Gravitational redshift from large-scale structure: nonlinearities, antisymmetries, and the dipole

Camille Bonvin; Lawrence Dam

REVIEW 2 major objections 4 minor 1 cited by

Gravitational redshift from large-scale structure: nonlinearities, antisymmetries, and the dipole

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A compact nonlinear streaming model reproduces the observed turnover of the gravitational-redshift dipole at roughly 20 h^-1 Mpc and traces it to the density-weighted pairwise potential difference rather than to lightcone effects.

desk verdict A careful, useful streaming-model framework for the gravitational-redshift dipole, with a convincing physical mechanism for the turnover but a validation that is partly calibrated to the same simulation it is tested against. read the letter →

arxiv 2506.22431 v2 pith:N7NHSATO submitted 2025-06-27 astro-ph.CO gr-qc

Lawrence Dam , Camille Bonvin This is my paper

classification astro-ph.COgr-qc

keywords gravitationalredshiftdipolecorrelationfunctionredshift-spacedistortionswide-angleeffectspairwisepotentialdifferencestreamingmodellightconecosmology

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 aims to provide a complete nonlinear model of the galaxy two-point correlation function in redshift space, keeping every relativistic effect of order H/k: gravitational redshift, redshift-space distortions, wide-angle effects to all orders, and lightcone and lookback-time effects. The model is a wide-angle streaming model whose displacement statistics are density-weighted cumulants of velocity and potential. Compared with N-body halo catalogues, it reproduces the observed turnover of the dipole at $s\simeq 20\,h^{-1}\mathrm{Mpc}$, a feature linear theory misses. The authors argue that the turnover is caused by a specific nonlinear shift term, $-\langle\!\langle\Delta\psi\rangle\!\rangle\,\mathrm{d}\xi_{AB}/\mathrm{d}s$, and that the key physical ingredient is the density-weighted pairwise potential difference, not the lightcone. If the model is right, it supplies the template needed to interpret upcoming gravitational-redshift dipole measurements.

What carries the argument

The machinery is the real-to-redshift radial displacement map $\delta\chi = u\cdot\hat{n} + \psi - \psi_O$, with $\psi = -H^{-1}\Psi$, inserted into a full-sky Gaussian streaming model. The transition probability $p(\chi|\chi')$ is built from density-weighted cumulants, notably the mean $m = m_{\mathrm{RSD}} + m_{\mathrm{grav}}$; decomposing $m$ under tracer exchange yields pairwise functions, of which the antisymmetric pairwise potential difference $\langle\!\langle\Delta\psi\rangle\!\rangle(r) = \langle\!\langle\psi(x_1)-\psi(x_2)\rangle\!\rangle$ carries the gravitational-redshift signal. A small-displacement expansion of the streaming integral produces a closed dipole formula, Eq. (63), whose shift term is the product of $\langle\!\langle\Delta\psi\rangle\!\rangle$ with the slope of the real-space cross-correlation. The halo model enters to supply the one-point values $\phi_A(0)$ and $\phi_B(0)$ through the one-halo term.

What would settle it

Compute the density-weighted pairwise potential difference $\langle\!\langle\Delta\psi\rangle\!\rangle$ directly from N-body halo catalogues at the same redshifts and compare it with the one-halo-plus-two-halo prediction; if the measured statistic differs significantly near $20\,h^{-1}\mathrm{Mpc}$, the claimed cause of the turnover is falsified.

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Extended reading notes

Core claim

The central claim is that a single compact formula, Eq. (27), captures all order-$H/k$ asymmetries in the redshift-space correlation function, and that the dipole turnover follows from the pairwise potential difference $\langle\!\langle\Delta\psi\rangle\!\rangle$ through an advection-like shift term. The paper shows that the mean displacement separates into a redshift-space-distortion part and a gravitational-redshift part, that only the gravitational part has a nonvanishing one-point function because density weighting prefers potential wells, and that the one-halo contribution to the pairwise potential difference dominates and is required to reproduce the turnover. In the perturbative expansion the dipole is given by Eq. (63), where the shift term $-\langle\!\langle\Delta\psi\rangle\!\rangle\,\mathrm{d}\xi_{AB}/\mathrm{d}s$ is negative and grows on small scales, canceling the positive wide-angle RSD contribution near $20\,h^{-1}\mathrm{Mpc}$. The lightcone and lookback-time effects are shown to be small and not the cause of the turnover.

