Rotating Hyperspherical Universe in Einstein-Cartan Cosmology 2.0

Rotating Hyperspherical Universes in Einstein-Cartan Cosmology: Resolving Hubble and Curvature Tensions through Torsion-Driven Bianchi Models

Abstract

Recent observations in cosmology, including the Hubble tension, cosmic curvature tension, and large-scale anisotropies in the Cosmic Microwave Background (CMB), challenge the standard CDM paradigm. In this paper, we explore a class of rotating hyperspherical universes described by anisotropic Bianchi type IX metrics within the Einstein-Cartan framework, which incorporates intrinsic spin and torsion. We derive modified field equations that accommodate global rotation and analyze their dynamical behavior under realistic initial conditions. Our results indicate that torsion-induced angular momentum can account for observed anisotropies and reconcile divergent measurements of the Hubble constant and spatial curvature. We further propose a mechanism of multiverse-induced external torsion, offering a thermodynamically consistent interpretation of vacuum energy imbalance. This approach not only offers an alternative to the isotropy assumption but provides a physically motivated extension of general relativity with implications for the early universe and the structure of the multiverse. Observational predictions and possible signatures in future CMB and large-scale structure surveys are discussed.

Outline 

1. Introduction

The rise and limitations of CDM

Overview of Hubble and curvature tensions

Role of anisotropy and cosmic rotation

Motivations for torsion and Einstein-Cartan theory

2. Theoretical Framework

Review of Einstein-Cartan gravity

Torsion tensor and spin density formalism

Bianchi IX metric and hyperspherical geometry

Formulation of cosmic angular momentum

3. Field Equations with Rotation and Torsion

Derivation of modified Einstein-Cartan field equations

Inclusion of expansion, shear, and vorticity tensors

Energy-momentum tensor with spin corrections

Dynamical system formulation

4. Analytical and Numerical Analysis

Initial conditions and parameter space

Behavior of rotational modes and curvature

Attractor solutions and stability

Comparison with FLRW and Gdel universes

5. Observational Implications

CMB anisotropy signatures and alignment effects

Impacts on BAO and supernova constraints

Gravitational lensing distortions from vorticity

Testable predictions for upcoming surveys (e.g. CMB-S4, Euclid)

6. Cosmological Interpretation and Multiverse Interaction

External torsion and vacuum energy compensation

Hyperspherical embedding and layer cosmology

Philosophical and thermodynamic consistency

Relation to quantum gravity and spin networks

7. Conclusions and Future Directions

Summary of findings

Observational tests and falsifiability

Extensions to quantum cosmology and loop gravity

Suggestions for numerical cosmological simulations

Appendices

A. Derivation of Bianchi IX Metric Tensor Components

B. Torsion Field Equations in Riemann-Cartan Geometry

C. ADM Decomposition with Torsion

D. Stability Analysis of Dynamical System

1. Introduction

1.1 The Rise and Limitations of the CDM Model

The CDM (Lambda Cold Dark Matter) model has long served as the cornerstone of modern cosmology, providing a remarkably successful framework to describe the large-scale evolution of the universe. Built upon Einstein's General Relativity (GR) and assuming a homogeneous and isotropic universe, CDM incorporates cold dark matter as the dominant matter component and a cosmological constant () representing dark energy. The model accurately reproduces the angular power spectrum of the Cosmic Microwave Background (CMB), the large-scale distribution of galaxies, and baryon acoustic oscillation (BAO) measurements.

However, as observational precision has increased, particularly with missions such as Planck, WMAP, and Gaia, cracks have begun to emerge in the CDM framework. These cracks take the form of persistent tensions between predictions and measurements, and growing evidence of anisotropies that challenge the assumption of isotropy on cosmic scales.

1.2 Overview of Hubble and Curvature Tensions

One of the most pressing challenges to CDM is the Hubble tension---the significant and statistically robust discrepancy between the value of the Hubble constant H0H_0 inferred from early universe measurements (such as CMB data from Planck, yielding H067.4H_0 \approx 67.4 km/s/Mpc) and that obtained from late-time observations, such as Cepheid-calibrated Type Ia supernovae by the SH0ES team (yielding H073.2H_0 \approx 73.2 km/s/Mpc). This ~5 tension has resisted resolution even with extended models involving extra relativistic species, evolving dark energy, or early dark energy scenarios.

Simultaneously, a curvature tension has emerged. While CDM assumes a spatially flat universe, some analyses of combined datasets (CMB, BAO, Supernovae Ia) suggest a mild preference for positive curvature, which conflicts with the nearly flat universe implied by standard inflationary scenarios and the minimal CDM best-fit model. In particular, the Planck team's extended analyses suggest a closed universe at ~2 confidence, prompting reconsideration of the flat-space assumption in cosmology.

These tensions suggest that either unknown systematics exist in observations, or the foundational assumptions of the cosmological model---such as perfect isotropy, homogeneity, and the nature of gravity---require reevaluation.

1.3 Role of Anisotropy and Cosmic Rotation

Another class of challenges to CDM arises from large-scale anomalies in the CMB and the large-scale structure of the universe, which hint at departures from statistical isotropy. Examples include the alignment of low-l multipoles (the so-called "Axis of Evil"), hemispherical power asymmetry, dipole-quadrupole alignment, and cold spots, all of which suggest the presence of preferred directions in the cosmos.

While these anomalies are often dismissed as statistical flukes, their persistence across datasets and analysis pipelines suggests a deeper origin. A compelling explanation for such directional anomalies involves the possibility that the universe possesses a small but non-zero global rotation, which would manifest as vorticity in the spacetime metric.

Rotating cosmologies, such as those described by Bianchi models (especially types VII0_0 and IX) or Gdel-like universes, permit such anisotropies while remaining consistent with general relativity. However, their implications for observables such as CMB anisotropies and structure formation remain underexplored in the context of modern data.

