Impact of Rastall gravity on hydrostatic mass of galaxy clusters

This study demonstrates that Rastall gravity offers a viable phenomenological framework for mitigating the hydrostatic mass bias in galaxy clusters by effectively aligning hydrostatic mass estimates with observed baryonic or lensing masses, although it does not universally outperform other modified gravity models under standard statistical criteria.

The Gist

Imagine the universe as a giant, cosmic construction site. The biggest buildings on this site are Galaxy Clusters—massive groups of thousands of galaxies held together by gravity.

For a long time, astronomers have been trying to weigh these cosmic buildings. They use two main methods:

1. The "Scale" Method (Hydrostatic Mass): They look at the hot gas inside the cluster. By measuring how hot and dense the gas is, they calculate how heavy the cluster must be to hold that gas down. This is like weighing a balloon by seeing how much air is inside it.
2. The "Bending Light" Method (Lensing Mass): They look at how the cluster bends light from objects behind it. This is like seeing how much a heavy bowling ball bends a rubber sheet; the more it bends, the heavier the ball.

The Problem:
Usually, the "Scale" method says the cluster is much lighter than the "Bending Light" method. It's like the bowling ball looks heavy when you bend the sheet, but light when you weigh the air inside. Astronomers call this the "Hydrostatic Mass Bias." It's a big mystery.

The Proposed Solution: Rastall Gravity
The authors of this paper ask: What if our rules for gravity are slightly wrong?

Standard physics (Einstein's General Relativity) says that energy and momentum are perfectly conserved, like money in a closed bank account. Rastall Gravity suggests that maybe, at very large scales, energy can "leak" or interact with the shape of space in a way we didn't expect. It's like saying the bank account rules change slightly when you have trillions of dollars.

The authors tested this new rulebook on galaxy clusters in two different scenarios:

Scenario 1: The "No Dark Matter" Test

• The Idea: What if there is no invisible "Dark Matter" holding these clusters together? What if the "missing weight" is just because our gravity rules are slightly off?
• The Analogy: Imagine you are trying to balance a scale. You think you are missing a heavy weight (Dark Matter) to make it balance. But what if the scale itself is slightly tilted (Rastall Gravity)?
• The Result: When they applied the Rastall rules, the "Scale" weight dropped down and matched the "visible" weight of the gas and stars almost perfectly.
  • The Score: They got a match of 1.07 (where 1.0 is a perfect match). This is a huge improvement over standard physics, which was way off.

Scenario 2: The "Dark Matter Exists" Test

• The Idea: What if Dark Matter is real, but the "Scale" method is still underestimating the total weight compared to the "Bending Light" method?
• The Analogy: Imagine you know there is a heavy backpack (Dark Matter) on the scale, but the scale still reads too low. Can tweaking the gravity rules help the scale read the correct total weight?
• The Result: Yes! By tweaking the Rastall rules, the "Scale" weight went up and matched the "Bending Light" weight almost perfectly.
  • The Score: They got a match of 0.99. This is incredibly close to a perfect 1.0. It suggests Rastall Gravity can fix the "bias" problem even if Dark Matter exists.

The Catch (The "Fine Print")

While the new gravity rules made the average match look great, the authors did a deeper statistical check (like checking if every single data point fits the line, not just the average).

• They found that while Rastall Gravity fixes the big picture trend, it doesn't perfectly explain every single galaxy cluster better than other theories.
• The Verdict: It's a very promising new tool that helps explain the mystery, but it's not a magic wand that solves everything instantly. It's a strong contender, but scientists need to test it on even more galaxy clusters to be sure.

Summary

Think of this paper as a mechanic trying to fix a car that runs too fast (too much mass) or too slow (too little mass).

• Standard Physics: The car is off by a lot.
• Rastall Gravity: The mechanic tweaks the engine (the gravity rules), and suddenly the car runs almost perfectly.
• Conclusion: The tweak works really well for the overall speed, but the mechanic needs to test it on more cars to see if it's the perfect fix for every single model.

In short: The authors found that if we slightly change how we think gravity works, the mystery of why galaxy clusters seem to weigh different amounts depending on how we measure them largely disappears.

Technical Summary

1. Problem Statement

Galaxy clusters are the largest virialized structures in the universe, dominated by dark matter (DM). A persistent discrepancy exists between the mass of these clusters derived from hydrostatic equilibrium (using X-ray or Sunyaev-Zel'dovich observations) and the mass derived from gravitational lensing. This is known as the hydrostatic mass bias, where lensing masses are systematically higher than hydrostatic masses.

Standard General Relativity (GR) and Newtonian approximations fail to resolve this bias without invoking complex astrophysical non-thermal pressure support or specific DM halo profiles. While modified gravity theories (e.g., non-local gravity, STVG, EiBI) have been proposed to explain the missing mass or reduce this bias, many struggle to produce a linear scaling relation between theoretical and observed masses with a slope close to unity ($M=1$).

2. Methodology

The authors investigate whether Rastall gravity—a theory where the energy-momentum tensor (EMT) is not conserved ($\nabla_\mu T^{\mu\nu} = \lambda \nabla^\nu R$)—can resolve these mass discrepancies. The study proceeds in two distinct scenarios:

• Scenario A (Absence of Dark Matter): The authors assume the cluster mass is composed entirely of baryonic matter. They derive the hydrostatic mass within Rastall gravity and compare it against observed baryonic masses. The goal is to find a Rastall parameter ($\lambda$) that yields a linear fit slope ($M$) close to 1, implying Rastall gravity alone can account for the mass without DM.
• Scenario B (Presence of Dark Matter): The authors assume DM exists but aim to alleviate the bias between hydrostatic and lensing masses. They compare the Rastall-modified hydrostatic mass against observed lensing masses to see if the theory reduces the discrepancy.

