Having covered some of the generalities of perturbation theories, we can now return to structure formation in the Universe. Linearised Newtonian physics was used in the prerequisite course PHAS3137 to understand how small density perturbations grow. Newton’s laws turn out to be an adequate approximation of general relativity when describing non-relativistic matter (i.e. when the pressure $P$ is much less than the energy density $\rho $, and the velocities $v$ are much less than the speed of light). It’s worth being aware that, in cosmological research, nearly all computer simulations of the non-linear growth of structure have been programmed in the Newtonian limit. It is therefore useful to understand the properties of the Newtonian solutions. Ultimately, though, the results will only be strictly justified once we rederive them (in the next Chapter) from relativity.
First, let’s set up the variables we need. We will work in a flat Universe with comoving Cartesian coordinates $\bm{x}$. The density $\rho (t,\bm{x})$ is made equal to a background value $\overline{\rho}(t)$ plus a small perturbation:
$$\rho (t,\bm{x})=\overline{\rho}(t)+\delta \rho (t,\bm{x})\equiv \overline{\rho}(t)\left(1+\delta (t,\bm{x})\right)\text{.}$$ | (317) |
The second expression defines the fractional overdensity $\delta $ (sometimes just referred to as the overdensity). We also allow the pressure to be perturbed,
$$P(t,\bm{x})=\overline{P}(t)+\delta P(t,\bm{x})\text{.}$$ | (318) |
This can be rewritten as:
$$P(t,\bm{x})=\overline{P}(t)+{c}_{s}^{2}\overline{\rho}(t)\delta (t,\bm{x}),$$ | (319) |
where ${c}_{s}$ is the fluid sound speed.
📝 Exercise 12A
Prove result (319) starting from (318) and using the definition of sound speed ${\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$c$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$s$}}^{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$2$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\equiv $}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{d}$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$P$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$/$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{d}$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\rho $}$.
Using these definitions and the Newtonian fluid equations, one arrives at the following equation:
$$\ddot{\delta}+2H\dot{\delta}-\frac{{c}_{s}^{2}{\nabla}^{2}\delta}{{a}^{2}}-4\pi G\overline{\rho}\delta =0\text{,}$$ | (320) |
where to make the expression concise I am suppressing the explicit functional dependencies, and $\nabla $ is the gradient with respect to the comoving coordinates $\bm{x}$.
😇 Exercise 12B
Work through the full derivation of this equation in the PHAS3137 notes (provided on Moodle).
The linear-theory result (320) is typical of equations for perturbed fields in that it contains the spatial differential operator ${\nabla}^{2}$ as well as time derivatives. As described in the previous Chapter, the usual trick to simplify such equations is to Fourier transform them in the spatial domain; that is, we multiply equation (320) by ${e}^{-i\bm{k}\cdot \bm{x}}$ and integrate over all $\bm{x}$. Only $\delta $ has any $\bm{x}$ dependence so all other quantities come out the front of the integral; furthermore the transform of ${\nabla}^{2}$ was discussed under Eq. (271) where we noted that one picks up a $-{k}^{2}$ factor. So the Fourier-domain equivalent of (320) is:
$$\ddot{\delta}(\bm{k},t)+2H\dot{\delta}(\bm{k},t)+\left(\frac{{c}_{s}^{2}{k}^{2}}{{a}^{2}}-4\pi G\overline{\rho}\right)\delta (\bm{k},t)=0\text{.}$$ | (321) |
This is now an ordinary differential equation in time $t$. Although the spatial dependence is still encoded in $\bm{k}$ there are no derivatives with respect to $\bm{k}$, making the solutions much easier to find.
By transforming from clock time $t$ to conformal time $\eta $, one can rewrite the above equation as
$${\delta}^{\prime \prime}(\bm{k},\eta )+\frac{{a}^{\prime}}{a}{\delta}^{\prime}(\bm{k},\eta )+\left({c}_{s}^{2}{k}^{2}-4\pi G{a}^{2}\overline{\rho}\right)\delta =0,$$ | (322) |
where primes denote derivatives with respect to $\eta $. This transformation is sometimes helpful for making contact with the equations from GR, where conformal time is a more natural choice of time coordinate. But for the remainder of this chapter we will continue to work with clock time $t$.
