[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Aims. We propose that the condition of relative motion between us and the objects that we observe in the Universe should generate relativistic aberration on the photons that such objects emit, varying the observed flux similarly to the cases of blazars and neutron stars, with instead a decrease of this radiative flux.

Methods. We follow some important papers and textbooks used in modern cosmology, adding the effects of relativistic aberration in the theoretical description of the Cosmos.

Results. New definitions for the luminosity distance and the angular distance arise, with the consequence of changing the cosmological parameters measured in the last years. A qualitative description of all of this is also performed on the multipole analysis of the CMB.

Key words. Cosmological parameters - Cosmology: theory - Cosmology: observations - Relativistic processes

From the observations of the redshifted spectra of Supernovae Type Ia, such as in (Perlmutter et al. 1999) and the homogeneity and isotropy of the CMB, as measured in (Planck Collaboration et al. 2016), the current understanding of the cosmos describes an homogeneous, isotropic and expanding Universe mostly made of dark energy and dark matter in which all of the COs are following the expansion, increasing the distance between them as cosmological time passes.

Starting from the work of (Hubble 1929), a hundred years of observations showed that all the spectra measured from very far away objects appear redshifted by a factor, which is the same for all the wavelengths as stated in (Ellis et al. 2012); in the same work it’s underlined how this redshift may be interpreted as an effect of time dilation given by the condition of relative motion between us and any object as a result of the expansion of the Universe. From now on, we will call Cosmological Object (CO) anyone of these far away objects or events that produce radiation.

The presence of this redshift was immediately interpreted as a recession velocity and a consequence of the expansion of the Universe, thanks to the previous work of (Friedmann 1922), (Friedmann 1924), (Lemaître 1927) and the end of the static-model of the Universe, as in (Robertson 1933). (Pauli & Sparzani 2008) attribute to Von Laue in 1931¹ the result that in a RW metric the product between the momentum of a photon and the scale parameter of the Universe is constant; this was then expanded to derive the variation of the wavelength and direction of the photons with the expansion of the Cosmos as in (Ehlers 1961) and (Ellis et al. 1983) and to describe the evolution of the anisotropies of the CMB as a sum of covariant spherical multipoles as in (Thorne 1981) and (Ellis et al. 1983).

The interpretation of redshift as a recession velocity created also some problems with some observed velocities of COs which were apparently superluminal, like in the case of 3C 279; these problems were resolved in (Dishon & Weber 1977) and (Horák 1978) with the introduction of relativistic Doppler effect instead of the classical one. In the last years many papers about the interpretation of redshift as a recession velocity were published, like (Chodorowski 2007), (Bunn & Hogg 2009), (Chodorowski 2011) and (Ter-Kazarian 2022).

The aim of this work is to find how the interpretation of the observed redshift as a recession velocity have some consequences on the photons emitted by the COs, moving at relativistic speeds: as the COs are in a condition of relative motion, we expect that relativistic aberration should affect the trajectories of the departing photons from these objects as observed in our reference frame. Recently, the work of (Maartens et al. 2023) also used relativistic aberration in order to explain the observed Hubble tension, but in a different manner than in this paper.

This work is divided as follows: in Sec. 2 we will give a full treatment on the effects of relativistic aberration on the observed distances and spectra of COs; in Sec. 3 we will talk about how the results of Sec. 2 may change our view of the Universe, with just a qualitative review of its effects on the CMB multipole expansion analysis, and in Sec. 4 we will give the conclusions of this paper.

¹ This work is practically unobtainable, as I searched for it for months and found nearly nothing. Instead of writing it in the bibliography, I decided to write the little information I was able to find here; the note that (Pauli & Sparzani 2008) cited was some publication/lecture of the "Preußischen Akademie der Wissenschaften" in 1931 and was called "Die Lichtfortpflanzung in Raumen mit zeitlich veranderlicher Krümmung nach der allgemeinen Relativitatstheorie. (Kl. 26. Febr.; SB.)"

Everywhere in modern cosmological literature, the geometry of the Universe is described by the FLRW metric, in which the infinitesimal line element \( ds^2 \) connecting two close events is expressed as

\[
ds^2 = dt^2 - R^2(t) \left( \frac{dr^2}{1-kr^2} + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2 \right)
\] (1)

we have clearly used natural unit \( c = 1 \). This metric was firstly proposed in (Friedmann 1922) to show the possibility of the existence of a non-static Universe even in the absence of the cosmological term in the Einstein equation, as stated in (Pauli & Sparzani 2008). The spectra of all the COs are redshifted by a factor \( z \), which is related to the scale factor \( R(t) \) of the metric by

\[
1 + z = \frac{R(t_0)}{R(t_1)}
\] (2)

in which \( t_0 \) is the cosmological time of observation of a photon, and \( t_1 \) is the cosmological time of its emission, as in (Weinberg 1972). As in (Ehlers 1961) and (Ellis et al. 2012), we interpret this redshift as the sum of two contributions, one is a "Doppler" term given by the expansion of the Universe, and the second is a "gravitational" term given by the variation of the four-velocity of the observer. In theory, this redshift could be given by a lot of other different contribution, but in a FLRW Universe the only contribution will be the relativistic Doppler effect. Other models were described such as in (Ellis et al. 1978) dominated by the gravitational redshift, but as stated in a footnote by the same author in (Ellis et al. 2012) "it is not a realistic Universe model".

As a consequence of all of this, we suppose that the origin of redshift will only be the Doppler effect; if that's not the case, it's sufficient to suppose that the expansion of the Universe is not changing its velocity too fast², so that we can ignore the presence of gravitational redshift. In fact, the covariant component of the velocity \( v_i \) of an object in a Riemannian space-time described by the metric tensor \( g_{ik} \) in the position \( x^i \) can be expressed as in (Pauli & Sparzani 2008) as

\[
v_i = \frac{dx_i}{d\tau} = \frac{d(g_{ik} x^k)}{d\tau} = \frac{d g_{ik}}{d\tau} x^k + g_{ik} \frac{dx^k}{d\tau}
\] (3)

so even if an object isn't changing its position, like galaxies in this description of the Universe, the evolution of the metric with the proper time of the said object can result in an observed relative velocity. All of this means that every CO is in free-fall following this expansion, and so that every one of them is in its own free-falling reference frame. The question now is how this will affect the observed trajectories of the emitted photons as seen in a different point of spacetime, which also corresponds to different reference frames.

As stated in a lot of sources in literature, such as (Weinberg 1972) or (Ellis et al. 2012), the Cosmological Principle imposes that any emitted photon following the expansion of the Universe will draw radially-directed geodesics with null length in the reference frame of the source: but different observers should see different initial directions for these geodesics as a consequence of the condition of relative motion in which they are with the COs.

One could think that we don't need any such treatment because relative to the Universe our instruments and all of the events that we observe are in first approximation points without dimensions, connected by one null-length geodesic: but the COs that we observe and the telescopes that we use have in fact physical dimensions even if they are extremely small compared to the Universe, and the difference of these dimensions as seen in their respective reference frame have an impact on measures, as we will soon show.

Thanks to the Equivalence Principle and the invariance of space we can treat this aberration: the Equivalence Principle was originally described in (Einstein 1916) and we can illustrate it as in (Pauli & Sparzani 2008): "For every space-time region infinitely small (...) there always exists a reference frame \( K_0 \) (...) in which every effect of gravity is absent on the motion of material points and every other physical phenomena.". Every \( K_0 \) is clearly a free-falling reference frame; we can individuate two free-fall reference frames in both our reference frame and in the reference frame in which any CO is, as both are following the expansion of the Universe (i.e. they move only under the effect of gravity so they are in free-fall). Then (Pauli & Sparzani 2008) continues, stating that "All the systems \( K_0 \), obtainable one from another with a Lorentz transformation, are equivalents." If we suppose that the variation of the velocity of expansion of the Universe is not too fast (i.e. is negligible), then the Lorentz transformation relating our reference frame with the CO reference frame won't have any type of dependency on accelerations and it will be the usual Lorentz transformation of (Einstein 1905); in other words, these two reference frames will be inertial and the transformation laws of Special Relativity can be applied.

