Subsequently, let us consider two time instants,
$t_{1}$ and $t_{2}$, such that

$V(t_{1})=0\quad\text{and}\quad\dot{V}(t_{2})=-W_{1}\,,$ (4.26)

and let us try to understand which happens first. Suppose that $t_{1}<t_{2}$,
so that $V$ vanishes while $\dot{V}$ still satisfies the inequality (4.24),
with $|\dot{V}|$ larger than the critical value $W_{1}$, in accordance with
eq. (4.25). Besides, the time $t_{1}$ cannot be infinite because the absolute
value of the time derivative is larger than $W_{1}$ and the function $V=V(t)$
reaches zero in a finite period of time. Therefore, we can approximate the
volume function for $t\lesssim t_{1}$ with expression:

$V\simeq\sigma\,(t_{1}-t)^{\lambda}\,,$ (4.27)

where $\sigma$ and $\lambda$ are positive constants. For $\lambda>1$, the
velocity becomes:

$\dot{V}\simeq-\lambda\,\sigma\,(t_{1}-t)^{\lambda-1}\to 0\quad{\rm for}\quad\
t\to t_{1}\,,$ (4.28)

which contradicts the condition (4.25). On the other hand, if $\lambda<1$, the
velocity in eq. (4.28) diverges for $t\to t_{1}$, which violates the bound
(4.24). The only possible choice left is $\lambda=1$, in which case we need
another term in the expansion around $t_{1}$, namely

$V\simeq\sigma\,(t_{1}-t)+\eta_{1}\,(t_{1}-t)^{\mu}\ ,$ (4.29)

where $\eta_{1}$ is a positive constant and $\mu>1$. Substituting the
associated expressions for $V$, $\dot{V}$, and $\ddot{V}$ into eq. (4.22), we
have:

$\mu\,\eta_{1}\,\left(\mu-1\right)\left[\sigma(t_{1}-t)+\eta_{1}(t_{1}-t)^{\mu}\right](t_{1}-t)^{\mu-2}\simeq-\left[\sigma+\mu\,\eta_{1}\,(t_{1}-t)^{\mu-1}-2\alpha_{0}\right]^{2}+\alpha_{0}^{2}-\beta_{0}^{2}\,.$
(4.30)

The leading term in the left-hand side above behaves as $(t_{1}-t)^{\mu-1}$,
which vanishes for $t\to t_{1}$. In the right-hand side, the leading term is
instead a constant, which should therefore vanish, so that

$\sigma^{2}-4\alpha_{0}\sigma+3\alpha_{0}^{2}+\beta_{0}^{2}=0\hskip
11.38109pt\text{for}\hskip 11.38109ptt\rightarrow t_{1}\,.$ (4.31)

One of the solutions of this equation is
$\boldsymbol{\sigma}\boldsymbol{=}\boldsymbol{W_{1}}$, which means that
$\dot{V}$ reaches the critical value $W_{1}$ at the same time when the volume
$V$ vanishes, so that $t_{1}=t_{2}$. Next, we equate terms of order
$(t_{1}-t)^{\mu-1}$ in eq. (4.30),

$\mu\,\eta_{1}(\mu-1)\,\sigma=-2\mu\,\eta_{1}(\sigma-2\alpha_{0})\,,\hskip
17.07164pt\text{which gives}\hskip
17.07164pt\mu=1+\frac{2\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}\,}}{2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}\,}}\,,$
(4.32)

where
$\boldsymbol{1}\boldsymbol{<}\boldsymbol{\mu}\boldsymbol{\leq}\boldsymbol{3}$.
As for the constant $\eta_{1}$, note that it takes different values depending
on the initial conditions. It is worth noting that we could also consider the
case of $\dot{V}$ reaching the critical value $-W_{1}$ at the moment
$t_{2}<t_{1}$ while the volume $V(t_{2})>0$. Nevertheless, a simple analysis
similar to that presented above, shows that this case is excluded. Thus, we
can say that for any contracting evolution, the universe hits the singularity
$V=0$ at some finite time $t_{1}$ when the velocity $\dot{V}$ reaches the
critical value $-W_{1}$.
From the (approximate) evolution law of the volume, we can determine the
anisotropy factors $\alpha(t)$ and $\beta(t)$. Using eqs. (4.17) and (4.20),
we directly find out that:

$\beta(t)=\beta_{0}\int\frac{dt}{V}\simeq-\frac{\beta_{0}\,\ln(t_{1}-t)}{2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\hskip
17.07164pt\text{and}\hskip
17.07164pt\alpha(t)\simeq\left(\frac{2}{3}-\frac{\alpha_{0}}{2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\right)\ln(t_{1}-t)\,.$
(4.33)

For definiteness, let us set $\beta_{0}\geq 0$. Using definitions (4.11), we
can write the three scale factors in the Kasner form:

$A(t)\\!\\!\sim\\!\\!\left(t_{1}-t\right)^{p_{1}}\,,\hskip
17.07164ptB(t)\\!\\!\sim\\!\\!\left(t_{1}-t\right)^{p_{2}}\,,\hskip
17.07164ptC(t)\\!\\!\sim\\!\\!\left(t_{1}-t\right)^{p_{3}}\,,$ (4.34)

where

$\displaystyle p_{1}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\alpha_{0}-\beta_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}{2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}<0\,,$
$\displaystyle p_{2}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\alpha_{0}+\beta_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}{2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}>0\,,$
(4.35) $\displaystyle p_{3}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}{2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}>0\,.$

It is straightforward to check that the exponents $p_{1}$, $p_{2}$, and
$p_{3}$ indeed satisfy the Kasner relations, i.e.

$p_{1}+p_{2}+p_{3}=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1\,,$ (4.36)

which means that the presence of the magnetic field does not change the
character of the singularity. Such a behaviour is plausible. In detail,
substituting expressions for $V$ and $\ddot{V}$ into eq. (4.21), one obtains
that the magnetic field contributes to Einstein’s equations the quantity:

$\frac{\mathcal{B}_{0}^{2}}{A^{2}B^{2}}\sim\frac{\mu\,\eta_{1}}{\sigma}\,(\mu-1)\,(t_{1}-t)^{\mu-3}\
,$ (4.37)

where $\mu-3>-2$. Therefore, the term (4.37) is weaker than the anisotropy,
which contributes a term of order $(t_{1}-t)^{-2}$ and dominates near the
singularity. Besides, the presence of matter less stiff than stiff matter is
well known to leave the Kasner type of singularity unaffected [34].

###### Expanding universe: small, large-scale limit and transition

Let us consider now the expanding universe. In this case, we have
$\alpha_{0}<0$ and therefore the time derivative $\dot{V}$ will be positive.
The expansion will last for $t\to\infty$ with the time derivative $\dot{V}$
approaching the critical value:

$W_{2}=-2\alpha_{0}+\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}>0.$ (4.38)

Thus, the behavior of $V=V(t)$ can be approximated in the limit $t\to\infty$
as:

$V\simeq W_{2}\,t-\eta_{2}\,t^{\nu}\ ,$ (4.39)

where $\eta_{2}$ is a positive constant and $0<\nu<1$ . Then, substituting the
associated expressions for $V$, $\dot{V}$, and $\ddot{V}$ in eq. (4.22), we
find once again that:

$\nu=1+\frac{2\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}\,}}{2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\
,$ (4.40)

with $\dfrac{1}{3}\leq\nu<1$. The anisotropy factors then read:

$\alpha=\left(\frac{2}{3}+\frac{\alpha_{0}}{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}-2\alpha_{0}}\right)\ln{t}\hskip
17.07164pt\text{and}\hskip
17.07164pt\beta=\frac{\beta_{0}\,\ln{t}}{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}-2\alpha_{0}}\,.$
(4.41)

Overall, the scale factors take once again the familiar Kasner form,

$A(t)\sim t^{p_{1}}\,,\quad B(t)\sim t^{p_{2}}\,,\quad C(t)\sim t^{p_{3}}\,,$
(4.42)

where

$\displaystyle p_{1}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}-\alpha_{0}+\beta_{0}}{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}-2\alpha_{0}}\,,$
$\displaystyle p_{2}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}-\alpha_{0}-\beta_{0}}{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}-2\alpha_{0}}\,,$
(4.43) $\displaystyle p_{3}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}-2\alpha_{0}}\,,$

which also satisfy Kasner relations (4.36). The presence of the magnetic field
does not influence the asymptotic structure of the metric at $t\to\infty$,
which therefore does not isotropize, unlike the Heckmann-Schucking solution
with dust [17, 21]. The reason for this phenomenon is clear. The energy
density of the magnetic field at $t\to\infty$ is given by:

$\frac{\mathcal{B}_{0}^{2}}{A^{2}B^{2}}\simeq\nu\,(\nu-1)\,\frac{\eta_{2}}{W_{2}}\,t^{\nu-3},$
(4.44)

where $\nu-3<-2$. Hence it remains weaker than the anisotropy term.
Subsequently, let us examine what happens with the expanding universe in its
distant past. One can suppose that it was born from the initial singularity at
$t=0$, when its volume was $V(0)=0$, and its time derivative had the smallest
critical value,

$\dot{V}(0)=W_{3}\equiv-2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}>0\,.$
(4.45)

In this case, the scale factors will be of the form of eq. (4.42), with the
Kasner exponents reading

$\displaystyle p^{\prime}_{1}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\alpha_{0}-\beta_{0}+\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}{2\alpha_{0}+\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\,,$
$\displaystyle p^{\prime}_{2}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\alpha_{0}+\beta_{0}+\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}{2\alpha_{0}+\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\,,$
(4.46) $\displaystyle p^{\prime}_{3}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}{2\alpha_{0}+\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\,.$

They again satisfy the Kasner relations (4.36).
Aiming to establish a relation between the set of Kasner indices at the
beginning and at the end of the evolution, it is convenient to use the
Lifshitz–Khalatnikov parametrization [35]. If the Kasner indices are ordered
as:

$p_{1}\leq p_{2}\leq p_{3},$ (4.47)

they can be represented by means of a real parameter $u\geq 1$ according to:

$\displaystyle p_{1}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{u}{1+u+u^{2}}\,,$ $\displaystyle p_{2}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle\frac{1+u}{1+u+u^{2}}\,,$ (4.48)
$\displaystyle p_{3}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{u\,(1+u)}{1+u+u^{2}}\,.$

The ordering (4.47) can be obtained, for example, by setting the anisotropy
parameters like:

$\alpha_{0}<0\,,\quad\beta_{0}<0\,,\quad|\beta_{0}|<\frac{3}{5}\,|\alpha_{0}|\,.$
(4.49)

In particular, this choice implies that the universe expands in the $y$ and
$z$ directions, but does so more rapidly along the direction $z$ of the
magnetic field, while it contracts along the $x$ axis. Combining eqs. (4.3.3)
and (4.49), we obtain:

$u^{\prime}=\frac{p_{3}^{\prime}}{p_{2}^{\prime}}=\frac{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}{|\alpha_{0}|+|\beta_{0}|-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\,.$
(4.50)

It is convenient to introduce the parameter
$\boldsymbol{\xi}\boldsymbol{\equiv}|\boldsymbol{\beta_{0}}|/|\boldsymbol{\alpha_{0}}|$,
which, when plugged into eq. (4.50), results in the relation:

$\xi=\frac{(u^{\prime}+1)^{2}-u^{\prime 2}}{(u^{\prime}+1)^{2}+u^{\prime
2}}\,.$ (4.51)

Note that if the parameter $\xi$ satisfies the conditions (4.49), then
$u^{\prime}$ satisfies the conditions $1<u^{\prime}<\infty$. Let us look now
at the Kasner exponents (4.43), describing the final stage of the cosmological
evolution. In this case, the order of the exponents is: $p_{3}\leq p_{1}\leq
p_{2}$. Therefore, their representation in terms of $u$ reads:

$\displaystyle p_{1}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle\frac{1+u}{1+u+u^{2}}\,,$ $\displaystyle p_{2}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle\frac{u\,(1+u)}{1+u+u^{2}}\,,$
(4.52) $\displaystyle p_{3}$ $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-\frac{u}{1+u+u^{2}}\,,$

where

$u=\frac{p_{2}}{p_{1}}=\frac{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}-\alpha_{0}-\beta_{0}}{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}-\alpha_{0}+\beta_{0}}\,.$
(4.53)

Substituting the formulae (4.50) and (4.51) into the equation above, we find:

$u=\frac{1+u^{\prime}}{u^{\prime}}<2\hskip 17.07164pt\text{or inversely}\hskip
17.07164ptu^{\prime}=\frac{1}{u-1}\,.$ (4.54)

