Pattern of perturbations from a coherent quantum inflationary horizon

Craig Hogan

REVIEW 1 major objections 1 minor 98 references

Pattern of perturbations from a coherent quantum inflationary horizon

T0 review · 1 major / 1 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read If the inflationary horizon is a coherent quantum object, the CMB temperature correlation must vanish exactly at 90 degrees of separation—a testable signature that standard quantum inflation can produce only by chance.

desk verdict Hogan's exact C_T(90°)=0 is a genuinely new and testable ansatz, but the paper assumes the very symmetry it claims to derive—still worth refereeing on empirical sharpness. read the letter →

arxiv 1908.07033 v5 pith:WQ6RZNBU submitted 2019-08-19 astro-ph.CO gr-qc

Craig Hogan This is my paper · ORCID

classification astro-ph.COgr-qc

keywords holographicinflationprimordialcurvatureperturbationsCMBangularcorrelationanomaliesinflationaryhorizonquantumcoherenceonnullsurfacesgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that primordial density perturbations do not have to be vacuum fluctuations of an inflaton field: if quantum geometry is coherent on null surfaces, Planck-scale holographic fluctuations of the inflationary horizon itself imprint the initial curvature pattern. Because the horizon is a nearly spherical coherent object that collapses onto causal-diamond boundaries around each observer, its fluctuations obey exact directional constraints that standard quantum inflation does not have. The paper's sharpest, most precisely defined prediction is that the two-point angular correlation function of CMB temperature vanishes exactly at 90 degrees of separation, $C_T(90^\circ)=0$. That number is measurable with existing full-sky maps, survives Doppler effects without reconstruction, and would be a direct relic of horizon-scale quantum coherence if confirmed. The same causal symmetries also offer a unified explanation of several long-noticed CMB anomalies, such as the low quadrupole, the aligned quadrupole and octopole axes, and odd/even parity asymmetry.

What carries the argument

The load-bearing object is the inflationary horizon $\mathcal{H}$, the past light cone of an observer at the end of inflation, modeled as a coherent, nearly spherical quantum surface like a single atom. The argument is carried by the causal-symmetry relation of Eq. (7): the sky average of $\Delta(\vec{\Omega})$ times the azimuthal mean of $\Delta$ on the great circle normal to $\vec{\Omega}$ is exactly zero, which is identical to $C_\Delta(90^\circ)=0$. The mechanism is the projection of quantum collapse onto spherical causal-diamond boundaries rather than infinite plane waves, so that incoming phase information from a polar axis cannot reach the equatorial belt until the end of inflation. A second supporting symmetry is constant variance on great circles, Eq. (11), which links the known quadrupole and octopole alignment to the holographic absence of one independent rotational degree of freedom.

What would settle it

Take a full-sky, foreground-cleaned CMB map, subtract the Doppler and kinematic contributions, and compute the even-$\ell$ part of the angular correlation function at 90 degrees via $\sum_{\ell\ \mathrm{even}}(2\ell+1)C_\ell P_\ell(0)$; if the result is not consistent with zero at the map resolution, the exact symmetry $C_T(90^\circ)=0$ is falsified. A reconstructed primordial-potential map showing $C_\Delta(90^\circ)\neq 0$ would falsify the curvature-level symmetry before Doppler corrections.

Watch

Extended reading notes

Core claim

The central claim is that holographic inflation—where the inflationary horizon $\mathcal{H}$ is the observer's past light cone at the end of inflation, treated as a coherent nonlocal quantum state rather than a set of plane-wave modes—produces primordial curvature perturbations with exact angular symmetries. The main derivation starts from the causal structure of information on the horizon: along any axis, incoming phase information from polar directions reaches the equatorial plane only at the end of inflation, so the product of a polar perturbation and the azimuthal mean on the great circle normal to it averages to zero over the sky. That equality, Eq. (7), is equivalent to $C_\Delta(90^\circ)=0$, and because the Doppler contribution vanishes at 90 degrees the paper derives the temperature-level prediction $C_T(90^\circ)=0$, Eq. (18), with no need to reconstruct the primordial potential. The same reasoning yields candidate equilateral symmetries at 30 degrees, a constant variance on all great circles, antipodal anticorrelation with a parity-breaking parameter $E$, and a vanishing intrinsic dipole. The paper further argues that these symmetries are properties of each realization, not ensemble averages, so they avoid the usual cosmic-variance penalty and can be tested against current CMB data; it reports that published maps are consistent with the 90-degree zero and with the known large-angle anomalies.

Load-bearing premise

The load-bearing premise is that incoming phase information from polar directions along any axis reaches the equatorial plane of the horizon only at the end of inflation, so the sky average of a polar perturbation times the azimuthal mean on its perpendicular great circle is exactly zero; the paper draws this causal constraint from a diagram rather than deriving it from the toy model.

Editorial extensions

If this is right

If the central claim is right, a high-resolution all-sky CMB map should show $C_T(90^\circ)=0$ at much better precision than standard quantum inflation can produce by chance; the comparison can be made without cosmic-variance penalty because the symmetry is exact for every realization.
The model would convert several recognized CMB anomalies—the unusually small quadrupole, the aligned quadrupole and octopole axes, and the odd/even parity imbalance—into a single physical consequence of coherent horizon state reduction.
Because Doppler contributions vanish exactly at 90 degrees, no reconstruction of the primordial potential is needed for the strongest test; for other predicted symmetries, such as the 30-degree zero, reconstruction from temperature and polarization maps would be required.
The same angular correlations should be present in the three-dimensional galaxy distribution, since the holographic pattern is imprinted on the horizon and later conserved in large-scale structure.
A confirmed 90-degree zero would support the broader hypothesis that scalar curvature perturbations originate from coherent holographic quantum geometry rather than from quantum field vacuum modes in a classical background.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the 90-degree zero is confirmed, the most natural next target is the predicted 30-degree zero after careful dipole subtraction, which tests the model's claim that the intrinsic dipole in the cosmic rest frame vanishes.
The same horizon-coherence reasoning applied in flat space implies that interferometric light paths with nontrivial three-dimensional or rotational geometry should show Planck-scale cross-correlated position noise; current null results cover only a coplanar radial configuration, leaving the other geometries open.
A sharper harmonic-space test than the raw correlation value would check the implied conspiracy of even-$\ell$ coefficients, $\sum_{\ell\ \mathrm{even}} (2\ell+1)C_\ell P_\ell(0)=0$ up to the map resolution; rejecting that combination would falsify the symmetry even if the low-$\ell$ correlation looks zero by eye.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague—this one is worth your attention, but not for the reasons the abstract suggests. The genuinely new thing is the exact, parameter-free prediction C_T(90°)=0 and the associated great-circle variance symmetry. That is sharper than Hogan's earlier directional-correlation proposals and is the kind of statement that could, in principle, be confirmed or killed by existing Planck/WMAP data. Credit where due: the paper asks a crisp question and gives a crisp answer.

