CMB sky for an off-center observer in a local void I: framework for forecasts

Giulia Cusin; Martin Kunz; Viraj Nistane

REVIEW 2 major objections 5 minor 44 references

CMB sky for an off-center observer in a local void I: framework for forecasts

T0 review · 2 major / 5 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 An off-center observer inside a large local void would see a statistically anisotropic CMB, and two concrete void models would imprint a detectable lensing-like signal in Planck data with signal-to-noise above 10.

desk verdict A sound framework paper whose headline S/N>10 applies only to already-ruled-out toy voids with all parameters fixed; the body is honest about this, the abstract is not. read the letter →

arxiv 1908.05484 v2 pith:YOKOHPHS submitted 2019-08-15 astro-ph.CO gr-qc

Viraj Nistane , Giulia Cusin , Martin Kunz This is my paper · ORCID

classification astro-ph.COgr-qc

keywords LTBvoidoff-centerobservercosmicmicrowavebackgroundstatisticalanisotropygravitationallensingFisherforecastsignal-to-noiseCopernicanprinciple

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 establishes a framework for predicting and forecasting the CMB temperature sky seen by an observer displaced from the center of a large local underdense region described by a Lemaitre-Tolman-Bondi metric. Because the geometry is not isotropic about the observer, the CMB correlation matrix acquires off-diagonal elements, produced by a lensing-like deflection that the paper models as a pure gradient. The core result is an analytic formula for these off-diagonal correlations, together with a Fisher forecast for the amplitude of the lensing contribution applied to two illustrative large void models. The authors find that for both models the signal would be detectable in Planck data with signal-to-noise ratio over 10, and the framework is designed to be applied systematically to other void profiles in a companion paper.

What carries the argument

The load-bearing object is the lensing-like deflection angle $\tilde{\Gamma}$ between the emission direction at last scattering and the arrival direction, which for this axial geometry is a pure gradient, so it is encoded in a lensing potential $\phi_{\ell 0}$ computed from the geodesic integration: $\phi_{\ell 0} = \frac{2\pi}{\ell(\ell+1)} \sqrt{\frac{2\ell+1}{4\pi}} \int_{-1}^{1} \partial_\xi P_\ell(\cos\xi)\, \tilde{\Gamma}(\xi)\, d\cos\xi$. This potential enters the two-point correlation function through Eq. (4.6), which combines it with Gaunt coefficients and the primary spectrum $C_\ell$ to produce off-diagonal temperature correlations. The Fisher forecast in Section 5 uses this covariance, treats the non-stochastic anisotropies as a known mean, and replaces the primary spectrum $C_\ell$ by a Planck-noise-augmented version $C_\ell^N$ to estimate the attainable signal-to-noise.

What would settle it

A full-sky CMB survey with noise lower than Planck could measure the off-diagonal temperature correlations $F_{\ell m}^{\ell' m'}$ for $|\ell-\ell'|>1$; if the observed amplitudes are significantly smaller than the prediction for an assumed void size, or if a non-zero curl component of the deflection is detected, the linear pure-gradient picture and the claimed detectability would fail.

Watch

Extended reading notes

Core claim

An off-center observer inside a local LTB void measures a CMB sky that is not statistically isotropic. Decomposing the observed temperature anisotropies into a primary, statistically isotropic component, geometry-induced non-stochastic anisotropies, and a lensing-like secondary component, the authors derive the off-diagonal part of the temperature two-point correlation matrix: in the frame aligned with the observer-void axis it takes the form $F_{\ell m}^{\ell' m'}|_\phi = \sum_{\ell_1} \phi_{\ell_1 0} \, C^{\, m \, m' \, 0}_{\ell \; \ell' \; \ell_1} \, (\alpha_+ C_\ell + \alpha_- C_{\ell'})$, with $\alpha_\pm = \frac{1}{2}[\ell_1(\ell_1+1) \pm (\ell'-\ell)(\ell'+\ell+1)]$ and $C^{...}_{...}$ the Gaunt coefficients. Using this correlation structure together with the primary spectrum from CLASS, a Fisher matrix forecast for the amplitude of the lensing-like contribution gives a cumulative signal-to-noise ratio exceeding 10 for both illustrative large void models in Planck temperature data. The authors also show that a kinematic boost induces off-diagonal correlations only between multipoles separated by $\pm 1$ at linear order, so the void-lensing signature is distinguishable from a boost of the observer.