Considering the temporal evolution towards the singularity (e.g. like in the
oscillating approach towards the cosmological singularity [9, 36]), we can see
that the universe passes through two transformations in the transition from
the parameter $u$ to the parameter $u^{\prime}$, according to eq. (4.54b). The
first transformation is characterised by the shift $u\to u-1$, in which the
roles of the $x$ and $z$ axes, corresponding to the exponents $p_{1}$ and
$p_{3}$ respectively, are exchanged. This transformation is called ‘change of
Kasner epoch’ [36]. As a result of this transformation, we arrive at a value
of the parameter $u-1<1$, see eq. (4.54). The next transformation is defined
by $u-1\to\dfrac{1}{u-1}$, which exchanges the roles of the axes $y$ and $z$,
and is called ‘change of Kasner era’.
We see that our solution displays a transition between two Kasner regimes
chacterized by the same law which is found in an empty Bianchi-II universe.
For a detailed description of the dynamics in the Bianchi-II universe in
General Relativity and other gravity models, see Ref. [37]. In Bianchi-VIII or
Bianchi-IX models, the universe passes through an infinite series of changes
of the Kasner epochs and eras. In our case, for the choice of parameters in
eq. (4.49), the law of transformation for the Kasner exponents (4.54b)
includes one change of Kasner epoch and one change of Kasner era.
On the other hand, it is worth considering the opposite relation between the
anisotropy parameters, to wit

$\alpha_{0}<0,\quad\beta_{0}<0,\quad\frac{3}{5}\,|\alpha_{0}|<|\beta_{0}|<1\,,$
(4.55)

or, in other words: $\dfrac{3}{5}<\xi<1$ . In this case, we have:

$u^{\prime}=\frac{p_{2}^{\prime}}{p_{3}^{\prime}}\,,\hskip
17.07164pt\text{corresponding to}\hskip
17.07164pt\xi=\frac{(u^{\prime}+1)^{2}-1}{(u^{\prime}+1)^{2}+1}\,,$ (4.56)

with $u=u^{\prime}+1$ or inversely $u^{\prime}=u-1$. The last two relations
show that we now have only a change of Kasner epoch.

#####  4.3.4 Exact evolution

Let us turn back to eq. (4.22), written now as:

$\frac{d}{dt}\left(V\,\dot{V}+4\alpha_{0}\,V\right)=-3\alpha_{0}^{2}-\beta_{0}^{2}\,,$
(4.57)

which can be analytically integrated through the variable change
$V(t)=X(t)\,t$. Using that, our equation recasts into the solvable form:

$\frac{X\,dX}{X^{2}+4\alpha_{0}\,X+3\alpha_{0}^{2}+\beta_{0}^{2}}=-\frac{dt}{t}\,.$
(4.58)

Upon integration of the above equation, we obtain:

$\displaystyle\frac{\alpha_{0}}{\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\left[\ln\biggl{(}1+\frac{X}{2\alpha_{0}+\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\biggr{)}-\ln\biggl{(}1+\frac{X}{2\alpha_{0}-\sqrt{\alpha_{0}^{2}-\beta_{0}^{2}}}\biggr{)}\right]$
$\displaystyle=-\frac{1}{2}\ln\left(1+\frac{X^{2}+4\alpha_{0}X}{3\alpha_{0}^{2}+\beta_{0}^{2}}\right)+\ln\left(\frac{t_{0}}{t}\right)\,,$
(4.59)

where integration constants are chosen so that $t_{0}>0$ is the time at which
the volume $V$ (hence $X$) vanishes. Unfortunately, eq. (4.3.4) cannot be
inverted to find $X$ as a function of $t$. Thus, we cannot make use of it to
find precise expressions for the anisotropy factors. However, we can use it to
study the approach to the singularity for $t\to t_{0}$ when $X\to 0$.
Expanding the left-hand side of eq. (4.3.4), we see that the leading term is
proportional to $X^{2}$, and the equation reduces to

$X^{2}(t)\simeq
2\left(3\alpha_{0}^{2}+\beta_{0}^{2}\right)\frac{t_{0}-t}{t_{0}}\,,$ (4.60)

which implies that

$V(t)\simeq\sqrt{2\bigl{(}3\alpha_{0}^{2}+\beta_{0}^{2}\bigr{)}t_{0}\,(t_{0}-t)}\hskip
17.07164pt\text{and}\hskip
17.07164pt\dot{V}\simeq-\sqrt{\frac{\bigl{(}3\alpha_{0}^{2}+\beta_{0}^{2}\bigr{)}t_{0}}{2(t_{0}-t)}}\to-\infty\,.$
(4.61)

Note that (4.61b) breaks the inequality (4.24). This means that solution
(4.60) corresponds to initial conditions which are incompatible with the
existence of a homogeneous magnetic field. However, solution (4.3.4) is
associated with a different regime, corresponding to a volume singularity
given by $t\to 0$, with $X(0)$ finite. In this case, the function $X(t)$ tends
to a positive constant which can be found from a quadratic equation, and
corresponds to the extremal points of the inequality (4.24). Hence, the regime
of singularity approach (vicinity of $t=0$) described above, is compatible
with the general solution of eq. (4.57).

