So far, we have discussed the evolution, but not the origin, of the primordial perturbations which provided seeds for cosmological structure. Most physical theories describe what happens to a system given some set of initial conditions, and that’s what we have so far: evolution equations. Inflation is remarkable because, at the classical level, it apparently removes any inhomgoeneities – but at the quantum level, it seemingly predicts a particular amplitude for inhomogeneities. In other words, it provides the initial conditions for the universe. To what extent this is a complete solution to the initial conditions problem is debated, but it’s widely agreed that it’s the best working model we have. It made predictions for the nature of large-scale-structure which have been confirmed over the last couple of decades.
We are most interested in the scalar perturbations to the metric, as these couple to the density of matter and radiation. In addition to scalar perturbations, however, inflation also generates tensor fluctuations. As we have seen, these are not coupled to the density at linear order and thus are not responsible for the large-scale structure of the universe. However, they do induce fluctuations in the cosmic microwave background (CMB). In fact, the effect these fluctuations has on polarisation turn out to be a unique signature of inflation which, assuming it can be measured by a forthcoming instrument, will offer the best window on the physics driving inflation.
We will study quantum tensor perturbations before quantum scalar perturbations. But before all of that we need a reminder of how to move from a classical picture to a quantum one. Famously, there is no complete theory of quantum gravity and so you might imagine this is an impossible or ill-defined task – but actually one just assumes that a final theory of gravity will behave like classical GR in the relevant limit, and everything else follows by consistency. So we don’t need a full theory of quantum gravity.
In order to compute the quantum fluctuations in the metric, we need to quantize the field. For both scalar and tensor perturbations, the easiest way to do this is to directly apply the uncertainty principle. Let’s see how this works for a simple harmonic oscillator.
We start with the Lagrangian
$$\mathcal{L}=\frac{1}{2}\left({\dot{x}}^{2}-{\omega}^{2}{x}^{2}\right).$$ | (382) |
The momentum is defined as
$$p\equiv \frac{\partial \mathcal{L}}{\partial \dot{x}}=\dot{x}.$$ | (383) |
We can derive the equation of motion from the Euler-Lagrange equation, which can be written as
$$\frac{\mathrm{d}p}{\mathrm{d}t}=\frac{\partial \mathcal{L}}{\partial x}\text{,}$$ | (384) |
and then expanded to the familiar $\ddot{x}=-{\omega}^{2}x$. A suitable solution is $x=A\mathrm{sin}(\omega t+{\varphi}_{0})$ for some amplitude $A$, giving momentum $p=A\omega \mathrm{cos}(\omega t+{\varphi}_{0})$, where ${\varphi}_{0}$ is an arbitrary, constant phase.
The uncertainty principle states that the minimum amplitude of fluctuations obeys (temporarily including $\mathrm{\hslash}$ for clarity, though in later sections we will set this to $1$):
$$\mathrm{\Delta}x\mathrm{\Delta}p\ge \frac{\mathrm{\hslash}}{2}\text{,}$$ | (385) |
where $\mathrm{\Delta}x$ and $\mathrm{\Delta}p$ can be interpreted as the r.m.s. uncertainty in the position and momentum respectively,
$\mathrm{\Delta}x$ | $={\u27e8{x}^{2}\u27e9}^{1/2}={\displaystyle \frac{A}{\sqrt{2}}}$ | |||
$\mathrm{\Delta}p$ | $={\u27e8{p}^{2}\u27e9}^{1/2}={\displaystyle \frac{A\omega}{\sqrt{2}}}\text{.}$ | (386) |
Substituting into the uncertainty relation (385), one gets the result
$${A}^{2}\ge \frac{\mathrm{\hslash}}{\omega}\text{,}$$ | (387) |
which sets a minimum bound on the energy of the oscillator:
$$E=\frac{{p}^{2}}{2}+\frac{{\omega}^{2}{x}^{2}}{2}=\frac{{A}^{2}{\omega}^{2}}{2}\ge \frac{\mathrm{\hslash}\omega}{2}\text{.}$$ | (388) |
This is the classic result from introductory quantum mechanics courses. The ground state is the state of minimum energy, $E=\mathrm{\hslash}\omega /2$.
