13 Relativistic Structure Formation

Relativistic perturbation theory supposes that the metric of spacetime can be decomposed into a background piece ${\overline{g}}_{\mu \nu }$ and a perturbation $\delta g_{\mu \nu }$:

$$\mathrm{d}{s}^2=({\overline{g}}_{\mu \nu }+\delta g_{\mu \nu })\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }\text{.}$$

(337)

Just as we did in the Newtonian case, we take ${\overline{g}}_{\mu \nu }$ to be the solution to Einstein’s equations for a homogeneous universe, then require $\delta g_{\mu \nu }$ to be small and find its approximate solution.

📝 Exercise 13A Show that, at linear order in $\delta g_{\mu \nu }$, the inverse metric $g^{\mu \nu }$ may be written $$g^{\mu \nu }={\overline{g}}^{\mu \nu }-\delta g^{\mu \nu }$$ (338) where $\delta g^{\mu \nu }={\overline{g}}^{\mu \alpha }{\overline{g}}^{\nu \beta }\delta g_{\alpha \beta }$. Note particularly the sign change in $\delta g$ when raising its indices (omitting the minus sign is a common source of errors in perturbation theory derivations).

Relative to the Newtonian case, relativistic perturbation theory suffers from added complexity because:

1. 1.

   There are, at first glance, ten variables in the metric perturbation $\delta g_{\mu \nu }$ (compared to just the single Newtonian potential $\mathrm{\Phi }$).

2. 2.

   There are conceptual complications arising from gauge freedom, which in turn arises from the lack of a preferred Newtonian frame.

Gauge freedom²⁶²⁶ 26 The word “gauge” is unhelpfully cryptic, and apparently derives from the problem of railway gauge (i.e. width), as follows: there’s nothing intrinsically better about a railway with rails separated by 4ft 1in rather than 4ft. But to make a coherent railway system, everyone has to choose the same width. Similarly, there’s nothing necessarily better about one “gauge choice” over another in physics, but the choice still needs to be made consistently throughout a given calculation. refers to the fact that two apparently different solutions can, in fact, describe the same physical behaviour. Electromagnetism is the most familiar theory with gauge freedom: for example, the electrostatic potential $\varphi$ can be replaced with $\varphi +{\varphi }_0$ for some constant ${\varphi }_0$ – and there is no physical difference in the behaviour of the system described.

In general relativistic perturbation theory, gauge effects arise because there is more than one way to define a background (homogeneous) metric ${\overline{g}}_{\mu \nu }$ for any given perturbed (inhomogeneous) metric $g_{\mu \nu }$. Figure 14 tries to make this clearer. How should the actual picture be decomposed into a smooth background and perturbations? There is more than one way to do it. In Newtonian perturbation theory, the choice is very natural because Newton’s theory contains a notion of absolute time. We average along slices of fixed time to get the “background”. But in Einstein’s theories, there is no definitive time to reference, so this option is no longer open.

It might seem like the situation is hopeless, but that’s not really the case. We just have to be extra careful about being self-consistent in developing our perturbation theory and distinguishing physical predictions (which must be independent of the gauge choice) from purely mathematical statements (which may be gauge-dependendent). There are essentially three ways to do this:

1. (i)

   1+3 gauge-invariant covariant perturbation theory constructs the entire dynamics of the spacetime in terms of physically observable quantities from the outset. This is very satisfying and beautiful, but is quite unwieldy for most practical purposes. We won’t use it for this course.²⁷²⁷ 27 But if you’re interested, take a look at these very well-written lecture notes that pursue the approach: http://arxiv.org/pdf/gr-qc/9812046v5.pdf.

2. (ii)

   Gauge-invariant perturbation theory (sometimes known as “Bardeen” perturbation theory) starts from gauge-dependent quantities, identifies combinations of these variables that are gauge-independent, then works with those combinations alone. This is closely related to the actual option we’ll pursue which is (iii):

3. (iii)

   Gauge-fixed perturbation theory decides from the outset to work in a specified gauge. This is simplest, but comes with dangers. For example, one can easily over-interpret the resulting gauge-specific equations and come to misleading conclusions. With sufficient care this is not a huge concern and we will not particularly discuss it further. Just bear in mind that, for complete safety, one should always construct physically observable quantities. The density contrast $\delta$, for example, is not strictly observable. An example of mapping the perturbation variables onto something observable will be given when we discuss the cosmic microwave background in Section 15.

One possible approach to gauge fixing is to realise that there is, after all, a uniquely sensible definition of cosmic time – specifically, the time measured by clocks that are freefalling. The gauge implied by this choice is known as the synchronous gauge²⁸²⁸ 28 Usually the freefalling clocks are assumed to coincide with the motion of dark matter on large scales, in which case strictly we have the comoving synchronous gauge. To implement the idea, we have to choose a coordinate system where our chosen family of free-falling particles have ${\dot{x}}^{\mu }=u^{\mu }=(1,0,0,0)$ everywhere, meaning that their spatial coordinates never change:

😇 Exercise 13B Suppose the background metric ${\overline{g}}_{\mu \nu }$ is given by equation (23), i.e. the standard flat FRW metric in cartesian coordinates. Show, by writing the geodesic equation (38) for $\mathrm{d}{x}^{\mu }/\mathrm{d}\lambda =u^{\mu }$ to first order in $\delta g$, that the perturbation to the metric in synchronous gauge must obey $$\delta {\dot{g}}_{00}=0\text{ and }2\delta {\dot{g}}_{0i}-\delta g_{00,i}=0.$$ (339) The simplest solution is $\delta g_{00}=\delta g_{0i}=0$.

