Pith. sign in

REVIEW 2 major objections 3 minor 2 cited by

Galaxy number counts at second order: an independent approach

T0 review · 2 major / 3 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 This paper derives the full second-order relativistic galaxy number-count overdensity in a flat, matter-dominated FLRW universe, assembling the redshift, angular diameter distance, and survey volume corrections needed for percent-level…

desk verdict The derivation is serious and the linear-order check is good, but Eq. (4.23) drops the background expansion term in the linearized Sachs shear equation, so the advertised second-order number counts in Eq. (5.17) are not reliable as written. read the letter →

arxiv 1908.08400 v4 pith:L7YECPTK submitted 2019-08-22 astro-ph.CO gr-qc

Jorge L. Fuentes , Juan Carlos Hidalgo , Karim A. Malik This is my paper · ORCID
classification astro-ph.COgr-qc PACS 98.80.Cq
keywords galaxynumbercountscosmologicalperturbationtheorysecondorderangulardiameterdistanceSachsequationlensinglarge-scalestructurerelativisticredshift
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

This paper computes the observed galaxy number-count overdensity up to second order in cosmological perturbation theory, an expression long and intricate enough that earlier groups disagree at this order. The authors build the observable from first principles in a flat universe filled with pressureless matter: the second-order redshift perturbations, the angular diameter distance from the Sachs optical equations, and the physical volume element of a survey. They carry out the distance calculation in a conformally mapped perturbed Minkowski background and convert final results from affine parameter to observed redshift. The main result is Eq. (5.17), $\Delta_g^{(2)}$, which, if correct, gives the complete relativistic correction up to second order and provides an independent cross-check of previously published second-order expressions.

What carries the argument

The load-bearing machinery is the Sachs propagation equations for a bundle of null geodesics, which relate the expansion of the bundle to the Ricci and Weyl terms and give the angular diameter distance through $d^2 d_A/d\lambda^2 = -\frac12(R_{\mu\nu}k^\mu k^\nu + \Sigma_{\mu\nu}\Sigma^{\mu\nu})d_A$. The paper supplements these with the Kristian\,–\,Sachs series expansion of $d_A^2$, which supplies the boundary conditions $d_A(\lambda_o)=0$ and $d_A'(\lambda_o)=-E_o$. To simplify integrations, a conformal transformation maps the perturbed FLRW metric to a perturbed Minkowski metric, where the background distance is simply $\lambda_o-\lambda_s$ and the area distance maps back as $\hat{d}_A=a\,d_A$. The volume element $dV=-E\,d_A^2\,d\lambda\,d\Omega$ then combines energy and distance, and perturbative inversion of the redshift\,–\,affine-parameter relation converts the final result into redshift space.

What would settle it

Evaluate the paper's Eq. (5.17) side by side with the independent second-order number-count results of Refs. [13], [14], [15], and [17] in an Einstein\,–\,de Sitter universe, translating all expressions to a common gauge and notation; any residual second-order difference would show that the claimed full expression is not gauge-complete. Alternatively, recompute the angular diameter distance in a model with a deliberately non-zero first-order shear at the observer and check whether $\Delta_g^{(2)}$ shifts.

Watch

Extended reading notes

Core claim

The central claim is that Eq. (5.17) is the full second-order galaxy number-count overdensity $\Delta_g^{(2)}$ in a flat FLRW universe with pressureless matter and scalar perturbations, allowing non-zero anisotropic stress. The derivation proceeds along an independent route: instead of computing luminosity distance, the authors use the volume measure $dV = -E\,d_A^2\,d\lambda\,d\Omega$, with the angular diameter distance $d_A$ obtained from the Sachs propagation equations. After a conformal transformation to a Minkowski background, they integrate the null geodesic equations, obtain the second-order redshift, distance, and volume, and then invert the affine parameter to observed redshift so that the overdensity is expressed in terms of observable quantities. The paper verifies that the linear-order limit of its expression agrees exactly with the established results of Refs. [13], [14], [15], [17], and the linear-order literature, while the full second-order comparison is left to a follow-up paper.

Load-bearing premise

The derivation assumes that setting the first-order shear perturbation of the light bundle to zero at the observer's position is a free choice, so the integrated shear in the Sachs equation carries no observer-side boundary term; if that choice is not genuinely general, the final number-count expression changes.