The soft spot is at the load-bearing step. Eq. (7) asserts that the product of a polar value and the azimuthal mean on the normal great circle averages to zero over the sky. Eq. (8) says that is C_Δ(90°)=0. Those are the same statement, so the headline "derivation" of the vanishing correlation function restates the causal assumption from §II.E.1 and Fig. 5. The appendix's spin-algebra toy model computes variances and uncertainties, not the 90° two-point function, so it doesn't supply the missing derivation. I read the paper honestly as proposing a candidate symmetry, not proving one. The saturation step for constant great-circle variance (Eqs. 10–11) is also asserted rather than derived, and the parity parameter E is read from the data, so that piece is more postdiction than prediction.

Empirically the paper is admirably direct about the tension: Planck's four foreground-removal methods give C_T(90°) slightly but significantly nonzero; the zero comes from a later reanalysis by the same group with different masking. That is not enough to call the prediction confirmed, but it is enough to say the prediction is alive and testable. If future data nail C_T(90°)=0, we will have validated an interesting phenomenological ansatz—not the holographic derivation, but something worth explaining.

Who should read this? Cosmologists working on large-angle CMB anomalies and anyone interested in holographic cosmology proposals. Readers looking for a derivation from quantum gravity will be disappointed; readers looking for a sharp, falsifiable signature of a radical idea will find value. I would not cite it as a derivation, but I would reference it as the source of the exact-zero ansatz if that becomes relevant.

Recommendation: send it to referees, not desk reject. The central claim is soft, but the empirical target is sharp, the literature is engaged, and the paper is honest about the mixed data. A good referee can force the central assumption into the open and ask for the derivation to be rebuilt from the model, or for the paper to be reframed as a phenomenological conjecture.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a 'holographic inflation' scenario in which primordial curvature perturbations are generated by coherent quantum fluctuations on the inflationary horizon rather than by quantized inflaton field modes. It argues that such a coherent horizon imprints directional symmetries on the CMB, the most striking being an exact vanishing of the temperature correlation function at 90 degrees angular separation, C_T(90°) = 0, which the author claims can be tested with existing data and which standard inflation cannot produce except by cosmic variance. The paper also suggests that the model explains several known CMB anomalies, including low quadrupole, parity asymmetry, and large-angle correlations, and it discusses interferometric tests of Planck-scale geometry.

Significance. If the central prediction were actually derived from a well-founded theory, the paper would offer a striking and falsifiable signature of quantum gravity on cosmological scales, potentially unifying several CMB anomalies. The manuscript is clearly written, openly discusses limitations, and proposes concrete observational tests, including a null test at 90 degrees. However, the significance is critically undermined by the fact that the headline prediction is not derived from holography but is assumed at the outset, as I detail below. The empirical support is also mixed, and the main supportive reanalysis comes from the same research group. The paper's value is therefore primarily as a speculative proposal and a stimulus for further data analysis, not as a validated derivation.

major comments (1)

[§II.E.1, Eqs. (6)–(8)] The theoretical foundation is acknowledged to be incomplete, yet the derivations depend on it. Appendix A begins by noting that 'there is as yet no consensus on the magnitude or physical effects of coherent, large-angle fluctuations of horizons,' and Appendix C states that the spin model 'does not address quantum dynamics at the Planck scale.' These are appropriate caveats, but they apply directly to the causal-independence assumption in Eq. (7): the assumption is neither derived from a known theory nor from the Appendix's toy model. The paper would need a concrete mechanism—for example, a calculation showing that the coherent horizon state implies vanishing two-point correlations at 90°—for the central claim to be more than an ansatz.

minor comments (1)

[General] The reference to Baumann's TASI lectures is listed with an incomplete page range; please verify the publication details.

Circularity Check

2 steps flagged · score 8.0 of 10

The headline C_T(90°)=0 prediction is not derived: Eq. (7), asserted from a causal diagram, is by definition C_Δ(90°)=0, so the result is an input assumption restated as an output; empirical support additionally rests on a same-author reanalysis.

self definitional [Section II.E.1, Eqs. (6)-(8); Section III.A, Eq. (18)]
"C∆(Θ)≡⟨ ∆(⃗Ω)⟨∆⟩Θ,⃗Ω⟩⃗Ω, (6) ... Uncorrelated incoming polar information produces the following exact global equatorial symmetry of relic curvature on each sphere around an observer: ⟨∆(⃗Ω)⟨∆⟩⊥⃗Ω⟩⃗Ω = 0. (7) ... Eq. (7) is equivalent to C∆(90◦) = 0. (8)"

By the paper's own definition (6), C_Δ(Θ) is the sky average of Δ(Ω) times the azimuthal mean of Δ on the circle at polar angle Θ about Ω. At Θ = 90°, that azimuthal mean is exactly ⟨Δ⟩_{⊥Ω}, the great-circle mean normal to Ω. Therefore Eq. (7), which asserts ⟨Δ(Ω)⟨Δ⟩_{⊥Ω}⟩_Ω = 0, is literally the statement C_Δ(90°) = 0. The paper then uses this equivalence to obtain the headline prediction, CT(90°) = C_Δ(90°) = 0 (Eq. 18). The predicted exact symmetry is thus not derived from holography plus calculation; it is the causal assumption in Eq. (7) restated in new notation. The Fig. 5 causal narrative is not formalized into a model that implies the vanishing, and the Appendix toy model computes variances and uncertainties (Eqs.
self citation load bearing [Section III.C.1, paragraph beginning 'Published WMAP and Planck plots differ in detail', reference [39]]
"In the case of Planck [8, 13], four different foreground removal techniques agree with each other, and CT(90◦) differs slightly but significantly from zero in all of them. Recently, these maps were re-analyzed ([39]) with uniform masking. The result of this analysis was that the least biased measurements of C(Θ) from the two satellites and most of the different foreground removal techniques are in good agreement with each other near Θ = 90◦, and fall into a remarkably narrow range around C(90◦) = 0."