What’s rather nice about this conclusion is that already we have reduced the number of degrees of freedom in the perturbed metric from $10$ to $6$ by removing the time-time and space-time parts. This eliminates part of the extra complexity of GR over Newtonian perturbation theory, but we can go even further.

After adopting the synchronous gauge, the GR gravitational field is represented by a symmetric $3\times 3$ matrix with $6$ independent components at each point in space. But Newtonian theory has just one number – the potential – at each point in space. What is all this extra complexity doing, and how can we tame it?

It helps to work in Fourier space from the outset. That is, we take each slice of the universe at fixed conformal time $\eta$ and express the metric perturbation $\delta g_{ab}(\eta ,𝒙)$ in terms of a superposition of fourier modes in the background comoving coordinates:

$$\delta g_{ab}(\eta ,𝒙)=\int \frac{{\mathrm{d}}^3𝒌}{{(2\pi )}^{3/2}}{h}_{ab}(\eta ,𝒌)e^{i𝒌\cdot 𝒙}{a}^2\text{.}$$

(340)

Note that I’ve included a factor of $a^2$ which makes some of the later equations a bit simpler. In the new view, the metric perturbations $h_{ab}$ are a function of fourier wavemode $𝒌$ and conformal time $\eta$. To get evolution equations for $h_{ab}$ we will always consider just one mode at a time, i.e. a fixed $𝒌$; because we are considering linear perturbations, the solution for the general case will just be the sum over all wavemodes. Just as with $\delta g_{\mu \nu }$, in synchronous gauge $h_{00}=h_{0i}=0$, so only the spatial part $h_{ab}$ need be considered.

So, let’s consider the behaviour of one fourier mode. Take it along the $z$ axis, so that $𝒌={(0,0,k)}^{\top }$ to be definite; obviously the coordinate system can be rotated to make this the case for any single initial $𝒌$. There is still a residual freedom in the orientation of the coordinate system: we can rotate around the $z$ axis without changing $𝒌$, so whatever conclusions we draw about the physics of $h_{ab}$ must be left invariant when we do so. We start by classifying the different 3$\times$3 symmetric matrices according to their transformation under this rotation:

☞ Exercise 13C Write down a $3\times 3$ rotation matrix $𝐑(\theta )$ for angle $\theta$ around the $z$ axis. Recalling that any matrix $𝐌$ transforms as ${\mathrm{𝐑𝐌𝐑}}^{\top }$, calculate the transformation under $𝐑(\theta )$ of the following: ${𝐌}_{S1}$ ${𝐌}_{S2}$ ${𝐌}_{V1}$ ${𝐌}_{V2}$ ${𝐌}_{T1}$ ${𝐌}_{T2}$ (341) Convince yourself also that any symmetric matrix $𝐡$ can be written as ${\sum }_i{h}_i{𝐌}_i$.

You should have found in the exercise above that the S1 and S2 matrices above do not change under a rotation around the z axis, which is the direction of propagation of our example wave. So waves proportional to S1 or S2 are called scalar degrees of freedom. Like you’d expect for vectors, V1 and V2 matrices return to their original form once $\theta$ reaches $2\pi$, and for this reason they are called vector degrees of freedom. The T1 and T2 matrices return to their original form once $\theta$ reaches $\pi$, and so they’re neither vectors nor scalars; they are called tensor degrees of freedom.

Our metric perturbation $h_{ab}$ for a single mode $𝒌$ can always be uniquely written as a superposition of all six $𝐌$ matrices above. By extension, we can uniquely decompose a metric into scalar, vector and tensor parts.

Still considering just a single mode, we can begin to see a glimmer of how the GR perturbation theory connects to the Newtonian perturbation theory:

• •

  The scalar part (2 degrees of freedom per mode) drives structure formation and is common to GR and Newtonian theories;

• •

  The vector part (2 degrees of freedom per mode) separates out more naturally in GR, but is also present in Newtonian theory in the form of fluid vorticity;

• •

  The tensor part (2 degrees of freedom per mode) is unique to GR and in fact corresponds to gravitational waves which do not appear in the Newtonian theory.

It’s important that all three parts evolve independently at linear order in perturbation theory, i.e. they do not affect each other. This can be explicitly demonstrated by deriving equations of motion for the superposition of all three types, though it’s actually a matter of logical consistency: all terms in an equation must transform in the same way under a rotation if the underlying physical law does not have a preferred direction. So the linear equations for the three types of perturbation simply must be independent of each other. By a similar argument, the classification of mode types must be gauge-invariant.

Having gauge-fixed, to work out how a given perturbation will evolve in GR is very similar to the Newtonian analysis. One expands all the evolution equations to first order in the perturbed quantities. The difficulty now is the sheer complexity of that manipulation; remember that the Einstein equations dicate the evolution and these involve the Ricci scalar and tensor, in turn calculated from the Riemann tensor, which itself is calculated from the Christoffel symbols, which have to first be calculated from derivatives of the perturbed metric…

Luckily this exercise only has to be done once to give you the evolution equations. In this course we’ll simply quote the results for the Einstein tensor without going through the intermediate derivations.

Before we derive equations that reveal the behaviour of perturbations in the Universe, it is helpful to discuss one piece of terminology: sub- and super-horizon.

In Section 1 we introduced the idea of two objects being either in causal contact or causally disconnected, according to whether their comoving distance is less than or greater than the comoving horizon respectively. When objects are causally disconnected, they cannot affect each other’s behaviour.

We can therefore anticipate that the evolution of a Fourier mode in the Universe will depend greatly on its wavelength and classify the modes accordingly:

• •

  If the wavelength is very large compared to the causal horizon, the mode is called “super-horizon”.