Editorial extensions

If this is right

  • The expression can be evaluated directly with metric potentials from field equations or simulations, giving predicted number counts as a function of line-of-sight position for future surveys.
  • Surveys reaching percent-level precision can test the relativistic corrections beyond linear order, since lensing, Doppler, and integrated Sachs\,–\,Wolfe contributions now appear consistently at second order.
  • Because the derivation is independent of the luminosity-distance route used by other groups, it provides a cross-check that can isolate where earlier second-order disagreements originate.
  • The result is expressed directly in observed redshift and comoving distance, so it can be implemented in survey analysis pipelines without further gauge translation.

Reading between the lines

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

  • Inference: the validity of setting the first-order shear at the observer to zero is testable; a non-zero boundary value would alter the integrated shear entering the angular diameter distance and hence Eq. (5.17), so the paper's claim of general validity rests on this choice being truly free.
  • Inference: the deferred comparison against the independent second-order results of Refs. [13], [14], [15], and [17] is the decisive checkpoint, because if those expressions do not match after a common gauge and variable translation, the difference will point to whether the redshift, distance, volume, or $\lambda$-to-$z$ conversion step contains the discrepancy.
  • Inference: the affine-parameter formulation suggests a numerical evaluation along simulated light cones without first inverting $\lambda(z)$, which would yield a direct observational test of the second-order corrections.
  • Inference: one could construct a toy model with a controlled non-zero observer shear to see how much $\Delta_g^{(2)}$ shifts, providing a quantitative measure of the boundary condition's impact on next-generation survey predictions.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper presents an independent derivation of the relativistic galaxy number-count overdensity to second order in cosmological perturbation theory. Working in a flat FLRW universe with pressureless matter and scalar perturbations in the longitudinal gauge, the authors compute the perturbed null geodesics, the observed redshift, the angular diameter distance (via a conformal transformation to a perturbed Minkowski spacetime), and the survey volume element, and combine these into the second-order galaxy overdensity Delta_g^(2) in Eq. (5.17). The linear-order result is compared with the works of Di Dio et al., Bertacca et al., and Yoo and Zaldarriaga, and found to agree. A comparison at second order is explicitly deferred to a companion paper, Ref. [37]. The tensor algebra was performed with xAct and xPand.

Significance. If the second-order expression were correct, the paper would provide a useful independent derivation of a quantity needed for next-generation surveys, and the use of the affine parameter as the integration variable is a genuine methodological difference from most earlier work. The linear-order agreement with three independent groups is a strong consistency check, and the derivation is self-contained rather than fitted to any target. However, the central second-order result is not verified against any existing calculation, and the derivation contains a serious error in the linearized null-shear equation that propagates directly into Eq. (5.17). The paper's contribution therefore remains potentially valuable but is not reliable as it stands.