Load-bearing premise

The load-bearing premise is that the halo-model estimate of the one-point potential values $\phi_A(0)$ and $\phi_B(0)$, with concentrations taken from the same catalogues used for comparison, is accurate; if this one-point function is wrong, the predicted turnover shifts in scale or disappears.

Editorial extensions

If this is right

The dipole's turnover near $20\,h^{-1}\mathrm{Mpc}$ becomes a predicted feature of the model, so future measurements can use it as a standard signature of gravitational redshift rather than an anomaly.
Because lightcone and lookback-time effects are subdominant, the turnover scale can be used to infer the density-weighted pairwise potential difference and hence the depth of potential wells at tracer positions.
The pairwise-function decomposition yields a compact formula for the wide-angle RSD dipole in terms of the pairwise mean velocity, making wide-angle corrections straightforward to add to standard streaming-model analyses.
The same formalism gives a physical origin for the empirical nonperturbative correction used in earlier quasi-linear dipole models.
The covariance contributions from gravitational redshift and velocity-potential cross-correlations are numerically negligible for the dipole, so the mean displacement supplies the dominant antisymmetric signal.

Reading between the lines

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

A testable extension the paper leaves implicit: replacing the halo-model one-point values with a direct simulation measurement of $\langle\!\langle\Delta\psi\rangle\!\rangle$ should predict the dipole with no free zero-point, sharpening forecasts for upcoming surveys.
The shift mechanism implies a second turnover near the halo-exclusion scale, where the pair probability and its slope change sign; the paper notes this expectation but leaves the quantitative prediction to future work.
The same pairwise-potential logic should generate predictable amplitudes in the octupole and higher odd multipoles, so measuring those would independently confirm that potential differences, rather than kinematics, drive the asymmetry.

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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

This paper gives the LSS community a compact streaming-model formula for the redshift-space correlation function that includes gravitational redshift, wide-angle effects, lightcone effects, and lookback time at order H/k. The genuinely new pieces are the pairwise decomposition into symmetric and antisymmetric functions and the identification of the shift term -<> d xi_AB/ds as the cause of the dipole turnover around 20 h^-1 Mpc. That is a real step forward: earlier models either used an EFT-like nuisance parameter fitted to simulations or added an ad-hoc potential correction, and this paper shows how that correction arises naturally from density weighting. The derivation is careful and self-contained. Equation (27) is compact, and the authors recover linear theory from it in a few lines. The checks that C_cross and C_grav do not matter for the dipole are useful. The discussion of the infrared divergence of potential correlations and the role of the local potential in Appendix C is thoughtful and physically clear. The comparison with RayGal is encouraging, and the perturbative expression Eq. (63) will be handy for forecasts. The main soft spot is exactly what the stress-test note says. The one-halo contribution to <> is computed with NFW concentrations estimated from the same RayGal catalogues used for validation. The paper is honest about this: Section VI.C.3 says the zero-point can be treated as a single free parameter, and Section VII.C says uncertainty in that quantity translates into uncertainty in whether, and where, the turnover occurs. So the headline claim that the turnover is "well accounted for" should be read as "the model can reproduce the turnover if the one-point function is chosen appropriately," not as an independent prediction. That does not sink the paper - the framework and the physical mechanism are still valuable - but it does mean the validation is conditional. A cleaner test would measure <> directly from the simulation, or fit the zero-point and show the turnover scale is not overly sensitive to it. This paper is for anyone building templates for DESI or Euclid dipole measurements and for people working on relativistic corrections in clustering. It deserves a serious referee, and the referee should push on the independence of the halo-model input and ask for error bars or a released implementation. I would accept it for peer review and would cite it if I worked on the dipole.

Referee Report

2 major / 4 minor

Summary. This paper develops a nonperturbative, wide-angle streaming model of the galaxy two-point correlation function in redshift space, extended beyond Doppler RSD to include gravitational redshift, lightcone corrections, and lookback-time effects. Starting from number conservation and the radial real-to-redshift map (Eq. 2), the authors derive the integral formula (Eq. 4) and, after truncating the displacement distribution at second cumulant, the Gaussian streaming model (Eq. 9); the lightcone-corrected complete expression is Eq. (27). The model is evaluated from seven linear-theory correlation functions plus a halo-model treatment of the density-weighted potential, and is compared with the RayGalGroup N-body halo catalogues. The central numerical claim is that the dipole turnover at s ≈ 20 h⁻¹ Mpc, absent in linear theory, is reproduced (Fig. 4). A small-displacement expansion yields the perturbative dipole formula (63), in which the turnover is attributed to the shift term −⟪Δψ⟫ dξ_AB/ds, driven by the pairwise potential difference ⟪Δψ⟫. The displacement statistics are decomposed into symmetric and antisymmetric pairwise functions (Section VI), identifying ⟪Δψ⟫ as the key nonlinear ingredient; its one-halo component dominates at all separations (Fig. 6). The paper also gives a compact wide-angle RSD dipole (Eq. G1), a covariant derivation of the lightcone effect, and a treatment of the local potential and the apparent IR divergence of ψ correlations (Appendix C).