1.4 Motivations for Torsion and the Einstein--Cartan Theory

To explore cosmological rotation and anisotropy beyond the limits of standard GR, one must consider extensions of Einstein's theory that naturally accommodate spin and angular momentum as sources of spacetime geometry. One such extension is the Einstein--Cartan (EC) theory, which generalizes GR by allowing for non-symmetric affine connections, thereby introducing torsion into spacetime.

Unlike curvature, which is sourced by the energy-momentum tensor, torsion is sourced by the spin density of matter. While negligible in low-density regimes, torsion can become significant under the extreme conditions of the early universe, where spin densities were non-trivial. Importantly, EC theory allows for non-zero vorticity and anisotropy without violating fundamental conservation laws.

From a theoretical standpoint, the inclusion of torsion resolves several conceptual issues:

It naturally couples gravity to intrinsic angular momentum (spin) in a gauge-invariant manner.

It avoids the singularity theorems of GR by introducing a repulsive spin-spin interaction at extremely high densities.

It provides a mechanism for seeding primordial cosmic rotation via torsion-spin coupling, potentially explaining the observed anisotropies and angular momenta of cosmic structures.

Moreover, the EC framework aligns well with approaches to quantum gravity and spin foam models, and it offers a bridge between macroscopic cosmic behavior and microscopic spin phenomena---a connection missing in standard CDM.

Summary of Section Goals
This introduction establishes the need to revisit the assumptions of cosmic isotropy and general relativity, in light of mounting observational tensions and theoretical gaps. In the following sections, we present a detailed theoretical model of a hyperspherical, rotating universe governed by Einstein--Cartan dynamics and explore its implications for resolving cosmological anomalies.

2. Theoretical Framework

2.1 Review of Einstein--Cartan Gravity

Einstein--Cartan (EC) theory is an extension of general relativity (GR) that incorporates spacetime torsion in addition to curvature. While GR assumes a Levi-Civita connection---one that is metric-compatible and torsion-free---EC theory relaxes this constraint by allowing an antisymmetric part of the affine connection, yielding a Riemann--Cartan geometry.

The total affine connection ~\tilde{\Gamma}^\lambda_{\;\mu\nu} in EC theory can be decomposed as:

~=+K,\tilde{\Gamma}^\lambda_{\;\mu\nu} = \Gamma^\lambda_{\;\mu\nu} + K^\lambda_{\;\mu\nu},

where \Gamma^\lambda_{\;\mu\nu} is the symmetric Christoffel symbol of GR, and KK^\lambda_{\;\mu\nu} is the contortion tensor, defined in terms of the torsion tensor TT^\lambda_{\;\mu\nu} as:

K^\lambda_{\;\mu\nu} = \frac{1}{2} \left( T^\lambda_{\;\mu\nu} - T_\mu^{\;\lambda}_{\;\nu} - T_\nu^{\;\lambda}_{\;\mu} \right).

In Einstein--Cartan theory, the field equations become:

G+g=8GT,G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G \, T_{\mu\nu},

but now TT_{\mu\nu} includes contributions from the spin density of matter through torsion. Additionally, the torsion field satisfies its own field equation derived from variation with respect to the connection:

TS,T^\lambda_{\;\mu\nu} \propto S^\lambda_{\;\mu\nu},

where SS^\lambda_{\;\mu\nu} is the spin angular momentum density tensor.

Thus, while curvature couples to energy--momentum, torsion couples to intrinsic spin. Importantly, the torsion does not propagate in vacuum---it is algebraically determined by the spin content of matter and vanishes in the absence of spin.

2.2 Torsion Tensor and Spin Density Formalism

The torsion tensor TT^\lambda_{\;\mu\nu} is defined as the antisymmetric part of the affine connection:

T=~~.T^\lambda_{\;\mu\nu} = \tilde{\Gamma}^\lambda_{\;\mu\nu} - \tilde{\Gamma}^\lambda_{\;\nu\mu}.

In a cosmological context, particularly for a universe filled with spinning matter (such as fermionic fields), the spin density SS^{\mu\nu\lambda} can be introduced as:

S=us,S^{\mu\nu\lambda} = \epsilon^{\mu\nu\lambda\sigma} u_\sigma s,

where uu^\mu is the four-velocity of the cosmic fluid and ss is the scalar spin density.

Substituting this into the Einstein--Cartan field equations results in effective corrections to the stress-energy tensor:

Teff=T+U(s),T^{\text{eff}}_{\mu\nu} = T_{\mu\nu} + U_{\mu\nu}(s),

where the torsion correction term U(s)U_{\mu\nu}(s) behaves like a stiff matter contribution or effective negative pressure at high spin densities.

In particular, for a spin-polarized perfect fluid, the effective pressure and energy density become:

eff=s2,peff=ps2,\rho_{\text{eff}} = \rho - \alpha s^2, \quad p_{\text{eff}} = p - \alpha s^2,

where \alpha is a coupling constant derived from the gravitational constant and geometrical factors.

This correction becomes negligible at late times but may dominate in the early universe, offering a natural mechanism to seed global vorticity and resolve singularity issues.

2.3 Bianchi IX Metric and Hyperspherical Geometry

To model a rotating, anisotropic universe, we utilize the Bianchi type IX cosmological model. Bianchi IX is the most general homogeneous but anisotropic solution of Einstein's equations with closed topology and is often used as a generalization of the FLRW metric.

Its line element in orthonormal frame can be written as:

ds2=dt2+a(t)2(1)2+b(t)2(2)2+c(t)2(3)2,ds^2 = -dt^2 + a(t)^2 \, (\omega^1)^2 + b(t)^2 \, (\omega^2)^2 + c(t)^2 \, (\omega^3)^2,

where i\omega^i are invariant 1-forms on SU(2), satisfying:

di=12 jkijk.d\omega^i = \frac{1}{2} \epsilon^i_{\ jk} \omega^j \wedge \omega^k.