Clearly this isn't exactly valid, as in (Weinberg 1972) can be found that in order to change position between two given points in an FLRW metric we must do it through "quasi-translations" and they aren't Lorentz transformations; the argument presented above is a way of justifying the fact that the condition of motion between two points in a Universe which is evolving must be considered in order to obtain correct observations: with this we mean that performing a quasi-translation only changes the position in an FLRW Universe, but not the reference frame. (Ellis et al. 2012) also showed that usual Lorentz contraction and time dilation are contained in the FLRW metric, and their presence immediately implies the existence of relativistic aberration in cosmology.

We can analyze the effect of aberration as seen in an inertial reference frame with a null relative velocity respect to us, very close to the CO, in a condition of motion relative to it; let's call it auxiliary reference frame. We suppose that the auxiliary reference frame is so close to the CO that the effects of general relativity are negligible as the Equivalence Principle states. Clearly this third reference frame will be inertial with both the other two. This enables us to describe the variation of the initial directions of photons as the usual special relativistic aberration. Then we will move from this auxiliary reference system to our actual position in an FLRW Universe, but not the reference frame. (Ellis et al. 2012) also showed that usual Lorentz contraction and time dilation are contained in the FLRW metric, and their presence immediately implies the existence of relativistic aberration in cosmology.

We can analyze the effect of aberration as seen in an inertial reference frame with a null relative velocity respect to us, very close to the CO, in a condition of motion relative to it; let's call it auxiliary reference frame. We suppose that the auxiliary reference frame is so close to the CO that the effects of general relativity are negligible as the Equivalence Principle states. Clearly this third reference frame will be inertial with both the other two. This enables us to describe the variation of the initial directions of photons as the usual special relativistic aberration. Then we will move from this auxiliary reference system to our actual po-

² as a "proof" of this, think about the fact that the Hubble parameter nowadays is nearly a constant.

of the redshift. As a result of the theorem of relativistic composition of velocities, an observer in our reference frame won't observe the radiation as emitted isotropically, but will see an increase of the radiative flux in the direction of motion of the CO, with a decrease of the flux in the opposite direction.

All of this affects only the initial direction of the departing photons, which will then travel through space along the usual null-length geodesics; with this we mean that each photon will travel along a different (relative to the canonical treatment) radially directed null-length geodesic centered at the CO, as no acceleration is present.

We will follow the treatment given in (Weinberg 1972), which in order to estimate the observed flux from any CO has considered just the quasi-translation in space and not the relative movement of the CO, so we will add this correction to understand how the number of observed photons change. Suppose that we have a telescope with a given mirror of radius b and we are observing a point-like CO. In (Weinberg 1972) it's shown that b is related to the angular dimension ε that the radius of the mirror implies in the sky as seen in the position of the CO as

sition³, with a null relative velocity with it, and see how this impact the observed fluxes. This translation will affect the direction of the photons only geometrically, as in the canonical work of (Weinberg 1972): these photons will proceed in their journey following the geodesics defined by general relativity, but will appear more numerous in the direction of motion of the CO rather than in the direction towards us.

With all of this we can now say that a decrease of the observed flux of the CO will be seen in our reference system. Fig. 2.1 helps understand this effect.

First of all we can link the redshift z and the module of the velocity v' of the reference frame in which the CO is relative to the auxiliary system thanks to the relativistic relation which, for example, can be found in (Einstein 1905) for the Doppler effect in the case of no transversal motion

\( z := \frac{\lambda_0 - \lambda}{\lambda} = \sqrt{\frac{1 + \beta}{1 - \beta}} \)    (4)

in which λ is the wavelength as observed in the CO reference frame, which is moving with a relative velocity ||v|| = cβ, while λ₀ is the observed wavelength in the auxiliary system; clearly λ₀ is also the wavelength as observed by us. We can easily reverse this relationship in order to find the relative motion of the CO as a function of the observed redshift of the spectra

\( \beta = \frac{(1 + z)^2 - 1}{(1 + z)^2 + 1} \)    (5)

this relationship can be found in (Dishon & Weber 1977), in which it was derived to explain the apparent superluminar motion of the blazar 3C279, but also in (Chodorowski 2007) and (Ter-Kazarian 2022).

Equation 5 clearly doesn't hold if the expansion of the Universe between us and the CO is faster than c, but in that case we simply can't detect any light signal from the CO so the problem doesn't arise.

Let's now see how the initial direction of the trajectories for the photons changes as seen in our reference frame as a function

\( b \approx R(t_0) r_1 \epsilon \)    (6)

here r₁ is the position of CO as seen by us, which, thanks to the fact that photons travel along radially directed null-length geodesic, will be given as in (Weinberg 1972) by

\( \int_{t_1}^{t_0} \frac{dt}{R(t)} = \int_0^{r_1} \frac{dr}{\sqrt{1 - kr^2}} = f(r_1) \)    (7)

where

\[
f(x) = \begin{cases}
\sin^{-1} x & \text{if } k = +1 \\
x & \text{if } k = 0 \\
\sinh^{-1} x & \text{if } k = -1
\end{cases}
\]    (8)

while R(t₀) is defined as in equation 2. Equation 6 doesn't take account of the aberration generated by the relative motion of the said CO: it's a result in which the object isn't moving relative to us, so the CO is in the same inertial reference frame as we are. Let's call ε′ the angular dimension of the telescope in the CO reference frame. We can easily find how this angle is related to ε, the angular dimension of the telescope as seen in the auxiliary reference frame: the transformation of a generic angle α under the change of reference frame is described by (Einstein 1905) with the formula

\(
\tan\alpha = \frac{\sin\alpha' \sqrt{1 - \beta^2}}{\cos\alpha' + \beta}
\)    (9)

in which the angle α is the angle formed by the direction of light and the positive direction of the x¹ (or x′¹) axis. In an expanding Universe, the CO is effectively receding from us, so the mirror will be placed in the opposite direction of the CO direction of motion. All of this means that instead of working with ε and ε′ we have to work with the angles π − ε and π − ε′:

\(
\tan(\pi - \epsilon) = \frac{\sin(\pi - \epsilon') \sqrt{1 - \beta^2}}{\cos(\pi - \epsilon') + \beta}
\)    (10)

³ mathematically it means using the usual quasi-translation described in (Weinberg 1972)

Article number, page 3 of 9

Thanks to the symmetries of the trigonometric functions and the fact that the angles ϵ and ϵ′ are very small, we can rewrite the expression 10 as

The physical meaning of this equation is that the photons captured by the telescope will fall in a solid angle with area πϵ′^2, which is smaller than the canonical πϵ^2: if the radiation is emitted isotropically in the CO reference frame, instead of capturing the fraction of photons emitted on the whole 4π surface

our telescope will detect the fraction of the total number of photons

which can be expressed as a function of the redshift z thanks to the equation 4

As we can see, we have a decrease by a factor (1 + z)^−2 more than in the canonical treatment. The effect of aberration in an expanding Universe is, as expected, a decrease of the measured flux which may have been interpreted as an overestimation of the cosmological distances. We will now derive an expression for the luminosity distance which will be used later: if we continue to follow (Weinberg 1972) adding the effects of relativistic aberration, the power P received by the mirror of our telescope in our reference frame is linked to the luminosity L of the CO in its reference frame by

Here t_0 and t_1 are defined as in equation 2, while A = πb^2 is the proper area of the mirror. The extra-factors R^2(t_1)/R^2(t_0) are given by the effects of both time-dilation and wavelength increase with the expansion of the Universe (change of reference frame) as in the canonical treatment. The radiative flux F passing through the mirror of the telescope as observed in our reference frame will be

and thanks to equation 2 which links z and R(t), the luminosity distance will immediately be given by

we notice that it has a factor R(t_0)/R(t_1) more than in the canonical treatment as a consequence of the presence of relativistic aberration on the direction of photons.

