Minimal Asymmetric Expansion of the Universe: A Friedmann-Sasaki-Randers Spacetime Framework Incorporating Dark Matter and Dark Energy

Abstract

In this work, we investigate the structure of the gravitational field and the Friedmann equations in a Sasaki–Randers spacetime, incorporating dark matter and dark energy within the Lorentz tangent bundle of spacetime. Additional terms arising in the geodesic equations and the Friedmann equations modify and generalize their standard forms in general relativity. Within this geometrical framework, the intuitive notion of distance as a symmetric and observer-independent quantity breaks down in relativistic and cosmological settings, where measurement procedures are time-extended, observer-dependent, and constrained by causal structure. Consequently, distance naturally acquires an asymmetric character, such that, in general, $F(A,B)\ne F(B,A)$. This intrinsic asymmetry admits a natural mathematical description in Finsler geometry, particularly through the Randers metric, where directional dependence introduces a preferred geometric flow. Within this framework, as developed in Finsler–Randers cosmology, anisotropy emerges not from matter distributions or initial conditions, but from the underlying structure of spacetime itself. The equivalence principle is also expanded with extra internal conditions on the tangent structure of spacetime and the Raychaudhuri equations are derived for this model. As a result, the universe appears observationally nearly isotropic, with only minimal anisotropies arising due to the direction-dependent nature of distance and cosmic expansion, while the standard notion of isotropy is recovered only as an approximation.

1. Introduction

On cosmological scales, observations of the Cosmic Microwave Background (CMB) indicate that the Universe is remarkably homogeneous and isotropic to a high degree of accuracy [1]. Nevertheless, on smaller scales, significant deviations from homogeneity and isotropy arise due to the presence of galaxies, galaxy clusters, super clusters, cosmic filaments, and large cosmic voids. Understanding how these structures emerge and evolve remains one of the central challenges of modern cosmology. Furthermore, large-scale observations of the Universe, particularly from CMB measurements and galaxy surveys, reveal small persistent deviations from statistical isotropy, e.g., [2,3,4,5], including hemispherical power asymmetry, which can be interpreted as a geometric imprint of anisotropic spacetime structure, and preferred directional alignments [6,7,8,9]. Dark matter and dark energy constitute two of the most profound unresolved problems in contemporary cosmology, together accounting for approximately 95% of the total energy content of the Universe. Dark matter, generally assumed to consist of particles beyond the Standard Model of particle physics, plays a fundamental role in structure formation, galactic dynamics, and the evolution of the large-scale cosmic web through its gravitational influence. Dark energy, on the other hand, raises fundamental questions concerning the nature of spacetime, possible modifications of General Relativity, and the properties of the quantum vacuum. Recent progress enabled by next-generation CMB observations, large-scale galaxy surveys, gravitational-wave astronomy, and other precision cosmological probes offers unprecedented opportunities to constrain the properties and interactions of these elusive components [2,7,10]. In this context, the investigation of extended Friedmann equations and higher-dimensional gravitational theories is essential for advancing our understanding of spacetime structure and cosmological evolution. Generalizations of the standard cosmological framework, including brane-world and dimensional models, provide promising approaches for addressing key problems such as primordial inflation, late-time cosmic acceleration, and the nature of dark matter and dark energy. These extended theories may offer a unified geometric framework capable of explaining observational phenomena that remain beyond the scope of the standard cosmological model.

Modified gravity theories do not replace General Relativity; rather, they extend its domain of applicability and explore regimes where the standard theory may be incomplete. These extensions are achieved through various geometric and dynamical modifications, leading to generalized field equations and novel cosmological consequences. Consequently, modified theories of gravity have become an essential component of modern cosmology, offering alternative explanations for observational phenomena that remain challenging within the standard framework [2,3]. Furthermore, if gravity is assumed to be geometrically incorporated within a metric spacetime extended to include acceleration degrees of freedom, its description can be formulated within the framework of generalized gravitational theories. Such an approach naturally leads to extensions of the conventional Riemannian geometry of spacetime. In these generalized geometric frameworks, including Finsler and Finsler-type geometries and tangent-bundle extensions of spacetime, anisotropic cosmological models emerge as natural consequences of the underlying geometric structure. These theories provide new perspectives for describing the gravitational field and of cosmic evolution with possible deviations from isotropy. We mention some remarkable works with different approaches and results in the field of applications of Finsler, Lagrange and Finsler-like geometries in gravitation and cosmology [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33].

The possibility that spacetime itself possesses anisotropic properties has attracted considerable interest in recent years [6,34]. A possible source of cosmic anisotropy is a primordial vector field, such as a cosmological magnetic field, which may induce directional dependence in the cosmic expansion rate. As the Universe evolves, possible internal dynamical mechanisms connected with thermodynamic variables could gradually suppress the bulk of anisotropies, leading to the nearly isotropic Universe observed today. Elucidating the relationship between primordial anisotropy and the observed anisotropies of the CMB remains a major challenge and may reveal previously unknown physical processes [35]. Anisotropy can also arise from several physical and geometrical mechanisms. The directional asymmetry of mass particles has also been investigated [9,36], while cosmic expansion itself may exhibit directional dependence influenced by primordial magnetic fields or asymmetric galaxy rotation [34,37]. Extra- dimensional theories extend General Relativity by introducing additional degrees of freedom, providing alternative explanations for cosmic evolution, gravity, and high-energy phenomena. The anisotropy observed in the Universe may also be directly related to its matter content [38]. In particular, anisotropies generated during cosmic expansion could contribute to particle creation processes through mechanisms activated under conditions of rapidly varying spacetime geometry, potentially leading to anisotropic dark-energy effects. Furthermore, anisotropy may provide valuable insights into the nature of dark matter and the formation of large-scale cosmic structures. In addition, anisotropy may also emerge through interactions between the internal and external structures of spacetime. In this framework, vector fields play a fundamental role in modifying the spacetime metric and influencing the geometric properties of the Universe. A primordial vector field embedded within spacetime may generate anisotropic effects and influence gravitational phenomena, including horizon structures. From a different perspective, several studies [6] have proposed ellipsoidal-universe models to explain cosmic anisotropies, relating CMB quadrupole anomalies and cosmic parallax to intrinsic geometric properties. In addition, the significant rotational asymmetry identified in galaxies observed by the Advanced Deep Extragalactic Survey [37] suggests that early-Universe structures may have influenced galaxy formation and could contribute to the observable Hubble tension.

Recent investigations of the Hubble constant ($H_0$) tension indicate that the continuous discrepancy between early- and late-Universe measurements may point toward new physics beyond the standard $\Lambda$-CDM cosmological model. Consequently, anisotropic cosmologies, modified gravity theories, and generalized spacetime geometries have become promising theories for addressing this problem and for advancing our understanding of the fundamental laws governing the Universe. In this framework one can proceed to more compelling geometrical modifications, namely use Finsler and Finsler-like geometries, which have richer structure than the Riemannian framework, and use them in order to construct gravitational and cosmological theories to interpret different kinds of physical phenomena [2,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58]. In recent years, research on the applications of Finsler, Lagrange and Finsler-like geometries to gravitation and cosmology has drawn the attention of many researchers from various remarkable perspectives; some of these are cited in Large-scale observations of the Universe, particularly from CMB measurements and galaxy surveys, reveal persistent deviations from statistical isotropy. These anomalies, confirmed by multiple independent analyses, challenge the assumptions of the standard $\Lambda$-CDM model and motivate the consideration of more general geometric frameworks in which anisotropy may be intrinsic rather than emergent.

Observational cosmology reveals large-scale small anomalies in the cosmic microwave background and directional features in galaxy surveys. While such effects can be partially explained within the standard isotropic FLRW framework through cosmic variance, structure formation, and observational biases, their persistence motivates the exploration of more general geometric descriptions of spacetime. In this context, anisotropic frameworks, such as Finsler–Randers geometry, provide a natural extension in which direction-dependent effects and time asymmetry [36] can be incorporated at a fundamental geometrical level rather than treated as perturbations.

A natural extension of the Riemannian structure of spacetime is also provided by the Finsler–Randers metric, originally introduced by Gunnar Randers in “On an Asymmetrical Metric in the Four-Space of General Relativity” (1941) [59]. In this formulation, spacetime is described by a metric with intrinsic asymmetry, motivated by the unidirectional character of time. The Randers metric represents the simplest nontrivial generalization of a Riemannian metric that incorporates such asymmetry.

This geometric framework acquires physical relevance in the presence of anisotropies or when velocity-dependent effects are significant, as in phase-space descriptions. Accordingly, the Randers metric was first applied in cosmology in [19], and later by other authors [26,30,54,56,57]. In this case, FR-spacetime describes the work/power or the energy that the Riemannian spacetime expends entirely in a definite direction.

A further generalization of the Finsler–Randers (FR) cosmological model can be formulated on the Lorentz tangent bundle of spacetime by introducing a Sasaki-type metric in conjunction with the underlying FR structure [18]. This extended geometric setting offers a consistent framework for addressing open problems in cosmology, including the phenomenology of dark matter and dark energy.

This work constitutes a continuation of previous works [11,18,21,44] investigating the generalized geometric structure of spacetime, presenting new effective cosmological elements and physical interpretations with more degrees of freedom in Friedmann equations, geodesics, curvatures, etc. Fundamental geometric and physical ingredients of this study are a Sasaki-type metric $\mathcal{G}$ on $TM$ correlated with a Randers metric on the internal (vertical) structure of spacetime which we consider to be related to a dark potential metric and the dark energy-momentum tensor of spacetime.

The plan of the work is as follows. In Section 2, we give some necessary fundamental elements from the geometrical theory of Finsler and tangent bundles, as well as present the generalized field equations which we have derived in a previous work. In Section 3, we extend the equivalence principle with extra internal conditions on the tangent structure of spacetime giving a physical interpretation of geometrical concepts. In Section 4, we investigate an FRW-like model with ordinary and dark matter and a weak Randers correction. We also derive the Friedmann equations and we find a solution for the anisotropic key concept B in our model. In addition, we investigate the influence of internal anisotropic terms within the framework of geodesics, Raychaudhuri equations, redshift and dipole in Finsler cosmology. Finally, in the conclusion we discuss the main results of our theory.

2. Elements of Gravitational Theory on the Lorentz Tangent Bundle

The Lorentz tangent bundle $TM$ of a 4-dimensional manifold M is a fibered 8-dimensional manifold with local coordinates $\{x^{\mu },y^{\alpha }\}$ where the indices of the x variables are $\kappa ,\lambda ,\mu ,\nu ,\dots =0,\dots ,3$ and the indices of the y variables are $\alpha ,\beta ,\dots ,\theta =4,\dots ,7$.