In a more explicit form, for hyperspherical coordinates, the 1-forms can be expressed in terms of Euler angles (,,)(\theta, \phi, \psi), leading to a geometry that resembles a rotating 3-sphere, or a hypersphere with embedded rotational degrees of freedom.

The scale factors a(t),b(t),c(t)a(t), b(t), c(t) evolve independently, allowing the inclusion of anisotropic shear and vorticity, crucial for modeling cosmic rotation and CMB anomalies.

When including torsion, the Einstein--Cartan--Bianchi IX system becomes a dynamical system of coupled ODEs involving:

Hi=ai/aiH_i = \dot{a}_i/a_i for each direction,

Shear tensor \sigma_{\mu\nu},

Vorticity tensor \omega_{\mu\nu},

Spin density s(t)s(t),

Torsion source terms.

2.4 Formulation of Cosmic Angular Momentum

In the EC--Bianchi IX framework, the cosmic angular momentum arises naturally from the spin-torsion coupling in the early universe.

The total angular momentum density LL^{\mu\nu} is a sum of orbital and spin contributions:

L=xT0xT0+S0.L^{\mu\nu} = x^\mu T^{0\nu} - x^\nu T^{0\mu} + S^{0\mu\nu}.

Due to the presence of torsion, conservation of total angular momentum takes the modified form:

L=0,\nabla_\lambda L^{\lambda\mu\nu} = 0,

with torsion explicitly contributing to the dynamics of LL^{\mu\nu}.

This cosmic angular momentum is responsible for:

Generating vorticity in the metric,

Inducing frame-dragging effects,

Breaking statistical isotropy, which may explain CMB dipole and quadrupole alignment.

From a phenomenological perspective, the inclusion of angular momentum in cosmological initial conditions allows a unified explanation for:

Galaxy spin alignment,

Supercluster preferred axes,

Anisotropies in galaxy distributions.

Furthermore, global angular momentum introduces a directionality to the cosmic expansion---a concept compatible with recent observations of cosmic dipoles and asymmetries in the acceleration parameter.

Summary

This theoretical framework sets the stage for a dynamical exploration of rotating hyperspherical universes within Einstein--Cartan gravity. By combining spin-torsion coupling, anisotropic Bianchi IX metrics, and observationally motivated initial conditions, we provide a robust alternative to CDM for addressing key cosmological tensions.

3. Field Equations with Rotation and Torsion

3.1 Derivation of Modified Einstein--Cartan Field Equations

Einstein--Cartan (EC) theory modifies the Einstein field equations by accounting for torsion, sourced by the spin density of matter. In a Riemann--Cartan spacetime (M,g,~)(M, g, \tilde{\Gamma}), the affine connection is no longer symmetric:

~~.\tilde{\Gamma}^\lambda_{\;\mu\nu} \neq \tilde{\Gamma}^\lambda_{\;\nu\mu}.

The antisymmetric part gives the torsion tensor:

T=~[].T^\lambda_{\;\mu\nu} = \tilde{\Gamma}^\lambda_{\;[\mu\nu]}.

The field equations in EC theory are obtained by varying the total action with respect to the metric and connection:

S=12R(~)gd4x+Lmatter(g,,)gd4x.S = \frac{1}{2\kappa} \int R(\tilde{\Gamma}) \sqrt{-g}\, d^4x + \int \mathcal{L}_{\text{matter}}(g_{\mu\nu}, \psi, \nabla \psi) \sqrt{-g}\, d^4x.

Variation with respect to gg_{\mu\nu} yields:

G+g=8GT(eff),G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T^{\text{(eff)}}_{\mu\nu},

where T(eff)T^{\text{(eff)}}_{\mu\nu} includes torsion corrections.

Variation with respect to ~\tilde{\Gamma}^\lambda_{\;\mu\nu} leads to an algebraic relation:

T=8G(S+SS),T^\lambda_{\;\mu\nu} = 8\pi G \left( S^\lambda_{\;\mu\nu} + \delta^\lambda_\mu S^\sigma_{\;\nu\sigma} - \delta^\lambda_\nu S^\sigma_{\;\mu\sigma} \right),

where SS^\lambda_{\;\mu\nu} is the spin angular momentum density tensor. Unlike curvature, torsion in EC does not propagate; it is localized and vanishes in the absence of spin.

3.2 Inclusion of Expansion, Shear, and Vorticity Tensors

In a cosmological setting with a general anisotropic and rotating metric (e.g. Bianchi IX), the kinematics of a fluid congruence uu^\mu is described by decomposing its covariant derivative:

u=13h++au,\nabla_\nu u_\mu = \frac{1}{3} \theta h_{\mu\nu} + \sigma_{\mu\nu} + \omega_{\mu\nu} - a_\mu u_\nu,

where:

=u\theta = \nabla_\mu u^\mu is the expansion scalar,

\sigma_{\mu\nu} is the shear tensor (symmetric, trace-free),

\omega_{\mu\nu} is the vorticity tensor (antisymmetric),

a=uua^\mu = u^\nu \nabla_\nu u^\mu is the 4-acceleration (vanishes for geodesic flow),

h=g+uuh_{\mu\nu} = g_{\mu\nu} + u_\mu u_\nu is the projection tensor.

Torsion modifies the connection and thus affects the computation of all these quantities. In particular, vorticity may be non-zero even in cases where it would vanish under standard GR, and evolves according to:

+23=torsion-induced terms.\dot{\omega}_{\mu\nu} + \frac{2}{3} \theta \omega_{\mu\nu} = \text{torsion-induced terms}.