major comments (2)
  1. [Section IV, Eqs. (4.22)-(4.24)] The linearized Sachs shear equation is missing the background expansion term. In the conformal Minkowski background, a bundle focused at the observer has theta_bar = -2/(lambda_o - lambda), so linearizing Eq. (4.6) gives d delta^(1)Sigma_ij/dlambda = -theta_bar delta^(1)Sigma_ij + S_ij = [2/(lambda_o - lambda)] delta^(1)Sigma_ij + S_ij, not d delta^(1)Sigma_ij/dlambda = S_ij as written in Eq. (4.22). The regular solution with delta^(1)Sigma_ij(lambda_o)=0 is delta^(1)Sigma_ij(lambda_s) = -(lambda_o - lambda_s)^{-2} integral_{lambda_s}^{lambda_o} (lambda_o - lambda)^2 S_ij(lambda) dlambda, which for constant trace-free S_ij equals -S_ij (lambda_o - lambda_s)/3 rather than the S_ij (lambda_o - lambda_s) of Eq. (4.23). The squared contraction in Eq. (4.24) is therefore wrong by a factor of 9. Since delta^(1)Sigma delta^(1)Sigma enters the second-order area-distance equation (4.21), the error propagates through Eq. (C9), the volume element (C10), and the main result Eq. (5.17). All downstream expressions must be recomputed with the corrected shear.
  2. [Section VII and Eq. (5.17)] The central second-order result is not compared with any existing second-order calculation; the authors state in Section VII that a full comparison is left to the companion paper, Ref. [37]. Given that the paper itself notes that previous second-order results are in disagreement, and given the shear error identified above, Eq. (5.17) is currently an unverified expression at exactly the order at which the paper claims novelty. The authors should provide the second-order comparison, or at minimum a check in a simplified limit (for example Einstein-de Sitter with no anisotropic stress), before the main claim can be reliably assessed.
minor comments (3)
  1. [Section IV, near Eq. (4.23)] The phrase 'Without loss of generality, we set the perturbation of the shear at the observer delta^(1)Sigma_mu_nu|_o = 0' is misleading: for a bundle focused at the observer, this is the regular caustic boundary condition, not a free gauge choice. With the corrected linearized equation, this condition selects the unique regular solution.
  2. [Eq. (5.17)] After the change of variable to the comoving distance chi, several integrals still have upper limit lambda_s (for example the terms written as integral_0^{lambda_s} dchi [Phi_1 (dPhi_1/dzeta - 2 dPsi_1/dzeta)]); the upper limit should be chi_s for dimensional consistency.
  3. [Appendix C, Eq. (C9)] The nested integrals in Eq. (C9) are extremely difficult to verify by eye; providing the xAct/xPand notebook or an intermediate derivation step for the shear contribution would substantially help referees and readers.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the second-order galaxy overdensity is derived directly from standard geodesic, Sachs, and volume relations, with no fitted parameters and no load-bearing self-citation.

full rationale

The derivation is self-contained. The central result, Eq. (5.17), is obtained by substituting previously derived pieces, Eq. (C9) for the second-order angular diameter distance, Eq. (C10) for the second-order volume perturbation, Eq. (C11) for the redshift-space density perturbation, and the second-order redshift (C8), into the exact identity Δ_g = δ_z + δV/V (5.7)–(5.9). Each of these pieces is computed from the perturbed metric (2.1), the null geodesic equations (3.4)–(3.5), (C1)–(C4), and the Sachs equations (4.5)–(4.7). No parameter is fitted to data, and no target expression is used as an input. The only self-referential element is Ref. [37], the planned companion paper, cited twice (Section VI and Section VII) for the deferred second-order comparison; this comparison is not used anywhere in the derivation, so it is not load-bearing. The first-order result is checked against the independent external results of Di Dio et al., Bertacca et al., and Yoo & Zaldarriaga (Section VI), which confirms that the pipeline reproduces a known result rather than being defined to match it. The boundary condition δ(1)Σ_μν|_o = 0 after Eq. (4.22) is an assumption about the observer shear, and the skeptic's complaint that the integrated shear solution in Eq. (4.23) omits the background expansion term in the linearized Sachs equation is a physical-correctness concern, not a circularity: the final expression is not wired to equal that assumption by construction. Likewise, the acknowledged absence of a full second-order comparison to other groups (explicitly deferred to the companion paper) is a completeness limitation and not an instance of the derivation reducing to its own inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The derivation uses standard general relativity and cosmological perturbation theory, with no free parameters fitted to data. The main physical assumptions are the FLRW background, pressureless matter, scalar perturbations, and the neglect of galaxy bias. The shear boundary condition at the observer is a non-generic choice that may affect second-order results, and the paper does not justify it fully.

assumptions (6)
  • domain assumption FLRW background with flat spatial curvature and a single pressureless matter component (rho proportional to (1+z)^3)
    Stated in Section VII and used to set the background metric (Eq. 2.1) and density evolution (Eq. 5.11).
  • domain assumption Only scalar metric perturbations in longitudinal gauge; vector and tensor modes are neglected
    Stated in Section II A, after Eq. (2.1).
  • standard math Photon propagation follows null geodesics and the Sachs optical equations
    Used in Sections III and IV to compute redshift, distance, and volume.
  • standard math The conformal transformation maps null geodesics and sets hat_d_A = a d_A and d_hat_lambda = a^2 d_lambda
    Invoked in Section IV to simplify the angular diameter distance calculation; standard result in conformal GR.
  • ad hoc to paper The shear perturbation at the observer is set to zero without loss of generality (delta^(1)Sigma_mu_nu|o = 0)
    Introduced in Section IV near Eq. (4.23); not proven to be a general gauge choice and could affect the second-order area distance.
  • domain assumption Galaxy bias and evolution bias are omitted; the computed quantity is the matter density perturbation, not the biased galaxy density
    The authors compare with Bertacca et al. by omitting the evolution bias be (Section VI B); the final expression (5.17) contains no bias parameters.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Galaxy number counts at second order: an independent approach." pith.science (2026). https://pith.science/paper/L7YECPTK

