1 Introduction↩︎

A large number multi-particle systems, such as accretion discs [1]–[4], ordinary and neutron stars, gas clouds as well as the universe as a whole [5], [6], can be described as kinetic gases [7]. The properties of a kinetic gas are encoded in the so-called 1-particle distribution function \(\varphi\) (1PDF), defined on the 1-particle phase space of spacetime, that is, on either the tangent or the cotangent bundle of spacetime [8]–[11]. The dynamics of a kinetic gas are described by the relativistic Boltzmann equation, a partial differential equation on phase space determining the behavior of the 1PDF along the trajectories of the constituents: \[\begin{align} \label{eq:relBoltz} \dot{x}^a\frac{\partial}{\partial x^a} \varphi(x,\dot{x}) +\ddot x^a \frac{\partial}{\partial \dot{x}^a} \varphi(x,\dot{x}) = C\,. \end{align}\tag{1}\] The source term \(C\) takes into account particle collisions and reactions. For a vanishing collision rate \(C=0\), this equation is known as Liouville equation.

In contrast to this phase space treatment of the dynamics of a kinetic gas, its gravitational field is usually derived from the Einstein equations sourced by an energy momentum tensor that lives on spacetime. To obtain it, one employs an averaging procedure; more precisely, the energy-momentum tensor is defined as the second moment of the 1PDF with respect to the normalized particle 4-velocities \[\begin{align} \label{eq:KGEMtens} T^{ab}_{KG} = \int_{\mathcal{S}_x} d\Sigma_x \frac{\dot{x}^a \dot{x}^b}{g(\dot{x}, \dot{x})} \varphi(x,\dot{x})\,, \end{align}\tag{2}\] where \(\mathcal{S}_x\) is the set of normalized \(4\)-velocities \(\dot{x}\) and \(d\Sigma_x\) is a suitable volume form [7]–[11]. From a phase space standpoint, the above gas-gravity coupling via averaging is rather specific – and leads to two immediate questions: Why is only the second moment of the 1PDF the source of the gravitational field of the gas? How do the infinitely many other moments of the 1PDF contribute to the gravitational field?

The latter question can be naturally answered if one considers an extension of general relativity, based on Finsler geometry [12]–[18]. Since the defining function of Finsler geometry naturally lives on the 1-particle phase space of spacetime, it can naturally be coupled to the 1PDF of a kinetic gas, without resorting to any averaging [19]. From the canonical action based Finsler gravity equation on the phase space, effective field equations on spacetime can be obtained for any moment of the 1PDF with respect to the velocities. Thus, Finsler gravity, which is a straightforward generalization of general relativity to phase space, offers the opportunity to understand the gravitational field of a kinetic gas from a new perspective.

The goal is to study if the contributions of the higher moments of kinetic gases to their gravitational field can be the source of the dark matter and dark energy phenomena [20]. In this article, we show that this is indeed the case, already for vacuum states of the kinetic gas universe. An exponential expansion of the vacuum universe is caused by the direct coupling of the 1PDF of kinetic gases to a Finslerian spacetime geometry. The obtained geometry is indistinguishable from FLRW pseudo-Riemannian spacetime geometry for massive particles and observers propagating slowly with respect to the preferred cosmological rest frame; deviations emerge for fast objects.

This remarkable result is obtained as follows.

In Section 2, we consider homogeneous and isotropic Finsler spacetimes of separated variables type, meaning that their time dependence can be absorbed completely in one conformal factor that can be interpreted as the scale factor of spacetime; in particular, this means the existence of a conformal cosmological time.

Using this Ansatz in the Finsler gravity equation sourced by the 1PDF of a kinetic gas, we find the conformal time canonical Finsler gravity Friedmann equation in Section 3. The name is justified, since its left-hand side consists of a sum of three terms proportional to the spatial curvature of the universe \(k\), the conformal time Hubble function squared \(\mathcal{H}^2\) and respectively, its conformal time derivative \(\dot{\mathcal{H}}\). The coefficients of these terms are velocity-dependent Finsler geometric expressions. We prove that, by taking any moment of this equation with respect to the unit-normalized 4-velocities, one obtains \((0,n)\)-tensor density equations on spacetime for any \(n\), and this is how all the moments of the 1PDF gravitate. All of them, and not only the second moment (\(n=2\)), are required to understand the complete gravitational field of a kinetic gas. This way, not only the matter source terms of the field equations are obtained via averaging, as in the classical approach to kinetic gases, but also the gravitational (geometric) terms, which are therefore put on the same footing. Since the 1PDF of the kinetic gas sources the gravitational field equation, and since it is integrable, also the geometric terms on the left-hand side of the equations must be integrable for compatible spacetime geometries.

In Section 4, we discuss how the evolution in time and the direction dependence are determined from the canonical Finsler gravity Friedmann equation. For a non-vanishing 1PDF as source, the dynamics of these two dependencies disentangle partly, while for the vacuum equation, they disentangle completely. The latter feature allows us to derive solutions of the canonical Finsler gravity Friedmann vacuum equation. We solve the time dependence of the kinetic gas vacuum universe in conformal time analytically in full generality. When passing to cosmological time in a particular case, we find an accelerated (exponential) expanding universe. The corresponding causal structure (velocity dependence) of the expanding universe Finsler Lagrangian is then determined by solving the remaining equation numerically.

Finally, in Section 5 we discuss the physical perspective of this new view on the gravitational field of the kinetic gas in more detail and we conclude the article in Section 6.

Overall, the approach which we present here, leads to an improved self-consistent coupling between physical systems described as kinetic gases and gravity described as the Finslerian geometry of spacetime. As the vacuum solution already leads to an exponential expansion of the universe, the prospect is to obtain the observed early and late time accelerated expansion of the universe, from the Finsler gravity equations sourced by the 1PDF of the cosmological kinetic gas filling our universe, with less, or even without, the need for dark energy.

Throughout this article we use the following notation: \(M\) is a \(4\)-dimensional manifold and \(TM\) is its (\(8\)-dimensional) tangent bundle. \((U, x^a)\) denotes a local chart on \(M\), that leads to induced local coordinates \((x^a,\dot{x}^a)\) on \(TU \subset TM\). Latin indices \(a,b,c\) run over the set \(0,1,2,3\), while Greek indices \(\alpha, \beta, \gamma\) can take the values \(1,2,3\). Moreover \(\partial_a = \frac{\partial}{\partial x^a}\) and \(\dot{\partial}_a = \frac{\partial}{\partial \dot{x}^a}\) denote the canonical coordinate bases of the tangent spaces to the tangent bundle.

2 Homogeneous and isotropic Finsler gravity↩︎

We start by briefly introducing the mathematical preliminaries we need, in order to discuss the canonical Finsler-Friedmann equation, that describes spacetime dynamics, as well as the Boltzmann or Liouville equation which encodes the dynamics of the kinetic gas.

Finsler geometry is a straightforward generalization of Riemannian geometry, in which the geometry of a manifold is derived from a general spacetime line element \(\sqrt{|L|}\) and its canonical Cartan nonlinear connection [21], [22], instead of a spacetime metric \(g\) and its Levi-Civita connection. In particular, Finsler spacetime geometry has been developed in detail [23]–[25], which paved the way to a Finslerian theory of gravity extending general relativity.

A Finsler spacetime \((M,L)\) is a \(4\)-dimensional manifold \(M\) equipped with a Finsler Lagrangian \(L\), i.e., a smooth mapping \(L:\mathcal{A}\to \mathbb{R}\) (where \(\mathcal{A} \subset TM\) is a conic subbundle) with the following properties:

• \(L\) is positively \(2\)-homogeneous in the \(\dot{x}\) variables, \(L(x,\lambda \dot{x})=\lambda^2 L(x,\dot{x}), \forall \lambda>0\) ;

• \(L\) possesses a nondegenerate Finsler metric \(g=g(x,\dot{x})\), given, in any chart of \(\mathcal{A}\), by: \[\begin{align} g_{ab}(x,\dot{x}) = \frac{1}{2}\dot{\partial}_a \dot{\partial}_b L(x,\dot{x})\,, \end{align}\] such that a well defined causal structure exists.

  Technically, the latter means that there exists a conic subbundle \(\mathcal{T}\subset \mathcal{A}\) with connected fibers \(\mathcal{T}_x = \mathcal{T}\cap T_xM\), \(\forall x\in M\), such that on each \(\mathcal{T}_x\) we have: \(L>0\), \(g_{ab}\) has Lorentzian signature \((+,-,-,-)\) and \(L\) can be continuously prolonged as 0 to the boundary \(\partial\mathcal{T}_x\) of \(\mathcal{T}_x\).

  Physically, the set \(\mathcal{T}\) is interpreted as the set of future pointing timelike directions.

Besides \(\mathcal{T}\), this definition of Finsler spacetime ensures the existence of the following additional relevant sets of directions:

• The conic set of admissible vectors \(\mathcal{A}\subset TM\), where \(L\) is smooth and all Finsler geometric objects are well defined.

• A cone of non-trivial null directions \(\mathcal{N} = \{(x,\dot{x})\in TM | L(x,\dot{x}) = 0\}\), describing the propagation of light.

• The set \(\mathcal{S} = \bigcup_{x\in M}\mathcal{S}_x\), with \(\mathcal{S}_x :=\{(x,\dot{x})\in \mathcal{T}_x | L(x,\dot{x}) = 1\}\), of all normalized future pointing timelike directions, identified as physical \(4\)-velocities.

These technical requirements ensure that we can consistently study physical systems and their evolution on a Finsler spacetime. For all details on the definition of a Finsler spacetime we refer to [23]–[25].

The canonical action based Finsler gravity equations are obtained as Euler-Lagrange equations from the canonical generalization of the Einstein-Hilbert action to Finsler spacetimes [13], [19], [26], as \[\begin{align} \label{eq:fgrav} 3\frac{ R}{L} -\frac{1}{2}g^{ab}\dot{\partial}_{a}\dot{\partial}_{b}R + g^{ab}\left(\nabla_{\delta_{a}}P_{b}-P_{a}P_{b}+\dot{\partial}_{a}(\nabla P_{b})\right) = \kappa \varphi \,. \end{align}\tag{3}\] Here, \(\kappa\) is the Finsler gravitational constant (to be determined from the appropriate general relativity or Newtonian limit), \(\varphi\) is the 1PDF of the kinetic gas which acts as a source of the gravitational field, \(R\) is the Finsler-Ricci scalar, \(P\) is the Landsberg tensor, \(\nabla = \dot{x}^a \nabla_{\delta_{a}}\) is the dynamical covariant derivative and \(\nabla_{\delta_{a}}\) the Chern-Rund-linear covariant derivative on \(TM\), the definition of these objects can be found in Appendix 7.

