The Lemaître–Tolman–Bondi Metric with a Central Pointlike Mass
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe paper considers a non-Copernican universe, which is isotropic but not homogeneous. It is based on the LTB metric, where a mass point Md is placed far beyond the horizon to preserve isotropy of the CMB. The equations of general relativity are solved, and Friedman-like equations are derived that look familiar to the FLRW Friedman equations but allow for spatial dependence of curvature and expansion rate. There are terms for Md which reveal it to be the source of dark matter. There is no dark energy, but accelerated expansion is permitted via a cosmological constant. The Hubble and S8 tensions are resolved due to the allowance of spatial dependence of expansion rate.
Like many readers will probably have, I have an issue with the mass point, which seems to define a special location. Furthermore, even in the absence of Md the observer on Earth also seems to have a special location, being at the center of an isotropic but spherically symmetric universe. Nevertheless, due to the Hubble tension, the S8 problem, and the difficult questions of deciding whether or not dark energy is evolving and dark matter is decaying, it may be necessary to consider radical modifications to our notions of the nature of the universe.
This paper and others like would benefit from analysis of cosmology data from the various probes (Sn1a, BAO, CMB), with appropriate limits set on the various parameters. The equations are much more complicated than cosmologists are familiar with, and it would be good to broaden their horizons as the difficult effort to understand the universe proceed.
Author Response
Response to Reviewer 1
We express our gratitude to the reviewer for the thoughtful and constructive evaluation of our manuscript. We are particularly encouraged by the recognition that “radical modifications to our notions of the nature of the universe” may be necessary to address the current cosmological tensions (H0 and S8 ).
1. Regarding the statement “no dark energy”:
The reviewer states that “there is no dark energy, but accelerated expansion is permitted via a cosmological constant”. The cosmological constant Λ incorporated in our Eq. (2) is the standard theoretical representation for dark energy that drives the accelerated expansion of the Universe. We have refined the text following Eq. (2) to clarify that the cosmological constant is
used to represent the effect of dark energy within the Einstein field equations. This addresses the reviewer’s distinction between the mathematical constant and the physical phenomenon of accelerated expansion.
2. Regarding the “Special Location” and the Copernican Principle:
The reviewer expresses concerns that the model defines a “special location” for the central pointlike mass Md and the observer. We would like to clarify this fundamental aspect of our LTB framework:
In our model, the geometric origin (Ld = 0) is occupied by the central point-like mass Md , which acts as a physical boundary condition for the manifold. Consequently, a physical observer cannot occupy this position. The terrestrial observer is necessarily situated at an off-center position, far away from this origin. This configuration strictly adheres to the Copernican Principle: the observer occupies a typical, non-privileged position within the inhomogeneous regime of the manifold. The observed large-scale isotropy of the CMB is maintained because the terrestrial observer is situated at a vast distance from the geometric origin Ld = 0. Furthermore, in the limit Md → 0, the LTB metric reduces to the standard FLRW case, where the choice of origin is merely a coordinate convention, confirming that no “special” physical location is required for the observer.
3. Regarding the Analysis of Cosmological Data (Sn1a, BAO, CMB):
We fully agree with the reviewer that the model would benefit from a rigorous quantitative analysis against various cosmological probes. However, as the reviewer correctly noted, the equations derived in this framework—incorporating scale-dependent Hubble parameters (H and $\widetilde{H}$) and relativistic shear—are significantly more complex than those of the standard FLRW paradigm.
The primary objective of the present work is to provide the foundational general relativistic derivation and the theoretical identification of Md as the material source of dark matter in cosmology. Establishing this robust theoretical framework is a necessary prerequisite for any subsequent numerical verification. As we emphasize in our manuscript: “Consequently, we propose that future analyses, particularly regarding large-scale surveys like DESI or Euclid, should prioritize spatially resolved observations without preemptive angular averaging.”
While a comprehensive parameter estimation is an essential next step for the community, it lies beyond the scope of this fundamental theoretical derivation. We believe that presenting the mathematical completion of the model first is the most rigorous approach to advancing the discussion on inhomogeneous cosmology.
Reviewer 2 Report
Comments and Suggestions for Authorshe authors present a general relativistic analysis of the Lemaître–Tolman–Bondi (LTB) metric, incorporating a cosmological constant Λ and a central pointlike mass Md at the geometric origin. The scale factor is a function of radial distance from the point mass. The title of the manuscript is general, but when I read the last paragraph of section 1, I was surprised. The most of the current problems of ΛCDM model can be solved by this work. I read carefully the text. This work is a completion of a previous work of the same authors (citation [10]) which has been published in a journal (Parana J. Sci. Educ.) without peer reviews. In LTB model the scale factor is a function of radial distance, but there is no significant difference between FRW model, where the scale factor is a function of time (in Cosmology the time is a function of distance). There are no proofs for the conclusions during the text, only assumptions which solve the Hubble tension or the S8 tension. I don't know if the authors can transform this work in possible publication form. I can't suggest any further consideration by the journal.