The empirical support for the central prediction is carried by reference [39], whose author list includes the present author (Hagimoto, Hogan, Lewin, and Meyer). The paper itself concedes that independent Planck analyses find CT(90°) significantly nonzero in all four foreground-removal techniques; only this same-group reanalysis yields the zero. Because the reanalysis is not machine-checked or code-reproduced in the paper, and because it is invoked to overturn the published Planck result, the cited support is a load-bearing self-citation rather than independent confirmation. This does not make the arithmetic circular, but it removes the independent evidentiary grounding that would ordinarily justify treating the predicted zero as an external check.

full rationale

The central claim of the paper is that holographic inflation predicts an exact vanishing of the angular correlation function at 90 degrees, CT(90°) = C_Δ(90°) = 0. Walking the derivation chain shows that this result is not obtained from the semiclassical model or the Appendix's spin-algebra calculations; it is inserted as Eq. (7), a 'causal constraint' stated after Fig. 5, and then identified with C_Δ(90°) = 0 in Eq. (8) by the definition in Eq. (6). The subsequent Doppler-cancellation argument that CT(90°) = C_Δ(90°) is physically conditional and not the source of the zero; the zero is entirely the content of Eq. (7). Thus the derivation reduces to itself: if the polar/great-circle average vanishes, the correlation at 90 degrees vanishes. The paper is honest that this is a candidate symmetry, but the abstract and conclusion present it as a prediction, which is circular in the specific sense that Eq. (7) and Eq. (8) are the same statement by construction. In addition, the empirical case leans on a same-author reanalysis of Planck data, while the paper acknowledges that the standard Planck analyses give a significantly nonzero value; this is a load-bearing self-citation rather than independent external validation. The score is 8 rather than 10 because Eq. (7) is framed as a physical causal input rather than a tautology, and the paper does offer a separate, if conjectural, argument that Doppler contributions cancel at 90 degrees, so the equality CT(90°) = C_Δ(90°) is not entirely empty. But the predicted exact zero itself is an assumption restated as a result.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the holographic-inflation hypothesis itself, the amplitude relation from the author's earlier work, and the specific causal constraint that polar and equatorial information are uncorrelated. The toy model is explicitly disclaimed as incomplete. E is a parameter set by the data it is meant to explain.

free parameters (2)

Symmetry-breaking parameter E = Eathered from data: ET ~ 0.3 at low ell, from Planck RTT as cited in Section II.E.3
Introduced in Eq. (16) to quantify the ratio of even to odd perturbation power; the paper notes a detection of ET around 0.3 from published CMB anisotropy data, so the parameter is informed by the data it is meant to explain.
Hubble expansion rate H during inflation = Set by matching Eq. (5), <Delta^2> = H tP, to the observed Delta^2 ~ 1e-9
The amplitude relation from the author's prior model fixes H from the observed perturbation amplitude; this is a normalization rather than a parameter-free prediction.

assumptions (5)

ad hoc to paper Null surfaces (light cones) are coherent quantum objects whose state reduction creates classical curvature perturbations on the horizon
Core hypothesis of holographic inflation, stated in Section II.B and Appendix A; not derived, but assumed as the foundation of the model.
domain assumption The dominant perturbation amplitude is <Delta^2> = H tP (Eq. 5)
Taken from the author's earlier work [18]; the paper adopts it without derivation here.
ad hoc to paper Incoming phase information from polar directions cannot influence the equatorial plane, giving Eq. (7)
The central causal constraint that yields C(90 degrees) = 0; asserted from Fig. 5 and causal reasoning, not derived from a fundamental theory.
ad hoc to paper The spin-algebra toy model (Eqs. 37-48) applies to inflationary curvature perturbations and yields the variance bound used for Eq. (10)
Flat-space toy model with tau0 = tP; the paper acknowledges it is not a fully consistent noncommutative quantum geometry.
domain assumption For a single realization, the average over directions of great-circle variance equals the sky variance, so the bound in Eq. (10) must saturate in all directions
Unstated; needed to derive Eq. (11); may not hold for a fixed map without ensemble averaging.

invented entities (1)

Coherent quantum inflationary horizon (holographic 'spooky' correlations on H) independent evidence
purpose: Generates primordial scalar curvature perturbations with amplitude H tP and imprints the predicted large-angle symmetries
Falsifiable via CT(90 degrees) = 0, great-circle-variance symmetry, and interferometer predictions (Eq. 21); no direct detection yet.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Pattern of perturbations from a coherent quantum inflationary horizon." pith.science (2026). https://pith.science/paper/WQ6RZNBU

@misc{pith2026190807033,
  author       = {Pith},
  title        = {Pith review of: Pattern of perturbations from a coherent quantum inflationary horizon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WQ6RZNBU}},
  note         = {Machine review of arXiv:1908.07033}
}

read the original abstract

It is proposed that if quantum states of space-time are coherent on null surfaces, holographic Planck-scale fluctuations of inflationary horizons dominate the formation of primordial scalar curvature perturbations. It is shown that the reduction of quantum states on nearly-spherical emergent horizon surfaces around each observer creates a distinctive pattern whose correlations in the angular domain differ from the standard quantum theory of inflation. Causal constraints are used in a semiclassical model to formulate candidate directional symmetries. It is suggested that this hypothesis could provide a physical explanation for several well known anomalies measured in CMB anisotropy. New exact symmetries are predicted, such as a vanishing temperature correlation function at 90 degrees angular separation, that can be tested with current data.

Figures

Figures reproduced from arXiv: 1908.07033 by the authors.

