Simulations of Galaxy Formation and Comparison with Observations

Understanding the formation and evolution of galaxies is a fundamental challenge in astrophysics, necessitating a comprehensive comparison between theoretical models and observational data. This paper reviews the critical role of numerical simulations in modeling the complex physical processes that govern galaxy formation over cosmic time. By employing advanced computational techniques, researchers can simulate the gravitational interactions of dark matter, the dynamics of baryonic matter, and the intricate feedback mechanisms that influence star formation and galaxy morphology. The theoretical foundations of galaxy formation are grounded in the Λ Cold Dark Matter (ΛCDM) cosmology, which provides a framework for understanding the initial conditions of the universe. The paper discusses the significance of initial density fluctuations, as described by the primordial power spectrum, and their evolution under gravitational collapse, leading to the formation of dark matter halos. The Press-Schechter formalism and its extensions are highlighted as essential tools for predicting halo abundance and mass distribution. In addition to dark matter dynamics, the paper delves into baryonic physics, emphasizing processes such as gas cooling, star formation, and chemical enrichment. The interplay between these processes is crucial for reproducing observed galaxy properties, and the paper outlines various hydrodynamical techniques employed in simulations, including Smoothed Particle Hydrodynamics (SPH) and Adaptive Mesh Refinement (AMR).Key simulation projects, such as the Millennium Simulation, EAGLE, Illustris, and FIRE, are systematically compared, showcasing their unique strengths and contributions to the field. The results of these simulations are juxtaposed with observational data from large surveys like the Sloan Digital Sky Survey (SDSS) and the Galaxy and Mass Assembly (GAMA) survey. The paper presents a detailed analysis of the galaxy stellar mass function, size-mass relation, and star formation main sequence, illustrating both the successes and the ongoing challenges in accurately modeling galaxy properties. Ultimately, this review underscores the importance of continued refinement in simulation methodologies and the integration of observational insights to enhance our understanding of galaxy formation and evolution. By addressing existing discrepancies and exploring future research directions, this work aims to contribute to the broader discourse on the nature of galaxies and their role in the universe.

Galaxies exhibit an astonishing diversity in morphology, size, stellar populations, and gas content. Large surveys such as the Sloan Digital Sky Survey (SDSS) [1] and the Galaxy and Mass Assembly (GAMA) survey [2] have provided extensive datasets to characterize galaxy properties across cosmic time. To interpret these observations, theoretical models grounded in the Λ Cold Dark Matter (ΛCDM) cosmology are essential.

Numerical simulations have become indispensable in this context. They allow us to model the gravitational collapse of dark matter, gas dynamics, star formation, and feedback processes in a self-consistent manner [3]. Such simulations generate synthetic galaxy populations that can be directly compared to observational datasets, enabling the testing and refinement of physical models.

Initial conditions and cosmology

The initial density fluctuations that seeded galaxy formation are well described by the primordial power spectrum, constrained by measurements of the Cosmic Microwave Background (CMB) [3]. These fluctuations grow under gravity, leading to structure formation.

Simulations typically initialize at high redshift (z ∼ 100) using transfer functions generated by Boltzmann codes such as CAMB [4]. The initial density fields are sampled as Gaussian random fields with statistical properties consistent with the concordance cosmological model.

Dark matter halo formation

Dark matter dominates the mass budget and forms the scaffolding of structure in the universe. Overdensities collapse to form bound structures called halos [5]. The abundance of halos as a function of mass and redshift is described by the Press Schechter formalism [6] and its extensions like the ShethTormen model [7]:

$n(M,z)=\frac{p}{M}\frac{dIn{\sigma }^{-1}}{dM}f(\sigma )\text{ (1)}$

where f(σ) is a fitting function calibrated by simulations.

Baryonic physics

Baryons fall into dark matter halos, cooling via radiative processes. Key physical processes include:

• Gas cooling: Atomic, molecular, and metal-line cooling, described by cooling functions [8].
• Star formation: Empirically modeled following the KennicuttSchmidt law [9]:a

${\displaystyle {\sum }_{SFR}}\text{=A (}\frac{{\displaystyle {\sum }_{gas}}}{1M_{\odot }p{c}^{-2}}{\text{)}}^{\text{n}}\text{ (2)}$

where A ∼ 2.5 × 10−4 and n ∼ 1.4.

• Feedback: Supernovae and Active Galactic Nuclei (AGN) inject energy, regulating further star formation [10]. Feedback processes are critical for reproducing observed galaxy properties.
• Chemical enrichment: Metals produced by stars enhance cooling efficiency and influence subsequent star formation [11].

