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arxiv: 2606.13356 · v1 · pith:BO6CPTJYnew · submitted 2026-06-11 · 🌀 gr-qc · astro-ph.CO

Bounds on Λ at the Galactic Center

Prajwal Hassan Puttasiddappa , Muzammil Mushtaq , Willian Ramirez , David F. Mota This is my paper

Pith reviewed 2026-06-27 06:13 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.CO
keywords cosmological constantgalactic centerSgr A*stellar orbitsSchwarzschild-de SitterBayesian MCMCS2 stargeodesics
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The pith

Observations of three stars orbiting Sgr A* yield upper bounds on the cosmological constant of 6.9×10^{-48} m^{-2} at 68% credibility and 1.0×10^{-38} m^{-2} at 95% credibility.

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

The paper models the orbits of the S2, S1, and S14 stars around the Milky Way's central black hole by numerically integrating their paths as timelike geodesics in Schwarzschild-de Sitter spacetime, incorporating relativistic redshift and time-delay effects. It then uses Bayesian MCMC sampling to infer the orbital parameters along with the value of the cosmological constant Lambda from astrometric and spectroscopic data. A sympathetic reader would care because this tests whether a term usually associated with the large-scale expansion of the universe can be detected or bounded in the strong-gravity environment near a supermassive black hole. If the resulting limits are correct, any local contribution from Lambda at galactic-center scales must be extremely small. The three independent orbital constraints are combined to produce the tightest reported bounds.

Core claim

The central claim is that the posterior distributions obtained from the MCMC analysis of the three stellar orbits in Schwarzschild-de Sitter spacetime place upper bounds on the magnitude of Lambda at the galactic center of Lambda ≲ 6.9×10^{-48} m^{-2} at 68% credibility and Lambda ≲ 1.0×10^{-38} m^{-2} at 95% credibility.

What carries the argument

Numerical integration of timelike geodesics in Schwarzschild-de Sitter spacetime together with Bayesian MCMC inference of orbital and spacetime parameters from astrometric and spectroscopic data.

If this is right

  • The bound on Lambda applies at the radial scale of the observed stellar orbits around Sgr A*.
  • Combining independent constraints from multiple stars produces a stronger limit than any single orbit alone.
  • The model must reproduce the observed relativistic redshift and time-delay effects only if Lambda remains below the stated thresholds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geodesic-fitting approach could be applied to additional stars with future high-precision data to tighten the limit further.
  • The derived upper bound remains many orders of magnitude above the value of Lambda inferred from cosmology, leaving room for consistency between local and global measurements.
  • If Lambda were found to differ between the galactic center and cosmological scales, that difference would require an explanation beyond standard general relativity with a constant term.

Load-bearing premise

The stellar motion is accurately described by numerically integrating timelike geodesics in Schwarzschild-de Sitter spacetime that include relativistic redshift and time-delay corrections.

What would settle it

A precise measurement of the S2 star's radial velocity or sky position that deviates from the geodesic prediction by an amount larger than the 95% upper limit on Lambda would falsify the reported bound.

Figures

Figures reproduced from arXiv: 2606.13356 by David F. Mota, Muzammil Mushtaq, Prajwal Hassan Puttasiddappa, Willian Ramirez.

Figure 1
Figure 1. Figure 1: Shadow radius rsh/M (in the units of c 2/G) as a function of the cosmological constant Λ for a static observer. The shaded bands represent the 1σ and 2σ observational constraints on the shadow size of Sgr A∗ . prediction, we define the dimensionless parameter fSP, which quantifies the deviation from the purely Schwarzschild precession (fSP = 1), fSP = 1 + Λc 4a 4 (1 − e 2 ) 3/2 6G2M2 . (10) We compare this… view at source ↗
Figure 2
Figure 2. Figure 2: The plot for fSP as a function of Λ. The shaded region denotes the constraint from GRAVITY observations [39] 0 10 20 30 40 50 r (geometric units) 0.5 0.0 0.5 1.0 1.5 V E 2 = 0 = 10 26 m 2 = 10 24 m 2 Event horizon (r = 2M) 10 15 20 25 30 35 40 45 r 0.004 0.002 0.000 0.002 0.004 V E 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Effective potential V (r) in SdS spacetime for different values of Λ. Bound motion occurs between the turning points rp and ra, which are highlighted with •. For sufficiently large Λ (yellow curve), the outer potential barrier disappears, leading to unbound trajectories. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The marginalized posterior distributions of the fitted parameters with their respective [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Astrometric and spectroscopic fits for the S2 orbit in SdS spacetime. The concate [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Marginalized posterior distribution contours with 68% (1 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Best-fit SdS orbit for S1. The figure shows the projected sky trajectory together [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as Fig. 6, but for the S14 star [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Marginalized posterior distributions for [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the inferred black-hole mass [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
read the original abstract