**Figure 3.** FIG. 3. Two dimensions of a spatial hypersurface at a sin [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** FIG. 4. Spherical footprints of horizons for three world [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. How azimuthal averages of emergent perturbations [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6. Classical gravitational shocks in nearly-flat space [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

98 extracted references · 57 canonical work pages

[1]

conspiracy

Azimuthal symmetries One consequence of emergent causality is that incom- ing phase information that determines polar values of po- tential along any given axis on the horizon only reaches the equatorial plane at the end of inﬂation, so it cannot aﬀect the potential for points in that plane (see Fig. 5). As shown in the examples given in the Appendix, the...
[2]

A separate symmetry may be associated with nonlocal cor- relations of variance normal to each axis

Constant variance on great circles The global equatorial symmetry still allows mean val- ues of ∆ on great circles to vary according to random incoming polar information associated with each axis. A separate symmetry may be associated with nonlocal cor- relations of variance normal to each axis. It is possible that the variance on great circles is a const...
[3]

cosmic variance

Antipodal anticorrelation and parity violation The inﬂationary horizon could also display antipodal anticorrelation, a tendency of opposite points in the sky to have opposite signs. It occurs in quantum models of eternal black hole horizons[19–21], where antipodes on the horizon are actually identiﬁed, and time-reversed conjugate particle states are entan...
[4]

Seven-year wilkinson microwave anisotropy probe (wmap) observations: Are there cos- mic microwave background anomalies?

C. L. Bennett, R. S. Hill, G. Hinshaw, D. Lar- son, K. M. Smith, J. Dunkley, B. Gold, M. Halpern, N. Jarosik, A. Kogut, E. Komatsu, M. Limon, S. S. Meyer, M. R. Nolta, N. Odegard, L. Page, D. N. Spergel, G. ˆA S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright, “Seven-year wilkinson microwave anisotropy probe (wmap) observations: Are there cos- mic mi...

2011
[5]

secondary

Axes deﬁned by the quadrupole and octopole are closely aligned. The WMAP all-sky maps[2–4] revealed a remarkably close agreement in direction for quadrupole (𝓁 = 2) and octopole ( 𝓁 = 3) harmonics. The aligned direction is deﬁned by the axis that maximizes the sum of the squares of a𝓁,𝓁 anda𝓁,−𝓁 spherical harmonic coeﬃ- cients, that is, maximizes polar as...
[6]

Relics of primordial symmetries on the inﬂationary horizon survive in ∆( θ,φ ) at last scat- tering, and can be reconstructed from measurements

can also be written in terms of an empirical estimator C∆(Θ) =⟨∆a∆b⟩∠ab=Θ, (17) an all-sky average over all pairs of points a,b at angular separation ∠ab = Θ. Relics of primordial symmetries on the inﬂationary horizon survive in ∆( θ,φ ) at last scat- tering, and can be reconstructed from measurements. In the particular case of angular separation Θ = 90 ◦...
[7]

An unex- pected lack of large angle correlation power has been ap- parent since the ﬁrst measurements with COBE[35]

The two-point temperature correlation function is small at large angular separation [8, 13, 34]. An unex- pected lack of large angle correlation power has been ap- parent since the ﬁrst measurements with COBE[35]. In the WMAP analysis of C(Θ), based on 7 years of data[4], the authors comment on the (true) fact that there is no signiﬁcant conﬂict with the ...
[8]

The quadrupole and other even harmonics are smaller than expected . As discussed above, an excess of odd over even ﬂuctuation power [8, 13, 34], measured in harmonic decomposition, shows signiﬁcant anomalous antipodal anticorrelation on angular scales much smaller than the dipole, which also appears as a signiﬁcant neg- ative correlation in CT (Θ) near 18...

Show all 98 references

[9]

Such a precise null test of equa- torial symmetry (Eq

Global azimuthal symmetries The exact value of C(Θ) has no particular signiﬁcance in the standard scenario, so a very close agreement with zero can indicate a likelihood with a strong preference for the holographic theory. Such a precise null test of equa- torial symmetry (Eq....
[10]

conspire

Intrinsic Dipole In emergent cosmology, it is plausible that (unlike the standard scenario) there is a vanishing intrinsic dipole, because enclosed matter and its bounding CMB hori- zon share a common causal past (see Fig. 2): they are bounded by the same causal diamond, whose...
[11]

anomalies

Future Holometer experiments Nonlocal coherent ﬂuctuations on causal diamonds, as posited here for inﬂation, should also aﬀect light prop- agating between massive bodies in ﬂat space on macro- scopic scales[41–46]. It has recently become possible to probe these coherent Planck...
[12]

Steven Weinberg, Cosmology (Oxford University Press, 2008)

2008
[13]

Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results,

C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik, G. Hinshaw, N. Odegard, K. M. Smith, R. S. Hill, B. Gold, M. Halpern, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, E. Wollack, J. Dunkley, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, and E. L. Wright, “Nine-year Wilki...

2013 arXiv
[14]

Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Ob- servations: Cosmological Parameter Results,

G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, and E. L. Wright, “Nine-year Wilki...

2013 arXiv
[15]

Planck 2015 results. XIII. Cosmological parameters,

P. A. R. Ade et al. (Planck), “Planck 2015 results. XIII. Cosmological parameters,” Astron. Astrophys. 594, A13 (2016)

2016
[16]

Planck 2015 results. XX. Constraints on inﬂation,

P. A. R. Ade et al. (Planck), “Planck 2015 results. XX. Constraints on inﬂation,” Astron. Astrophys. 594, A20 (2016)

2016
[17]

Improved Constraints on Cosmology and Foregrounds from BI- CEP2 and Keck Array Cosmic Microwave Background Data with Inclusion of 95 GHz Band,

P. A. R. Ade et al. (BICEP2, Keck Array), “Improved Constraints on Cosmology and Foregrounds from BI- CEP2 and Keck Array Cosmic Microwave Background Data with Inclusion of 95 GHz Band,” Phys. Rev. Lett. 116, 031302 (2016)

2016
[18]

Planck 2015 results. XVI. Isotropy and statistics of the CMB,

P. A. R. Ade et al. (Planck), “Planck 2015 results. XVI. Isotropy and statistics of the CMB,” Astron. Astrophys. 594, A16 (2016), arXiv:1506.07135 [astro-ph.CO]

2016 arXiv
[19]