Significance. If correct, the framework supplies a complete, directly evaluable template for odd multipoles in the mildly nonlinear regime, with immediate application to DESI BGS and Euclid gravitational-redshift measurements. The derivation is unusually transparent: linear theory is recovered in a few lines (Section III.C.1), the lightcone effect is derived from first principles (Section III.B.1), and the perturbative dipole (63) is obtained without fitted nuisance parameters. The algebraic results in Appendices F–H are checkable, the DESI wide-angle forecasts (Appendix E) are quantitative, and the predicted second turnover at the halo-exclusion scale (Section VII.C) is falsifiable. The authors are also candid about the model's approximations (Gaussian truncation, linear bias and linear evolution for all but one correlation). The principal weakness is that the validation of the headline turnover is conditional: the one-halo input (Eq. 45) is computed with concentrations from the same RayGal catalogues used for comparison, and the total zero-point is degenerate with a single free parameter. The agreement at 20 h⁻¹ Mpc is therefore encouraging but not yet an independent test of the model.

major comments (2)

[§V.C.2, Table II; §VI.C.3; §VII.C] The numerical validation of the headline claim is conditional on halo-model inputs derived from the same simulation against which the model is compared. The one-halo term (45) is evaluated with NFW profiles whose concentrations c_A and c_B are estimated from the RayGalGroup catalogues (Table II), and Section V.C.2 reports a value φ_A(0) = 0.036 h⁻¹ Mpc that is about 1.6 times the two-halo value ξ^HM_{ψδA}(0) = 0.022 h⁻¹ Mpc. Section VII.C itself states that uncertainty in the one-point function “translates to uncertainty in the scale at which the turnover occurs—or whether it occurs at all,” and Section VI.C.3 notes that the total zero-point may be treated as a single free parameter. This is not circular in the strict sense, since the dipole measurement is not used as a fitting target, but it does compromise the independence of the test. I recommend adding a sensitivity analysis: vary the concentrations over plausible ranges (or adopt an external concentration–mass relation), and report the predicted turnover scale as a function of the degenerate zero-point. If the turnover scale is robust to these variations, the claim should be restated with that evidence; if not, the abstract and conclusions should be softened accordingly.
[§VII.B, Eq. (63), Fig. 7] The physical attribution of the turnover to the shift term (64) rests on the perturbative dipole (63), but the agreement between Eq. (63) and the full nonperturbative model (27) is not quantified. Eq. (63) is obtained from Eq. (H2) after dropping the disconnected term ∂_a∂_b(m^a_[AB] m^b_(AB)), and the four contributions plotted in Fig. 7 are compared with the full streaming model only visually. The authors should state the size and origin of any residual difference between the total of Eq. (63) and Eq. (27) on the scales of the turnover, and report the turnover position and amplitude predicted by Eq. (63) versus the full model, since the interpretation of the turnover as an advection-like effect is built on this expansion.

minor comments (4)

[§V.C.2, Appendix G] The comparison in Fig. 4 uses the opposite dipole sign convention (the model is multiplied by −1), but this is stated only in the last sentence of Appendix G; the convention should be stated where the comparison is presented.
[Fig. 7 caption] The “streaming model (full sky)” reference curve should specify which covariance components are retained (the caption mentions only the connected piece) and whether the lightcone-corrected weighting (28) is used.
[Table III] Table III sets the NFW concentration to c = 9 for all masses at z = 0.2, whereas Section V uses the catalogue-dependent concentrations of Table II; given the role of φ_A(0) in the one-halo term, the sensitivity of the Table III values to c should be stated.
[§V.C.1, §VIII] The RayGalGroup comparison retains only Doppler and gravitational-redshift contributions and excludes the lightcone and lookback-time effects, so the validation exercises only part of Eq. (27); the conclusions should state this scope explicitly even though Fig. 5 shows the excluded effects to be small for the dipole.