When we analyse Fourier components of a field (below), it is far simpler to work with a complex Fourier transform rather than attempt to keep all degrees of freedom real. For that reason, let’s repeat the analysis just given but with a complex solution to the harmonic oscillator, ${x}_{c}={A}_{c}{e}^{i\omega t}$ (using the subscript $c$ to denote the complex solution). This is already somewhat neater because the phase factor ${\varphi}_{0}$ can be absorbed into the arbitrary amplitude ${A}_{c}$.
However this way of working does require us to keep a clear head about what we mean. If we’re really imaging a “mass on a spring” or similar example, a complex value for $x$ is physical nonsense. What is intended is that the physical degree of freedom $x$ is the real part of the mathematical solution ${x}_{c}$. A suitable Lagrangian for ${x}_{c}$ can then be written
$${\mathcal{L}}_{c}=\frac{1}{2}\left({|{\dot{x}}_{c}|}^{2}-{\omega}^{2}{|{x}_{c}|}^{2}\right)\text{,}$$ | (389) |
where the absolute value squared expands to ${|{x}_{c}|}^{2}={x}_{c}{x}_{c}^{*}={x}^{2}+{\left(\mathrm{Im}{x}_{c}\right)}^{2}$ ($*$ indicates a complex conjugate). So in effect this Lagrangian models two harmonic oscillators, one described by the real part and the other by the imaginary part of ${x}_{c}$. These two degrees of freedom can equally well be described by the variables ${x}_{c}$ and ${x}_{c}^{*}$.
The conjugate momentum to ${x}_{c}$ is now
$$p=\frac{\partial {\mathcal{L}}_{c}}{\partial {\dot{x}}_{c}}=\frac{1}{2}{\dot{x}}_{c}^{*}\text{;}$$ | (390) |
note in particular the factor of $1/2$ which we would miss if we naively used a complex solution paired to the original Lagrangian.
The uncertainty principle remains the same, but with
$\mathrm{\Delta}x$ | $={\u27e8{|{x}_{c}|}^{2}\u27e9}^{1/2}=A$ | |||
$\mathrm{\Delta}p$ | $={\u27e8{|{p}_{c}|}^{2}\u27e9}^{1/2}={\displaystyle \frac{A\omega}{2}}\text{.}$ | (391) |
from which we obtain once again
$${|{A}_{c}|}^{2}\ge \frac{\mathrm{\hslash}}{\omega}\text{.}$$ | (392) |
So we reach the same conclusion about the amplitude $|{A}_{c}|$ as we did about the real amplitude $A$; but along the way factors of $2$ turned up in slightly different places. If one mixes up elements of the two derivations by mistake it is all-too-easy to get out by a factor of $2$.
Finally, we want the energy just of the original degree of freedom $x=\mathrm{Re}{x}_{c}$; you can verify that one reaches the correct conclusion
$$E\ge \frac{\mathrm{\hslash}\omega}{2}$$ | (393) |
as before.
The moral: if one naively switches between real and complex numbers, one can very easily lose or gain extra factors of $1/2$. We should always keep a clear head about whether we’re working with a real or complex quantity and how that relates to the actual physical degrees of freedom.
The uncertainty principle can be applied in this way only when one has decoupled degrees of freedom. For example, if we have two quantum oscillators
$$\mathcal{L}=\frac{1}{2}\left({\dot{x}}^{2}-{\omega}^{2}{x}^{2}+{\dot{y}}^{2}-{\omega}^{2}{y}^{2}\right),$$ | (394) |
it’s perfectly OK to separately quantise the $x$ and $y$ oscillators. On the other hand if the degrees of freedom are coupled – for example
$$\mathcal{L}=\frac{1}{2}\left({r}^{2}{\dot{\theta}}^{2}+{\dot{r}}^{2}-{\omega}^{2}{r}^{2}\right)\text{,}$$ | (395) |
the conjugate momenta involve multiple variables; in this specific example, $\partial \mathcal{L}/\partial \dot{\theta}={r}^{2}\dot{\theta}$ which simultaneously involves the $r$ and $\theta $ solutions. This makes quantization harder and so is a situation to avoid. In this case the easiest approach to quantisation would be by first substituting variables – most obviously, one could let $x=r\mathrm{cos}\theta $ and $y=r\mathrm{sin}\theta $ and recover the Lagrangian (394).