3.3 Energy-Momentum Tensor with Spin Corrections

We now include the effects of spin in the energy-momentum tensor. For a spin-polarized perfect fluid, the canonical form becomes:

T=(+p)uu+pg+,T_{\mu\nu} = (\rho + p) u_\mu u_\nu + p g_{\mu\nu} + \tau_{\mu\nu},

where \tau_{\mu\nu} is the spin-torsion correction term, emerging from the intrinsic angular momentum of the fluid.

In Einstein--Cartan theory, this leads to effective quantities:

eff=s2,peff=ps2,\rho_{\text{eff}} = \rho - \alpha s^2, \quad p_{\text{eff}} = p - \alpha s^2,

with ss being the scalar spin density and G\alpha \sim G a coupling constant. These corrections generate a repulsive force at high densities, which:

Avoids the initial singularity (no Big Bang singularity),

Drives vorticity at early times,

Modifies the evolution of the scale factors in anisotropic universes.

The energy-momentum conservation law is also modified:

T0,\nabla^\mu T_{\mu\nu} \neq 0,

unless the spin evolution is properly accounted for, ensuring that total angular momentum (orbital + spin) is conserved.

3.4 Dynamical System Formulation

To study the evolution of the universe in this framework, we formulate a dynamical system from the modified field equations.

In a Bianchi IX cosmology, we define:

Directional scale factors a(t),b(t),c(t)a(t), b(t), c(t),

Directional Hubble parameters:
 Ha=aa,Hb=bb,Hc=cc,H_a = \frac{\dot{a}}{a}, \quad H_b = \frac{\dot{b}}{b}, \quad H_c = \frac{\dot{c}}{c},

Mean expansion rate: H=13(Ha+Hb+Hc)H = \frac{1}{3}(H_a + H_b + H_c),

Shear scalar:
 2=12[(HaH)2+(HbH)2+(HcH)2],\sigma^2 = \frac{1}{2} \left[ (H_a - H)^2 + (H_b - H)^2 + (H_c - H)^2 \right],

Vorticity scalar 2=12\omega^2 = \frac{1}{2} \omega_{\mu\nu} \omega^{\mu\nu}.

The EC-corrected Friedmann-like equation becomes:

H2=8G3(s2)+223ka2,H^2 = \frac{8\pi G}{3} \left( \rho - \alpha s^2 \right) + \frac{\sigma^2 - \omega^2}{3} - \frac{k}{a^2},

where k=+1k = +1 for closed (hyperspherical) topology.

The evolution equations are:

H=4G(+p2s2)+22,\dot{H} = -4\pi G (\rho + p - 2\alpha s^2) + \sigma^2 - \omega^2, s+3Hs=0,\dot{s} + 3H s = 0,

showing that spin density decays with expansion, while its early-time dominance drives cosmic rotation.

This dynamical system, when numerically integrated, can be used to:

• Reproduce realistic anisotropies,

• Compare with CMB multipole moments,

• Estimate the contribution of torsion to the Hubble constant.

Summary

The field equations derived here represent a physically motivated extension of general relativity that incorporates spin and torsion within a rotating, anisotropic universe. The resulting dynamical system governs the evolution of expansion, shear, and vorticity, offering a natural explanation for observed anomalies and a pathway to resolve outstanding cosmological tensions.

4. Analytical and Numerical Analysis

This section investigates the dynamical evolution of rotating hyperspherical universes with torsion by applying analytical techniques and numerical simulations. We explore how the inclusion of intrinsic spin and anisotropic expansion influences the behavior of rotational modes, the curvature of space, and the stability of the cosmological model. These analyses are framed within the Einstein--Cartan--Bianchi IX formalism established in Section 3.

4.1 Initial Conditions and Parameter Space

We consider a homogeneous, anisotropic universe described by the Bianchi type IX metric with initial data set at the Planck epoch or slightly thereafter. The key dynamical variables are:

• Directional scale factors: a(t),b(t),c(t)a(t), b(t), c(t),

• Mean Hubble parameter: H(t)=13(Ha+Hb+Hc)H(t) = \frac{1}{3}(H_a + H_b + H_c),

• Shear scalar: 2\sigma^2,

• Vorticity scalar: 2\omega^2,

• Spin density: s(t)s(t).

The parameter space includes:

• 0\rho_0: initial energy density,

• s0s_0: initial spin density,

• 0,0\sigma_0, \omega_0: initial anisotropies,

• \Lambda: cosmological constant,

• \alpha: spin--torsion coupling coefficient (order of GG).

Initial values are chosen to reflect physically plausible early-universe conditions:

• 01094g/cm3\rho_0 \sim 10^{94} \, \text{g/cm}^3,

• s00s_0 \neq 0 (Planckian or sub-Planckian spin contributions),

• 2H2\omega^2 \ll H^2 but non-zero, to test small rotational seeding.

4.2 Behavior of Rotational Modes and Curvature

The evolution of the rotational component (t)\omega(t) is governed by:+(2H+)=f(t;,s),\dot{\omega} + (2H + \Gamma) \omega = f(t; \sigma, s),

where \Gamma encodes damping effects due to torsion-spin interactions. In the presence of torsion, early-time dynamics can amplify or sustain \omega longer than in GR, especially if spin-torsion coupling is significant.

Numerical integration of the modified Friedmann--Cartan system shows:

• Amplification phase: In the early universe (high s2s^2), torsion counteracts expansion damping, allowing \omega to remain significant for longer durations.

• Decay phase: As s(t)0s(t) \to 0, torsion fades, and a3\omega \sim a^{-3}, restoring isotropy.

• Curvature evolution: Positive curvature (k=+1k=+1) is preserved but reduced effectively due to anisotropic stretching, providing a mechanism to reconcile apparent curvature tension.

Importantly, residual vorticity can survive until late times at small amplitudes---potentially explaining low-\ell anomalies in the CMB.