Circularity Check

0 steps flagged · score 2.0 of 10

Derivation is self-contained; the turnover prediction is conditional on a halo-model input estimated from the same RayGal catalogues, but that is a modeling caveat rather than a reduction-by-construction circularity.

full rationale

The central derivation is not circular. Equation (27) follows from number conservation together with a Gaussian truncation of the displacement cumulants, and the dipole formula (63) is obtained by expanding the resulting correlation function, so the shift term -⟨⟨Δψ⟩⟩ dξ_AB/ds is a derived consequence of the model rather than an input. The paper also checks its compact RSD dipole expression against the established linear formula (G2), which is an external consistency check. The one genuinely load-bearing numerical ingredient is the pairwise potential difference ⟨⟨Δψ⟩⟩: its one-halo contribution is evaluated using NFW profiles with concentrations listed in Table II as 'estimated mean halo concentration' from the same RayGalGroup catalogues against which the dipole is compared (Section V.C.2, 'we compute the nonlinear one-point correlation functions of Eq. (30) according to the specifications of the halo catalogue'). This weakens the independence of the validation, but it is not a circular reduction: the dipole itself is not used to fit those concentrations. The paper is also transparent about the sensitivity of the result to this input: Section VI.C.3 states that the total zero-point ϕA(0)+ξψδA(0)-ϕB(0)-ξψδB(0) 'may be treated as a single unknown quantity', and Section VII.C cautions that 'uncertainty in this quantity translates to uncertainty in the scale at which the turnover occurs—or whether it occurs at all.' These are explicit limitations rather than hidden assumptions. The self-citation to Ref. [60] for the wide-angle streaming model is supporting, and the relevant equations are re-derived here, so it is not load-bearing. Overall: no significant circularity in the derivation chain, with a validation caveat that is honestly disclosed.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The model introduces no new particles or fields. Its main external input is the halo-model machinery for the nonlinear one-point function of the gravitational potential, which is the principal non-derivation input.

free parameters (1)

Halo concentration parameters c_A, c_B = 8.2, 6.6, 6.1 (Table II)
Used in the NFW profile to compute the central potentials phi(0) that control the one-halo contribution to the pairwise potential difference. Estimated from the same RayGal halo catalogues used for validation; the text (Section VI.C.3) notes the zero-point can alternatively be treated as a free parameter.

assumptions (5)

domain assumption Gaussian truncation of the displacement distribution after the second cumulant
The streaming model approximates p(chi - chi') by a Gaussian with mean m and covariance C, dropping non-Gaussian cumulants (Section II.A).
domain assumption Linear bias and linear perturbation theory for all correlations except the one-point potential
The seven correlation functions entering m and C are evaluated at linear order with linear bias; only the density-weighted potential one-point function is treated nonperturbatively (Section IV).
domain assumption Halo model: all matter in NFW halos with Sheth-Tormen mass function and peak-background split bias
Used to compute the one-halo and two-halo terms of the pairwise potential difference (Sections IV.A and Appendix F).
domain assumption Haloes are spherical and can be treated as isolated bodies for the one-halo term
Newton's shell theorem justifies the 1/r potential outside R_vir and the form of phi_A(0) (Section VI.C.3).
domain assumption Ideal survey: magnification and evolution bias neglected
The authors explicitly restrict to an ideal survey to simplify the model, noting these H/k terms can be added (Section II footnote 1).

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Cite this review

Pith. "Pith review of Gravitational redshift from large-scale structure: nonlinearities, antisymmetries, and the dipole." pith.science (2026). https://pith.science/paper/N7NHSATO

@misc{pith2026250622431,
  author       = {Pith},
  title        = {Pith review of: Gravitational redshift from large-scale structure: nonlinearities, antisymmetries, and the dipole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7NHSATO}},
  note         = {Machine review of arXiv:2506.22431}
}

abstract

Gravitational redshift imprints a slight asymmetry in the observed clustering of galaxies, producing odd multipoles (e.g. the dipole) in the cross-correlation function. But there are other sources of asymmetry which must also be considered in any model which aims to measure gravitational redshift from large-scale structure. In this work we develop a nonlinear model of the redshift-space correlation function complete down to these subleading, asymmetric effects. In addition to gravitational redshift and the well-known redshift-space distortions, our model, given by a compact nonperturbative formula, also accounts for wide-angle effects (to all orders), lightcone effects, and other kinematic contributions. We compare our model with $N$-body simulations and find good agreement; in particular we find that the observed turnover in the dipole moment around a separation of $20\,h^{-1}\mathrm{Mpc}$ (a feature absent in the linear predictions) is well accounted for. By examining the exchange properties of distinct tracers, we identify the pairwise potential difference as the key physical ingredient of the dipole. Several new insights related to the theory of redshift-space distortions are also given.

