Level 4: Structure — \(2\text{M}\) years
Elements
Size: \(300 \text{pm}\)
Initial state: Light atoms and molecules (first period)
Production process: Stellar nucleosynthesis
Meta-stable equilibrium: Heavy atoms
Stable future: Iron-56 nucleus
Cosmic foam
Size: \(100 \text{MPc}\)
Initial state: Matter-dominated, transparent universe
Production process: Gravitational collapse
Meta-stable equilibrium: Large-scale structures
Stable future: Distorted structural ruins
We have left the universe in the darkness that characterized its early times in the matter-dominated era. But we know that is not how the universe looks today. What we see when we raise our heads are countless stars! As observations have improved, humans could distinguish that those stars make up distinct galaxies, which in turn assemble into groups, clusters, superclusters, and lastly filaments. All of this fractal organization amounts to what we may call a cosmic foam: at each scale, denser and more complex areas form a thin lineament around enormous voids.
In terms of our levels of organization, (4\(\Omega\)) is indeed zooming from the whole matter-dominated universe (which, remember, is not a snapshot, but a spatiotemporal extension) to that "foam", i.e. to the multiple large-scale structures that appear within that object.
It’s important to note that, even though we’ll evoke smaller objects, like the first stars or quasars in this section, the reference scale is still larger, since those structures are more temporary, and the ones we want to "zoom" on later, such as our Sun and it planetary system, are a different breed of astral objects, born in galaxies, which will be the object of our next section.
The dark epoch : \(2 \text{M}\) to \(100 \text{M}\) years
The main storyline of structure formation might seem quite intuitive: with gravitation now the dominant force, matter gathers into dense clumps, to a point where it can collapse into a ball that’s hot enough to trigger nuclear fusion... aka, a star.
But the details are far from obvious: some matter clouds can be stelliferous, while others will never be. Galaxies only seem to exist within a narrow size range. Protostars can only evolve under specific conditions...
To gain some fascinating insights into this topic, we need to delve in basic thermodynamics. It is quite remarkable that equations and analytical methods that were developed to study steam engines in the nineteenth century can be almost directly ported to to the study of primordial gas clouds in the early universe! Although some of the inputs (like the gas cooling rate) incorporate results from quantum mechanics, the general concepts are the same as in classical studies of thermal systems.
Where did we stand right after recombination, when the world went dark? The contents of the universe were as follows:
Radiation was mostly made up of the products from neutrino and photon decoupling. The latter, the Cosmic Microwave Background had already been redshfited deep into infrared territory by that point.
Matter had settled into a ratio of about 3/4 dark matter to 1/4 ordinary matter.
Since dark matter is not subject to electromagnetic interaction and cannot find itself in a thermal equilibrium with radiation, as hadrons and leptons had, its behavior had been dominated by gravity for a long time. As we’ve seen, the tiny quantum fluctuations that had been blown up by inflation could then act as gravity wells for dark matter, in what would later become the filaments that make up the cosmic foam.
After decoupling, ordinary matter found itself dominated by gravity for the first time, and started to concentrate in these filaments. As we’ve seen, it had settled into a cold, gaseous molecular cloud of mostly dihydrogen, plus some helium.
Dark energy, a cosmological constant term representing the energy from a constant curvature term, cannot be neglected at the cosmic level (it is the reason the universe is near-flat), but doesn’t have any local impact on structure formation.
The dark epoch has elapsed in that monotonous way, a story of cooling and accretion. Right after recombination, the amplitude of overdensities was still very small for regular matter (around \(10^{-6}\)), but much larger already for dark matter, since accretion was already well on its way (on the order of \(1000\)).
Now that we’re looking at longer timescales, it is crucial to take into account that the universe is expanding. It can be summed up with one figure, the scaling factor \(a\), which effectively dilutes distances. Another metric is often used: the redshift \(z\), whose decrease follows the decrease in frequency of a free photon in the expanding universe. For high values, it moves in inverse proportion to the scaling factor: \(a= (1+z)^{-1}\).
