Formation of the large-scale structures of the Universe

Until now, we have studied the evolution of a homogeneous Universe, at least on very large scales beyond about $100\,\Mpc$. Today, however, we can see that matter is agglomerated in the form of planets, stars, galaxies, clusters of galaxies and superclusters of galaxies. The question that arises in this chapter is: how do these structures form in an expanding Universe? While the formation of the smallest stars involves many physical processes and is highly dependent on local initial conditions, it is possible to develop a simple linear model of the evolution of the largest $\sim 50\,\Mpc$ structures, such as galaxy clusters or superclusters. To do this, we simply use a Newtonian theory of linear perturbations, and calculate the evolution of these perturbations in an expanding Universe.

1Validity regime¶

However, let us state the conditions necessary for the validity of the Newtonian perturbation theory. The first obvious condition is that the perturbation of the metric $g_{\mu\nu}$ and the energy-momentum tensor $T^{\mu\nu}$ must be small. In the case of non-relativistic matter, this means that the perturbations of the density $\delta \rho$ must remain small in front of the mean density $\rho_0$ i.e. $\delta \rho/\rho_0 \ll 1$. The perturbations of the gravity field $\delta \phi$ are also weak compared to the mean gravity field $\phi_0$ i.e. $\delta \phi / \phi_0 \ll 1$.

But in order to return to Newtonian gravity in a homogeneous expanding space-time and study only density perturbations, there is another condition. The size of the regions showing a deviation from the average properties of the universe (or the wavelength of the modes considered) must be much smaller than the Hubble radius $R_H={c / H}$, because in Newtonian gravity the interaction propagates instantaneously. If this is not the case, by the time the information from a collapsing region has travelled to the other side of the region, the expansion factor has changed significantly: a situation that obviously cannot be dealt with using Newtonian gravity in a homogeneously expanding space-time.

2Preamble about classical acoustics¶

The non-relativistic matter field is described by a perfect fluid model. To look at the evolution of disturbances in a perfect fluid is to be interested in the field of acoustics. Here are a few useful concepts.

In an Eulerian description of fluids, for a fluid at rest and perfect, we define its velocity field $\vec v(t, \vec r)$, its pressure field $P(t, \vec r)$ and its mass density field $\rho(t, \vec r)$ as constant:

On top of this, we superimpose low-amplitude perturbations:

In acoustics, it is assumed that compressions are adiabatic and reversible, i.e. if the size of the disturbance is greater than the mean free path of the fluid particles, then heat conduction can be neglected. On the other hand, overpressure generates pressure work which changes the internal energy of a disturbance and therefore its temperature, and this work allows acoustic energy to propagate. We therefore use isentropic compressibility $\chi_S$ instead of isothermal compressibility $\chi_T$ :

We deduce the link between pressure and density perturbations via isentropic compressibility:

There is also local conservation of mass:

As the fluid is perfect, its dynamics is governed by the Euler equation (Navier-Stokes without viscosity):

where gravitation is usually neglected. By combining these three equations, we obtain a d’Alembert equation:

where $c_s$ is the sound speed. For a perfect gas and isentropic transformations, we have Laplace’s Law $PV^\gamma = \cst$ with $\gamma=C_P / C_V$ the ratio of heat capacities at constant pressure $C_P$ and volume $C_V$, also called the adiabatic index. If the fluid particles have $N_{dof}$ degrees of freedom (translations, rotations, vibrations) to store energy in kinetic (thermal) form, then the adiabatic index is $\gamma = 1+2/N_{dof}$. For a diatomic gas such as air at the usual temperature, there are $N_{dof}=5$ degrees of freedom (3 translations and 2 rotations) so $\gamma = 7/5$. From this we derive a formula for the sound velocity as a function of temperature and particle mass^[1] :

with $M$ the molar mass of the fluid and $m_{\text{air}}$ the effective mass of a particle of air if this is the medium under consideration, calculated by :

3Acoustics in an expanding space¶

The CMB emission shows that the Universe is slightly inhomogeneous, with small overdensities of the order of $\delta \rho / \rho_0 \sim 10^{-5}$. These fluctuations are likely to increase over time, leading to a Universe where the density contrast is of the order of $\delta \rho / \rho_0 \sim 1$^[2] (Figure 1,Figure 2). What are the mechanisms behind this growth?

The formation of the large-scale structures of the Universe can be understood by a linear theory of perturbations in the perfect fluid constituted by non-relativistic matter after recombination. To study the evolution of these density perturbations in the matter field, we need to rewrite the previous acoustic equations in the specific context of an expanding medium subject to the law of gravity.

Coordinate systems¶

The equations of classical physics are valid using physical (non comoving) quantities: proper distance, proper time, etc... However, using physical distances as a coordinate system is impractical in an expanding Universe. A more practical method is to write the equation in comoving space. Note

• $\vec{r}$ the physical position vector (whose norm is the proper distance $D_p(t)$),
• $\vec{\sigma}$ the comoving position vector (whose norm is the comoving distance χ),
• $\vec{v}_r$ the physical velocity,
• $\vec{v}_{\sigma}$ the comoving velocity.

