pith. sign in

arxiv: 2606.24886 · v1 · pith:DLDG6MICnew · submitted 2026-06-23 · 🌌 astro-ph.CO · gr-qc· hep-th

The Topology of the Universe

Craig J. Copi , Deyan P. Mihaylov , Anna Negro , Amirhossein Samandar , Glenn D. Starkman , Yashar Akrami , George Alestas , Stefano Anselmi
show 9 more authors
Javier Carr\'on Duque Fernando Cornet-Gomez Linn Htat Lu Andrew H. Jaffe Arthur Kosowsky Mikel Martin Barandiaran Thiago S. Pereira Catherine Petretti Andrius Tamosiunas
This is my paper

Pith reviewed 2026-06-25 22:56 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-th
keywords cosmic topologyCMBmatched circlesBayesian analysisstatistical isotropyuniverse geometryPlanckLiteBIRD
0
0 comments X

The pith

No definitive evidence for non-trivial cosmic topology has been found in CMB observations, but detectable signals could still exist even at large scales.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews three decades of searches for whether the universe has a non-trivial topology that would make it finite yet without edges. Such a topology would produce repeating patterns in the cosmic microwave background and in the distribution of galaxies. Analyses of data from WMAP and Planck using matched circle searches and Bayesian methods based on topology-specific covariance matrices have turned up no clear confirmation. Only limited ranges of topologies, scales, and observer locations have been ruled out so far. The review notes that signals could remain observable even if the topology scale is larger than the visible universe, and that upcoming polarization measurements and galaxy surveys offer new ways to test this possibility.

Core claim

Although searches using matched circle pairs and full Bayesian likelihood analysis based on topology-dependent covariance matrices have yielded no definitive evidence for non-trivial topology, current constraints exclude only some topologies, parameter ranges, and observer positions. Recent advances show that detectable signals may persist even when the topology scale exceeds the size of the visible Universe. Planned CMB experiments, including LiteBIRD and Taurus, and high-precision galaxy and line intensity-mapping surveys, could expand the detectable parameter space by exploiting polarisation data, and by exploring topology-induced correlations at all accessible redshifts.

What carries the argument

Matched circle pairs in the CMB sky together with Bayesian likelihood analysis that uses topology-dependent covariance matrices to identify repeating patterns from global connectivity.

If this is right

  • Current data exclude only some topologies, parameter ranges, and observer positions.
  • Detectable signals may persist even when the topology scale exceeds the size of the visible Universe.
  • Polarization data from planned CMB experiments can expand the detectable parameter space.
  • High-precision galaxy and line intensity-mapping surveys can explore topology-induced correlations at all accessible redshifts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A positive detection would establish that the universe is finite in at least one direction while remaining locally flat.
  • Persistent non-detection across future datasets would tighten limits on possible global structures without necessarily proving the universe is infinite.
  • The same search techniques could be adapted to test specific candidate topologies predicted by some models of the early universe.

Load-bearing premise

Non-trivial cosmic topology would leave imprints on the CMB and matter distribution that can be distinguished from ordinary statistical isotropy by existing or planned analysis techniques.

What would settle it

A consistent set of matched circle pairs appearing in multiple independent CMB maps at the expected angular separation for a specific topology model, or a null result from combined polarization and redshift surveys that rules out all remaining topologies with scales comparable to or smaller than the observable universe.

Figures

Figures reproduced from arXiv: 2606.24886 by Amirhossein Samandar, Andrew H. Jaffe, Andrius Tamosiunas, Anna Negro, Arthur Kosowsky, Catherine Petretti, Craig J. Copi, Deyan P. Mihaylov, Fernando Cornet-Gomez, George Alestas, Glenn D. Starkman, Javier Carr\'on Duque, Linn Htat Lu, Mikel Martin Barandiaran, Stefano Anselmi, Thiago S. Pereira, Yashar Akrami.