The Finsler gravity equation can compactly be written as \[\begin{align} \label{eq:fgrav2} \mathcal{G}(x,\dot{x}) = \kappa \varphi(x,\dot{x})\,. \end{align}\tag{4}\] It shows how all the information contained in the 1PDF contributes to the gravitational field of the kinetic gas, described by the Finslerian geometry of spacetime. Applying an integral over the set of all unit timelike directions \(\mathcal{S}_x\) at \(x \in M\), generates effective gravitational field equations on spacetime for each moment of the 1PDF: \[\begin{align} \label{eq:moments95general} &G^{a_1...a_n}(x) =\\ & \int_{\mathcal{S}_x} \frac{\dot{x}^{a_1} ...\dot{x}^{a_n}}{L^{\frac{n}{2}}}\;\mathcal{G}(x,\dot{x}) \;d\Sigma_x\; = \kappa \int_{\mathcal{S}_x} \frac{\dot{x}^{a_1} ...\dot{x}^{a_n}}{L^{\frac{n}{2}}} \;\varphi(x,\dot{x})\;d\Sigma_x\\\ &=\kappa T^{a_1...a_n}(x)\,. \end{align}\tag{5}\] For simplicity, it is often assumed that \(\varphi\) is compactly supported on the set of unit timelike directions [10], [11], at least it must be integrable, since the zeroth moment of the 1PDF gives the particle number density on spacetime, which is finite.

For \(n=2\), the equations resemble the form of the Einstein equations \[\begin{align} G^{ab}(x) = \kappa T^{ab}(x)\,. \end{align}\] Despite this formal resemblance, the left-hand side of 5 is generally expected to differ from the Einstein tensor, in the sense that it acquires additional and different terms through the integration procedure. It is still under investigation under which conditions the Einstein tensor is reproduced. One conjecture is that it might emerge when the integral is only taken around unit timelike directions that describe massive objects with small velocities or at rest.

The gravitational field equations are complemented by the Finsler generalization of the relativistic Boltzmann equation 1 , which takes the form \[\begin{align} \label{eq:FinBolz} \nabla \varphi = C\,; \end{align}\tag{6}\] this becomes the so-called Finsler Liouville equation for \(C=0\). The scalar \(C\) is called the collision rate and its vanishing means that the number of particles is conserved, as proven in detail in [27]. Equations 3 and 6 are the Finsler extension of the Einstein-Vlasov equations, which treat gravity and matter on the same footing. They take all properties of the kinetic gas into account, when deriving its gravitational field, and do not ignore any moments. Physically speaking, the Finsler gravity equation determines not only the evolution and the position dependence of the gravitational field on spacetime (encoded in the \(x\)-dependence of the geometry defining Finsler Lagrangian \(L\)), but also the causal structure of spacetime such as light cones and unit timelike directions (encoded in the \(\dot{x}\)-dependence of the geometry defining Finsler Lagrangian \(L\)).

To demonstrate explicitly how to find physically viable solutions to the Finsler gravity equations and how averaging over unit timelike velocities leads to effective field equations on spacetime, we study the equations in cosmological, homogeneous and isotropic symmetry. Homogeneous and isotropic Finsler spacetime functions \(L\) are defined in spherical local coordinates \((x^a, \dot{x}^a) = (t,r,\theta,\phi,\dot{t}, \dot{r}, \dot{\theta}, \dot{\phi})\) by, see [28], \[\begin{align} \label{eq:homisoL} L(t,r,\theta,\phi,\dot{t}, \dot{r}, \dot{\theta}, \dot{\phi}) = L(t,\dot{t}, w)\,,\quad w^2 = \frac{\dot{r}^2}{1-k r^2}+r^2\big(\dot{\theta}^2+\sin^2(\theta) \dot{\phi}^2\big)\,, \end{align}\tag{7}\] where \(k\) is the spatial curvature parameter of the universe. Due to the homogeneity property of \(L\), it can conveniently be expressed in terms of a new variable \(s=w/\dot{t}\) as \[\begin{align} \label{eq:homIsoL} L(t,\dot{t}, w) = \dot{t}^2 L(t,1, s) =: \dot{t}^2 h(t,s)^2\,. \end{align}\tag{8}\] In general, it is unclear how to display the Finsler gravity equation 3 for general homogeneous and isotropic Finsler Lagrangians \(L\) in a readable and compact way, as one can see from the Finsler geometric objects displayed in Appendix 8. Therefore, we continue with a very promising subclass, that in particular contains the homogeneous and isotropic Landsberg spacetimes, i.e.those Finsler spacetimes for which the Landsberg tensor \(P\) is identically zero [29] and the Finsler gravity equation thus simplifies.

3 The canonical Finsler Gravity Friedmann Equations↩︎

We turn our discussion to the dynamical equations of homogeneous and isotropic Finsler gravity. We first discuss the gravitational field equations before we turn to the Boltzmann and Liouville equations at the end of this section.

The function \(h(t,s)\) defining the homogeneous and isotropic Finsler Lagrangian 8 encodes, from a spacetime perspective, in principle infinitely many degrees of freedom; this is seen as, for each value of \(s\), there is one free function of \(t\). Only when the \(s\)-dependence is fixed by some principle or through some equation, the number of \(t\)-dependent functions will be specified. This is a vast generalization, compared to the situation of Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime geometry, where only the scale factor \(a(t)\) and the value of the spatial curvature \(k\) are present as degrees of freedom, whereas the \(s\)-dependence is a priori fixed.

In order to find Finsler Lagrangians close to pseudo-Riemannian geometry, we consider a conformal time, separated variable Ansatz of the type \[\begin{align} \label{eq:LHomIso} L(\eta, s) = \dot{\eta}^2 a(\eta)^2 f(s)^2\,, \end{align}\tag{9}\] for which we renamed the general time coordinate \(t\) to \(\eta\), as it resembles conformal time in FLRW geometry. The conformal time scale factor \(a(\eta)\) encodes the spacetime evolution of the gravitational field, while the function \(f(s)\) determines the causal structure of spacetime (in other words, the 4-velocity dependence of the Finsler Lagrangian). We note that the canonical, unique Landsberg-type generalization of FLRW spacetime geometry is of this form and that the explicit expression of \(f(s)\) for this case has been derived in [29]. In this language, pseudo-Riemannian FLRW spacetime geometry in conformal time is given by choosing \(f(s) = \sqrt{1-s^2}\), since then \[\begin{align} L &= \dot{\eta}^2 a(\eta)^2 (1 - s^2) = a(\eta)^2 (\dot{\eta}^2 - w^2)\\ &= a(\eta)^2 \left(\dot{\eta}^2 - \frac{\dot{r}^2}{1-kr^2} - r^2\dot{\theta}^2 - r^2 \sin^2(\theta) \dot{\phi}^2\right)\,. \end{align}\] From the phase space perspective that we take in this article, \(f(s)\) is not chosen a priori but determined from dynamical equations.

For a general Finsler Lagrangian of the form 9 , the Finsler gravity equation 3 nicely splits into terms that also have separated variables. We will call the obtained equation, the Finsler Gravity Friedmann equation \[\begin{align} \label{eq:FinsFried} \frac{k}{a(\eta)^2}\;\mathcal{G}_k(s) + \frac{\mathcal{H}^2(\eta)}{a(\eta)^2}\;\mathcal{G}_\mathcal{H}(s) + \frac{\dot{\mathcal{H}}(\eta)}{a(\eta)^2}\;\mathcal{G}_{\dot{\mathcal{H}}}(s) = \kappa\;\varphi(\eta,s)\,, \end{align}\tag{10}\] where \(\mathcal{H}(\eta) = \frac{\dot{a}(\eta)}{a(\eta)}\) (here "dot" means derivative w.r.t. \(\eta\)), and the functions \(\mathcal{G}_k\), \(\mathcal{G}_\mathcal{H}\) and \(\mathcal{G}_{\dot{\mathcal{H}}}\) are built from \(f\) and its derivatives. Their explicit form, shown in Appendix 9, is derived from the objects displayed in Appendix 8. In homogeneous and isotropic symmetry, the 1PDF can only depend on \(\eta\) and \(s\).

In 10 , we clearly see the form of a Friedmann-type equation emerging, where the functions \(\mathcal{G}_k,\) \(\mathcal{G}_\mathcal{H}\) and \(\mathcal{G}_{\dot{\mathcal{H}}}\) are the prefactors of the spatial curvature term \(k\), the Hubble function squared \(\mathcal{H}^2\) and, respectively, its derivative \(\dot{\mathcal{H}}\). Assuming that the matter content of the universe is regarded as a kinetic gas, described by the 1PDF \(\varphi\), this equation determines the gravitational field without losing any information about the gas and its imprint on spacetime geometry – and thus, on gravity.

Due to the specific form of equation 10 , we find the following necessary conditions, which partly separate the time evolution and the causal structure (direction dependence) of the geometry \[\begin{align} \label{eq:splitting-eta-s} 0 = \frac{d}{d\eta}\left(\frac{\frac{d}{d\eta}\left(\frac{1}{2 \mathcal{H} \dot{\mathcal{H}}}\frac{d}{d\eta} \left(\kappa\;a^2 \varphi(\eta,s)\right)\right)}{\frac{d}{d\eta}\left(\frac{\ddot {\mathcal{H}}}{2 \mathcal{H} \dot{\mathcal{H}} }\right)}\right)\,,\quad\quad 0 = \frac{d}{ds}\left( \frac{\frac{d}{ds} \left( \frac{ \frac{d}{ds}\left( \frac{\varphi(\eta, s)}{\mathcal{G}_k}\right) }{ \frac{d}{ds}\left( \frac{\mathcal{G}_\mathcal{H}}{\mathcal{G}_k}\right) } \right) }{ \frac{d}{ds} \left( \frac{\frac{d}{ds}\left( \frac{\mathcal{G}_{\dot{\mathcal{H}}}}{\mathcal{G}_k} \right) }{ \frac{d}{ds} \left( \frac{\mathcal{G}_\mathcal{H}}{\mathcal{G}_k} \right) } \right) } \right)\,. \end{align}\tag{11}\]

