Bianchi I Solutions in f(T) Gravity

• The paper presents f(T) modifications to Bianchi I models, extending GR by incorporating nonlinear torsion terms and new dynamical degrees of freedom.
• Analytic methods reveal vacuum, perfect-fluid, and bouncing scenarios where anisotropy decays, ultimately guiding the universe toward isotropic, de Sitter–like acceleration.
• The study underscores the role of remnant gauge symmetry in teleparallel theory, highlighting both its potential for modeling cosmic acceleration and challenges in physical viability.

Bianchi I cosmological solutions in $f(T)$ gravity comprise a class of spatially homogeneous, anisotropic models constructed in the teleparallel framework with generalized torsion-based dynamics. These solutions extend the classical Bianchi I models of General Relativity (GR) by promoting the gravitational Lagrangian from the torsion scalar $T$ to a nonlinear function $f(T)$, introducing new degrees of freedom and distinctly nontrivial dynamical features. The underlying geometry is characterized by distinct scale factors along independent spatial directions, allowing the early universe's anisotropies and their decay to be tracked in detail. The $f(T)$ paradigm remains second order in derivatives, in contrast to $f(R)$ models, but leads to important subtleties in gauge structure and degrees of freedom.

1. Formulation: Teleparallel Gravity and Bianchi I Ansatz

The Bianchi I metric in $(1+d)$-dimensions is given by

$$ds^2 = -N(t)^2 dt^2 + \sum_{i=1}^{d} a_i(t)^2 (dx^i)^2,$$

with associated diagonal tetrad

$$e^A_\mu = \mathrm{diag}(N(t), a_1(t), a_2(t), \ldots, a_d(t)),$$

where $a_i(t)$ are the directional scale factors and $N(t)$ the lapse. The torsion scalar—central to $f(T)$ models—is

$${\mathbb T} = \frac{1}{N^2} \sum_{i=1}^d \frac{\dot{a}_i}{a_i} \sum_{j \neq i} \frac{\dot{a}_j}{a_j} = 2\sum_{i<j} H_i H_j,$$

where $H_i = \dot{a}_i / a_i$ are the directional Hubble parameters (Golovnev et al., 21 Aug 2025).

For standard $f(T)$ gravity, the action

$$S = \frac{1}{16\pi G} \int d^{d+1} x\, N\, \left[\prod_{i} a_i\right]\, f({\mathbb T})$$

is varied with respect to the tetrad, yielding two principal equations: a Hamiltonian (temporal) constraint,

$$2{\mathbb T}f'({\mathbb T}) - f({\mathbb T}) = 16\pi G\,\rho,$$

and spatial equations for each direction $i$,

$$-\left(\dot{H}_i + H_i\sum_{j\neq i}H_j\right) f'({\mathbb T}) - \left(\sum_{j\neq i}H_j\right)f''({\mathbb T})\dot{\mathbb T} = 8\pi G (\rho + p_i),$$

where $p_i$ are the directional pressures (Golovnev et al., 21 Aug 2025).

2. Phase Structure: Vacuum, Matter, and Radiation-Dominated Solutions

The functional form of $f(T)$ and the matter content determine the evolution structure:

• Vacuum ($\rho=p_i=0$): The constraint enforces constant torsion scalar ($\dot{\mathbb T}=0$). For $f(T)=T$, Bianchi I reduces to the Kasner solution:

$$a_i(t) \propto t^{p_i}, \qquad \sum_i p_i = 1, \quad \sum_i p_i^2 = 1,$$

with the standard Kasner exponents (Paliathanasis et al., 2016, Skugoreva et al., 2017).

• Power-law $f(T)$ corrections: For $f(T) = (-T)^n$, the Kasner relations generalize to

$\sum_i p_i = 2n-1, \qquad \sum_i p_i^2 = 2n-1,$

so the GR limit is recovered for $n=1$. Solutions with $n \neq 1$ yield new evolutions where anisotropies and the decay thereof can differ substantially (Paliathanasis et al., 2016).

• Perfect fluids: The system admits analytic matter- and radiation-dominated solutions. For dust ($\omega=0$) and radiation ($\omega=1/3$), conservation equations and scale factor relations yield reconstructions of $f(T)$ corresponding to those phases (Sharif et al., 2011). Mixtures allow interpolation between epochs.
• Dark energy ($\omega=-1$): The conservation law yields constant energy density and, crucially, a constant $F(T)$ solution:

$F(T) = 2\kappa \rho_d.$

This acts as an effective cosmological constant, driving late-time acceleration and providing a teleparallel analogue of $\Lambda$CDM (Sharif et al., 2011).

3. Isotropization, Late-Time Acceleration, and Attractor Behavior

A consistent finding across analytic and numerical studies is the emergence of a future isotropic attractor. Regardless of the initial anisotropy, the system's trajectories converge toward an isotropic, accelerating (de Sitter–like) solution in the presence of appropriately chosen $f(T)$. In particular:

• The anisotropy parameter (e.g., $\Delta(t)$ or combinations of Hubble deviations (Rodrigues et al., 2013, Tretyakov, 2021)) decays with time, signaling an approach to isotropy.
• At late times, directional EoS skewness parameters and normalized shear also vanish, substantiating isotropic attractor behavior even out of initial Bianchi I anisotropy (Rodrigues et al., 2013, Rodrigues et al., 2014).
• The equation of state parameter $\omega(t)$ for the effective fluid can settle at $-1$ (cosmological constant limit), or vary in the regime $-1<\omega<0$ (quintessence-like), depending on the model and parameters. Transitions from radiation or matter dominance to dark energy-dominated acceleration are readily modeled (Sharif et al., 2011, Rodrigues et al., 2014).