@misc{pith2026190808400,
  author       = {Pith},
  title        = {Pith review of: Galaxy number counts at second order: an independent approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L7YECPTK}},
  note         = {Machine review of arXiv:1908.08400}
}
read the original abstract

Next generation surveys will be capable of determining cosmological parameters beyond percent level. To match this precision, theoretical descriptions should look beyond the linear perturbations to approximate the observables in large scale structure. A quantity of interest is the Number density of galaxies detected by our instruments. This has been focus of interest recently, and several efforts have been made to explain relativistic effects theoretically, thereby testing the full theory. However, the results at nonlinear level from previous works are in disagreement. We present a new and independent approach to computing the relativistic galaxy number counts to second order in cosmological perturbation theory. We derive analytical expressions for the full second order relativistic observed redshift, for the angular diameter distance and for the volume spanned by a survey. Finally, we compare our results with previous works which compute the general distance-redshift relation, finding that our result is in agreement at linear order.

Figures

Figures reproduced from arXiv: 1908.08400 by the authors.

Figure 1
Figure 1. FIG. 1: Affine parameter convention of a light ray in a radio observation. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The lightray bundle going from the source, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: When a light-ray bundle travelling from a source, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Volume corresponding to an infinitesimal change in the affine parameter from [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Observable Gravitational Wave Strain at Second Order

    gr-qc 2025-12 conditional novelty 7.0 of 10

    At second order, the gravitational-wave strain measured by geodesic observers exchanging light pulses is the transverse-traceless metric perturbation in the Newton gauge (h_N^(2)).

  2. Tetrad formalism for exact cosmological observables

    astro-ph.CO 2019-08 accept novelty 7.0 of 10

    The paper derives exact, coordinate-independent equations for cosmological observables on a new 'observer space-time' manifold, using tetrads to track the observer frame.

Reference graph

Works this paper leans on

45 extracted references · 25 canonical work pages · cited by 2 Pith papers

  1. [37]

    cosmic rulers

    we perform a comparison of the leading terms of the second order expansion of the galaxy number counts. 20 Our result, as given in Eq. (5.7) is ∆(1) g (ni,z ) = [δ(1)ρ(ni,z ) ¯ρ(z) + 3Φ1 ] − 2 (Φ1 + Ψ1) + 1 H ( Ψ1 ′−∂χΦ1 + d ( v1ini) dς ) + (H′ H2 + 2 Hχ )[ Φ1− ( v1ini) + ∫ χ 0 d˜χ ( Φ1 ′ + Ψ1 ′)] − 4 χ ∫ χ 0 d˜χΨ1 − 1 χ ∫ χ 0 d˜χ (˜χ−χ) ˜χ [ ∇2 (Φ1 + Ψ1)...

    work page 2016

  2. [1]

    Geodesic Equation Solving Eq. (2.8) at second order gives dδ(2)ν dλ =−1 2 [dΦ2 dλ + dΨ2 dλ ] + 2dδ(1)n i dλ δ(1)ni− 2dδ(1)ν dλ δ(1)ν− dδ(2)n i dλ ni (C1) − 4dδ(1)ν dλ [ Φ1 + Ψ1 ] − 4δ(1)ν [dΦ1 dλ + dΨ1 dλ ] − 4Φ1 [dΦ1 dλ + dΨ1 dλ ] − 4dΦ1 dλ [ Φ1 + Ψ1 ] , dδ(2)n i dλ = 2dΨ2 dλ ni− [ Φ2 i , + Ψ2 i , ] − 4Φ1Ψ1 ′ni + 4δ(1)n i Ψ1 ′ + 4Φ1Φ1 i , (C2) − 2δ(1)n j...