Figure 1
Figure 1. Figure 1: Equivalent representations of a two-dimensional torus. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A single cactus in a universe with non-trivial topology. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Matched-circle pairs in the CMB Shown here as they would appear in CMB maps in the simple 3-torus topology E1, with a cubic domain of dimension equal to 80% of the diameter of the last scattering surface (LSS). On the top left, the LSS is shown inside a cubic fundamental domain representing the periodic domain picture of cosmic topology as described in the text. In the top centre, the overlap between the (… view at source ↗
Figure 4
Figure 4. Figure 4: CMB temperature correlation matrices Absolute values of the rescaled CMB temperature (T T) correlation matrices for topologies in which the length scale is chosen such that the diameter of the circles passing through the observer is 5% larger than the diameter of the LSS: the covering space (also known as E18, top left panel), E1 topology (top right panel); E3 topology with the observer on the axis of rota… view at source ↗
read the original abstract

Is the Universe infinite in all directions? The only way to know is to look. A non-trivial cosmic topology would imprint subtle signatures on the cosmic microwave background (CMB) and on the three-dimensional distribution of matter, breaking statistical isotropy and, potentially, homogeneity at the largest scales. If the topology scale is small enough, these signatures would be observable. Over the past three decades, successive space missions, most notably WMAP and $\textit{Planck}$, have enabled sophisticated searches for these signatures, using methods ranging from looking for matched circle pairs to full Bayesian likelihood analysis based on topology-dependent covariance matrices. Although these searches have yielded no definitive evidence for non-trivial topology, current constraints exclude only some topologies, parameter ranges, and observer positions. Recent advances show that detectable signals may persist even when the topology scale exceeds the size of the visible Universe. Planned CMB experiments, including LiteBIRD and $\textit{Taurus}$, and high-precision galaxy and line intensity-mapping surveys, could expand the detectable parameter space by exploiting polarisation data, and by exploring topology-induced correlations at all accessible redshifts. Whether cosmic topology is observable remains uncertain, but current and future data offer an unprecedented opportunity to probe the global structure of the Universe.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript is a review summarizing three decades of observational searches for non-trivial cosmic topology in the CMB and large-scale structure. It covers methods including matched-circle pair searches and Bayesian analyses using topology-dependent covariance matrices from WMAP and Planck data, notes the absence of definitive detections, states that existing constraints exclude only limited topologies/parameter ranges/observer positions, and highlights recent advances indicating that detectable signals may persist even for topology scales exceeding the visible Universe. It discusses prospects for future experiments such as LiteBIRD, Taurus, and intensity-mapping surveys to expand the searchable space via polarization and redshift-dependent correlations.

Significance. As a review consolidating the literature on cosmic topology searches, the paper is significant for providing a clear overview of the field's status and future opportunities. It explicitly credits the cited body of work (e.g., matched-circle methods and covariance-matrix likelihoods) for the constraints and methods summarized, and correctly frames the central statements as resting on published results rather than new derivations.

minor comments (2)
  1. [Abstract] The abstract and introduction could more explicitly distinguish between the review's role as a synthesis versus any implied forward-looking claims; for instance, the statement that 'recent advances show that detectable signals may persist...' would benefit from a specific citation or section reference to the relevant recent papers.
  2. [Introduction] Notation for topology scales and observer positions is used consistently but could include a brief glossary or table in an early section to aid readers unfamiliar with the subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, accurate summary of its scope, and recommendation to accept. We appreciate the recognition that the review consolidates existing literature without introducing new derivations.

Circularity Check

0 steps flagged

Review paper with no new derivations or predictions

full rationale

This document is a review summarizing three decades of published searches for cosmic topology in the CMB and large-scale structure. It introduces no original equations, derivations, fitted parameters, or predictions. All statements, including the claim that detectable signals may persist beyond the visible Universe, are explicitly attributed to prior literature rather than derived within the paper. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain; the text is self-contained as a survey of external results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a review paper, the text introduces no new free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5824 in / 965 out tokens · 24895 ms · 2026-06-25T22:56:29.879132+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

121 extracted references · 1 canonical work pages

  1. [1]

    Three-body models have been frequently used (see e.g

    is expected to deliver substantial improvements over Planckfor multipoles ℓ≲ 200, especially in polarisation. These gains should make scalar E-mode polarisation auto- correlation and cross-correlation with temperature more useful to searches for cosmic topology. Assessing the extent of this improvement and its implications for dis- covering or constrainin...

    work page doi:10.13039/501100011033 2026

  2. [2]

    Schwarzschild, K.Ueber das zulassige Krummungsmaass des Raumes(Deutsche Nationalbibliothek, 1900)

    1900

  3. [3]

    Sommerville, D. M. Y.The elements of non-Euclidean geometry(G. Bell and sons, Ltd., 1914)

    1914

  4. [4]

    Third PaperMon

    de Sitter, W.On Einstein’s Theory of Gravitation and its Astronomical Consequences. Third PaperMon. Not. Roy. Astron. Soc.78, 3–28 (1917)

    1917

  5. [5]

    ¨Uber die M¨ oglichkeit einer Welt mit kon- stanter negativer Kr¨ ummung des RaumesZ