2.1. Adapted Basis and Nonlinear Connection

In order to take a horizontal and vertical basis on a tangent bundle $TM$, we need to define a nonlinear connection $\mathbf{N}$ to unequivocally divide the bundle to a horizontal and vertical sub-bundle. The nonlinear connection defines a split of the total space $T\left(TM\right)$ into a horizontal subspace $T_{H}TM$ and a vertical subspace $T_{V}TM$. The total space is the Whitney sum [11,48]:

$$T\left(TM\right)=T_{H}TM\oplus T_{V}TM$$

(1)

We consider a vector field $X=X^{\mu }{\displaystyle \frac{\partial }{\partial x^{\mu }}}$ on a base Riemannian manifold M along a curve $x^{\mu }\left(t\right)$. If the vector field $X^{\mu }$ coincides with tangent vector of the curve $X^{\mu }={\dot{x}}^{\mu }\left(t\right)$ then the geodesic equation on M is the standard equation using the Levi-Civita connection. It is written as:

$${\displaystyle \frac{d^2{x}^{\mu }}{d{s}^2}}+{\Gamma }_{\nu \kappa }^{\mu }{\dot{x}}^{\nu }{\dot{x}}^{\kappa }=0$$

(2)

The parallel transport of X is given by the equation of geodesics (2):

$${\displaystyle \frac{d{X}^{\mu }}{ds}}+{\Gamma }_{\nu \kappa }^{\mu }{X}^{\nu }{X}^{\kappa }=0$$

(3)

where ${\Gamma }_{\nu \kappa }^{\mu }$ is the metrical connection.

We can extend the vector field X to the tangent bundle $TM$ of M as

$$\overline{X}=X^{\mu }{\displaystyle \frac{\partial }{\partial x^{\mu }}}+{\displaystyle \frac{d{X}^{\alpha }}{dt}}{\displaystyle \frac{\partial }{\partial y^{\alpha }}}$$

(4)

where the coefficients $X^{\alpha }$ are equal, one by one, to the coefficients $X^{\mu }$. Thus, $X^{\mu }$ and ${\displaystyle \frac{d{X}^{\alpha }}{dt}}$ are the coefficients of the extended vector $\overline{X}$ on $TM$, the basis for the horizontal and vertical subspace of the tangent bundle.

We can substitute rel. (3) to rel. (4) and find:

$$\overline{X}=X^{\mu }{\displaystyle \frac{\partial }{\partial x^{\mu }}}-\left({\Gamma }_{\nu \kappa }^{\alpha }{X}^{\nu }{X}^{\kappa }\right){\displaystyle \frac{\partial }{\partial y^{\alpha }}}$$

(5)

By assuming a Cartan-type connection ${\Gamma }_{\beta \mu }^{\alpha }{y}^{\beta }=N_{\mu }^{\alpha }$ [48] in a Finsler connection, where $N_{\mu }^{\alpha }$ are the coefficients of a nonlinear connection, we find:

$$\overline{X}=X^{\mu }\left({\displaystyle \frac{\partial }{\partial x^{\mu }}}-N_{\mu }^{\alpha }{\displaystyle \frac{\partial }{\partial y^{\alpha }}}\right)=X^{\mu }{\delta }_{\mu }$$

(6)

where we defined:

$${\delta }_{\mu }(x,y)={\displaystyle \frac{\partial }{\partial x^{\mu }}}-N_{\mu }^{\alpha }{\displaystyle \frac{\partial }{\partial y^{\alpha }}}$$

(7)

to be an adapted basis for the tangent bundle. Therefore, the nonlinear connection induces the basis $\left\{E_A\right\}=\{{\delta }_{\mu },{\dot{\partial }}_a\}$ on the total space, with

$${\delta }_{\mu }={\displaystyle \frac{\delta }{\delta x^{\mu }}}={\displaystyle \frac{\partial }{\partial x^{\mu }}}-N_{\mu }^{\alpha }(x,y){\displaystyle \frac{\partial }{\partial y^{\alpha }}}$$

(8)

and

$${\dot{\partial }}_{\alpha }={\displaystyle \frac{\partial }{\partial y^{\alpha }}}$$

(9)

The anholonomy coefficients of the nonlinear connection are defined as

$${\mathsf{\Omega }}_{\nu \kappa }^{\alpha }={\displaystyle \frac{\delta N_{\nu }^{\alpha }}{\delta x^{\kappa }}}-{\displaystyle \frac{\delta N_{\kappa }^{\alpha }}{\delta x^{\nu }}}$$

(10)

2.2. Metric Tensor

A generalized Sasaki-type metric $\mathcal{G}$ on $TM$ is:

$$\mathcal{G}=g_{\mu \nu }(x,y)d{x}^{\mu }\otimes d{x}^{\nu }+v_{\alpha \beta }(x,y)\delta y^{\alpha }\otimes \delta y^{\beta }$$

(11)

where $\delta y^{\alpha }=d{y}^{\alpha }+N_{\lambda }^{\alpha }(x,y)d{x}^{\lambda }$. We define the metrics $g_{\mu \nu }$ and $v_{\alpha \beta }$ to be pseudo-Finslerian.

A pseudo-Finslerian metric $f_{\alpha \beta }(x,y)$ is defined as one that has a Lorentzian signature of $(-,+,+,+)$ and that also obeys the following form:

$$\begin{array}{c}\hfill f_{\alpha \beta }(x,y)=\pm {\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2{F}^2}{\partial y^{\alpha }\partial y^{\beta }}}\end{array}$$

(12)

where the function F satisfies the following conditions:

1.

F is continuous on $TM$ and smooth on $\widetilde{TM}\equiv TM\setminus \left\{0\right\}$, i.e., the tangent bundle minus the null set $\left\{\right(x,y)\in TM|F(x,y)=0\}$.

2.

F is positively homogeneous of first degree on its second argument:

$$F(x^{\mu },k{y}^{\alpha })=kF(x^{\mu },y^{\alpha }),k>0$$

(13)

3.

The form

$$f_{\alpha \beta }(x,y)=\pm {\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2{F}^2}{\partial y^{\alpha }\partial y^{\beta }}}$$

(14)

defines a non-degenerate matrix:

$$\det \left[f_{\alpha \beta }\right]\ne 0$$

(15)

where the sign in (12) is chosen so that the metric has the correct signature.

At this point, it is important to make a distinction between several definitions concerning the metric.

A Sasaki metric is a natural choice of Riemannian metric on the tangent bundle of a Riemannian manifold. It is a convenient choice when one wants to study classic Riemannian geometry on the tangent bundle and takes the following form:

$$\mathcal{G}=g_{\mu \nu }\left(x\right)d{x}^{\mu }\otimes d{x}^{\nu }+g_{\alpha \beta }\left(x\right)\delta y^{\alpha }\otimes \delta y^{\beta }$$

(16)

We notice that the Riemannian metric $g_{\mu \nu }\left(x\right)$—without explicit dependence on y—appears on both the horizontal and vertical subspaces.

A Sasaki-type metric is given in (11). This is an extension of the simpler Sasaki metric in two ways. First, there are two distinct metrics in the horizontal and vertical subspaces of the tangent bundle. Second, the metric tensors $g_{\mu \nu }(x,y)$ and $v_{\alpha \beta }(x,y)$ now can generally depend on both x and y explicitly.

A pseudo-Finsler metric tensor is one defined by Equation (12). A horizontal metric tensor $g_{\mu \nu }(x,y)$ or a vertical metric tensor $v_{\alpha \beta }(x,y)$ can be a pseudo-Finsler one if it is derived by definition Equation (12). Dependence on y by itself does not guarantee that a metric tensor is a pseudo-Finsler one.

A Finsler metric structure on the tangent bundle is one defined by a Sasaki-type metric in Equation (11) with the additional condition that the metric tensors $g_{\mu \nu }(x,y)$ and $v_{\alpha \beta }(x,y)$ are pseudo-Finsler tensors.

A Finsler-Randers metric structure is a subcase of Finsler metric structures, where the Finsler function $F(x,y)$ in Equation (12) takes the specific form of a Finsler-Randers function, specified later in this work (Equation (62)).

2.3. Connection, Curvature and Torsion

In this work, we consider a distinguished connection (d-connection) D on $TM$. This is a linear connection with coefficients $\left\{{\Gamma }_{BC}^A\right\}=\{L_{\nu \kappa }^{\mu },L_{\beta \kappa }^{\alpha },C_{\nu \gamma }^{\mu },C_{\beta \gamma }^{\alpha }\}$ which preserves the horizontal and vertical distributions:

$$\begin{array}{cc}\hfill D_{{\delta }_{\kappa }}{\delta }_{\nu }=L_{\nu \kappa }^{\mu }(x,y){\delta }_{\mu }& ,D_{{\dot{\partial }}_{\gamma }}{\delta }_{\nu }=C_{\nu \gamma }^{\mu }(x,y){\delta }_{\mu }\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill D_{{\delta }_{\kappa }}{\dot{\partial }}_{\beta }=L_{\beta \kappa }^{\alpha }(x,y){\dot{\partial }}_{\alpha }& ,D_{{\dot{\partial }}_{\gamma }}{\dot{\partial }}_{\beta }=C_{\beta \gamma }^{\alpha }(x,y){\dot{\partial }}_{\alpha }\hfill \end{array}$$

(18)

From these, the definitions for partial covariant differentiation follow as usual, e.g., for $X\in T\left(TM\right)$ we have the definitions for covariant h-derivative

$$X_{|\nu }^A\equiv D_{\nu }{X}^A\equiv {\delta }_{\nu }{X}^A+L_{B\nu }^A{X}^B$$

(19)

and covariant v-derivative

$$X^A{|}_{\beta }\equiv D_{\beta }{X}^A\equiv {\dot{\partial }}_{\beta }{X}^A+C_{B\beta }^A{X}^B$$

(20)

The d-connection is metric-compatible when the following conditions are met:

$$D_{\kappa }{g}_{\mu \nu }=0,D_{\kappa }{v}_{\alpha \beta }=0,D_{\gamma }{g}_{\mu \nu }=0,D_{\gamma }{v}_{\alpha \beta }=0$$

(21)

A d-connection can be uniquely defined given that the following conditions are satisfied:

• The d-connection is metric compatible
• Coefficients $L_{\nu \kappa }^{\mu },L_{\beta \kappa }^{\alpha },C_{\nu \gamma }^{\mu },C_{\beta \gamma }^{\alpha }$ depend solely on the quantities $g_{\mu \nu }$, $v_{\alpha \beta }$ and $N_{\mu }^{\alpha }$
• Coefficients $L_{\kappa \nu }^{\mu }$ and $C_{\beta \gamma }^{\alpha }$ are symmetric on the lower indices, i.e., $L_{\left[\kappa \nu \right]}^{\mu }=C_{\left[\beta \gamma \right]}^{\alpha }=0$

We use the symbol $\mathcal{D}$ instead of D for a connection satisfying the above conditions, and call it a canonical and distinguished d-connection. The coefficients of canonical and distinguished d-connection are

$$\begin{array}{cc}\hfill L_{\nu \kappa }^{\mu }& ={\displaystyle \frac{1}{2}}{g}^{\mu \rho }\left({\delta }_k{g}_{\rho \nu }+{\delta }_{\nu }{g}_{\rho \kappa }-{\delta }_{\rho }{g}_{\nu \kappa }\right)\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill L_{\beta \kappa }^{\alpha }& ={\dot{\partial }}_{\beta }{N}_{\kappa }^{\alpha }+{\displaystyle \frac{1}{2}}{v}^{\alpha \gamma }\left({\delta }_{\kappa }{v}_{\beta \gamma }-v_{\delta \gamma }{\dot{\partial }}_{\beta }{N}_{\kappa }^{\delta }-v_{\beta \delta }{\dot{\partial }}_{\gamma }{N}_{\kappa }^{\delta }\right)\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill C_{\nu \gamma }^{\mu }& ={\displaystyle \frac{1}{2}}{g}^{\mu \rho }{\dot{\partial }}_{\gamma }{g}_{\rho \nu }\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill C_{\beta \gamma }^{\alpha }& ={\displaystyle \frac{1}{2}}{v}^{\alpha \delta }\left({\dot{\partial }}_{\gamma }{v}_{\delta \beta }+{\dot{\partial }}_{\beta }{v}_{\delta \gamma }-{\dot{\partial }}_{\delta }{v}_{\beta \gamma }\right)\hfill \end{array}$$

(25)

Curvatures and torsions on $TM$ can be defined by the multilinear maps:

$$\mathcal{R}(X,Y)Z=[{\mathcal{D}}_X,{\mathcal{D}}_Y]Z-{\mathcal{D}}_{[X,Y]}Z$$

(26)

and

$$\mathcal{T}(X,Y)={\mathcal{D}}_{X}Y-{\mathcal{D}}_{Y}X-[X,Y]$$

(27)

where $X,Y,Z\in T\left(TM\right)$. We use the following definitions for the curvature components:

$$\begin{array}{cc}\hfill \mathcal{R}({\delta }_{\lambda },{\delta }_{\kappa }){\delta }_{\nu }=R_{\nu \kappa \lambda }^{\mu }{\delta }_{\mu }& ,\mathcal{R}({\delta }_{\lambda },{\delta }_{\kappa }){\dot{\partial }}_{\beta }=R_{\beta \kappa \lambda }^{\alpha }{\dot{\partial }}_{\alpha }\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \mathcal{R}({\dot{\partial }}_{\gamma },{\delta }_{\kappa }){\delta }_{\nu }=P_{\nu \kappa \gamma }^{\mu }{\delta }_{\mu }& ,\mathcal{R}({\dot{\partial }}_{\gamma },{\delta }_{\kappa }){\dot{\partial }}_{\beta }=P_{\beta \kappa \gamma }^{\alpha }{\dot{\partial }}_{\alpha }\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \mathcal{R}({\dot{\partial }}_{\delta },{\dot{\partial }}_{\gamma }){\delta }_{\nu }=S_{\nu \gamma \delta }^{\mu }{\delta }_{\mu }& ,\mathcal{R}({\dot{\partial }}_{\delta },{\dot{\partial }}_{\gamma }){\dot{\partial }}_{\beta }=S_{\beta \gamma \delta }^{\alpha }{\dot{\partial }}_{\alpha }\hfill \end{array}$$

(30)

In addition, we use the following definitions for the torsion components:

$$\begin{array}{cc}\hfill \mathcal{T}({\delta }_{\kappa },{\delta }_{\nu })=& {\mathcal{T}}_{\nu \kappa }^{\mu }{\delta }_{\mu }+{\mathcal{T}}_{\nu \kappa }^{\alpha }{\dot{\partial }}_{\alpha }\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \mathcal{T}({\dot{\partial }}_{\beta },{\delta }_{\nu })=& {\mathcal{T}}_{\nu \beta }^{\mu }{\delta }_{\mu }+{\mathcal{T}}_{\nu \beta }^{\alpha }{\dot{\partial }}_{\alpha }\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill \mathcal{T}({\dot{\partial }}_{\gamma },{\dot{\partial }}_{\beta })=& {\mathcal{T}}_{\beta \gamma }^{\mu }{\delta }_{\mu }+{\mathcal{T}}_{\beta \gamma }^{\alpha }{\dot{\partial }}_{\alpha }\hfill \end{array}$$

(33)

The h-curvature tensor of the d-connection in the adapted basis and the corresponding h-Ricci tensor have, respectively, the components given by (28):

$$\begin{array}{cc}\hfill & R_{\nu \kappa \lambda }^{\mu }={\delta }_{\lambda }{L}_{\nu \kappa }^{\mu }-{\delta }_{\kappa }{L}_{\nu \lambda }^{\mu }+L_{\nu \kappa }^{\rho }{L}_{\rho \lambda }^{\mu }-L_{\nu \lambda }^{\rho }{L}_{\rho \kappa }^{\mu }+C_{\nu \alpha }^{\mu }{\mathsf{\Omega }}_{\kappa \lambda }^{\alpha }\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & R_{\mu \nu }=R_{\mu \nu \kappa }^{\kappa }={\delta }_{\kappa }{L}_{\mu \nu }^{\kappa }-{\delta }_{\nu }{L}_{\mu \kappa }^{\kappa }+L_{\mu \nu }^{\rho }{L}_{\rho \kappa }^{\kappa }-L_{\mu \kappa }^{\rho }{L}_{\rho \nu }^{\kappa }+C_{\mu \alpha }^{\kappa }{\mathsf{\Omega }}_{\nu \kappa }^{\alpha }\hfill \end{array}$$

(35)

The v-curvature tensor of the d-connection in the adapted basis and the corresponding v-Ricci tensor have, respectively, the components (30):

$$\begin{array}{cc}\hfill S_{\beta \gamma \delta }^{\alpha }& ={\dot{\partial }}_{\delta }{C}_{\beta \gamma }^{\alpha }-{\dot{\partial }}_{\gamma }{C}_{\beta \delta }^{\alpha }+C_{\beta \gamma }^{ϵ}{C}_{ϵ\delta }^{\alpha }-C_{\beta \delta }^{ϵ}{C}_{ϵ\gamma }^{\alpha }\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill S_{\alpha \beta }& =S_{\alpha \beta \gamma }^{\gamma }={\dot{\partial }}_{\gamma }{C}_{\alpha \beta }^{\gamma }-{\dot{\partial }}_{\beta }{C}_{\alpha \gamma }^{\gamma }+C_{\alpha \beta }^{ϵ}{C}_{ϵ\gamma }^{\gamma }-C_{\alpha \gamma }^{ϵ}{C}_{ϵ\beta }^{\gamma }\hfill \end{array}$$

(37)

The generalized Ricci scalar curvature in the adapted basis is defined as

$$\mathcal{R}=g^{\mu \nu }{R}_{\mu \nu }+v^{\alpha \beta }{S}_{\alpha \beta }=R+S$$

(38)

where

$$\begin{array}{c}\hfill R=g^{\mu \nu }{R}_{\mu \nu },S=v^{\alpha \beta }{S}_{\alpha \beta }\end{array}$$

(39)

2.4. Field Equations

A Hilbert-like action on $TM$ can be defined as [11]

$$K={\int }_{\mathcal{N}}{d}^8\mathcal{U}\sqrt{\left|\mathcal{G}\right|}\mathcal{R}+2\kappa {\int }_{\mathcal{N}}{d}^8\mathcal{U}\sqrt{\left|\mathcal{G}\right|}{\mathcal{L}}_M$$

(40)

for some closed subspace $\mathcal{N}\subset TM$, where $\left|\mathcal{G}\right|$ is the absolute value of the metric determinant, ${\mathcal{L}}_M$ is the Lagrangian of the matter fields, $\kappa$ is a constant and

$$d^8\mathcal{U}=d{x}^0\wedge \dots \wedge d{x}^3\wedge d{y}^4\wedge \dots \wedge d{y}^7$$

(41)

Variation with respect to $g_{\mu \nu }$, $v_{\alpha \beta }$ and $N_{\kappa }^{\alpha }$ leads to the following field equations:

$$\begin{array}{cc}\hfill {\overline{R}}_{\mu \nu }-{\displaystyle \frac{1}{2}}(R+S)g_{\mu \nu }+\left({\delta }_{\nu }^{(\lambda }{\delta }_{\mu }^{\kappa )}-g^{\kappa \lambda }{g}_{\mu \nu }\right)\left({\mathcal{D}}_{\kappa }{\mathcal{T}}_{\lambda \beta }^{\beta }-{\mathcal{T}}_{\kappa \gamma }^{\gamma }{\mathcal{T}}_{\lambda \beta }^{\beta }\right)& =\kappa T_{\mu \nu }\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill S_{\alpha \beta }-{\displaystyle \frac{1}{2}}(R+S)v_{\alpha \beta }+\left(v^{\gamma \delta }{v}_{\alpha \beta }-{\delta }_{\alpha }^{(\gamma }{\delta }_{\beta }^{\delta )}\right)\left({\mathcal{D}}_{\gamma }{C}_{\mu \delta }^{\mu }-C_{\nu \gamma }^{\nu }{C}_{\mu \delta }^{\mu }\right)& =\kappa Y_{\alpha \beta }\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill g^{\mu [\kappa }{\dot{\partial }}_{\alpha }{L}_{\mu \nu }^{\nu ]}+2{\mathcal{T}}_{\mu \beta }^{\beta }{g}^{\mu [\kappa }{C}_{\lambda \alpha }^{\lambda ]}& ={\displaystyle \frac{\kappa }{2}}{\mathcal{Z}}_{\alpha }^{\kappa }\hfill \end{array}$$