The scaling factor itself, in a matter-dominated universe, increases with time following a power law: \[a \propto t^{2/3}\]
\(a\) being a scaling factor, it is expected that the average density evolves as: \(\overline{\rho} \propto a^{-3}\), since it can be seen as the ratio of a constant mass over and increasing volume.
Interestingly, we can also demonstrate that small over-densities (roughly, below \(1\)) evolve linearly with \(a\): \[\frac{\delta\rho}{\overline{\rho}} \propto a\]
At the start of the dark epoch, dark matter is already way beyond that point. But although we’ll focus on ordinary matter structures in this section, it is interesting to rewind and observe what happened to dark matter. Indeed it’s a simpler version of what occurs for ordinary matter, which crucially has the extra ability to react electromagnetically and from a nuclear perspective.
For any volume in the universe (a "cloud" of dark matter), we start with a certain average density and temperature, as well as local perturbations. Then what? The key point, which is the bedrock of structure formation and of all astrophysics really, is as follows: what determines the evolution of any cloud is the balance of power between two opposite actions:
Gravity pulls stuff in: it acts as an attractive force, and over-densities naturally tend to draw everything around them.
Pressure pushes stuff out. In the case of dark matter, the dynamics are collisionless: there are no shocks between particles; however there is gravitational scattering.
We can prove rigorously, but also understand intuitively that, all else being equal:
When the temperature \(T\), the measure of thermal agitation, increases, pressure wins out.
When the density \(\rho\) increases, the cloud’s mass increases and gravity wins out (that is because the cloud’s gravitational energy is about a mass interacting with itself, a \(\rho^2\) term, whilst thermal energy is a linear function of density).
When the length of the cloud \(L\) increases, the volume, and thus the mass of the cloud also increases, and gravity wins out.
For a spherical cloud of diameter \(L\), density \(\rho\) and temperature \(T\), we can define a Jeans parameter, with \(m\) the mass a single gas molecule or atom: \[J = L \sqrt{\frac{G\rho m}{ k T}}\]
When \(J<1\), pressure wins, and the cloud settles in an oscillating state, with pressure waves similar to the ones from acoustic phenomena. When \(J>1\), gravity wins and the cloud experiences collapse: it starts getting continuously smaller and denser.
Equivalently, we can consider the characteristic times for each process. Considering gravity (and ignoring pressure), the free-fall time \(t_{ff}\)(the time it would take for particles on the edge of the cloud to reach its center) defines the scale of the collapse’s timeframe.
\[t_{ff} \approx \frac{1}{\sqrt{G\rho}}\]
Considering pressure (and ignoring gravity), the "speed of sound" \(c_s\) represents the speed of pressure waves in the gaseous medium:
\[c_s \approx \sqrt{\frac{P}{\rho}} = \sqrt{\frac{kT}{m}}\]
The "sound-crossing time" for a cloud of length \(L\) is:
\[t_{sc} = \frac{L}{c_s} = L \sqrt{\frac{m}{KT}}\]
We can see that
\[J=\frac{t_{sc}}{t_{ff}}\]
If the free-fall time is shorter than the sound-crossing time: \(t_{ff}<t_{sc}\), then the cloud collapses faster than the medium is able to redistribute the pressure via "sound waves"; indeed, \(J>1\).
What happens concretely? It’s probably best to work with a simple numerical example to understand the various forces at play.
The first time that structures can form is when dark matter decouples from all forms of radiation and is only influenced by gravitational forces. Of course, we don’t know what dark matter is. For illustration, let’s make one hypothesis that is at least plausible, and assume dark matter is made of supersymmetric neutralinos of mass: \[m_{\tilde{N}^0_1}=1000 \text{GeV/c}^2= 1.7\times 10^{-24} \text{kg}\]
For ordinary particles, decoupling happens when they stop absorbing and emitting photons, so much so that there can be no thermal equilibrium with photons. Of course, dark matter never interacted with photons in the first place. Decoupling simply means that it stops transforming into other particles and becomes inert from the point of view of all interactions except gravitation.