    Friedmann, A. ¨Uber die M¨ oglichkeit einer Welt mit kon- stanter negativer Kr¨ ummung des RaumesZ. Phys.21, 326–332 (1924)

    1924

  6. [6]

    Stoops (1958)

    Lemaˆ ıtre, G.La Structure et l’´Evolution de l’UniversIn 11th Solvay Physics Conference, Brussels: La Structure et l’ ´Evolution de l’Univers, R. Stoops (1958)

    1958

  7. [7]

    John Wiley & Sons (1962)

    Witten, L.Gravitation: An Introduction to Current Re- 12 search. John Wiley & Sons (1962)

    1962

  8. [8]

    B., Sokoloff, D

    Zeldovich, Y. B., Sokoloff, D. D. and Starobinskii, A.150 Years of the Lobachevsky geometry (in Russian)Institute of Scientific and Technic Information, Moscow (1977)

    1977

  9. [9]

    R.Theory and Observational Limits in Cos- mologySpecola Vaticana (1987)

    Stoeger, W. R.Theory and Observational Limits in Cos- mologySpecola Vaticana (1987)

    1987

  10. [10]

    Fang, L.-Z.Global topology of the universeInThe First William Fairbank Meeting: Relativistic Gravitational Ex- periments in Space(1990)

    1990

  11. [11]

    V.Compactification of Friedmann’s Hyper- bolic ModelPhys

    Fagundes, H. V.Compactification of Friedmann’s Hyper- bolic ModelPhys. Rev. Lett.51, 517–518 (1983)

    1983

  12. [12]

    Weeks, J. R. and Thurston, W. P.The Mathematics of Three-Dimensional ManifoldsSci. Am.251, 108–120 (1984)

    1984

  13. [13]

    and Luminet, J.-P.Cosmic topology Phys

    Lachieze-Rey, M. and Luminet, J.-P.Cosmic topology Phys. Rep.254, 135-214 (1995)

    1995

  14. [14]

    Fagundes, H. V. and Wichoski, U. F.A Search for QSOs to Fit a Cosmological Model with Flat, Closed Spatial SectionsAstrophys. J.322, L5 (1987)

    1987

  15. [15]

    and Luminet, J.-P.Cos- mic crystallographyAstron

    Lehoucq, R., Lachieze-Rey, M. and Luminet, J.-P.Cos- mic crystallographyAstron. Astrophys.313, 339–346 (1996)

    1996

  16. [16]

    and Luminet, J.-P.A new crys- tallographic method for detecting space topologyAstron

    Uzan, J.-P., Lehoucq, R. and Luminet, J.-P.A new crys- tallographic method for detecting space topologyAstron. Astrophys.351, 766 (1999)

    1999

  17. [17]

    Astrophys.363, 1 (2000)

    Lehoucq, R., Uzan, J.-P., Luminet, J.-P.Limits of crys- tallographic methods for detecting space topologyAstron. Astrophys.363, 1 (2000)

    2000

  18. [18]

    I., Teixeira, A

    Gomero, G. I., Teixeira, A. F. F., Reboucas, M. J. and Bernui, A.Spikes in cosmic crystallographyInt. J. Mod. Phys. D11, 869–892 (2002)

    2002

  19. [19]

    A.New restrictions on spatial topology of the universe from microwave background temperature fluctuationsJ

    Starobinsky, A. A.New restrictions on spatial topology of the universe from microwave background temperature fluctuationsJ. Exp. Theor. Phys. Lett.57, 622–625 (1993)

    1993

  20. [20]

    and Smoot, G

    de Oliveira-Costa, A. and Smoot, G. F.Constraints on the topology of the universe from the 2-year COBE data Astrophys. J.448, 477 (1995)

    1995

  21. [21]

    de Oliveira-Costa, A., Smoot, G. F. and Starobin- sky, A. A.Can the lack of symmetry in the COBE / DMR maps constrain the topology of the universe?Astrophys. J.468, 457 (1996)

    1996

  22. [22]

    R., Pogosian, D

    Bond, J. R., Pogosian, D. and Souradeep, T.Constraints on compact hyperbolic spaces from COBEIn18th Texas Symposium on Relativistic Astrophysics, 297–299 (1996)

    1996

  23. [23]

    J., Spergel, D

    Cornish, N. J., Spergel, D. and Starkman, G.Can COBE see the shape of the universe?Phys. Rev. D57, 5982– 5996 (1998)

    1998

  24. [24]