The treatment will be done following that of (Ellis et al. 2012) and adding the correction of the relativistic aberration. Thanks to the expansion of the Universe (the change of reference frame), if the CO has a luminosity L in its reference frame, the observed flux F passing through a surface dS in our reference frame will be

two factors (1 + z)^−1 come from the different energies and times measured by a receding observer, while the other (1 + z)^−2 factor comes as a result of relativistic beaming, as the transformation law for the angular dimension of the telescope placed opposite to the direction motion of the CO, remembering that dΩ ∝ ϵ^2, will be

which is valid only for small solid angles. There is also another way of deriving equation 18: as in (Thorne 1981), the invariant integration element on the light cone dV_p in the local rest frame of the observer who is looking for photons with a given frequency ν can be expressed as

but in the rest frame of the CO emitting these same photons will have a different frequency ν′, so as widely acknowledged if we measure photons with a given frequency ν in the CO reference frame they will have a frequency such that

this equation with equation 19 and the invariance of dV_p gives the transformation law 18 for observing the same photons. Let’s now continue the treatment defining the galaxy area distance r_G by the relation

in which the values dS_G and dΩ_G are respectively the cross-sectional area of the telescope as seen in the auxiliary reference frame and the solid angle implied by the telescope in the same condition. With this equation, we express equation 17 as

which we can rewrite as an inverse square law

Here the denominator is another way in which we can define the luminosity distance

---

**Equations:**

1. ϵ = (√(1 - β^2) / (1 - β)) ϵ′    (11)

2. ϵ = (1 + z) ϵ′    (12)

3. πϵ^2 / 4π = πb^2 / (4π R^2(t_0) r_1^2)

4. πϵ′^2 / 4π = πϵ^2 / 4π(1 + z)^2 = πb^2 / [4π R^2(t_0) r_1^2 (1 + z)^2]    (13)

5. P = L [R^2(t_1) / R^2(t_0)] [A / (4π R^2(t_0) r_1^2 (1 + z)^2)]    (14)

6. F = P/A = [L R^2(t_1)] / [4π R^4(t_0) r_1^2 (1 + z)^2]    (15)

7. d_L := (L / 4πF)^(1/2) = R^3(t_0) / R^2(t_1) r_1    (16)

8. FdS = [L / 4π(1 + z)^4] dΩ    (17)

9. dΩ′ = dΩ / (1 + z)^2     (18)

10. dV_p = ν dΩ dν     (19)

11. ν′ = (1 + z) ν    (20)

12. dS_G = r_G^2 dΩ_G    (21)

13. F = L / [4π (1 + z)^4 r_G^2]    (22)

14. F = F_G / d_L^2   ;   F_G := L / 4π    (23)

15. d_L := (1 + z)^2 r_G   (24)

---

Article number, page 4 of 9

Equation 23 will give us the same old module distance law, relating the apparent magnitude \( m \) of the CO, its absolute magnitude \( M \) and its luminosity distance \( d_L \), but with a different relation between \( d_L \) and \( z \)

\( m - M = 5 \cdot \log_{10} \left( \frac{d_L(z)}{\text{Mpc}} \right) + 25 \qquad (25) \)

Clearly this is an "ideal world" equation: in the real world when it's time to measure some luminosity distances, one should continue to include in equation 25 the terms for the k-correction, the extinction, ecc.

At this point, an apparent problem for this model would be that the change in the definition of the luminosity distance destroys the Etherington relation, which is supported by a lot of observational data such as (Liao et al. 2016), (Rana et al. 2017), but there are a lot of other experimental evidences in literature; a closer look at the problem shows that these difficulties don’t arise.

If we continue following (Ellis et al. 2012), we define the surface area \( dS_O \) of the CO as seen in our reference frame as

\( dS_O := r_O^2 d\Omega_O \qquad (26) \)

in which \( d\Omega_O \) is the solid angle implied by the CO as seen in our reference frame and \( r_O \) is the distance binding these two quantities; this solid angle will also be affected by relativistic aberration. Contrary to what one could expect from a quick look back at equation 12, the solid angle of the source \( d\Omega_O \) in our reference frame is actually bigger than it would appear in a reference frame at rest with the CO.

The condition of relative motion, the invariance of c and the fact that photons coming from different points of an extended CO have different directions and screen spaces produce a bigger angle implied by the CO, as seen in Fig. 2.3. If we position our telescope orthogonal to the photons coming from the center of a galaxy, the photons coming from its edge will be slightly inclined relative to this direction; it may be an extremely small angle, but for high redshift, as we can see from equation 12 this effect becomes important even if the angles involved are small.

All of this could seem contradictory with what stated for the angular dimension of the telescope as seen in the CO reference frame, but it’s not just the trajectories of the photons which are important, but also their directions: effectively, the dimension of the mirror in the CO reference frame will appear bigger, but the photons which will hit it will be of a lower number.

Let's see more into detail why we should see bigger angular sizes for the COs: as for the luminosity distance, we construct an inertial reference frame at rest with the CO, but this time we put it at our position; it will clearly move relative to us with a velocity given by equation 5, but it’s important to remember that this reference frame is receding away from us; we will call it assistant reference frame in order to distinguish it from the other two and also from the auxiliary one. Again, it’s a reference frame at our position which is inertial to us, so every effect given by general relativity is canceled as the Equivalence Principle states. This means that we can use the usual Lorentz transformations in order to work between these two reference frames. The assistant reference frame will travel relative to us with a velocity \( v = (-c\beta, 0, 0) \) where \( \beta \) is given by equation 5.

Now, let's consider a photon in the assistant reference frame which travels orthogonal to the \( x' = 0 \) plane: we may think of this as the CO covers half of the sky in this reference frame, with the center of the CO placed in the \( x' = 0 \) plane and the photon originated at the edge of it. We choose this photon to have a velocity \( u' = (0, -c, 0) \) in the assistant reference frame, but in our reference frame we will observe a different velocity for this photon as a result of the relativistic theorem of velocities composition for a purely radial motion: from (Einstein 1905)

\[
u_x = \frac{u'_x + v_x}{1 + v_x u'_x / c^2} \ ; \quad u_y = \frac{u'_y \sqrt{1 - \beta^2}}{1 + v_x u'_x / c^2} \qquad (27)
\]

we ignore the z spatial component because it’s null in both the reference frames. In our reference frame we will see the photon hit the \( x = 0 \) plane with a velocity \( u = (-c\beta, -c\sqrt{1 - \beta^2}, 0) \). This is a qualitative way that helps us understand that the angle implied by the CO in the sky as measured by us will be bigger than in a reference frame at rest with it, because we measure it from the negative x axis. For small angular sized COs it can be easily checked that the angle transformation law is still given by equations 12 and 18, but with swapped superscripts.

If we see a bigger source for high redshifts, this doesn’t necessarily mean that the object is closer to us because the angular distance has a minimum, as stated in (Sandage 1961), in agreement with the correction of the luminosity distance 16 and 24. All of this said, let’s correct equation 12 with the observed quantity as seen in our reference frame, instead that in the assistant reference frame, with the use of equation 26

\( dS_O = r_O^2 (1 + z)^2 d\Omega_O \qquad (28) \)

this yields a natural definition for the angular distance

\( d_A := r_O (1 + z) \qquad (29) \)

Article number, page 5 of 9

From conservation of the number of photons in the space between us and the CO, we can prove as in (Ellis et al. 2012) that there exists a conserved quantity along the path of the photons

This relation won’t be affected by aberration, but only because we have already taken it into account and we are now doing the computations in a fixed reference frame, we just have to choose one and use it. Substituting equations 21 and 28 in it and using the definitions for luminosity distance 24 and for the angular distance 29 we immediately obtain the Etherington relation, which still holds

We can link the luminosity distance 24 and the angular distance 29 with the cosmological parameters appearing in the Friedmann equations, the usual matter density Ω_M, the vacuum density Ω_Λ and the curvature parameter Ω_k as expressed in (Carroll et al. 1992)

where ρ_{M0} is the present-day baryons + dark matter density, H_0 is the present-day Hubble constant, G is the gravitational constant, R_0 = R(t_0), k is the normalized constant curvature and Λ is the cosmological constant.

We will follow the treatment given in (Carroll et al. 1992), expressing the computations in detail showing how and where we need to add the correction for relativistic aberration.