Another aspect of 10 is that one can obtain infinitely many Finsler Friedmann tensor-density equations on spacetime by considering the \(n\)-th moment of the original field equation with respect to the 4-velocities: \[\require{physics} \begin{align} \label{eq:FinsFriedInt} &\tfrac{k}{a^2} \int_{\mathcal{S}_x} \left(\tfrac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{\frac{n}{2}}} \mathcal{G}_k(s) \dd \Sigma_x\right)|_{\mathcal{S}_x} + \tfrac{\mathcal{H}^2}{a^2} \int_{\mathcal{S}_x} \left(\tfrac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{\frac{n}{2}}} \mathcal{G}_\mathcal{H}(s) \dd \Sigma_x\right) |_{\mathcal{S}_x} + \tfrac{\dot{\mathcal{H}}}{a^2} \int_{\mathcal{S}_x} \left(\tfrac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{\frac{n}{2}}} \mathcal{G}_{\dot{\mathcal{H}}}(s) \dd \Sigma_x\right)|_{\mathcal{S}_x}\nonumber\\ &= \kappa \int_{\mathcal{S}_x} \left(\tfrac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{\frac{n}{2}}} \varphi(\eta,s) \dd \Sigma_x\right)|_{\mathcal{S}_x}\,, \end{align}\tag{12}\] here, \(\mathcal{S}_x\) is the set of unit-normalized future pointing \(4\)-velocities introduced in Section 2 and \(\require{physics} \dd \Sigma_x\) is the canonical Finslerian volume form on \(\mathcal{S}_x\), expressed as \[\require{physics} \begin{align} \dd \Sigma_x = \frac{-\det g}{L^2} i_\mathbb{C}(d^4x\wedge d^4\dot{x})\,, \quad \mathbb{C} = \dot{x}^a \dot{\partial}_a\,. \end{align}\] Thus, from the pure spacetime point of view, the integrated field equations 12 can be expressed as tensor density equations \[\begin{align} \label{eq:FriedTensDen} \frac{k}{a^2} G_k^{a_1...a_n} + \frac{\mathcal{H}^2}{a^2}G_{{\mathcal{H}}}^{a_1...a_n} + \frac{\dot{\mathcal{H}}}{a^2} G_{\dot{\mathcal{H}}}^{a_1...a_n} = \kappa T^{a_1...a_n}\,, \end{align}\tag{13}\] where the coefficient tensor-densities can symbolically be written as \[\require{physics} \begin{align} G_k^{a_1...a_n} &= \int_{\mathcal{S}_x} \left(\tfrac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{\frac{n}{2}}} \mathcal{G}_k(s) \dd \Sigma_x\right)|_{\mathcal{S}_x}\,,\\ G_{{\mathcal{H}}}^{a_1...a_n} &= \int_{\mathcal{S}_x} \left(\tfrac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{\frac{n}{2}}} \mathcal{G}_\mathcal{H}(s) \dd \Sigma_x\right) |_{\mathcal{S}_x}\,, \\ G_{\dot{\mathcal{H}}}^{a_1...a_n}&= \int_{\mathcal{S}_x} \left(\tfrac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{\frac{n}{2}}} \mathcal{G}_{\dot{\mathcal{H}}}(s) \dd \Sigma_x\right)|_{\mathcal{S}_x}\,. \end{align}\] In general, one cannot guarantee that the individual integrals are finite. However, the integrability of the 1PDF ensures that their weighted sum, as displayed in 13 , is always finite. In Appendix 10, we show the details of how such integrals can be calculated explicitly.

A consistent solution of equations 13 for every \(n\), which is equivalent to a solution of the original non-averaged field equation 10 , gives the complete gravitational field of a kinetic gas.

Let us have a closer look at the second moment equation, since it resembles the Einstein equations in a bit more detail, \[\begin{align} \label{eq:FriedTensDenSecond} \frac{k}{a^2} G_k^{ab} + \frac{\mathcal{H}^2}{a^2} G_{{\mathcal{H}}}^{ab} + \frac{\dot{\mathcal{H}}}{a^2} G_{\dot{\mathcal{H}}}^{ab} = \kappa T^{ab}\,. \end{align}\tag{14}\] Just considering this equation alone, certainly does not capture all the aspects of the system. In particular, it fails to take into account all properties of the matter field encoded in the 1PDF \(\varphi\) (from the phase space point of view), respectively, in the matter tensor densities \(T^{a_1...a_n}\) (from the spacetime point of view).

In conformal time, the Einstein equations for a FLRW spacetime metric \(\tilde{g}\) coupled to a perfect fluid, characterized through its energy density \(\rho\) and its pressure \(p\), take the form 14 , with \[\begin{align} (G_k^{ab}) = (G_\mathcal{H}^{ab}) = \begin{pmatrix} -3 & 0 & 0 & 0\\ 0 & -(1-k r^2) & 0 & 0\\ 0 & 0 & -\frac{1}{r^2} & 0\\ 0 & 0 & 0 & -\frac{1}{r^2 \sin^2\theta} \end{pmatrix}\,, \qquad (G_{\dot{\mathcal{H}}}^{ab}) = \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & -2(1-k r^2) & 0 & 0\\ 0 & 0 & -\frac{2}{r^2} & 0\\ 0 & 0 & 0 & -\frac{2}{r^2 \sin^2\theta} \end{pmatrix}\,, \end{align}\] and \[\begin{align} (T_{KG}^{ab}) = \begin{pmatrix} \frac{\rho}{a^2} & 0 & 0 & 0\\ 0 & \frac{(1-k r^2) p}{a^2} & 0 & 0\\ 0 & 0 & \frac{p}{r^2 a^2} & 0\\ 0 & 0 & 0 & \frac{p}{r^2 \sin^2\theta}\,. \end{pmatrix}\,. \end{align}\] In contrast to the equations 12 which we derived from a phase space perspective, only the right-hand side of 14 , i.e., the energy momentum tensor, is obtained through averaging [7], [9], [9], [11], as \[\require{physics} \begin{align} T^{ab}_{KG} = \int_{\tilde{\mathcal{S}}_x} \dd \tilde{\Sigma}_x \frac{\dot{x}^a \dot{x}^b}{\tilde{g}(\dot{x}, \dot{x})} \varphi(x,\dot{x})\,, \end{align}\] where \(\require{physics} \dd \tilde{\Sigma}_x\) is the canonical volume form on the set of unit timelike future pointing directions \(\tilde{\mathcal{S}}_x\) defined by the spacetime metric \(\tilde{g}\).

The relativistic Boltzmann equation for a Finsler function of the type 9 becomes \[\begin{align} \label{eq:Boltz} \nabla \varphi(\eta,s)|_{\mathcal{S}} = \frac{1}{a(\eta)f(s)} \left(\frac{\partial}{\partial \eta} \varphi - \mathcal{H}(\eta) \frac{f'}{f''} \frac{\partial}{\partial s} \varphi(\eta,s)\right) = C\,, \end{align}\tag{15}\] which can easily be seen by setting \(h(t,s)\to a(\eta) f(s)\) in equation 36 , and using that on \(\mathcal{S}\) we have \(\dot{\eta}= 1/(a(\eta)f(s))\). In particular, for a collisionless gas (\(C=0\)) 15 becomes the Liouville equation. Its integration leads to a 1PDF of the form: \[\begin{align} \varphi(\eta,s) = \varphi(a(\eta) f'(s))\,. \end{align}\] These findings generalize the results presented in [5], [11] for kinetic gases on FLRW spacetime.

The phase space coupling between the matter degrees of freedom of the kinetic gas and the geometric gravitational degrees of freedom considers both of them on the same footing, in a very natural way.

Having clarified the conceptual aspects of the canonical Finsler-Friedmann equation, we turn towards finding solutions and to their physical consequences.

4 The exponentially expanding vacuum solution↩︎

On the basis of general relativity and of the Einstein equations, the only way to obtain a non-trivial homogeneous and isotropic vacuum solution is to introduce a cosmological constant, which serves as the effective source of the vacuum dynamics. Usually, it is interpreted as one possible source of dark energy.

Here, we find that the vacuum state of a gravitating kinetic gas universe, for which the whole 1PDF is considered as the source of a Finsler spacetime geometry, naturally allows for an accelerated expansion from its dynamical equation, without the need of introducing any cosmological constant or any further quantities. Thus, we can interpret the new Finsler gravity tangent bundle degrees of freedom of the gravitational field as a potential source of dark energy. This perspective is not available from the pure spacetime point of view.

Before we derive the exponentially expanding Finsler gravity kinetic gas universe in Section 4.2, we briefly discuss, for completeness and future investigation, how the zeroth and first moments of the canonical Finsler-Friedmann equation can be related to the standard form of the Friedmann equations on spacetime.

4.1 The averaged equations on spacetime↩︎

Due to homogeneous and isotropic symmetry, we find two independent equations on spacetime from the zeroth and the first moment of the Finsler gravity equation. The derivation works as outlined in 13 (and using the techniques from 10):

• The number density equation, see also 10.1, \[\require{physics} \begin{align} \label{eq:NumbDense} \kappa \hat{N}(x) &= \kappa \int_{\mathcal{S}_x} \left(\varphi_L(\eta,s) \dd\Sigma_x\right)|_{\mathcal{S}_x}\\ &= \frac{k}{a^2} G_k + \frac{\mathcal{H}^2}{a^2}G_{{\mathcal{H}}} + \frac{\dot{\mathcal{H}}}{a^2} G_{\dot{\mathcal{H}}}\,, \end{align}\tag{16}\]

• The only non-vanishing component particle-current density equation, see also 10.2, \[\require{physics} \begin{align} \label{eq:ParticleCurrentDense} \kappa \hat{J}^0(x) &= \kappa \int_{\mathcal{S}_x} \left( \dot{\eta} \varphi_L(\eta,s)\dd\Sigma_x\right)|_{\mathcal{S}_x}\\ &= \frac{k}{a^2} G_k^0 + \frac{\mathcal{H}^2}{a^2}G_{{\mathcal{H}}}^0 + \frac{\dot{\mathcal{H}}}{a^2} G_{\dot{\mathcal{H}}}^0\,. \end{align}\tag{17}\]

This system of equations can be solved for \[\begin{align} \frac{\mathcal{H}^2}{a^2} &= \frac{k}{a^2} \left(\frac{G_k G_{\dot{\mathcal{H}}}^0 - G_k^0 G_{\dot{\mathcal{H}}} }{G_{\dot{\mathcal{H}}} G_{{\mathcal{H}}}^0 - G_{\dot{\mathcal{H}}}^0 G_{{\mathcal{H}}}}\right) + \kappa \left(\frac{ G_{\dot{\mathcal{H}}} \hat{J}^0 - G_{\dot{\mathcal{H}}}^0 \hat{N} }{G_{\dot{\mathcal{H}}} G_{{\mathcal{H}}}^0 - G_{\dot{\mathcal{H}}}^0 G_{{\mathcal{H}}}} \right) \,,\\ \frac{\dot{\mathcal{H}}}{a^2} &= \frac{k}{a^2} \left(\frac{G_{{\mathcal{H}}} G_{k}^0 - G_{{\mathcal{H}}}^0 G_{k} }{G_{\dot{\mathcal{H}}} G_{{\mathcal{H}}}^0 - G_{\dot{\mathcal{H}}}^0 G_{{\mathcal{H}}}} \right) + \kappa \left(\frac{ G_{{\mathcal{H}}}^0 \hat{N} - G_{{\mathcal{H}}} \hat{J}^0 }{G_{\dot{\mathcal{H}}} G_{{\mathcal{H}}}^0 - G_{\dot{\mathcal{H}}}^0 G_{{\mathcal{H}}}} \right) \,, \end{align}\] which are the counterparts to the usual Friedmann equations in general relativity, which, in conformal time, take the form \[\begin{align} \frac{\mathcal{H}^2}{a^2} &= - \frac{k}{a^2} + \frac{\kappa}{3} \rho\,,\\ \frac{\dot{\mathcal{H}}}{a^2} &= -\frac{\kappa}{6} (3 p+ \rho)\,. \end{align}\] All further non-vanishing moment equations that can be obtained from 13 determine the tensor components \(G_I^{a_1...a_n},\;I=k, \mathcal{H}, \dot{\mathcal{H}}\) from the higher moments of the 1PDF of the kinetic gas, and thus the function \(f(s)\) in 9 , which encodes the causal structure of spacetime.