    Ellis, G. F. R.Topology and cosmologyGen. Relativ. Gravit.2, 7-21 (1971)

    1971

  25. [25]

    m´ etaphys

    Poincar´ e, H.La mesure du tempsRev. m´ etaphys. morale 6, 1–13 (1898)

  26. [26]

    Phys.322, 891-921 (1905)

    Einstein, A.Zur Elektrodynamik bewegter K¨ orperAnn. Phys.322, 891-921 (1905)

    1905

  27. [27]

    and Peter, P

    Uzan, J.-P., Luminet, J.-P., Lehoucq, R. and Peter, P.. Twin paradox and space topologyEur. J. Phys.23, 277 (2002)

    2002

  28. [28]

    Barrow, J. D. and Levin, J. J.The Twin paradox in compact spacesPhys. Rev. A63, 044104 (2001)

    2001

  29. [29]

    V.Closed spaces in cosmologyGen

    Fagundes, H. V.Closed spaces in cosmologyGen. Rel. Grav.24, 199 (1992)

    1992

  30. [30]

    Barrow, J. D. and Levin, J. J.The Copernican principle in compact space-timesMon. Not. Roy. Astron. Soc.346, 615 (2003)

    2003

  31. [31]

    and Weeks, J

    Riazuelo, A., Uzan, J.-P., Lehoucq, R. and Weeks, J. R. Simulating cosmic microwave background maps in multi- connected spacesPhys. Rev. D69, 103514 (2004)

    2004

  32. [32]

    and Weeks, J

    Gausmann, E., Lehoucq, R., Luminet, J.-P., Uzan, J.-P. and Weeks, J. R.Topological lensing in spherical spaces Class. Quant. Grav.18, 5155–5186 (2001)

    2001

  33. [33]

    J.Topology and the cosmic microwave back- groundPhys

    Levin, J. J.Topology and the cosmic microwave back- groundPhys. Rept.365, 251–333 (2002)

    2002

  34. [34]

    P.The geometry and topology of three- manifolds: With a preface by Steven P

    Thurston, W. P.The geometry and topology of three- manifolds: With a preface by Steven P. KerckhoffAmer- ican Mathematical Society (2022)

    2022

  35. [35]

    T.Computation of eigenmodes on a com- pact hyperbolic spaceClass

    Inoue, K. T.Computation of eigenmodes on a com- pact hyperbolic spaceClass. Quant. Grav.16, 3071–3094 (1999)

    1999

  36. [36]

    Cornish, N. J. and Spergel, D. N.On the eigenmodes of compact hyperbolic 3-manifoldsarXiv Mathematics e-prints, math/9906017 (1999)

    Pith/arXiv arXiv 1999

  37. [37]

    and Steiner, F.The Cosmic microwave back- ground for a nearly flat compact hyperbolic universeMon

    Aurich, R. and Steiner, F.The Cosmic microwave back- ground for a nearly flat compact hyperbolic universeMon. Not. Roy. Astron. Soc.323, 1016–1024 (2001)

    2001

  38. [38]

    and Weeks, J

    Uzan, J.-P., Riazuelo, A., Lehoucq, R. and Weeks, J. R. Cosmic microwave background constraints on lens spaces Phys. Rev. D69, 043003 (2004)

    2004

  39. [39]

    and Then, H.Hy- perbolic universes with a horned topology and the CMB anisotropyClass

    Aurich, R., Lustig, S., Steiner, F. and Then, H.Hy- perbolic universes with a horned topology and the CMB anisotropyClass. Quant. Grav.21, 4901–4926 (2004)

    2004

  40. [40]

    A new analysis of Poincar´ e dodecahe- dral space modelAstron

    Caillerie, S.et al. A new analysis of Poincar´ e dodecahe- dral space modelAstron. Astrophys.476, 691 (2007)

    2007

  41. [41]

    and Lustig, S.Cosmic Topology of Polyhedral Double-Action ManifoldsClass

    Aurich, R. and Lustig, S.Cosmic Topology of Polyhedral Double-Action ManifoldsClass. Quant. Grav.29, 235028 (2012)

    2012

  42. [42]

    and Lustig, S.Cosmic Topology of Prism Double-Action ManifoldsClass

    Aurich, R. and Lustig, S.Cosmic Topology of Prism Double-Action ManifoldsClass. Quant. Grav.29, 215005 (2012)

    2012

  43. [43]

    R., Uzan, J.-P., Lehoucq, R

    Riazuelo, A., Weeks, J. R., Uzan, J.-P., Lehoucq, R. and Luminet, J.-P.Cosmic microwave background anisotropies in multi-connected flat spacesPhys. Rev. D 69, 103518 (2004)