So, our goal is to take the Lagrangian describing the physics of the early Universe and cast it into variables which are decoupled. Those variables are the ones to which we can apply the uncertainty principle, which will give us the amplitude of perturbations.
Recall that in Section 4 we found the equation for the amplitude of a single gravitational wave:
$${A}^{\prime \prime}+2\mathscr{H}{A}^{\prime}+{k}^{2}A=0$$ | (396) |
To perform a successful quantisation based on the ideas above, we must now ask how, precisely, $A$ appears in the Lagrangian $\mathcal{L}$. Calculating this is far from straight-forward and is best attempted with the aid of a computer algebra package. To get the correct Lagrangian for $A$ you need to include both the Einstein-Hilbert and the matter part of the Lagrangian^{36}^{36} 36 At first this might come as a bit of a surprise – after all, gravitational waves don’t interact with perturbations in the energy-momentum at linear order. The reason to include the matter part of the Lagrangian nonetheless is to correctly reproduce the effects of the background expansion (which certainly does care about the energy-momentum) on the gravitational wave.. This therefore involves calculating the metric determinant and the Ricci curvature to second order in a perturbed metric. The easiest approach is to include a single gravitational wave travelling down (say) the $\widehat{z}$ axis with a fixed polarisation ${M}_{ab}$.
Actually we previously did precisely that, in Eq. (344). The only difference now is that we need to make a specific choice to normalise the polarisation matrix. This arbitrary normalisation will ultimately drop out; it renormalises $A$ in the Lagrangian but also gets divided out when we transform back to the physical variable ${h}_{ab}$. So we will just make a definite choice of $M=\mathrm{diag}(0,2,-2,0)$.
With sufficient patience^{37}^{37} 37 Performing the derivation of the Lagrangian is non-examinable, and I will not present the steps in class since they really require a computer algebra system to verify (I used Mathematica). Note the appearance of absolute values in the Lagrangian; this is because complex modes are “paired up” to create a real metric fluctuation, similar to what we discussed in Section 2. you can verify that everything boils down to (in natural units where $8\pi G=1$):
$\mathcal{L}$ | $=\left({|{A}^{\prime}|}^{2}-{k}^{2}{|A|}^{2}\right){a}^{2}$ | (397) | ||
$={\left|{\stackrel{~}{A}}^{\prime}+{\displaystyle \frac{1}{\eta}}\stackrel{~}{A}\right|}^{2}-{k}^{2}{|\stackrel{~}{A}|}^{2}$ | (398) |
where I have defined $\stackrel{~}{A}(\eta )=A(\eta )a(\eta )$, and used the idealised inflationary solution $a=-1/(H\eta )$ (Table 2).
Why define $\stackrel{~}{A}$ in this way? The conjugate momentum to $A$ would have been ${a}^{2}{A}^{\prime}$ which therefore means $a$ and $A$ are not independent degrees of freedom. As discussed in Section 2, life is much simpler when quantising decoupled variables. Switching to the modified variable $\stackrel{~}{A}$ makes things work out nicely:
Now we’re ready to apply the uncertainty principle; this states $\mathrm{\Delta}\stackrel{~}{A}\mathrm{\Delta}p={\u27e8{|\stackrel{~}{A}|}^{2}\u27e9}^{1/2}{\u27e8{|p|}^{2}\u27e9}^{1/2}\ge 1/2$ (taking $\mathrm{\hslash}=1$). Applying this in the sub-horizon regime ($|k\eta |\gg 1$), where $p\simeq {({\stackrel{~}{A}}^{*})}^{\prime}$, and assuming we have a vacuum (i.e. ground state) so that the inequality is saturated, we obtain
$${|{\stackrel{~}{A}}_{0}|}^{2}=\frac{1}{2k}\text{.}$$ | (402) |
The result that we’re interested in is for the opposite limit – the super-horizon case, $|k\eta |\ll 1$ – but ${\stackrel{~}{A}}_{0}$ is by assumption constant as the wave evolves from one limit to the other so our calculation is valid.