(44)

with

$$\begin{array}{cc}\hfill T_{\mu \nu }& \equiv -{\displaystyle \frac{2}{\sqrt{\left|\mathcal{G}\right|}}}{\displaystyle \frac{\Delta \left(\sqrt{\left|\mathcal{G}\right|}{\mathcal{L}}_M\right)}{\Delta g^{\mu \nu }}}=-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\Delta \left(\sqrt{-g}{\mathcal{L}}_M\right)}{\Delta g^{\mu \nu }}}\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill Y_{\alpha \beta }& \equiv -{\displaystyle \frac{2}{\sqrt{\left|\mathcal{G}\right|}}}{\displaystyle \frac{\Delta \left(\sqrt{\left|\mathcal{G}\right|}{\mathcal{L}}_M\right)}{\Delta v^{\alpha \beta }}}=-{\displaystyle \frac{2}{\sqrt{-v}}}{\displaystyle \frac{\Delta \left(\sqrt{-v}{\mathcal{L}}_M\right)}{\Delta v^{\alpha \beta }}}\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\mathcal{Z}}_{\alpha }^{\kappa }& \equiv -{\displaystyle \frac{2}{\sqrt{\left|\mathcal{G}\right|}}}{\displaystyle \frac{\Delta \left(\sqrt{\left|\mathcal{G}\right|}{\mathcal{L}}_M\right)}{\Delta N_{\kappa }^{\alpha }}}=-2{\displaystyle \frac{\Delta {\mathcal{L}}_M}{\Delta N_{\kappa }^{\alpha }}}\hfill \end{array}$$

(47)

where ${\mathcal{L}}_M$ is the Lagrangian of the matter fields, ${\delta }_{\nu }^{\mu }$ and ${\delta }_{\beta }^{\alpha }$ are the Kronecker symbols, $\left|\mathcal{G}\right|$ is the absolute value of the determinant of the total metric (11), and

$${\mathcal{T}}_{\nu \beta }^{\alpha }={\dot{\partial }}_{\beta }{N}_{\nu }^{\alpha }-L_{\beta \nu }^{\alpha }$$

(48)

are torsion components, where $L_{\beta \nu }^{\alpha }$ is defined in (23). From the form of (11) it follows that $\sqrt{\left|\mathcal{G}\right|}=\sqrt{-g}\sqrt{-v}$, with $g,v$ the determinants of the metrics $g_{\mu \nu },v_{\alpha \beta }$ respectively.

3. Geometric Foundations of the Equivalence Principle on the Tangent Bundle of Spacetime

The equivalence principle in General Relativity can be generalized in the framework of the tangent bundle $TM$ of a spacetime manifold M, equipped with a Sasaki-type metric of the general form

$$\mathcal{G}=g_{\mu \nu }\left(x\right)d{x}^{\mu }\otimes d{x}^{\nu }+v_{\alpha \beta }(x,y)\delta y^{\alpha }\otimes \delta y^{\beta }$$

(49)

If we consider the space in the form $(TM,\mathcal{G}(x,y),N_{\mu }^{\gamma }(x,y))$ with $g_{\mu \nu }\left(x\right)$ and $v_{\alpha \beta }(x,y)$ Lagrange or Finsler-like metrics induced by a regular Lagrangian $L(x,y)$ and $N_{\mu }^{\gamma }(x,y)$ a canonical nonlinear connection, then for any point $u_o=(x_o,y_o)\in TM$, a local adapted coordinate system exists, in which

$$N_{\mu }^{\gamma }\left(u_o\right)=0,L_{\nu \kappa }^{\mu }\left(u_o\right)=0,C_{\nu \gamma }^{\mu }\left(u_o\right)=0,{\partial }_{\lambda }{g}_{\mu \nu }\left(x_o\right)=0$$

(50)

with

$$\begin{array}{cc}\hfill L_{\nu \kappa }^{\mu }& ={\displaystyle \frac{1}{2}}{g}^{\mu \rho }\left({\delta }_k{g}_{\rho \nu }+{\delta }_{\nu }{g}_{\rho \kappa }-{\delta }_{\rho }{g}_{\nu \kappa }\right)\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill C_{\nu \gamma }^{\mu }& ={\displaystyle \frac{1}{2}}{g}^{\mu \rho }{\dot{\partial }}_{\gamma }{g}_{\rho \nu }\hfill \end{array}$$

(52)

and ${\delta }_{\mu }={\partial }_{\mu }-N_{\mu }^{\alpha }{\dot{\partial }}_{\alpha }$.

In standard GR, the equivalence principle states that locally one can eliminate gravity by choosing a freely falling frame at a point $x_o$,

$${\Gamma }_{\nu \lambda }^{\mu }\left(x_o\right)=0$$

(53)

by suitable coordinates of the Riemannian geometry of spacetime M. On the bundle geometrical structure of spacetime, gravity is described by positions and velocities as well as by a nonlinear connection $N_{\nu }^{\gamma }(x,y)$ instead of just the Christoffel symbols. Therefore, on the tangent-bundle geometric structure of spacetime, gravity and anisotropy vanish at the $(x_o,y_o)$ point.

This recovers the (EP) of GR using the Cartan nonlinear connection

$$N_{\mu }^{\alpha }={\Gamma }_{\mu \nu }^{\alpha }{y}^{\nu }$$

(54)

In this consideration, the equivalence principle is extended on $TM$.

On the tangent bundle frame of spacetime, autoparallels with respect to the standard Levi-Civita connection satisfy the condition

$${\displaystyle \frac{d{y}^{\alpha }}{ds}}+N_{\nu }^{\alpha }{y}^{\nu }=0$$

(55)

At (EP) frame we get

$${\displaystyle \frac{d{y}^{\alpha }}{ds}}=0\Rightarrow {\displaystyle \frac{d^{2}x}{d{s}^2}}=0$$

(56)

3.1. General Comment

Field Equations (42) and (43) in a torsionless vacuum ($T_{\mu \nu }=0$, ${\mathcal{T}}_{\kappa \gamma }^{\alpha }=0$, $C_{\nu \gamma }^{\mu }=0$):

$$\begin{array}{c}\hfill {\overline{R}}_{\mu \nu }-{\displaystyle \frac{1}{2}}(\overline{R}+S)g_{\mu \nu }=0\end{array}$$

(57)

$$\begin{array}{c}\hfill S_{\alpha \beta }-{\displaystyle \frac{1}{2}}(\overline{R}+S)v_{\alpha \beta }=0\end{array}$$

(58)

imply that $S_{\alpha \beta }=0$.

In our approach, spacetime behaves like an anisotropic medium in velocity space. The universe is considered the spacetime with its internal motion and acceleration. The direction-dependent geometry does not change matter. The inertial mass depends on the direction of motion, inertia becomes direction-sensitive in phase space. In our consideration, different paths “feel” different gravitational strength because the connection coefficients depend on y. Therefore, particles with different velocities follow different effective geodesics in the same spacetime, and effective gravitational coupling becomes anisotropic.

In differential geometry, the acceleration bundle $T\left(TM\right)$ (often identified as the double tangent bundle) is the geometric framework used to describe the second-order dynamics of a manifold [60]. In the Riemannian spacetime of general relativity, it can represent the geometrical properties of the gravitational field.

The elements of an acceleration bundle have coordinates $(x,\dot{x},\ddot{x})$ on $T\left(TM\right)$, denoting the position, velocity, and acceleration of a particle. The base of this bundle is the tangent bundle $TM$ of spacetime. While the tangent bundle describes states (position and velocity), the acceleration bundle describes how those states change. Therefore, the gravitational field is naturally accommodated within such a spacetime tangent bundle form equipped with anisotropic, directional, or velocity-dependent terms.

Adopting this framework for the investigation of the gravitational field leads us to consider generalized geometric structures and to explore extended Einstein equations and generalized Friedmann equations with additional degrees of freedom. The study of these equations can further our understanding of the evolution and acceleration of the universe, including the problems of dark matter and dark energy. A fundamental role in this approach is played by the connections, nonlinear connection, and curvature tensors (including anisotropic terms), as will be discussed in the next chapter. In particular, the curvatures of the external and internal structures of the spacetime tangent bundle can be linked to the total matter content—incorporating baryonic matter, dark matter, and dark energy—as well as to the deflection of geodesic flows and the geodesic spray G.

3.2. Physical Interpretations of Geometrical Concepts on Tangent and Acceleration Bundle of Spacetime

Ordinary General Relativity (GR) lives on spacetime M, while generalized Finsler/Lagrange gravity lives on the tangent bundle $TM$ where each spacetime point x carries additional internal variables y, interpreted physically as velocity, direction, or acceleration degrees of freedom. As is known, GR geometrizes spacetime, while Finsler-type gravity geometrizes spacetime together with the observer’s kinematics. In this framework gravity becomes direction-dependent with y. With the observer’s 4-velocity, $TM$ represents external spacetime along the internal motion space; the nonlinear connection ${N_{\mu }}^{\alpha }$ couples the two subbundles $T\left(TM\right)=T_{H}TM\oplus T_{V}TM$, where the horizontal sector H constitutes ordinary spacetime directions, and the vertical sector V contains internal velocity and acceleration directions. In other words, nonlinear connection represents an interaction between ordinary gravitation of spacetime and internal kinematics of velocity dependence and acceleration structure. The “acceleration bundle” $T\left(TM\right)$ emerges when the fiber variables are interpreted not merely as velocities but as dynamical accelerative degrees of freedom [60]. Thus the bundle geometry contains information about acceleration fields. The physical interpretation becomes:

• Gravitation is encoded not only by curvature of spacetime, but also by curvature of motion-space (internal space), geodesics depend on velocity, tidal effects depend on anisotropy, and cosmological expansion can also emerge from vertical curvature terms. where the metric has a horizontal spacetime part and a vertical internal part. This means the gravity has extra degrees of freedom associated with internal anisotropy. The nonlinear connection modifies the parallel transport and geodesics. So physically the gravitational field becomes a coupled spacetime–kinematical field.

3.2.1. Cosmological Interpretation

Based on the aforementioned physical interpretations we can conclude the following:

• the vertical geometry contributes effective energy terms1;
• acceleration of the universe can arise geometrically, because of its internal mechanisms associated with the external and internal curvatures of spacetime;
• dark-energy-like behavior may emerge from anisotropic bundle curvature.