Load-bearing premise

The forecast depends on the assumption that the CMB covariance in the void is just the standard statistically isotropic primary spectrum plus a small, linear-order, pure-gradient lensing-like correction, with all void and observer parameters fixed and known; if the void changes the primary anisotropies themselves, if the deflection has non-negligible curl or higher-order parts, or if the parameters are not precisely known, the predicted off-diagonal correlations and the claimed signal-to-noise of over 10 would change.

Editorial extensions

If this is right

The forecast method can be applied directly to any LTB void profile to predict whether its lensing-like CMB distortion is measurable with a given experiment's noise.
A kinematic boost and void lensing produce different selection rules on multipole separations, so a measured off-diagonal correlation matrix can in principle separate the two effects.
For Planck-like noise the cumulative signal-to-noise saturates near $\ell\approx 1600$, meaning higher resolution is what would further improve detectability.
The framework treats non-stochastic anisotropies as a known mean, so in practice unrealistic parameter uncertainties would enlarge the error bars, limiting the guaranteed signal-to-noise.

Reading between the lines

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

The same covariance formalism could be extended to CMB polarization, where the lensing-like deflection would create E/B-mode correlations that give an independent check of the off-center interpretation.
A natural next step is to search for the curl (magnetic) component of the deflection at higher order; detecting it would probe deviations from the pure-gradient assumption and the smoothness of the void edge.
Because the off-diagonal signal is anisotropic in a preferred direction (the observer-void axis), the framework could be used to reconstruct the position of the observer relative to the void center from future high-resolution CMB data, not just test detectability.
The two models considered were chosen for simplicity and have already been ruled out as dark-energy alternatives; the more realistic models with a cosmological constant, planned for the companion paper, could yield a different (possibly lower) signal-to-noise.

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

The short version: this is a genuine methods contribution. It extends the Cusin-Pitrou-Uzan framework from a singular shell to continuous LTB profiles and runs the first Fisher forecast on the lensing-like amplitude. The derivation follows the established formalism, the correlation function (4.6) is explicit, and they validate their Fisher code against the known boost result from Amendola et al. That is real, reproducible work, and the paper deserves a serious referee.

What is actually new: the off-diagonal CMB temperature correlation structure for general LTB voids with continuous density transitions, the lensing potential computed from geodesics, and the forecast for the amplitude A_V with Planck-like noise. The split between non-stochastic anisotropies and stochastic lensing is careful, and the comparison with kinematic effects in Section 4.1 is a useful sanity check.

The soft spots are real but not fatal. The two illustrative voids are explicitly ruled out as dark energy alternatives (footnote 1, refs. 11-14), so the S/N>10 result is a demonstration of the method, not a detection claim about our Universe. The paper says this in the body. The bigger issue is that the forecast fixes all void and observer parameters and treats the non-stochastic quadrupole as a perfectly known mean. Figure 5 shows that quadrupole is comparable to the primary quadrupole, so that assumption is not a minor detail. The paper acknowledges this in Section 5 — 'A more realistic analysis would vary the void/observer parameters' — but the abstract and conclusions present S/N>10 without that caveat. That is a fair criticism, and it should be fixed. Marginalizing over the shared parameters could plausibly drop the signal below detection threshold, so the headline is an upper limit, not a robust claim.

Minor: no code or detailed numerical recipes, so reproduction is harder than it should be. Also, one author of the foundational paper is a co-author here, but the work extends rather than repeats that paper, and the formalism is independent enough that this is not a problem.

