Modelling the expulsion of baryons from haloes from first principles: the role of feedback and of the cosmological constant

1. Problem Statement

The distribution of baryons within dark matter (DM) haloes is a central issue in galaxy formation. While massive galaxy clusters retain a baryon mass fraction ($f_b$) consistent with the cosmic average ($f_{b, \text{cosmic}} = \Omega_b/\Omega_m$), lower-mass haloes ($M_{200} \lesssim 10^{12.8} M_\odot$) are significantly deficient in baryons. This "missing baryon problem" is attributed to astrophysical feedback (stellar winds, supernovae, AGN jets) expelling gas beyond the virial radius.

Previous numerical studies (e.g., Ayromlou et al. 2023) identified an empirical relationship between the closure radius ($R_{\text{closure}}$)—the distance from a halo center where the enclosed baryon fraction returns to the cosmic value—and the halo mass/baryon fraction. However, these relationships were purely empirical fits lacking a physical derivation from first principles. Furthermore, the dependence of this phenomenon on cosmological parameters, specifically the cosmological constant ($\Lambda$), had not been systematically investigated, as most simulations fix cosmology to match the observed Universe.

2. Methodology

Analytical Model Development

The authors derived a first-principles analytical model for the gas density profile and closure radius by balancing three forces acting on baryons:

1. Gravitational Collapse: The self-gravity of the halo.
2. Hydrostatic Pressure: The pressure gradient of the gas.
3. Cosmic Expansion/Dark Energy: The repulsive force due to $\Lambda$.

Key Assumptions & Derivations:

• Force Balance: The model assumes spherical symmetry and solves the equation of motion for a gas parcel, incorporating the cosmological constant term ($\Lambda c^2 r / 3$).
• Equation of State: The gas is treated as an ideal polytropic fluid ($P \propto \rho^{n+1}$).
• Density Profile: In the limit where gas density $\rho \ll \rho_{\text{crit}}$ (far from the halo), the authors derive a power-law density profile: $\rho(r) \propto r^{-\eta}$.
• Closure Radius Definition: $R_{\text{closure}}$ is defined as the radius where $M_b(<R)/M(<R) = \Omega_b/\Omega_m$.
• Final Formula: By integrating the density profile from the virial radius ($R_{200}$) to $R_{\text{closure}}$, they derived an expression (Equation 22) relating $R_{\text{closure}}$ to $R_{200}$, the halo baryon fraction ($f_{b, \text{halo}}$), the gas fraction ($f_{\text{gas}}$), the polytropic index ($n$ or slope $\eta$), and cosmological parameters ($\Omega_m, \Omega_\Lambda, z$).

Numerical Validation

To test the model, the authors utilized a suite of EAGLE cosmological hydrodynamical simulations (Barnes et al. 2018) with varying cosmological constants:

• Cosmologies: $\Lambda = 0$ (Einstein-de Sitter), $\Lambda_0$ (observed value), $10\Lambda_0$, $30\Lambda_0$, and $100\Lambda_0$.
• Feedback: All simulations used identical subgrid physics for star formation and feedback (stellar and AGN), calibrated to the observed Universe, isolating the effect of varying $\Lambda$.
• Analysis: They computed the closure radius for haloes with $M_{200} > 10^{11} M_\odot$ across redshifts $0 < z < 3$. They performed $\chi^2$ minimization to fit the analytical model parameters ($\eta$ and $f_{\text{gas}}$) to the simulation data.

3. Key Contributions

1. First-Principles Derivation: The paper provides the first analytical derivation of the closure radius based on the balance of gravity, pressure, and dark energy, moving beyond purely empirical fits.
2. Cosmological Dependence: It systematically quantifies how the cosmological constant ($\Lambda$) influences baryon evacuation, a factor previously neglected in closure radius studies.
3. Physical Explanation of Empirical Fits: The authors demonstrate that the empirical linear relation proposed by Ayromlou et al. (2023) is a limiting case of their more general first-principles model.
4. Quantification of Dark Energy's Role: The study isolates the specific contribution of dark energy to baryon loss, distinguishing it from astrophysical feedback.

4. Key Results

• Model Accuracy: The analytical model shows excellent agreement with EAGLE simulation data for haloes with $M_{200} > 10^{11} M_\odot$ across all tested cosmologies ($0 \le \Lambda \le 30\Lambda_0$) and redshifts ($0 < z < 3$).
• Parameter Evolution:
  • The gas density slope ($\eta$) and gas fraction ($f_{\text{gas}}$) exhibit redshift evolution.
  • Effect of $\Lambda$: Increasing $\Lambda$ leads to a lower best-fit $\eta$, indicating a more gradual decline in gas density with radius (gas is more spread out). It also leads to a lower $f_{\text{gas}}$, meaning haloes retain less gas in their inner regions.
• Linear Approximation: A linear approximation of the model (valid at fixed redshift) yields parameters ($\eta_{\text{approx}} \approx 1.8$, $f_{\text{gas, approx}} \approx 0.002$) that are remarkably constant across redshift and cosmology. This suggests that the primary driver of closure radius variations is the change in cosmological parameters ($\Omega_m, \Omega_\Lambda$) rather than changes in the intrinsic thermodynamic properties of the gas.
• Impact of Dark Energy:
  • In the fiducial universe ($\Lambda_0$), dark energy contributes to baryon evacuation by approximately 30% relative to the effects of astrophysical feedback and gas pressure (comparing $\Lambda_0$ to an EdS universe).
  • In high-$\Lambda$ universes ($\Lambda \ge 10\Lambda_0$), dark energy becomes the dominant factor in baryon evacuation. The closure radius can extend to tens of virial radii, and the accelerated expansion prevents baryons from re-accreting.
• Comparison with Ayromlou et al. (2023): While the A23 formula fits well for $\Lambda = \Lambda_0$, it deviates significantly for higher $\Lambda$ values. The authors' model generalizes the A23 formula, providing a physically motivated explanation for the empirical correlation.

5. Significance and Implications

• Theoretical Robustness: The model proves that the relationship between closure radius and halo mass is robust against variations in the cosmological constant, provided the underlying astrophysical feedback physics remains consistent.
• Anthropic Reasoning: The results have implications for anthropic arguments. In universes with high $\Lambda$, the efficient stripping of baryons from haloes would severely suppress star formation and the formation of heavy elements, potentially rendering such universes inhospitable to life. The study quantifies the "habitability window" regarding the retention of baryons.
• Observational Guidance: The model provides predictive formulas for the closure radius as a function of halo mass, redshift, and cosmology. This offers a framework for interpreting future observations from instruments like JWST and the ELT, which aim to map the distribution of baryons in the circumgalactic and intergalactic media.
• Limitations: The study notes that uncertainties in fitting parameters increase for high-$\Lambda$ simulations due to the scarcity of massive haloes in smaller simulation boxes. Future work requires larger volumes and higher particle counts to refine these constraints.

In summary, this paper successfully bridges the gap between numerical simulations and analytical theory, demonstrating that dark energy plays a non-negligible, and in extreme cases dominant, role in the "missing baryon" problem by accelerating the evacuation of gas from galactic haloes.