Therefore we can calculate that
$$\u27e8{|\stackrel{~}{A}|}^{2}\u27e9\to \frac{{|{\stackrel{~}{A}}_{0}|}^{2}}{{(k\eta )}^{2}}=\frac{1}{2{k}^{3}{\eta}^{2}}\mathit{\hspace{1em}}\text{as}\mathit{\hspace{1em}}k\eta \to 0\text{.}$$ | (403) |
Now we have all the ingredients we need to construct the power spectrum of gravitational waves ${h}_{ij}$. We can work with a scalar quantity ${h}_{ij}{h}^{ij*}$ (summing over $i$ and $j$) to summarise this power; note that ${M}_{ij}{M}^{ij*}=8$ with our definitions. The overall power will still need to be multiplied by two to account for the second polarisation state. Putting all this together, (316) tells us^{38}^{38} 38 Recall this is basically just the power spectrum definition (278) but with modifications to consider just a single mode at a time. See Section 4. the power is then:
${P}_{h}(k)$ | $\equiv \underset{\mathrm{polarisation}}{\underset{\u23df}{2\times}}{\displaystyle \frac{{k}^{3}}{4{\pi}^{2}}}\u27e8{h}_{ij}{h}^{ij*}\u27e9$ | |||
$=2{M}_{ij}{M}^{ij*}{\displaystyle \frac{{k}^{3}}{4{\pi}^{2}}}{\displaystyle \frac{1}{{a}^{2}}}\u27e8{|\stackrel{~}{A}|}^{2}\u27e9$ | ||||
$\to 8{\left({\displaystyle \frac{H}{2\pi}}\right)}^{2}\text{as}\eta \to 0\text{,}$ | (404) |
where in the final step we need to recall that $\eta =-1/aH$. Note that through the slow-roll Friedmann equation ($3{H}^{2}=V$) we could re-express this result in terms of the inflationary potential $V$ (rather than the expansion rate) at horizon exit: ${P}_{h}(k)=2V/(3{\pi}^{2})$.
Earlier we adopted natural units, $8\pi G=\mathrm{\hslash}=c=1$; if you reinstate the units you find that
$${P}_{h}(k)=\frac{8{\mathrm{\hslash}}^{2}}{{M}_{\mathrm{Pl}}^{2}{c}^{4}}{\left(\frac{H}{2\pi}\right)}^{2}.$$ | (405) |
where the squared Planck mass ${M}_{\mathrm{Pl}}^{2}=\mathrm{\hslash}c/(8\pi G)$.
This result is important. A detection of primordial gravitational waves could be made by studying the polarisation of the cosmic microwave background (more later). The result (405) tells us that such a detection would reveal the value of $H$ relative to the Planck scale ${M}_{\mathrm{Pl}}$ in the inflationary era – and since $3{H}^{2}=V$ during slow-roll inflation, it would be telling us the energies at which this exotic physics operates. As yet there is no such detection which in itself tells us that the energy scale of inflation must be well below the Planck scale.
Although the primordial spectrum we have derived appears to be scale-free (i.e. no $k$-dependence, so modes are equally excited at all wavelengths), there are two important complications.
As discussed in Section 4, the gravitational waves decay once they re-enter the horizon in the radiation-dominated or matter-dominated era. So in practice the observed amplitude is always greatest on the large scales which have only recently come back inside the horizon.
We have implicitly assumed in deriving equation (405) that $H$ is constant (e.g. when we substitute $\mathscr{H}=-1/\eta $). In reality $H$ changes, alebit very gradually, during slow-roll evolution. When worked out with this complication, it turns out that one can retain (405) on the understanding that $H$ is evaluated at the time of horizon exit for the wavenumber $k$ i.e. when $|k\eta |=1$ (see Section 3). This is because $A=\text{const}$ is a solution to equation 396 when $k/\mathscr{H}\to 0$, even if $H$ isn’t constant (provided it only varies slowly).