Who this is for: people working on LTB voids, CMB statistical anisotropy, and Copernican-principle tests. It is a framework paper, with the systematic exploration promised for the companion. A serious referee should engage with it, mainly to ensure the parameter-freezing idealization is clearly labeled as such throughout, not just in one section. Send it to peer review.

Referee Report

2 major / 5 minor

Summary. The paper constructs a framework for predicting the CMB temperature two-point correlation function seen by an off-center observer inside an LTB void, and uses it to forecast the detectability of the associated lensing-like distortion. After solving the LTB photon geodesics, the authors split the observed anisotropies into primary, non-stochastic geometry-induced, and lensing-induced parts, derive the off-diagonal correlation matrix from a lensing potential, and compute a Fisher forecast for the lensing amplitude. For two illustrative void profiles, they claim a Planck signal-to-noise ratio above 10. The paper also discusses the effect of a peculiar velocity and validates the Fisher machinery against the boost forecast of reference [43].

Significance. If the derivation is correct, the paper provides a useful and fairly general tool for computing off-diagonal CMB correlations in LTB models, extending earlier work to continuous matter profiles. It correctly identifies the lensing-like gradient signal as a distinctive signature separate from a boost, and it validates the numerical pipeline against an independent result. The main caveat is that the quantitative S/N>10 claim is obtained in an idealized setting: all void and observer parameters are fixed, the non-stochastic mean is treated as known, and the two illustrative models are already disfavored by other observations (as the paper itself notes). The significance is therefore that of a proof-of-concept framework rather than a robust detection claim.

major comments (2)

[§5, Eq. (5.4)–(5.6), Fig. 5, §6] The headline claim that the two void models would leave a detectable signal in Planck with S/N>10 is computed with all void/observer parameters fixed and with the non-stochastic anisotropies Θ_NS treated as a perfectly known mean in the likelihood (5.1). Figure 5 shows that the non-stochastic quadrupole is comparable to the primary quadrupole, and the same parameters (r0, x0, Δx, Δα) control both Θ_NS and the lensing template δC entering Eq. (5.6). Thus the quoted S/N is an upper limit. The paper acknowledges this in §5 ('A more realistic analysis would vary the void/observer parameters'), but the abstract and Section 6 state the detectability without this qualification. I recommend either adding the caveat explicitly to the abstract/conclusions or providing a simple demonstration that the S/N remains above threshold after marginalizing over the void parameters.
[§3.1, Table 1, Table 2, Eq. (4.6)] The primary C_l used in the correlation matrix is taken from CLASS with the fiducial H0=67.56 (Table 2), while the LTB models have h_out=0.51 in the outer region (Table 1). The angular diameter distance to last scattering in the LTB background is therefore very different from that in the CLASS universe, so the primary spectrum in the void is not automatically the standard CLASS spectrum. Since the forecast's inverse covariance and the template δC in Eq. (4.6) both depend on the actual primary C_l, the numerical S/N reported in Fig. 8 should be justified as an input assumption or recomputed with a C_l consistent with the LTB background. A brief discussion of why the standard C_l is a good proxy would suffice if the intent is only to illustrate the framework.

minor comments (5)

[Abstract and §6] The phrase 'would leave a detectable signal' in the conclusions is stronger than the abstract's 'potentially capable to detect'; since the forecast fixes all parameters, I suggest wording such as 'in the idealized fixed-parameter setting' in both places.
[Eq. (A.9)] The factor '(2l2+2)' in the square root appears to be a typo for '(2l2+1)'.
[Fig. 4 and surrounding text] The axis labels and numerical ranges in the text around Fig. 4 are corrupted ('10 14 10 12 ...'), making the figure description hard to read; please check the typesetting.
[§5, Eq. (5.13)] Replacing C_l by C_l^N=(C_l+N_l)/f_sky in both occurrences in Eq. (5.12) effectively multiplies the S/N by f_sky^2; the standard partial-sky treatment typically multiplies the mode count by f_sky. Since f_sky=0.85 the quantitative effect is small, but the prescription should be clarified or justified.
[§3.2 and §4] Key derivations of the lensing potential and of Eq. (4.6) are quoted from appendices D.1 and H of reference [10]; the present paper would be more self-contained if the essential steps were summarized in an appendix.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lensing template is computed from the LTB geodesics, the primary spectrum is external (CLASS), and the forecast is validated against an independent boost result; the one co-authored citation [10] is an independent derivation, not a fitted input.