A Comparison of Cosmolog- ical Parameters Determined from CMB Temperature Power Spectra from the South Pole Telescope and the Planck Satellite,

K. Aylor et al. (SPT), “A Comparison of Cosmolog- ical Parameters Determined from CMB Temperature Power Spectra from the South Pole Telescope and the Planck Satellite,” Astrophys. J. 850, 101 (2017), arXiv:1706.10286 [astro-ph.CO]

2017 arXiv
[20]

Planck 2018 results. I. Overview and the cosmological legacy of Planck,

Y. Akrami et al. (Planck), “Planck 2018 results. I. Overview and the cosmological legacy of Planck,” (2018), arXiv:1807.06205 [astro-ph.CO]

2018 arXiv
[21]

Planck 2018 results. VI. Cosmological parameters,

N. Aghanim et al. (Planck), “Planck 2018 results. VI. Cosmological parameters,” (2018), arXiv:1807.06209 [astro-ph.CO]

2018 arXiv
[22]

Planck 2018 results. X. Con- straints on inﬂation,

Y. Akrami et al. (Planck), “Planck 2018 results. X. Con- straints on inﬂation,” (2018), arXiv:1807.06211 [astro- ph.CO]

2018 arXiv
[23]

Planck 2018 results. VII. Isotropy and Statistics of the CMB,

Y. Akrami et al. (Planck), “Planck 2018 results. VII. Isotropy and Statistics of the CMB,” (2019), arXiv:1906.02552 [astro-ph.CO]

2019 arXiv
[24]

Precision of inﬂaton potential reconstruction from CMB using the general slow-roll approximation,

Kenji Kadota, Scott Dodelson, Wayne Hu, and Ewan D. Stewart, “Precision of inﬂaton potential reconstruction from CMB using the general slow-roll approximation,” Phys. Rev. D72, 023510 (2005)

2005
[25]

Inﬂation,

Daniel Baumann, “Inﬂation,” in Physics of the large and the small, TASI 09 (2011) pp. 523–686, arXiv:0907.5424 [hep-th]

2011 arXiv
[26]

The Quest for 11 B Modes from Inﬂationary Gravitational Waves,

Marc Kamionkowski and Ely D. Kovetz, “The Quest for 11 B Modes from Inﬂationary Gravitational Waves,” Ann. Rev. Astron. Astrophys. 54, 227–269 (2016)

2016
[27]

The holographic spacetime model of cosmology,

Tom Banks and W. Fischler, “The holographic spacetime model of cosmology,” Int. J. Mod. Phys. D27, 1846005 (2018), arXiv:1806.01749 [hep-th]

2018 arXiv
[28]

Nonlocal entanglement and directional correlations of primordial perturbations on the inﬂation- ary horizon,

Craig Hogan, “Nonlocal entanglement and directional correlations of primordial perturbations on the inﬂation- ary horizon,” Phys. Rev. D 99, 063531 (2019)

2019
[29]

The Quantum Black Hole as a Hy- drogen Atom: Microstates Without Strings Attached,

Gerard ’t Hooft, “The Quantum Black Hole as a Hy- drogen Atom: Microstates Without Strings Attached,” (2016), arXiv:1605.05119 [gr-qc]

2016 arXiv
[30]

Black hole unitarity and antipo- dal entanglement,

Gerard ’t Hooft, “Black hole unitarity and antipo- dal entanglement,” Found. Phys. 46, 1185–1198 (2016), arXiv:1601.03447 [gr-qc]

2016 arXiv
[31]

Virtual black holes and space–time structure,

Gerard ’t Hooft, “Virtual black holes and space–time structure,” Foundations of Physics48, 1134–1149 (2018)

2018
[32]

Quantum-ﬁrst gravity,

Steven B. Giddings, “Quantum-ﬁrst gravity,” Found. Phys. 49, 177–190 (2019), arXiv:1803.04973 [hep-th]

2019 arXiv
[33]

Black holes in the quantum universe,

Steven B. Giddings, “Black holes in the quantum universe,” Proceedings, Topological avatars of new physics: London, United Kingdom, March 4-5, 2019 , Phil. Trans. Roy. Soc. Lond. A377, 20190029 (2019), arXiv:1905.08807 [hep-th]

2019 arXiv
[34]

Holographic discreteness of inﬂationary perturbations,

Craig J. Hogan, “Holographic discreteness of inﬂationary perturbations,” Phys. Rev. D66, 023521 (2002)

2002
[35]

Discrete spectrum of inﬂationary ﬂuc- tuations,

Craig J. Hogan, “Discrete spectrum of inﬂationary ﬂuc- tuations,” Phys. Rev. D70, 083521 (2004)

2004
[36]

Observing the beginning of time: New maps of the cosmic background radiation may display ev- idence of the quantum origin of space and time,

Craig J. Hogan, “Observing the beginning of time: New maps of the cosmic background radiation may display ev- idence of the quantum origin of space and time,” Amer- ican Scientist 90, 420–427 (2002)

2002
[37]

Black Hole Remnants in the Early Universe,

Fabio Scardigli, Christine Gruber, and Pisin Chen, “Black Hole Remnants in the Early Universe,” Phys. Rev. D83, 063507 (2011), arXiv:1009.0882 [gr-qc]

2011 arXiv
[38]

Gauge-invariant cosmological per- turbations,

James M. Bardeen, “Gauge-invariant cosmological per- turbations,” Phys. Rev. D 22, 1882–1905 (1980)

1980
[39]

Perturbations of a Cosmo- logical Model and Angular Variations of the Microwave Background,

R. K. Sachs and A. M. Wolfe, “Perturbations of a Cosmo- logical Model and Angular Variations of the Microwave Background,” Astrophys. J. 147, 73 (1967)

1967
[40]

Cosmic microwave back- ground anisotropies,

Wayne Hu and Scott Dodelson, “Cosmic microwave back- ground anisotropies,” Ann. Rev. Astron. Astrophys. 40, 171–216 (2002), arXiv:astro-ph/0110414 [astro-ph]

2002 arXiv
[41]

Theoretical overview of cosmic microwave background anisotropy,

E. L. Wright, “Theoretical overview of cosmic microwave background anisotropy,” in Measuring and modeling the universe. Proceedings, Symposium, Pasadena, USA, November 17-22, 2002 (2003) pp. 291–308, arXiv:astro- ph/0305591 [astro-ph]