The null-length radially directed geodesic equation in the FLRW metric can be characterized as

with the infinitesimal cosmological time dt which can be linked to z, Ω_M and Ω_Λ through the Friedmann equation as in the canonical treatment offered in (Carroll et al. 1992) by

using this value in equation 33, rearranging the terms, multiplying both sides by R(t_0) and integrating yields

and f(x) defined as in the equation 8. Using the definition 32 for Ω_k immediately allows us to express the proper distance r_1 between us and the CO as a function of redshift as in the canonical treatment

the final step will be to derive the dependence of luminosity distance d_L from redshift z inserting equation 37 and 2 into equation 16, which gives us

We have inserted in both the right hand sides of equations 39 and 40 a factor c in order to express d_L in Mpc and H_0 in Km/s/Mpc without the natural unit c = 1 used almost everywhere in this paper.

We can think of this correction in an easy way:

If we compare the luminosity distance 39 with the canonical one used to measure the cosmological parameters with Supernovae type Ia in (Goobar & Perlmutter 1995), (Perlmutter et al. 1997), (Riess et al. 1998), (Perlmutter et al. 1999) we see that the cosmological parameters measured there may be wrong, including the curvature of the Universe. As we will see, following this treatment the measure of cosmological parameters through the CMB analysis used up until now should also be corrected for relativistic aberration.

It’s easy to see that the shape of the spectra and the measures of fluxes and intensities are unaffected by this correction, with the only consequence of changing their relationship with cosmological parameters as stated in the last Subsect.

We continue to follow the treatment of (Ellis et al. 2012), but little to no modifies are needed: the only things that change are the formulae for d_L and d_A, but this won’t affect the shape of the spectra.

---

Equations and math expressions:

dS_G dΩ_O = dS_O dΩ_G (1 + z)^2  (30)

\[
\frac{d_L}{d_A (1+z)^2} = 1
\]  (31)

Ω_M := \frac{8πG}{3H_0^2}ρ_{M0} ; Ω_Λ := \frac{Λ}{3H_0^2} ; Ω_k := -\frac{k}{R_0^2 H_0^2}  (32)

\[
\frac{dr}{dt} = \frac{\sqrt{1 - kr^2}}{R(t)}
\]  (33)

\[
dt = \frac{1}{H_0 (1 + z) \sqrt{(1 + z)^2(1 + Ω_M z) - z(2 + z)Ω_Λ}} dz
\]  (34)

\[
R(t_0) f(r_1) = \frac{1}{H_0} \int_0^z \frac{dz'}{\sqrt{H(z', Ω_M, Ω_Λ)}}
\]  (35)

H(z', Ω_M, Ω_Λ) := (1 + z'^2)(1 + Ω_M z') - z'Ω_Λ(2 + z')  (36)

\[
r_1(z, Ω_k, Ω_Λ, Ω_M) = F\left( \sqrt{|Ω_k|} \int_0^z \frac{dz'}{\sqrt{H(z', Ω_M, Ω_Λ)}} \right)
\]  (37)

\[
F(x) =
\begin{cases}
  \sin x & \text{for a closed Universe } Ω_k < 0 \\
  x & \text{for a flat Universe } Ω_k = 0 \\
  \sinh x & \text{for an open Universe } Ω_k > 0
\end{cases}
\]  (38)

\[
d_L = \frac{c(1 + z)^2}{H_0 \sqrt{|Ω_k|}} \cdot F\left( \sqrt{|Ω_k|} \int_0^z \frac{dz'}{\sqrt{H(z', Ω_M, Ω_Λ)}} \right)
\]  (39)

using the Etherington relation 31, we can also get the angular distance d_A as a function of z

\[
d_A = \frac{c}{H_0 \sqrt{|Ω_k|}} \cdot F\left( \sqrt{|Ω_k|} \int_0^z \frac{dz'}{\sqrt{H(z', Ω_M, Ω_Λ)}} \right)
\]  (40)

We define a function \(\mathcal{I}(\nu)\) such that \(L\mathcal{I}(\nu)d\nu\) is the rate at which the radiation is emitted by the CO between frequencies \(\nu\) and \(\nu + d\nu\). The function is normalized such that its integral over all frequencies is 1. We can then rewrite the inverse square law 23 as
\[
\frac{F d_L^2}{F_G} = \int_0^{\infty} \mathcal{I}(\nu_G)d\nu_G = (1 + z) \int_0^{\infty} \mathcal{I}(\nu_O(1 + z))d\nu_O\qquad (41)
\]

clearly here \(d_L\) is defined as in 39 and 24 instead of that in the canonical treatment. Then the observed flux between frequencies \(\nu\) and \(\nu + d\nu\) will be
\[
F_\nu d\nu = \frac{F_G(1 + z)}{d_L^2} \mathcal{I}(\nu(1 + z))d\nu \qquad (42)
\]
again, with the different definition of \(d_L\). A little precaution goes into the specific intensity \(I_\nu\), but the result is the same. An object with a surface area \(dS'_O\) as defined in 28 will have an observed specific intensity
\[
I_\nu d\nu := \frac{F_\nu d\nu}{d\Omega_O} = I_G \frac{\mathcal{I}(\nu(1 + z))}{(1 + z)^3}d\nu \qquad ; \qquad I_G := \frac{F_\nu}{dS'_O} \qquad (43)
\]
where \(I_G\) is the surface brightness of the CO as seen in the assistant reference frame, which is also the one that we observe. This result still holds because the Etherington relation 31 remains valid. It’s easy to see from 43 that any point-like CO that emits as a blackbody with a temperature \(T_G\) is seen by any observer as a blackbody but with an observed temperature \(T_O\) as in the canonical treatment
\[
\frac{T_G}{T_O} = 1 + z \qquad (44)
\]

The fact that we have modified the definition of luminosity distance with the experimental evidence that the CMB is still a blackbody in our reference frame is another justification of the fact that we also need to modify the angular distance.

Let’s now study, from a qualitative point of view, what happens at the measures of cosmological parameters 32, and in particular we start with the case of a flat Universe: we easily see from equation 25 that the numerical value of \(d_L\) taken from the measure of the distance modulus are correct, but the presence of the new factor \((1 + z)\) in its expression means that until now we could have possibly overestimated the integral in 39 and 37: in order for the integral to be smaller we should need a bigger denominator 36, which happens if we decrease \(\Omega_\Lambda\) and we increase \(\Omega_M\). So for a flat Universe, up until now we could have underestimated the mass component and overestimated the dark energy component: we don’t anymore need an acceleration of the expansion of the Universe as strong as measured in order to explain the increased distance between us and any CO: it’s the aberration which does the trick of decreasing the observed flux, making us think that the COs are more distant from us than they really are.

All of this would be a beautiful result, but only measures will say if it’s true; more importantly, we also have the effect of the curvature of the Universe to consider, which deeply impact the shape of \(d_L(z)\).

In a curved space-time, if we measure a \(d_L\) and we increase by a factor \((1 + z)\) the RHS of 39, it means that the function \(F\) should decrease and/or that \(|\Omega_k|\) should be above 1: not only we have different matter density and cosmological constant, but also a different value of the \(F\) function, suggesting that the curvature of the Universe is yet to be determined.

All of this might have an impact on the age of the Universe and its end. Fortunately the age of Universe vary very little with the variation of cosmological parameters and depends strongly only on \(H_0\), as shown in (Carroll et al. 1992), but the values of \(\Omega_\Lambda\) and \(\Omega_M\) have an impact on the deceleration parameter \(q_0\) as stated, in the same paper, and might produce a different end for the Universe.

First of all, let us state that it’s evident how all said up until now is also valid for the fluxes of gravitational waves. It’s possible it will have an impact on the measures of masses of merging bodies, as a Fourier transform of the observed fluxes is needed in order to measure them. This article can be easily experimentally disproved if the correction proposed applied to the data analysis of a gravitational wave generated by two merging neutron stars will produce two measures of their original masses above the Volkov-Oppenheimer limit.

It seems like the cosmological parameters observed through the CMB should be correct, but the fact that it’s an extended object has an impact on its observation. As we said in Subsect. 2.5, a generic point-like source emitting as a blackbody will continue to appear as a blackbody with a lower temperature to every observer, but the CMB is an object that cover the whole sky so another effect will be present as equations 12 and 18 aren’t exactly valid anymore.