    2004

  44. [44]

    and Weeks, J

    Lehoucq, R., Uzan, J.-P. and Weeks, J. R.Eigenmodes of lens and prism spacesKodai Math. J.26, 119 – 136 (2003)

    2003

  45. [45]

    R., Uzan, J.-P., Gausmann, E

    Lehoucq, R., Weeks, J. R., Uzan, J.-P., Gausmann, E. and Luminet, J.-P.Eigenmodes of three-dimensional spherical spaces and their application to cosmologyClass. Quant. Grav.19, 4683–4708 (2002)

    2002

  46. [46]

    Promise of Future Searches for Cosmic TopologyPhys

    Akrami, Y.et al. Promise of Future Searches for Cosmic TopologyPhys. Rev. Lett.132, 171501 (2024)

    2024

  47. [47]

    Eskilt, J. R.et al. Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Eu- clidean manifoldsJ. Cosmol. Astropart. Phys.03, 036 (2024)

    2024

  48. [48]

    Copi, C. J.et al. Cosmic topology. Part IIb. Eigen- modes, correlation matrices, and detectability of non- orientable Euclidean manifoldsarXiv Astrophysics e- prints, 2510.05030 (2025)

    arXiv 2025

  49. [49]

    Cosmic topology

    Samandar, A.et al. Cosmic topology. Part IIIb. Eigen- modes and correlation matrices of spin-2 perturbations in orientable Euclidean manifoldsJ. Cosmol. Astropart. Phys.08, 015 (2025)

    2025

  50. [50]

    Cosmic topology

    Samandar, A.et al. Cosmic topology. Part IIIa. Mi- crowave background parity violation without parity- violating microphysicsJ. Cosmol. Astropart. Phys.11, 020 (2024)

    2024

  51. [51]

    Higuchi, A.Symmetric Tensor Spherical Harmonics on 13 the N Sphere and Their Application to the De Sitter Group SO(N, 1)J. Math. Phys.28, 1553 (1987)

    1987

  52. [52]

    R., Pogosian, D

    Bond, J. R., Pogosian, D. and Souradeep, T.Comput- ing CMB anisotropy in compact hyperbolic spacesClass. Quant. Grav.15, 2671–2687 (1998)

    1998

  53. [53]

    R., Pogosian, D

    Bond, J. R., Pogosian, D. and Souradeep, T.CMB anisotropy in compact hyperbolic universes. 1. Computing correlation functionsPhys. Rev. D62, 043005 (2000)

    2000

  54. [54]

    R., Pogosian, D

    Bond, J. R., Pogosian, D. and Souradeep, T.CMB anisotropy in compact hyperbolic universes. 2. COBE maps and limitsPhys. Rev. D62, 043006 (2000)

    2000

  55. [55]

    J.524, 497–503 (1999)

    Aurich, R.The Fluctuations of the cosmic microwave background for a compact hyperbolic universeAstrophys. J.524, 497–503 (1999)

    1999

  56. [56]

    J., Spergel, D

    Cornish, N. J., Spergel, D. N. and Starkman, G. D. Circles in the Sky: Finding Topology with the Microwave Background Radiation(1996). arXiv General Relativity and Quantum Cosmology e-prints, gr-qc/9602039 (1996)

    Pith/arXiv arXiv 1996

  57. [57]

    J., Spergel, D

    Cornish, N. J., Spergel, D. N. and Starkman, G. D. Circles in the sky: Finding topology with the microwave background radiationClass. Quant. Grav.15, 2657–2670 (1998)

    1998

  58. [58]

    J., Spergel, D

    Cornish, N. J., Spergel, D. N. and Starkman, G. D. Measuring the topology of the universeProc. Nat. Acad. Sci.95, 82 (1998)

    1998

  59. [59]

    R.Reconstructing the global topology of the universe from the cosmic microwave backgroundClass

    Weeks, J. R.Reconstructing the global topology of the universe from the cosmic microwave backgroundClass. Quant. Grav.15, 2599–2604 (1998)

    1998

  60. [60]

    J., Spergel, D

    Cornish, N. J., Spergel, D. N., Starkman, G. D. and Komatsu, E.Constraining the topology of the universe Phys. Rev. Lett.92, 201302 (2004)

    2004

  61. [61]