We have already seen that on sub-horizon scales the classical equation of motion for scalar field perturbations is (373). Note that this is identical to equation (396) with the replacement $A\to \delta \phi $. It should therefore come as no surprise that the Lagrangian in terms of this variable looks very much like the gravitational wave Lagrangian,
$$\mathcal{L}=\frac{1}{2}\left({|\delta {\phi}^{\prime}|}^{2}-{k}^{2}{|\delta \phi |}^{2}\right){a}^{2}\text{.}$$ | (408) |
(Incidentally this Lagrangian is far easier to verify for yourself, starting from Equation (225). You can assume an unperturbed, flat FRW background because you already know the gravitational terms are negligible on subhorizon scales). There are two key differences
This expression is only valid on sub-horizon scales; this is not a huge issue but does mean we will need to handle the transition to the super-horizon regime differently.
There is a factor of $1/2$ that was missing in the gravitational wave case, which will lead us to a different normalisation.
Specifically, with $\delta \stackrel{~}{\phi}\equiv a\delta \phi $ we find that $\partial L/\partial (\delta {\stackrel{~}{\phi}}^{\prime})\simeq \delta {\stackrel{~}{\phi}}^{\prime *}/(2{a}^{2})$ for sub-horizon modes and consequently the minimum uncertainty principle gets us to an amplitude
$${|\delta {\stackrel{~}{\phi}}_{0}|}^{2}=\frac{1}{k}\text{,}$$ | (409) |
a factor two larger than the corresponding result (402). As this leaves the horizon, we have
$$\u27e8{|\delta \stackrel{~}{\phi}|}^{2}\u27e9=\frac{1}{{k}^{3}{\eta}^{2}}\text{,}$$ | (410) |
and so we find that
$${P}_{\phi}(k)\equiv \frac{{k}^{3}}{4{\pi}^{2}}\u27e8{|\delta \phi |}^{2}\u27e9=\frac{{k}^{3}}{4{\pi}^{2}}\frac{1}{{k}^{3}{\eta}^{2}}={\left(\frac{H}{2\pi}\right)}^{2}\text{,}$$ | (411) |
provided that this power spectrum is evaluated for modes which are only modestly super-horizon – as the mode gets carried even further outside the horizon, the approximate equation of motion/Lagrangian that we are using breaks down (Section 9).
In Section 11 we showed that the subsequent evolution through to the end of slow-roll conserves $\mathcal{R}=-\delta \phi \mathscr{H}/{\overline{\phi}}^{\prime}$ – so at the end of slow-roll we are best-placed to express the power spectrum of $\mathcal{R}$ rather than $\phi $ itself:
$${P}_{\mathcal{R}}(k)={\left(\frac{H}{2\pi}\frac{\mathscr{H}}{{\overline{\phi}}^{\prime}}\right)}_{|k\eta |=1}^{2}$$ | (412) |
where the expression on the right should be evaluated at horizon exit ($|k\eta |=1$) for each $k$; it’s only at this time that both evolution equations (375) and (373) are valid, allowing $\mathcal{R}$ to be calculated from $\delta \phi $. Note that one can equivalently use the condition $k/(aH)=1$ since $\eta \simeq -1/(aH)$.
By applying the slow-roll relations we get
$${P}_{\mathcal{R}}(k)={\left(\frac{H}{2\pi}\right)}^{2}\frac{1}{2{\u03f5}_{V}}=\frac{8}{3{\u03f5}_{V}}{\left(\frac{{V}^{1/4}}{\sqrt{8\pi}}\right)}^{4}$$ | (413) |
The strange business of taking ${V}^{1/4}$ and then re-raising it to the fourth power is a convention that helps us understand the energy scale of inflation. The potential $V$ has dimensions of an energy density, $[E]/{[L]}^{3}$. Because $\mathrm{\hslash}=c=1$, lengths have dimensions of ${[E]}^{-1}$. So overall, the potential has dimensions of an energy to the fourth power, ${[E]}^{4}$. So, rearranging the equation for ${P}_{\mathcal{R}}$ above, the energy scale of inflation is given by
$${V}^{1/4}={\left(\frac{3{\u03f5}_{V}}{8}\right)}^{1/4}\sqrt{8\pi}{P}_{\mathcal{R}}^{1/4}.$$ | (414) |
But this is no practical use because it is dimensionless. To get recognisable units, we need to find the right combination of $\mathrm{\hslash}$, $c$ and $8\pi G$ to give an energy. The only combination with the right dimensions is ${(\mathrm{\hslash}{c}^{5}/8\pi G)}^{1/2}\simeq 2.4\times {10}^{18}\mathrm{GeV}$, so the above equation should be multiplied by these units. Finally, the large-angle CMB observations constrain ${P}_{\mathcal{R}}(k)\sim 2\times {10}^{-9}$ on current Hubble scales. Putting everything together gives