Qualitative speaking, from equations 12 and 18 we see that the effect of relativistic aberration is in fact a spread of the CMB on the sky: from observations \(z_{CMB} \approx 1000\), so in this small angle approximation a small anisotropy will imply a solid angle \(10^{6}\) larger in our reference frame than in the assistant reference frame. The problem is that this effect happen for every point of the sky, so this also spreads the points near the anisotropies. The overall result will be a “mixing” of the CMB spectra measured in a point of the sky with its adjacent regions; we don’t have to confuse this mixing with the one described in literature as (Thorne 1981) and (Challinor 2000), because while there relativistic aberration is presented as a source that generates a dipole in the CMB and increases the intensity of high-order multipoles, here is exactly the opposite: every anisotropy of the CMB is mixed with its more uniform surrounding regions, making it appear more homogeneous and isotropic. It’s interesting to notice that the effect on the surface area of every point of the CMB is increasing it by \(10^{6}\), while the temperature anisotropies of the CMB are in the order of \(10^{-6}\) K as measured in (Planck Collaboration et al., 2016).

This will have an impact on the measures of the cosmological parameters from the analysis of the high-order multipoles of the CMB, which should be corrected and cleaned by this effect. The problem presented is extremely complex, and we will eventually discuss it in another paper.

The CMB has an extremely homogeneous spectrum, as showed in the observations of (Planck Collaboration et al. 2016), but it's possible that this homogeneity may be the result of this mixing and the Universe may not be so much homogeneous and isotropic as we think. How much the intrinsic homogeneity and isotropy of the Universe and relativistic aberration are responsible for it, only time will tell, as for now we lack the complete mathematical description and analysis of it complete with the correction proposed in this paper.

As a result of this, there’s even the possibility that the Cosmological Principle may, in fact, not be entirely true, as the CMB is one of the major clues for its validity, and breaking down its homogeneity will have a deep impact on Cosmology. But again, these last few lines are only speculation and thoughts for now.

\[
\frac{H_{0,\mathrm{can}}}{H_0} = \frac{\sqrt{|\Omega_k|} \, r_1(z, 0, \Omega_{\Lambda,\mathrm{can}}, \Omega_{M,\mathrm{can}})}{(1+z) \, r_1(z, \Omega_k, \Omega_\Lambda, \Omega_M)} \tag{47}
\]

clearly for low redshifts we won’t observe any difference; this can be checked looking at this equation, remembering that at low redshift \(z \approx 0\) and the curvature of the universe becomes negligible, just like the influence of cosmological parameters over \(r_1\) and, as expected and as one could see from a quick look at equation 12, the effect of relativistic aberration also becomes negligible, so for \(z \approx 0\)

\[
\frac{H_{0,\mathrm{can}}}{H_0} \approx 1 \tag{48}
\]

### 3.4. The measures of the Hubble constant

One of the problems of modern cosmology is the so called Hubble Tension, as explained in great detail in (Di Valentino et al. 2021), with a discrepancy between late-Universe and early-Universe measurement of \(H_0\) which is now around \(4.4\sigma\), as in (Riess et al. 2019), and isn’t anymore negligible. As said in Sect. 1, relativistic aberration was also used in a (Maartens et al. 2023) to resolve this problem, but with a different approach, so there one can find different results than here.

So, what should we do with all said to try to correct the Hubble tension? If we compare the luminosity distance found in this paper with the canonical one used to measure the cosmological parameters with Supernovae type Ia in (Carroll et al. 1992), (Goobar & Perlmutter 1995), (Perlmutter et al. 1997), (Riess et al. 1998), (Perlmutter et al. 1999) it’s easy to see that the measures of the Hubble constant for low redshifts \(z \approx 0\) using the distance luminosity are essentially the same, while for what said in Subsect. 3.3 about the multipoles of the CMB, the indirect measures obtained from it should be corrected. With all of this in mind, it’s also possible that the late Universe direct measures should be corrected if one looks too far in the past of the Universe.

Let’s consider the example of the measure of the Hubble constant which involves the use of the luminosity distance; as described in Subsect. 2.2, the equation for the module distance 25 isn’t modified by relativistic aberration, but only its dependency from the redshift \(z\) is changed. This means that the measures of the module distances are correct, so

\[
d_L = 10^{m-M-25} = const. \tag{45}
\]

is equal in both the canonical treatment and the luminosity distance derived in this paper; let’s respectively call them \(d_{L,\mathrm{can}}\) and \(d_L\). Imposing \(d_{L,\mathrm{can}} = d_L\), using equation 39 and the expression for the luminosity distance given in (Perlmutter et al. 1999) we get

\[
\frac{c(1+z)^2}{H_0 \sqrt{|\Omega_k|}} r_1(z, \Omega_k, \Omega_\Lambda, \Omega_M) = \frac{c(1+z)}{H_{0,\mathrm{can}}} r_1(z, 0, \Omega_{\Lambda,\mathrm{can}}, \Omega_{M,\mathrm{can}}) \tag{46}
\]

in which we have called \(\Omega_{\Lambda,\mathrm{can}}\) and \(\Omega_{M,\mathrm{can}}\) the value of the cosmological parameters as measured in (Perlmutter et al. 1999), while \(r_1\) is defined as in equation 37. The RHS describes a flat Universe as measured in the paper just cited. Rearranging the factors in this relation and eliminating common factors produces

and the presence of relativistic aberration won’t impact the measures of \(H_0\). But what the equation 47 tells us? It says that if we are looking too far in the past, and this too far is extremely small as a new term \(\propto z^2\) is present in equation 39, in order to obtain the correct values of the cosmological parameters 32 we also have to ensure that the value that we use for \(H_0\) is correct and vice versa. There are also a lot of other ways of measuring \(H_0\), as explained in (Di Valentino et al. 2021), but the aim of this article is not the solution of the Hubble tension problem so we rather stop here: this subsec. is just the proposal of something new for this question with little to no computations, but with all said about the measure of the cosmological parameters 32 it’s also important to underlie the possibility of the existence of wrong measures of \(H_0\).

### 3.5. Other needed modifications on the measures

It’s evident how all of this will have an impact on other aspects of cosmology, for example the theory and use of the Malmquist Bias and the Number of counts, which both need to be corrected in order to take into account relativistic aberration. If relativistic aberration will change these, it’s possible that we will need to re-evaluate a lot of observational datas used up until now for our galaxy formation and evolutionary models.

## 4. Conclusions

In this paper we tried to argue that if the redshifts that we observe from Cosmological Objects, namely COs, can be described as an apparent 4-velocity of the sources relative to us, then a modification of the luminosity distance and angular distance are necessary in order to take relativistic aberration into account for the measurements of COs at high redshifts. In particular, this addition may have generated an overextimation of the cosmological distances, which may have led to the measures of wrong cosmological parameters both from Supernovae type Ia as in (Perlmutter et al. 1999). and from the CMB power spectrum analysis of the Plank Collaboration (Planck Collaboration et al. 2016). It’s possible that the proposed correction may also have an impact on other measures, such as the Hubble constant \(H_0\) and a lot of datas used in modeling galaxy formation and evolution.

Acknowledgements. Thanks to professors A. Cimatti and F.R. Ferraro of the University of Bologna, I was studying their lessons while having the idea for this paper. Also, thanks to Yè for the beautiful music that I listened to while writing this article.

Article number, page 8 of 9

‘3C 279’ on page 1

Weinberg, S. 1972, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley & Sons, Inc.)

Bunn, E. F. & Hogg, D. W. 2009, American Journal of Physics, 77, 688


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
In the era of LLMs, ICL has become a popular prompting strategy to elicit the power of LLMs on a wide range of tasks. In ICL, a few demonstrations are given in the input context during inference while not involving parameter tuning (Dong et al., 2022; Min et al., 2022).

As a major natural language processing (NLP) task, there have been several works exploring in-context example selection strategy for MT. Agrawal et al. (2023) propose R-BM25 on the basis of BM25 (Robertson et al., 1994) to enhance word overlap. M et al. (2023) propose CTQScorer combining multiple features. Zhang et al. (2023) conduct a comprehensive study on prompting LLMs for MT and claim that templates, number of examples, features and the quality of example databases all matter for ICL on MT.