• •

  If the wavelength is very small compared to the comoving horizon, the mode is called “sub-horizon”.

Note that the horizon expands with time (at least, if we temporarily ignore inflation), and therefore a mode that is super-horizon at one moment may become sub-horizon at a later time.

It is useful to formulate what we mean by sub- and super-horizon in mathematical terms. Since the conformal time $\eta$ can be regarded as the comoving horizon scale, we might say that sub-horizon means the comoving wavelength $\lambda$ obeys $\lambda \ll \eta$; conversely super-horizon could mean $\lambda \gg \eta$. Because we more typically work with $k=2\pi /\lambda$, we normally reformulate this as:

$$\text{Sub-horizon: }|k\eta |\gg 1\mathit{\hspace{1em}}\text{Super-horizon: }|k\eta |\ll 1\text{.}$$

(342)

You may have noticed that, technically, a factor of $2\pi$ has gone missing in the conversion between wavelength $\lambda$ and wavenumber $k$; this has been absorbed into the idea of ”much larger” and ”much smaller”, where factors $\sim 6$ are supposed to be negligible. Of course there is a broad “in between” range of scales where $k\eta$ is comparable to unity; in that case, the mode is neither sub- nor super-horizon.

There’s another complication in interpreting the above definitions. If inflation took place, the true comoving horizon is vastly greater than $\eta$ would suggest – recall we set the zero point of $\eta$ to be the beginning of radiation domination for convenience. But if we were to try setting the zero-point of $\eta$ more carefully, it’s not likely to be what we care about; things being in contact a long time ago is unlikely to mean they are physically influencing each other today.

The way around both problems is to think back to Section 2 where we defined the idea of the Hubble radius ${(aH)}^{-1}$ – this is the comoving distance that is in recent causal contact. We can redefine our notion of sub-horizon and super-horizon to look at this ’recent communication’ scale rather than the true horizon:

$$\text{Sub-horizon: }k/(aH)\gg 1\mathit{\hspace{1em}}\text{Super-horizon: }k/(aH)\ll 1\text{.}$$

(343)

Now refer back to Table 2. There we saw that, in radiation domination, matter domination or even during inflation, $aH\propto |{\eta }^{-1}|$. Again working on the principle that factors of proportionality can be lost in the definition of ‘much greater than’, conditions (343) are actually equivalent to (342), despite being motivated by a subtly different comparison.

The upshot: we can use conditions (343) and (342) interchangably as a good translation of the intuitive meaning of sub- and super-horizon limits. The simplicity of these conditions hides away sloppiness in the language, but that sloppiness is rarely a problem in practice.

Practically speaking, the behaviour of perturbations that we care about does change quite significantly depending on the magnitude of $|k\eta |$. The time at which $|k\eta |=1$ is known as horizon crossing – or horizon entry, assuming the mode is coming into the horizon (as in the radiation or matter domianted universes). But recall that during inflation, modes are transported out of the Hubble radius; in this case the transition goes in the opposite direction and the value of $\eta$ for which $|k\eta |=1$ is known as horizon exit.

Tensor perturbations, also known as gravitational waves, have no equivalent in Newtonian theory. We start with them because their evolution is actually far simpler to calculate than for the scalar perturbations.

A gravitational wave of comoving wavenumber $k$ propagating along the $\widehat{z}$ direction has metric perturbation of the form

$$\delta g_{ab}\equiv h_{ab}(\eta )a^2=A(\eta )e^{ikz}{M}_{ab}{a}^2,$$

(344)

where $M_{ij}$ is some fixed linear combination of ${𝐌}_{T1}$ and ${𝐌}_{T2}$ from above – like a “template” perturbation to the metric, which is then modulated in time and space by $A(\eta )$ and $e^{ikz}$ to form the final perturbation $\delta g_{ab}$. Inserting this perturbation into the Einstein equations, we obtain

	${\text{Einstein}}_0^0:$		$8\pi G\delta T_0^0$	$=\delta G_0^0=0$		(345a)
	${\text{Einstein}}_i^0:$		$8\pi G\delta T_i^0$	$=\delta G_i^0=0$		(345b)
	${\text{Einstein}}_j^i:$		$8\pi G\delta T_j^i$	$=\delta G_j^i=-{\displaystyle \frac{1}{a^2}}\left(A^{\prime \prime }+2\mathscr{H}{A}^{\prime }+k^{2}A\right)M_j^i$		(345c)

where primes ${}^{\prime }$ denote derivatives with respect to conformal time $\eta$ (recall $\partial /\partial \eta =a\partial /\partial t$) and $\mathscr{H}=a^{\prime }/a=aH$ is the conformal Hubble parameter.

These equations provide us with an immediate example of how perturbations of different types can’t mix. We have started with a tensor perturbation, so the time-time and time-space components of the Einstein equation (which transform as scalars and vectors respectively under rotations around the $\widehat{z}$ axis) have to vanish. Assuming the matter content of the universe to be a perfect fluid, the tensor part of $\delta T_j^i$ also vanishes at linear order, so the entire Einstein equations boil down to the background evolution plus the new equation governing gravitational waves:

$$\boxed{A^{\prime \prime }+2\mathscr{H}{A}^{\prime }+k^{2}A=0}\text{.}$$

(346)

The solution of this equation depends on the dominant content of the universe through $\mathscr{H}(\eta )$. For radiation domination, $\mathscr{H}=aH=1/\eta$ (Table 2) and you can verify that the solution is

$$A\propto \frac{1}{\eta }\sin k\eta \text{ or }\frac{1}{\eta }\cos k\eta \text{ (radiation domination).}$$

(347)

Since $\eta \propto a$, the overall behaviour described by equation (347) is of an oscillation with a decaying amplitude $\propto a^{-1}$. For matter domination, $\mathscr{H}=2/\eta$ and we have the more complicated solution

$$A\propto \frac{\cos k\eta +k\eta \sin k\eta }{{\eta }^3}\text{ or }\frac{\sin k\eta -k\eta \cos k\eta }{{\eta }^3}\text{ (matter domination).}$$

(348)

For sub-horizon modes $k\eta \gg 1$ [see Eq. (342)], and the behaviour is just like that in the radiation-dominated universe – oscillation with a decaying amplitude $\propto a^{-1}$ (recalling that $\eta \propto a^{1/2}$ in matter domination).