####  4.4 Magnetised Bianchi I cosmology with non-diagonal metric

In the present section we address the following question: What happens to
cosmic evolution if we consider a magnetic field oriented along a general
spatial direction, instead of $z$?. To begin with, let us remark that in
general, the anisotropic-magnetic pressure $\pi_{ab}=-B_{\langle
a}B_{b\rangle}$, contributes some non-diagonal terms in the energy-momentum
tensor. Therefore, it is clear that a magnetic field of arbitrary direction is
incompatible with the diagonal Bianchi I metric, which predicts, via Einstein
equations, a diagonal energy-momentum tensor. Wishing to consider an arbitrary
magnetic field, we study the non-diagonal Bianchi I case.

#####  4.4.1 Metric, energy-momentum tensor and magnetic field

In general, a homogeneous-anisotropic (Bianchi type) spacetime model can be
described by the synchronous frame metric [7]:

$ds^{2}=dt^{2}-\gamma_{\alpha\beta}(t)dx^{\alpha}dx^{\beta}\,,$ (4.62)

where $\gamma_{\alpha\beta}(t)$ represents the spatial metric. The dot
derivative of the latter defines the extrinsic curvature tensor
$\kappa_{\alpha\beta}\equiv\dot{\gamma}_{\alpha\beta}$ (with
$\kappa\equiv\kappa^{\alpha}_{\alpha}$ we denote its trace). Taking now into
account that the contravariant component of the magnetic field is proportional
to the inverse square root of the spatial metric’s determinant [38], we
determine its density evolution rate444Note that in this general (non-
diagonal) case, we determine evolution rates in terms of the metric
determinant, which is now used as a scale factor.:

$B^{a}\propto\frac{\mathcal{B}_{0}}{\sqrt{\gamma}}\hskip
17.07164pt\text{and}\hskip
17.07164pt\mathcal{B}^{2}=B^{a}B_{a}=g_{ab}B^{a}B^{b}\propto\frac{\mathcal{B}_{0}}{\gamma}\,.$
(4.63)

Assuming that the spacetime in question is filled with the magnetic field
$B^{a}$, Einstein equations for the Bianchi I model read (refer to [7] for
details):

$R^{0}_{0}-\frac{1}{2}R\delta^{0}_{0}=\frac{\mathcal{B}^{2}}{2}\hskip
5.69054pt\rightarrow\hskip
5.69054pt\kappa^{2}-\kappa_{\alpha}^{\beta}\kappa_{\beta}^{\alpha}=4\mathcal{B}^{2}=\frac{4\mathcal{B}^{2}_{0}}{\gamma}$
(4.64)

and

$R_{\alpha}^{\beta}-\frac{1}{2}R\delta_{\alpha}^{\beta}=T_{\alpha}^{\beta}\hskip
5.69054pt\rightarrow\hskip
5.69054pt\dot{\kappa}_{\alpha}^{\beta}-\dot{\kappa}\delta_{\alpha}^{\beta}+\frac{\kappa}{2}\kappa_{\alpha}^{\beta}-\frac{1}{4}\left(\kappa^{2}+\kappa_{\gamma}^{\delta}\kappa_{\delta}^{\gamma}\right)\delta_{\alpha}^{\beta}=\frac{1}{2}B_{\alpha}B^{\beta}+\frac{1}{4}\mathcal{B}^{2}\delta_{\alpha}^{\beta}\,,$
(4.65)

where we have considered the magnetic density $\mathcal{B}^{2}/2$, isotropic
and anisotropic pressure, $\mathcal{B}^{2}/6$ and
$-B_{a}B_{b}+(1/3)\mathcal{B}^{2}h_{ab}$, respectively. Taking now the trace
of (4.65), we find out that

