12 Newtonian Structure Formation Recap

For most purposes, CDM and baryons are treated together as a single, pressureless matter²⁵²⁵ 25 Note that the pressure in the baryons might be higher than that of the CDM, but we work in a limit where we ignore the pressures altogether. That limit certainly fails on small scales (below the so-called Jeans’ length) or at early times when radiation and baryons are coupled (this point becomes important in Section 15). Nonetheless we will show (Section 4) that quickly after recombination, the fractional overdensity in the baryons approaches that in the CDM. So the matter at these times does behave like a single pressure-free fluid with total density contrast ${\delta }_m$..

Consider the case where the universe is matter-dominated (in our own Universe, this is approximately true for the redshift interval ). Since $H^2\propto \overline{\rho }\propto a^{-3}$, we have $a\propto t^{2/3}$ and so $H=2/(3t)$ and $4\pi G\overline{\rho }=2/(3t^2)$. Equation (321) then gives the evolution of the density fluctuations in the pressure-free matter as

$${\ddot{\delta }}_m+\frac{4}{3t}{\dot{\delta }}_m-\frac{2}{3t^2}{\delta }_m=0.$$

(323)

📝 Exercise 12C By substituting ${\delta }_m\propto t^p$ (or otherwise), show that there are two independent solutions: ${\delta }_m\propto t^{-1}$ and ${\delta }_m\propto t^{2/3}\propto a$.

The growing-mode solution of the density contrast grows in direct proportion to the scale factor. Note that here, in an expanding universe, gravitational attraction has given rise to power-law growth of $\delta$ to be compared to the exponential growth predicted in a non-expanding model (try setting $H=0$ in equation (321) to verify this).

We can also consider perturbations to the Newtonian potential. These are sourced from the normal Newtonian Poisson equation ${\nabla }^2\mathrm{\Phi }=4\pi G\rho$; however here the ${\nabla }^2$ refers to physical Newtonian coordinates, not to the comoving coordinates $𝒙$. Since physical distances are related to comoving distances by the scalefactor $a$, the actual perturbations in the Newtonian potential ${\varphi }_N$ obey

$$a^{-2}{\nabla }^2{\varphi }_N=4\pi G\overline{\rho }\delta$$

(324)

This reveals that the gravitational potential perturbations are constant in time since, in Fourier space,

$$-k^2{\varphi }_N=4\pi G{a}^2\underset{\propto a^{-3}}{\overline{\underbrace{\rho }}}\underset{\propto a}{\underbrace{\delta }}=\mathrm{const}.$$

(325)

At late times, the dominant clustered component is the matter and we have

$${\ddot{\delta }}_m+2H{\dot{\delta }}_m-4\pi G{\overline{\rho }}_m{\delta }_m=0.$$

(326)

In matter domination, this reduces to Eq. (323) and ${\delta }_m$ grows like $a$, but when $\mathrm{\Lambda }$ comes to dominate $a\propto e^{t\sqrt{\mathrm{\Lambda }/3}}$ and $H\approx \mathrm{const}.$ It follows that $4\pi G{\overline{\rho }}_m\ll H^2$ (currently $4\pi G{\overline{\rho }}_m/H^2\sim 0.37$) and

$${\ddot{\delta }}_m+2H{\dot{\delta }}_m\approx 0.$$

(327)

The solutions of this are ${\delta }_m=\mathrm{const}.$ or ${\delta }_m\propto e^{-2Ht}\propto a^{-2}$, so $\mathrm{\Lambda }$ suppresses the growth of structure. Note also that a constant density contrast implies that the gravitational potential decays as $a^2{\overline{\rho }}_m\propto a^{-1}$. This leaves an imprint in the CMB called the integrated Sachs-Wolfe effect (Chapter 15).

During the epoch of radiation domination, a Newtonian analysis is highly questionable. However we will see in Chapter 13 that the equation (321) continues to correctly describe the evolution of the pressureless matter component, so let’s temporarily take that on trust and see what the result is.

The CDM sector feels the gravity of all clustered components; Eq. (320) immediately generalises to the $i$th component of a set of non-interacting (except through gravity) fluids as

$${\ddot{\delta }}_i+2H{\dot{\delta }}_i-4\pi G\sum _j{\overline{\rho }}_j{\delta }_j-\frac{c_s^2}{a^2}{\mathbf{\nabla }}^2{\delta }_i=0.$$

(328)

Specialising to pressure-free CDM we have

$${\ddot{\delta }}_c+2H{\dot{\delta }}_c-4\pi G\sum _j{\overline{\rho }}_j{\delta }_j=0.$$

(329)

We now steal a result from relativistic perturbation theory (Chapter 13), namely that radiation fluctuations with wavelengths smaller than the horizon at any given time oscillate – and their density contrast becomes negligible when averaged over a sufficiently long time period. It follows that the CDM behaves as though the radiation is unclustered; consequently its equation of motion is

$${\ddot{\delta }}_c+\frac{1}{t}{\dot{\delta }}_c-4\pi G{\overline{\rho }}_c{\delta }_c=0,$$

