Gravitational Instability for a Multifluid Medium in an Expanding Universe

This paper presents exact and approximate solutions for the gravitational clustering of a multifluid medium in an expanding Newtonian universe, demonstrating how the evolving ratio between fluid perturbations leads to distinct epochs of nonlinear clustering and galaxy formation for components like primordial hydrogen, helium, and massive neutrinos.

The Gist

Imagine the early universe not as a static, empty stage, but as a giant, stretching balloon filled with two different types of "cosmic soup." This paper, written by physicist D. Fargion in 1983 (and published in 1996), explores how these two soups interact as the balloon inflates, eventually clumping together to form the stars and galaxies we see today.

Here is the story of the paper, broken down into simple concepts and everyday analogies.

1. The Setting: A Stretching Dance Floor

Think of the universe as a massive dance floor that is constantly expanding. On this floor, there are two groups of dancers:

• Group 1: Maybe this is Hydrogen gas.
• Group 2: Maybe this is Helium gas, or perhaps heavy, slow-moving neutrinos (ghostly particles).

In a calm universe, these dancers are spread out perfectly evenly. But sometimes, a little ripple happens—a small crowd gathers here, and a gap opens there. In physics, we call these ripples perturbations.

2. The Problem: Who Clumps First?

Gravity is the invisible force that wants to pull these dancers together into tight groups (clusters). However, the dance floor is stretching (expanding), which tries to pull them apart.

The big question Fargion asks is: If you have two different types of fluids (dancers) mixed together, how do they behave when they try to clump up while the universe expands?

In a simple world with just one type of fluid, we know how they clump. But in a "multifluid" world (like our real universe with different gases and particles), it's more complicated. They pull on each other. Group 1's gravity helps Group 2 clump, and vice versa.

3. The Discovery: The "Race" to Form Galaxies

The most fascinating finding in this paper is about timing.

Imagine two runners on a track. Even if they start at the exact same line and run at the same speed initially, the paper shows that because of how they interact with the "stretching floor" and each other, their relationship changes over time.

• The Metaphor: Think of the two fluids as two different types of snowflakes falling in a windstorm. Even if they start falling together, one type might be heavier or react differently to the wind. Eventually, one type will start clumping into snowballs before the other type does.
• The Result: The paper proves mathematically that even if the two fluids start with identical "ripples" (disturbances), the ratio between them changes constantly. One fluid will become unstable and start collapsing into galaxies earlier than the other.

This means that in the real universe, different types of matter might form structures at different times. Hydrogen might start forming the first stars, while heavier neutrinos or helium might take longer to settle into their own structures.

4. The Math: A Symphony of Equations

The author uses a set of complex equations (labeled as System 3 and 8 in the text) to describe this.

• The "Dot" Notation: In the math, a dot over a letter (like $\dot{\delta}$) just means "how fast this is changing over time."
• The Expansion Factor ($R$): This represents the size of the universe. As $R$ gets bigger, the equations show how the "pull" of gravity fights against the "stretch" of the universe.
• Sound Speeds: The paper also considers how fast sound travels through these fluids. Think of this as how quickly the dancers can react to a change in the crowd. If they react too fast (high sound speed), they resist clumping. If they react slowly, they clump easily.

5. Why Does This Matter?

Before this paper, scientists mostly looked at the universe as having just one type of matter. Fargion showed that because the universe is a "multifluid" (a mix of different things), the story of galaxy formation is more nuanced.

The Takeaway:
The universe isn't a single, uniform event where everything happens at once. It's a layered process. Because different fluids interact and react to the expansion of the universe differently, galaxy formation happens in distinct "epochs" or stages. Some parts of the cosmic soup condense into stars first, while others wait their turn.

Summary in One Sentence

Just as different ingredients in a soup might settle at the bottom at different rates when the pot is shaken, different types of cosmic matter clump together at different times as the universe expands, leading to a staggered, multi-stage history of galaxy formation.

Technical Summary

1. Problem Statement

The paper addresses the problem of gravitational clustering (instability) within a multicomponent fluid medium situated in an expanding universe. While previous studies (notably by Grishchuk and Zel'dovich) had exhaustively analyzed these instabilities in a stationary universe, and Wasserman had studied the limit of zero wavenumber in an expanding universe, there was a need for exact analytic solutions for a general expanding universe containing multiple interacting fluid components.

The specific physical context involves:

• Multifluid Systems: Scenarios such as primordial hydrogen and helium, or cosmological massive neutrinos with different masses and velocity distributions.
• Mutual Interactions: The gravitational coupling between different fluid components is explicitly considered, rather than treating them as isolated.
• Cosmological Evolution: The analysis focuses on matter-dominated epochs (post-recombination) where the expansion of the universe significantly affects the growth of density perturbations.