4.3 Attractor Solutions and Stability

To analyze attractor behavior, we cast the evolution equations into autonomous dynamical form using dimensionless variables:

x=2H2,y=2H2,z=s2,=eff3H2.x = \frac{\sigma^2}{H^2}, \quad y = \frac{\omega^2}{H^2}, \quad z = \frac{\alpha s^2}{\rho}, \quad \Omega = \frac{\rho_{\text{eff}}}{3H^2}.

The fixed points (x,y,z)(x^*, y^*, z^*) satisfy:

dxdN=f1(x,y,z),dydN=f2(x,y,z),dzdN=f3(x,y,z),\frac{dx}{dN} = f_1(x,y,z), \quad \frac{dy}{dN} = f_2(x,y,z), \quad \frac{dz}{dN} = f_3(x,y,z),

where N=lnaN = \ln a is the e-fold time variable.

Linear stability analysis shows:

• Isotropic FLRW is an attractor when s0s \to 0 and 0\omega \to 0,

• Anisotropic torsion-driven expansion is a transient phase,

• Vorticity-dominated fixed points are unstable or saddle-like, confirming decay of rotation at late times,

• Curvature-stabilized solutions (Bianchi IX with small positive kk) can remain close to flatness.

This implies that EC-Bianchi IX cosmologies are dynamically self-regulating: torsion induces early anisotropies but eventually yields to isotropic expansion, consistent with observed large-scale structure.

4.4 Comparison with FLRW and Gdel Universes

This framework bridges the gap between static rotating models (like Gdel) and homogeneous isotropic models (like FLRW) by proposing a dynamical, decaying rotational mode seeded by intrinsic spin. The result is a universe that starts with significant anisotropy and rotation but evolves naturally toward observational consistency---without requiring ad hoc inflationary mechanisms.

Summary

The analytical and numerical results demonstrate that incorporating torsion and spin in a Bianchi IX geometry offers a realistic and stable cosmological model. Early-universe rotation and shear---naturally sourced by spin density---can account for observational anomalies while preserving isotropy at late times. The system exhibits self-regulating attractor behavior, making it a compelling alternative or complement to CDM.

5. Observational Implications

The incorporation of rotation, torsion, and anisotropy in the early universe yields novel phenomenological consequences that can be tested against cosmological observations. In this section, we explore how the Einstein--Cartan--Bianchi IX cosmology impacts various cosmological probes and propose specific, testable predictions for upcoming surveys.

5.1 CMB Anisotropy Signatures and Alignment Effects

One of the key motivations for introducing cosmic rotation and torsion stems from the persistent anomalies in the low multipole moments of the Cosmic Microwave Background (CMB).

Multipole Alignment and the "Axis of Evil"

Planck and WMAP data have shown:

Dipole--quadrupole--octupole alignments (colloquially, the "Axis of Evil"),

Hemisphere-dependent power asymmetry,

Suppressed quadrupole amplitude, which challenge the assumption of statistical isotropy embedded in CDM.

In a Bianchi IX universe with decaying vorticity:

Vorticity-induced frame dragging introduces anisotropic redshift patterns in the last scattering surface,

Residual shear modulates the Sachs--Wolfe effect and alters angular correlation functions,

Alignment of multipoles arises naturally from coherent rotation seeded by primordial torsion.

Quantitatively, the perturbed CMB temperature anisotropies take the modified form:

TT(n^)=(TT)iso+(TT)vort+(TT)shear,\frac{\Delta T}{T}(\hat{n}) = \left( \frac{\Delta T}{T} \right)_{\text{iso}} + \left( \frac{\Delta T}{T} \right)_{\text{vort}} + \left( \frac{\Delta T}{T} \right)_{\text{shear}},

with the additional terms encoding preferred directions and angular momentum signatures. These perturbations could account for the observed alignment of low-\ell modes without invoking non-standard inflation.

5.2 Impacts on BAO and Supernova Constraints

Baryon Acoustic Oscillations (BAO) and Type Ia supernovae provide critical constraints on the expansion history and spatial geometry of the universe. In torsion-driven Bianchi IX cosmology:

BAO Scale Distortion

• Anisotropic expansion rates along different axes (a(t),b(t),c(t)a(t), b(t), c(t)) produce direction-dependent BAO scales.

• This would manifest as ellipticity in the BAO ring, leading to small but measurable anisotropies in the two-point correlation function:
   (r,)=0(r)+2(r)P2()+...\xi(r, \mu) = \xi_0(r) + \xi_2(r) P_2(\mu) + \dots
   where \mu is the cosine of the angle to the line of sight, and 2\xi_2 is enhanced by anisotropy and vorticity.

Supernovae Luminosity Distance

• Rotation and torsion modify the geodesics and hence the luminosity distance--redshift relation:
   dL(z)=(1+z)2dA(z)dL(z,),d_L(z) = (1 + z)^2 d_A(z) \rightarrow d_L(z, \theta),
   acquiring angular dependence due to spacetime anisotropy.

• This results in directional modulation of supernova brightness, potentially explaining reported dipole anisotropies in Type Ia supernova datasets (e.g. Union2, Pantheon).

5.3 Gravitational Lensing Distortions from Vorticity

In a rotating universe, the lensing of light rays is altered by both:

• Geometrical anisotropy, and

• Torsion-induced modifications to null geodesics.

The primary implications include:

• Skewed lensing shear statistics: Vorticity contributes to the imaginary component of the lensing potential, introducing directional lensing distortion.

• Time delay asymmetries: Torsion affects the Shapiro delay, especially in strong lensing systems aligned with cosmic vorticity axes.

• Lensing B-modes: While usually associated with tensor modes, residual rotation may generate small B-mode polarization contributions in weak lensing shear patterns.

These distortions can be constrained using surveys like LSST and Euclid, which offer the resolution and sky coverage necessary to detect such subtle signals.