Figures

Figures reproduced from arXiv: 2506.22431 by the authors.

**Figure 2.** FIG. 2. Spacetime diagram for two neighbouring galaxies (red and blue) as they cross the lightcone. In the left panel the galaxies are at [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3. Comparison between the Gaussian streaming model in the exact wide-angle regime (solid black) and in the usual distant-observer [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. Comparison between model prediction ( [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: FIG. 5. Dipole of the streaming model (solid blue) and from linear [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: FIG. 6. The contributions to the pairwise potential di [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: FIG. 7. The four contributions to the dipole from the perturbative [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]

**Figure 10.** Figure 10: FIG. 10. Forecasted uncertainties for BGS compared with the size of the wide-angle corrections defined as [PITH_FULL_IMAGE:figures/full_fig_p035_10.png]

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p037_11.png]

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Works this paper leans on

122 extracted references · 30 canonical work pages · cited by 1 Pith paper

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A better approach would be to use, for example, convolu- tion Lagrangian perturbation theory [54, 79] to compute all the density-weighted correlations

An obvious starting point is the extension of the gravita- tional evolution and tracer bias beyond linear theory. A better approach would be to use, for example, convolu- tion Lagrangian perturbation theory [54, 79] to compute all the density-weighted correlations
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This implies an important change in the modelling of the pairwise potential differ- ence

This work used haloes as tracers, as per the RayGal cat- alogues, but a more realistic application should extend the modelling to galaxies. This implies an important change in the modelling of the pairwise potential differ- ence. Since galaxies do not necessarily lie at the centre of haloes ⟪∆ψ⟫ needs to be recalculated to account for this offset (e.g. th...
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Inclusion of magnification and evolution bias is another obvious area of improvement. As we showed in pre- vious work [60], these selection e ffects enter through a change in the density weighting in a similar way to the lightcone correction so should be straightforward to include in the model
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Testing the law of gravity with novel large-scale structure ob- servables

It would be interesting to investigate the impact of non-Gaussian cumulants on the dipole (in analogy with Ref. [82]). If the first cumulant (the mean) is impor- tant for the anti-symmetric correlations, as we found in this work, then it might reasonably be expected that the third cumulant (the skew) is the next most relevant. In this context, numerical s...

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24 Note that in pure vector formχ2 1χ2 2 = (χTAχ)2 with A =  0 1 /2 1/2 0 

Redshift-space distortions as a dynamical equation and the λ trick In Sections II C and VII we alluded to a dynamical way of viewing RSD; here we give further support for this picture. 24 Note that in pure vector formχ2 1χ2 2 = (χTAχ)2 with A =  0 1 /2 1/2 0 . 28 Introduce an auxiliary variable λ and consider the map s(x,λ ) = x +λδx(x) whic...
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On correlations of the potential Without loss of generality consider the off-diagonal component of Cgrav (neglecting the density weighting for simplicity): ⟨[ψ(x1)−ψO][ψ(x2)−ψO]⟩ =⟨ψ(x1)ψ(x2)⟩−⟨ ψ(x1)ψO⟩−⟨ ψ(x2)ψO⟩ +⟨ψ2 O⟩ =⟨ψ(x1)ψ(x2)⟩− σ2 ψ−⟨ψ(x1)ψO⟩ +σ2 ψ−⟨ψ(x2)ψO⟩ +σ2 ψ = ˜ξψψ(r)− ˜ξψψ(χ1)− ˜ξψψ(χ2) (C1) where in the last line we have identified ˜ξψψ ...
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The subtraction of the one-point function ξψψ(0) = σ2 ψ from ξψψ(r) renormalizes what is otherwise a divergent integral

IR divergence. The subtraction of the one-point function ξψψ(0) = σ2 ψ from ξψψ(r) renormalizes what is otherwise a divergent integral. Without this subtraction ξψψ(r) =⟨ψ(x1)ψ(x2)⟩∝ Z ∞ 0 k2dk 2π2 H k !4 j0(kr)Pδδ(k)∼ Z kIR 0 dk kns−2 is infrared divergent for ns < 1 (e.g. for ΛCDM which has ns≈ 0.96). This divergence is related to the fact that ψ and th...
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Observer dependence. The presence of the observer-dependent termψO breaks statistical homogeneity. As a consequence the two-point function depends also on χ1 and χ2 (the distance to each tracer). Due to the second and third terms in Eq. (C1),⟨(ψ1−ψO)(ψ2−ψO)⟩ acquires a multipole structure; see Figure 9. In the distant-observer limit χ1,χ 2→∞ (ϑ = 0), Eq. ...