During inflation, $\mathscr{H}=-1/\eta$ ($\eta$ is negative and approaches $0$ from below as $a\to \mathrm{\infty }$). The solution is then

$$A\propto \cos k\eta +k\eta \sin k\eta \text{ or }k\eta \cos k\eta -\sin k\eta \text{ (slow-roll inflation).}$$

(349)

The second of these two solutions decays as $\eta \to 0$, so is not relevant to the universe after inflation ends. But the first tends to a constant, which means that primordial gravitational waves can be generated and survive through to the present day. By matching the first solution of equation (349) onto the first solution of (347), we obtain a history of a gravitational wave that is generated during inflation () and survives through to the present-day universe ($\eta >0$) – see Figure 15.

We are about to look at scalar perturbations, which are the most important for structure formation in our Universe. However it will be convenient to do that not in synchronous gauge but in conformal Newtonian gauge (sometimes just called Newtonian gauge), because it will make the connection to Newtonian perturbation theory clearer.

To derive and understand this gauge, let’s think a bit more about the scalar modes discussed above. Unlike the vector and tensor matrices, where for example ${𝐌}_{V1}\to {𝐌}_{V2}$ under a suitable rotation — and so V1 and V2 are essentially equivalent — the ${𝐌}_{S1}$ and ${𝐌}_{S2}$ matricies cannot be transformed into each other. There are genuinely two separate scalar degrees of freedom in a GR gravitational field, compared with just one in Newtonian physics. Where has this extra freedom come from?

One way to think about it is that Newtonian time is absolute whereas GR time can be distorted, providing the extra degree of freedom. This is obscured in the synchronous gauge where we’ve removed the time freedom by anchoring everything to free-falling particles. The definition of the conformal Newtonian gauge is that the two scalar degrees of freedom appear as

$$\mathrm{d}{s}^2=a^2(\eta )\left[(1+2\mathrm{\Phi }(𝒌,\eta )e^{i𝒌\cdot 𝒙})\mathrm{d}{\eta }^2-(1-2\mathrm{\Psi }(𝒌,\eta )e^{i𝒌\cdot 𝒙}){\delta }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j\right]$$

(350)

again for one Fourier mode (in general there will be a superposition of many such modes). One advantage of the Newtonian gauge is that, provided only scalar modes are present, the space is locally isotropic. This contrasts with the synchronous case where there is a preferred direction defined by the freefalling test particles.

We will shortly see that for perfect fluids and scalar fields, the two potentials $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are actually equal; a single Newtonian potential re-emerges. This is why the Newtonian gauge is so useful and for our analysis of scalar perturbations we will stick to it.

Just in case it’s not yet clear: the same physics can be expressed in the synchronous gauge. The perturbation to the $\mathrm{d}{\eta }^2$ part of the metric will apparently disappear, but the physics encoded in $\mathrm{\Phi }$ will turn up elsewhere in the metric. We don’t have time to do it in this course, but this relationship between gauges can be explicitly written down without too much difficulty. One uses a coordinate transformation in the perturbed universe to change the implied mapping onto the homogeneous background. The new perturbations are then given in terms of the old ones plus a correction due to the coordinate transformation. An extensive discussion of the relationship between Newtonian and synchronous gauges can also be found at http://arxiv.org/pdf/astro-ph/9401007v1.pdf

Let’s now look at scalar peturbations in the conformal Newtonian gauge. These are much more messy, mainly because they couple to the matter in a non-trivial way. We get three equations from the time-time, space-time, and space-space Einstein equations:

	${\text{Einstein}}_0^0:$		$4\pi G{a}^2\delta T_0^0$	$=-k^2\mathrm{\Psi }-3\mathscr{H}({\mathrm{\Psi }}^{\prime }+\mathscr{H}\mathrm{\Phi })$		(351a)
	${\text{Einstein}}_i^0:$		$4\pi G{a}^2\delta T_i^0$	$=i{k}_i({\mathrm{\Psi }}^{\prime }+\mathscr{H}\mathrm{\Phi })$		(351b)
	${\text{Einstein}}_j^i:$		$4\pi G{a}^2\delta T_j^i$	$=-\left[{\mathrm{\Psi }}^{\prime \prime }+\mathscr{H}{(2\mathrm{\Psi }+\mathrm{\Phi })}^{\prime }+(2{\mathscr{H}}^{\prime }+{\mathscr{H}}^2)\mathrm{\Phi }-{\displaystyle \frac{1}{2}}{k}^2(\mathrm{\Phi }-\mathrm{\Psi })\right]{\delta }_{ij}-{\displaystyle \frac{1}{2}}(\mathrm{\Phi }-\mathrm{\Psi })k_i{k}_j$		(351c)

Don’t panic! These nasty-looking equations will boil back down to something manageable faster than you might imagine. Before moving on, let’s just recap exactly what we are looking at above. In each case the right-hand-side is the perturbation to (half) the Einstein tensor $G_{\beta }^{\alpha }/2$, calculated through the long process described at the end of Section 2, and the left-hand side is the perturbation to ($4\pi G$ times) the energy-momentum tensor $T_{\beta }^{\alpha }$ which has not yet been expanded into more useful quantities like density and so on (we’ll do this below). Be aware that both sides are gauge-dependent, and in particular the right-hand-side assumes the conformal Newtonian gauge.