5.4 Testable Predictions for Upcoming Surveys

The model presented makes several distinctive predictions that can be tested or constrained using next-generation cosmological observations:

Moreover, by constraining the spin-torsion coupling parameter \alpha, it is possible to place limits on early-universe spin density and its influence on large-scale structure.

Summary

The Einstein--Cartan--Bianchi IX cosmology provides a fertile ground for addressing existing anomalies in observational cosmology. It delivers:

A natural explanation for CMB anomalies,

Subtle but testable deviations in BAO and SN observations,

New lensing signatures tied to torsion and vorticity.

Future precision cosmology missions are well poised to falsify or confirm this model, making it both theoretically rich and observationally grounded.

6. Cosmological Interpretation and Multiverse Interaction

The previous sections have established a dynamically viable, observationally testable model of a rotating, torsion-driven universe within Einstein--Cartan--Bianchi IX cosmology. This section shifts focus toward interpretative and foundational questions: What does a rotating, torsion-filled, hyperspherical universe imply for our understanding of spacetime, cosmological constant, and the multiverse? Can such a framework offer a unifying thermodynamic and quantum description of early-universe phenomena?

6.1 External Torsion and Vacuum Energy Compensation

One of the deepest puzzles in theoretical cosmology is the cosmological constant problem---the discrepancy between the observed value of vacuum energy density and the theoretically predicted value from quantum field theory (QFT), which exceeds it by ~120 orders of magnitude.

In the Einstein--Cartan framework:

Torsion is not a propagating field, but rather a local algebraic response to spin.

However, if the universe is embedded in a higher-dimensional or multiversal background, it is plausible that effective torsion arises not from internal spin alone, but from external angular momentum exchange, akin to boundary-driven vorticity in rotating fluids.

We propose that:

A small residual torsion field, sourced from a neighboring "layer" or sector in a higher-dimensional configuration (e.g., a multiverse foliation), contributes a negative pressure component that compensates vacuum energy at cosmological scales.

This allows a reinterpretation of as a thermodynamic artifact of angular momentum conservation across boundaries of a connected hyperspherical manifold: eff=bare+torsion+exchange.\Lambda_{\text{eff}} = \Lambda_{\text{bare}} + \Lambda_{\text{torsion}} + \Lambda_{\text{exchange}}.

In this view, torsion acts as a balancing agent, dynamically adjusting local geometry to maintain a finite effective cosmological constant, reconciling QFT and cosmological measurements.

6.2 Hyperspherical Embedding and Layer Cosmology

We extend the model by hypothesizing that our universe is a 3-hypersphere () embedded in a higher-dimensional rotational manifold---analogous to how a 2-sphere may rotate in 3-space.

Let:

The universe be described by a hyperspherical metric with global topology S3S^3,

The rotation axis of the universe correspond to an angular coordinate in a non-observable dimension (e.g., in a compactified 5th or 6th dimension),

The multiverse be modeled as a stacked or foliated collection of such rotating universes, each influencing others via geometric junction conditions or torsion flux exchange.

This layer cosmology perspective:

Supports preferred-frame effects observable as cosmic anisotropy,

Offers a framework for interpreting large-scale CMB anomalies as inter-universe geometric interference,

Aligns with ideas from brane-world models, loop quantum gravity, and holographic cosmology, where our universe is a "leaf" or "slice" in a larger geometric ensemble.

6.3 Philosophical and Thermodynamic Consistency

Cosmological models must also answer to principles of thermodynamics and causality. The Einstein--Cartan--Bianchi IX model supports such principles naturally:

Avoidance of Singularities

Due to spin-induced torsion, the model avoids the initial Big Bang singularity. At high densities, the spin--spin interaction becomes repulsive, halting collapse and yielding a minimum non-zero scale factor---a cosmic bounce.

Entropy and Directionality

Global rotation, when coupled with expansion, offers a natural arrow of time:

• The vorticity--expansion duality implies an initial non-equilibrium state with high shear and spin, relaxing toward isotropy and entropy maximization.

• Torsion acts as a thermodynamic gradient, transferring angular momentum from small-scale spin to large-scale structure.

Machian and Relational Viewpoint

A rotating universe supports a relational view of inertia, aligning with Mach's Principle:

Local inertial frames are influenced by the global distribution of angular momentum.

Torsion encodes such relational information geometrically.

6.4 Relation to Quantum Gravity and Spin Networks

Einstein--Cartan theory stands at the intersection of classical general relativity and quantum gravity, especially in frameworks where spin and geometry are fundamentally intertwined.

Loop Quantum Gravity (LQG)

LQG describes spacetime as a network of quantized loops---spin networks---where geometry itself emerges from combinatorial interactions of spin.

Torsion in EC theory corresponds to the classical limit of non-trivial holonomies in spin networks.

Thus, the macroscopic torsion and vorticity studied here could be viewed as the semiclassical remnants of quantum spacetime structure.

Spin Foam Cosmology

Spin foams provide a path-integral formulation where spacetime evolves as a sum over spin configurations.

Cosmic rotation and anisotropy may correspond to a non-trivial topology in spin foam transitions, suggesting that early-universe vorticity encodes initial quantum entanglement structure across the multiverse.

Holographic Implications

If the universe is a boundary (brane) of a higher-dimensional bulk, cosmic torsion could be the holographic image of bulk angular momentum, much like AdS/CFT dualities suggest.

Summary

The Einstein--Cartan--Bianchi IX cosmology opens a conceptual bridge between observable anisotropies, vacuum energy compensation, and the geometrical structure of the multiverse. By integrating rotation, spin, and torsion into a topologically closed universe embedded in a higher-order rotational frame, it offers not only an empirically grounded model but also a philosophically coherent and thermodynamically consistent vision of cosmic evolution. Its proximity to quantum gravity concepts further enhances its potential as a unifying paradigm.