$${V}^{1/4}\sim 6\times {10}^{16}{\u03f5}_{V}^{1/4}\mathrm{GeV}.$$ | (415) |
Again, this describes the overall energy scale of inflation, and since ${\u03f5}_{V}\ll 1$, the energy scale is at least two orders of magnitude below the Planck scale ($\sim {10}^{19}\mathrm{GeV}$). It is, however, possible that inflation occurred around the GUT scale, $\sim {10}^{16}\mathrm{GeV}$ if minimal inflation implies that the large-angle scales we see today actually left the horizon just as slow roll was ending. Note that
$$r\equiv \frac{{P}_{h}(k)}{{P}_{\mathcal{R}}(k)}\approx 16{\u03f5}_{V},$$ | (416) |
which defines the dimensionless tensor-to-scalar ratio $r$. It follows that $r\approx -8{n}_{t}$ in slow-roll inflation which is an example of a slow-roll consistency relation between the spectra and amplitude of perturbations. Violation of this consistency relation would disprove single-field slow-roll inflation, although so many more complicated variations of inflation exist that it sadly wouldn’t rule out inflation itself. Still, it would tell us a lot about physics at high energies. Even the existence of primordial gravitational waves has not been demonstrated as yet^{39}^{39} 39 In 2016, the LIGO consortium detected gravitational waves from merging black holes but these have vastly smaller wavelengths than the primordial waves that we would like to measure. While the primordial waves are generated at all wavelengths, they decay after entering the horizon so only the longest wavelength modes would remain detectable today., so measuring their spectral index ${n}_{t}$ is a long way off.
Slow-roll inflation produces a spectrum of curvature perturbations that is almost scale-invariant. We can quantify the small departures from scale-invariance by forming the scalar spectral index ${n}_{s}(k)$. Generally, this is a scale-dependent quantity defined by
$${n}_{s}(k)-1\equiv \frac{d\mathrm{ln}{P}_{\mathcal{R}}(k)}{d\mathrm{ln}k},$$ | (417) |
where the $-1$ is a bit of a nuisance, but a convention that we’re stuck with. It means that a scale-free spectrum has ${n}_{s}=1$.
If we observe the large-scale distribution of matter, we don’t directly see the curvature fluctuations $\mathcal{R}$ left over by inflation. There have been 14 billion years of intervening time during which a lot has happened. But the large-scale structure of the universe can nonetheless be understood from ${P}_{\mathcal{R}}$ with just a few further steps.
We start this story at the end of reheating, the beginning of the radiation-dominated phase. As the universe evolves and the horizon begins to expand again, modes are unfrozen at different times depending on $k$ as we discussed extensively in Section 13. The amplitudes of the modes are therefore changed by a $k$-dependent amount. But if linear theory is valid, we must be able to write
$${\delta}_{m}(\eta ,\bm{k})=T(\eta ,k)\mathcal{R}(0,\bm{k}),$$ | (419) |
for some function $T(\eta ,k)$ known as a transfer function^{40}^{40} 40 Note that sometimes the transfer function $T\mathit{}\mathrm{(}k\mathrm{)}$ is expressed relative to the initial density fluctuations, rather than our convention here of expressing it relative to the curvature fluctuations $\mathcal{R}$. This involves dividing $T\mathit{}\mathrm{(}k\mathrm{)}$ by ${k}^{\mathrm{2}}$ to compensate for the ${k}^{\mathrm{2}}$ factor appearing in the relationship between $\mathcal{R}$ and $\delta \mathit{}\rho $.. This function relates the primordial curvature perturbation to the synchronous-gauge matter perturbation, and $\eta =0$ corresponds to the end of slow-roll inflation and the start of the radiation-dominated era.
Modes that remain outside the horizon throughout radiation domination have the simplest behaviour. The physical scale of a mode just entering the horizon at matter-radiation equality is approximately ${r}_{\mathrm{eq}}\simeq H{(z={z}_{\mathrm{eq}})}^{-1}$, where ${z}_{\mathrm{eq}}$ is the redshift of matter-radiation equality.