full rationale

The paper's central off-diagonal correlation formula, Eq. (4.6), is taken from the prior work [10], whose author list overlaps with the present paper. This is a self-citation, and it is load-bearing for the form of the lensing-like covariance. However, it is not circular: [10] is an external, parameter-free derivation of the lensing correlation in an LTB geometry, and its stated assumptions do not include the detectability claim made here. The lensing potential phi_l0 is computed from the geodesic deflection via Eq. (3.11), with the deflection Gamma obtained by solving the geodesic equations (2.14)-(2.25) for the specified LTB metric; it is not fitted to CMB data. The primary C_l is taken from the independent CLASS code, and the Fisher forecast in Eq. (5.6) uses these derived inputs rather than any quantity fitted to the target signal. The claimed S/N > 10 is an idealized forecast with all void/observer and cosmological parameters fixed, and the paper explicitly flags this in Section 5 ('A more realistic analysis would vary the void/observer parameters') and the Introduction ('parameter degeneracies can increase the error bars on the amplitude'). Treating the non-stochastic anisotropies as a known mean subtracted in the likelihood is an idealization that affects the robustness of the forecast, but it is not a circular step. The method is also checked against the independent boost results of [43]. No fitted parameter is renamed as a prediction, and no equation reduces to its own input by construction. The only self-citation, [10], provides independent support rather than circularity.

Assumptions & free parameters 12 free parameters · 7 assumptions · 0 invented entities

The central claim depends on the LTB void model, the standard primary C_l, the gradient-only lensing approximation, and the assumption that non-stochastic anisotropies are a known mean. The void parameters are taken from previous fits in [7,8] and the observer offset is chosen by hand. No new physical entities are introduced.

free parameters (12)

A_V (lensing amplitude) = 1 (fiducial)
The amplitude of the lensing-like correlation, varied in the Fisher forecast with fiducial value 1.
Model A: Delta alpha = 0.9
Density contrast between void center and exterior; taken from Alnes et al. [7,8] fits to SN and CMB distances.
Model A: x0 = 1.450 Gpc
Void transition radius, from [7,8].
Model A: Delta x / x0 = 0.40
Transition width, from [7,8].
Model A: h_in = 0.65
Hubble parameter inside the void, from [7,8] fit to low-z SN.
Model A: h_out = 0.51
Hubble parameter outside the void, from [7,8] fit to CMB.
Model B: Delta alpha = 0.78
Density contrast for model B, from [7,8].
Model B: x0 = 1.804 Gpc
Void transition radius for model B, from [7,8].
Model B: Delta x / x0 = 0.03
Transition width for model B, from [7,8].
Model B: h_in = 0.63
Inside Hubble parameter for model B, from [7,8].
Observer offset r0 = 200 Mpc
Physical distance of observer from void center, chosen for illustration.
Planck noise parameters = theta0=7 arcmin, sigma=2e-6, fsky=0.85
Instrument noise model from [43], approximating Planck's best channel.

assumptions (7)

