Reconciling the Fundamental Plane of Early-Type Galaxies with hydrodynamical simulations: The case of IllustrisTNG100-1

This study demonstrates that the long-standing discrepancy between the observed Fundamental Plane of Early-Type Galaxies and hydrodynamical simulations like IllustrisTNG100-1 can be largely resolved by adopting observationally motivated measurement techniques and accounting for mass-dependent Initial Mass Function variations, highlighting the critical role of observational realism and stellar population modeling in interpreting galaxy scaling relations.

The Gist

Imagine you are trying to understand how a specific type of car, let's call them "Early-Type Galaxies" (ETGs), are built and how they move. Astronomers have discovered a very tight rule, called the Fundamental Plane (FP), that links three things about these galaxies: how big they are, how bright they are, and how fast their stars are moving inside them.

Think of this like a rule for cars: "If a car is heavier and faster, it must also be a certain size." This rule is so consistent in the real universe that it acts like a fingerprint for how these galaxies form.

However, when scientists tried to recreate these galaxies using super-computer simulations (like the IllustrisTNG100-1 project), the rule didn't quite work. The simulated galaxies didn't line up with the real ones. It was like building a virtual car that was the right weight and speed, but the wrong size. Scientists thought this meant their computer models of physics (how gas cools, stars form, and black holes explode) were broken.

This paper says: "Wait a minute. Maybe the physics isn't broken; maybe we just measured the virtual cars wrong."

Here is a breakdown of what the authors found, using simple analogies:

1. The "Blurry Lens" Problem (Resolution)

In computer simulations, you can't see every single star perfectly. There is a limit to how small a detail the computer can "see," called the softening length. It's like looking at a high-resolution photo through a slightly blurry lens. If you try to measure the speed of stars right in the center of a galaxy (where the blur is worst), the computer underestimates how fast they are moving.

• The Old Way: Previous studies just took the speed numbers the computer gave them directly. Because of the "blur," these numbers were too low.
• The New Way: The authors created a "virtual catalog" where they applied a correction. They used a mathematical trick to guess what the speed should be if the lens weren't blurry. They also used a more realistic way to measure the galaxy's size and brightness (using a method called Sérsic profiles, which is like fitting a smooth curve to the light rather than just counting pixels).

The Result: When they used these "corrected" measurements, the simulated galaxies suddenly lined up perfectly with the real ones. The "tilt" in the rule disappeared. It turns out the simulation physics were actually doing a decent job; the error was in how the scientists were reading the data.

2. The "Star Recipe" Problem (The IMF)

There is another factor: the Initial Mass Function (IMF). This is essentially the "recipe" for how many big stars vs. small stars are born in a galaxy.

• The Standard Assumption: Most simulations assume every galaxy uses the exact same recipe (a "Chabrier" recipe), producing a standard mix of stars.
• The Reality: Real galaxies seem to change their recipes. Massive galaxies might have a "bottom-heavy" recipe (lots of tiny, dim stars that add a lot of mass but not much light).

The authors tested what would happen if they changed the recipe in their simulations after the fact (a process called "forward modeling"):

• Top-Heavy Recipe (More big stars): This made the simulated galaxies drift further away from reality.
• Bottom-Heavy Recipe (More small stars): This made the simulated galaxies fit the real-world rule even better.

The Big Takeaway

The paper concludes that the long-standing mystery of why computer simulations didn't match the "Fundamental Plane" of real galaxies wasn't necessarily because the physics engines were broken. Instead, it was because:

1. We were measuring the virtual galaxies with "blurry" tools (ignoring resolution limits).
2. We assumed all galaxies use the same "star recipe," when in reality, massive galaxies might have a different mix of stars.

By fixing how we measure the data and allowing for different star recipes, the simulations finally match the real universe. The authors suggest that while the underlying physics of galaxy formation might still need some tweaking, a huge part of the problem was simply how we interpreted the numbers coming out of the computer.

In short: The computer simulation wasn't necessarily failing to build the galaxy correctly; we just needed to learn how to "read the manual" (the data) more accurately to see that it actually did a great job.

Technical Summary

Technical Summary: Reconciling the Fundamental Plane of Early-Type Galaxies with Hydrodynamical Simulations

Problem Statement
The Fundamental Plane (FP) of Early-Type Galaxies (ETGs) represents a tight empirical correlation between effective radius ($R_e$), central velocity dispersion ($\sigma_e$), and mean surface brightness ($\langle I_e \rangle$). While cosmological hydrodynamical simulations, such as IllustrisTNG100-1, successfully reproduce many galaxy properties, they have historically struggled to match the observed "tilt" of the FP (the specific slopes of the relation). Previous discrepancies were often attributed to deficiencies in baryonic physics (e.g., feedback mechanisms) or insufficient resolution. However, a systematic assessment of how galaxy observables are extracted and interpreted within simulations—specifically regarding "observational realism"—has been lacking. A critical issue identified is the underestimation of central velocity dispersions in simulations due to the gravitational softening length, which artificially shallows the potential well at small scales.