Thus, the “acceleration bundle” is not only kinematical but also becomes a source of effective gravitation. The tangent bundle represents an enlarged gravitational framework.

• fiber variables represent physical motion states;
• the nonlinear connection couples spacetime and acceleration/velocity structure;
• gravity becomes locally anisotropic;
• acceleration and gravitation are encoded in a unified geometric framework;
• When gravity is formulated on $TM$, field equations split into horizontal spacetime components and vertical (velocity/fiber) components.

3.2.2. Examples

In classical mechanics, the form $\ddot{x}=f(x,\dot{x})$ is a vector field on $T\left(TM\right)$, where M represents a 4-dimensional spacetime manifold. In flat spacetime, a charged particle moving inside an electromagnetic field is subject to the Lorentz force:

$${\ddot{x}}^{\mu }=q{F^{\mu }}_{\nu }{\dot{x}}^{\nu }$$

(59)

This equation is re-written as:

$${\ddot{x}}^{\mu }-q{F^{\mu }}_{\nu }{\dot{x}}^{\nu }=0$$

(60)

Inspecting the last equation, we notice that the EM field has a geometrical interpretation on $T\left(TM\right)$.

A physical interpretation of the geometrical structure of Sasaki–Randers spacetime, which incorporates a coupling of a one-form field to the gravitational one, can be formulated within the framework of the tangent and acceleration bundles of spacetime. In this approach, the nonlinear connection ${N_{\alpha }}^{\mu }(x,\dot{x})$ plays a fundamental role. As we can notice from relations (104), (8) and (11), the EM field can be connected with the Sasaki-Randers spacetime, offering a generalized geometric form of the EM-field. In particular, the splitting of the acceleration bundle $T\left(TM\right)$ into horizontal and vertical subbundles allows for a distinction between external (observable) and internal (or “dark”) sectors of spacetime.

Within this framework, the nonlinear connection provides a natural generalization of gravity in anisotropic spacetime, incorporating kinematical dependence on velocity. It encodes how velocities evolve under dynamical processes (i.e., acceleration) in the presence of cosmic anisotropy. Consequently, Finsler and Finsler-like geometric structures offer a consistent description of gravitational phenomena and their evolution on the total spacetime bundle.

Furthermore, based on the aforementioned geometric structures, effects such as anisotropic expansion, time-delay phenomena, and contributions associated with dark energy and dark matter may naturally emerge. In particular, scalar curvature S of the internal velocity space can act effectively as a dark energy component through its coupling to spacetime geometry.

Within this context, the resulting modified Friedmann equations acquire additional terms, which can significantly influence the dynamical evolution of the universe.

4. FRW-like Model with Ordinary and Dark Matter and a Weak Randers Correction

4.1. Metric Tensor

We consider a Sasaki-type metric:

$$G=g_{\mu \nu }\left(x\right)d{x}^{\mu }\otimes d{x}^{\nu }+v_{\alpha \beta }(x,y)\delta y^{\alpha }\otimes \delta y^{\beta }$$

(61)

and a Finsler metric function

$$F(x,y)=L_R+B_{\gamma }\left(x\right)y^{\gamma }$$

(62)

where $L_R=\sqrt{-g_{\alpha \beta }\left(x\right)y^{\alpha }{y}^{\beta }}$ is the Riemannian norm.

The horizontal metric tensor $g_{\mu \nu }\left(x\right)$ is defined by the line element

$$d{s}^2=-d{t}^2+a^2\left(t\right)\left(d{x}_1^2+d{x}_2^2+d{x}_3^2\right)$$

(63)

The vertical metric tensor $v_{\alpha \beta }(x,y)$ is derived from the Finsler metric function (62) according to (12):

$$v_{\alpha \beta }(x,y)=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2{F}^2}{\partial y^{\alpha }\partial y^{\beta }}}$$

(64)

We find

$$v_{\alpha \beta }(x,y)=g_{\alpha \beta }\left(x\right)+w_{\alpha \beta }(x,y)$$

(65)

where we have set

$$\begin{array}{c}\hfill w_{\alpha \beta }={\displaystyle \frac{1}{L_R}}(B_{\beta }{g}_{\alpha \gamma }{y}^{\gamma }+B_{\gamma }{g}_{\alpha \beta }{y}^{\gamma }+B_{\alpha }{g}_{\beta \gamma }{y}^{\gamma })+{\displaystyle \frac{1}{L_R^3}}{B}_{\gamma }{g}_{\alpha ϵ}{g}_{\beta \delta }{y}^{\gamma }{y}^{\delta }{y}^{ϵ}\end{array}$$

(66)

We make the assumption that $|B_{\gamma }{y}^{\gamma }|\ll 1$, so $w_{\alpha \beta }$ is considered a perturbation over the Riemannian metric $g_{\alpha \beta }\left(x\right)$.

4.2. Energy-Momentum Tensor of Ordinary and Dark Matter

The energy-momentum tensor of our model can be considered that of a perfect fluid:

$$T_{\mu \nu }=\left(\rho \left(x\right)+p\left(x\right)\right)u^{\mu }{u}^{\nu }+p\left(x\right)g_{\mu \nu }$$

(67)

with $\rho \left(x\right)={\rho }^{\left(O\right)}\left(x\right)+{\rho }^{\left(D\right)}\left(x\right)$ the total energy density content of the universe, $p\left(x\right)=p^{\left(O\right)}\left(x\right)$ the isotropic pressure of the fluid, ${\rho }^{\left(O\right)}\left(x\right)$ is the energy density of the ordinary matter fields, $p^{\left(O\right)}\left(x\right)$ is the isotropic pressure of the ordinary matter fields, ${\rho }^{\left(D\right)}\left(x\right)$ is the energy density of the dark matter fields and $u^{\mu }$ the fluid 4-velocity [12]. Thus, we can write:

$$T_{\mu \nu }=T_{\mu \nu }^{\left(O\right)}+T_{\mu \nu }^{\left(D\right)}$$

(68)

with

$$T_{\mu \nu }^{\left(O\right)}=\left({\rho }^{\left(O\right)}\left(x\right)+p^{\left(O\right)}\left(x\right)\right)u^{\mu }{u}^{\nu }+p^{\left(O\right)}\left(x\right)g_{\mu \nu }$$

(69)

the energy-momentum component contributed by the ordinary matter fields and

$$T_{\mu \nu }^{\left(D\right)}={\rho }^{\left(D\right)}\left(x\right)u^{\mu }{u}^{\nu }$$

(70)

the component contributed by the dark matter field(s). We assume the same 4-velocity $u^{\mu }$ for the ordinary and the dark matter fluids.

The presence of ordinary matter alone, expressed by $T_{\mu \nu }^{\left(O\right)}$, corresponds to an h-metric solution $g_{\mu \nu }^{\left(O\right)}$. Similarly, a universe with only dark matter content $T_{\mu \nu }^{\left(D\right)}$ corresponds to an h-metric solution $g_{\mu \nu }^{\left(D\right)}$. The solution $g_{\mu \nu }$ of the field equations for the total energy-momentum tensor (67) will have contributions from both ordinary and dark matter fields. A goal of Finsler-like gravitational models is to account for some or all of the dark matter content of the universe by means of their extended geometrical structure. Thus, one would expect for the contribution of $T_{\mu \nu }^{\left(D\right)}$ to be partially or completely replaced by the richer geometric dynamics of these generalized models.

4.3. Field Equations

The S curvature scalar is given by

$$S=v^{\alpha \beta }{S}_{\alpha \beta }=v^{\alpha \beta }\left({\dot{\partial }}_{\gamma }{C}_{\alpha \beta }^{\gamma }-{\dot{\partial }}_{\beta }{C}_{\alpha \gamma }^{\gamma }+C_{\alpha \beta }^{ϵ}{C}_{ϵ\gamma }^{\gamma }-C_{\alpha \gamma }^{ϵ}{C}_{ϵ\beta }^{\gamma }\right)$$

(71)

according to Equations (37) and (38).

For a Finsler metric (64), ${\dot{\partial }}_{\gamma }{C}_{\alpha \beta }^{\gamma }-{\dot{\partial }}_{\beta }{C}_{\alpha \gamma }^{\gamma }=0$. Additionally, $C_{\alpha \beta }^{ϵ}{C}_{ϵ\gamma }^{\gamma }$ is second order in $w_{\alpha \beta }$. From these, we get at first order:

$$S=0$$

(72)

The torsion components appearing in the field Equation (42) are

$${\mathcal{T}}_{\kappa \beta }^{\alpha }={\displaystyle \frac{1}{2}}{v}^{\alpha \gamma }\left(v_{\delta \gamma }{\dot{\partial }}_{\beta }{N}_{\kappa }^{\delta }+v_{\beta \delta }{\dot{\partial }}_{\gamma }{N}_{\kappa }^{\delta }-{\delta }_{\kappa }{v}_{\beta \gamma }\right)$$

(73)

The nonlinear connection is given by

$$N_{\mu }^{\alpha }={\displaystyle \frac{1}{2}}{y}^{\gamma }{g}^{\alpha \beta }{\partial }_{\mu }{g}_{\beta \gamma }$$

(74)

After calculations, we get

$${\mathcal{T}}_{\nu \alpha }^{\alpha }=-{\displaystyle \frac{1}{2}}{\delta }_{\nu }w=-{\displaystyle \frac{5}{2}}{\delta }_{\nu }\left(B_{\gamma }{\displaystyle \frac{y^{\gamma }}{L_R}}\right)$$

(75)

with $w=w_{\alpha }^{\alpha }$. One can show that

$${\delta }_{\nu }\left({\displaystyle \frac{y^{\gamma }}{L_R}}\right)=-E_{\beta \nu }^{\gamma }{\displaystyle \frac{y^{\beta }}{L_R}}$$

(76)

where

$$E_{\beta \mu }^{\gamma }\left(x\right)\equiv {\displaystyle \frac{1}{2}}{g}^{\alpha \gamma }{\partial }_{\mu }{g}_{\alpha \beta }$$

(77)

Applying (76) to (75), we get

$${\mathcal{T}}_{\nu \alpha }^{\alpha }=-{\displaystyle \frac{5}{2}}{K}_{\nu \gamma }{\displaystyle \frac{y^{\gamma }}{L_R}}$$

(78)

where

$$K_{\nu \gamma }\left(x\right)\equiv {\partial }_{\nu }{B}_{\gamma }-B_{\beta }{E}_{\gamma \nu }^{\beta }$$

(79)