This happens very shortly after the end of cosmic inflation. For the sake of argument, let’s start from there. So early after the Big Bang, it is clear that basic calculations straight out of Newtonian physics are a shamefully rough approximation. Yet remarkably, this approach provides a perfectly valid picture of the structure formation process, at least as a "toy model".
In cosmology, we can extrapolate long-past values of the main thermodynamic parameters using the relations we’ve already mentioned between \(t\), \(a\), and \(\rho\). We can add the temperature \(T\), a length \(L\) and a volume \(V\) to that group.
\[\begin{aligned} a \propto & \quad t^{q} \\ L \propto & \quad a \\ V \propto & \quad a^3\\ T \propto & \quad a^{-1}\\ \rho_{\text{m}} \propto & \quad a^{-3} \end{aligned}\]
For the radiation-dominated era, \(q=1/2\), but when matter dominates, \(q=2/3\). What I’ve called \(\rho_\text{m}\) is the density of matter (either ordinary or dark).
Through observation, we know what these values are for the observable universe today, \(13.8\) billion years after the Big Bang. In the below, we call \(L_\Omega\) the diameter of the observable universe (we don’t know the size of the whole universe and it might very well be infinite). \(\rho_{DM}\) is the density of dark matter specifically (3/4 that of all matter).
\[\begin{aligned} t =& \quad 13.8 \text{Gy}\\ a =& \quad 1\\ L_{\Omega} =& \quad 31 \text{Gpc}\\ T =& \quad 2.7 \text{K}\\ \rho_{\text{DM}}=& \quad 2.2 \times 10^{-27} \text{kg/m}^3 \end{aligned}\]
Now, we can use the formulas above to "rewind" to any point in time and get the approximate value of those parameters. We’ve assumed that dark matter decoupled right after cosmic inflation ends. We know that happened roughly \(10^{-32}\text{s}\) after the Big Bang and can deduce the other parameters.
\[\begin{aligned} t =& \quad 10^{-32}\text{s}\\ a =& \quad 2 \times 10^{-26}\\ L_{\Omega} =& \quad 19 \text{m}\\ T =& \quad 1.4 \times 10^{26} \text{K}\\ \rho_{\text{DM}}=& \quad 2.7 \times 10^{50} \text{kg/m}^3 \end{aligned}\]
The observable universe was just a volume nineteen meters wide at the time, of course incredibly dense, since all the matter now in large-scale structure was tightly packed in that volume.
For now, let’s assume the distribution of matter and temperature to be entirely homogeneous. Since \(\rho\) and \(T\) are then given, we can compute the Jean length, the minimum cloud size \(L_J\) for which \(J>1\).
\[\begin{aligned} L_J =& \quad 0.48 \mu\text{ m}\\ M_J =& \quad 62 M_\odot\\ t_{ff} =& \quad 1.5 \times 10^{-20} \text{s} \end{aligned}\]
The outcome is that \(L_J\) is less than a micron. Despite the small size of the observable universe at the time, the possibility of cloud collapse is there. Good news for us: complexity could never emerge if the universe’s matter just settled in an eternal oscillating state; it would be the end of our story. The corresponding cloud mass is \(62\) solar masses.
Cloud of size roughly \(L_J\), called proto-haloes then start to separate and attract the matter around them, all the while collapsing. We have computed above the characteristic free-fall time of \(1.5 \times 10^{-20} \text{s}\): it is huge as compared to the age of the Universe at that point, but of course infinitesimal compared to what we know of actual structure formation. What gives?
In a gravitational collapse, the cloud’s radius diminishes in \(\cos (t/t_ff)\). This means that originally \(L_{halo}\) decreases very slowly. But all the while, the universe expands, which mechanically makes \(L_{halo}\) bigger. The halo’s enlargement from cosmic expansion massively outweighs the halo’s diminution from gravitational collapse. This means that the halo’s density diminishes (in \(1/L^3\)) and that \(t_{ff}\) recedes in the future (it increases in \(L^{3/2}\)).