We will now quickly summarise some results which you have seen from your previous courses.
For most purposes, CDM and baryons are treated together as a single, pressureless matter^{25}^{25} 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 $$). Since ${H}^{2}\propto \overline{\rho}\propto {a}^{-3}$, we have $a\propto {t}^{2/3}$ and so $H=2/(3t)$ and $4\pi G\overline{\rho}=2/(3{t}^{2})$. Equation (321) then gives the evolution of the density fluctuations in the pressure-free matter as
$${\ddot{\delta}}_{m}+\frac{4}{3t}{\dot{\delta}}_{m}-\frac{2}{3{t}^{2}}{\delta}_{m}=0.$$ | (323) |
The growing-mode solution of the density contrast grows in direct proportion to the scale factor. Note that here, in an expanding universe, gravitational attraction has given rise to power-law growth of $\delta $ to be compared to the exponential growth predicted in a non-expanding model (try setting $H=0$ in equation (321) to verify this).
We can also consider perturbations to the Newtonian potential. These are sourced from the normal Newtonian Poisson equation ${\nabla}^{2}\mathrm{\Phi}=4\pi G\rho $; however here the ${\nabla}^{2}$ refers to physical Newtonian coordinates, not to the comoving coordinates $\bm{x}$. Since physical distances are related to comoving distances by the scalefactor $a$, the actual perturbations in the Newtonian potential ${\varphi}_{N}$ obey
$${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{\underset{\u23df}{\rho}}}\underset{\propto a}{\underset{\u23df}{\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,$$ | (330) |
where we used $a\propto {t}^{1/2}$ and so $H=1/(2t)$. It turns out that we can neglect the last term – we’ll justify why in the exercise below, but it comes down to the fact that ${\overline{\rho}}_{r}\gg {\overline{\rho}}_{c}$. So, ignoring the last term in Eq. (330), we have solutions with ${\delta}_{c}=\mathrm{const}.$ and ${\delta}_{c}\propto \mathrm{ln}t$. Overall, this can conveniently be rewritten as
$${\delta}_{c}=A\mathrm{ln}\frac{t}{{t}_{0}}$$ | (331) |
for constant $A$ and ${t}_{0}$. This very slow growth of matter perturbations on sub-horizon scales during radiation domination has an important observational fingerprint in the large scale structure, distorting the matter power spectrum in a way that we will cover later.
Before decoupling, the baryon dynamics is linked to that of the radiation by efficient Compton scattering. On sub-Hubble scales, ${\delta}_{b}$ oscillates like the radiation but, after matter-radiation equality, ${\delta}_{c}$ grows like $a$. It follows that just after decoupling, ${\delta}_{c}\gg {\delta}_{b}$. Subsequently, the baryons fall into the potential wells sourced mainly by the CDM and ${\delta}_{b}\to {\delta}_{c}$ as we shall now show.
Ignoring baryon pressure and $\mathrm{\Lambda}$, the coupled dynamics of the baryon and CDM fluids after decoupling is approximately given by
${\ddot{\delta}}_{b}+{\displaystyle \frac{4}{3t}}{\dot{\delta}}_{b}$ | $=4\pi G({\overline{\rho}}_{b}{\delta}_{b}+{\overline{\rho}}_{c}{\delta}_{c})$ | (332) | ||
${\ddot{\delta}}_{c}+{\displaystyle \frac{4}{3t}}{\dot{\delta}}_{c}$ | $=4\pi G({\overline{\rho}}_{b}{\delta}_{b}+{\overline{\rho}}_{c}{\delta}_{c}).$ | (333) |
We can decouple these equations by transforming to coordinates ${\delta}_{m}$ and $\mathrm{\Delta}$ where
$${\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.