We constrain the cosmological constant $\Lambda$ using astrometric and spectroscopic observations of the S2, S1, and S14 stars orbiting Sgr A$^*$. The stellar motion is modelled by numerically integrating timelike geodesics in Schwarzschild-de Sitter spacetime, including relativistic redshift and time-delay corrections. Orbital and spacetime parameters are inferred using a Bayesian MCMC analysis. The resulting posterior distributions place upper bounds on the magnitude of $\Lambda$ at the Galactic Center (GC). Combining the independent constraints from the S2, S1, and S14 orbits yields upper bounds of $\Lambda \lesssim 6.9\times10^{-48} \mathrm{m}^{-2}$ at 68\% credibility and $\Lambda \lesssim 1.0\times10^{-38} \mathrm{m}^{-2}$ at 95\% credibility.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to constrain the cosmological constant Λ at the Galactic Center by numerically integrating timelike geodesics in Schwarzschild-de Sitter spacetime for the S2, S1, and S14 stars, incorporating relativistic redshift and time-delay effects, and performing Bayesian MCMC inference on astrometric and spectroscopic data. It reports upper bounds obtained by combining constraints from the three orbits: Λ ≲ 6.9×10^{-48} m^{-2} at 68% credibility and Λ ≲ 1.0×10^{-38} m^{-2} at 95% credibility.

Significance. If the statistical procedure is corrected, the work would offer a useful test of whether a non-zero Λ is consistent with stellar dynamics at the scale of Sgr A*, providing local bounds that can be compared to cosmological values. The modeling choice of Schwarzschild-de Sitter geodesics with standard relativistic corrections is a clear strength, as is the use of multiple independent stellar datasets.

major comments (1)
  1. [Abstract] Abstract: the procedure of 'combining the independent constraints from the S2, S1, and S14 orbits' is not statistically valid. Because the central mass M, distance D, and Λ are shared across all three stars, separate per-star MCMC runs (each marginalizing over its own M and D) followed by multiplication of the resulting one-dimensional Λ posteriors undercounts the joint uncertainty on the common parameters and produces an overly tight combined upper limit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and for identifying a statistical issue in our analysis. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the procedure of 'combining the independent constraints from the S2, S1, and S14 orbits' is not statistically valid. Because the central mass M, distance D, and Λ are shared across all three stars, separate per-star MCMC runs (each marginalizing over its own M and D) followed by multiplication of the resulting one-dimensional Λ posteriors undercounts the joint uncertainty on the common parameters and produces an overly tight combined upper limit.

    Authors: We agree with the referee that multiplying one-dimensional Λ posteriors obtained from separate per-star MCMC runs is not statistically rigorous, as it fails to properly marginalize over the shared parameters M and D and can produce an overly tight combined constraint. In the revised manuscript we will replace this procedure with a single joint MCMC analysis that simultaneously models the three stellar datasets while sharing M, D, and Λ. The resulting joint posterior on Λ will be reported in place of the current combined bounds. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds obtained via MCMC fitting to external astrometric/spectroscopic data.

full rationale

The paper's derivation consists of numerically integrating timelike geodesics in Schwarzschild-de Sitter spacetime and performing Bayesian MCMC inference on observed stellar positions, velocities, and redshifts for S2/S1/S14. No step reduces a claimed prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames a known result. The combination of per-orbit posteriors is a statistical procedure applied to external observations rather than an internal redefinition; the result is therefore not equivalent to its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Only abstract available; ledger entries are inferred at high level from the described modeling choices.

free parameters (2)
  • orbital elements and spacetime parameters for S2, S1, S14
    Inferred via Bayesian MCMC from astrometric and spectroscopic data
  • Λ magnitude
    Upper-bounded as the target parameter
axioms (2)
  • domain assumption Spacetime around Sgr A* is exactly Schwarzschild-de Sitter
    Used to integrate timelike geodesics
  • domain assumption Relativistic redshift and time-delay corrections are sufficient and correctly implemented
    Included in the numerical model

pith-pipeline@v0.9.1-grok · 5680 in / 1345 out tokens · 19819 ms · 2026-06-27T06:13:22.660231+00:00 · methodology

discussion (0)

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Reference graph

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