2003
[42]

CMB to- mography: Reconstruction of adiabatic primordial scalar potential using temperature and polarization maps,

Amit P. Yadav and Benjamin D. Wandelt, “CMB to- mography: Reconstruction of adiabatic primordial scalar potential using temperature and polarization maps,” Phys. Rev. D 71, 123004 (2005), arXiv:astro-ph/0505386 [astro-ph]

2005 arXiv
[43]

All-sky reconstruction of the primordial scalar poten- tial from WMAP temperature data,

Sebastian Dorn, Maksim Greiner, and Torsten A. Enßlin, “All-sky reconstruction of the primordial scalar poten- tial from WMAP temperature data,” JCAP 1502, 041 (2015), arXiv:1412.8315 [astro-ph.CO]

2015 arXiv
[44]

CMB Anomalies after Planck,

Dominik J. Schwarz, Craig J. Copi, Dragan Huterer, and Glenn D. Starkman, “CMB Anomalies after Planck,” Class. Quant. Grav. 33, 184001 (2016)

2016
[45]

Two-point correlations in the [ITAL]COBE[/ITAL] DMR four-year anisotropy maps,

G. Hinshaw, A. J. Banday, C. L. Bennett, K. M. G´ orski, A. Kogut, C. H. Lineweaver, G. F. Smoot, and E. L. Wright, “Two-point correlations in the [ITAL]COBE[/ITAL] DMR four-year anisotropy maps,” The Astrophysical Journal 464, L25–L28 (1996)

1996
[46]

Alignments of parity even/odd-only mul- tipoles in CMB,

Pavan K. Aluri, John P. Ralston, and Amanda Weltman, “Alignments of parity even/odd-only mul- tipoles in CMB,” MNRAS 472, 2410–2421 (2017), arXiv:1703.07070 [astro-ph.CO]

2017 arXiv
[47]

Parity asymmetry in the CMBR temperature power spectrum,

Pavan K. Aluri and Pankaj Jain, “Parity asymmetry in the CMBR temperature power spectrum,” MNRAS 419, 3378–3392 (2012), arXiv:1108.5894 [astro-ph.CO]

2012 arXiv
[48]

Large-scale alignments from WMAP and Planck,

Craig J. Copi, Dragan Huterer, Dominik J. Schwarz, and Glenn D. Starkman, “Large-scale alignments from WMAP and Planck,” MNRAS 449, 3458–3470 (2015), arXiv:1311.4562 [astro-ph.CO]

2015 arXiv
[49]

Symmetries of CMB temperature correlation at large angular separations,

Ray Hagimoto, Craig Hogan, Collin Lewin, and Stephan S. Meyer, “Symmetries of CMB temperature correlation at large angular separations,” The Astrophys- ical Journal 888, L29 (2020)

2020
[50]

Bulk ﬂow in the combined 2MTF and 6dFGSv surveys,

Fei Qin, Cullan Howlett, Lister Staveley-Smith, and Tao Hong, “Bulk ﬂow in the combined 2MTF and 6dFGSv surveys,” Mon. Not. Roy. Astron. Soc. 477, 5150–5166 (2018), arXiv:1804.05520 [astro-ph.CO]

2018 arXiv
[51]

Interferometers as Probes of Planckian Quantum Geometry,

Craig J. Hogan, “Interferometers as Probes of Planckian Quantum Geometry,” Phys. Rev. D85, 064007 (2012)

2012
[52]

Interferometric Tests of Planckian Quantum Geometry Models,

Ohkyung Kwon and Craig J. Hogan, “Interferometric Tests of Planckian Quantum Geometry Models,” Class. Quant. Grav. 33, 105004 (2016)

2016
[53]

Statistical mea- sures of planck scale signal correlations in interferome- ters,

Craig J Hogan and Ohkyung Kwon, “Statistical mea- sures of planck scale signal correlations in interferome- ters,” Classical and Quantum Gravity34, 075006 (2017)

2017
[54]

Exotic Rotational Correlations in Quan- tum Geometry,

C. J. Hogan, “Exotic Rotational Correlations in Quan- tum Geometry,” Phys. Rev. D 95, 104050 (2017)

2017
[55]

Statisti- cal Model of Exotic Rotational Correlations in Emergent Space-Time,

C. J. Hogan, O. Kwon, and J. Richardson, “Statisti- cal Model of Exotic Rotational Correlations in Emergent Space-Time,” Class. Quantum Grav. 34, 135006 (2017)

2017
[56]

Observa- tional Signatures of Quantum Gravity in Interferome- ters,

Erik P. Verlinde and Kathryn M. Zurek, “Observa- tional Signatures of Quantum Gravity in Interferome- ters,” (2019), arXiv:1902.08207 [gr-qc]

2019 arXiv
[57]

Interferometric Constraints on Quantum Geometrical Shear Noise Correlations,

A. Chou, H. Glass, H. R. Gustafson, C. J. Hogan, B. L. Kamai, O. Kwon, R. Lanza, L. McCuller, S. S. Meyer, J. Richardson, C. Stoughton, R. Tomlin, and R. Weiss (Holometer Collaboration), “Interferometric Constraints on Quantum Geometrical Shear Noise Correlations,” Class. Quan...

2017
[58]

Problems and prospects in ele- mentary particle research,

John Archibald Wheeler, “Problems and prospects in ele- mentary particle research,” Proceedings of the American Philosophical Society 90, 36–47 (1946)

1946
[59]

Experiment and the foundations of quantum physics,

Anton Zeilinger, “Experiment and the foundations of quantum physics,” Rev. Mod. Phys. 71, S288–S297 (1999)

1999
[60]

What is observable in classical and quan- tum gravity?