domain assumption The local underdense region is described by an LTB metric matched smoothly to an outer FLRW universe (Eqs. 2.1, 2.9, 2.10).
This is the physical model underlying the entire calculation; if the void is not LTB-like or not smoothly embedded, the signal differs.
domain assumption Primary CMB anisotropies are statistically isotropic with power spectrum C_l from CLASS with Planck-like parameters (Table 2).
Used as the diagonal covariance in Eq. (4.4); the void is assumed not to modify the primary spectrum.
domain assumption The lensing-like deflection is a pure gradient mode so a lensing potential exists (Eq. 3.10), and radial modulation is subdominant (Section 3.2).
From [10]; if curl modes or radial modulation are non-negligible, Eq. (4.6) misses terms.
domain assumption The non-stochastic anisotropies are treated as a known mean and do not contribute to the covariance (Section 5, Eq. 5.1).
The Fisher forecast assumes the mean is perfectly known; parameter uncertainties would add variance.
standard math The off-diagonal part of the covariance is small compared to the diagonal, allowing the inverse approximation in Eq. (5.9).
Standard perturbation expansion used to simplify the Fisher matrix; valid when delta C/C << 1.
domain assumption All void and observer parameters (except A_V) are fixed in the forecast.
The authors state this; parameter degeneracies would increase the error on A_V.
domain assumption The void models A and B, although ruled out as dark energy alternatives, are used as illustrative fiducial models.
The paper acknowledges this in footnote 1; the detectability claim is conditional on these models.

how reviews work

0 comments

Cite this review

Pith. "Pith review of CMB sky for an off-center observer in a local void I: framework for forecasts." pith.science (2026). https://pith.science/paper/YOKOHPHS

@misc{pith2026190805484,
  author       = {Pith},
  title        = {Pith review of: CMB sky for an off-center observer in a local void I: framework for forecasts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOKOHPHS}},
  note         = {Machine review of arXiv:1908.05484}
}

read the original abstract

The Universe is not perfectly homogeneous, the large scale structure forms overdense regions and voids. In this paper, we consider the possibility that we occupy a special position in our Universe, close to the center of a local underdense region that we model as an LTB void embedded in a homogeneous and isotropic Universe. The CMB sky measured by an off-center observer in this void is not statistically isotropic. In addition to the non-stochastic CMB anisotropies due to the geometry of the model, we also observe a lensing-like distortion of the CMB anisotropies. In this article, we propose a framework to forecast the precision with which we can measure the amplitude of the lensing-like deformation of the CMB temperature anisotropies. For illustrative purposes, we apply this method to a couple of large-scale void models differing over the matter density profile and we show that the CMB temperature data from the Planck satellite is potentially capable to detect the effect for the large voids chosen here. A companion paper will be dedicated to a systematic exploration of different realistic void models.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 14 canonical work pages

[43]

Amendola, R

L. Amendola, R. Catena, I. Masina, A. Notari, M. Quartin, and C. Quercellini, Measuring our peculiar velocity on the CMB with high-multipole oﬀ-diagonal correlations , JCAP 1107 (2011) 027, [arXiv:1008.1183]

arXiv 2011
[1]

Distances and lensing in cosmological void models

K. Tomita, Distances and lensing in cosmological void models , Astrophys. J. 529 (2000) 38, [astro-ph/9906027]

work page Pith review arXiv 2000
[2]

Analyses of Type Ia Supernova Data in Cosmological Models with a Local Void

K. Tomita, Analyses of type Ia supernova data in cosmological models with a local void , Prog. Theor. Phys. 106 (2001) 929–939, [ astro-ph/0104141]

work page Pith review arXiv 2001
[3]

A Local Void and the Accelerating Universe

K. Tomita, A local void and the accelerating universe , Mon. Not. Roy. Astron. Soc. 326 (2001) 287, [astro-ph/0011484]

work page Pith review arXiv 2001
[4]

Lemaˆ ıtre,L’Univers en expansion, Annales de la Soci´ et´ e Scientiﬁque de Bruxelles53 (1933)

G. Lemaˆ ıtre,L’Univers en expansion, Annales de la Soci´ et´ e Scientiﬁque de Bruxelles53 (1933)

work page 1933
[5]

R. C. Tolman, Eﬀect of Inhomogeneity on Cosmological Models , Proceedings of the National Academy of Science 20 (Mar., 1934) 169–176

work page 1934
[6]

Bondi, Spherically symmetrical models in general relativity , Mon

H. Bondi, Spherically symmetrical models in general relativity , Mon. Not. Roy. Astron. Soc. 107 (1947) 410–425

work page 1947
[7]

