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REVIEW 3 major objections 6 minor 46 references

Constraining the Cosmological Constant from Stellar Orbits Around Sgr A* Using Physics-Informed Neural Networks

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A physics-informed neural network trained on S2's astrometry infers the orbit's total precession, subtracts the Schwarzschild contribution, and attributes the residual to the cosmological constant, obtaining Λ ≤ 5.67 × 10^-40 m^-2.

desk verdict Likely a fit residual, not a Λ measurement—honest PINN work that needs a zero-Λ mock test before the bound can be taken seriously. read the letter →

arxiv 2508.19719 v1 pith:JDQPTUQY submitted 2025-08-27 astro-ph.CO astro-ph.GA

Shinsei Eyama , Youhei Masada This is my paper
classification astro-ph.COastro-ph.GA
keywords cosmologicalconstantstellarorbitsSgrA*orbitalprecessionSchwarzschildphysics-informedneuralnetworksinversePINNGalacticCenter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that the cosmological constant Λ can be constrained from the orbit of a single star near the Milky Way's central black hole, without relying on cosmology-wide data. Using 25 years of S2 astrometric data, a physics-informed neural network infers the orbital parameters and the total precession of the orbit, then subtracts the known Schwarzschild precession. The leftover is assigned to the cosmological precession caused by Λ, yielding an upper bound Λ ≤ 5.67 × 10^-40 m^-2, roughly two orders of magnitude tighter than the previous machine-learning-based estimate. The paper argues the bound holds for S2 because S2 is the only star in the sample with a fully observed orbit; extending the same analysis to two longer-period stars, S1 and S9, gives numerically tighter bounds, but the paper concludes those are unreliable because only part of each orbit has been observed.

What carries the argument

Key machinery is the inverse physics-informed neural network (iPINN). A fully connected network maps the orbital phase angle φ to the inverse radius u = 1/r, trained by minimizing a weighted sum of a regression loss against astrometric data and a physics loss enforcing the Schwarzschild-metric equations in Darwin variables (eqs. 5–7). The physical model's unknown parameters e and p are optimized along with the network weights, acting as an inverse solver. The trained network is then evaluated over two full orbital periods, the two minima of u(φ) locate successive periapsis passages, and their angular separation defines the total precession δφ_Reg. Subtracting the analytic Schwarzschild prece

What would settle it

Fit the same S2 astrometric data with a conventional orbit model that adds the extended cluster mass and black-hole spin as free parameters alongside Λ; if the best-fitting Λ is inconsistent with zero or with the quoted 5.67 × 10^-40 m^-2 upper bound at the 3σ level, the residual is not measuring Λ. A second independent check is to recompute δφ_Reg with a network ensemble and a bootstrap of the data: if the scatter between realizations exceeds the quoted σ_Reg, the bound is an artifact of a single trained network.

Watch

Extended reading notes

Core claim

The central discovery the paper argues for is that stellar orbits around Sgr A* can act as a local probe of the cosmological constant. With the inverse PINN framework, the authors recover the eccentricity and semi-latus rectum of the S2 orbit to within about 0.3% of the values reported from direct orbital fits, and read off a total relativistic precession of about 12.149 arcminutes per orbit. Subtracting the Schwarzschild prediction leaves a tiny residual, which the analytic formula for cosmological precession converts into the upper bound Λ ≤ 5.67 × 10^-40 m^-2. The authors present this as roughly a hundredfold tightening of the previous PINN-based limit, and argue that the S2-based constra

Load-bearing premise

The load-bearing premise is that the precession the trained network measures is exactly the sum of Schwarzschild and cosmological precession, with no other source—extended stellar mass, black-hole spin, or alternative gravity—contributing, and that the residual after subtracting Schwarzschild is wholly due to Λ.