(330)

where we used $a\propto t^{1/2}$ and so $H=1/(2t)$. It turns out that we can neglect the last term – we’ll justify why in the exercise below, but it comes down to the fact that ${\overline{\rho }}_r\gg {\overline{\rho }}_c$. So, ignoring the last term in Eq. (330), we have solutions with ${\delta }_c=\mathrm{const}.$ and ${\delta }_c\propto \ln t$. Overall, this can conveniently be rewritten as

$${\delta }_c=A\ln \frac{t}{t_0}$$

(331)

for constant $A$ and $t_0$. This very slow growth of matter perturbations on sub-horizon scales during radiation domination has an important observational fingerprint in the large scale structure, distorting the matter power spectrum in a way that we will cover later.

😇 Exercise 12D Why was it OK to neglect the last term in equation (330)? Hint: Substitute the logarithmic solution for ${\delta }_c$ into the expression. Using the radiation domination solution, express $t$ in terms of $H$ and then (using the Friedmann equation) replace $H$ with ${\overline{\rho }}_r$’s. Hence argue that the last term must always be negligible.

Before decoupling, the baryon dynamics is linked to that of the radiation by efficient Compton scattering. On sub-Hubble scales, ${\delta }_b$ oscillates like the radiation but, after matter-radiation equality, ${\delta }_c$ grows like $a$. It follows that just after decoupling, ${\delta }_c\gg {\delta }_b$. Subsequently, the baryons fall into the potential wells sourced mainly by the CDM and ${\delta }_b\to {\delta }_c$ as we shall now show.

Ignoring baryon pressure and $\mathrm{\Lambda }$, the coupled dynamics of the baryon and CDM fluids after decoupling is approximately given by

	${\ddot{\delta }}_b+{\displaystyle \frac{4}{3t}}{\dot{\delta }}_b$	$=4\pi G({\overline{\rho }}_b{\delta }_b+{\overline{\rho }}_c{\delta }_c)$		(332)
	${\ddot{\delta }}_c+{\displaystyle \frac{4}{3t}}{\dot{\delta }}_c$	$=4\pi G({\overline{\rho }}_b{\delta }_b+{\overline{\rho }}_c{\delta }_c).$		(333)

We can decouple these equations by transforming to coordinates ${\delta }_m$ and $\mathrm{\Delta }$ where

$${\delta }_m\equiv \frac{{\overline{\rho }}_b{\delta }_b+{\overline{\rho }}_c{\delta }_c}{{\overline{\rho }}_b+{\overline{\rho }}_c},$$

(334)

and $\mathrm{\Delta }\equiv {\delta }_c-{\delta }_b$. Expressed in these new coordinates, the equations become

$$\ddot{\mathrm{\Delta }}+\frac{4}{3t}\dot{\mathrm{\Delta }}=0\mathit{\hspace{1em}}\Rightarrow \mathit{\hspace{1em}}\mathrm{\Delta }=\text{const. or}\mathrm{\Delta }\propto t^{-1/3},$$

(335)

while ${\delta }_m$ follows Eq. (323) and has solutions $\propto t^{-1}$ and $t^{2/3}$. Since

$$\frac{{\delta }_c}{{\delta }_b}=\frac{{\overline{\rho }}_m{\delta }_m+{\overline{\rho }}_b\mathrm{\Delta }}{{\overline{\rho }}_m{\delta }_m-{\overline{\rho }}_c\mathrm{\Delta }}\to \frac{{\delta }_m}{{\delta }_m}=1,$$

(336)

we see that ${\delta }_b$ approaches ${\delta }_c$.

The non-zero initial value of ${\delta }_b$ at decoupling, and, more importantly ${\dot{\delta }}_b$, leaves a small imprint in the late-time ${\delta }_m$. The effect fluctuates with scale because the coupled baryon-photon fluid (before decoupling) undergoes oscillations, as we will see in the next Chapter. The residual baryon acoustic oscillations (BAO) are detectable in the distribution of galaxies and gas through the present-day universe. We will be able to calculate the comoving scale of the oscillations, giving us a standard ruler for observations.

Newtonian physics permits a perturbation theory analysis of the growth of structure in non-relativistic matter, from which we have already extracted some important results. However, these must be taken with the caveat that they need re-checking with relativistic perturbation theory because we now understand Newtonian physics as an approximation to general relativity. We will do this in the next chapter.

Some important results include:

• •

  the growth of density fluctuations (but fixed potentials) in a matter-dominated universe;

• •

  the decay of potentials in a $\mathrm{\Lambda }$-dominated universe;

• •

  the very slow growth, in a radiation-dominated universe, of matter density fluctuations with wavelengths shorter than the horizon; this is due to rapid oscillations in the radiation density and therefore the gravitational potential;

• •

  the equalisation of baryon and cold-dark-matter overdensities after baryons decouple from radiation.