How Inflation Might Explain Large-Scale Anisotropy

Abstract: Several recent observational analyses have suggested the possible existence of a large-scale anisotropy in our Universe. It is often thought that this would contradict the inflationary picture of the earliest times. We show that this is not the case: if we relax the artificial restriction that spacetime torsion must be zero, then such anisotropies arise naturally, indeed almost inevitably, in a theory of torsional Inflation. This is seen most clearly if one uses a ``$1\,+\,3\,$'' analysis of a torsional spacetime, leading naturally to concepts of intrinsic torsion and extrinsic torsion. An anisotropic quantum fluctuation of the inflaton leads to an extrinsic torsion field which is anisotropic and which grows throughout the inflationary era, even though the intrinsic geometry of the spatial sections is isotropised in the conventional manner. The competition between the intrinsic torsion (which does inflate away) and the extrinsic torsion leads to a short interval, early in the inflationary era, during which there is a torsion-induced energy flux, which leaves an anisotropic signal in superhorizon modes and thus ultimately at large scales in the present Universe.

• The paper demonstrates in Einstein–Cartan theory that a spin-bearing inflaton generates anisotropic extrinsic torsion and a transient directional energy flux without altering the FRW metric.
• The model predicts that the flux peaks within roughly 0.69–0.94 late e-folds after inflation begins, allowing it to affect superhorizon modes that later appear as large-scale anisotropy.
• The mechanism depends on uncertain parameters including the spin-to-energy scale ratio, reheating conditions, and total inflation duration, leaving observational tests—such as reheating gravitational waves—as important open avenues.

Motivation and central claim

A growing body of observational work has reported hints of anisotropy in the cosmic expansion and in cosmological parameters on large scales, while other analyses find no such evidence. The standard objection is theoretical: in conventional inflationary cosmology, "isotropisation" drives away any anisotropic perturbation inside the horizon, so a persistent large-scale anisotropy would appear to contradict the inflationary picture. Brett McInnes argues in this paper (2608.19765) that this objection fails once the implicit assumption of vanishing spacetime torsion is dropped. In Einstein–Cartan theory, with a spin-bearing inflaton, an anisotropic extrinsic torsion field arises almost inevitably, grows throughout the inflationary era, and produces a torsion-induced energy flux concentrated in the early inflationary period — precisely where it can imprint anisotropy on superhorizon modes and hence on present-day large scales only.

The paper builds directly on the author's earlier "$1+3$" analysis of inflationary spacetime (McInnes, 9 Aug 2025), in which isotropisation must act on both the intrinsic metric $g$ and the second fundamental form $h$ of a distinguished foliation. In zero-torsion general relativity, symmetries of $g$ force those of $h$, so isotropising $g$ suffices to suppress anisotropy. Torsion breaks that control.

The geometric role of inflation: converting intrinsic to extrinsic curvature

As motivation, the paper reviews what inflation does to spatial curvature in the torsion-free case. In creation-from-nothing scenarios, the initial hypersurface has zero extrinsic curvature but enormous positive intrinsic curvature (a three-sphere). Today the situation is reversed: near-zero intrinsic curvature but substantial extrinsic curvature (the Hubble parameter). Inflation accomplishes this conversion exactly: for de Sitter spacetime foliated by three-spheres, the intrinsic curvature decays as $\mathrm{sech}^2(ct/L)$ while the extrinsic curvature grows as $\tanh^2(ct/L)$, their sum remaining constant — as required by the Gauss formula, since de Sitter spacetime has constant curvature. Equivalently, the Hamiltonian constraint relates the (constant) inflaton energy density to the scalar curvature plus terms quadratic in $h$; if ${Scal}_g$ decays, the $g$0-terms must grow. This framing sets up the torsional analogue: inflation should convert intrinsic torsion into extrinsic torsion.

Intrinsic versus extrinsic torsion in Einstein–Cartan theory

The spacetime torsion splits, on vectors tangent to the spatial sections, into an intrinsic part $g$1 (torsion of the induced connection) and an extrinsic part determined by the antisymmetric component of $g$2:

$g$3

where $g$4 is the extrinsic torsion form. A key geometric fact drives the entire argument: any non-zero two-form on an odd-dimensional manifold is intrinsically anisotropic. Extrinsic torsion therefore cannot be isotropic; its matrix has one real eigenvector with zero eigenvalue defining a distinguished direction, and its magnitude is parametrised by a single time-dependent scalar $g$5. Via the Hubble tensor's polar decomposition, $g$6 controls the difference between expansion rates parallel and perpendicular to that direction.