C Rovelli, “What is observable in classical and quan- tum gravity?” Classical and Quantum Gravity 8, 297– 316 (1991)

1991
[61]

Time Dilation in Quantum Systems and De- coherence,

Igor Pikovski, Magdalena Zych, Fabio Costa, and ˇCaslav Brukner, “Time Dilation in Quantum Systems and De- coherence,” New Journal of Physics 19, 025011 (2017)

2017
[62]

Information loss,

William G Unruh and Robert M Wald, “Information loss,” Reports on Progress in Physics 80, 092002 (2017)

2017
[63]

Eﬀective ﬁeld theory, black holes, and the cosmological constant,

A. G. Cohen, D. B. Kaplan, and A. E. Nelson, “Eﬀective ﬁeld theory, black holes, and the cosmological constant,” Phys. Rev. Lett. 82, 4971 (1999)

1999
[64]

Entanglement entropy of black holes,

Sergey N. Solodukhin, “Entanglement entropy of black holes,” Living Rev. Rel. 14, 8 (2011)

2011
[65]

Tabletop experiments for quantum gravity: a user’s manual,

Daniel Carney, Philip C E Stamp, and Jacob M Tay- lor, “Tabletop experiments for quantum gravity: a user’s manual,” Classical and Quantum Gravity 36, 034001 (2019). 12

2019
[66]

Thermodynamics of Spacetime: The Ein- stein Equation of State,

T. Jacobson, “Thermodynamics of Spacetime: The Ein- stein Equation of State,” Phys. Rev. Lett. 75, 1260 (1995)

1995
[67]

Entanglement Equilibrium and the Ein- stein Equation,

Ted Jacobson, “Entanglement Equilibrium and the Ein- stein Equation,” Phys. Rev. Lett. 116, 201101 (2016)

2016
[68]

On the Origin of Gravity and the Laws of Newton,

E. Verlinde, “On the Origin of Gravity and the Laws of Newton,” JHEP 1104, 029 (2011)

2011
[69]

General Relativity from a Thermo- dynamic Perspective,

T. Padmanabhan, “General Relativity from a Thermo- dynamic Perspective,” Gen. Rel. Grav. 46, 1673 (2014)

2014
[70]

Dimensional reduction in quantum gravity,

Gerard ’t Hooft, “Dimensional reduction in quantum gravity,” Conference on Highlights of Particle and Con- densed Matter Physics (SALAMFEST) Trieste, Italy, March 8-12, 1993 , Conf. Proc. C930308, 284–296 (1993)

1993
[71]

The world as a hologram,

L. Susskind, “The world as a hologram,” J. Math. Phys. 36, 6377 (1995)

1995
[72]

The Holographic principle,

Raphael Bousso, “The Holographic principle,” Rev.Mod.Phys. 74, 825–874 (2002)

2002
[73]

Holographic derivation of entanglement entropy from AdS/CFT,

Shinsei Ryu and Tadashi Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006)

2006
[74]

Aspects of Holo- graphic Entanglement Entropy,

Shinsei Ryu and Tadashi Takayanagi, “Aspects of Holo- graphic Entanglement Entropy,” JHEP 08, 045 (2006)

2006
[75]

AdS/CFT Duality User Guide,

Makoto Natsuume, “AdS/CFT Duality User Guide,” Lect. Notes Phys. 903, pp.1–294 (2015)

2015
[76]

Soft Hair on Black Holes,

Stephen W. Hawking, Malcolm J. Perry, and Andrew Strominger, “Soft Hair on Black Holes,” Phys. Rev. Lett. 116, 231301 (2016), arXiv:1601.00921 [hep-th]

2016 arXiv
[77]

Black Hole Entropy and Soft Hair,

Sasha Haco, Stephen W. Hawking, Malcolm J. Perry, and Andrew Strominger, “Black Hole Entropy and Soft Hair,” JHEP 12, 098 (2018), arXiv:1810.01847 [hep-th]

2018 arXiv
[78]

Spacetime and Universal Soft Modes — Black Holes and Beyond,

Yasunori Nomura, “Spacetime and Universal Soft Modes — Black Holes and Beyond,” (2019), arXiv:1908.05728 [hep-th]

2019 arXiv
[79]

The gravitational shock wave of a massless particle,

Tevian Dray and Gerard ’t Hooft, “The gravitational shock wave of a massless particle,” Nuclear Physics B 253, 173 – 188 (1985)

1985
[80]

On the gravitational ﬁeld of a massless particle,

P. C. Aichelburg and R. U. Sexl, “On the gravitational ﬁeld of a massless particle,” General Relativity and Grav- itation 2, 303–312 (1971)

1971
[81]

Natural Covariant Planck Scale Cut- oﬀs and the Cosmic Microwave Background Spectrum,

Aidan Chatwin-Davies, Achim Kempf, and Robert T. W. Martin, “Natural Covariant Planck Scale Cut- oﬀs and the Cosmic Microwave Background Spectrum,” Phys. Rev. Lett. 119, 031301 (2017), arXiv:1612.06445 [gr-qc]

2017 arXiv
[82]

Measurement of Quantum Fluctua- tions in Geometry,

Craig J. Hogan, “Measurement of Quantum Fluctua- tions in Geometry,” Phys. Rev. D77, 104031 (2008), arXiv:0712.3419 [gr-qc]

2008 arXiv
[83]

Indeterminacy of Holographic Quan- tum Geometry,

Craig J. Hogan, “Indeterminacy of Holographic Quan- tum Geometry,” Phys. Rev. D78, 087501 (2008), arXiv:0806.0665 [gr-qc]

2008 arXiv
[84]

A new type of isotropic cosmological models without singularity,

A.A. Starobinsky, “A new type of isotropic cosmological models without singularity,” Physics Letters B 91, 99 – 102 (1980)

1980
[85]

Polchinski, String Theory (Cambridge University Press, 1998)

J. Polchinski, String Theory (Cambridge University Press, 1998)

1998
[86]

Introduction to superstring theory,

J. H. Schwarz, “Introduction to superstring theory,” (2000), arXiv:hep-ex/0008017

2000 arXiv
[87]

Rovelli, Quantum Gravity (Cambridge University Press, 2004)

C. Rovelli, Quantum Gravity (Cambridge University Press, 2004)

2004
[88]

P. A. M. Dirac, Quantum Mechanics (Oxford University Press, Oxford, UK, 1958)

1958
[89]

Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1996)

S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1996)

1996
[90]

and Wittke, J

Dicke, R.H. and Wittke, J. P., Introduction to Quantum Mechanics (Reading: Addison-Wesley, 1960)

1960
[91]

sum over histories

Landau, L. D. and Lifshitz, E. M., Quantum Mechanics: Non-Relativistic Theory (Oxford: Pergamon, 1977). APPENDIX A. Theoretical motivation for a coherent quantum inﬂationary horizon The main hypothesis of holographic inﬂation is that the boundary between classical and quantum ...