    F., Lew, B., Cechowska, M., Marecki, A

    Roukema, B. F., Lew, B., Cechowska, M., Marecki, A. and Bajtlik, S.A Hint of Poincare dodecahedral topology in the WMAP First Year Sky mapAstron. Astrophys. 423, 821 (2004)

    2004

  62. [62]

    J., Spergel, D

    Shapiro Key, J., Cornish, N. J., Spergel, D. N. and Starkman, G. D.Extending the WMAP Bound on the Size of the UniversePhys. Rev. D75, 084034 (2007)

    2007

  63. [63]

    and Banday, A

    Bielewicz, P. and Banday, A. J.Constraints on the topol- ogy of the Universe derived from the 7-year WMAP data Mon. Not. Roy. Astron. Soc.412, 2104 (2011)

    2011

  64. [64]

    Bielewicz, P., Banday, A. J. and Gorski, K. M.Con- straints on the topology of the Universe derived from the 7-year WMAP CMB data and prospects of constraining the topology using CMB polarisation mapsProceedings of the XLVIIth Rencontres de MoriondAuge, 91 (2012)

    2012

  65. [65]

    M., Starkman, G

    Vaudrevange, P. M., Starkman, G. D., Cornish, N. J. and Spergel, D. N.Constraints on the Topology of the Universe: Extension to General GeometriesPhys. Rev. D86, 083526 (2012)

    2012

  66. [66]

    J., Scannapieco, E

    Levin, J. J., Scannapieco, E. and Silk, J.The Topology of the universe: The Biggest manifold of them allClass. Quant. Grav.15, 2689–2698 (1998)

    1998

  67. [67]

    J., Barrow, J

    Levin, J. J., Barrow, J. D., Bunn, E. F. and Silk, J. Flat spots: Topological signatures of an open universe in COBE sky mapsPhys. Rev. Lett.79, 974–977 (1997)

    1997

  68. [68]

    J., Scannapieco, E., de Gasperis, G., Silk, J

    Levin, J. J., Scannapieco, E., de Gasperis, G., Silk, J. and Barrow, J. D.How the universe got its spotsPhys. Rev. D58, 123006 (1998)

    1998

  69. [69]

    Scannapieco, E., Levin, J. J. and Silk, J..Temperature correlations in a finite UniverseMon. Not. Roy. Astron. Soc.303, 797 (1999)

    1999

  70. [70]

    Levin, J. J. and Barrow, J. D.Fractals and scars on a compact octagonClass. Quant. Grav.17, L61–L68 (2000)

    2000

  71. [71]

    and Starkman, G

    Olson, D. and Starkman, G. D.The Angular scale of topologically induced flat spots in the cosmic microwave background radiationClass. Quant. Grav.17, 3093–3100 (2000)

    2000

  72. [72]

    I., Reboucas, M

    Gomero, G. I., Reboucas, M. J. and Tavakol, R. K. Detectability of cosmic topology in almost flat universes Class. Quant. Grav.18, 4461–4476 (2001)

    2001

  73. [73]

    and Steiner, F.Betti Functionals as Probes for Cosmic TopologyUniverse10, 190 (2024)

    Aurich, R. and Steiner, F.Betti Functionals as Probes for Cosmic TopologyUniverse10, 190 (2024)

    2024

  74. [74]

    J., Copi, C

    Schwarz, D. J., Copi, C. J., Huterer, D. and Stark- man, G. D.CMB Anomalies after PlanckClass. Quant. Grav.33, 184001 (2016)

    2016

  75. [75]

    Ade, P. A. R.et al. Planck 2013 results. XXIII. Isotropy and statistics of the CMBAstron. Astrophys.571, A23 (2014)

    2013

  76. [76]

    Ade, P. A. R.et al. Planck 2015 results. XVI. Isotropy and statistics of the CMBAstron. Astrophys.594, A16 (2016)

    2015

  77. [77]

    Planck 2018 results

    Akrami, Y.et al. Planck 2018 results. VII. Isotropy and Statistics of the CMBAstron. Astrophys.641, A7 (2020)

    2018

  78. [78]

    Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomaliesJ

    Abdalla, E.et al. Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomaliesJ. High Energy Astrophys.34, 49–211 (2022)

    2022

  79. [79]

    and Peiris, H

    Pontzen, A. and Peiris, H. V.The cut-sky cosmic mi- crowave background is not anomalousPhys. Rev. D81, 103008 (2010)

    2010

  80. [80]

    Bennett, C. L.et al. Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Are There Cosmic Microwave Background Anomalies?Astrophys. J. Suppl.192, 17 (2011)

    2011

Showing first 80 references.