The model assumes FRW metric geometry throughout (anisotropies live only in the extrinsic geometry), a maximally symmetric fully antisymmetric intrinsic torsion proportional to the volume form, $g$7, and an approximately constant inflaton spin density $g$8. The Einstein–Cartan relation $g$9 then fixes $h$0 at the onset of inflation, and constancy of $h$1 forces

$h$2

so $h$3 grows monotonically toward $h$4 as the intrinsic torsion dilutes. The modified Friedmann equation becomes

$h$5

which admits an exact solution for $h$6: monotonic, effectively exponential at late times, confirming that "torsional inflation" is genuinely inflationary, with a de Sitter-like length scale $h$7 combining the energy-density length $h$8 and spin-density length $h$9. The metric Kretschmann scalar diverges at $g$0, so the analysis is truncated at some minimum scale factor $g$1; the singularity itself is not treated.

The presence of extrinsic torsion at reheating contributes to $g$2 alongside the radiation density, so fixing $g$3 by the inflaton decay rate constrains $g$4: demanding radiation domination before BBN imposes a hard lower bound on $g$5. The paper concedes that reheating-temperature constraints are currently too weak and model-dependent to give precise numerical bounds, though a detection of stochastic gravitational waves from reheating could change this.

The torsion-induced energy flux

The physical manifestation of the anisotropy comes from combining the Einstein–Cartan field equation with a torsional Codazzi relation (proved in an appendix):

$g$6

The final algebraic term, involving the product of the second fundamental form and the intrinsic torsion, has no zero-torsion counterpart and is universally present in any metric-compatible torsional geometry. It yields an energy flux within each spatial section,

$g$7

even when nothing depends on spatial position. Rotational symmetry about the distinguished axis leaves a single non-zero component along it:

$g$8

This flux vanishes at the start of inflation, rises to a maximum $g$9, i.e. $h$0, and then declines toward zero as the decaying intrinsic torsion overwhelms the growing extrinsic torsion. The competition between the two torsions thus produces a transient directional energy flux — an anisotropic disturbance in the inflaton fluctuations that seeds superhorizon perturbations.

Timing: why only large scales are affected

The crucial quantitative result concerns when the flux peaks. Writing $h$1 in units of $h$2 ("late" e-folds), the exact solution gives $h$3 as a monotonically decreasing function of $h$4, asymptotic to approximately 0.69. Since BBN constraints forbid very small $h$5, and values below unity would violate the Dominant Energy Condition (with the caveat, acknowledged in the paper, that non-minimally coupled scalars can evade this condition without acausality, and that the relevant theorem may not extend to torsional theories), the paper adopts $h$6, giving

$h$7

The maximum flux therefore occurs no more than roughly one late e-fold after inflation begins. If $h$8 is close to unity, the flux is also not small relative to the energy density, making it a significant seed — yet bounded, consistent with the fact that observed anisotropy, if real, is inconspicuous. The paper links BBN physics to large-scale structure through this bound, which is one of its more striking claims.

This timing resolves the apparent paradox that extrinsic torsion is largest at the end of inflation while the claimed anisotropies are large-scale only: the observable signal comes from the early-time flux, affecting modes that left the horizon then, not from the late-time magnitude of $h$9. However, this interpretation leans on the assumption of "just enough" inflation (total e-folds comparable to the minimum), motivated partly by hints of small positive spatial curvature in recent data. The paper admits that the current uncertainty in total inflationary duration prevents specific predictions for large-scale observables, and notes an unresolved tension: the suggested positive curvature parameter conflicts with the spherical spatial geometry used here.

Limitations and open questions

Several assumptions constrain the scope of the results. The analysis is restricted to Einstein–Cartan theory for tractability; the paper argues the qualitative conclusions survive in more general torsional theories (where torsion propagates) but offers no quantitative demonstration. The spin density is assumed constant, justified only heuristically (e.g., a pseudoscalar fermionic condensate); additional spacetime torsion components beyond the intrinsic/extrinsic pair are set to zero, though their inclusion is argued not to change the conclusions. Whether the metric geometry isotropises simultaneously with the extrinsic geometry — certainly true in reality — is deliberately left aside. Most significantly, the mechanism's observational viability hinges on poorly constrained quantities: the ratio $g$0, the reheating temperature, and the total number of e-folds. The paper also notes that photon trajectories are not directly affected by torsion, so CMB consequences would be indirect, via large-scale structure.

Conclusion

The paper demonstrates, through exact calculations in Einstein–Cartan theory, that relaxing the zero-torsion assumption allows inflation to generate a naturally anisotropic extrinsic torsion whose competition with the decaying intrinsic torsion produces a directional energy flux peaking within roughly one e-fold of the start of inflation. Large-scale anisotropy, should it be confirmed observationally, would therefore not necessarily contradict inflation; it could instead be a signature of spin-torsion coupling in the early universe. The main open questions are whether the bounds on $g$1 can be sharpened — potentially via stochastic gravitational-wave observations of reheating — and how the mechanism's predictions depend on the total duration of inflation.