    work page

  3. [2]

    Energy The perturbed energy in terms of the metric potentials is given by δ(1)E = ( −2Φ1 ⏐⏐⏐ s o + ∫ λs λo ( Φ1 ′ + Ψ1 ′) dλ ) + Φ1−v1ini, (C5) δ(2)E = 1 2 [ 2 {( −2Φ1 ⏐⏐⏐ s o + ∫ λs λo ( Φ1 ′ + Ψ1 ′) dλ ) + Φ1−v1ini } Φ1 (C6) − Φ1 2− 5Ψ1 2− 6Φ1Ψ1 + 1 2 (Φ2− Ψ2)−niv2i − 4 (Φ1 + Ψ1) Φ1 ⏐⏐⏐ s o + 2 (Φ1 + Ψ1) ∫ λs λo ( Φ1 ′ + Ψ1 ′) dλ − 4niv1iΨ1 ⏐⏐⏐ s o + 2v...

    work page

  4. [3]

    Observed Redshift At second order the redshift is δ(2)z = [ Φ1 2 + ( v1ini)2 − 2Φ1 ( v1ini)] o − Φ1|oδ(1)ν|s +δ(1)ν|s ( v1ini) o− Φ1|sΦ1|o (C7) + Φ1|s ( v1ini) o + ( v1ini) s Φ1|o− ( v1ini) s ( v1ini) o + 1 2 [ 2δ(1)EΦ1− Φ1 2− 5Ψ1 2 − 6Φ1Ψ1 +δ(2)ν + 1 2 (Φ2− Ψ2) + 2δ(1)ν (Φ1 + Ψ1)− 2δ(1)n i v1i−niv2i ] s − 1 2 [ Φ1 2− 5Ψ1 2− 6Φ1Ψ1 + 1 2 (Φ2− Ψ2)− (v1ini)Φ...

    work page

  5. [4]

    Angular Diameter Distance Using Eqs. (3.4), (3.5), (C3), (C4), (C6), (4.18), (B1), (B2), (B3) and (4.24) we find that the second order pertur- bation to the angular diameter distance becomes δ(2)dA(λs) ¯dA(λs) = 1 2 [ Φ1 2− 5Ψ1 2− 6Φ1Ψ1 + 1 2 (Φ2− Ψ2)− (v1ini)Φ1−niv2i ] o (C9) − 1 λo−λs ∫ λs λo dλ ∫ λs λs d˜λ { 4 ( −2Φ1 ⏐⏐⏐ s s + ∫ λs λo ( Φ1 ′ + Ψ1 ′) dλ ...

    work page

  6. [5]

    Physical Volume The second order perturbation to the physical volume is dδ(2)V =− ¯E ¯d2 A [(δ(1)dA ¯dA )2 + (δ(1)E ¯E )(δ(1)dA ¯dA ) + δ(2)dA ¯dA + 1 2 δ(2)E ¯E ] dλdΩ (C10) =a2 (λs) (λs−λo)2 {( Φ1|o− (v1ini)o− Φ1|s o− 3 2 (Ψ1− Φ1) ⏐⏐⏐ s o + ∫ λs λo dλ ∫ λs λs d˜λ { (Φ1 + Ψ1)′′−∇ 2 (Φ1 + Ψ1) } − 1 λo−λs [ 2 ∫ λs λo dλ ∫ λs λs d˜λ (Φ1 + Ψ1)′− ∫ λs λo dλ ∫...

    work page

  7. [6]

    (3.14), (C7) and (5.15) in Eq

    Redshift Density Using Eqs. (3.14), (C7) and (5.15) in Eq. (5.10) we find that the redshift density perturbation at second order is given by δ(2) z (ni,z ) = 1 2 δ(2)ρ(ni,z ) ¯ρ(z) + 3 2(1 + ¯z)δ(2)z(ni,z ) + 3 (1 +z)2 [ δ(1)z(ni,z ) ]2 (C11) = 1 2 δ(2)ρ(ni,z ) ¯ρ(z) + 3 2(1 + ¯z) ([ Φ1 2 + ( v1ini)2 − 2Φ1 ( v1ini)] o − Φ1|o [ −2Φ1 + ∫ λs λo ( Φ1 ′ + Ψ1 ′)...