Thankfully, calculating the perturbation to the energy-momentum tensor is much easier than for the Einstein tensor. Let’s do it now for a perfect fluid. We’ll then be ready to boil things down to a single equation that tells us about the evolution of perturbations, just as equation (320) does for the Newtonian theory.

😇 Exercise 13D Let’s calculate the perturbation to the energy-momentum tensor. • First remind yourself (Exercise 1) that for a homogeneous universe, the background 4-velocity ${\overline{U}}^{\alpha }$ must have components ${\overline{U}}^{\alpha }=(a^{-1},0,0,0)$ and hence that ${\overline{U}}_{\alpha }=(a,0,0,0)$. • Next show that, to linear order, $\delta U_0=-a^2\delta U^0$. (Hint: remember both $U^{\alpha }$ and ${\overline{U}}^{\alpha }$ should be unit-normalised, and do not allow yourself to assume $\delta U^{\alpha }$ can have its index lowered by a metric — it can’t, because it’s the difference between tensors that live in spaces with different metrics.) Recall that a perfect fluid has energy-momentum tensor given by equation (89). Use the result you derived above to show that, to first order, $\delta T_0^0$ $=\delta \rho ;$ (352a) $\delta T_i^0$ $={\displaystyle \frac{1}{a}}(\overline{\rho }+\overline{P})\delta U_i;$ (352b) $\delta T_j^i$ $=-\delta P{\delta }_j^i.$ (352c)

With these in hand, we can put everything together to produce the perturbation equations for fluids in a relativistic universe. First, combining eqs. (351c) and (352c), we immediately notice that any off-diagonal terms ($i\ne j$) must vanish. This means

$$k_i{k}_j(\mathrm{\Phi }-\mathrm{\Psi })=0\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em} }(i\ne j).$$

(353)

We are not interested in the case $𝒌=0$ (which corresponds to a homogeneous perturbation that can be absorbed into a change in the background). So we can read off $\mathrm{\Phi }=\mathrm{\Psi }$, which immediately makes everything simpler because we have only one potential to worry about.

Putting together the full set of Einstein equations we now get:

	$-k^2\mathrm{\Phi }-3\mathscr{H}({\mathrm{\Phi }}^{\prime }+\mathscr{H}\mathrm{\Phi })$	$=4\pi G{a}^2\delta \rho ;$		(354a)
	$i{k}_i({\mathrm{\Phi }}^{\prime }+\mathscr{H}\mathrm{\Phi })$	$=4\pi Ga(\overline{\rho }+\overline{P})\delta U_i;$		(354b)
	${\mathrm{\Phi }}^{\prime \prime }+3\mathscr{H}{\mathrm{\Phi }}^{\prime }+(2{\mathscr{H}}^{\prime }+{\mathscr{H}}^2)\mathrm{\Phi }$	$=4\pi G{a}^2\delta P.$		(354c)

To keep things as simple as possible, let’s assume we have a single fluid²⁹²⁹ 29 We will discuss generalisation to multiple fluids in Section 2 with equation of state $P=w\rho$. Then we also have $\delta P=w\delta \rho$ and can combine equations (354a) and (354c) to give:

$${\mathrm{\Phi }}^{\prime \prime }+3(1+w)\mathscr{H}{\mathrm{\Phi }}^{\prime }+w{k}^2\mathrm{\Phi }+\left[2{\mathscr{H}}^{\prime }+(1+3w){\mathscr{H}}^2\right]\mathrm{\Phi }=0\text{.}$$

(355)

The term in square brackets vanishes:

😇 Exercise 13E Verify that $2{\mathscr{H}}^{\prime }+(1+3w){\mathscr{H}}^2=0$ for this case, for example by starting from the Friedmann equation (recalling also that $\overline{\rho }\propto a^{-3(1+w)}$).

Accordingly, the equation starts to look much more tractable:

$$\boxed{{\mathrm{\Phi }}^{\prime \prime }+3(1+w)\mathscr{H}{\mathrm{\Phi }}^{\prime }+w{k}^2\mathrm{\Phi }=0}\text{.}$$

(356)

Note the similarity to equation (346) for the amplitude of a gravitational wave; although there are some key differences in the specific factors that appear, you might (rightly) expect that a similar story to the one presented in Figure 15 can be assembled for scalar perturbations. We’ll put off examining this in detail until Section 14.

Recall that, in addition to the Einstein equations, there is also a conservation equation for energy-momentum ${\nabla }_{\mu }{T}_{\nu }^{\mu }=0$ (Section 2).

If the Universe contains just one fluid, the conservation equation does not add any information to the Einstein equations, since the conservation is already built in geometrically. However, if we have more than one fluid and the fluids do not interact with each other (except through gravity) the fluids individually satisfy energy-momentum conservation. In this case, there is additional information available by looking at the conservation equation for each individual fluid. As we’ve already seen, dark matter, dark energy, baryons and photons are all independent for most of the history of the Universe.

On top of that, we previously discussed that energy-momentum conservation encodes some fluid dynamics familiar from Newtonian physics: the conservation and Euler equations (see Section 2). For this reason, even if we have only one fluid, the conservation equation can provide intuitive insight into the behaviour of GR. So let’s examine its implications.