Subgrid physics: In astrophysical simulations, particularly those focused on galaxy formation, the resolution of the computational grid often limits the ability to directly model certain physical processes. These processes, which occur on scales smaller than the resolution of the simulation, are referred to as "subgrid" processes. Key subgrid processes include star formation, black hole growth, and feedback mechanisms from supernovae and active galactic nuclei (AGN). To address these limitations, subgrid models are employed to approximate the effects of these unresolved processes on the larger-scale dynamics of galaxies.

Importance of subgrid models: Subgrid models are essential for several reasons:

• Capturing complex physics: Many physical processes, such as star formation and feedback, occur on scales that are not resolved in cosmological simulations. For instance, the typical resolution of a simulation may be on the order of kiloparsecs, while star formation occurs in molecular clouds on scales of parsecs or less. Subgrid models allow simulations to incorporate the effects of these unresolved processes, enabling a more accurate representation of galaxy evolution.
• Calibration against observations: Subgrid models are calibrated to reproduce key galaxy observables, such as the stellar mass function, galaxy sizes, and star formation rates. By tuning the parameters of these models based on observational data, researchers can ensure that the simulations yield results that are consistent with what is observed in the universe [9,10].
• Facilitating comparisons with observational data: The ability to simulate galaxy populations that closely match observed properties is crucial for testing theoretical models. Subgrid models play a vital role in bridging the gap between simulation results and observational data, allowing for meaningful comparisons that can validate or challenge existing theories of galaxy formation.

Key subgrid processes: Several key processes are typically included in subgrid models:

• Star formation: The Star Formation Rate (SFR) is often modeled using empirical relations, such as the Kennicutt-Schmidt law, which relates the SFR to the gas surface density. This relationship is crucial for determining how efficiently gas is converted into stars within a galaxy. Subgrid models may also incorporate factors such as gas cooling, turbulence, and the influence of magnetic fields to refine star formation predictions.
• Black hole growth: The growth of Supermassive Black Holes (SMBHs) is another critical subgrid process. Models often include mechanisms for gas accretion onto black holes, as well as feedback processes that regulate star formation and gas dynamics in the surrounding environment. The interplay between black hole growth and galaxy evolution is complex, and accurate modeling of this relationship is essential for understanding the co-evolution of galaxies and their central black holes.
• Feedback mechanisms: Feedback from supernovae and AGN is a vital component of galaxy evolution. These processes inject energy and momentum into the surrounding gas, influencing star formation and the overall dynamics of the galaxy. Subgrid models must account for the efficiency and impact of feedback processes, which can vary significantly depending on the local environment and the properties of the galaxy.

Calibration methodologies: Calibration of subgrid models is a critical step in ensuring that simulations produce realistic galaxy populations. This process typically involves:

• Parameter tuning: Researchers adjust the parameters of subgrid models to match observed galaxy properties, such as the stellar mass function and the size-mass relation. This tuning process often relies on a combination of observational data and theoretical predictions to find the optimal set of parameters.
• Validation against observations: Once calibrated, subgrid models are validated by comparing simulation results with observational datasets. This validation process helps identify discrepancies and areas for improvement in the models, guiding further refinements.
• Iterative refinement: The calibration of subgrid models is often an iterative process, where feedback from observational comparisons leads to adjustments in the models. This cycle of refinement is essential for improving the accuracy and predictive power of simulations.

Major simulation projects

Numerous simulation projects have significantly advanced our understanding of galaxy formation and evolution. Each project employs different methodologies and focuses on various aspects of galaxy formation, contributing unique insights to the field.

Millennium simulation: The Millennium Simulation is one of the largest dark matter-only simulations ever conducted, designed to study the large-scale structure of the universe. It was carried out by the Virgo Consortium and utilized a particle mesh method to simulate the evolution of dark matter over cosmic time.

• Key features: The simulation covers a volume of 2 Gpc³ and follows the evolution of 10 billion particles, each representing a dark matter particle. The Millennium Simulation has provided valuable insights into the formation of cosmic structures, including galaxy clusters and the distribution of dark matter.
• Impact on the field: The Millennium Simulation has been instrumental in validating the ΛCDM model and has served as a benchmark for subsequent simulations. Its results have been widely cited in the literature and have influenced our understanding of galaxy formation and clustering.