Editorial extensions

If this is right

  • If the bound is right, a single well-observed stellar orbit around a supermassive black hole can constrain Λ to 10^-40 m^-2, a regime between cosmological and Solar System limits that no other single probe covers.
  • The method's sub-percent recovery of e and p from noisy, unevenly sampled astrometry suggests PINN-based inference can extract dynamical parameters from sparse data, which is directly relevant for future S-star monitoring.
  • The S1 and S9 tests imply that apparently tighter bounds from longer-period stars should not be trusted until their orbits are phase-complete; full orbital coverage, not just more data points, is the deciding factor.
  • Tuning the loss-weight w is identified as a necessary step: the paper finds w = 6.0 × 10^-5 minimizes the physics loss and argues that earlier PINN analyses, by not scanning w, left the inferred bound partly uncontrolled.
  • The framework extends to future ELT/VLT astrometry to tighten the Λ bound and to test alternative gravity theories by searching for precession residuals beyond Schwarzschild plus Λ.

Reading between the lines

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

  • Because the residual is attributed to Λ after assuming no other precession sources, the number can also be read as an upper bound on any unmodeled precession such as an extended stellar cluster, black-hole spin, or modified gravity; a dedicated multi-parameter fit with those terms free would be needed to tell which interpretation survives.
  • The subtraction is not fully independent: δφ_Reg and δφ_SP both come from the same trained network and the same fitted e and p, so the quoted error bars on δφ_Λ likely understate systematic correlations. A bootstrap or ensemble-of-networks estimate would test this.
  • A similar residual analysis applied to future stars with complete orbits should produce a sequence of bounds that scale with the semimajor axis and eccentricity as the cosmological-precession formula predicts; if they do not, the attribution of the residual to Λ is suspect.
  • The apparently tighter S1 and S9 bounds provide a cautionary template: any machine-learning constraint whose nominal precision improves with less complete data should be re-examined for overfitting before being quoted.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper applies a physics-informed neural network (iPINN) to astrometric data of the S-stars around Sgr A*, focusing on S2, and infers the orbital elements and the total periapsis precession δφReg. The Schwarzschild precession δφSP is computed from the inferred semi-latus rectum, and the difference δφReg − δφSP is attributed to the cosmological precession. Using the Kerr et al. (2003) formula, the paper derives an upper bound Λ ≤ 5.67×10^-40 m^-2, about two orders of magnitude tighter than the previous PINN-based estimate of Galikyan et al. (2023). The same procedure is applied to S1 and S9, yielding numerically tighter but less reliable bounds; the authors conclude that the S2 constraint is the most robust.

Significance. If the derivation were valid, the paper would present a novel, local dynamical constraint on the cosmological constant and demonstrate an interesting use of PINNs for extracting relativistic precession from sparse astrometric data. The exploration of the loss-weight hyperparameter and the honest discussion of partial-orbit coverage for S1/S9 are useful contributions, and the sub-percent recovery of the S2 orbital elements is a positive result for the machine-learning approach. However, the central constraint is not supported: the residual used to isolate Λ is an internal self-consistency error of a Schwarzschild-constrained fit rather than a measured non-Schwarzschild precession, no mock-injection or zero-Λ validation is provided, the quoted uncertainties have no documented derivation, and important astrophysical precession sources are omitted. The headline bound therefore cannot be accepted as a physical constraint.