Now we define

$$Z_{\mu \nu }=\left({\delta }_{\nu }^{(\lambda }{\delta }_{\mu }^{\kappa )}-g^{\kappa \lambda }{g}_{\mu \nu }\right)\left({\mathcal{D}}_{\kappa }{\mathcal{T}}_{\lambda \beta }^{\beta }-{\mathcal{T}}_{\kappa \gamma }^{\gamma }{\mathcal{T}}_{\lambda \beta }^{\beta }\right)$$

(80)

which is the last term of the lhs of (42). The product ${\mathcal{T}}_{\kappa \gamma }^{\gamma }{\mathcal{T}}_{\lambda \beta }^{\beta }$ is second order in w so we discard it. After calculations, we get

$$Z_{\mu \nu }=-{\displaystyle \frac{5}{2}}\left({\delta }_{\nu }^{(\lambda }{\delta }_{\mu }^{\kappa )}-g^{\kappa \lambda }{g}_{\mu \nu }\right)\left({\partial }_{\kappa }{K}_{\lambda \gamma }-E_{\gamma \kappa }^{\beta }{K}_{\lambda \beta }-{\Gamma }_{\kappa \lambda }^{\sigma }{K}_{\sigma \gamma }\right){\displaystyle \frac{y^{\gamma }}{L_R}}$$

(81)

where ${\Gamma }_{\mu \nu }^{\lambda }$ are the Christoffel symbols:

$${\Gamma }_{\nu \kappa }^{\mu }={\displaystyle \frac{1}{2}}{g}^{\mu \rho }\left({\partial }_{\kappa }{g}_{\rho \nu }+{\partial }_{\nu }{g}_{\rho \kappa }-{\partial }_{\rho }{g}_{\nu \kappa }\right)$$

(82)

The field Equation (42) now take the form

$$R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }R+Z_{\mu \nu }=\kappa T_{\mu \nu }$$

(83)

where $R_{\mu \nu }$ and R are the standard Riemannian Ricci tensor and scalar of GR respectively. Differentiating (83) with respect to $y^{\gamma }$ gives

$$\left({\delta }_{\nu }^{(\lambda }{\delta }_{\mu }^{\kappa )}-g^{\kappa \lambda }{g}_{\mu \nu }\right)\left({\partial }_{\kappa }{K}_{\lambda \gamma }-E_{\gamma \kappa }^{\beta }{K}_{\lambda \beta }-{\Gamma }_{\kappa \lambda }^{\sigma }{K}_{\sigma \gamma }\right)=0$$

(84)

which is equivalent to

$$Z_{\mu \nu }=0$$

(85)

Now, Equation (83) combined with Equation (85) leads to the Friedmann equations:

$$\begin{array}{c}{\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2={\displaystyle \frac{\kappa }{3}}\rho \end{array}$$

(86)

$$\begin{array}{c}{\displaystyle \frac{\ddot{a}}{a}}=-{\displaystyle \frac{\kappa }{6}}\left(\rho +3p\right)\end{array}$$

(87)

Remark 1.

The Friedmann equations derived in this section take a similar form to the ones recovered in classic GR, but are not the same regarding their physical content. Generally, we expect distinct dynamics for point particles in our model, since matter generally does not flow on the classic geodesic curves in generalized Finsler-like spacetimes but on trajectories the equations of which depend on both the classic scale factor $a\left(t\right)$ and the one-form field $B_{\gamma }\left(t\right)$. Thus, the standard interpretation of $a\left(t\right)$ as an indicator of the universe’s expansion does not hold equally in the present model and one will have to look at equations describing particle dynamics for a more complete picture, such as the Raychaudhuuri equations for our model derived in the following paragraph. Related to the above is the observation that the standard spacetime distance measured by the metric $g_{\mu \nu }$ does not hold the classic physical meaning regarding particle trajectories, since one expects that the distance measured on the vertical space by the metric $v_{\alpha \beta }$ will play a crucial role in defining the equations of particle motion, an idea that we follow later on in this work.

Now, Equation (84) governs the evolution of the Randers one-form $B_{\gamma }$ of the vertical subspace of the Lorentz tangent bundle. Thus, Equation (84) provides an extension to the standard form Friedmann equations, manifesting the departure from classic GR. We write Equation (84) again for the reader’s convenience:

$$\left({\delta }_{\nu }^{(\lambda }{\delta }_{\mu }^{\kappa )}-g^{\kappa \lambda }{g}_{\mu \nu }\right)\left({\partial }_{\kappa }{K}_{\lambda \gamma }-E_{\gamma \kappa }^{\beta }{K}_{\lambda \beta }-{\Gamma }_{\kappa \lambda }^{\sigma }{K}_{\sigma \gamma }\right)=0$$

By computing specific components of this equation, we get a system of differential equations for $B_{\gamma }$:

• $\mu =0,\nu =0,\gamma =4$

  $$-3B_0\left(t\right)\dot{a}\left(t\right)+a\left(t\right){\dot{B}}_0\left(t\right)=0$$

  (88)
• $\mu =1,\nu =1,\gamma =4$

  $$3B_0\left(t\right)\left(\dot{a}{\left(t\right)}^2+a\left(t\right)\ddot{a}\left(t\right)\right)+a\left(t\right)\left(\dot{a}\left(t\right){\dot{B}}_0\left(t\right)-a\left(t\right){\ddot{B}}_0\left(t\right)\right)=0$$

  (89)
• $\mu =0,\nu =0,\gamma =5$

  $$-3B_1\left(t\right)\dot{a}\left(t\right)+a\left(t\right){\dot{B}}_1\left(t\right)=0$$

  (90)
• $\mu =1,\nu =1,\gamma =5$

  $$-2\dot{a}\left(t\right){\dot{B}}_1\left(t\right)-3B_1\left(t\right)\ddot{a}\left(t\right)+a\left(t\right){\ddot{B}}_1\left(t\right)=0$$

  (91)
• $\mu =0,\nu =0,\gamma =6$

  $$-3B_2\left(t\right)\dot{a}\left(t\right)+a\left(t\right){\dot{B}}_2\left(t\right)=0$$

  (92)
• $\mu =1,\nu =1,\gamma =6$

  $$-2\dot{a}\left(t\right){\dot{B}}_2\left(t\right)-3B_2\left(t\right)\ddot{a}\left(t\right)+a\left(t\right){\ddot{B}}_2\left(t\right)=0$$

  (93)
• $\mu =0,\nu =0,\gamma =7$

  $$-3B_3\left(t\right)\dot{a}\left(t\right)+a\left(t\right){\dot{B}}_3\left(t\right)=0$$

  (94)
• $\mu =1,\nu =1,\gamma =7$

  $$-2\dot{a}\left(t\right){\dot{B}}_1\left(t\right)-3B_3\left(t\right)\ddot{a}\left(t\right)+a\left(t\right){\ddot{B}}_3\left(t\right)=0$$

  (95)

All other component combinations either do not give independent equations or give trivial equations.

The solution to these leads to the vertical subspace one-form:

$$B_{\gamma }\left(t\right)=a^3\left(t\right)\left(\begin{array}{c}0\\ c_1\\ c_2\\ c_3\end{array}\right)$$

(96)

where $c_1,c_2,c_3$ are constants bound by the condition $|B_{\gamma }{y}^{\gamma }|\ll 1$.

From Equations (86), (87) and (96), we get the Friedmann-like equations for the evolution of $B_{\gamma }\left(t\right)$:

$$\begin{array}{c}{\dot{B}}_{\gamma }={\left(3\kappa \rho \right)}^{1/2}{B}_{\gamma }\end{array}$$

(97)

$$\begin{array}{c}{\ddot{B}}_{\gamma }={\displaystyle \frac{3\kappa }{2}}\left(\rho -p\right)B_{\gamma }\end{array}$$

(98)

Remark 2.

Anisotropy in this model manifests in Friedmann equations because of the internal covector B. These equations in our model are not the same as the ones of General Relativity (GR) although they have similar form. Restricted to the horizontal background, Friedmann equations coincide with those of Λ-CDM, in that case, the models are background-degenerate. In the Finsler–Randers/Sasaki geodesic equations, curvature structure, vertical dynamics, or matter/geometry coupling differ from GR. The solution $B_{\gamma }$ (Rel. (96)) comes from Rel. (84), a term which generalizes the standard Einstein field equations [11] where $C_i$ is a component of a constant spatial vector $C=\left(C_i\right)$, $i=1,2,3$. In this model, the cosmological evolution contains both ordinary expansion $a\left(t\right)$ and the direction of Finsler expansion $B_{\gamma }(t,C)$. Therefore, these field equations provide an extension to the standard form Friedmann Equations (86) and (87). This means that the Finslerian contribution is dynamically coupled to the cosmic expansion of the ‘bridge’ $B_{\gamma }$ while its magnitude and direction depend on $C_i$. Therefore, the total acceleration in our model arises from the base Riemannian manifold of spacetime, as well as from the contributions of $B_{\gamma }$ on the tangent bundle since the gravitational field is considered to be accommodated on the tangent bundle instead of the background manifold. This combined contribution to particle acceleration manifests in the geodesic equations and the Raychaudhuri equations and creates modified acceleration/deceleration cosmic dynamics and a small asymmetry. This internal perturbation differentiates the Riemannian background evolution of the universe; consequently, the theories are not dynamically equivalent and can in principle be distinguished by non-background cosmological observables.

4.4. Geodesics

To derive the geodesic equations for the proposed model, we assume that the Finsler function Equation (62) plays the role of the Lagrangian for the derivation. The resulting equation is given by:

$${\displaystyle \frac{d{\dot{x}}^{\lambda }}{ds}}+{\Gamma }_{\mu \nu }^{\lambda }{\dot{x}}^{\mu }{\dot{x}}^{\nu }+k{\left(-g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }\right)}^{{\displaystyle \frac{1}{2}}}{F^{\lambda }}_{\kappa }{\dot{x}}^{\kappa }=0$$

(99)

where $\dot{x}$ denotes the tangent vector along the geodesic, ${\Gamma }_{\mu \nu }^{\lambda }$ are the Christoffel symbols, and ${F^{\lambda }}_{\kappa }$ represents a force field defined as:

$$F_{\lambda \kappa }={\partial }_{\lambda }{B}_{\kappa }-{\partial }_{\kappa }{B}_{\lambda }$$

(100)

Introducing the specific form of $B_{\gamma }$ as articulated in Equation (96), we arrive at the following system of geodesic equations:

$$\begin{array}{cc}\hfill \ddot{t}+a\dot{a}{\delta }_{ij}{\dot{x}}^i{\dot{x}}^j& =-3a^2\dot{a}{c}_i{\dot{x}}^i=-{\dot{B}}_i{\dot{x}}^i,\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill {\ddot{x}}^i+2{\displaystyle \frac{\dot{a}}{a}}\dot{t}{\dot{x}}^i& =-3\dot{a}{c}^i\dot{t},\hfill \end{array}$$

(102)

where $i,j$ represent spatial indices of the horizontal subspace and take values from the range $\{1,2,3\}$. Here, $\ddot{t}$ and ${\ddot{x}}^i$ represent the second derivatives with respect to the affine parameter s, while $a\left(t\right)$ is the scale factor of the universe. The terms $\dot{a}$ and $c_i$ encapsulate the dynamics of the expansion and the influence of perturbations, respectively. The nonzero constant k in Rel. (99) can be set to 1 by rescaling the constants $c_1,c_2,c_3$. For our calculations, we have set

$$g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-1$$

(103)

This condition sets the parameter along the geodesics up to affine transformations.

