Cosmic topology. Part IIc. Detectability with non-standard primordial power spectrum

1. Problem Statement

The global topology of the Universe remains an open question in cosmology. While General Relativity dictates local geometry, it does not fix global connectivity. Current observations favor a spatially flat, simply-connected (infinite) universe ($E_{18}$), but a multiply-connected (finite) topology is not ruled out.

The primary challenge in detecting cosmic topology via the Cosmic Microwave Background (CMB) is the degeneracy between topological signatures and uncertainties in the primordial power spectrum (PPS).

• Standard Assumption: Previous studies assumed a standard, nearly scale-invariant power-law PPS ($P_R(k) \propto k^{n_s-1}$).
• The Gap: Non-trivial topologies impose boundary conditions that could theoretically alter the PPS itself. Furthermore, independent physical mechanisms (e.g., inflationary features) could modify the PPS at large scales (low $k$) regardless of topology.
• Core Question: How do deviations from the standard PPS (cutoffs, enhancements, oscillations) impact the detectability of non-trivial topologies? Do these deviations hide topological signatures or enhance them?

2. Methodology

The authors employed a dual approach combining theoretical information theory (KL Divergence) and machine learning (CatBoost) to assess detectability.

A. Topological Models

The study focuses on the six fully compact, orientable Euclidean topologies ($E_1$–$E_6$):

• $E_1$: 3-torus (cubic and non-cubic).
• $E_2$–$E_5$: Manifolds with screw motions (half-turn, quarter-turn, etc.).
• $E_6$: Hantzsche-Wendt manifold.
• Observer Positions: Both "on-axis" (rotation axis passes through the observer) and "off-axis" positions were simulated to account for inhomogeneity in non-cubic manifolds.

B. Modifications to the Primordial Power Spectrum (PPS)

The authors modeled three phenomenological modifications in the infrared (IR) regime (large scales), parameterized by a dimensionless wavenumber $x = k L_{mod} / 2\pi$:

1. Cutoff: Suppression of power at large scales ($P_R^{mod} = P_R(1 - ae^{-x/x_c})$).
2. Enhancement: Amplification of power at large scales ($P_R^{mod} = P_R(1 + ae^{-x/x_c})$).
3. Oscillations: Periodic modulations ($P_R^{mod} = P_R(1 + a \sin(fx)e^{-x/x_c})$).

These were tested in two scenarios:

• Topology-Induced: Modifications appear only in the non-trivial topology ($E_i$), not in the covering space ($E_{18}$).
• Topology-Independent: Modifications appear in both $E_i$ and $E_{18}$ (simulating universal inflationary features).

C. Analytical Metric: Kullback-Leibler (KL) Divergence

The KL divergence ($D_{KL}$) was used to quantify the information loss when modeling a non-trivial topology ($E_i$) as a trivial one ($E_{18}$).

• Forward KL ($D_{KL}(E_i || E_{18})$): Measures distinguishability if the universe is truly non-trivial.
• Backward KL ($D_{KL}(E_{18} || E_i)$): Measures the risk of false positives (fitting trivial data with a non-trivial model).
• Threshold: $D_{KL} \geq 1$ is considered the threshold for detectability.

D. Machine Learning Classification

• Algorithm: CatBoost (gradient-boosted decision trees) was used to classify simulated CMB temperature maps.
• Input: Spherical harmonic coefficients ($a_{\ell m}$) for $\ell \in [2, 30]$.
• Tasks:
  • Binary Classification: Distinguishing a specific topology (e.g., $E_1$) from the covering space ($E_{18}$).
  • Multi-class Classification: Distinguishing between multiple topologies and observer configurations simultaneously.
• Interpretability: SHAP (SHapley Additive exPlanations) values were used to identify which $a_{\ell m}$ coefficients drive the classification decisions.