4.2 Vacuum solutions↩︎

To demonstrate the impact of the change of perspective in the understanding of the gravitational field of a kinetic gas, we solve the Finsler-Friedmann equation in vacuum. In this case, it simplifies to \[\begin{align} \label{eq:FinsFriedVac} k \mathcal{G}_k(s) + \mathcal{H}^2(\eta)\;\mathcal{G}_\mathcal{H}(s) + \dot{\mathcal{H}}(\eta)\;\mathcal{G}_{\dot{\mathcal{H}}}(s) = 0\,, \end{align}\tag{18}\] and determines the scale factor \(a\) as well as the direction dependence of the Finsler Lagrangian encoded in \(f(s)\). In vacuum, we can decouple the \(\eta\) and \(s\) dependencies as follows.

Taking a first \(\eta\) derivative yields \[\begin{align} 2 \mathcal{H} \dot{\mathcal{H}} \;\mathcal{G}_\mathcal{H}(s) + \ddot {\mathcal{H}}\;\mathcal{G}_{\dot{\mathcal{H}}}(s) = 0 \Rightarrow 2 \mathcal{G}_\mathcal{H}(s) = - \frac{\ddot {\mathcal{H}}\;\mathcal{G}_{\dot{\mathcal{H}}}(s)}{\mathcal{H} \dot{\mathcal{H}} } \end{align}\] which, via another \(\eta\) derivative then implies \[\begin{align} \label{eq:dH61} \frac{d}{d\eta} \frac{\ddot {\mathcal{H}}}{{\mathcal{H}} \dot{\mathcal{H}} } = 0 \Leftrightarrow \dot{\mathcal{H}} = c_1 {\mathcal{H}}^2 + c_2\,. \end{align}\tag{19}\] Using this expression in 18 , we find the equivalent conditions \[\begin{align} k \mathcal{G}_k(s) + c_2 \mathcal{G}_{\dot{\mathcal{H}}}(s) &= 0 \tag{20}\\ \mathcal{G}_\mathcal{H}(s) + c_1 \mathcal{G}_{\dot{\mathcal{H}}}(s) &= 0 \tag{21}\,. \end{align}\]

Equation 19 completely determines the time evolution of the vacuum state of the homogeneous and isotropic kinetic gas universe, while 20 and 21 specify its causal structure by determining \(f\), i.e., the velocity dependence of the geometry defining a Finsler Lagrangian.

In the following, we identify self-consistent physical solutions of these equations. Most remarkably, we find an accelerated expanding universe.

4.2.1 The time evolution of the vacuum state of the kinetic gas universe↩︎

In order to find the evolution of the kinetic gas vacuum universe in cosmological time, we use the relation \(a(\eta)d\eta=dt\), to express the conformal time Hubble function \[\begin{align} \mathcal{H}(\eta(t)) = \frac{\dot{a}(\eta(t))}{a(\eta(t))} = \frac{da(\eta(t))}{dt}\,, \end{align}\] and the time-evolution equation 19 as \[\begin{align} a \frac{d^2 a}{dt^2} = c_1 \left(\frac{d a}{dt} \right)^2 + c_2\,. \end{align}\] Finding a closed general solution of this equation, using elementary functions, is not possible. However, it is possible to find solutions for specific values of \(c_1\) and \(c_2\). Among them, we find extraordinarily interesting cases when we choose \(c_2 = 0\). There exist two branches for this case:

• For \(c_1=1\), the cosmological time scale factor becomes: \[\begin{align} \label{eq:expuni} a(t)=d_2 e^{d_1t}\,,\quad d_1, d_2 \in\mathbb{R}\,, \end{align}\tag{22}\] which is a most remarkable result. The Finsler gravity Friedmann equation in vacuum, i.e. the vacuum state of a gravitating kinetic gas, naturally leads to an exponentially expanding universe. This is a first direct indication that there are contributions of the kinetic gas to the gravitational field (even of the vacuum state), which are missing in the usual Einstein-Vlasov treatment of the gravitational field of a kinetic gas and that can explain, at least in part, the dark energy content of the universe.

• If \(c_1\neq 1\), then, \[\begin{align} \label{eq:c2610-2} a(t)= \left(d_1t+d_2\right)^{\frac{1}{1-c_1}}\,,\quad d_1, d_2 \in\mathbb{R}\,, \end{align}\tag{23}\] which reveals that the kinetic gas vacuum offers a variety of interesting dynamics for the universe.

Although we could not present a closed, general solution for arbitrary \(c_1\) and \(c_2\) of equation 19 in cosmological time, we find the scale factor \(a(\eta)\) in conformal time in full generality. The different branches that emerge are:

• For \(c_2 >0\):

  \(c_1\)	\(>0\)	\(=0\)	\(<0\)
  \(\mathcal{H}(\eta)\)	\(\tfrac{\sqrt{c_2} \tan \left(\sqrt{c_1} \sqrt{c_2} (\eta+c_3)\right)}{\sqrt{c_1}}\)	\(c_3 + c_2 \eta\)	\(\tfrac{\sqrt{c_2} \tanh \left(\sqrt{|c_1|} \sqrt{c_2} (\eta+c_3)\right)}{\sqrt{|c_1|}}\)
  \(a(\eta)\)	\(\tfrac{c_4}{|\cos\left(\sqrt{c_1} \sqrt{c_2} (\eta+c_3)\right)|^{1/c_1}}\)	\(c_4 e^{\frac{c_2 \eta^2}{2}+c_3 \eta}\)	\(\tfrac{c_4}{(-1)^{1/(2 c_1)}\cosh\left(\sqrt{|c_1|} \sqrt{c_2} (\eta+c_3)\right)^{1/c_1}}\)

• For \(c_2 = 0\), we do not need to conduct a case-by-case study for different choices of \(c_1\), as we needed to do in cosmological time (where we found the solutions 22 and 23 ); in conformal time, the general solution is: \[\begin{align} \label{eq:c2610} \mathcal{H}(\eta) = -\frac{1}{c_1 \eta+ c_3}\,,\quad a(\eta) = \pm \frac{c_4^2}{(c_3+c_1 \eta)^{1/c_1}},. \end{align}\tag{24}\]

• For \(c_2< 0\):

  \(c_1\)	\(>0\)	\(=0\)	\(<0\)
  \(\mathcal{H}(\eta)\)	\(- \tfrac{\sqrt{|c_2|} \tanh \left(\sqrt{c_1} \sqrt{|c_2|} (\eta+c_3)\right)}{\sqrt{c_1}}\)	\(c_3 + c_2 \eta\)	\(- \frac{\sqrt{|c_2|} \tan \left(\sqrt{|c_1|} \sqrt{|c_2|} (\eta+c_3)\right)}{\sqrt{|c_1|}}\)
  \(a(\eta)\)	\(\tfrac{c_4}{(-1)^{1/(2 c_1)}\cosh\left(\sqrt{c_1} \sqrt{|c_2|} (\eta+c_3)\right)^{1/c_1}}\)	\(c_4 e^{\frac{c_2 \eta^2}{2}+c_3 \eta}\)	\(\frac{c_4}{|\cos\left(\sqrt{|c_1|} \sqrt{|c_2|} (\eta+c_3)\right)|^{1/c_1}}\)

Having identified a solution of the canonical Finsler-Friedmann equation that leads to an accelerated expansion of the universe, the remaining question to answer is: what is the corresponding spacetime causal structure encoded in the remaining free function \(f(s)\)?

4.2.2 The causal structure of the exponentially expanding vacuum universe↩︎

Equations 20 and 21 , determine the function \(f(s)\) and with it, the causal structure of spacetime. In contrast to the time evolution equations, which could be solved in conformal time in all generality, we are not able to do the same for these sixth order nonlinear ordinary differential equations for \(f.\)

In order to find a physically interesting and viable solution, that can be compared to classical FLRW cosmology, we focus on the case \(c_2=k=0\). The first choice \(c_2=0\) is motivated from our finding of the accelerated expanding vacuum universe 22 and the second one, by the current observational constraint that the spatial curvature \(k\) of the universe is very close to zero.

For this choice of parameters, equation 20 becomes an identity, thus it only remains to solve equation 21 . As this is, still, a 6-th-order ordinary differential equation, we are not able to find a solution in closed form in terms of elementary functions. Therefore, we will proceed perturbatively and numerically to find the exponentially expanding Finsler FLRW spacetime geometry.

Taking a power series Ansatz in \(s\) \[\begin{align} \label{eq:power} f(s) = \sum_{i=0}^\infty f_i s^i\,, \end{align}\tag{25}\] we determine the first six non-vanishing coefficients of \(f(s)\), which then serve as initial conditions to numerically determine the solution to all orders.

An immediate observation from using 25 in 21 is that all odd powers of \(s\) in \(f\) must vanish. Hence, a refined Ansatz is \[\begin{align} f(s(X)) = \sum_{i=0}^\infty f_{2i} S^{2i} = \sum_{i=0}^\infty Q_{i} X^{i} = Q(X(s)) \,. \end{align}\] Then, it turns out that the coefficients \(Q_0\) and \(Q_1\) can be chosen freely, while the next four coefficients \(Q_2\) to \(Q_5\) are obtained by solving equation 21 order by order in \(X\), \[\begin{align} \label{eq:bdry} Q_2 &= -\frac{(c_1+3) Q_1^2}{10 Q_0}\\ Q_3 &= \frac{\left(16 c_1^3-75 c_1^2-922 c_1-1779\right) Q_1^3}{1050 (c_1-9) Q_0^2}\\ Q_4 &= -\frac{\left(490 c_1^5-6359 c_1^4-22227 c_1^3+321773 c_1^2+1778345 c_1+2576298\right) Q_1^4}{189000 \left(c_1^2-19 c_1+90\right) Q_0^3}\\ Q_5 &=\frac{Q_1^5}{363825000(c_1-9)^2 \left(3 c_1^2-65 c_1+350\right) Q_0^4}\left( \right.511320 c_1^8-16231154 c_1^7 +125704897 c_1^6+ \\ & \left.+582150849 c_1^5-23003963752 c_1^3 +214489583169 c_1^2 +1071379945593 c_1 +1431584382546\right)\,. \end{align}\tag{26}\] To determine suitable values for the coefficients \(Q_0\) and \(Q_1\) we compare the power series expansion of \(f(s)\) to their values in FLRW geometry, for which \[\begin{align} \label{eq:FLRW} f(s) = \sqrt{1-s^2} \sim 1 - \frac{1}{2}s^2 + \mathcal{O}(s^4)\,. \end{align}\tag{27}\] Therefore, in order to obtain a causal structure which for slow velocities looks like FLRW geometry, we choose \[\begin{align} \label{eq:Q0-Q1} Q_0 = 1\,,\quad Q_1 = -\frac{1}{2}\,. \end{align}\tag{28}\] We like to point out that deviations in the causal structure generically appear from order \(s^4\) on; a peculiar case is \(c_1=-3\), which would lead to a causal structure which differs from FLRW geometry only at order \(s^6\). However, the most interesting case, from our point of view, is \(c_1=1\), since it directly leads to an exponentially expanding universe as we saw in 22 .