Alnes and M

H. Alnes and M. Amarzguioui, CMB anisotropies seen by an oﬀ-center observer in a spherically symmetric inhomogeneous Universe, Phys. Rev. D74 (2006) 103520, [ astro-ph/0607334]

arXiv 2006

Show all 44 references

[8]

Alnes, M

H. Alnes, M. Amarzguioui, and O. Gron, An inhomogeneous alternative to dark energy? , Phys. Rev. D73 (2006) 083519, [ astro-ph/0512006]

2006 arXiv
[9]

Fanizza and F

G. Fanizza and F. Nugier, Lensing in the geodesic light-cone coordinates and its (exact) illustration to an oﬀ-center observer in Lemaˆ ıtre-Tolman-Bondi models, JCAP 1502 (2015), no. 02 002, [ arXiv:1408.1604]

2015 arXiv
[10]

Cusin, C

G. Cusin, C. Pitrou, and J.-P. Uzan, Are we living near the center of a local void? , JCAP 1703 (2017), no. 03 038, [ arXiv:1609.02061]

2017 arXiv
[11]

J. P. Zibin and A. Moss, Linear kinetic Sunyaev-Zel’dovich eﬀect and void models for acceleration, Class. Quant. Grav. 28 (2011) 164005, [ arXiv:1105.0909]

2011 arXiv
[12]

Zhang and A

P. Zhang and A. Stebbins, Conﬁrmation of the Copernican Principle at Gpc Radial Scale and above from the Kinetic Sunyaev Zel’dovich Eﬀect Power Spectrum , Phys. Rev. Lett. 107 (2011) 041301, [arXiv:1009.3967]

2011 arXiv
[13]

P. Bull, T. Clifton, and P. G. Ferreira, The kSZ eﬀect as a test of general radial inhomogeneity in LTB cosmology, Phys. Rev. D85 (2012) 024002, [ arXiv:1108.2222]

2012 arXiv
[14]

Chluba, Tests of the CMB temperature?redshift relation, CMB spectral distortions and why adiabatic photon production is hard , Mon

J. Chluba, Tests of the CMB temperature?redshift relation, CMB spectral distortions and why adiabatic photon production is hard , Mon. Not. Roy. Astron. Soc. 443 (2014), no. 3 1881–1888, [arXiv:1405.1277]

2014 arXiv
[15]

Valkenburg, V

W. Valkenburg, V. Marra, and C. Clarkson, Testing the Copernican principle by constraining spatial homogeneity, Mon. Not. Roy. Astron. Soc. 438 (2014) L6–L10, [ arXiv:1209.4078]

2014 arXiv
[16]

Valkenburg, Complete solutions to the metric of spherically collapsing dust in an expanding spacetime with a cosmological constant, Gen

W. Valkenburg, Complete solutions to the metric of spherically collapsing dust in an expanding spacetime with a cosmological constant, Gen. Rel. Grav. 44 (2012) 2449–2476, [arXiv:1104.1082]

2012 arXiv
[17]

Valkenburg, Perceiving the equation of state of Dark Energy while living in a Cold Spot , JCAP 1201 (2012) 047, [ arXiv:1106.6042]

W. Valkenburg, Perceiving the equation of state of Dark Energy while living in a Cold Spot , JCAP 1201 (2012) 047, [ arXiv:1106.6042]

2012 arXiv
[18]

A. E. Romano, M. Sasaki, and A. A. Starobinsky, Eﬀects of inhomogeneities on apparent cosmological observables: ’fake” evolving dark energy , Eur. Phys. J. C72 (2012) 2242, [arXiv:1006.4735]

2012 arXiv
[19]

Nistane, G

V. Nistane, G. Cusin, and M. Kunz, CMB sky for an oﬀ-center observer in a local void II: observability, In Prep. (2019)

2019
[20]

Hajian and T

A. Hajian and T. Souradeep, Measuring statistical isotropy of the CMB anisotropy , Astrophys. J. 597 (2003) L5–L8, [ astro-ph/0308001]. – 21 –