3. Key Contributions

1. Systematic PPS Sensitivity Analysis: The first comprehensive study quantifying how IR modifications to the PPS affect topological detectability metrics (KL divergence and ML accuracy).
2. Distinction of Modification Origins: Explicitly separating the effects of topology-induced PPS changes from universal (topology-independent) PPS changes.
3. Validation of ML Robustness: Demonstrating that ML classifiers trained on standard power spectra remain robust (or even improve) when tested against non-standard spectra, provided the modifications are topology-specific.
4. Observer Position Dependence: Detailed analysis of how off-axis observer positions in inhomogeneous manifolds ($E_2$–$E_6$) alter detectability compared to on-axis positions.

4. Key Results

A. Impact of PPS Modifications on KL Divergence

• Topology-Induced Modifications (Only in $E_i$):
  • Enhancement & Oscillations: Significantly increase $D_{KL}$, making the topology more detectable than in the standard case.
  • Cutoff: Also increases $D_{KL}$ relative to the standard case, as the suppression of power in the covering space ($E_{18}$) makes the residual power in the topology stand out more.
• Topology-Independent Modifications (In both $E_i$ and $E_{18}$):
  • Cutoff: Decreases $D_{KL}$ significantly. Suppressing large-scale power removes the very modes that carry topological information (off-diagonal correlations), rendering the topology indistinguishable from the trivial case.
  • Enhancement: Increases $D_{KL}$, but the effect is less pronounced than when the modification is topology-specific.
  • Oscillations: Show complex behavior but generally track the standard case closely.

B. Machine Learning Performance

• Accuracy Trends: ML test accuracy closely mirrors the KL divergence trends.
  • When power is suppressed in both topologies (Topology-Independent Cutoff), classification accuracy drops below the baseline, confirming that large-scale modes are critical for detection.
  • When power is enhanced or modified only in the non-trivial topology, accuracy improves.
• Feature Importance (SHAP):
  • Classification relies heavily on low-$\ell$ coefficients (large angular scales), consistent with the theoretical expectation that topological signatures manifest on the largest scales.
  • The distribution of important features ($a_{\ell m}$) varies distinctly between different topologies (e.g., $E_1$ vs. $E_6$), allowing the ML to learn specific topological "fingerprints."
• Multi-class Results: The classifier successfully distinguishes between different topologies ($E_1, E_3, E_4, E_6$) and observer positions with high accuracy (>70% for 7 classes), even when power spectrum modifications are present.

C. Observer Position and Geometry

• Off-axis Observers: In inhomogeneous manifolds ($E_2$–$E_6$), off-axis observers generally yield higher KL divergences and classification accuracies than on-axis observers because they see more "matched circles" (clones) at equal distances, enhancing correlations.
• Critical Size: Detectability drops sharply when the fundamental domain size $L$ exceeds the diameter of the last scattering surface ($L_{LSS}$). However, PPS enhancements can extend the detectable range slightly beyond this limit.

5. Significance and Conclusions

• Robustness of Searches: The study concludes that uncertainties in the primordial power spectrum are a critical factor in cosmic topology searches. Ignoring potential IR features could lead to false negatives (missing a topology because power was suppressed) or false positives.
• Enhanced Detectability: If the non-trivial topology itself induces a modification to the PPS (e.g., a large-scale enhancement), the universe becomes easier to detect than previously thought under standard assumptions.
• Suppression Risk: Conversely, if large-scale power is universally suppressed (e.g., by a short inflation period), topological signatures may be erased, making detection impossible even for small fundamental domains.
• Future Outlook: The authors emphasize the need for future analyses to include realistic noise, sky masking, and polarization data (e.g., from the LiteBIRD satellite). They also suggest that Bayesian frameworks jointly fitting topology and PPS parameters will be necessary to break degeneracies in real data analysis.

In summary, this work demonstrates that the detectability of cosmic topology is not a fixed property of the universe's geometry but is dynamically coupled to the shape of the primordial power spectrum. Accurate detection requires accounting for these spectral uncertainties.