In previous studies, for both statistical MT and neural MT, syntax plays an important role to improve model performance (Williams and Koehn, 2014; Wu et al., 2017). However, in case of in-context example selection for MT, previous works focus on superficial word-level matching (like BM25 and R-BM25) or combining multiple straightforward features (like CTQScorer). To the best of our knowledge, no research dig out syntax-level features for ICL demonstrations. To this end, we propose a syntax-based in-context example selection strategy for MT, which selects examples most similar to the test source in syntax based on similarity of dependency trees. In addition, word-level and syntax-level features complement each other and combining both would further elicit LLMs’ power on MT. Therefore, we propose an ensemble strategy, concatenating examples selected by BM25 and our syntax-based strategy.

In-context learning (ICL) is the trending prompting strategy in the era of large language models (LLMs), where a few examples are demonstrated to evoke LLMs’ power for a given task. How to select informative examples remains an open issue. Previous works on in-context example selection for machine translation (MT) focus on superficial word-level features while ignoring deep syntax-level knowledge. In this paper, we propose a syntax-based in-context example selection method for MT, by computing the syntactic similarity between dependency trees using Polynomial Distance. In addition, we propose an ensemble strategy combining examples selected by both word-level and syntax-level criteria. Experimental results between English and 6 common languages indicate that syntax can effectively enhancing ICL for MT, obtaining the highest COMET scores on 11 out of 12 translation directions.

Experimental results between English and 6 common languages indicate that syntax helps find better in-context examples and thus improves LLMs’ ability in MT. Comparing with various baselines, our proposed methods obtain the highest COMET scores on 11 out of 12 translation directions.

Our contributions can be summarized as follows:

National Key Laboratory for Multimedia Information Processing, Peking University
MOE Key Laboratory of Computational Linguistics, Peking University
School of Computer Science, Peking University
{tangchenming@stu, ekko@stu, wuyf@}pku.edu.cn

1. Generally, the syntactic structures of sentences are represented by syntax trees, and thus it is natural to measure the syntactic similarity based on similarity or distance between syntax trees. Tree similarity can be measured by Edit Distance (de Castro Reis et al., 2004), Polynomial Distance (Liu et al., 2022), Tree Kernel (Collins and Duffy, 2002; Vishwanathan et al., 2004; Moschitti, 2006), etc.

2. Compared to other algorithms mentioned above, Polynomial Distance (Liu et al., 2022) is convenient to implement and relatively time-saving to run. So, in this work, we adopt Polynomial Distance between dependency trees to measure the syntactic similarity.

3. Concretely, given d as the number of dependency labels, which indicate the grammatical relations between dependents and their heads, we transform dependency trees into polynomials recursively based on two variable sets: X = {x₁, x₂, ... x_d} and Y = {y₁, y₂, ... y_d}. Considering a node with Label l as nˡ, its corresponding polynomial is represented as:

4.
$$
P(n^l) =
\begin{cases}
x_l, & n^l \text{ is leaf,} \\
y_l + \prod_{i=1}^k P(n_i), & n^l \text{ is non-leaf,}
\end{cases}
$$

5. where n₁, ..., nₖ are all child nodes of nˡ if it is non-leaf. Then, given a root node r, we obtain the polynomial representing the whole tree by computing P(r). A detailed demonstration from Liu et al. (2022) can be found in Appendix A.

6. Next, we compute the distance between polynomials. Note that the polynomial representing a dependency tree can be described term by term. For each term $c x_1^{e_{x1}} x_2^{e_{x2}} ... x_d^{e_{xd}} y_1^{e_{y1}} y_2^{e_{y2}} ... y_d^{e_{yd}}$ in the polynomial, where e_{xi}, e_{yi} and c are the exponent of variable xᵢ, yᵢ and the coefficient of the term respectively, we denote it as a term vector:

7.
$$
t = [e_{x1}, e_{x2}, ..., e_{xd}, e_{y1}, e_{y2}, ..., e_{yd}, c]
$$

8. In this way, a polynomial P can be written as a set of term vectors V_P. Then, we compute the distance between two polynomials (P and Q) as:

9.
$$
d(P, Q) = \frac{\sum_{s \in V_P} \min_{t \in V_Q} \| s - t \|_1 + \sum_{t \in V_Q} \min_{s \in V_P} \| s - t \|_1}{|V_P| + |V_Q|}
$$

10. where ∥s − t∥₁ is the Manhattan distance (Craw, 2017) between term vector s and t.

11. In recent years, ICL has been proved effective for improving LLMs' performance on various tasks without training or finetuning (Brown et al., 2020; Von Oswald et al., 2023).

12. To design instructions for MT, we first inform the language pair and the template of presenting the result. Then, we provide several in-context examples selected using our proposed strategy. Last, the source sentence of the test sample is concatenated at the end. An example of our prompt is shown in Figure 1, and we draw inspiration from the work of Agrawal et al. (2023).

13. We parse all our datasets with spaCy (Honnibal et al., 2020) to get dependency trees. Appendix B lists spaCy models we use for different languages.

14. For each test sample, we compute its Polynomial Distance from each instance of the example database on the source side, and then select the top-k most similar examples. The selected examples serve as demonstrations for ICL, as shown in Figure 1.

15. We hypothesize that a combination of word-level closeness and syntax-level similarity would make the LLM generate results good on both counts. So we propose an ensemble strategy, where examples selected by different selection strategies are concatenated. In this work, we explore the ensemble of BM25 and Polynomial Distance, where half of the final examples are from BM25 and the other half are from Polynomial Distance.

16. Following Agrawal et al. (2023), we adopt XGLM₇.₅ᴮ (Lin et al., 2022) for all our experi-

ments, which is a decoder-only multilingual generative language model supporting 30 languages. XGLM₇․₅B has 32 layers, a hidden dimension of 4096 and 7.5B parameters in total.

The first 4 examples are the top-4 from BM25 and the rest 4 are the top-4 from Polynomial. Polynomial + BM25: The first 4 are from Polynomial and the rest 4 are from BM25.

---

Example Database  We adopt WikiMatrix v1 (Schwenk et al., 2021) as our example database, which has 135M parallel sentences for 1620 language pairs. Detailed statistics of WikiMatrix for each language can be found in Appendix C.

---

Test Set  We perform our evaluation on the devtest set of FLORES+ ¹ (Costa-jussà et al., 2022), which has 1012 sentences for around 200 languages. We experiment between English and 6 common languages including German, Spanish, French, Japanese, Russian and Chinese.

---

Experimental results on all 6 languages (into and out of English) are shown in Table 1, where the number of in-context examples is set to 8.

---

Without the help of word-level closeness, Polynomial itself cannot reach perfection in many cases. However, it performs competitively on DE-EN translation, outperforming Random by 1.58 points and BM25 by 3.53 points.

---

With a combination of word-level closeness and syntax-level similarity, our ensemble strategies take the lead in most cases, outperforming Random by around 1 point on average.

---

To sum up, the highest scores of 11 out of 12 translation directions are achieved by our proposed methods, which indicates the effectiveness of syntax in in-context example selection for MT.

---

Evaluation Metrics  We report COMET (Rei et al., 2020) from wmt20-comet-da, which is considered a superior metric for MT nowadays (Kocmi et al., 2021). As a complement, we report BLEU from sacreBLEU (Post, 2018) in Appendix D.

---

Surprisingly, the performance of R-BM25 is poor under our experimental settings. This might be due to differences in example database, data preprocessing and the design of prompt templates. We leave exploration of the cause to future work.

---

Cleaning  We remove sentences longer than 100 or shorter than 4 tokens and those cannot be identified as their corresponding languages by Langid.py ⁵. Sentence pairs with a source/target length ratio exceeding 1.5 are also removed. We have around 85% sentences remaining after preprocessing for all languages except Japanese, of which the percentage is 67%. See Appendix C for detailed statistics.