In Section 2 we showed how energy-momentum conservation implies the continuity equation in the homogeneous limit (taking $\nu =0$). Now that we have perturbed fluid variables and a perturbed metric (which enters through the covariant derviative), at first order we obtain (after simplification) an additional equation, here expressed in Newtonian gauge:

$$\delta {\rho }^{\prime }=-3\mathscr{H}(\delta \rho +\delta P)-(\overline{\rho }+\overline{P})(i{k}_i\delta U_i-3{\mathrm{\Psi }}^{\prime })\text{.}$$

(365)

In the unperturbed universe, the spatial components of energy-momentum conservation vanished identically. However, when applied to the perturbed universe at linear order one gets

$$\delta U_i^{\prime }=-\mathscr{H}\delta U_i-\frac{{\overline{P}}^{\prime }\delta U_i+i{k}_i\delta P}{\overline{P}+\overline{\rho }}-i{k}_i\mathrm{\Phi },$$

(366)

which is the relativistic version of the Euler equation.

We won’t apply these immediately, since for the moment we can get everything we need from the Einstein equations. But we’ll return to them when discussing the CMB in Section 15, where it is essential to consider multiple fluids.

Before the radiation-domination era, there may have been a period of slow-roll inflation (Section 3). Assuming this is driven by a single scalar field, we can continue to trace our perturbations back in time to their origin by looking at how they behave in this non-fluid setting. As when dealing with scalar field/slow-roll cosmology before, we set $8\pi G=1$ in this section.

Start from the energy-momentum tensor for the scalar field, equation (243), then write $\phi =\overline{\phi }(\eta )+\delta \phi (𝒙,\eta )$; finally expand all expressions to first order. This exercise generates

	$\delta T_0^0$	$={\displaystyle \frac{\mathrm{d}V}{\mathrm{d}\phi }}\delta \phi \text{;}$		(367a)
	$\delta T_i^0$	$={\displaystyle \frac{i{k}_i}{a^2}}{\overline{\phi }}^{\prime }\delta \phi \text{;}$		(367b)
	$\delta T_j^i$	$=\left[{\displaystyle \frac{\mathrm{d}V}{\mathrm{d}\phi }}\delta \phi +{\displaystyle \frac{2\mathrm{\Phi }}{a^2}}{\overline{\phi }}^{\prime 2}+{\displaystyle \frac{2}{a^2}}{\overline{\phi }}^{\prime }\delta {\phi }^{\prime }\right]{\delta }_j^i\text{.}$		(367c)

We can get the equation of evolution for the scalar field by applying the Euler-Lagrange equations (232) to the Lagrangian (229), giving us

$$-{a^2\frac{{\mathrm{d}}^{2}V}{\mathrm{d}{\phi }^2}|}_{\overline{\phi }}\delta \phi =-4\mathrm{\Phi }\mathscr{H}{\overline{\phi }}^{\prime }-2\mathrm{\Phi }{\overline{\phi }}^{\prime \prime }+k^2\delta \phi +2\mathscr{H}\delta {\phi }^{\prime }-{\overline{\phi }}^{\prime }{\mathrm{\Phi }}^{\prime }+3{\overline{\phi }}^{\prime }{\mathrm{\Psi }}^{\prime }+\delta {\phi }^{\prime \prime },$$

(368)

where we’ve already made the substitution ${\nabla }^2\to -k^2$ (i.e. $\delta \phi \propto e^{i𝒌\cdot 𝒙}$) by assuming we’re dealing with just one $k$-mode at a time, just as in the fluid case. We now need to understand the relationship between $\mathrm{\Phi }$, $\mathrm{\Psi }$ and $\phi$ to make further progress. In fact, for the sub-horizon case, we’ll be able to show that $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are negligible as follows.

Equation (367a) substituted into (351a) is the Poisson-like equation that will give us the relationship we need once suitably simplified. Just as with the fluid case, we first spot that (367c) is diagonal (zero for $i\ne j$) which implies that $\mathrm{\Phi }=\mathrm{\Psi }$. Then we can use the $0i$ equations from equation (367b) substituted into (351b) to re-express the ${\mathrm{\Psi }}^{\prime }+\mathscr{H}\mathrm{\Phi }$ term in (351a), giving us

$$-k^2\mathrm{\Phi }=\frac{1}{2}\left(\frac{\mathrm{d}V}{\mathrm{d}\phi }{a}^2+3\mathscr{H}{\overline{\phi }}^{\prime }\right)\delta \phi \text{,}$$

(369)

remembering that we are setting $8\pi G=1$ when dealing with scalar fields. Next, we recall that the background evolves according to

$${\overline{\phi }}^{\prime \prime }+2\mathscr{H}{\overline{\phi }}^{\prime }+a^2\frac{\mathrm{d}V}{\mathrm{d}\phi }=0\text{,}$$

(370)

which you showed³³³³ 33 It can also be derived directly from the Euler-Lagrange equations written in conformal time $\eta$ instead of clock time $t$ in an exercise leading to equation (251) (remember that $\mathscr{H}\equiv aH$). This allows us to simplify (369) further to

$$k^2\mathrm{\Phi }=\frac{1}{2}\left({\overline{\phi }}^{\prime \prime }-\mathscr{H}{\overline{\phi }}^{\prime }\right)\delta \phi =\frac{\ddot{\overline{\phi }}{a}^2}{2}\delta \phi \text{,}$$

(371)

where the last step has re-expressed the conformal time derivatives as physical time derivatives: the double overdot denotes ${\mathrm{d}}^2/\mathrm{d}{t}^2$ as normal.