major comments (3)
  1. [§2.3, Eq. (10); §3.2, Eq. (20); §4.2] The decomposition δφReg = δφSP + δφΛ is circular as implemented. The PINN output u(φ) is trained by minimizing Eq. (20), whose physics loss enforces the Schwarzschild equations (5)–(7), which contain no Λ term. The 'total precession' δφReg is read off from that same fitted orbit, while δφSP is computed from the same inferred p via Eq. (9). To the extent the physics loss is minimized, δφReg is driven toward the Schwarzschild precession for the fitted (e, p), so δφReg − δφSP is a self-consistency error of the fit, not an independent measurement of a non-Schwarzschild precession. The reported residual ≈6×10^-4 arcmin is within 1.5σ of σReg = 4.1×10^-4 arcmin and is statistically indistinguishable from zero. The paper needs a mock-injection test, including Λ = 0, demonstrating that the residual tracks an injected cosmological precession; no such validation is presented.
  2. [§4.2, Table 1] The uncertainties σReg and σSP are not derived anywhere in the manuscript. No ensemble size, bootstrap procedure, or error-propagation formula is given. In particular, p is an inferred iPINN parameter, but no uncertainty on p is reported, so the formal σSP ≈ 1.6×10^-9 arcmin appears to assume a delta-function p. The 3σ upper-limit construction in §4.2 therefore lacks a validated statistical basis. The authors should specify exactly how σReg was obtained (e.g., from retraining with different seeds), how the Gaussian assumption is justified, and how hyperparameter choice enters the quoted uncertainty.
  3. [§2.3, Eq. (10); §4.2] Equation (10) omits known physical contributions to the S2 periastron shift. An extended mass distribution in the Galactic center and black-hole spin (Lense–Thirring) produce precessions that are not included in the model. The paper cites Galikyan et al. (2024) for a cluster-density constraint but does not incorporate such a term. At the claimed precision of ~10^-4 arcmin, a quantitative bound on these contaminants is required before the residual can be attributed to Λ. Without it, the extracted δφΛ is degenerate with unmodeled Newtonian/post-Newtonian effects.
minor comments (6)
  1. [§4.2, Table 1] The text states that inferred orbital parameters agree with observational values to within 0.3%, but for p the deviation is (228 − 225.25)/228 ≈ 1.2%. Please correct this statement.
  2. [Eq. (12)] The cosmological precession formula is typeset ambiguously. Please write it as a clear fraction, verify dimensional consistency, and confirm the sign/factor against Kerr et al. (2003), since a factor error here would directly rescale the bound.
  3. [§3.1–§3.2] The variable φ is defined as the true anomaly in §2.1 but is later called the 'declination of the elliptical orbit' in §3.2. Please standardize the notation.
  4. [§4.3, Fig. 7] The text says w = 10^-3 is adopted for both S1 and S9, while Fig. 7 reports optimal w values that minimize LossPhys. Please clarify which w values were actually used and how they were selected.
  5. [Figs. 3 and 4] The loss curves are said to be normalized by their values in the first epoch. This normalization is not defined in the captions or in the main text, making it difficult to interpret the minima. Please specify the procedure and the number of training epochs.
  6. [Data Availability] The data-availability statement says data will be shared on reasonable request, but no statement is made about code availability. Given the central role of the PINN implementation, releasing code would substantially improve reproducibility.

Circularity Check

1 steps flagged · score 8.0 of 10

The Λ bound is derived from δφReg−δφSP, but both quantities are outputs of the same Schwarzschild-constrained PINN fit; the residual is a fit self-consistency error, not an independent cosmological signal.

  1. fitted input called prediction [§2.3 Eq. (10); applied in §4.2]
    "The values of δφSP and δφReg are computed using Eq.(9) and (11), respectively. The cosmological precession δφΛ is then calculated as the difference δφReg−δφSP, in accordance with Eq.(10)."

    Eq. (10) defines δφΛ as δφReg−δφSP. But δφReg is not an independent measurement: it is the periapsis shift of the PINN orbit, and the PINN's physics loss (Eq. 20, with w=6×10−5 adopted in §4.1) enforces the Schwarzschild equations (5)–(7), which contain no Λ term. In the limit of zero physics residual, the predicted orbit is a Schwarzschild solution and δφReg equals δφSP≡6πGM/(c²p) for the same inferred p. Thus δφReg−δφSP is the self-consistency error of the fit (the extent to which the network cannot simultaneously satisfy the data and the Λ-free Schwarzschild equations), not a measurement of Λ-induced precession. Converting this residual to Λ via Eq. (12) re-labels a fit inconsistency as a physical signal, so the headline bound reduces by construction to the model's own residual.