The first six coefficients in the power series expansion \(Q_0\) to \(Q_5\) are used as boundary conditions to solve equation 21 numerically. For \(c_1=1, Q_0 = 1\) and \(Q_1 = -\frac{1}{2}\) we have that \[\begin{align} \label{eq:Q2-Q5} Q_2 = -\frac{1}{10},\quad Q_3 = -\frac{23}{560},\quad Q_4 = -\frac{269}{12600},\quad Q_5 = -\frac{97167659}{7761600000}\,. \end{align}\tag{29}\] Employing the \(NDSolve\) algorithm of Mathematica [30], with these initial conditions, leads to the Finsler FLRW solution displayed in Figure 1, where we compare the all order numerical solution for \(f(s^2)\) with the approximate Taylor series solution and up to order \(s^{20}\) and the non-approximate FLRW geometry given by 27 .

The difference between the numerical and the perturbative solution up to order \(s^{20}\) is shown in Figure 2.

The point \(s_0\) where \(f(s_0)=0\) marks the lightcones of the geometry, as is clear from the conformal Ansatz of our geometry 9 . While for FLRW geometry in this parametrization, this is given by \(s_0=\pm 1\), for the numerical solution the lightcone is reached by \(s_0\sim \pm 1.07517\). Thus, the lightcones in the Finslerian geometry are wider as displayed on the right in Figure 3.

In the range we are interested in, \(s\in (0,1.07517)\), the numerical all-order solution for \(f(s)\) and the approximate Taylor series solution are in close agreement with each other, and nearly indistinguishable for \(s<1\). Hence, we use the approximate solution to display the future-pointing unit timelike cones on the left of Figure 3.

Similarly to the lightcones, we find that the set of all unit timelike directions is opened more widely, compared to FLRW geometry. It is no longer a hyperboloid, but a deformation thereof. Near \(s=0\), it is close to the hyperbolic shape of FLRW geometry and, the larger the value of \(s\), the more different these shapes become. Physically speaking, this means observers (massive objects) propagating with a trajectory whose tangent is close to the canonical cosmological time direction \(V = \partial_t\) see basically a FLRW geometry, while only observers propagating with a spatial velocity large against \(V\) experience relativistic effects that deviate from pseudo-Riemannian geometry.

At this point, we have found a Finsler modification of FLRW geometry which represents the gravitational field of the vacuum of a kinetic gas universe. Its causal structure is undistinguishable from FLRW geometry for small spatial velocities \(s\) of massive objects / observers, relative to the canonical cosmological propagation direction \(V=\partial_t\) - and only differs from FLRW geometry for massive objects propagating on trajectories that correspond to a large \(s\). While the difference between the causal structures is rather mild, the difference in the evolution of the universe is striking.

The gravitational degrees of freedom, which only become visible from the phase space (tangent bundle/cotangent bundle) point of view, that are sourced by the 1PDF of a kinetic gas, lead naturally to an exponential expansion of the universe, which quantifies the conjecture we made in [20] for the first time: the usually neglected parts of the 1PDF of a kinetic gas are a source of dark energy.

5 Discussion↩︎

The origin of dark matter and dark energy belongs to the most puzzling questions in modern fundamental physics. Even without knowing their origin, the so far most successful theoretical model \(\Lambda CDM\), which assumes a cosmological constant and a specific cold form of dark matter, is currently challenged by the discrepancies of measurements of the value of the "present-day" Hubble function and other cosmological parameters, [31], [32].

In the \(\Lambda CDM\) model, the matter content of the universe as a whole is described in terms of a (perfect) fluid energy-momentum tensor. It determines the cosmological evolution of the universe through the Einstein equations. The (perfect) fluid energy-momentum tensor is, however, only an effective description of a multi-particle system, capturing just some of its properties. Thus, when used as source for the gravitational field of the system in the Einstein equations, also only parts of the gravitational field of the system are derived, namely those which are sourced by the properties of the system captured in the energy-momentum tensor. Additionally, potentially relevant contributions to the gravitational field cannot be taken into account.

In this article we discussed and demonstrated explicitly how the gravitational field of a physical multi-particle system can be derived from a more fundamental point of view, namely, instead of from its energy-momentum tensor through the Einstein equations, directly from its 1PDF, which acts as a source of the Finsler gravity equation. Since the 1PDF of a kinetic gas contains a multitude of more information about the system compared to the corresponding energy-momentum tensor (that is usually obtained from an averaging procedure), we quantified how these additional aspects of the gas, in particular its velocity distribution, contribute to its gravitational field. The consequences of this new point of view are striking.

When applied to homogeneous and isotropic symmetry, and considering such Finsler spacetime geometries for which the existence of a conformal cosmological time is ensured, we found the Finsler gravity Friedmann equation 10 , that has the 1PDF of a homogeneous and isotropic kinetic gas as a source term. This equation generalizes the Friedmann equations, that are sourced by the components of the fluid energy momentum tensor. We discussed in detail how the Finsler gravity Friedmann equation on phase space takes all the properties of the gas that are encoded in the 1PDF into account for the gravitational field. Moreover, we showed that this phase space equation is in general equivalent to infinitely many tensor density equations on spacetime 13 and not only to a 2-tensor equation as the Einstein equations are. This comparison makes it very visible that one neglects important aspects of the kinetic gas when deriving its gravitational field through the Einstein equations from an energy-momentum tensor. Namely, only specific parts of the velocity distribution of matter are taken into account, although its full velocity distribution is available. More specifically, in the Einstein equations only the second moment of the 1PDF is taken into account to derive the gravitational field of the gas. In the Finsler gravity equation, or the corresponding infinitely many effective equations on spacetime, all moments of the 1PDF are taken into account. No moments are lost – and all of them contribute to the gravitational field.

The deeper physical meaning behind the phase space / tangent bundle dynamics of the 1PDF of the kinetic gas coupled through the Finsler gravity equation to the Finsler geometry of spacetime is that it not only determines the evolution of the universe in time, but also its causal structure, meaning the precise shape of the convex cones of past and future pointing unit-timelike directions. Mathematically, they are encoded in the direction/velocity dependence of the geometry defining a Finsler Lagrangian \(L\). These degrees of freedom of the geometry of spacetime are fixed in the Einstein equations by assuming pseudo-Riemannian spacetime geometry; in contrast, in the Finsler gravity setting, they are subject to dynamical equations [13], [19], [24], [33]. In this way, we presented how to realize Wheeler’s famous quote

"Space-time tells matter how to move; matter tells space-time how to curve" ,

for kinetic gases and their 1PDFs.

In homogeneous and isotropic symmetry, which we investigated, the causal structure of spacetime is determined from the \(s = w/\dot{t}\) dependence. We found in Equation 11 that the Finsler Friedmann equation allows us to partially separate the time evolution (the dependence of the spacetime geometry on conformal time \(\eta\)), and the causal structure (the dependence of the spacetime geometry on \(s\)). Even more, in vacuum, i.e., for a vanishing 1PDF, the time evolution of the scale factor and the causal structure determination decouple completely. The former is determined by Equation 19 , while the latter is determined by the Equations 20 and 21 .

The time evolution equation can be solved analytically in conformal time in all generality. The most exciting finding is that there exists a choice of integration constants such that the corresponding cosmological time evolution gives an exponential expansion of the universe 22 , without the need of a cosmological constant, that is, without the need of dark energy. This is a purely geometric finding for the vacuum state of the kinetic gas spacetime geometry. We thus found the first direct evidence for the conjecture that the dark energy phenomenon is, at least partly, sourced by the contribution of the velocity distribution of many particle systems to their gravitational field.

For the moment, we were not able to solve the corresponding dynamical equation for the causal structure 21 analytically. Yet, we could find a spacetime solution numerically and perturbatively. Both the numerical and the perturbative solutions (see Figure 1) show a behavior that deviates from pseudo-Riemannian FLRW geometry, but in a very controlled way. The deviation is negligible around the canonical cosmological time direction, i.e.for massive objects at rest or with small velocities. The deviation increases the faster the objects are, the closer they propagate with a velocity near the speed of light, or near the lightlike directions. This finding allows the interpretation that pseudo-Riemannian geometry sourced by an energy-momentum tensor is a small velocity approximation of a more general Finslerian spacetime structure sourced by a kinetic gas.

6 Conclusion and Outlook↩︎

In conclusion, we found that taking the full 1PDF of kinetic gases as source of their gravitational field, requires an extended notion of spacetime geometry beyond pseudo-Riemannian geometry. The most natural candidate to derive the contribution of all moments of the 1PDF to the gravitational field of the gas is Finslerian spacetime geometry. Already in vacuum, Finsler geometry gives rise to at least one exponentially expanding universe as a solution. This finding does not require any exotic form of matter or energy, it just requires that we use the spacetime geometry that can take into account as many properties of the matter as possible. Here, this means not fixing the sets of causal directions (lightcones and unit-timelike directions) a priori, but let it be dynamically determined by the source matter.

These very promising vacuum solutions are only the first step to a realistic kinetic gas universe description of our universe. The next important step is to find a realistic Ansatz for the 1PDF that describes the cosmological kinetic gas, and to solve the Finsler gravity Friedmann equation for this case, at least, numerically. The question is, how does a realistic 1PDF of the kinetic gas that fills the universe look like?

We expect that, just as for the vacuum case, solutions to the non-vacuum Finsler gravity Friedmann equation will naturally lead to an exponential expansion of the universe, that can reduce the need for dark energy, or even make it superfluous. In both cases, the answer to what produces dark energy would be the forgotten moments of kinetic gas matter.

The authors would like to acknowledge networking support by the COST Actions CA23130 “Bridging high and low energies in search of quantum gravity (BridgeQG)” and CA21136 "Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse)". CP acknowledges support by the excellence cluster QuantumFrontiers of the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers – 390837967 and was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project Number 420243324 - and by the Transilvania Fellowships for Visiting Professors grant 2024 of the Transilvania University of Brasov.

7 Some basic notions of Finsler geometry↩︎

In this Appendix, we briefly introduce the Finsler geometric objects that represent the building blocks of the Finsler gravity equation 7 . For further details, we refer to the textbooks [21], [22] or the articles [16], [24].

• The Cartan tensor and its trace: \[\begin{align} \label{eq:Cartan} C_{abc} = \frac{1}{4}\dot{\partial}_a \dot{\partial}_b \dot{\partial}_c L = \frac{1}{2} \dot{\partial}_c g_{ab}, \quad C_a = g^{bc}C_{abc}\,. \end{align}\tag{30}\] The Cartan tensor measures, at each point \(x \in M\), how much the Finslerian metric tensor \(g_{ab}(x,\dot{x})\) differs from a pseudo-Riemannian (i.e., quadratic in \(\dot{x}\)) one.