EAGLE: The EAGLE (Evolution and Assembly of GaLaxies and their Environments) project is a hydrodynamical simulation that aims to reproduce the observed properties of galaxies across cosmic time. It incorporates both dark matter and baryonic physics, allowing for a more comprehensive study of galaxy formation.

• Key features: EAGLE employs a subgrid model for star formation and feedback, calibrated to match the observed galaxy stellar mass function. The simulation covers a volume of 100 Mpc³ and achieves a resolution of 1 kpc, enabling detailed studies of galaxy morphology and evolution [9].
• Impact on the field: EAGLE has successfully reproduced key galaxy observables, such as the stellar mass function and the size-mass relation, providing a robust framework for understanding galaxy formation. Its results have been used to inform observational campaigns and have contributed to the development of new theoretical models.

Illustris and TNG

The Illustris project and its successor, IllustrisTNG (The Next Generation), are large-volume, high-resolution hydrodynamical simulations that capture the co-evolution of dark and baryonic matter in the universe.

• Key Features: Illustris covers a volume of 106.5 Mpc³ and achieves a resolution of 1 kpc, while IllustrisTNG further improves upon this with enhanced physics and resolution. Both projects incorporate sophisticated subgrid models for star formation, feedback, and black hole growth.
• Impact on the field: The Illustris and TNG simulations have provided groundbreaking insights into galaxy morphology, quenching mechanisms, and the role of feedback processes. Their results have been widely disseminated and have significantly influenced the field of galaxy formation [10].

FIRE

The FIRE (Feedback In Realistic Environments) project focuses on high-resolution zoom-in simulations that emphasize the role of stellar feedback in galaxy formation.

• Key Features: FIRE simulations achieve resolutions as high as 0.1 kpc, allowing for detailed studies of star formation and feedback processes in individual galaxies. The project employs a novel approach to modeling feedback, incorporating both supernova and stellar winds.
• Impact on the Field: FIRE has demonstrated the critical importance of stellar feedback in regulating star formation and shaping galaxy properties. Its results have provided valuable insights into the interplay between star formation and feedback, highlighting the need for accurate modeling of these processes in simulations.

Simulations employ various approaches:

• Smoothed Particle Hydrodynamics (SPH) [12]: Describes fluids as particles smoothed over a kernel.
• Adaptive Mesh Refinement (AMR) [13]: Uses a dynamic grid to resolve highdensity regions.
• Moving mesh [14]: Combines advantages of SPH and grid methods; exemplified by the AREPO code.

The comparison of simulation results with observational data is a critical aspect of validating theoretical models of galaxy formation and evolution. By juxtaposing the outcomes of numerical simulations with empirical observations, researchers can assess the accuracy and reliability of their models, identify discrepancies, and refine their understanding of the underlying physical processes. This section discusses the methodologies employed in these comparisons, highlights key findings, and addresses the ongoing challenges faced in reconciling simulations with observations.

Methodologies for comparison

Data sources: Observational data used for comparison typically come from large-scale surveys that provide comprehensive datasets on galaxy properties. Notable surveys include:

• Sloan Digital Sky Survey (SDSS): One of the most extensive astronomical surveys, SDSS has provided detailed photometric and spectroscopic data for millions of galaxies, enabling the characterization of their morphology, stellar populations, and kinematics.
• Galaxy and Mass Assembly (GAMA) Survey: GAMA focuses on the study of galaxy formation and evolution, providing high-quality data on galaxy properties across a wide range of redshifts.
• Hubble Space Telescope (HST): HST has contributed significantly to our understanding of galaxy morphology and structure through its high-resolution imaging capabilities.

Key observables: When comparing simulations with observations, several key galaxy properties are typically analyzed [15]:

• Stellar Mass Function (SMF): The SMF describes the distribution of galaxies as a function of their stellar mass. It is a fundamental observable that reflects the underlying processes of galaxy formation and evolution.
• Size-Mass relation: This relation characterizes how the effective radius of galaxies correlates with their stellar mass. Understanding this relationship is crucial for studying galaxy morphology and the effects of feedback processes.
• Star Formation Rate (SFR): The SFR quantifies the rate at which stars are formed in galaxies. It is a critical parameter for understanding the growth and evolution of galaxies over time.
• Galaxy morphology: The structural properties of galaxies, including their shapes and distributions of stars, gas, and dark matter, provide insights into the formation processes and evolutionary histories of galaxies.