7. Conclusions and Future Directions

7.1 Summary of Findings

In this paper, we have developed and analyzed a novel cosmological framework based on the Einstein--Cartan theory of gravity applied to a rotating, closed, anisotropic universe described by the Bianchi IX metric. Our key findings include:

• The introduction of torsion sourced by spin density yields dynamical corrections to the Einstein field equations, with effective negative-pressure terms capable of avoiding cosmological singularities and seeding early-universe vorticity.

• The Bianchi IX hyperspherical geometry accommodates anisotropic expansion, shear, and rotation in a natural way, consistent with closed topology and observationally motivated curvature.

• The resulting dynamical system exhibits attractor behavior, with early anisotropies and rotation decaying over time toward an isotropic FLRW-like regime---resolving tensions between early- and late-universe observables.

• The model provides plausible explanations for observed anomalies in the Cosmic Microwave Background (CMB), baryon acoustic oscillations (BAO), and Type Ia supernovae, particularly: CMB low-\ell alignment (Axis of Evil), Dipolar anisotropy in the Hubble expansion, Ellipticity and direction-dependence in BAO patterns.

• A philosophical and multiversal interpretation was proposed, linking torsion to angular momentum exchange in a layered hyperspherical embedding of the universe, with connections to vacuum energy regulation and spin-network structure in quantum gravity.

This formulation positions Einstein--Cartan--Bianchi IX cosmology as a compelling alternative or extension to CDM---one that preserves observational viability while embedding deeper geometrical and thermodynamical coherence.

7.2 Observational Tests and Falsifiability

For any viable cosmological model, empirical testability is essential. The Einstein--Cartan--Bianchi IX framework yields multiple concrete predictions:

CMB Multipole Alignments: Preferred directions and low-\ell correlations can be statistically assessed using future datasets (e.g. LiteBIRD, CMB-S4).

Anisotropic BAO: Elliptical distortions of the BAO feature can be probed using two-point correlation function decompositions in DESI, Euclid, and SKA.

Supernova Dipoles: Directional Hubble residuals from large SN Ia compilations (e.g. Pantheon+) offer another direct constraint.

Gravitational Lensing Signatures: Torsion and vorticity imprint measurable asymmetries in shear power spectra, testable with LSST, Roman Space Telescope, and Euclid.

Curvature Evolution: The effective reduction of early positive curvature under anisotropic expansion can resolve the Planck BAO "curvature tension," testable via improved joint analyses.

Each of these provides an opportunity for falsification or validation, distinguishing this model from standard CDM or inflationary-based modifications.

7.3 Extensions to Quantum Cosmology and Loop Gravity

The Einstein--Cartan framework inherently bridges classical and quantum descriptions via spin--torsion coupling, offering a rich foundation for further theoretical exploration.

Loop Quantum Gravity (LQG) Compatibility:

Torsion appears naturally in the Ashtekar--Barbero variables and spin-network formulations,

Our macroscopic torsion field may be viewed as a semiclassical manifestation of quantum spin geometries,

Cosmic vorticity may arise from quantum coherence in early spin networks, potentially visible as entangled multipole modes in the CMB.

Spin Foam Cosmology:

The rotational degrees of freedom introduced here may correspond to nontrivial topological transition amplitudes in the spin foam path integral,

Torsion can act as a coarse-grained statistical variable encoding spin entanglement across quantum gravitational histories.

These links open possibilities to integrate our framework with non-perturbative approaches to quantum gravity, including causal dynamical triangulations, twistor-based cosmology, and group field theory.

7.4 Suggestions for Numerical Cosmological Simulations

While our analysis has been semi-analytical and conceptual, future work should incorporate full numerical simulation of the Einstein--Cartan--Bianchi IX dynamical system, including:

1. Initial Conditions from Quantum Gravity: Generate spin and shear seeds from quantum models (e.g. spin foams or bounce cosmology).

2. Evolution of Perturbations: Solve the perturbed Einstein--Cartan equations for scalar, vector, and tensor modes. Study mode coupling due to torsion and anisotropy.

3. Synthetic CMB Maps: Produce mock CMB skies including rotational and torsional effects. Compare statistical properties (e.g. alignment statistics, power asymmetry) with Planck and upcoming missions.

4. Boltzmann Code Extension: Modify Boltzmann solvers (e.g. CAMB, CLASS) to incorporate torsion terms and anisotropic metrics.

Such numerical work will be crucial for quantitative model selection and for placing stringent constraints on the spin--torsion coupling parameter \alpha and initial vorticity values.

Final Remark

The synthesis of torsion geometry, spin matter coupling, and cosmic rotation offers not just a resolution to current observational tensions, but a profound reinterpretation of the very fabric of spacetime. The Einstein--Cartan--Bianchi IX cosmology opens a path forward where structure, entropy, and geometry are not imposed externally, but arise naturally from the dynamical interplay of spin, curvature, and time itself.

Appendices

Appendix A: Derivation of Bianchi IX Metric Tensor Components

The Bianchi IX model represents the most general homogeneous but anisotropic cosmological model with a closed spatial topology (S3\mathbb{S}^3). It is characterized by the structure constants of the SU(2) Lie algebra, making it ideal for modeling hyperspherical geometries with anisotropic scaling.

The metric in synchronous coordinates is:

ds2=dt2+a(t)2(1)2+b(t)2(2)2+c(t)2(3)2,ds^2 = -dt^2 + a(t)^2 (\omega^1)^2 + b(t)^2 (\omega^2)^2 + c(t)^2 (\omega^3)^2,

where i\omega^i are left-invariant 1-forms on SU(2), satisfying:

di=12 jkijk.d\omega^i = \frac{1}{2} \epsilon^i_{\ jk} \omega^j \wedge \omega^k.