😇 Exercise 13H Show that, in terms of the flatness parameters ${ϵ}_V$ and ${\eta }_V$ introduced in Section 3, the previous equation may be expressed as $$k^2\mathrm{\Phi }=\frac{1}{2}{\mathscr{H}}^2\sqrt{2{ϵ}_V}\left({\eta }_V-{ϵ}_V\right)\delta \phi \text{.}$$ (372)

The exercise above shows that the gravitational potential is suppressed both by the slow-roll factors and, on sub-horizon scales, by a factor ${\mathscr{H}}^2/k^2$. For perturbations within the horizon and even somewhat outside, we can consequently ignore the gravitational effects. We discard all such terms from equation (368) and one additional term which is $\sim a^2\delta \phi {\mathrm{d}}^{2}V/\mathrm{d}{\phi }^2=3{\eta }_V\delta \phi {\mathscr{H}}^2$. Finally, since we assume we are in a period of inflation with $w=-1$, we can replace $\mathscr{H}$ with $-1/\eta$ (Table 2). The equation of motion for the perturbations then boils down to something much more palatable:

$$\delta {\phi }^{\prime \prime }-\frac{2}{\eta }\delta {\phi }^{\prime }+k^2\delta \phi \simeq 0$$

(373)

Solutions to this equation are relatively easy to find; in fact we’ve already stated them in (349) for the gravitational wave, where the e.o.m. was given by equation (346). Note the e.o.m. is identical once you replace $A\to \delta \phi$ and $\mathscr{H}=-1/\eta$. So, one of the solutions (349) decays as $a\to \mathrm{\infty }$ (i.e. as inflation proceeds it is erased) but the other is more interesting:

$$\delta \phi =C\left[\cos \left(k\eta \right)+k\eta \sin \left(k\eta \right)\right],$$

(374)

for arbitrary constant $C$. There is a plot of this function in Figure 15 (though for scalar field perturbations you should only look at the part for , as we’ll need to handle the post-inflationary universe differently).

☞ Exercise 13I Verify that equation (374) solves equation (373) and show that as $a\to \mathrm{\infty }$, $\delta \phi \to C$.

This tendency of $\delta \phi$ to “freeze out” at a fixed value is in contrast to its behaviour on strongly sub-horizon scales where the mode oscillates and decays, just like we saw with gravitational waves.

We are getting very close to being able to predict how inflation seeds structure in the universe. However, the behaviour described by equation 374 is only valid up to the mildly super-horizon regime – i.e. until the increasing ${\mathscr{H}}^2/k^2$ overcomes the suppression by the slow-roll factors. Eventually the potential (372) must become large and we need a different approximation to understand the super-horizon evolution.

In the section above, we threw away a number of terms to make progress. However for super-horizon modes and as slow-roll inflation comes to an end, the resulting approximation is no longer valid and we need to look again.

Before we do so, there is an additional consideration. When inflation ends, a radiation-dominated universe is generated through reheating. This is a complex, still poorly-understood process that takes the energy from the scalar field and dumps it into particles from the standard model and dark sectors. How can we relate perturbations before reheating to perturbations afterwards?

One might imagine that keeping track of fluctuations as an unknown physical process proceeds would be impossible, or at least require numerical integration. But we are saved by two neat properties:

1. 1.

   all scales which will later be of cosmological interest are super-horizon during reheating – we can treat the universe as a series of separate homogeneous patches evolving independently of each other;

2. 2.

   the slow-roll trajectory is an attractor (Section 3). It removes any slight differences between the patches by the time we hit reheating, so each patch goes through an identical process.

Working on super-horizon scales because of the first consideration, the second consideration turns out to imply the following:

$$\frac{\partial ℛ}{\partial \eta }=0\text{ where }ℛ\equiv -\mathrm{\Psi }+\frac{\delta \rho }{3(\overline{\rho }+\overline{P})}\text{.}$$

(375)

It looks like I’ve plucked this out of thin air, so let’s first verify that it is true and then explain why. We will make use of the continuity and Euler equations given³⁴³⁴ 34 Note that, despite being couched in terms of fluid variables, these are just as valid for a scalar field. The conservation equations from which they derive are implied by the Einstein equations so must hold true with the appropriate definitions of $\rho ={\dot{\phi }}^2+V(\phi )$ etc (see Section 9). in Section 8.

😇 Exercise 13J Use the perturbed continuity equation (365) together with the background continuity equation (96) to show that equation (375) is true in the super-horizon limit $k\to 0$. You can assume${}^{35}$ that $\delta P={\overline{P}}^{\prime }/{\overline{\rho }}^{\prime }\delta \rho$.

But what is $ℛ$ and why is it conserved? Some people are satisfied by the proof in Exercise 10; but if you would like to deepen your understanding of where $ℛ$ comes from, read on (otherwise skip to Section 11).

Suppose we have some solution $a(t)$ that describes the universe running through slow-roll, reheating, and then into radiation domination. Consider a completely homogeneous universe and plot the scalefactor as a function of time (upper left panel of Figure 16). During slow-roll inflation, the scalefactor expands $\propto \exp (Ht)$. Then there is a period where slow-roll ends and the universe re-heats; we don’t really know what happens in detail. Finally, the energy in the field is dumped into relativistic particles and radiation domination begins, so the scalefactor starts expanding $\propto t^{1/2}$.