Key findings from comparisons

Stellar mass function: Comparative studies of the stellar mass function have revealed both successes and challenges in simulations [16]. For instance, many simulations, such as EAGLE and Illustris, have successfully reproduced the overall shape of the observed SMF, particularly at intermediate masses. However, discrepancies often arise at the low and high mass ends (Figure 1):

Figure 1: Galaxy stellar mass function: The simulated galaxy stellar mass function (blue solid line) reproduces the broad shape of the observed distribution (red dashed line). Notably, discrepancies at the low and high mass ends highlight ongoing challenges in modeling feedback and quenching.

• Low-Mass galaxies: Simulations frequently underpredict the number of low-mass galaxies compared to observations. This discrepancy may be attributed to insufficient modeling of star formation and feedback processes, which can suppress star formation in low-mass halos.
• High-Mass galaxies: Conversely, simulations may overpredict the abundance of high-mass galaxies. This overabundance can result from overly efficient star formation and insufficient feedback, leading to the formation of too many massive galaxies.

Size-Mass relation: The size-mass relation is another critical area of comparison. Simulations like Illustris and EAGLE have shown reasonable agreement with observational data regarding the effective radii of galaxies [17]. However, some challenges persist (Figure 2):

Figure 2: Size mass relation: Simulated galaxy sizes (blue) are in reasonable agreement with observational data (red). Slight overestimation at higher masses suggests improvements in angular momentum preservation are needed.

• Overestimation of sizes: Many simulations tend to overestimate the sizes of high-mass galaxies. This suggests that the angular momentum conservation during galaxy formation may not be accurately captured in the simulations, leading to larger-than-observed effective radii.
• Environmental effects: The influence of the environment on galaxy sizes is an area of ongoing research. Observations indicate that galaxy sizes can vary significantly based on their local environment, a factor that may not be fully accounted for in some simulations.

Star formation main sequence: The Star Formation Main Sequence (SFMS) describes the correlation between the star formation rate and stellar mass of galaxies [18-19]. Simulations have generally succeeded in reproducing the tight correlation observed in empirical data (Figure 3):

Figure 3:

• Consistency with observations: Both EAGLE and Illustris simulations have shown that the SFMS is well-represented, indicating that the implemented physical models for star formation are effective.
• Discrepancies in SFR: Despite the overall agreement, some simulations may struggle to match the observed SFRs for specific galaxy populations, particularly at the extremes of the mass range. This highlights the need for further refinement of feedback processes and star formation prescriptions.

Ongoing challenges

While significant progress has been made in comparing simulation results with observations, several challenges remain:

• Modeling feedback processes: Accurately modeling the effects of feedback from supernovae and AGN is crucial for reproducing observed galaxy properties. Current models may not fully capture the complexity of these processes, leading to discrepancies in galaxy formation predictions.
• Resolution limitations: The resolution of simulations can impact the accuracy of the results. Processes such as star formation and feedback often occur on scales smaller than the simulation resolution, necessitating the use of subgrid models that may introduce uncertainties.
• Data quality and completeness: The quality and completeness of observational data can also affect comparisons. Incomplete datasets or biases in selection can lead to misleading conclusions about the agreement between simulations and observations.

The study of galaxy formation and evolution is a cornerstone of modern astrophysics, providing insights into the fundamental processes that shape the universe. This paper has reviewed the theoretical foundations, numerical methods, and major simulation projects that contribute to our understanding of galaxy formation. By comparing simulation results with observational data, we have highlighted both the successes and the ongoing challenges faced by researchers in this dynamic field.

One of the primary achievements of contemporary simulations is their ability to reproduce many observed properties of galaxies, such as the stellar mass function, size-mass relation, and star formation main sequence. Projects like EAGLE, Illustris, and FIRE have demonstrated that sophisticated hydrodynamical techniques and well-calibrated subgrid models can yield results that closely align with empirical observations. These simulations have not only validated the Λ Cold Dark Matter (ΛCDM) cosmology but have also provided a framework for understanding the complex interplay between dark matter, baryonic physics, and feedback processes. The ability to generate synthetic galaxy populations that mirror observed distributions is a testament to the advancements in computational astrophysics and the refinement of theoretical models.

However, despite these successes, significant discrepancies remain, particularly at the low and high mass ends of the stellar mass function. Simulations often struggle to accurately predict the abundance of low-mass galaxies, which may be attributed to insufficient modeling of star formation and feedback processes. Conversely, the overabundance of high-mass galaxies in some simulations suggests that feedback mechanisms may not be adequately represented. These challenges underscore the need for continued refinement of subgrid models and feedback prescriptions to enhance the accuracy of simulations. Addressing these discrepancies is crucial for developing a more comprehensive understanding of galaxy formation and the factors that govern the evolution of galaxies over cosmic time.