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Tidal effects and the equivalence principle. The physical content of Eq. (C1) and its expression into multipoles can be better understood by taking the viewpoint of an observer locally comoving with the fluid flow (such an observer might be considered properly inertial if not ...
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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

A better approach would be to use, for example, convolu- tion Lagrangian perturbation theory [54, 79] to compute all the density-weighted correlations

An obvious starting point is the extension of the gravita- tional evolution and tracer bias beyond linear theory. A better approach would be to use, for example, convolu- tion Lagrangian perturbation theory [54, 79] to compute all the density-weighted correlations

[2] [2]

This implies an important change in the modelling of the pairwise potential differ- ence

This work used haloes as tracers, as per the RayGal cat- alogues, but a more realistic application should extend the modelling to galaxies. This implies an important change in the modelling of the pairwise potential differ- ence. Since galaxies do not necessarily lie at the centre of haloes ⟪∆ψ⟫ needs to be recalculated to account for this offset (e.g. th...

[3] [3]

Inclusion of magnification and evolution bias is another obvious area of improvement. As we showed in pre- vious work [60], these selection e ffects enter through a change in the density weighting in a similar way to the lightcone correction so should be straightforward to include in the model

[4] [4]

Testing the law of gravity with novel large-scale structure ob- servables

It would be interesting to investigate the impact of non-Gaussian cumulants on the dipole (in analogy with Ref. [82]). If the first cumulant (the mean) is impor- tant for the anti-symmetric correlations, as we found in this work, then it might reasonably be expected that the third cumulant (the skew) is the next most relevant. In this context, numerical s...

2020

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24 Note that in pure vector formχ2 1χ2 2 = (χTAχ)2 with A =  0 1 /2 1/2 0 

Redshift-space distortions as a dynamical equation and the λ trick In Sections II C and VII we alluded to a dynamical way of viewing RSD; here we give further support for this picture. 24 Note that in pure vector formχ2 1χ2 2 = (χTAχ)2 with A =  0 1 /2 1/2 0 . 28 Introduce an auxiliary variable λ and consider the map s(x,λ ) = x +λδx(x) whic...

[6] [6]

On correlations of the potential Without loss of generality consider the off-diagonal component of Cgrav (neglecting the density weighting for simplicity): ⟨[ψ(x1)−ψO][ψ(x2)−ψO]⟩ =⟨ψ(x1)ψ(x2)⟩−⟨ ψ(x1)ψO⟩−⟨ ψ(x2)ψO⟩ +⟨ψ2 O⟩ =⟨ψ(x1)ψ(x2)⟩− σ2 ψ−⟨ψ(x1)ψO⟩ +σ2 ψ−⟨ψ(x2)ψO⟩ +σ2 ψ = ˜ξψψ(r)− ˜ξψψ(χ1)− ˜ξψψ(χ2) (C1) where in the last line we have identified ˜ξψψ ...

[7] [7]

The subtraction of the one-point function ξψψ(0) = σ2 ψ from ξψψ(r) renormalizes what is otherwise a divergent integral

IR divergence. The subtraction of the one-point function ξψψ(0) = σ2 ψ from ξψψ(r) renormalizes what is otherwise a divergent integral. Without this subtraction ξψψ(r) =⟨ψ(x1)ψ(x2)⟩∝ Z ∞ 0 k2dk 2π2 H k !4 j0(kr)Pδδ(k)∼ Z kIR 0 dk kns−2 is infrared divergent for ns < 1 (e.g. for ΛCDM which has ns≈ 0.96). This divergence is related to the fact that ψ and th...

[8] [8]

The presence of the observer-dependent termψO breaks statistical homogeneity

Observer dependence. The presence of the observer-dependent termψO breaks statistical homogeneity. As a consequence the two-point function depends also on χ1 and χ2 (the distance to each tracer). Due to the second and third terms in Eq. (C1),⟨(ψ1−ψO)(ψ2−ψO)⟩ acquires a multipole structure; see Figure 9. In the distant-observer limit χ1,χ 2→∞ (ϑ = 0), Eq. ...