• The geodesic spray coefficients \(G^a\), the canonical nonlinear connection coefficients \(N^a{}_b\) and the horizontal derivatives \(\delta_a\): \[\begin{align} \label{eq:geodspray} G^a = \frac{1}{4} g^{ab}(\dot{x}^c \partial_c \dot{\partial}_b L - \partial_b L)\,,\quad N^a{}_b = \dot{\partial}_b G^a\,,\quad \delta_a = \partial_a - N^b{}_a \dot{\partial}_b\,. \end{align}\tag{31}\]

• The curvature of the canonical nonlinear connection and the canonical curvature scalar (or Finsler-Ricci scalar: \[\begin{align} \label{eq:curv} R^c{}_{ab}\dot{\partial}_c = [\delta_a,\delta_b]\,,\quad R = R^a{}_{ab}\dot{x}^b \,. \end{align}\tag{32}\]

• The Chern-Rund covariant derivatives, defined on the adapted basis \(\{\delta_a,\dot{\partial}_a\}\) of \(T_{(x,\dot{x})}TM\) as: \[\begin{align} \nabla_{\delta_{a}}\delta_{b}&=\Gamma^{c}{}_{ab}\delta_{c}=\frac{1}{2}g^{cd}\left(\delta_{a}g_{bd}+\delta_{b}g_{ad}-\delta_{d}g_{ab}\right)\delta_{c}\,, & & \nabla_{\delta_{a}}\dot{\partial}_{b}=\Gamma^{c}{}_{ab}\dot{\partial}_{c}\,,\\ \nabla_{\dot{\partial}_{a}}\delta_{b}&=0, & & \nabla_{\dot{\partial}_{a}}\dot{\partial}_{b}=0\,. \end{align}\] We note that, in the particular case of pseudo-Riemannian geometry, \(\Gamma^{c}{}_{ab}\) become the usual Christoffel symbols, whereas the Finsler-Ricci scalar \(R=R^{}{^a}_{bac}\dot{x}^b\dot{x}^c\) is obtained from the usual Ricci tensor by contracting it twice by \(\dot{x}\).

• The dynamical covariant derivative - measuring the rate of variation of tensor fields along geodesics of \((M,L)\), is obtained from the Chern-Rund one by contraction with \(\dot{x}^a\) and expressed by the symbol \(\nabla:=\dot{x}^a \nabla_{\delta_{a}}\).

• The Landsberg tensor and its trace \[\begin{align} \label{eq:lands} P_{abc} = \nabla C_{abc}\,,\quad P_a = g^{bc}\nabla C_{abc}\,, \end{align}\tag{33}\] which keep track of how much the "non-Riemannianity" of our Finsler space varies as one moves along its geodesics.

8 Homogeneous and isotropic Finsler geometry↩︎

In this appendix, we explicitly list the geometric tensors of Finsler geometry in homogeneous and isotropic symmetry. They were obtained in [29] and serve as the foundation for deriving the conformal homogeneous and isotropic Finsler Friedmann equation 10 . We first display them here for general homogeneous and isotropic symmetry, for further use in the future. Then, for the derivations in this article, we set \(t\to \eta\) and \(h(t,s) \to a(\eta) f(s)\).

8.1 Finsler metric and Cartan tensor↩︎

We begin with the Finsler metric and the Cartan tensor for cosmologically symmetric Finsler Lagrangians \(L\), which are all described by: \[\begin{align} L = \dot{t}^2 h(t,s)^2\,, \quad s=\frac{w}{\dot{t}}\,,\quad w^2 = \frac{\dot{r}^2}{1-k r^2}+r^2\big(\dot{\theta}^2+\dot{\phi}^2\sin^2\theta \big)\,. \end{align}\] We note that \(w^2\) is the Finsler function of the metric on the spatial slices \(t=const.\), which, just as in the FLRW case, is actually a 3-dimensional Riemannian metric of constant scalar curvature (the difference, in the Finslerian case, is the much more general intertwining between space and time, encoded in the general, non-quadratic dependence of \(h\) on \(s\)). Introducing the notations, \[\begin{align} w_{\alpha\beta} = \frac{1}{2}\dot{\partial}_\alpha \dot{\partial}_\beta w^2 = \textrm{diag}\left(\frac{1}{1-kr^2},r^2,r^2 sin^2\theta\right)\,, \quad \dot{\partial}_{\alpha} w = w_\alpha = \frac{1}{w} w_{\alpha\beta}\dot{x}^\beta\,, \quad w_a = (-s, w_\alpha)\,, \end{align}\] we first obtain: \[\begin{align} \dot{\partial}_0 s = -\frac{s}{\dot{t}}\,,\quad \dot{\partial}_\alpha s = \frac{w_\alpha}{ \dot{t}}\,. \end{align}\] Denoting by primes derivatives with respect to \(s\), we find:

• The components of the Finsler metric and its inverse: \[\begin{align} \label{metric95comps95cosmo} g_{00}&=h^{2}-2shh'+s^{2}(h'{}^{2}+h h'')\,,\\ g_{0\alpha}&=w_{\alpha}\left(h h'-s(h'{}^{2}+h h'')\right)\,,\\ g_{\alpha\beta}&=h h' \frac{w_{\alpha\beta}}{s}+\left(s h'{}^{2}+s h h''-h h'\right)\frac{w_{\alpha}w_{\beta}}{s}\,, \end{align}\tag{34}\] and \[\begin{align} \label{eq:inverse95metric} g^{00}&=A\,, & A&=\frac{h'^2+ h h''}{h^3 h''}\,, \nonumber\\ g^{0\alpha }&=B\frac{\dot{x}^{\alpha }}{w}\,, & B&=\frac{s (h'^2+ h h'') - h h'}{h^3 h''}\,,\\ g^{\alpha \beta }&=C w^{\alpha \beta }+\frac{D}{w^{2}}\dot{x}^{\alpha }\dot{x}^{\beta }\,, & C&=\frac{s}{ hh{' }}\,, D=\frac{(h-sh')\left(hh'-s(h'{}^{2}+hh'')\right)}{h^{3}h'h''}\,,\nonumber \end{align}\tag{35}\] where \(w^{\alpha\beta}=\text{diag}(1-kr^2,\frac{1}{r^2},\frac{1}{r^2sin^2\theta})\) is the inverse of \(w_{\alpha\beta}\).

• The Cartan tensor: \[\begin{align} C_{000} &= -\frac{1}{2} \frac{s^3}{\dot{t}} ( h h''' + 3 h' h'')\,,\\ C_{00\alpha} &= w_\alpha \frac{s^3}{2 w} ( h h''' + 3 h' h'')\,,\\ C_{0\alpha\beta} &= \frac{w_{\alpha\beta}}{2 w}\left( h (h'-s h'')- s h'^2 \right) - \frac{w_\alpha w_\beta}{2 w}\left( h(h'-sh'')- s h'^2 + s^2 (h h''' + 3 h' h'') \right)\,,\\ C_{\alpha\beta\gamma} &= - \frac{(w_{\beta\gamma} w_{\alpha} + w_{\alpha\gamma} w_{\beta} + w_{\alpha\beta} w_{\gamma} - 3 w_{\alpha} w_{\beta} w_{\gamma}) \left( h (h' - s h'') - s h'^2\right)}{2 s w} \nonumber\\ &+ \frac{s w_{\alpha} w_{\beta} w_{\gamma} ( h h''' + 3 h' h'')}{2 w} \,. \end{align}\] Introducing the scalar variable \(T = \dot{t} w\), this can be combined to a full Finsler spacetime tensor as \[\begin{align} C_{abc} &= \frac{1}{2} \left( h (h' - s h'') - s h'^2\right) T_{abc} + \frac{\left( h h''' + 3 h' h''\right)s}{2 w} w_a w_b w_c\,, \end{align}\] where: \[\begin{align} T_{abc} &:= \dot{\partial}_a \dot{\partial}_b \dot{\partial}_c T\\ &= (\delta_c^0 \delta_b^\beta \delta_a^\alpha + \delta_a^0 \delta_b^\beta \delta_c^\alpha +\delta_b^0 \delta_c^\beta \delta_a^\alpha) \left(\tfrac{w_{\alpha\beta}-w_\alpha w_\beta}{w}\right)\nonumber\\ &+ \frac{\dot{t}}{w^2}\delta_c^\gamma \delta_b^\beta \delta_a^\alpha (3 w_\alpha w_\beta w_\gamma - w_{\alpha\beta} w_\gamma - w_{\alpha\gamma} w_\beta - w_{\gamma\beta} w_\alpha)\,. \end{align}\]

8.2 Geodesic spray components, nonlinear connection coefficients and Landsberg tensor↩︎

We continue by deriving the geodesic spray and the canonical Cartan non-linear connection. Dots over functions will mean derivatives with respect to \(t\), in order to distinguish them from derivatives w.r.t. \(s\), which are denoted by primes. The dot over a function should not be confused with the symbols for the velocity coordinates \(\dot{x}\) or the corresponding derivatives \(\dot{\partial}\).

From the definition of the geodesic spray 31 we find: \[\begin{align} G^0 = \dot{t}^2 \frac{\bigl(h'' \dot{h} - h' \dot{h}'\bigr)}{2 h h''} \end{align}\] and \[\begin{align} G^\alpha = \tilde{G}^\alpha + \frac{1}{2} \dot{t} \dot{x}^{\alpha} \frac{ \Bigl(s h'' \dot{h} + \bigl(h - s h'\bigr) \dot{h}'\Bigr) }{s h h''}\,, \end{align}\] where the quantities \(\tilde{G}^\alpha\) are the geodesic spray coefficients of the 3-dimensional Finsler function \(w^2\), given by \[\begin{align} \tilde{G}^\alpha = \frac{1}{4}w^{\alpha\beta}\left( \dot{x}^\gamma \dot{\partial}_\beta \partial_{\gamma} w^2- \partial_\beta w^2\right)\,. \end{align}\] The nonlinear connection coefficients are then obtained as \(N^a{}_b = \dot{\partial}_b G^a\), as follows: \[\begin{align} N^0{}_0 &= \frac{\dot{t}}{2 h^2 h''^2}\left[ h''^2(sh'+2h)\dot{h} - h'\bigl((sh'+2)h''+shh'''\bigr) \dot{h}'+sh'h''\dot{h}''\right]\,,\\ N^0{}_\alpha &= \frac{\dot{t} w_{\alpha} h'}{2 s h^2 h''^2} \left[ -h''^2\dot{h} + (h'h''+hh''')\dot{h}'-hh''\dot{h}''\right]\,,\\ N^\alpha{}_0 &= \frac{\dot{x}^{\alpha}}{2 h^2 h''^2} \biggl[ sh''^2(h+sh')\dot{h} + \Bigl( h''\bigl( 2h^2-sh'(h+sh')\bigr)+shh'''(h-sh')\Bigr)\dot{h}'\nonumber\\ &- sh^2h''(h-sh')\dot{h}''\biggr],\\ N^\alpha{}_\beta &= \Gamma^{\alpha}{}_{\beta\gamma} \dot{x}^{\gamma} + \frac{\dot{t} \delta^{\alpha}_{\beta}}{2 s h h''} \left[s h'' \dot{h} + \bigl(h - s h'\bigr) \dot{h}'\right] - \frac{w_{\beta} \dot{x}^{\alpha}}{2 s^2 h''^2} \left[(h''+sh''')\dot{h}'-s\dot{h}'' \right] \nonumber\\ &+ \frac{w_{\beta} \dot{x}^{\alpha} h'}{2 h^2 h''^2} \left[-h''^2\dot{h} + (h'h''+hh''')\dot{h}'-hh''\dot{h}''\right]. \end{align}\] where \(\Gamma^{\alpha}{}_{\beta\gamma}\) are the coefficients of the Levi-Civita connection of the spatial metric with components \(w_{\alpha\beta}\).