Gravitational collapse is playing catch up with cosmic expansion for a long, long time!
A quick numerical simulation shows that, taking only graviational energy into account, the halo’s size peaks at \(0.8\) parsec, then complete collapse happens around \(t=2.8\text{Myr}\), almost three million years after its start. Of course, in a physical situation, complete collapse (such that could give rise to a black hole) does not necessarily happen, and the halo usually virializes (reach thermal equilibrium again due to pressure forces).
That does not sound right either. From actual over-densities in ordinary matter, we know that dark matter must have formed structures already before photon decoupling, creating the gravity wells that would later welcome ordinary gas.
The big assumption we’ve made is that dark matter distribution was quasi-homogeneous. What we observe today shows it cannot have been: quantum fluctuations from the quantum gravity era have likely been frozen in the inflation process, generating energy clumps that immediately translated into over-densities after dark matter decoupling.
These over-densities are generally represented as a gaussian random field: local over-densities of all scales exist (\(10^{-6}\), \(1\), \(10^{6}\)...), but the higher the over-density, the rarer and smaller the corresponding volumes are.
We can see that there is an interplay with the Jeans length. Consider an extreme over-density of one million. There are a large number of such over-dense areas (\(\rho>10^{6} \overline{\rho}\)), but their median volume is tiny. The Jeans length for a dense area is much shorter than for an average area, but whatever its value it still acts as a cut-off: most over-dense volumes as too short and will just dissipate via thermal equilibrium; some of them, however, are large enough to undergo gravitational collapse.
Naturally, the free-fall time is shorter, the larger the over-density is. Before, we’ve simulated an effective free-fall time of two million years, starting with no over-density at all. For a Jeans volume with over-density \(1\) starting in the same era, the effective free-fall time is \(166,000\) years. For an over-density of \(100\), it is only \(8,000\) years.
We don’t know exactly the nature of the initial over-density distribution, and if it is indeed Gaussian, we don’t know its characteristic parameters. But we can see qualitatively that a bottom-up scheme is promoted here:
Tiny, hyper-dense regions undergo a quick collapse, and virialize into an even tinier halo.
Larger, denser regions undergo a slower collapse, resulting in larger halos.
On a very large scale, there’s almost no over-density and the parameters of the collapse correspond to our homogeneous scenario.
It’s important to understand that haloes of different sizes are not distinct regions of the universe. It’s a Russian doll system where the tiny haloes are born within larger ones. What we’ve been describing as the thermodynamics of a gas cloud can perhaps be more usefully represented in terms of gravitational mechanics: smaller haloes (dense clumps bathing in a less dense gas) start attracting each other and slowly coalescing, thus creating an over-density over a larger area, boosting the accretion of the surrounding gas, and accelerating the collapse.
When the dark ages start, at the two million-year mark, we can then expect that dark matter as a whole is virialized and that structure formation is more of less stabilized: the cosmic web structure is then already extremely mature. We’re facing the following structures then:
Over-dense regions where dark matter has accrued appear as large surfaces called walls (thus called because today they appear like a wall of galaxies. The largest wall in our observable universe is about 3 billion parsecs wide).
Regions where walls intersect are called filaments.
Regions where filaments intersect are the future loci for galaxy clusters.
Volumes inside the space delineated by the walls are simply called voids.
We have seen that ordinary matter fully decoupled at the start of the golden epoch. Whatever the original over-density distribution, it would be impossible to reach current over-density levels for ordinary matter if dark matter didn’t exist. The gravitational inhomogeneity from the cosmic web leads ton an incommensurable acceleration of that process: hydrogen and helium just slide into pre-existing gravity wells.