---

We also compare with CTQScorer (M et al., 2023) in Table 2, where the number of examples is set to 4 and we focus on only 3 language pairs, to be in line with their work. Our proposed methods secure all the highest scores on all 6 translation directions. Note that M et al. (2023) use Europarl (Koehn, 2005) and ParaCrawl (Bañón et al., 2020) as example database, and experiment on FLORES-101, which is an earlier version of FLORES+. These factors may lead to an unequal comparison.

---

Tokenization  We tokenize Chinese with Jieba ², Japanese with Mecab ³ and other languages with Sacremoses ⁴.

---

Random: We report the average result of 3 different random seeds. BM25: The BM25 scores are computed using Rank-BM25 (Brown, 2020). R-BM25: We evaluate R-BM25 ⁶ (Agrawal et al., 2023) on our datasets.

---

Polynomial: Our syntax-based selection method with Polynomial Distance. BM25 + Polynomial:

---

We carry out experiments under different numbers of in-context examples. The average COMET scores of our ensemble strategy and other baselines on all 12 translation directions under different numbers of examples are shown in Figure 2. Please refer to Appendix E for our full results. Our proposed ensemble strategy constantly outperforms baselines under different numbers of examples.

In this work, we investigate whether syntactic information can help find better in-context examples for MT. We propose selecting examples based on similarity of dependency trees, and present a simple but effective ensemble method by selecting examples from both word-overlap-based and syntax-similarity-based selection. Our proposed methods obtain the highest COMET scores on 11 out of 12 translation directions, indicating that injecting syntax information during in-context example selection is helpful for MT. We call on the NLP community to pay more attention to syntactic knowledge for syntax-rich tasks like MT.

We give an instance to compare the examples selected by different strategies, as shown in Table 3. BM25 selects sentences with more word overlapping ("cannot be purchased") while Polynomial selects sentences with the similar syntactic structure (perfect tense, "that" clause, negative sentence, passive voice). For closely related language pairs like EN-DE, when the source sentences share similar syntax or patterns, it is likely that the target sentences also share these features. See Appendix F for an instance of model outputs.

Table 3: Top-1 examples of EN-DE translation selected by BM25 and Polynomial.

First, due to limited time and computational resources, we have not evaluated our methods on low-resource languages and other LLMs. Second, in our ensemble strategy, we have only explored combining word-level selection and syntax-level selection. Other types of selection strategies (e.g., selection based on semantics) are not explored in the ensemble. Third, we have not explored methods based on constituent trees and other algorithms of tree similarity besides Polynomial distance. Last, our selection is based on the similarity on the source side. However, similarity on the target side or between two sides is also worth trying.

Computational Budget  We run pre-processing and in-context example selection on Intel® Xeon® Gold 5218 CPU and the LLM's inference on NVIDIA A40. Table 4 shows the average computation time.

Reproducibility  All the experiments are completely reproducible since our selection methods are deterministic and sampling is disabled during LLM generation.

Scientific Artifacts  We cite all the creators of scientific artifacts we use in this paper. Licenses of these scientific artifacts are shown in Table 5.

Sweta Agrawal, Chunting Zhou, Mike Lewis, Luke Zettlemoyer, and Marjan Ghazvininejad. 2023. In-context examples selection for machine translation. In Findings of the Association for Computational Linguistics: ACL 2023, pages 8857–8873, Toronto, Canada. Association for Computational Linguistics.

Marta Bañón, Pinzhen Chen, Barry Haddow, Kenneth Heafield, Hieu Hoang, Miquel Esplà-Gomis, Mikel L. Forcada, Amir Kamran, Faheem Kirefu, Philipp Koehn, Sergio Ortiz Rojas, Leopoldo Pla Sempere, Gema Ramírez-Sánchez, Elsa Sarrías, Marek Strelec, Brian Thompson, William Waites, Dion Wiggins, and Jaume Zaragoza. 2020. ParaCrawl: Web-scale acquisition of parallel corpora. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 4555–4567, Online. Association for Computational Linguistics.

Dorian Brown. 2020. Rank-BM25: A Collection of BM25 Algorithms in Python.

Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. 2020. Language models are few-shot learners. In Advances in Neural Information Processing Systems, volume 33, pages 1877–1901. Curran Associates, Inc.

Michael Collins and Nigel Duffy. 2002. New ranking algorithms for parsing and tagging: Kernels over discrete structures, and the voted perceptron. In Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics, pages 263–270, Philadelphia, Pennsylvania, USA. Association for Computational Linguistics.

Marta R Costa-jussà, James Cross, Onur Çelebi, Maha Elbayad, Kenneth Heafield, Kevin Heffernan, Elahe Kalbassi, Janice Lam, Daniel Licht, Jean Maillard, et al. 2022. No language left behind: Scaling human-centered machine translation. arXiv preprint arXiv:2207.04672.

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## A Demonstration of Polynomial Distance

The original demonstration of converting dependency trees to polynomials taken from Liu et al. (2022) is shown in Figure 3.

## B The spaCy Models We Use for Parsing

The spaCy models we use for dependency parsing are listed in Table 6.

## C Data Statistics

Statistics on the size of WikiMatrix for each language are shown in Table 7.

## D BLEU Results

BLEU scores of translation with 8 in-context examples are shown in Table 8.

COMET scores of translation with 2, 4, and 16 in-context examples are shown in Table 9, 10 and 11, respectively. Note that when there are 16 in-context examples, the context sometimes exceeds the LLM’s maximum length, which may hurt the precision of the result.

case shown in Table 12, BM25 omits a constituent (“de plus” in the source sentence) while Polynomial translates not accurately enough. With the help of both word-level and syntax-level demonstrations, our ensemble strategy “BM25 + Polynomial” produces a relatively high-quality translation. This indicates that both word-level closeness and syntax-level similarity matters in in-context example selection for MT.

In addition to Section 5.3, we display an example output with different selection strategies. In the

Table 7: Size of WikiMatrix before and after pre-processing and the percentage of reserved sentences after pre-processing for 6 languages. Each entry refers to the data between English and the corresponding language.

| Strategy          | Output                                                                                |
| ----------------- | ------------------------------------------------------------------------------------------------------------ |
| Source (FR)       | De plus, le Centre d'alerte des tsunamis dans le Pacifique a déclaré qu'il n'y avait aucun signe de tsunami. |
| Reference (EN)    | Also the Pacific Tsunami Warning Center said that there was no Tsunami indication.                           |
| BM25              | The Pacific Tsunami Warning Center said there was no sign of a tsunami.                                      |
| Polynomial        | In addition, the Pacific Tsunami Warning Center has declared there is no sign of a tsunami.                  |
| BM25 + Polynomial | In addition, the Pacific Tsunami Warning Center said there was no sign of a tsunami.                         |

Table 12: Outputs of FR-EN translation under different selection strategies.


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Abstract. In this note, we provide a very simple proof of the uniformization theorem of Riemann surfaces by Ricci flow. The argument builds on a refinement of Hamilton’s isoperimetric estimate for the Ricci flow on the two-sphere.

MSC. primary, 53E20; secondary, 30F10

Keywords. Ricci flow, Isoperimetric inequality, Riemann surfaces, Uniformization

The classical uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. To prove this theorem by Ricci flow, one needs to show that: on any compact surface with a Riemannian metric, the (normalized) Ricci flow converges smoothly to a metric with constant curvature. The main difficulty lies in the genus zero (sphere) case. There have been two ways to settle this.

The first way was established by Richard Hamilton [2], improved by Bennett Chow [3, 4] and completed by Chen-Lu-Tian [6]. Hamilton first showed that the normalized Ricci flow with positive Gaussian curvature converges to a gradient shrinking Ricci soliton on the two-sphere, then Ben Chow removed the assumption of positive curvature; later Chen-Lu-Tian confirmed that any gradient shrinking Ricci soliton on the two-sphere must have constant curvature.

The second way, also opened up by Hamilton [5] and simplified by Andrews-Bryan [7], makes use of the isoperimetric data. Hamilton derived the evolution equation for the isoperimetric ratio and obtained its monotonicity to rule out the possibility of a ‘Type II’ singularity; combined with his earlier work in classifying both Type I and Type II singularities, he got a proof of the desired convergence. Andrews-Bryan’s proof is more straightforward, utilizing a comparison theorem for the isoperimetric profiles of solutions of the normalized Ricci flow with the ones of positively curved axisymmetric metric. They got rid of the blowup or compactness arguments, Harnack estimates, or any classification of behavior near singularities employed earlier by Hamilton and others; however, they invoked the Rosenau solution as the model metric to guarantee the exponential convergence of the Gaussian curvature, which seems a little bit unnatural to us.