We note that the dynamical covariant derivative acts on functions \(\varphi(t,s)\) of \(t\) and \(s\) as \[\begin{align} \label{eq:nablah} \nabla \varphi(t,s) = \dot{t}\left( \partial_t \varphi - \frac{\dot{h}'}{h''} \partial_s \varphi\right)\,. \end{align}\tag{36}\]

The Landsberg tensor 33 can now be displayed as \[\begin{align} \label{eq:landshomiso} 4 P_{abc} &= \left( \nabla\left( h (h' - s h'') - s h'^2\right) + \dot{t} q(t,s) \left( h (h' - s h'') - s h'^2\right) \right)T_{abc} \\ &+ \left(\nabla \left( \tfrac{( h h''' + 3 h' h'')}{\dot{t}}\right) + 3 \left( h h''' + 3 h' h''\right)p(t,s) \right)w_a w_b w_c\,, \end{align}\tag{37}\] where we used that \[\begin{align} \label{eq:nablaw} \nabla w_a &= p(t,s) w_a \dot{t}\,,\quad \nabla T_{abc} = q(t,s) \dot{t} T_{abc}\,, \end{align}\tag{38}\] with \[\begin{align} p(t,s) = \frac{1}{2 h h''^2}\bigl[ -h''^2\dot{h} + (h'h''+hh''')\dot{h}'-hh''\dot{h}''\big]\,, \end{align}\] and \[\begin{align} q(t,s) = \frac{1}{2 sh h''^2}\Bigl[ -sh''^2\dot{h}+\bigl( h''(2h+sh')+shh'''\bigr)\dot{h}'-shh''\dot{h}''\Bigr]\,. \end{align}\]

8.3 The curvature↩︎

The last ingredient to evaluate the Finsler Friedmann equation 10 is the Finsler-Ricci curvature scalar \(R\), given by \[\begin{align} & \frac{R}{w^2} = \frac{R^a{}_{ab}\dot{x}^b }{w^2} = -2 k - \frac{3 h'''^2 \dot{h}'^2}{s^2 4 h''^4} - \frac{9 \bigl(h'' \dot{h} - h' \dot{h}'\bigr)^2}{4h^2 s^2 h''^2} \\ &+ \frac{ \dot{h}' \Bigl(s h'''' \dot{h}' + h''' \bigl(2 \dot{h}' + 3 s \dot{h}''\bigr)\Bigr)}{2s^3 h''^3} + \frac{ 2 \ddot h' + s \ddot h''}{2s^3 h''}\\ &+ \frac{3 \biggl(- h' h''' \dot{h}'^2 + 2 h' h'' \dot{h}' \dot{h}'' + h''^3 \ddot h - h''^2 \Bigl(\dot{h}'^2 + h' \ddot h'\Bigr)\biggr)}{2h s^2 h''^3}\\ &- \frac{ -6 \dot{h}'^2 + 4 s \dot{h}' \bigl(2 \dot{h}'' + s \dot{h}'''\bigr) + s^2 \Bigl(\dot{h}''^2 + 2 h''' \ddot h'\Bigr)}{4s^4 h''^2}\,. \end{align}\]

9 The coefficients of the Finsler gravity equations↩︎

We display the explicit form of the coefficient functions introduced in 10 . They are obtained from the geometric objects introduced in the previous Appendix 8 with help of the computer algebra program xAct [34] and by redenoting \(t\) by \(\eta\) as well as \(h(t,s)\) by \(a(\eta)f(s)\). The notation is that \('\) denotes the derivative of \(f\) with respect to \(s\).

The coefficient function in front of the spatial curvature \(k\) is given by \[\begin{align} \mathcal{G}_k = \frac{2 \bigl(f - s f'\bigr) \Bigl(- s f'^2 + f \bigl(f' + 2 s f''\bigr)\Bigr)}{f^3 f' f''}\,. \end{align}\]

The coefficient function in front of the square of the conformal time Hubble function \(\mathcal{H^2}\) is \[\begin{align} \mathcal{G}_{\mathcal{H}} &=\frac{3 }{4}\frac{f'^4(4ff'f'''s-15ff''^2s+2sf'^2f''+4ff'f'')}{sf^5f''^4}\\ &+\frac{1 }{4}\frac{f'^2(-98s^2f''^4+4sf'f''^2(11f''+13sf''')-6f''^2-18s^2f'''^2-sf''(16f'''-7sf''''))}{s^2f^3f''^5}\\ &+\frac{1}{2}\frac{33sf''^2-26sf'f''-22f'f''}{sf^2f''^2}+\frac{1}{4}\frac{f'^2(14f''^2+55s^2f'''^2+sf''(32f'''-27sf''''))}{s^2f^2f''^4}\\ &-\frac{1}{4}\frac{f'^3(24s^2f'''^2-3sf''f'''(-4f'''+7sf'''')+f''^2(4f'''-4sf''''+3s^2f'''''))}{s^2f^2f''^6}\\ &-\frac{1}{2}\frac{2s^2f''^4-2s^2f'f''^2f'''-11s^2f'^2f''f''''+17s^2f'^2f'''^2-2sf'f''^3-5f'^2f''^2}{s^2ff'^2f''^3}\\ &+\frac{1}{2}\frac{f'(-8f''^3+27s^3f'''^3-28s^3f''f'''f''''+f''^2(-6sf'''+5s^3f'''''))}{s^3ff''^5}\\ &+\frac{1}{4}\frac{f'^2(6f''^4-30s^4f'''^4+42sf''f'''^2f''''+f''^2(6s^2f'''^2-6s^4f''''^2-9s^4f'''f'''''))}{s^4ff''^7}\\ &+\frac{1}{4}\frac{f'^2(8f'''-2sf''''+s^3f'''''')}{s^3ff''^4}. \end{align}\]

And the coefficient function in front of the conformal time derivative of the conformal time Hubble function \(\dot{\mathcal{H}}\) is \[\begin{align} \mathcal{G}_{\dot{\mathcal{H}}} &=\frac{1}{4}\frac{f'^2(3sf'^2f''+sff'f'''-4sff''^2-8ff'f'')}{sf^4f''^3}\\ &+\frac{1}{4}\frac{20sf'f''^3-6s^2f''^4-f'^2(2f''^2+s^2f'''^2+sf''(4f'''-sf''''))}{s^2f^2f''^4}\\ &+\frac{1}{2}\frac{2s^2f''^4+s^2f'f''^2f'''-2s^2f'^2f''f''''+2s^2f'^2f'''^2-2sf'f''^2-2f'^2f''^2}{s^2ff'^2f''^3}\\ &+\frac{1}{4}\frac{f'(4f''^3-3s^3f'''^3+4s^3f''f'''f''''+f''^2(2sf'''-s^3f'''''))}{s^3ff''^5}. \end{align}\]

10 The moment integrals↩︎

The Finsler Friedmann tensor-density equations 12 are constructed from integrals of the type \[\require{physics} \begin{align} \label{eq:int} \int_{\mathcal{S}_x} \left(\frac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{n/2}} \varphi(\eta,s) \dd\Sigma_x\right)\Big|_{\mathcal{S}_x}\,, \end{align}\tag{39}\] where the volume element is given by \[\require{physics} \begin{align} d\Sigma_x &= \frac{-\det g}{L^2}\; \;i_{\mathbb{C}}\left(\dd \dot{\eta}\wedge \dd \dot{r} \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\right)\\ &= \frac{-\det g}{ a(\eta)^4 f^4 }\;\;\frac{i_{\mathbb{C}}\left(\dd \dot{\eta}\wedge \dd \dot{r} \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\right)}{\dot{\eta}^4}\,, \end{align}\] and \(\mathbb{C} = \dot{x}^a \dot{\partial}_a\) is the generator of the vector rescalings \(\dot{x}\mapsto e^{\lambda}\dot{x}\), known in the Finsler geometry literature as the Liouville vector field. Let us focus on the differential form part of the volume form \[\require{physics} \begin{align} \label{eq:differentials} &i_{\mathbb{C}}\left(\dd \dot{\eta}\wedge \dd \dot{r} \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\right)\\ &= \dot{\eta}(\dd \dot{r} \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}) - \dot{r} (\dd \dot{\eta}\wedge \dd \dot{\theta} \wedge \dd \dot{\phi}) + \dot{\theta} (\dd \dot{\eta}\wedge \dd \dot{r} \wedge \dd \dot{\phi}) - \dot{\phi} (\dd \dot{\eta}\wedge \dd \dot{r} \wedge \dd \dot{\theta})\,. \end{align}\tag{40}\] Since we are working on homogeneous and isotropic Finsler spacetimes, it is convenient to employ the following coordinate change \[\begin{align} \label{eq:dotr} \dot{r}^{\pm} = \pm \sqrt{(1- kr^2)(s^2 \dot{\eta}^2 - r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2))}\,. \end{align}\tag{41}\] Moreover, on \(\mathcal{S}_x\) we have that \(L=1\) and thus, by the assumed conformal separated variable form of the Finsler Lagrangian 9 , we have \[\require{physics} \begin{align} \label{eq:dt} \dot{\eta} = \pm \frac{1}{a f} \Rightarrow \dd \dot{\eta} = \mp \left(\frac{\dot{a}}{a^2 f} \dd \eta + \frac{f'}{a f^2} \dd s\right)\,. \end{align}\tag{42}\] This coordinate transformation is well defined for \(s\neq s_I\), where \(s_I,\;I=1,2,3,...\) are the roots of \(L\) \[\begin{align} L(\eta,\dot{\eta}, s_I) = \dot{\eta}^2 a(\eta)^2 f(s_I)^2 = 0 \Leftrightarrow f(s_I) = 0\,, \end{align}\] hence, in particular, it is well defined on \(\mathcal{S}_x\).