full rationale

The central claim (Λ ≤ 5.67×10−40 m−2, Eq. 21) rests entirely on Eq. (10), which decomposes the PINN's periapsis shift δφReg into Schwarzschild (δφSP) and cosmological (δφΛ) parts. The circularity is that δφReg and δφSP are both outputs of the same fit: the network is trained with Loss = w LossReg + (1−w) LossPhys, where LossPhys penalizes residuals of Eqs. (5)–(7), the Λ-free Schwarzschild equations. Therefore the trained orbit is constructed to be Schwarzschild-like, and its periapsis shift is not an independently measured total precession; it is nearly equal to 6πGM/(c²p) for the fitted p. The residual δφReg−δφSP is the physics-loss residual (model inconsistency), and the reported magnitude (~6×10−4 arcmin) is comparable to σReg=4.1×10−4 arcmin. No mock data test with Λ=0 is provided to show that the pipeline recovers zero Λ; without such a test the residual cannot be interpreted as cosmological precession. I found no load-bearing self-citation: the cited PINN prior work (Galikyan et al. 2023) is by different authors and is used as a baseline, not as the evidential support for the decomposition. The defect is therefore not self-citation but definitional reduction of the prediction to the fit residual, giving a score of 8.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The analysis introduces no new physical entities. It relies on the standard Schwarzschild and Kerr et al. formulas, plus two inferred orbital parameters (e, p) that are fitted by the network. The most important free choice is the loss weight w, which is manually tuned and directly affects the reported bound. The key circular step is the definition of δφΛ as the residual between two quantities that both come from the same fitted Schwarzschild model.

free parameters (3)
  • Loss weight w = 6.0e-5
    Chosen by minimizing LossPhys over a grid of w (Fig. 3). Controls the balance between data fit and physics constraint; the reported Λ bound depends directly on this manual choice.
  • Orbital eccentricity e (inferred) = 0.8861 (S2)
    Inferred by the iPINN rather than taken from Gillessen et al. (2017); used in computing δφSP and δφΛ via eqs. (9) and (12).
  • Semi-latus rectum p (inferred) = 225.25 au (S2)
    Inferred by the iPINN; directly sets δφSP = 6πGM/(c^2 p), and enters the cosmological precession formula (12). The Λ upper bound is a residual of the fit using these parameters.
assumptions (4)
  • domain assumption The total precession of S2 is exactly the sum of the Schwarzschild precession and the cosmological precession, with no other contributions (eq. 10).
    Invoked in Section 2.3 to define δφΛ as the residual δφReg - δφSP. Neglects precession from the extended stellar cluster, black hole spin, and other S-stars, which the paper does not model.
  • domain assumption The cosmological precession formula of Kerr et al. (2003), δφΛ = ... (eq. 12), is correct and applies to the S2 orbit.
    Used to convert the residual precession into an upper bound on Λ. The formula is taken from the cited literature without re-derivation.
  • ad hoc to paper The PINN output u(φ), trained with the Schwarzschild physics loss, provides an unbiased estimate of the true orbital precession.
    The paper's central inference depends on interpreting the periapsis shift of the fitted orbit as a measured total precession, even though the physics loss forces the orbit to obey the Schwarzschild equations. This makes the residual a self-consistency error of the fit rather than an independent observable.
  • standard math The Darwin-variable expansion of the Schwarzschild orbit to first order in μ (eq. 8) is accurate enough for the claimed 3σ bound.
    Used to derive δφSP = 6πμ. This is standard first-order post-Newtonian precession and is not the main concern.

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Cite this review

Pith. "Pith review of Constraining the Cosmological Constant from Stellar Orbits Around Sgr A* Using Physics-Informed Neural Networks." pith.science (2026). https://pith.science/paper/JDQPTUQY

@misc{pith2026250819719,
  author       = {Pith},
  title        = {Pith review of: Constraining the Cosmological Constant from Stellar Orbits Around Sgr A* Using Physics-Informed Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JDQPTUQY}},
  note         = {Machine review of arXiv:2508.19719}
}
abstract