Here we provide a new simple and direct proof of Hamilton and Andrews-Bryan’s result; the only ingredients are Hamilton’s evolution equation for isoperimetric ratio, the isoperimetric inequality on curved surfaces [1], and the maximum principle. We will derive the time-dependent lower bound for the isoperimetric ratio from the maximum principle; then through the isoperimetric inequality on curved surfaces, this lower bound renders the Gaussian curvature to converge exponentially fast to constant, which is enough for the convergence of the Ricci flow.

Acknowledgement. The author is partially funded by National Financial Support Program for Postdocs of China, under grant GZC20230533.

For a compact surface with Riemannian metric \((M,g)\), we denote by \(K(x)\) its Gaussian curvature at point \(x\) and by \(\kappa\) the maximum of \(K(x)\) on \(M\). The normalized Ricci flow is the equation

\[
(2.1)\qquad \frac{\partial g}{\partial t} = 2(1-K)g
\]

if we normalize the initial total area to be \(4\pi\). It is well-known that the total area \(\int_M dV_g\) remains being \(4\pi\) along (2.1).

Consider a curve \(\Lambda\) with length \(L\) on \(S^2\) dividing the total area into two parts \(A\) and \(4\pi - A\). We denote such curve on the \emph{round} sphere (with the areas enclosed by it also equal \(A\) and \(4\pi-A\)) by \(\bar{\Lambda}\) and its length by \(\bar{L}\). Following Hamilton [5], the isoperimetric ratio is defined as

\[
(2.2)\qquad I_A := \frac{\inf_\Lambda L}{\inf_{\bar{\Lambda}} \bar{L}}
\]

Now we need the following isoperimetric inequality due to Chavel-Feldman [1]:

**Theorem 2.1.** We have
\[
(2.3)\qquad L^2 \geq 4\pi A - \kappa A^2
\]
with the equality achieved iff the area enclosed by \(\Lambda\) is a geodesic disk with constant Gaussian curvature \(\kappa\).

It is obvious that the equality can be achieved for \(\forall A \in (0,4\pi)\) on the round sphere; for arbitrary metrics, the equality might not be achieved but the inequality is tight for \(A \to 0\) (to see this, take the geodesic disk centered at the point \(x\) where \(K(x)\) attains maximum \(\kappa\); let the radius approach \(0\), then (2.3) approaches equality on this disk). Hence from Theorem 2.1 we can easily see that

\[
(2.4)\qquad I_A^2 \geq \frac{4\pi A - \kappa A^2}{4\pi A - A^2}
\]

with the equality being approached for \(A \to 0\). Due to the Gauss-Bonnet formula \(\int_M K(x) dV_g = 4\pi\), we have \(\kappa > 1\) for any non-standard metric on \(S^2\). Thus the tight lower bound \(\frac{4\pi A - \kappa A^2}{4\pi A - A^2}\) is less than 1 for non-standard metrics.

**Remark 2.2.** The first order expansion \(L = \sqrt{4\pi A - \kappa A^2} = \sqrt{4\pi A - \frac{\kappa A^3}{4\sqrt{\pi}}} + o(A^2)\) as \(A \to 0\) of (2.3) was adopted by Andrews-Bryan [7, Theorem 2] to estimate the isoperimetric profiles; yet we only need the original version in our argument.

Last but not least, we recall the evolution equation for \(I_A\) derived by Hamilton [5, Formula 2.1]:

**Theorem 2.3.** \(I_A\) satisfies
\[
(2.5)\qquad \frac{\partial}{\partial t}\ln I_A^2 = \frac{\partial^2}{\partial r^2}\ln I_A^2 + \frac{A^2 + (4\pi - A)^2}{A(4\pi - A)}[1-I_A^2]
\]
here \(r\) denotes the normal direction of the curve \(\Lambda\).

Note that Hamilton proved Theorem 2.3 under the unnormalized Ricci flow; nevertheless, we can easily verify that (2.5) also holds under the normalized Ricci flow (2.1) by simply repeating his calculations in [5, Section 2] with mild modifications.

3. THE PROOF

**Proposition 3.1.** There exist constants \( B > 0 \) and \( C \) which possibly depend on \( A \), such that
\[
I_A^2 \geq \frac{1}{1 + e^{-Bt - C}}
\]
for all \( t \in [0, +\infty) \).

**Proof.** From (2.4) we know that the lower bound for \( I_A^2 \) is no greater than 1, so we pick \( C \) such that \( I_A^2 \geq \frac{1}{1 + e^{-C}} \) at \( t = 0 \). We solve the ODE
\[
\frac{\partial}{\partial t} \ln f = B[1 - f]
\]
with \( B := \frac{A^2 + (4\pi - A)^2}{A(4\pi - A)} \) and the initial value \( f(0) = \frac{1}{1 + e^{-C}} \). The solution is \( f(t) = \frac{1}{1 + e^{-Bt - C}} \). By the maximum principle of heat-type equations, we conclude that the solution \( I_A^2 \) of the PDE (2.5) satisfies
\[
I_A^2 \geq \frac{1}{1 + e^{-Bt - C}}
\]
for all \( t \in [0, +\infty) \). □

We can prove the uniformization theorem now.

**Theorem 3.2.** *The normalized Ricci flow on \( S^2 \) converges to the metric with constant curvature.*

**Proof.** We will show that \( \kappa \) satisfies
\[
\kappa(t) - 1 \leq (\kappa(0) - 1)e^{-2t}
\]
for all \( t \in [0, +\infty) \). We argue by contradiction: suppose (3.2) does not hold, then there exists \( T > 0 \) such that \( \kappa(T) - 1 > (\kappa(0) - 1)e^{-2T} \).

On the other hand, we restrict the domain for \( A \) in \( (0, \min\{\frac{4\pi}{\kappa(0)}, \frac{4\pi}{\kappa(T)}\}) \), and take \( e^{-C} := \frac{(\kappa(0) - 1)A^2}{4\pi A - \kappa(0)A^2} \), then at \( t = 0 \):
\[
I_A^2 \geq \frac{4\pi A - \kappa(0)A^2}{4\pi A - A^2} = \frac{1}{1 + e^{-C}};
\]
by Proposition 3.1, at \( t = T \) we have \( I_A^2 \geq \frac{1}{1 + e^{-BT - C}} \); since the inequality (2.4) is tight for \( A \to 0 \), we can choose \( A \) sufficiently small such that
\[
\frac{4\pi A - \kappa(T)A^2}{4\pi A - A^2} \geq \frac{1}{1 + e^{-BT - C}};
\]
putting \( e^{-C} = \frac{(\kappa(0) - 1)A^2}{4\pi A - \kappa(0)A^2} \) back into the above inequality and rearranging the terms yield
\[
e^{-BT} \cdot \frac{4\pi A - \kappa(T)A^2}{4\pi A - \kappa(0)A^2} \geq \frac{\kappa(T) - 1}{\kappa(0) - 1}.
\]
Since \( B = \frac{A^2 + (4\pi - A)^2}{A(4\pi - A)} \geq 2 \) and \( \kappa(T) - 1 > (\kappa(0) - 1)e^{-2T} \), let \( A \to 0 \) in (3.3) we get
\[
e^{-2T} \geq e^{-BT} \cdot 1 \geq \frac{\kappa(T) - 1}{\kappa(0) - 1} > e^{-2T},
\]
a contradiction.

Thus (3.2) shows that \( \kappa - 1 \) converges exponentially fast to 0. By the same arguments as in [7, Section 5], the normalized Ricci flow converges smoothly to the metric with constant Gaussian curvature \( K \equiv 1 \), i.e. the round metric. □

Fudan University, 220 Handan Rd, Yangpu District, Shanghai 200433, China

Research Institute of Intelligent Complex Systems

Email address: jiyucheng1991@gmail.com