Author Response
Response to Reviewer 2
We thank the reviewer for their critical reading of our manuscript. However, we would like to address several points where the reviewer’s comments stem from a misunderstanding of the formal and physical nature of our work.
1. Regarding the peer-review status of Ref. [10]:
The reviewer states that Ref. [10] was published in a journal without peer review. We would like to clarify that the Parana Journal of Science and Education (PJSE) is indeed a peer-reviewed journal, as explicitly stated in its editorial policies and on its official website. To ensure complete clarity for the reader, we have updated the manuscript in the Introduction (Sec. 1)
to explicitly refer to it as our “previous peer-reviewed work [10].”
2. Regarding the distinction between the FLRW and LTB models:
The reviewer suggests that there is “no significant difference” between the LTB and FLRW models because time and distance are coupled in cosmology. We respectfully emphasize that from a general relativistic standpoint, this is not the case. The FLRW metric is a highly constrained, globally homogeneous subcase of the more general LTB metric. While FLRW mandates a single, purely time-dependent expansion rate H(t), the LTB metric introduces radial gradients and relativistic shear. This allows for a non-trivial ratio between the expansion rates H(L, t) and $\widetilde{H}(L, t)$, which both evolve as explicit functions of the radial coordinate L.
The FLRW model only emerges from our LTB framework in the limit Md → 0 or L → ∞. Consequently, the FLRW model must be ruled out as a global template because it cannot physically account for the observed H0 and S8 tensions without the introduction of undetected particle species. Our LTB framework provides the necessary geometric degrees of freedom to
resolve these tensions qualitatively and reasonably.
3. Regarding the nature of the proofs:
The reviewer notes a lack of quantitative proofs. We would like to clarify that the present work provides a foundational general relativistic derivation by obtaining the exact local Friedmann equations directly from the Einstein field equations. Demonstrating that a self-consistent geometric mechanism exists to resolve the current cosmological anomalies constitutes a theoretical proof.
As argued in the text, many current cosmological data are subjected to isotropic sky-averaging, which masks the underlying radial gradients our model predicts. Therefore, providing a rigorous theoretical completion of the model is the essential first step before any subsequent numerical calibration can be performed.
4. Conclusion:
While the reviewer suggests no further consideration, we believe that providing a mathematically rigorous alternative to the standard ΛCDM paradigm is of significant value to the community, especially given the ongoing empirical discrepancies in modern cosmology.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe paper study cosmological model based on BLT metric, describing an isotropic but not homogeneous universe with a central point. In the source-free case, the only solution is the Schwarzschild solution. After a special substitution, one can also find homogeneous FLRW metrics describing a spatially flat or curved (open or closed) universe. However, typically, this metric is used for the description of gravitational collapse. The central assumption made in the paper concerns the source energy-momentum tensor consisting of dust (pressureless baryonic) matter and radiation homogeneously distributed, i.e., admitting only time dependence. Then the authors discuss cosmological consequences of such a scenario, arguing that one can explain in this way both the dark matter (central mass) and cosmological tension problems. The paper is interesting enough to be published.
Author Response
Response to Reviewer 3
We would like to thank Reviewer 3 for the positive evaluation and the recommendation for publication. To ensure the theoretical consistency of the discourse, we would like to provide a brief clarification regarding several technical points mentioned in the report:
- Vacuum Solution (Kottler vs. Schwarzschild): The reviewer notes that the source-free case leads to the Schwarzschild solution. However, within our framework, which accounts for the cosmological constant Λ, the appropriate vacuum limit is the Kottler metric (Schwarzschild–de Sitter). It is important to emphasize that the Kottler metric is static, whereas our model employs comoving coordinates to correctly describe the dynamical evolution of the expanding Universe.
- Incompatibility with the FLRW Metric: The reviewer suggests that the FLRW metric can be recovered through a “special substitution”. We wish to clarify that our inhomogeneous LTB metric, characterized by a non-zero central pointlike mass, is mathematically distinct from the homogeneous FLRW metric. A transformation between the two is not possible under these conditions. The FLRW metric emerges only as a limiting case when the central pointlike mass vanishes (Md = 0) or L → ∞.
- Isotropy and Inhomogeneity: Our model describes a Universe that is isotropic only with respect to the center, but is generally inhomogeneous. This specific architecture is fundamental to our proposed explanation of dark matter (via the central pointlike mass) and the resolution of current cosmological tensions.
We appreciate the reviewer’s recognition that the paper provides an interesting perspective on these fundamental problems.
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsI read the authors' response to the comments I had made during the first phase of the evaluation. The authors avoided responding to the substance of the comments I made. I do not know the reason, but their responses do not satisfy me. Consequently, I cannot recommend further evaluation by the journal.