Ordinary matter follows the same type of evolution as was just described, , grounded in the gravity/pressure balance, with a major difference: ordinary matter can cool through the dissipation of energy. All that dark matter can do is to swap its thermal agitation for gravitational energy. But an ordinary gas, when dense enough, can also dissipate energy via radiation, thanks to its interactions with the electromagnetic field. Simply put, the kinetic energy of ions, atoms or molecules can be carried away by a photon!
For effective cooling to take place, pressure and temperature need to be high enough, so ordinary matter doesn’t radiate at all when it first starts accreting into the dark matter structures. And when it does, the resulting photons are not in the visible spectrum, so "cosmic dawn" is not on yet!
The cosmic dawn epoch : \(100\) to \(250 \text{M}\) years
At first, cooling then has little impact, and ordinary matter doesn’t behave very differently from dark matter. As density increases, however, the cooling rate becomes more and more significant, and more energetic photons are emitted, resulting in a warm ambient light around the dark matter filaments: that is cosmic dawn.
Just like the free-fall time is a characteristic yardstick for gravitational collapse, we can now define a cooling time \(t_{cool}\). It is inversely proportional to the cooling rate (which only depends on temperature), but also to the gas density. In addition, the cooling rate and cooling time depend on the chemical composition of the gas.
If we consider a Jeans cloud of ordinary matter, its collapse will then largely depend on the cooling time:
If \(t_{cool}>>t_{ff}\): cooling is not important and can be ignored: the cloud evolves like a similar cloud of dark matter.
If \(t_{cool}>t_{ff}\): cooling cannot be ignored. As pressure increases, temperature is not as high as in a dark matter scenario, due to dissipation. This accelerates the collapse (\(t_{ff}\) does not depend on the temperature because it’s a theoretical metric that ignores pressure; thhe actual collapse time depends on it). However cooling is slow enough that the system always finds itself in thermic equilibrium.
If \(t_{cool}<t_{ff}\): cooling is catastrophic. Temperature drops fast on a time scale shorter than the collapse time. The Jeans length then decrases too, and the gas quickly clumps into smaller and smaller, denser and denser volumes.
Thus, the composition and characteristics of an ordinary matter cloud strongly determine its history. Dissipation-less dark matter is much more amorphous: the density distribution can determine the pace and order of structure formation, but the landing zone is known. For any given pressure/density couple, there is a certain volume which will collapse, and then on the long run, aggregates with others, etc. For a molecular cloud, the pressure and density similarly determine a Jeans length, but also a cooling time which influences everything that follows.
At the start of the dark epoch, the gas cloud is too cold for cooling to be efficient, at around \(1000\text{K}\). As it gravitates to the dark matter structures over dozens of millions of years, temperature, pressure and density increase, up to a point when the cooling time of some volumes becomes shorter than their free-fall time.
High-density cores then start appearing, accreting more and more matter, some of them becoming the first proto-stars. As I had mentioned, dark matter only interacts gravitationally, whilst ordinary gases are subject to all four interactions: electromagnetism (e.g. through dissipation), the weak nuclear force (but we know that one is short-range only), and the strong nuclear force.
The strong interaction is now ready to make a comeback! Some protostars never fullfil their destiny and just become brown dwarves (hot astral bodies in a long thermal equilibrium). But some become stars: very hot, very luminous astral bodies that are characterized by nuclear fusion.
As the protostar’s gravity is "pushing in", temperature is rising dramatically, even as cooling is preventing it from becoming too high (which would disperse the matter back into a volatile hot cloud). The star is now a "time machine" that brings us back to previous states of matter in the universe: first, \(\text{H\_2}\) molecules dissociate; then, electrons break free, turning the gas into a plasma of protons and helium nuclei. Finally, the temperature gets high enough that fusion starts. With a similar process to that of the primordial nucleosynthesis, hydrogen nuclei start to turn into helium.
These primordial stars are called (confusingly) "Population III" stars. Although their initial mass is a strong determining factor of their lifecycle, stars’ formation involve more complex factors than the simple cloud dynamics we’ve evoked in this section, such as magnetic fields, angular momentum, etc.