We remark that the generalized geodesic Equations (101) and (102) introduce force-like terms on their rhs, induced by the Randers one-form $B_{\gamma }$ of the vertical subspace.

Example

From relations (99), (100) and (103) and by interpreting $F_{\mu \nu }$ as the electromagnetic field strength tensor, one gets the nonlinear connection in the following form:

$$N_{\mu }^{\alpha }={\Gamma }_{\beta \mu }^{\alpha }{y}^{\beta }-{F^{\alpha }}_{\mu }$$

(104)

where $B_{\gamma }\equiv A_{\gamma }$ represents the electromagnetic vector potential for $F_{\mu \nu }$.

4.5. Raychaudhuri Equation

Raychaudhuri equation in GR describes the evolution of a geodesic congruence in terms of its expansion, shear and vorticity. An extension of these concepts can be made in the framework of the Lorentz tangent bundle.

We define the horizontal subspace separation vector W of the geodesics as one that satisfies the commutation relation:

$${[U,W]}^{\left(H\right)}=0$$

(105)

i.e., the horizontal part of the commutator between the horizontal 4-velocity $U={\dot{x}}^{\mu }{\delta }_{\mu }$ and the horizontal separation vector W vanishes. This condition is necessary so that the tangent to the geodesic 4-velocity vector field and the separation vector can form a local adapted basis adjusted to the congruence.

After calculations, we get:

$$[U,W]=\left(U^{\nu }{D}_{\nu }{W}^{\mu }-W^{\nu }{D}_{\nu }{U}^{\mu }\right){\delta }_{\mu }+U^{\mu }{W}^{\nu }{\mathsf{\Omega }}_{\mu \nu }^{\alpha }{\dot{\partial }}_{\alpha }$$

(106)

which, along with Equation (105), gives

$$U^{\nu }{D}_{\nu }{W}^{\mu }-W^{\nu }{D}_{\nu }{U}^{\mu }=0$$

(107)

This relation is useful for calculating the directional derivative of the separation vector:

$$U^{\nu }{D}_{\nu }{W}^{\mu }=W^{\nu }{D}_{\nu }{U}^{\mu }={B^{\mu }}_{\nu }{W}^{\nu }$$

(108)

where we defined

$${B^{\mu }}_{\nu }=D_{\nu }{U}^{\mu }$$

(109)

This is the expansion tensor and it measures the failure of the separation vector W to be parallel transported along the geodesic. Differentiating the expansion tensor along the direction of $U^{\mu }$ gives

$$U^{\kappa }{D}_{\kappa }{B^{\mu }}_{\nu }=-R_{\lambda \kappa \nu }^{\mu }{U}^{\lambda }{U}^{\kappa }-{B^{\kappa }}_{\nu }{B^{\nu }}_{\kappa }+U^{\kappa }\left({\mathcal{T}}_{\kappa \nu }^{\lambda }{D}_{\lambda }{U}^{\mu }+{\mathsf{\Omega }}_{\kappa \nu }^{\alpha }{D}_{\alpha }{U}^{\mu }\right)$$

(110)

Now, one can take the trace of ${B^{\mu }}_{\nu }$ with the aid of the projection tensor ${P^{\mu }}_{\nu }={\delta }_{\nu }^{\mu }+U^{\mu }{U}_{\nu }$:

$$P^{\mu \nu }{B}_{\mu \nu }=D_{\nu }{U}^{\nu }\equiv \mathsf{\Theta }$$

(111)

where $\mathsf{\Theta }$ is the expansion scalar of the congruence. The symmetric traceless part of ${B^{\mu }}_{\nu }$ is

$${\sigma }_{\mu \nu }=B_{\left(\mu \nu \right)}-{\displaystyle \frac{1}{3}}\mathsf{\Theta }{P}_{\mu \nu }$$

(112)

and is called the shear tensor of the congruence, while the antisymmetric part of ${B^{\mu }}_{\nu }$ is

$${\omega }_{\mu \nu }=B_{\left[\mu \nu \right]}$$

(113)

and is called the vorticity tensor of the congruence.

Taking the trace of (110) and using the above definitions, we get:

$$\dot{\mathsf{\Theta }}=-{\displaystyle \frac{1}{3}}{\mathsf{\Theta }}^2-R_{\mu \nu }{U}^{\mu }{U}^{\nu }-{\sigma }_{\mu \nu }{\sigma }^{\mu \nu }+{\omega }_{\mu \nu }{\omega }^{\mu \nu }+U^{\kappa }\left({\mathcal{T}}_{\kappa \nu }^{\lambda }{D}_{\lambda }{U}^{\nu }+{\mathsf{\Omega }}_{\kappa \nu }^{\alpha }{D}_{\alpha }{U}^{\nu }\right)+D_{\nu }{I}^{\nu }$$

(114)

where

$$I^{\nu }=L_{\kappa \lambda }^{\nu }{U}^{\kappa }{U}^{\lambda }-2G^{\nu }$$

(115)

with $G^{\nu }$ the geodesic spray. For an ideal fluid Equation(67), we get

$$\dot{\mathsf{\Theta }}=-{\displaystyle \frac{1}{3}}{\mathsf{\Theta }}^2-{\displaystyle \frac{\kappa }{2}}\left({\rho }^{\left(O\right)}+{\rho }^{\left(D\right)}+3P^{\left(O\right)}\right)-{\sigma }_{\mu \nu }{\sigma }^{\mu \nu }+{\omega }_{\mu \nu }{\omega }^{\mu \nu }+U^{\kappa }\left({\mathcal{T}}_{\kappa \nu }^{\lambda }{D}_{\lambda }{U}^{\nu }+{\mathsf{\Omega }}_{\kappa \nu }^{\alpha }{D}_{\alpha }{U}^{\nu }\right)+D_{\nu }{I}^{\nu }$$

(116)

For the model under consideration, we have

$$I^{\nu }=-{F^{\nu }}_{\kappa }{U}^{\kappa }$$

(117)

and

$$D_{\nu }{I}^{\nu }=3a\left({\dot{a}}^2+a\ddot{a}\right)\left(c_1{U}^1+c_2{U}^2+c_3{U}^3\right)=3\left(\dot{H}+2H^2\right)B_{\mu }{U}^{\mu }$$

(118)

We also get

$${\mathcal{T}}_{\kappa \nu }^{\lambda }=0$$

$${\mathsf{\Omega }}_{\kappa \nu }^{\alpha }=0$$

while the expansion scalar $\mathsf{\Theta }$ is calculated as

$$\mathsf{\Theta }={\partial }_{\nu }{U}^{\nu }+3H{U}^0$$

(119)

The term $D_{\nu }{I}^{\nu }$ in Equation (114), taking the form of Equation (118), is of particular interest, as it can influence the acceleration or deceleration rate of the universe. Assuming a non-negative $\ddot{a}$ and non-comoving matter, we notice that the orientation of $U^{\mu }$ relative to $B_{\gamma }$ determines the sign of $D_{\nu }{I}^{\nu }$ and by extension the accelerating or decelerating contribution of this term. A positive $B_{\mu }{U}^{\mu }$ contributes to the acceleration of the expansion of the congruence, while a negative $B_{\mu }{U}^{\mu }$ contributes to the deceleration of the expansion.

4.6. Redshift

From the definition of the redshift,

$$1+z={\displaystyle \frac{a_0}{a\left(t\right)}},$$

(120)

where we have normalized the present-day scale factor to

$$a_0=1,$$

(121)

and using the relation

$$B_i\left(z\right)=c_i{a}^3\left(z\right),$$

(122)

we obtain

$$B_i\left(z\right)=c_i{\left({\displaystyle \frac{a_0}{1+z}}\right)}^3={\displaystyle \frac{B_{i0}}{{(1+z)}^3}},$$

(123)

where

$$B_{i0}=c_i{a}_0^3$$

(124)

represents the present-day value of the anisotropic factor of the model.

The above relation implies that for large redshifts ($z\gg 1$), the Randers anisotropy is strongly suppressed, whereas as $z\to 0$, the Randers anisotropy approaches its present-day value, $B_{i0}$.

This relation is particularly interesting since it follows directly from the field equations of the model. It is apparent that the anisotropy is not fixed but evolves together with the cosmological volume. Consequently, the model provides a direct geometrical connection between the expansion of the Universe and the directional asymmetry. Within this framework, the anisotropy may be interpreted as a wave asymmetry originating in the early Universe.

In Randers cosmology, the correction to the redshift is usually obtained through the projection of the Randers vector $\beta$ along the direction of light propagation. Denoting by $n^i$ the unit vector in the direction of propagation, one has

$$\beta =B_i{n}^i,$$

(125)

or, equivalently,

$$\beta \left(z\right)={\displaystyle \frac{B_{i0}{n}^i}{{(1+z)}^3}},$$

(126)

where the quantity

$$D=B_{i0}{n}^i$$

(127)

plays the role of the cosmic dipole.

4.7. Dipole in Finsler Cosmology

The cosmic dipole is the largest-scale anisotropy observed in the Universe. It is usually associated with the dipolar variation in the Cosmic Microwave Background (CMB) temperature, in addition similar dipoles are also investigated in the distribution of galaxies, radio sources, quasars, and other cosmological observables.