Theoretical Framework:

1. Formalism: The authors derive the hydrostatic equilibrium condition in Rastall gravity by starting with the Tolman-Oppenheimer-Volkoff (TOV) equation modified for Rastall theory. They then take the weak-field (Newtonian) limit to derive a modified mass equation:
   $$m(r) = -\frac{k_B}{\mu m_p} \frac{\kappa\lambda - 1}{4\kappa\lambda - 1} \tau(r)r \left( \frac{d \ln \rho}{d \ln r} + \frac{d \ln \tau}{d \ln r} \right) - \frac{\kappa\lambda}{4\kappa\lambda - 1} \left( \frac{\kappa r^3 \rho}{2} + \frac{c^2 r}{G} \frac{d \ln \rho}{d \ln r} \right)$$
   where $\lambda$ is the Rastall parameter, $\kappa$ is the coupling constant, and $\tau, \rho$ are temperature and density profiles.
2. Data & Profiles: They utilize observational data for 10 galaxy clusters (for Scenario A) and 9 clusters (for Scenario B). Gas density and temperature profiles are modeled using standard $\beta$-models and cooling flow models (parameters from Ref. [33]).
3. Statistical Analysis:
   • Linear Fitting: They perform linear regression between theoretical and observed masses to determine the slope $M$. Ideally, $M=1$.
   • Goodness-of-Fit: They calculate the Chi-squared ($\chi^2$) and reduced Chi-squared ($\chi^2_\nu$) to evaluate statistical consistency, accounting for the extra degree of freedom introduced by $\lambda$.
   • Likelihood: Relative likelihood functions are computed to assess the probability of the slope being unity.

3. Key Contributions

• Derivation of Rastall Hydrostatic Mass: The paper provides the first explicit derivation of the hydrostatic mass equation for galaxy clusters specifically within the Rastall gravity framework, bridging the gap between the relativistic TOV equation and Newtonian cluster analysis.
• Dual-Scenario Constraint: It uniquely constrains the Rastall parameter $\lambda$ in two regimes: one attempting to replace Dark Matter entirely, and another attempting to correct the hydrostatic bias in a DM-dominated universe.
• Comparative Statistical Evaluation: Unlike previous studies that focused solely on linear slopes, this work rigorously evaluates the goodness-of-fit ($\chi^2_\nu$) to determine if the improved slopes come at the cost of statistical consistency.

4. Key Results

Scenario A: Absence of Dark Matter (Comparison with Baryonic Mass)

• Standard Gravity: The Newtonian hydrostatic mass is significantly higher than the baryonic mass, yielding a slope $M = 8.24 \pm 0.86$.
• Rastall Gravity: By tuning the Rastall parameter, the hydrostatic mass is effectively reduced.
  • Best fit achieved at $\lambda = 1.14 \times 10^{-1}$.
  • Resulting slope: $M = 1.07 \pm 0.11$.
  • Significance: The value $M=1$ lies within the uncertainty range and the high-likelihood region. This suggests Rastall gravity can phenomenologically account for the cluster mass using only baryonic matter, outperforming other modified gravity models like EiBI ($M \approx 0.12$) and non-local gravity ($M \approx 0.84$) in terms of slope proximity to unity.

Scenario B: Presence of Dark Matter (Comparison with Lensing Mass)

• Standard Gravity: The slope between Newtonian hydrostatic mass and lensing mass is $M = 0.88 \pm 0.23$.
• Rastall Gravity: The theory helps alleviate the bias.
  • Best fit achieved at $\lambda = 1.29 \times 10^{-3}$.
  • Resulting slope: $M = 0.99 \pm 0.26$.
  • Significance: This is extremely close to unity, indicating that Rastall gravity effectively reduces the hydrostatic mass bias, bringing hydrostatic estimates in line with lensing measurements.

Goodness-of-Fit Analysis (Crucial Caveat)

• While Rastall gravity produces slopes closer to unity than standard gravity or other modified theories, the reduced Chi-squared ($\chi^2_\nu$) values indicate that the model does not universally outperform others.
• For the baryonic mass comparison, Rastall gravity ($\chi^2_\nu \approx 2.50 - 6.16$) showed a worse fit than non-local gravity ($\chi^2_\nu \approx 1.74$), despite non-local gravity having a slope further from unity ($0.84$).
• This highlights a trade-off: Rastall gravity captures the global scaling trend (slope) well but does not necessarily minimize the scatter of individual data points better than other models.

5. Significance and Conclusion

The study demonstrates that Rastall gravity provides a viable phenomenological framework for addressing the hydrostatic mass bias in galaxy clusters.

• It successfully modifies the hydrostatic mass equation such that the theoretical mass aligns closely with observed baryonic or lensing masses (slopes $\approx 1$).
• It offers a potential alternative to Dark Matter in the "no-DM" scenario or a mechanism to correct systematic biases in the "DM" scenario.
• Limitation: The improvement in the linear scaling relation does not automatically translate to a superior statistical fit ($\chi^2$) compared to all other modified gravity theories. The authors conclude that while Rastall gravity improves the scaling relations, larger datasets and more refined modeling are required to fully validate its superiority over standard GR or other modified theories.

The paper concludes that Rastall gravity is a promising candidate for resolving mass discrepancies at the cluster scale, but its acceptance depends on future statistical evaluations with larger samples.