    work page

  8. [7]

    Green et al., ArXiv e-prints (2012), 1208.4012

    J. Green et al., ArXiv e-prints (2012), 1208.4012

    arXiv 2012

Show all 45 references
  1. [8]

    The BOSS Collaboration, The Astronomical Journal 145, 10 (2013), 1208.0022. 35

    2013 arXiv

  2. [9]

    Zhao et al., Mon

    G.-B. Zhao et al., Mon. Not. Roy. Astron. Soc. 457, 2377 (2016), 1510.08216

    2016 arXiv

  3. [10]

    Scaramella et al

    R. Scaramella et al. (Euclid), IAU Symp. 306, 375 (2015), 1501.04908

    2015 arXiv

  4. [11]

    M. G. Santos, M. Cluver, M. Hilton, M. Jarvis, G. I. G. Jozsa, L. Leeuw, O. Smirnov, R. Taylor, F. Abdalla, J. Afonso, et al., Meerklass: Meerkat large area synoptic survey (2017), 1709.06099

    2017 arXiv

  5. [12]

    Maartens, F

    R. Maartens, F. B. Abdalla, M. Jarvis, and M. G. Santos, Cosmology with the ska – overview (2015), 1501.04076

    2015 arXiv

  6. [13]

    L. S. Collaboration, P. A. Abell, J. Allison, S. F. Anderson, J. R. Andrew, J. R. P. Angel, L. Armus, D. Arnett, S. J. Asztalos, T. S. Axelrod, et al., Lsst science book, version 2.0 (2009), 0912.0201

    2009 arXiv

  7. [14]

    Bertacca, R

    D. Bertacca, R. Maartens, and C. Clarkson, Journal of Cosmology and Astroparticle Physics 2014, 013 (2014), URL http://stacks.iop.org/1475-7516/2014/i=11/a=013

    2014

  8. [15]

    Jeong, F

    D. Jeong, F. Schmidt, and C. M. Hirata, Phys. Rev. D85, 023504 (2012), URL https://journals.aps.org/prd/abstract/ 10.1103/PhysRevD.85.023504

    2012 doi

  9. [16]

    Bonvin and R

    C. Bonvin and R. Durrer, Phys. Rev. D 84, 063505 (2011), URL https://journals.aps.org/prd/abstract/10.1103/ PhysRevD.84.063505

    2011

  10. [17]

    Challinor and A

    A. Challinor and A. Lewis, Phys. Rev. D 84, 043516 (2011), URL https://journals.aps.org/prd/abstract/10.1103/ PhysRevD.84.043516

    2011

  11. [18]

    Chen and D

    S. Chen and D. J. Schwarz, Phys. Rev. D91, 043507 (2015), 1407.4682

    2015 arXiv

  12. [19]

    J. Yoo, A. L. Fitzpatrick, and M. Zaldarriaga, Phys. Rev. D 80, 083514 (2009), 0907.0707

    2009 arXiv

  13. [20]

    Bertacca, R

    D. Bertacca, R. Maartens, and C. Clarkson, Journal of Cosmology and Astroparticle Physics 2014, 037 (2014), URL http://stacks.iop.org/1475-7516/2014/i=09/a=037

    2014

  14. [21]

    Marozzi, Class

    G. Marozzi, Class. Quant. Grav. 32, 045004 (2015), [erratum: Class. Quant. Grav.32,179501(2015)], 1406.1135

    2015 arXiv

  15. [22]

    Di Dio, R

    E. Di Dio, R. Durrer, G. Marozzi, and F. Montanari, JCAP 1412, 017 (2014), [Erratum: JCAP1506,no.06,E01(2015)], 1407.0376

    2014 arXiv

  16. [23]

    Yoo, Phys

    J. Yoo, Phys. Rev. D 82, 083508 (2010), URL http://link.aps.org/doi/10.1103/PhysRevD.82.083508

    2010 doi

  17. [24]

    Yoo and M

    J. Yoo and M. Zaldarriaga, Phys. Rev. D90, 023513 (2014), 1406.4140

    2014 arXiv

  18. [25]