The comparison of simulation results with observational data also reveals the importance of interdisciplinary collaboration between theorists and observational astronomers. As new observational campaigns, such as those conducted by the James Webb Space Telescope (JWST) and the Euclid mission, provide increasingly detailed data on galaxy properties, the insights gained from these observations can inform and refine theoretical models. Conversely, simulations can guide observational strategies by predicting where to look for specific galaxy populations or phenomena. This reciprocal relationship between theory and observation is essential for advancing our understanding of the universe and the processes that shape it.

Looking ahead, several key areas warrant further investigation. First, improving the modeling of feedback processes from supernovae and AGN is critical for reconciling discrepancies in galaxy properties. Enhanced feedback models that account for the complex interactions between stars, gas, and black holes will be essential for accurately capturing the dynamics of galaxy formation. Additionally, the integration of machine learning techniques into simulation frameworks presents a promising avenue for optimizing subgrid model parameters and improving predictive capabilities. By leveraging large datasets from both simulations and observations, researchers can refine their models and enhance their understanding of the underlying physical processes.

Furthermore, increasing the resolution of simulations will allow for a more detailed examination of the intricate processes that govern galaxy formation. High-resolution zoom-in simulations, such as those conducted in the FIRE project, have already demonstrated the importance of stellar feedback in regulating star formation. Continued efforts to balance resolution with computational efficiency will be crucial for advancing the field and addressing the complexities of galaxy evolution.

In conclusion, the study of galaxy formation through simulations is a rapidly evolving field that has made significant strides in recent years. While substantial progress has been made in reproducing observed galaxy properties, ongoing challenges remain that require further research and refinement. By fostering collaboration between theorists and observational astronomers, improving feedback modeling, and embracing new computational techniques, the astrophysics community can continue to deepen its understanding of galaxy formation and evolution. As we look to the future, the insights gained from both simulations and observations will undoubtedly shape our understanding of the cosmos and our place within it.

As the field of galaxy formation continues to evolve, several key areas present opportunities for future research and advancements. Addressing the challenges identified in current simulations will enhance our understanding of galaxy evolution and improve the fidelity of theoretical models. The following directions are proposed for future investigations:

Improving feedback mechanisms

One of the most significant challenges in simulating galaxy formation is accurately modeling feedback processes from supernovae and active galactic nuclei (AGN). Future research should focus on refining feedback prescriptions to better account for the impact of these processes on star formation rates and galaxy morphology, particularly in high-mass galaxies. Enhanced feedback models will help reconcile discrepancies observed in the stellar mass function and the size-mass relation.

Incorporating machine learning techniques

The integration of machine learning tools into simulation frameworks offers a promising avenue for improving subgrid model calibration. By leveraging large datasets from both simulations and observations, machine learning algorithms can optimize parameters in subgrid models, leading to more accurate representations of physical processes such as star formation and black hole growth. This approach can facilitate the exploration of complex parameter spaces and enhance predictive capabilities.

Increasing resolution in simulations

Higher resolution in zoom-in simulations is essential for studying the intricate details of galaxy formation, particularly regarding angular momentum transfer and the formation of galactic structures. Future projects should aim to increase spatial resolution while maintaining a balance with computational efficiency, allowing for a more detailed examination of the processes that govern galaxy evolution.

Exploring interdisciplinary connections

Future research should emphasize the interdisciplinary nature of astrophysics by exploring how simulations can inform observational campaigns, such as those conducted by the James Webb Space Telescope (JWST) and the Euclid mission. Collaborative efforts between theorists and observational astronomers can lead to a more comprehensive understanding of galaxy formation and evolution, as well as the development of new observational strategies that target specific theoretical predictions.

Addressing computational limitations

Explicitly stating the computational challenges faced in current simulations, such as trade-offs between resolution and volume, will be crucial for guiding future research. Developing new algorithms and computational techniques that can efficiently handle large-scale simulations while maintaining high resolution will be essential for advancing the field.

Public data accessibility

Promoting the availability of simulation data, such as the public release of the IllustrisTNG dataset, will encourage reproducibility and facilitate further research. Future projects should prioritize making simulation outputs accessible to the broader scientific community, fostering collaboration and innovation in the study of galaxy formation.

By pursuing these future directions, researchers can continue to refine our understanding of galaxy formation and evolution, ultimately contributing to a more comprehensive picture of the universe's structure and dynamics.

The author thanks colleagues sharing ideas during preparation of the paper.

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