2003 arXiv
[21]

Hajian and T

A. Hajian and T. Souradeep, The Cosmic microwave background bipolar power spectrum: Basic formalism and applications , astro-ph/0501001

arXiv
[22]

Basak, A

S. Basak, A. Hajian, and T. Souradeep, Statistical isotropy of cmb polarization maps , Phys. Rev. D74 (2006) 021301, [ astro-ph/0603406]

2006 arXiv
[23]

A. R. Pullen and M. Kamionkowski, Cosmic Microwave Background Statistics for a Direction-Dependent Primordial Power Spectrum, Phys. Rev. D76 (2007) 103529, [arXiv:0709.1144]

2007 arXiv
[24]

L. G. Book, M. Kamionkowski, and T. Souradeep, Odd-parity bipolar spherical harmonics , Phys.Rev.D 85 (Jan., 2012) 023010, [ arXiv:1109.2910]

2012 arXiv
[25]

A. F. Heavens and A. N. Taylor, A Spherical Harmonic Analysis of Redshift Space , Mon. Not. Roy. Astron. Soc. 275 (1995) 483–497, [ astro-ph/9409027]

1995 arXiv
[26]

A. J. S. Hamilton and M. Culhane, Spherical redshift distortions, Mon. Not. Roy. Astron. Soc. 278 (1996) 73, [ astro-ph/9507021]

1996 arXiv
[27]

A. S. Szalay, T. Matsubara, and S. D. Landy, Redshift space distortions of the correlation function in wide angle galaxy surveys , Astrophys. J. 498 (1998) L1, [ astro-ph/9712007]

1998 arXiv
[28]

Szapudi, Wide angle redshift distortions revisited , Astrophys

I. Szapudi, Wide angle redshift distortions revisited , Astrophys. J. 614 (2004) 51–55, [astro-ph/0404477]

2004 arXiv
[29]

Papai and I

P. Papai and I. Szapudi, Non-Perturbative Eﬀects of Geometry in Wide-Angle Redshift Distortions, Mon. Not. Roy. Astron. Soc. 389 (2008) 292, [ arXiv:0802.2940]

2008 arXiv
[30]

Bertacca, R

D. Bertacca, R. Maartens, A. Raccanelli, and C. Clarkson, Beyond the plane-parallel and Newtonian approach: Wide-angle redshift distortions and convergence in general relativity , JCAP 1210 (2012) 025, [ arXiv:1205.5221]

2012 arXiv
[31]

Shiraishi, N

M. Shiraishi, N. S. Sugiyama, and T. Okumura, Polypolar spherical harmonic decomposition of galaxy correlators in redshift space: Toward testing cosmic rotational symmetry , Phys. Rev. D95 (2017), no. 6 063508, [ arXiv:1612.02645]

2017 arXiv
[32]

Bartolo, A

N. Bartolo, A. Kehagias, M. Liguori, A. Riotto, M. Shiraishi, and V. Tansella, Detecting higher spin ﬁelds through statistical anisotropy in the CMB and galaxy power spectra , Phys. Rev. D97 (2018), no. 2 023503, [ arXiv:1709.05695]

2018 arXiv
[33]

N. S. Sugiyama, M. Shiraishi, and T. Okumura, Limits on statistical anisotropy from BOSS DR12 galaxies using bipolar spherical harmonics , arXiv:1704.02868

arXiv
[34]

Tansella, C

V. Tansella, C. Bonvin, G. Cusin, R. Durrer, M. Kunz, and I. Sawicki, Redshift-space distortions from vector perturbations II: Anisotropic signal , Phys. Rev. D98 (2018), no. 10 103515, [arXiv:1807.00731]

2018 arXiv
[35]

Garcia-Bellido and T

J. Garcia-Bellido and T. Haugboelle, Confronting Lemaˆ ıtre-Tolman-Bondi models with Observational Cosmology, JCAP 0804 (2008) 003, [ arXiv:0802.1523]