    Yoo and F

    J. Yoo and F. Scaccabarozzi, JCAP 1609, 046 (2016), 1606.08453

    2016 arXiv

  19. [26]

    Ben-Dayan, G

    I. Ben-Dayan, G. Marozzi, F. Nugier, and G. Veneziano, JCAP 1211, 045 (2012), 1209.4326

    2012 arXiv

  20. [27]

    Fanizza, M

    G. Fanizza, M. Gasperini, G. Marozzi, and G. Veneziano, JCAP 1311, 019 (2013), 1308.4935

    2013 arXiv

  21. [28]

    Durrer, The Cosmic Microwave Background (Cambridge University Press, 2008), ISBN 9780521847049, URL https: //books.google.co.uk/books?id=WZkN1QDOD1sC

    R. Durrer, The Cosmic Microwave Background (Cambridge University Press, 2008), ISBN 9780521847049, URL https: //books.google.co.uk/books?id=WZkN1QDOD1sC

    2008

  22. [29]

    J. T. Nielsen and R. Durrer, JCAP 1703, 010 (2017), 1606.02113

    2017 arXiv

  23. [30]

    Fanizza, J

    G. Fanizza, J. Yoo, and S. G. Biern (2018), 1805.05959

    2018 arXiv

  24. [31]

    Yoo, Class

    J. Yoo, Class. Quant. Grav. 31, 234001 (2014), 1409.3223

    2014 arXiv

  25. [32]

    Ellis, R

    G. Ellis, R. Maartens, and M. MacCallum, Relativistic Cosmology, Relativistic Cosmology (Cambridge University Press, 2012), ISBN 9780521381154, URL https://books.google.co.uk/books?id=FPRFi3L0h9kC

    2012

  26. [33]

    K. A. Malik and D. Wands, Physics Reports 475, 1 (2009), ISSN 0370-1573, URL http://www.sciencedirect.com/ science/article/pii/S0370157309000635

    2009

  27. [34]

    O. Umeh, C. Clarkson, and R. Maartens, Classical and Quantum Gravity 31, 205001 (2014), URL http://stacks.iop. org/0264-9381/31/i=20/a=205001

    2014

  28. [35]

    Clarkson, G

    C. Clarkson, G. F. R. Ellis, A. Faltenbacher, R. Maartens, O. Umeh, and J.-P. Uzan, Mon. Not. Roy. Astron. Soc. 426, 1121 (2012), 1109.2484

    2012 arXiv

  29. [36]

    R. K. Sachs, Proc. Roy. Soc. Lond. A264, 309 (1961)

    1961

  30. [39]

    Kristian and R

    J. Kristian and R. K. Sachs, The Astrophysical Journal 143, 379 (1966)

    1966

  31. [40]

    Ibison, J

    M. Ibison, J. Math. Phys. 48, 122501 (2007), 0704.2788

    2007 arXiv

  32. [41]

    Aretakis, Lecture notes on general relativity

    S. Aretakis, Lecture notes on general relativity. Columbia University. (2013), https://web.math.princeton.edu/ ~aretakis/columbiaGR.pdf, URL https://web.math.princeton.edu/~aretakis/columbiaGR.pdf

    2013

  33. [42]

    O. Umeh, C. Clarkson, and R. Maartens, Classical and Quantum Gravity 31, 202001 (2014), URL http://stacks.iop. org/0264-9381/31/i=20/a=202001

    2014

  34. [43]

    Campos, Proc

    A. Campos, Proc. Int. Sch. Phys. Fermi 132, 123 (1996), astro-ph/9510051

    1996 arXiv

  35. [44]

    J. L. Fuentes, J. C. Hidalgo, and K. A. Malik, In preparation (2019)

    2019

  36. [45]

    J. M. Mart´ ın-Garc´ ıa,xAct: Efficient tensor computer algebra for the Wolfram Language (2016), http://www.xact.es/, URL http://www.xact.es/

    2016

  37. [46]

    Pitrou, X

    C. Pitrou, X. Roy, and O. Umeh, Class. Quant. Grav. 30, 165002 (2013), 1302.6174

    2013 arXiv

Pith tools

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