$\dot{\kappa}+\frac{1}{8}\kappa^{2}+\frac{3}{8}\kappa_{\alpha}^{\beta}\kappa_{\beta}^{\alpha}=-\frac{3\mathcal{B}^{2}}{8}=-\frac{3\mathcal{B}^{2}_{0}}{8\gamma}\,.$
(4.66)

Then, multiplying (4.64) by $3/8$ and adding the result to (4.66), we arrive
at the equation:

$\dot{\kappa}+\frac{1}{2}\kappa^{2}=\frac{b^{*}}{\gamma}\hskip
5.69054pt\rightarrow\hskip 5.69054pt\gamma\hat{y}+y=\frac{2b^{*}}{\gamma}\,,$
(4.67)

where $b^{*}\equiv(9\mathcal{B}^{2}_{0})/8$ is a constant, $y\equiv\kappa^{2}$
and the hat denotes differentiation with respect to $\gamma$.

#####  4.4.2 Evolution formula and limits

Note that (4.67) is a linear differential equation in reference to $\gamma$,
the integration of which directly leads to:

$y\equiv\kappa^{2}=\frac{1}{\gamma}\ln{\left(c_{1}\gamma\right)}^{2b^{*}}\hskip
14.22636pt\text{or equivalently to}\hskip
14.22636pt\dot{\gamma}=\sqrt{2b^{*}\gamma}\sqrt{\ln{\left(c_{1}\gamma\right)}}\,.$
(4.68)

The last equation takes the integrable form:

$\frac{d\gamma}{\sqrt{2b^{*}\gamma}\sqrt{\ln{\left(c_{1}\gamma\right)}}}=dt\hskip
5.69054pt\rightarrow\hskip
5.69054ptt=\frac{1}{\sqrt{b^{*}}}\int{\frac{dx}{\sqrt{\ln{\left(\sqrt{c_{1}}|x|\right)}}}}\,,$
(4.69)

where $x\equiv\sqrt{\gamma}\geq 0$. Subsequently, under the variable change
$\omega\equiv\sqrt{\ln{\left(\sqrt{c_{1}|x|}\right)}}$, our integral
eventually recasts into the Gaussian form:

$t=\sqrt{\frac{\pi}{b^{*}c_{1}}}\int{\frac{2e^{\omega^{2}}}{\sqrt{\pi}}~{}d\omega}\hskip
5.69054pt\rightarrow\hskip 5.69054ptt\approx
e^{\omega^{2}}\approx\sqrt{\gamma}\hskip 11.38109pt\rightarrow\hskip
5.69054pt\boldsymbol{\gamma}\mathbf{\propto t^{2}}\hskip
11.38109pt(\text{for}~{}t\rightarrow 0~{}\text{or}~{}\infty)\,.$ (4.70)

The above shows that the model’s scale factor (i.e. the metric determinant)
increases parabolically with cosmic time at the limits. In deriving the last
expression, we have made use of the following approximation at the limits:

$t=c\int{e^{\omega^{2}}~{}d\omega}\approx\int{e^{\omega^{2}+\ln{\omega}+\ln{c}}~{}d\omega}\approx\int{\omega~{}e^{\omega^{2}}~{}d\omega}\approx
e^{\omega^{2}}\hskip 11.38109pt\text{for}~{}~{}\omega\rightarrow\pm\infty\,.$
(4.71)

In comparison to the diagonal magnetised model, we observe that once again the
behaviour at the limits remains of Kasner type. According to eq. (4.70), in
the non-diagonal case, cosmic time is an error function of the metric
determinant:

$t=\sqrt{\frac{\pi}{b^{*}c_{1}}}\text{erfi}\left(\sqrt{\ln{\left(\sqrt{c_{1}\gamma}\right)}}\right)\,,$
(4.72)

where $k_{1}$, $k_{2}$ are constants. Subsequently, let us check whether it is
possible to get a particular solution for the model’s metric. In the first
place, taking the dot derivative of (4.68a), we deduce an expression for
$\dot{\kappa}$ and, via eq. (4.66), an expression for
$\kappa_{\alpha}^{\beta}\kappa_{\beta}^{\alpha}$:

$\dot{\kappa}=-\frac{1}{2\gamma}\ln{\left(c_{1}\gamma\right)}^{2b^{*}}+\frac{b^{*}}{c_{1}\gamma}\hskip
17.07164pt\text{and}\hskip
17.07164pt\kappa_{\alpha}^{\beta}\kappa_{\beta}^{\alpha}=\ln{\left(C\gamma\right)}^{\frac{2b^{*}}{\gamma}}\,,$
(4.73)

with $c_{1}$ and $C$ constants. From the above formulae we find out that the
extrinsic curvature has the following form:

$\kappa_{\alpha}^{\beta}=\sqrt{\ln{\left(C\gamma\right)}^{\frac{2b^{*}}{\gamma}}}\lambda_{\alpha}^{\beta}\hskip
14.22636pt\text{with}\hskip
14.22636pt\lambda_{\alpha}^{\beta}\lambda_{\beta}^{\alpha}=1\hskip
14.22636pt\text{and}\hskip 14.22636pt\text{Tr}~{}\lambda=1\,,$ (4.74)

where $\lambda_{\alpha}^{\beta}$ is a $3\times 3$ symmetric matrix with real
constant elements. Given that any real symmetric matrix is diagonalisable, we
can write the diagonal form of $\lambda_{\alpha}^{\beta}$ as
$\text{diag}(p_{1},p_{2},p_{3})$, where $p_{1},p_{2},p_{3}$ (adopting the
Kasner notation) are its eigenvalues, satisfying the constraints:

$p_{1}+p_{2}+p_{3}=1\hskip 17.07164pt\text{and}\hskip
17.07164ptp^{2}_{1}+p^{2}_{2}+p^{2}_{3}=1\,.$ (4.75)

The aforementioned conditions are a direct consequence of (4.74c) and (4.74b).
Now lowering indices in (4.74a), using the $3$-D metric
$\gamma_{\alpha\beta}$, and deploying its diagonal form, we get:

$\kappa_{\alpha\alpha}\equiv\dot{\gamma}_{\alpha\alpha}=p_{\alpha}\sqrt{\ln{\left(C\gamma\right)}^{\frac{2b^{*}}{\gamma}}}\gamma_{\alpha\alpha}=p_{\alpha}\sqrt{\frac{2b^{*}}{\gamma}}\sqrt{\ln{\left(C\gamma\right)}}\gamma_{\alpha\alpha}\,,$
(4.76)

where the repeated index $\alpha$ does not imply summation here, it just
denotes the diagonal components. Although we can not solve (4.72) for $\gamma$
in the general case, we can make use of the small and large-scale limit, so
that (4.76) transforms into

$\frac{d\gamma_{\alpha\alpha}}{\gamma_{\alpha\alpha}}=\frac{\mathcal{D}}{t}\sqrt{\ln{\left(Ct^{2}\right)}}~{}dt\,,\hskip
5.69054pt\text{which upon integration yields to}\hskip
5.69054pt\gamma_{\alpha\alpha}=\gamma_{0}e^{\frac{\mathcal{D}}{3}\ln^{3/2}{\left(Ct^{2}\right)}}\,,$
(4.77)

where $\mathcal{D}\equiv p_{\alpha}\sqrt{\frac{2b^{*}}{\gamma_{0}}}$ is a
constant. The above approximate solution determines the rate of change of the
diagonal metric components at the limits. The question regarding the non-
diagonal components remains open.

#####  4.4.3 Concluding remarks

Throughout our analysis we have seen that the magnetic field affects cosmic
evolution in a double manner; first, directly as an energy density source;
second, indirectly as a source of anisotropy (due to its vector nature),
contributing to the evolution of the anisotropy metric parameters $\alpha$ and
$\beta$. In other words, it manifests its presence in both temporal (as energy
density) and spatial Einstein equations (via its anisotropic pressure).
In the diagonal case-compatible with magnetic field along a sole spatial
direction-, we have derived the full general solution under a clarifying and
convenient parametrisation. More importantly, our detailed qualitative
analysis of the small and large-scale cosmic limit has revealed the model’s
Kasner type behaviour at both limits, as well as the transition between those
via a ‘change of Kasner epoch’ and a subsequent ‘change of Kasner era’.
In the non-diagonal case-allowing for a generally oriented magnetic field-,
the general solution is given by cosmic time as a kind of error function of
the average scale factor (recall eq. (4.72)).

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### Last word

Wherever they appear in our universe, either in astrophysical or in
cosmological environments, electromagnetic fields are impelled by gravity to
manifest their extraordinary features. The phenomena arising from the gravito-
electromagnetic coupling await our exploration.