We have the following relationships:

Although the Hubble flow (first term of (11)) should not be considered as a real velocity, it appears in the relationship between $\vec{v}_r$ and $\vec{v}_{\sigma}$. This would be a problem if it could take values of the order of $c$ (or larger). But the condition that $\vec{r} \ll R_H$ guarantees that $\dot{a}\vec{\sigma} \ll c$ (see section Section 1).

How do you go from the physical coordinate system to the comoving coordinates system? The conversion of spatial partial derivatives is fairly straightforward:

The partial time derivative requires closer examination. Let’s write the differential of any function of time and space $f(t,\vec{r})$ :

So by identification :

Conservation of mass¶

In physical space, the continuity equation is written :

Using the comoving density $\rho_{\sigma}=\rho a^3$, we can express this equation in comoving space:

By introducing the classic vector calculus formula

where ϕ is a scalar and $\vec{A}$ a vector, the above equation reduces to :

The continuity equation therefore takes exactly the same form in comoving space, using comoving variables. However, this is not the case if we use the particular velocity $a \vec{v}_{\sigma}$.

Euler equation¶

In physical space, the Euler equation for a perfect (non-viscous) self-gravitating fluid takes the usual form:

In comobile space, it becomes :

with $\phi_\sigma = a \phi$ the comobile potential. Ensuring that :

the Euler equation in comoving coordinates reduces to :

We can see two additional terms compared to the version of the Euler equation written in physical coordinates. The extra term $2 H \vec{v}_{\sigma}$ is a drag force created by the expansion on the moving velocities.

The last term seems strange. It seems to cancel out the Newtonian force created at $\vec{\sigma}$ by a non-zero homogeneous density background. It could be argued that this force should be zero, for reasons of symmetry. However, by applying Gauss’s theorem, we can find a non-zero force directed towards the origin of the coordinates (wherever that may be!). The fact is that Poisson’s equation behaves badly (and should not be used) for a source field whose $L^2$ norm is not finite. But of course this implies scales beyond $R_H$ where we don’t expect our theory to hold up. As far as density perturbations are concerned, we will subtract the non-zero mean value and thus get rid of this large-scale inconsistency.

Poisson equation¶

In a universe with no cosmological constant, from general relativity we derive the usual Poisson equation in the limit of weak fields (35) :

where ϕ is the gravitational potential. Given the relationship between the potential and the Newtonian force, the comoving potential is naturally defined as $\phi_{\sigma}=\phi a$, and Poisson’s equation is also unchanged in comoving space:

4Newtonian perturbation theory¶

Solution of order 0¶

The conservation of mass equation in comoving space (18) has the following 0th order solution for a homogeneous and isotropic Universe:

At order 0, the mass density remains stationary and homogeneous and there is no particular velocity. Integration of the Poisson equation (24) in spherical coordinates ^[3] gives :

If we use Friedmann’s equations (59) :

for a Universe dominated by matter with $\Lambda=0$ and $\epsilon = c^2 \rho_{\sigma,0} a^{-3} \gg P$, we obtain the gravitational force:

The gradient of the gravitational field gives a centrifugal force of order 0.

From Euler’s equation (22) and the previous results, we arrive at :

Linear solution of order 1¶

Consider small perturbations around the zeroth-order solution:

The quantity δ is called the density contrast and is widely used in the theory of structure formation. Note that it has the same value in physical space as in comobile space. By injecting these expressions into the continuity, Poisson and Euler equations, the zeroth-order solution cancels out (as expected) and by removing the second-order terms, we obtain the linearised set of equations:

By taking the divergence of the linearised Euler equation (34), the time derivative of the linearised mass conservation equation (33) and reinjecting the Poisson equation (35), we finally obtain :

We introduce the speed of sound $c_s$ in the medium for adiabatic compression:

Then we obtain the equation for the growth of perturbations in comoving space:

Sound speed¶

In a Universe dominated by radiation, the properties of relativistic gas are :

hence the sound speed in relativistic plasma:

In a Universe dominated by non-relativistic matter, the thermodynamic properties of baryons are :

with $m_b=(1-Y_p)m_\mathrm{H} + Y_p m_{\mathrm{He}} \approx 1.72 m_p$ the effective mass of the baryons in a gas mixture of atomic hydrogen and helium (note the similarity with (10)), and $\gamma_b=5/3$ the adiabatic index of the baryons because the hydrogen is in atomic form (and the helium, of course). For a reversible adiabatic transformation, with Laplace’s law we have $P_b \rho_b^{-\gamma_b}=\cst$, the temperature of the baryons $T_b$ evolves as :

Now $n_b \propto a^{-3}$ therefore

The sound speed in the baryon gas is :

Just after the photons and electrons have decoupled, the temperature of the baryons is $T_{\mathrm{dec}} = 3055\,\kelvin$ so the speed of sound in the baryon gas is about $1.5\times 10^{-5}c \approx 5\,\mathrm{km}/\mathrm{s}$. The change is of several orders of magnitude, so the evolution of the perturbations changes drastically at this instant. Then the temperature of the baryons decreases in $a^{-2}$, at least until heating by interstellar radiation changes the situation (around $z\sim 10$), so the sound speed decreases in $a^{-1}$.