Because everything starts with a slow-roll attractor phase, there are only a very limited range of alternative histories $\tilde{a}(t)$ which could fit the same physics. In fact if we want to keep everything else constant (e.g. the time $t$ of reheating), the only possibility is to rescale, $\tilde{a}=a{e}^{ℛ}$ for constant $ℛ$ (centre panel of Figure 16). Note that for small rescalings $ℛ\ll 1$, we have $\tilde{a}\simeq a(1+ℛ)$. We will shortly see that this is exactly the same $ℛ$ as I wrote in equation (375).

The constant $ℛ$ rescaling might appear, depending on your perspective, as either a time delay or a scalefactor shift, or some combination of both (Figure 16, right panel). Under a time shift $t\to t+\delta t$, the scalefactor maps to $a\to a+\dot{a}\delta t=a+aH\delta t$. If we now return to perturbation theory with the conformal Newtonian gauge metric (350), the scalefactor shift from the perturbation is described by $a\to (1-\mathrm{\Psi })a$ at fixed $t$, so we have

	$(1+ℛ+H\delta t)a(\eta )=\tilde{a}(t)$	$=(1-\mathrm{\Psi })a(t)$
	$\Rightarrow ℛ$	$=-\mathrm{\Psi }-H\delta t\text{.}$		(376)

In fact, once we’ve decided on a specific gauge, we don’t get left with a choice about how to decompose $ℛ$ into the scalefactor shift $\mathrm{\Psi }$ and time shift $\delta \eta$. We arranged our setup so that reheating happens at the same time $t$ in the different patches and at the end of reheating one ends up with identical densities. So the only way of generating a density perturbation is by making use of the time shift, specifically

$$\delta \rho =\dot{\overline{\rho }}\delta t=-3H(\overline{\rho }+\overline{P})\delta t,$$

(377)

using the conservation equation (146). Consequently we have the final version of our expression for $ℛ$ in the conformal Newtonian gauge:

$$\Rightarrow \boxed{ℛ=-\mathrm{\Psi }+\frac{\delta \rho }{3(\overline{\rho }+\overline{P})}}\text{.}$$

(378)

which of course is the same definition of $ℛ$ from the start of the section.

The value of $ℛ$ is that it allows us to connect the value of $\delta \phi$ as it exits the horizon in slow-roll inflation to the perturbations that will later re-enter the horizon during radiation domination.

During inflation, the density and pressure are specified by equations (246) and (247). Consequently $\rho +P={\dot{\phi }}^2$ and

$$\delta \rho \simeq \frac{\partial V}{\partial \phi }\delta \phi \text{,}$$

(379)

where I have used ${\dot{\phi }}^2\ll V$. We also saw in Exercise 9 that $\mathrm{\Phi }=\mathrm{\Psi }$ is negligible provided we consider only mildly super-horizon modes during slow-roll. Consequently, equation 378 becomes

$$\boxed{ℛ=\frac{-H\delta \phi }{\dot{\phi }}=\frac{-\mathscr{H}\delta \phi }{{\phi }^{\prime }}}\mathit{\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}}\text{ (mildly super-horizon, slow-roll inflation),}$$

(380)

which is the desired relation between $ℛ$ and $\delta \phi$ that we will use as perturbations cross the horizon.

😇 Exercise 13K Show that during radiation domination on super-horizon scales ($k^2\ll {\mathscr{H}}^2$), and assuming that $\mathrm{\Phi }$ is constant, equation (354a) boils down to $-2\mathrm{\Phi }={\delta }_{\gamma }$. Use this result and the fact that $\mathrm{\Phi }=\mathrm{\Psi }$ to show that $$\boxed{\mathrm{\Phi }=-\frac{2ℛ}{3}}\mathit{\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}}\text{ (super-horizon, radiation domination).}$$ (381)

Equations (380) and (381) are the take-away point from this discussion: the potential $\mathrm{\Phi }$ on entry to radiation domination is set by $ℛ$ by equation (381); in turn, $ℛ$ is obtained from the field fluctuations back in the slow-roll regime using equation (380).

At this point it’s helpful to step back and take a look at what the previous sections have established, and where all this is headed.

Overall, this section has been looking at the growth of inhomogeneous perturbations around the homogeneous background universe. The full set of results is summarised in Table 3. Most importantly, we’ve established that:

• •

  there are three different types of perturbations in relativistic cosmology: scalar, vector and tensor;

• •

  tensor perturbations are actually gravitational waves, and we discussed their behaviour;

• •

  vector perturbations are not of interest because they decay rapidly, and scalar perturbations correspond to density waves that generate structure in the Universe;

• •

  for cold dark matter, densities evolve identically to the Newtonian case – in particular, there is a growing mode with $\mathrm{\Phi }$ constant and $\delta$ growing, $\delta \propto a$;

• •

  more generally, the evolution of perturbations depends on whether they are sub-horizon or super-horizon, with equivalent conditions for this provided by Eqs. (342) and (343);

• •

  for a universe dominated by radiation (recall this used to be true of our own universe), super-horizon scalar modes have a fixed potential $\mathrm{\Phi }$ while sub-horizon modes fluctuate and decay ($\propto {\eta }^{-2}$);

• •

  earlier still, in the slow-roll inflationary era, density perturbations can be characterised by the comoving curvature perturbation $ℛ$ which is fixed if the mode is outside the horizon; or, if inside the horizon, it’s more convenient to work directly with the scalar field perturbation $\delta \phi$, which oscillates and converges to a fixed (non-zero) value as the mode is stretched by inflation.

We have written down a relationship between $\delta \phi$, $ℛ$ and $\mathrm{\Phi }$ so all these results can be stitched together into a single history for structure in our universe. There is one important missing piece of information: what generates the fluctuations in $\delta \phi$ in the first place? That is our next topic.