Since we are interested in the integral over \(\mathcal{S}_x\) at a fixed spacetime point \(x\), we employ that \(\require{physics} \dd\eta= \dd r = \dd\theta = \dd\phi =0\). Moreover, we focus on the integral over the future pointing unit-timelike directions, thus \(\dot{\eta}>0\) on \(\mathcal{S}_x\). Therefore, depending on which sign of \(\dot{r}\) in 41 is chosen, the volume form on \(\mathcal{S}_x\) is given by the following two expressions, \[\require{physics} \begin{align} d\Sigma_x^\pm &= \frac{-\det g}{ a(\eta)^4 f^4 }\;\;\frac{i_{\mathbb{C}}\left(\dd \dot{\eta}\wedge \dd \dot{r} \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\right)}{\dot{\eta}^4}\\ &= \pm \frac{-\det g}{ a^3 f^3 }\;\;\frac{\sqrt{1-kr^2} s}{\sqrt{\frac{s^2}{f^2 a^2}-r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2)}} \dd s \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\,. \end{align}\] Finally, the integral 39 becomes a sum of two parts, namely integration over \(\mathcal{S}_x^+=\mathcal{S}_x|_{\dot{r}>0}\) and \(\mathcal{S}_x^-=\mathcal{S}_x|_{\dot{r}<0}\), and can be expressed as \[\require{physics} \begin{align} \label{eq:int2} &\int_{\mathcal{S}_x} \left(\frac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{n/2}} \varphi(\eta,s) \dd\Sigma_x\right)\Big|_{\mathcal{S}_x}\nonumber\\ &= \int_{\mathcal{S}^+_x} \left(\frac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{n/2}} \varphi(\eta,s) \dd\Sigma^+_x\right)\Big|_{\mathcal{S}^+_x} + \int_{\mathcal{S}^-_x} \left(\frac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{n/2}} \varphi(\eta,s) \dd\Sigma^-_x\right)\Big|_{\mathcal{S}^-_x}\nonumber\\ &= \int_{\mathcal{S}^+_x} \left( (\dot{x}^{a_1}...\dot{x}^{a_n})|_{\mathcal{S}^+_x}\; \varphi(\eta,s) \frac{-\det g}{ a^3 f^3 } \frac{ \sqrt{1-kr^2}\;s}{\sqrt{\frac{s^2}{f^2 a^2}-r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2)}} \dd s \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\right)\nonumber\\ &- \int_{\mathcal{S}^-_x} \left( (\dot{x}^{a_1}...\dot{x}^{a_n})|_{\mathcal{S}^-_x}\;\varphi(\eta,s) \frac{-\det g}{ a^3 f^3 } \frac{ \sqrt{1-kr^2}\;s}{\sqrt{\frac{s^2}{f^2 a^2}-r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2)}} \dd s \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\right)\nonumber\\ &= \int_{\mathcal{S}^+_x} \left( (\dot{x}^{a_1}...\dot{x}^{a_n})|_{\mathcal{S}^+_x}\; \varphi(\eta,s) \frac{-\det g}{ a^3 f^3 } \frac{ \sqrt{1-kr^2}\;s}{\sqrt{\frac{s^2}{f^2 a^2}-r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2)}} \dd s \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\right)\nonumber\\ &+ \int_{\mathcal{S}^+_x} \left( (\dot{x}^{a_1}...\dot{x}^{a_n})|_{\mathcal{S}^-_x}\;\varphi(\eta,s) \frac{-\det g}{ a^3 f^3 } \frac{ \sqrt{1-kr^2}\;s}{\sqrt{\frac{s^2}{f^2 a^2}-r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2)}} \dd s \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\right)\nonumber\\ &= \int_{\mathcal{S}^+_x} \left( \left( (\dot{x}^{a_1}...\dot{x}^{a_n})|_{\mathcal{S}^+_x}+(\dot{x}^{a_1}...\dot{x}^{a_n})|_{\mathcal{S}^-_x}\right)\; \varphi(\eta,s) \frac{-\det g}{ a^3 f^3 } \frac{ \sqrt{1-kr^2}\;s}{\sqrt{\frac{s^2}{f^2 a^2}-r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2)}} \dd s \wedge \dd \dot{\theta} \wedge \dd \dot{\phi}\right)\,. \end{align}\tag{43}\]

In local coordinates \((\eta,r,\theta,\phi)\), we can express the 4-velocity factor in 43 by using 41 and 42 as \[\begin{align} (\dot{x}^{a_1}...\dot{x}^{a_n})|_{\mathcal{S}^\pm_x} &= (\dot{\eta}^I \dot{r}^J \dot{\theta}^K \dot{\phi}^L )|_{\mathcal{S}^\pm_x}\\ &= (a f)^{-I} (\pm1)^J(1- kr^2)^{\frac{J}{2}}\left(\tfrac{s^2}{a^2 f^2} - r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2)\right)^{\frac{J}{2}} \dot{\theta}^K \dot{\phi}^L \,, \end{align}\] with \(I + J + K + L = n>0\), \(I,J,K,L \in \mathbb{N}\). For the integration, we are interested in the sum of the terms on \(\mathcal{S}^+_x\) and \(\mathcal{S}^-_x\), which becomes \[\begin{align} \label{eq:dotxn} &(\dot{x}^{a_1}...\dot{x}^{a_n})|_{\mathcal{S}^+_x} + (\dot{x}^{a_1}...\dot{x}^{a_n})|_{\mathcal{S}^-_x}\nonumber\\ &= (a f)^{-I} (1 + (-1)^J)(1- kr^2)^{\frac{J}{2}}\left(\tfrac{s^2}{a^2 f^2} - r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2)\right)^{\frac{J}{2}} \dot{\theta}^K \dot{\phi}^L\,. \end{align}\tag{44}\] In order to further evaluate the integrals 43 it is convenient to introduce new coordinates \(\dot{v}\) and \(\dot{\alpha}\) for \(\dot{\theta}\) and \(\dot{\phi}\) by \[\begin{align} r \dot{\theta} = \dot{v} \cos \dot{\alpha}\,,\quad r \sin\theta \dot{\phi} = \dot{v} \sin \dot{\alpha}\,,\quad \frac{s}{f a}>\dot{v}>0\,, \dot{\alpha} \in (0,2\pi)\,. \end{align}\] They lead to \[\require{physics} \begin{align} r^2(\dot{\theta}^2 + \sin^2\theta \dot{\phi}^2) = \dot{v}^2\,,\quad \dd \dot{\theta} \wedge \dd \dot{\phi} = \dot{v} \dd \dot{v} \wedge \dd \dot{\alpha}\,. \end{align}\] Moreover, \(s\in (s_1,s_2)\), where \(s_1<0\) and \(0<s_2\) are the roots of \(L\) closest to \(0\).

Using 44 and the new coordinates, the integral 43 can be written as \[\require{physics} \begin{align} \label{eq:momentIntGen} &\int_{\mathcal{S}_x} \left(\frac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{n/2}} \varphi(\eta,s) \dd\Sigma_x\right)\Big|_{\mathcal{S}_x}\nonumber\\ &= \int_{0}^{s_2} \int_0^{\frac{s}{f a}} \int_0^{2\pi} \mathcal{I}(\eta,r, s, \dot{v}, \dot{\alpha})\;\varphi(\eta,s)\;\dd s \wedge \dd \dot{v} \wedge \dd \dot{\alpha}\,, \end{align}\tag{45}\] with \[\begin{align} \label{eq:momentIntGen2} \mathcal{I}(\eta,r, s, \dot{v}, \dot{\alpha}) &= - s \det g \nonumber \\ &\times \left( \frac{(1 + (-1)^J)}{(a f)^{I+3}} (1- kr^2)^{\frac{J+1}{2}}\left(\tfrac{s^2}{a^2 f^2} - \dot{v}^2\right)^{\frac{J-1}{2}} \frac{\dot{v}^{K+L+1}}{r^{K+L}} \cos^K(\dot{\alpha}) \sin^L(\dot{\alpha})\right)\,. \end{align}\tag{46}\] The \(\dot{\alpha}\) and \(\dot{v}\) integration in 45 can be carried out in terms of Euler \(\Gamma\) functions \[\require{physics} \begin{align} \label{eq:momentIntGen3} &\int_{\mathcal{S}_x} \left(\frac{\dot{x}^{a_1}...\dot{x}^{a_n}}{L^{n/2}} \varphi(\eta,s) \dd\Sigma_x\right)\Big|_{\mathcal{S}_x}\nonumber\\ &= \left(1 + (-1)^J\right) \left(1+(-1)^K\right)\left(1+(-1)^{K+L}\right) \frac{\Gamma\left(\tfrac{(1+K)}{2}\right)\Gamma\left(\tfrac{(1+L)}{2}\right)\Gamma\left(\tfrac{(1+J)}{2}\right)}{4 \Gamma\left(\frac{(3+J+K+L)}{2}\right)}\nonumber\\ &\times\int_{0}^{s_2} \mathcal{J}(\eta,r, s)\;\varphi(\eta,s)\;\dd s \,, \end{align}\tag{47}\] with \[\begin{align} \label{eq:momentIntGen3b} \mathcal{J}(\eta,r, s) &= \det(w_{\alpha\beta}) \frac{(1- kr^2)^{\frac{J+1}{2}} }{r^{K+L}} a^{8} f^{5} f'^2 (-f'') \frac{s^{J+K+L}}{(a f)^{4+I+J+K+L}} \,. \end{align}\tag{48}\]

The expressions 47 and 48 are the key to evaluate the different moment integrals 39 by fixing the exponent \(I\) of \(\dot{\eta}\), \(J\) of \(\dot{r}\), \(K\) or \(\dot{\theta}\) and \(L\) or \(\dot{\phi}\).

10.1 The scalar integral↩︎

For the scalar integral, which can be used to evaluate 16 , we set \(n=0\) in 43 or \(I=J=K=L=0\) in 47 and 48 , to find \[\require{physics} \begin{align} \label{eq:int3} \hat{N} &= \int_{\mathcal{S}_x} \left(\varphi(\eta,s) \dd\Sigma_x\right)|_{\mathcal{S}_x}\nonumber\\ &=4\pi a^{4}\sqrt{1-kr^2} \det(w_{\alpha\beta}) \int_{s_0}^{s_1} \dd s \left( f f'^2 (-f'')\;\;\varphi(\eta,s) \right)\,. \end{align}\tag{49}\]

10.2 The vector integral↩︎

Setting \(n=1\) in 43 or one of the exponents \(I\), \(J\), \(K\) or \(L\) to \(1\) in 47 and 48 , we find the four first moments 17 : \[\require{physics} \begin{align} \label{eq:intVec1} \hat{A}^0 &= \int_{\mathcal{S}_x} \left(\dot{\eta}\varphi(\eta,s) \dd\Sigma_x\right)\Big|_{\mathcal{S}_x}\nonumber\\ &=4\pi a^{3} \sqrt{1-kr^2} \det w_{\alpha\beta} \int_{s_0}^{s_1} \dd s \left( \varphi(\eta,s) f'^2 (-f'') \right)\,. \end{align}\tag{50}\] For the first moments with either \(J=1\), \(K=1\), or \(L=1\) and all other exponents being zero, it is clear from 47 that they all vanish \[\require{physics} \begin{align} \label{eq:intVec2} \hat{A}^1 &= \int_{\mathcal{S}_x} \left(\dot{r} \varphi(\eta,s) \dd\Sigma_x\right)\Big|_{\mathcal{S}_x} = 0\,,\\ \hat{A}^2 &= \int_{\mathcal{S}_x} \left(\dot{\theta} \varphi(\eta,s) \dd\Sigma_x\right)\Big|_{\mathcal{S}_x} = 0\,,\\ \hat{A}^3 &= \int_{\mathcal{S}_x} \left(\dot{\phi} \varphi(\eta,s) \dd\Sigma_x\right)\Big|_{\mathcal{S}_x} = 0\,. \end{align}\tag{51}\]

The higher moment integrals can be discussed in an analogue way.

References↩︎