Methodology
The authors utilize a "virtual-ETG" catalogue derived from the IllustrisTNG100-1 simulation, containing 10,121 local ($z \le 0.1$) central ETGs with stellar masses between $10.47 \le \log M_*/M_\odot \le 11.95$. The study employs a multi-faceted approach to reconcile simulated data with observations:

1. Observationally Motivated Parameter Extraction:

   • Photometry: Effective radii ($R_e$) and mean surface brightness ($\langle I_e \rangle$) are derived using Sérsic profile modeling applied to synthetic stellar populations (generated via Flexible Stellar Population Synthesis with a Chabrier IMF and dust attenuation).
   • Kinematics: Two distinct velocity dispersion definitions are tested to address softening effects:
     • $\sigma_{e,corr}$: A weighted line-of-sight (LOS) velocity dispersion calculated above the softening length (1 kpc) and statistically corrected to account for the missing central contribution.
     • $\sigma_{e,dyn}$: A dynamically inferred dispersion obtained by solving the isotropic Jeans equation using the simulated 3D stellar and dark matter density profiles, explicitly excluding scales below the softening length.

2. Fitting Techniques:
   The FP parameters ($a, b, c$ in $\log R_e = a \log \sigma_e + b \log \langle I_e \rangle + c$) are determined using both "direct" fits (minimizing residuals in the dependent variable, $R_e$) and "orthogonal" fits (minimizing residuals perpendicular to the plane).

3. IMF Forward Modeling:
   To test the impact of non-universal Initial Mass Functions (IMF), the authors apply a posteriori prescriptions to the simulation data. They emulate two scenarios based on observational constraints:

   • Bottom-Heavy (BoH): Increases the fraction of low-mass stars, modifying the stellar mass-to-light ratio ($M_*/L$) via a mismatch factor ($\delta_{IMF}$) while keeping structural properties fixed.
   • Top-Heavy (ToH): Increases massive stars, altering feedback and luminosity, leading to changes in $L$ and $R_e$ while keeping total stellar mass largely unchanged.

Key Results

• Resolution of the Tilt via Kinematic Definitions: The study finds that the choice of velocity dispersion definition is the primary driver of FP slope discrepancies.
  • Using the standard weighted correction ($\sigma_{e,corr}$), the simulated FP slopes ($a \approx 1.04, b \approx -0.56$ for direct fits) remain significantly offset from observational medians.
  • Using the dynamically inferred dispersion ($\sigma_{e,dyn}$), the simulated FP slopes shift to $a \approx 1.18$ and $b \approx -0.71$ (direct fit), bringing them into agreement with observational data (e.g., SDSS samples) within $1\sigma$.
• Impact of IMF Variations:
  • Bottom-Heavy IMF: Applying a BoH IMF further refines the agreement. The resulting FP coefficients ($a \approx 1.11, b \approx -0.73$ for direct; $a \approx 1.45, b \approx -0.77$ for orthogonal) are fully consistent with observational constraints, achieving the lowest Z-scores (median deviation $< 1\sigma$ for orthogonal fits).
  • Top-Heavy IMF: Conversely, a ToH IMF moves the FP coefficients further away from observational values, particularly worsening the agreement for the $b$ parameter.
• Virial Consistency: The corrected FP slopes are dynamically consistent with the virial theorem when accounting for weak non-homology and mild luminosity-dependent mass-to-light ratios. The results suggest that the current TNG feedback implementation is broadly compatible with observed ETG structures, provided observables are extracted realistically.

Significance and Claims
The paper argues that the long-standing tension between simulated and observed Fundamental Planes does not primarily stem from flawed feedback physics or baryonic modeling in the IllustrisTNG100-1 simulation. Instead, the discrepancy arises largely from how galaxy observables are extracted and interpreted.

The authors claim that:

1. Observational Realism is Critical: Adopting observationally motivated definitions, particularly for velocity dispersions that minimize softening-length effects, substantially reduces the mismatch between simulation and reality.
2. IMF Variations are a Key Degeneracy: Variations in the IMF (specifically a bottom-heavy trend in massive galaxies) provide a mechanism to fully reconcile the simulated FP with observations without altering the underlying hydrodynamical feedback models.
3. Re-evaluation of Simulation Limits: The findings suggest that hydrodynamical simulations may already capture the essential physics of ETG formation, but their predictive power is currently limited by the "observational realism" of the post-processing analysis rather than the simulation physics itself.

The study concludes that future improvements in resolving the FP tilt should focus on refining the extraction of observables and incorporating self-consistent, non-universal IMF variations, rather than assuming fundamental failures in current feedback prescriptions.