Notably, in vacuum, the Kasner solution is an exact solution if $T=0$ (the "Kasner branch" in $f(T)$ gravity), while for nonzero constant $T$ (the other constraint branch in polynomial models) the Kasner regime appears asymptotically near singularities (Skugoreva et al., 2017).

4. Bouncing Cosmologies and Non-Singular Solutions

Bianchi I cosmological equations in $f(T)$ admit families of bouncing solutions—that is, cosmologies which connect a contracting phase to an expanding one through a non-singular bounce (the moment when the averaged Hubble parameter $H$ crosses zero) (Tretyakov, 2021). These features arise naturally:

• The presence of matter with $w>-1$ ensures that at the bounce, the time derivative of the energy density vanishes, and the field equations remain well-defined.
• Numerical studies reveal that bounce solutions are robust to variations in $f(T)$ (including quadratic corrections) and in the equation of state. Post-bounce, the universe isotropizes and enters an accelerated expansion ("future attractor") (Tretyakov, 2021).
• Analytical singularity analysis, using the method of movable singularities, confirms that while the FLRW case admits Laurent expansions at singularities, for Bianchi I the Kasner solution acts as a fixed singular regime rather than a movable singularity (Paliathanasis et al., 2016).

5. Influence of Model Construction: Linear vs. Nonlinear $f(T)$, Spinor Fields, and Cosmic Strings

• In the linear model ($f(T) = T$, i.e., TEGR), the field equations reduce to those of GR, with anisotropic initial conditions decaying toward isotropy and mimicking dark energy behavior (Rodrigues et al., 2013, Rodrigues et al., 2014). Cosmic string networks can be incorporated and are shown to disappear dynamically as the universe expands (Shekh et al., 2020).
• Nonlinear $f(T)$ models (e.g., $f(T) = T_0 + T_1 e^{T}$ or quadratic/power-law forms) further suppress early-time anisotropies and yield effective torsion contributions that facilitate earlier isotropization, aligning better with CMB and large-scale structure observations (Rodrigues et al., 2013, Rodrigues et al., 2014, Tretyakov, 2021).
• Spinor fields, when coupled in $f(T)$ gravity, can reconstruct a wide suite of dark energy scenarios, including Chaplygin gas and $\Lambda$CDM-like expansions, with implications for anisotropic stress control in Bianchi I backgrounds (Myrzakulov et al., 2013).

6. Gauge Structure, TP Connection Unpredictability, and Physical Viability

A critical structural issue is the unpredictability or underdetermination of the teleparallel connection in $f(T)$ gravity. For diagonal metrics and tetrads—even in isotropic FLRW spacetimes—one can apply arbitrary time-dependent spatial rotations to the tetrad without affecting the metric, torsion scalar, or dynamical equations (Golovnev et al., 21 Aug 2025). This reflects a "remnant symmetry" beyond what is found in TEGR:

• The antisymmetric part of the field equations identically vanishes for these transformations, so the dynamical system cannot fix the evolution of these extra degrees of freedom.
• As a consequence, the teleparallel connection lacks uniqueness, potentially undermining the physical predictivity of $f(T)$ gravity for cosmological modeling unless further gauge-fixing or additional constraints are imposed.
• This is not merely a technical redundancy: uncontrolled propagation of these extra modes may lead to pathologies such as strong coupling or instabilities that are absent in TEGR or $f(R)$ gravity (Golovnev et al., 21 Aug 2025).

7. Observational and Theoretical Implications

Bianchi I $f(T)$ cosmologies provide a flexible framework for modeling early-universe anisotropies, cosmic acceleration, and transitions through multiple cosmological epochs. Their relevance to observation includes:

• Natural isotropization mechanisms compatible with the small observed CMB quadrupole and EoS parameter constraints (Rodrigues et al., 2013, Rodrigues et al., 2014).
• Late-time acceleration with or without a prescribed cosmological constant (Sharif et al., 2011, Rodrigues et al., 2014, Tretyakov, 2021).
• Quintessence regions and the entire spectrum of $\omega$ evolution, depending on the functional form of $f(T)$ (Sharif et al., 2011, Shekh et al., 2020).
• Viable descriptions of string-dominated early epochs and their smooth decay, aligning with CMBR data (Shekh et al., 2020).
• A large class of exact analytical solutions, including de Sitter and Kasner regimes, as well as the possibility of extra-dimensional models with stabilized compact dimensions (Golovnev et al., 21 Aug 2025).

However, all physical predictions must be scrutinized in light of the indeterminacy of the teleparallel connection, which currently poses one of the most significant challenges for the viability of $f(T)$ gravity as an alternative cosmological paradigm.

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In summary, Bianchi I cosmological solutions in $f(T)$ gravity serve as a rich testing ground for anisotropic dynamics, cosmic acceleration mechanisms, and the detailed structure of teleparallel modified gravity. They combine analytic solvability with phenomenological versatility but reveal fundamental issues associated with gauge structure and the uniqueness of the gravitational dynamics that must be addressed in ongoing research (Sharif et al., 2011, Rodrigues et al., 2013, Myrzakulov et al., 2013, Rodrigues et al., 2014, Paliathanasis et al., 2016, Skugoreva et al., 2017, Shekh et al., 2020, Golovnev et al., 2020, Tretyakov, 2021, Golovnev et al., 21 Aug 2025).