1977
[92]

Gravitational shock waves of point particles with Planck momentum in ﬂat space-time Consider the classical gravitational eﬀect of a null point particle[19–21, 69, 70], as shown in Fig. (6). A localized pointlike momentum impulse p with impact parameter x⊥ creates an invariant ...
[93]

[19–21], with the grav- itational eﬀect based on the gravitational shock wave in the Rindler frame

Classical tides from point masses added and subtracted from a black hole The back-reaction of quantum point particles enter- ing and leaving a black hole horizon, including antipodal correlations, was studied in refs. [19–21], with the grav- itational eﬀect based on the gravit...
[94]

For coherent quadrupolar vibrational distortions of a black hole horizon of magnitude hhorizon∼δR/R, (25) we have approximately ¨Iij∼h2 horizon(R/G)

Virtual quadrupolar distortion of a black hole necessary to radiate classically at the Hawking rate In linear gravity, a source emits gravitational radiation according to the standard classical quadrupole formula, hwave = (G/r)¨Iij, (24) where hwave is strain of the wave at di...
[95]

[78], §75): ∆(x) = 2δ(xµxµ)x0/|x0|, (32) where xµ = (t,⃗ r) represents 4-position

Dirac light cone function Locality on light cones is conventionally deﬁned by the covariant four-dimensional generalization of the one- dimensional Dirac δ-function (ref. [78], §75): ∆(x) = 2δ(xµxµ)x0/|x0|, (32) where xµ = (t,⃗ r) represents 4-position. It vanishes at the orig...
[96]

collapse

Locality of quantum ﬁeld states The standard model of locality for ﬁeld states is based on point localization in a classical space-time back- ground. To quantize ﬁelds[78, 79], the light cone function is used to write covariant commutation relations for ﬁeld operators ˆA(x), [...
[97]

Model of coherent causal diamond ﬂuctuations A simple model of a geometrical quantum system can be built from new geometrical operators ˆτκ, whose ﬂuc- tuations will ultimately be identiﬁed as perturbations in potential. A contraction with the antisymmetric Levi- Civita 4-tens...
[98]

Consider projection onto a 3D spacelike surface of con- stant x0⁄= 0

Nonlocal information, projection and uncertainty Even though the degrees of freedom represented by the ˆτκ’s have no local or dynamical eﬀects, their ﬂuctuations in time and direction aﬀect correlations in nonlocal mea- surements. Consider projection onto a 3D spacelike surfac...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

conspiracy

Azimuthal symmetries One consequence of emergent causality is that incom- ing phase information that determines polar values of po- tential along any given axis on the horizon only reaches the equatorial plane at the end of inﬂation, so it cannot aﬀect the potential for points in that plane (see Fig. 5). As shown in the examples given in the Appendix, the...

[2] [2]

A separate symmetry may be associated with nonlocal cor- relations of variance normal to each axis

Constant variance on great circles The global equatorial symmetry still allows mean val- ues of ∆ on great circles to vary according to random incoming polar information associated with each axis. A separate symmetry may be associated with nonlocal cor- relations of variance normal to each axis. It is possible that the variance on great circles is a const...

[3] [3]

cosmic variance

Antipodal anticorrelation and parity violation The inﬂationary horizon could also display antipodal anticorrelation, a tendency of opposite points in the sky to have opposite signs. It occurs in quantum models of eternal black hole horizons[19–21], where antipodes on the horizon are actually identiﬁed, and time-reversed conjugate particle states are entan...

[4] [4]

Seven-year wilkinson microwave anisotropy probe (wmap) observations: Are there cos- mic microwave background anomalies?

C. L. Bennett, R. S. Hill, G. Hinshaw, D. Lar- son, K. M. Smith, J. Dunkley, B. Gold, M. Halpern, N. Jarosik, A. Kogut, E. Komatsu, M. Limon, S. S. Meyer, M. R. Nolta, N. Odegard, L. Page, D. N. Spergel, G. ˆA S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright, “Seven-year wilkinson microwave anisotropy probe (wmap) observations: Are there cos- mic mi...

2011

[5] [5]

secondary

Axes deﬁned by the quadrupole and octopole are closely aligned. The WMAP all-sky maps[2–4] revealed a remarkably close agreement in direction for quadrupole (𝓁 = 2) and octopole ( 𝓁 = 3) harmonics. The aligned direction is deﬁned by the axis that maximizes the sum of the squares of a𝓁,𝓁 anda𝓁,−𝓁 spherical harmonic coeﬃ- cients, that is, maximizes polar as...

[6] [6]

Relics of primordial symmetries on the inﬂationary horizon survive in ∆( θ,φ ) at last scat- tering, and can be reconstructed from measurements

can also be written in terms of an empirical estimator C∆(Θ) =⟨∆a∆b⟩∠ab=Θ, (17) an all-sky average over all pairs of points a,b at angular separation ∠ab = Θ. Relics of primordial symmetries on the inﬂationary horizon survive in ∆( θ,φ ) at last scat- tering, and can be reconstructed from measurements. In the particular case of angular separation Θ = 90 ◦...

[7] [7]

An unex- pected lack of large angle correlation power has been ap- parent since the ﬁrst measurements with COBE[35]

The two-point temperature correlation function is small at large angular separation [8, 13, 34]. An unex- pected lack of large angle correlation power has been ap- parent since the ﬁrst measurements with COBE[35]. In the WMAP analysis of C(Θ), based on 7 years of data[4], the authors comment on the (true) fact that there is no signiﬁcant conﬂict with the ...

[8] [8]

The quadrupole and other even harmonics are smaller than expected . As discussed above, an excess of odd over even ﬂuctuation power [8, 13, 34], measured in harmonic decomposition, shows signiﬁcant anomalous antipodal anticorrelation on angular scales much smaller than the dipole, which also appears as a signiﬁcant neg- ative correlation in CT (Θ) near 18...