2. Methodology

The author employs a Newtonian approximation for gravitational instability, following the framework suggested by Bonnor, Grishchuk, and Zel'dovich. The methodology proceeds as follows:

• Perturbation Setup: The system starts with two initially homogeneous and stationary fluid densities ($\rho_1, \rho_2$) and pressures ($p_1, p_2$). Small perturbations are introduced as plane waves with wave vector $\vec{K}$:
  $$\frac{\delta\rho_i}{\rho_i} = \delta_i(t) e^{i\vec{K}\cdot\vec{r}}$$
• Governing Equations: A system of coupled differential equations is derived for the time evolution of the density contrasts $\delta_1(t)$ and $\delta_2(t)$. This system includes:
  • Hubble drag terms ($2\frac{\dot{R}}{R}\dot{\delta}$).
  • Pressure support terms ($v_{is}^2 K^2$).
  • Self-gravity terms ($-4\pi G \rho_i$).
  • Cross-gravity terms ($-4\pi G \rho_j \delta_j$), representing the mutual gravitational attraction between the two fluids.
• Cosmological Assumptions:
  • The universe is approximated as a flat Friedmann universe ($\Omega \approx 1$) for matter-dominated epochs.
  • The scale factor evolves as $R(t) \propto t^{2/3}$, leading to the Hubble parameter $H = \frac{2}{3t}$.
  • Sound speeds are modeled as power laws of time based on the adiabatic index $\gamma_i$: $v_{is} \propto t^{1-\gamma_i}$.
  • The comoving wave vector scales as $K^2 \propto t^{-4/3}$.
• Analytic Approach: The coupled equations are transformed into a system of ordinary differential equations with time-dependent coefficients (Eq. 8). The author seeks exact analytic solutions for specific physically interesting cases where the general system is too complex for closed-form solutions.

3. Key Contributions

• Extension to Expanding Universe: The paper successfully extends the classic stationary universe instability analysis to a dynamic, expanding cosmological background.
• Exact Analytic Solutions: The primary contribution is the derivation of exact solutions for the coupled perturbation equations in two specific regimes:
  1. Identical Fluids: $\gamma_1 = \gamma_2 > 4/3$ and $v_1 = v_2$.
  2. Radiation-like/Specific Case: $\gamma_1 = \gamma_2 = 4/3$ with distinct sound speeds ($v_1 \neq v_2$).
• Multicomponent Generalization: It demonstrates that the fundamental laws governing two-component gravitational instability hold true and can be extended to multicomponent media (e.g., H, He, and massive neutrinos).

4. Results

• Evolution of Perturbation Ratios: A critical finding is that the ratio between the density perturbations of the two fluids ($\delta_1 / \delta_2$) is not constant, even if they start with equal amplitudes.
  • The ratio changes monotonically with time.
  • This implies that different fluid components grow at different rates due to their distinct thermodynamic properties (sound speeds) and gravitational coupling.
• Nonlinear Clustering Epochs: Because the growth rates differ, the fluids reach the threshold for nonlinear clustering (where $\delta \sim 1$) at different times.
  • This leads to the prediction of different epochs of galaxy formation for different components. For instance, a component with lower sound speed (less pressure support) might collapse earlier than a hotter component.
• Instability Criteria: The paper confirms that for an expanding universe, the instability conditions are analogous to the stationary case but modified by the expansion rate and the time-evolving sound speeds. The instability occurs for wave vectors satisfying specific inequalities involving the gravitational potential and pressure terms.

5. Significance

• Structure Formation Theory: The results provide a theoretical basis for understanding how different baryonic and non-baryonic components (like massive neutrinos) contribute to the formation of cosmic structures at different times.
• Galaxy Formation: The finding that different fluids cluster at different epochs offers a mechanism to explain variations in the timing of galaxy formation or the segregation of different matter components in the early universe.
• Analytical Benchmark: By providing exact solutions for specific cases, the paper offers a valuable benchmark for testing numerical simulations of multifluid cosmological evolution.
• Limitations: The author notes that these equations apply primarily to the matter-dominated era. In the radiation-dominated era (earlier than recombination), radiation drag effectively suppresses non-relativistic instability, making this analysis less relevant for that specific epoch.

In summary, Fargion's work bridges the gap between stationary fluid instability theory and realistic expanding cosmology, highlighting that mutual gravitational interactions in a multifluid universe lead to asynchronous structure formation, a crucial insight for modeling the early universe and the distribution of matter.