We present a novel analytical framework employing Physics-Informed Neural Networks (PINNs) to constrain the cosmological constant $\Lambda$ through the analysis of stellar orbits around the supermassive black hole (SMBH) Sgr A* at the Galactic center. Focusing on the well-observed S2 star, we use an inverse PINN (iPINN) architecture to infer orbital elements and estimate the total precession angle from astrometric data. By isolating the contribution from $\Lambda$, which is defined as the difference between the total precession and the Schwarzschild precession, we derive a stringent upper bound of $\Lambda \leq 5.67 \times 10^{-40}, \mathrm{m}^{-2}$, which is approximately two orders of magnitude tighter than previous estimates obtained using similar data-driven methods. Extension of our analysis to two additional long-period S-stars, S1 and S9, reveals that while the cosmological precession becomes relatively more prominent in such systems, limited orbital coverage introduces significant uncertainties in parameter estimation. Among the cases examined, the constraint derived from S2 remains the most robust. Our results highlight the potential of PINN-based approaches for extracting physical insights from sparse or noisy astronomical data. Future applications to next-generation observational data and further methodological improvements in machine learning are expected to refine the cosmological constraints and enable broader tests of gravitational theories.

Figures

Figures reproduced from arXiv: 2508.19719 by the authors.

Figure 1
Figure 1. Schematic representation of the orbital geometry. The central mass is located at the intersection of the dashed lines, corresponding to one of the two foci of the ellipse. The radial distance 𝑟 is measured from the focus to the orbiting object (black dot), and the corresponding inverse radius 𝑢 = 1/𝑟 is the quantity predicted by the PINN model. The angular coordinate 𝜙 denotes the true anomaly, defined as the angle … view at source ↗
Figure 2
Figure 2. Architecture of the PINN used in this study. The input to the network, i.e., 𝑥, is the angular coordinate 𝜙, while the output, i.e., 𝑦, is the predicted inverse radial distance 𝑢ˆ(𝜙) = 1/𝑟ˆ(𝜙) in this work. The training process simultaneously minimizes two loss terms: the regression loss LossReg, which quantifies the discrepancy between predicted and observed data points (𝑢𝑖 , 𝜙𝑖 ), and the physical loss LossPhys, w… view at source ↗
Figure 3
Figure 3. Response of Lossphys to variations in the weight parameter 𝑤. (a) Behavior of Lossphys over a broad range of 𝑤, spanning from 10−6 ≲ 𝑤 ≲ 10−1 . (b) Zoomed-in view of the region 10−5 ≲ 𝑤 ≲ 10−4 , highlighting the detailed structure of the response. The red dot indicates the value of 𝑤 yielding the smallest Lossphys within the range examined [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Similar to Fig.3 but for LossReg. physical constraints, or conversely, enforce the governing equations too strongly at the cost of observational accuracy. To identify an appropriate value for 𝑤, we systematically analyze the behavior of each loss term as a function of …
Figure 5
Figure 5. Figure 5: Predicted orbital trajectory of the S2 star, reconstructed using PINNs. The blue solid curve represents the orbit predicted by the PINN, while the colored dots correspond to the observed astrometric data used for training. The color of the data points indicates the pha…
Figure 6
Figure 6. Figure 6: Prediction of inverse radius 𝑢 = 1/𝑟 for the S2 orbit as a function of phase angle 𝜙. The blue curve shows the predicted trajectory from the trained PINN, while the dots represent the observed data. The prediction is extended over the interval 0 ≤ 𝜙 ≤ 4𝜋, and is used t…
Figure 7
Figure 7. Figure 7: Loss responses as a function of the weight parameter 𝑤 for (a) S1 and (b) S9. The red dot marks the 𝑤 value that yields the minimum LossPhys and is used in the subsequent analysis [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: Predicted orbital trajectories for (a) S1 and (b) S9 obtained using the optimal 𝑤 values identified in [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: Relative error (%) in eccentricity 𝑒 (blue) and semi-latus rectum 𝑝 (orange) for each star. While S2 shows sub-percent-level precision, S1 and S9 exhibit substantially larger uncertainties. ing Λ, unless their orbits are sufficiently sampled with high phase coverage. 5…

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    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.