Source:Large-scale structures

Jeans instability¶

The solutions to the perturbation evolution equation (38) will be studied next semester, but we can already study their behaviour from their dispersion relation. Let’s look for plane wave solutions of the form :

and study the solutions in the Fourier domain. The dispersion relation is written :

. By neglecting the damping term (of the order of the age of the Universe) in front of the characteristic time of evolution of the perturbations ($\omega \gg H$), we end up with :

where $k_\sigma = 2 \pi / \chi = a k_r$. This is the same dispersion relation as for electromagnetic waves in a plasma. We define the Jeans wavelength (The Royal Society (1902)) $k_J$ and the Jeans length $\lambda_J$ by :

• If $k_r > k_J \Leftrightarrow \lambda < \lambda_J$, then $\omega^2 > 0$ so we have an oscillating solution i.e. an acoustic wave which propagates: the perturbations of small size compared to the length of Jeans oscillate thanks to the pressure force.
• If $k_r < k_J \Leftrightarrow \lambda > \lambda_J$, $\omega^2 < 0$ then we have a non-oscillating solution in exponential: the large structures evolve only under the effect of gravity and increase (and the voids decrease).

As the λ scale evolves with the expansion ($\lambda \propto a$) as well as the Jeans length, the discussion is not easy to conduct along the history of the Universe. Let’s define $M$ the mass in a sphere of radius λ, conserved with expansion, and compare it with the mass of Jeans $M_J$.

• If $M < M_J$, then the structure is too light and the pressure can compensate for the force of gravity: the structure oscillates.
• If $M > M_J$, then the structure is too heavy and collapses gravitationally.

Plotting the Jeans mass as a function of the scale factor $a$ allows us to predict which structures can collapse gravitationally and which are prevented from growing Figure 4 (Weinberg (1989) p. 565).

Source:Large-scale structures

Describing the growth of structures before recombination contradicts the non-relativistic hypothesis on which our study is based. However, the sound speed scale remains a relevant scale and moreover non-relativistic matter dominates after equivalence. Knowing this (and that the relativistic study does not give fundamentally different results from what follows), we will take the liberty of discussing the growth of large structures in the primordial and recent Universe (although it is non-linear today).

From the Figure 4, we can see that before decoupling, structures with the mass of a galaxy or a cluster of galaxies are not heavy enough to collapse (or only in the first instants of the Universe). Pressure waves travel through them and they oscillate. However, after decoupling, the speed of sound drops by 5 orders of magnitude. The decoupling of photons freezes the pressure waves and structures the mass of galaxies and even dwarf galaxies can begin to grow.

Acoustic waves in the primordial Universe¶

Before decoupling, acoustic waves travel through the primordial plasma. Let’s define the comoving sound horizon as :

the maximum comoving distance travelled by an acoustic wave since the beginning of the Universe. At the moment of decoupling, it is $150\,\Mpc/a_\mathrm{dec}$ or $r_s = a_\mathrm{dec} r_s^c \approx 0.13\,\Mpc$. This is therefore the distance that a wave from an overdensity present at the Big Bang was able to travel. This wave propagation results in a positive correlation on the presence of matter on this fundamental spatial scale. This scale is converted into an angular separation on the sky, printed on the CMB temperature anisotropy map and given by :

A scale in distance space gives a sinusoidal power spectrum in reciprocal space. Measuring the position of maxima in the CMB temperature anisotropy power spectrum gives an extremely accurate Planck Collaboration et al., 2020 sound horizon:

which is one of the 6 free parameters of the ΛCDM model.

Optical depth¶

The optical depth τ is defined as the ratio of the number of photons received on Earth without having undergone any Thomson scattering $N(t_0)$ to the number of photons $N(t)$ emitted at a time $t$:

with

The time $t_{\mathrm{ls}}$ for which $\tau=1$ is called the time of last diffusion. This is the time since which a CMB photon has not interacted with an electron. More precisely

This is one of the six parameters of the standard ΛCDM model. After the emission of the cosmic microwave background, we enter the Dark Ages of the Universe, when the Universe is transparent but no stars are yet emitting light. But with the birth of the first stars and galaxies, perhaps 150 million years after the Big Bang, the neutral medium is ionised once again. Although very sparse, CMB photons again interact with electrons by Thomson scattering, which reduces the amplitude of the small-scale anisotropies in the CMB power spectrum, and introduces new anisotropies in the polarisation anisotropies. It is the least well measured parameter of the ΛCDM model at the moment (Figure 6) $\tau = 0.058 \pm 0.006$ Tristram, M. et al., 2024, but it informs about the appearance of the first luminous stars (Figure 7).

Source:Thermal history