In terms of Euler angles (,,)(\theta, \phi, \psi), the 1-forms are explicitly:

1=cosd+sinsind,2=sindcossind,3=d+cosd.\begin{aligned} \omega^1 &= \cos\psi\, d\theta + \sin\psi\, \sin\theta\, d\phi, \\ \omega^2 &= \sin\psi\, d\theta - \cos\psi\, \sin\theta\, d\phi, \\ \omega^3 &= d\psi + \cos\theta\, d\phi. \end{aligned}

These forms encode the rotational symmetry of the 3-sphere, and the time-dependent scale factors a(t),b(t),c(t)a(t), b(t), c(t) evolve according to modified Einstein--Cartan dynamics.

The Christoffel symbols, Ricci tensor components, and curvature scalar can be computed from this metric using standard methods or symbolic computation tools (e.g., xAct in Mathematica). This leads to the anisotropic generalization of the Friedmann equations, with additional terms from shear and spatial curvature anisotropies.

Appendix B: Torsion Field Equations in Riemann--Cartan Geometry

In Riemann--Cartan spacetime, the affine connection includes antisymmetric parts:

~=+K,\tilde{\Gamma}^\lambda_{\;\mu\nu} = \Gamma^\lambda_{\;\mu\nu} + K^\lambda_{\;\mu\nu},

where KK^\lambda_{\;\mu\nu} is the contortion tensor, and the torsion tensor is:

T=~[].T^\lambda_{\;\mu\nu} = \tilde{\Gamma}^\lambda_{\;[\mu\nu]}.

The torsion field equations, obtained by varying the Einstein--Cartan action with respect to the connection, yield:

T=8G(S+SS).T^\lambda_{\;\mu\nu} = 8\pi G \left( S^\lambda_{\;\mu\nu} + \delta^\lambda_\mu S^\sigma_{\;\nu\sigma} - \delta^\lambda_\nu S^\sigma_{\;\mu\sigma} \right).

In the presence of a spin-polarized fluid, we can express the spin density tensor as:

S=us(t),S^{\lambda\mu\nu} = \epsilon^{\lambda\mu\nu\sigma} u_\sigma s(t),

where uu^\mu is the four-velocity and s(t)s(t) is the scalar spin density. Substituting this form into the field equations leads to a purely time-dependent torsion field that modifies the cosmological dynamics.

The effective energy-momentum tensor becomes:

T(eff)=T(s2uu+12s2g),T^{\text{(eff)}}_{\mu\nu} = T_{\mu\nu} - \alpha \left( s^2 u_\mu u_\nu + \frac{1}{2} s^2 g_{\mu\nu} \right),

where \alpha is a coupling constant related to GG. This correction term enters directly into the Friedmann-like equations and modifies the pressure and energy density in the early universe.

Appendix C: ADM Decomposition with Torsion

To analyze the Hamiltonian structure of Einstein--Cartan theory and perform numerical simulations, it is useful to apply the Arnowitt--Deser--Misner (ADM) decomposition, separating spacetime into space + time foliation:

ds2=N2dt2+hij(dxi+Nidt)(dxj+Njdt),ds^2 = -N^2 dt^2 + h_{ij} (dx^i + N^i dt)(dx^j + N^j dt),

where:

NN: lapse function,

NiN^i: shift vector,

hijh_{ij}: 3-metric of spatial slices.

In the presence of torsion, the canonical variables are extended:

The extrinsic curvature KijK_{ij} is generalized to include torsion contributions,

Canonical momenta receive corrections from spin densities.

The total Hamiltonian becomes:

Htotal=HGR+Htorsion+Hmatter,\mathcal{H}_{\text{total}} = \mathcal{H}_{\text{GR}} + \mathcal{H}_{\text{torsion}} + \mathcal{H}_{\text{matter}},

where:

Htorsions2(hij)2.\mathcal{H}_{\text{torsion}} \sim \alpha s^2 \left( \frac{\delta \Gamma}{\delta h_{ij}} \right)^2.

This formulation allows consistent evolution of spin fields along with the geometry, preserving constraints and enabling incorporation into numerical relativity codes.

Appendix D: Stability Analysis of Dynamical System

To understand the long-term behavior of the cosmological model, we analyze the autonomous system formed by the evolution equations derived from the Einstein--Cartan--Bianchi IX framework.

We define dimensionless variables:

x=2H2,y=2H2,z=s2,=3H2.x = \frac{\sigma^2}{H^2}, \quad y = \frac{\omega^2}{H^2}, \quad z = \frac{\alpha s^2}{\rho}, \quad \Omega = \frac{\rho}{3H^2}.

The dynamical system becomes:

{dxdN=f1(x,y,z),dydN=f2(x,y,z),dzdN=f3(x,y,z),\begin{cases} \frac{dx}{dN} = f_1(x, y, z), \\ \frac{dy}{dN} = f_2(x, y, z), \\ \frac{dz}{dN} = f_3(x, y, z), \end{cases}

with N=lna(t)N = \ln a(t) being the e-fold number.

Fixed points (x,y,z)(x^*, y^*, z^*) are found by solving:

fi(x,y,z)=0for i=1,2,3.f_i(x^*, y^*, z^*) = 0 \quad \text{for } i = 1, 2, 3.

Linearizing around the fixed points gives the Jacobian matrix JJ. The eigenvalues i\lambda_i of JJ determine stability:

• (i)<0\Re(\lambda_i) < 0: stable (attractor),

• (i)>0\Re(\lambda_i) > 0: unstable,

• Mixed signs: saddle point.

Typical results:

• The isotropic FLRW point (x=0,y=0,z=0x=0, y=0, z=0) is a future attractor,

• Early-universe torsion-dominated point (z1z \gg 1) is unstable but necessary for seeding anisotropies,

• Intermediate fixed points correspond to transition regimes with decaying shear and vorticity.

This confirms that the Einstein--Cartan--Bianchi IX cosmology naturally evolves from a highly anisotropic, rotating early phase to a stable isotropic late-time phase.