Although that is still a very speculative area of research, those primordial stars are believed to have been super-massive, at least one thousand solar masses, and thus short-lived, going through their lifecycle in a matter of a few million years.
Unlike today’s massive stars, which morph into giants and produce heavier elements like carbon or oxygen in a stable fusion reaction, it is thought that primordial stars undergo complete core collapse before they’ve moved to that phase.
The specific collapse that’s suspected to happen is called "pair instability" and happens when electron-positron pair productions temporarily reduce the radiation pressure, resulting in a sudden and irreversible "win" for gravity. The star implodes, and the sudden burst of energy results into accelerated nuclear fusion that produces heavier elements (for primordial stars: up to iron, but not heavier). Those then get ejected into a diffuse nebula following an explosion, called supernova, leaving behind a neutron star or a spacetime singularity called black hole.
Coming back to our specific case: a few million years into the cosmic dawn epoch, a first generation of primordial start must have come and gone. Even though neutron stars and black holes do take away some density from the molecular cloud the process still continues. The ambient chemical composition is changing enough, though, that a new type of stars may emerge.
In astronomy, metallicity designates the relative density of elements heavier than helium. Primordial stars had quasi-zero metallicity, but the fusion process that stems from their explosion peppers heavier elements in the molecular clouds, resulting in new stars with higher and higher metallicity in some areas.
When that metallicity reaches \(10^{-5}\) (for comparison the metallicity of our Sun is around \(1\%\)), the resulting star is deemed to be part of Population II. Because chemical composition affects the cooling function and multiple nuclear processes, these stars are smaller and longer-lived than primordial stars. Both populations have kept developing in a positive feedback loop.
The reionization epoch : \(250\) to \(400\text{M}\) years
Hower, that equilibrium was to be broken in a key, if still poorly understood, event: re-ionization.
You remember that ordinary matter had become neutral when the universe’s baryonic content turned from an ionized plasma (an electron, nuclei and photons "soup") to a gas of neutral atoms or molecules.
During cosmic dawn, the inside of stars has reverted to a plasma, but the giant surrounding cloud is still molecular. About a quarter billion years after the Big Bang, that is changing however, as the high-energy photons from the billions of hyper-luminescent Population II and III stars are re-ionizing the molecular gas.
A key difference with earlier events is that the original "de-ionization" made the universe transparent, but this re-ionization is not making it opaque all over again: in the meanwhile it has expanded enormously. Ordinary matter can then be back to a plasma state, but it is not dense enough to meaningfully interact with photons, which thereby stay free.
A direct consequence, however, is the end of the first stelliferous era. Re-ionization makes cooling very inefficient, both by virtue of the higher temperature (remember, cooling is efficient within a "sweet spot", and neither at very high nor very low temperatures), and of the medium’s composition (molecules cool much better).
The former molecular cloud then settles into a hot plasma state that doesn’t cool, whose Jeans length is very large, and whose free-fall time is very long (billions of years). This is enough to hinder star production for the time being.
The stable future
The small-scale structures at Level 4 are heavy nuclei and atoms. We had already discussed the fact that no nucleus is truly stable in the very long run, save for Iron, which then constitutes the sole stable state.
Heavier elements are either radioactive (and become lighter through beta decay), or can still undergo \(\alpha\) decay, double-beta decay or spontaneous fission, all extremely infrequent nuclear processes. Elements lighter than Iron can undergo cold fusion to grow heavier. The point of convergence of either process is Iron-56, which cannot undergo a fusion nor a fission.
The large-scale structures from Level 4, which make up the cosmic foam, also have a clear end state, which is simply their demise. Continued expansion of the universe combined with the inevitable increase in entropy entails that the gravitational liens that keep matter tightly knit will stretch to a point where any hint of a structure becomes undetectable. Technically, over-densities will still exist, since expansion will prevent entropy from fully re-homogeneizing the universe’s matter content. But such structures will be loose, decaying, distorted ruins of what they used to be.