4.7.1. Intrinsic Cosmic Dipole

Possible causes for intrinsic cosmic dipole include primordial anisotropy, anisotropic expansion, anisotropic dark energy, or large void models. In this case, the decomposition

$$D=D_{kin}+D_{intrinsic}$$

(128)

is a useful phenomenological framework for comparing observations. In Finsler-Randers cosmologies, the metric contains a preferred one-form; the vector field $B_{\gamma }$ naturally introduces a preferred cosmic direction. Instead of $T\left(\theta \right)=T_0$, one obtains

$$T\left(\theta \right)=T_0[1+D\cos \theta +\dots ]$$

(129)

where $D=D\left(B_{\gamma }\right)$ and $\theta$ is the angle with the direction of motion. Thus the dipole becomes a geometric property of spacetime rather than merely an observer effect.

4.7.2. Connection with Redshift

In anisotropic cosmology, the observed redshift may depend on direction,

$$1+z=a_0/a_e[1+D\cos \theta ]$$

(130)

Consequently, luminosity distance, angular diameter distance, Hubble parameter, gravitational redshift may all exhibit dipolar anisotropies. This is one of the motivations for testing anisotropic cosmological models using observations of Type Ia supernovae, baryon acoustic oscillations, and the CMB.

4.7.3. Connection with $TM$-Bundle Gravity

In tangent bundle gravity, such as the approach developed, the additional fiber variables $y^{\alpha }$ represent local anisotropic degrees of freedom. The generalized metric $G_{AB}(x,y)$ of spacetime depends on both spacetime position and tangent-space coordinates, so a preferred direction can emerge dynamically. In this framework we can get some consequences.

• the cosmic dipole may arise from the anisotropic geometry of the tangent bundle,
• the field $B_{\gamma }\left(x\right)$ can induce direction-dependent cosmological redshifts,
• asymmetric light propagation may modify the past and future light cones,
• dipole anisotropies in redshift, luminosity distance, and the Hubble diagram become natural observables rather than purely kinematic effects.

Symbolically, we can write,

$$B_{\gamma }\left(x\right)\to \mathrm{anisotropic}\mathrm{metric}\to \mathrm{anisotropic}\mathrm{null}\mathrm{geodesics}\to z\left(\theta \right)\to \mathrm{dipole}.$$

This provides a geometric interpretation that differs from the standard $\Lambda$-CDM picture, in which the CMB dipole is almost entirely attributed to the observer’s peculiar velocity.

4.7.4. Current Status

The standard cosmological model explains the CMB dipole as predominantly kinematic and is consistent with a broad range of observations. However, measurements of dipoles in galaxy and quasar catalogs continue to be investigated to determine whether they are fully consistent with the kinematic prediction or whether an intrinsic cosmological component is present. At present, there is no consensus that an intrinsic cosmic dipole has been established; it remains an open area of observational and theoretical research. For researchers working on anisotropic cosmologies, including Finsler–Randers geometry and tangent-bundle gravity, the cosmic dipole is an especially important observable because it provides a direct test of whether the large-scale geometry of spacetime possesses a preferred direction. Such models make quantitative predictions that can be compared with CMB maps, galaxy surveys, supernova Hubble diagrams, and future high-precision cosmological datasets.

4.8. Volume

From the previous relations, one obtains

$${\dot{B}}_i=3{\displaystyle \frac{\dot{a}}{a}}{B}_i=\mathsf{\Theta }{B}_i,$$

(131)

where

$$\mathsf{\Theta }=3H=3{\displaystyle \frac{\dot{a}}{a}}$$

(132)

is the expansion scalar.

Consequently, the Randers one-form can be written as

$$\beta =a^3\left(t\right)c_{i}d{x}^i.$$

(133)

The above relation shows that the anisotropic one-form $\beta$ evolves directly with the cosmological expansion. In addition, the anisotropic field $\beta$ may affect geodesic deviation, induce anisotropic redshifts, and modify the directional Hubble expansion. Within this framework, $\beta$ can be interpreted as a geometric source of asymmetry between different spatial directions, or as a possible contributor to modified redshift relations and anisotropic geodesic deviation.

The corresponding comoving volume is

$$V_c=c_1{c}_2{c}_3,$$

(134)

or, equivalently,

$${\overrightarrow{V}}_c=(c_1,c_2,c_3).$$

(135)

Furthermore,

$$\mathsf{\Theta }={\displaystyle \frac{{\dot{B}}_i}{B_i}}=3{\displaystyle \frac{\dot{a}}{a}}=3H,$$

(136)

or, equivalently,

$$H={\displaystyle \frac{1}{3}}{\displaystyle \frac{{\dot{B}}_i}{B_i}}.$$

(137)

Remark

In cosmology, if $a\left(t\right)$ denotes the scale factor, then $a^3\left(t\right)$ naturally represents the evolution of a comoving volume element. In particular, the physical volume corresponding to a fixed comoving volume $V_c$ is given by

$$V_{\exp }\left(t\right)=a^3\left(t\right)V_c,$$

(138)

which implies

$$V\left(t\right)\propto a^3\left(t\right).$$

(139)

This relation is directly connected to the expansion scalar,

$$\mathsf{\Theta }={\displaystyle \frac{\dot{V}}{V}}=3{\displaystyle \frac{\dot{a}}{a}}=3H,$$

(140)

for a standard FLRW universe. The above relation is particularly important because it connects the cosmological expansion with the Raychaudhuri equation and the evolution of geodesic congruences.

In a locally anisotropic spacetime, the previous relation between the volumes can be generalized to

$$V_{\mathrm{eff}}(t,y)=a^3\left(t\right)\varphi \left(y\right),$$

(141)

where $\varphi \left(y\right)$ encodes the dependence on the anisotropic (fiber) coordinates y and directional anisotropy. For example, this function can introduce Finsler or tangent bundle corrections. In these generalized types of spacetime, this connection becomes particularly interesting when combined with the Raychaudhuri equation on the tangent bundle and the geodesic deviation theory developed in [46], where the evolution of volume elements is related to the deviation of geodesics (Jacobi fields), anisotropic expansion, and cosmological asymmetry in the Raychaudhuri equations.

In our model, from relations (96) and (139), we can write

$$B_i\left(t\right)=c_i{a}^3\left(t\right)\propto c_{i}V\left(t\right),$$

(142)

where $V\left(t\right)$ denotes the physical volume of the universe.

This implies that the anisotropic 1-form $\beta$ grows proportionally to the cosmological volume. Geometrically, the preferred direction $c_i$ remains fixed, while its contribution to the Randers structure evolves with cosmic expansion.

5. Discussion and Concluding Remarks

In recent years, a number of cosmological observations have motivated the investigation of gravitational theories beyond the standard isotropic cosmological model. In particular, reported indications of large-scale anisotropies, preferred directions, and dipolar asymmetries have stimulated the development of generalized geometric frameworks, including Finsler geometry and Lorentz tangent bundle gravity, in which anisotropy is regarded as an intrinsic geometric property of spacetime rather than as a perturbation of an isotropic background. Within this natural geometric framework, intrinsic spacetime anisotropy can be investigated systematically and confronted with cosmological observations, including large-scale directional asymmetries and dipole signatures. Although the current observational evidence remains inconclusive, several anomalies in the cosmic microwave background and other large-scale surveys continue to motivate the study of anisotropic cosmological models. The tangent bundle formulation, in which both the spacetime position and the directional variables are treated as fundamental coordinates, provides a natural extension of classical gravitation. Instead of describing gravity solely on a four-dimensional Lorentzian manifold $(M,g)$, the geometric setting is enlarged to the tangent bundle $TM$, allowing the introduction of generalized metric structures, nonlinear connections, adapted frames, distinguished connections, generalized curvature tensors, and Einstein-type field equations defined directly on $TM$. These geometric objects constitute the fundamental mathematical framework for describing gravitational interactions in anisotropic spacetime.

Within this class of models, anisotropic cosmology exhibits several distinctive features:

• anisotropy modifies the cosmological evolution equations;
• directional dependence affects the propagation of light and the observed redshift;
• cosmological observables may contain measurable signatures of intrinsic spacetime anisotropy.

An additional important development is the extension of several classical geometric results—including the Jacobi equation, the geodesic deviation equation, and the Raychaudhuri equation—to Finsler and tangent bundle geometry. These generalized equations provide powerful tools for investigating

• gravitational focusing;
• anisotropic tidal effects;
• gravitational-wave propagation in generalized geometries;
• geometric interpretations of dark matter and dark energy;
• CMB anisotropies as well as connections to multipoles.

The principal contribution of the present work is the formulation of gravitation directly on the tangent bundle. In this framework, the Einstein field equations are generalized to incorporate both horizontal and vertical curvature components, thereby introducing additional geometric degrees of freedom that may possess physical significance. Consequently, the fundamental geometric structure is no longer the Lorentzian manifold $(M,g)$ but the triple $(TM,G,N)$, where G denotes the metric on the tangent bundle and N is the associated nonlinear connection.

A distinctive aspect of our approach is that anisotropy is not introduced as a small perturbation of an isotropic cosmological background. Instead, anisotropy is built into the geometry itself through the dependence of the metric on both the spacetime coordinates and the directional variables. As a result, nonlinear connections, adapted frames, and distinguished connections emerge naturally as intrinsic geometric structures of the theory. This comes in contrast with General Relativity, where the anisotropy and non-homogeneities are considered within the same geometric structure of the Riemannian spacetime. In the present work, we investigate a minimal asymmetric expansion of the Universe within a Friedmann–Sasaki–Randers cosmological model that incorporates dark matter and dark energy as geometric sectors of the tangent bundle. The generalized gravitational field equations derived in our previous works are specialized to this cosmological model, leading to modified Friedmann equations that establish an explicit relationship between the cosmological scale factor $a\left(t\right)$ and the anisotropic quantity $B_{\gamma }$ arising from the Finsler–Randers metric.

Following an approach analogous to our previous works [12,18], we have investigated the gravitational influence of dark matter and dark energy on the geometric sector of spacetime and its tangent bundle for this model coupled with the total matter sector. The resulting cosmological dynamics show that even a small intrinsic spacetime anisotropy influences the behavior of null and timelike geodesics, the Raychaudhuri equation, the cosmic dipole, and the cosmological redshift. Additionally, the evolution of a geodesic congruence under the effect of the Randers function $B_{\gamma }$, which acts as a factor of anisotropy, reveals an accelerating or a decelerating contribution of the geometric anisotropy to the expansion of the universe, dependent on the relative orientation of the 4-velocity with respect to $B_{\gamma }$. Whether this geometric effect has observational significance remains an open question, to be addressed through future theoretical developments and increasingly precise cosmological observations. This issue will be addressed in a future work.