2008 arXiv
[36]

Biswas, A

T. Biswas, A. Notari, and W. Valkenburg, Testing the Void against Cosmological data: ﬁtting CMB, BAO, SN and H0 , JCAP 1011 (2010) 030, [ arXiv:1007.3065]

2010 arXiv
[37]

Enqvist, Lemaˆ ıtre-Tolman-Bondi model and accelerating expansion, Gen

K. Enqvist, Lemaˆ ıtre-Tolman-Bondi model and accelerating expansion, Gen. Rel. Grav. 40 (2008) 451–466, [ arXiv:0709.2044]

2008 arXiv
[38]

J. W. Moﬀat, Cosmic microwave background, accelerating Universe and inhomogeneous cosmology, JCAP 0510 (2005) 012, [ astro-ph/0502110]

2005 arXiv
[39]

Clarkson, B

C. Clarkson, B. Bassett, and T. H.-C. Lu, A general test of the Copernican Principle , Phys. Rev. Lett. 101 (2008) 011301, [ arXiv:0712.3457]

2008 arXiv
[40]

Nadathur and S

S. Nadathur and S. Sarkar, Reconciling the local void with the CMB , Phys. Rev. D83 (2011) 063506, [arXiv:1012.3460]. – 22 –

2011 arXiv
[41]

Lesgourgues, The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview , arXiv:1104.2932

J. Lesgourgues, The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview , arXiv:1104.2932

arXiv
[42]

D. Blas, J. Lesgourgues, and T. Tram, The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes , JCAP 1107 (2011) 034, [ arXiv:1104.2933]

2011 arXiv
[44]

Aghanim et al., Planck 2018 results

Planck Collaboration, N. Aghanim et al., Planck 2018 results. V. CMB power spectra and likelihoods, arXiv:1907.12875. – 23 –

2018 arXiv

Pith tools

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

[1] [43]

Amendola, R

L. Amendola, R. Catena, I. Masina, A. Notari, M. Quartin, and C. Quercellini, Measuring our peculiar velocity on the CMB with high-multipole oﬀ-diagonal correlations , JCAP 1107 (2011) 027, [arXiv:1008.1183]

arXiv 2011

[2] [1]

Distances and lensing in cosmological void models

K. Tomita, Distances and lensing in cosmological void models , Astrophys. J. 529 (2000) 38, [astro-ph/9906027]

work page Pith review arXiv 2000

[3] [2]

Analyses of Type Ia Supernova Data in Cosmological Models with a Local Void

K. Tomita, Analyses of type Ia supernova data in cosmological models with a local void , Prog. Theor. Phys. 106 (2001) 929–939, [ astro-ph/0104141]

work page Pith review arXiv 2001

[4] [3]

A Local Void and the Accelerating Universe

K. Tomita, A local void and the accelerating universe , Mon. Not. Roy. Astron. Soc. 326 (2001) 287, [astro-ph/0011484]

work page Pith review arXiv 2001

[5] [4]

Lemaˆ ıtre,L’Univers en expansion, Annales de la Soci´ et´ e Scientiﬁque de Bruxelles53 (1933)

G. Lemaˆ ıtre,L’Univers en expansion, Annales de la Soci´ et´ e Scientiﬁque de Bruxelles53 (1933)

work page 1933

[6] [5]

R. C. Tolman, Eﬀect of Inhomogeneity on Cosmological Models , Proceedings of the National Academy of Science 20 (Mar., 1934) 169–176

work page 1934

[7] [6]

Bondi, Spherically symmetrical models in general relativity , Mon

H. Bondi, Spherically symmetrical models in general relativity , Mon. Not. Roy. Astron. Soc. 107 (1947) 410–425

work page 1947

[8] [7]

Alnes and M

H. Alnes and M. Amarzguioui, CMB anisotropies seen by an oﬀ-center observer in a spherically symmetric inhomogeneous Universe, Phys. Rev. D74 (2006) 103520, [ astro-ph/0607334]

arXiv 2006