📝 Exercise 14F
Show that $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$H$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$($}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$z$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$)$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\simeq $}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$H$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\sqrt{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$2$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{\Omega}$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$m$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$,$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$($}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$1$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$+$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$z$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$)$}}^{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$3$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$/$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$2$}}$ and hence that
$${\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\simeq $}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$H$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\sqrt{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$2$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{\Omega}$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$m$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$,$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$0$}}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$($}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$1$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$+$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$z$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$)$}}^{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$1$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$/$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$2$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\text{.}$}$$ | (420) |
(Hint: remember that $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}$ is a comoving wavenumber. You can ignore factors of $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$2$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\pi $}$ as we are only looking at the approximate way the quantities scale.)
For ${\mathrm{\Omega}}_{m,0}=0.25$ and ${z}_{\mathrm{eq}}=3000$, this gives ${k}_{\mathrm{eq}}\simeq 0.01h{\mathrm{Mpc}}^{-1}$ where $h\simeq 0.7$ is the Hubble parameter in units of $100\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$.
First consider modes that were outside the horizon throughout radiation domination, $k\ll {k}_{\mathrm{eq}}$. For such modes, $\mathrm{\Phi}(\eta ,\bm{k})/\mathcal{R}(0,\bm{k})$ remains independent of $k$. When a mode is super-horizon, the mix between CDM and radiation is fixed and ${\delta}_{c}\propto {\delta}_{r}$. In turn, we can calculate the perturbations in the synchronous-gauge radiation ${\delta}_{r}$ using the Poisson equation (362), yielding
$${\delta}_{c}\propto {\delta}_{r}=\frac{-{k}^{2}\mathrm{\Phi}}{4\pi G{a}^{2}{\overline{\rho}}_{R}}\propto {k}^{2}{a}^{2}\mathrm{\Phi}.$$ | (421) |
There is an overall growth factor scaling with ${a}^{2}$, but we’re most interested in the $k$-dependence from which we conclude
$$T(k)\propto {k}^{2}\mathit{\hspace{1em}\hspace{1em}}k\ll {k}_{\mathrm{eq}}.$$ | (422) |
For $k>{k}_{\mathrm{eq}}$, the mode entered the horizon during radiation domination. As we saw in Section 3, ${\delta}_{c}$ then grows only logarithmically with time. It’s probably easiest to think of this as “losing” the growth that one would have got by staying super-horizon (which is $\propto {a}^{2}$, as we saw above). See Figure 17 for a visual explanation that might help, showing how modes with $k>{k}_{\mathrm{eq}}$ lose growth relative to those with $$.
There is a small compensating growth described by equation (331); in principle one needs to match the amplitude and growth rates at the start of radiation domination to establish $A$ and ${t}_{0}$ but we can roughly write ${t}_{0}={t}_{\mathrm{eq}}$ and $A$ of course is just an overall scaling. After matter-radiation equality, all scales grow at an equal rate (see Section 1) so the loss needs to be calculated between the time of horizon re-entry, ${a}_{\mathrm{enter}}$ and matter-radiation equality ${a}_{\mathrm{eq}}$. Overall, then:
$$T(k)\propto {k}^{2}{\left(\frac{{a}_{\mathrm{entry}}}{{a}_{\mathrm{eq}}}\right)}^{2}\mathrm{ln}\frac{{t}_{\mathrm{eq}}}{{t}_{\mathrm{entry}}}\mathit{\hspace{1em}\hspace{1em}}k\gg {k}_{\mathrm{eq}}.$$ | (423) |
Finally, we need to express ${a}_{\mathrm{entry}}$ and ${t}_{\mathrm{entry}}$ in terms of $k$.
☞ Exercise 14G
Use Table 2’s radiation domination solutions with the horizon entry condition $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$|$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\eta $}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$|$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$=$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$1$}$ to show that ${\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{entry}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$/$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$a$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$=$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$/$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}$ and ${\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$t$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$/$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$t$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{entry}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$=$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$($}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$/$}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$)$}}^{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$2$}}$. Putting together all results, show that the $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}$-dependence of $\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$T$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$($}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$)$}$ may be written:
$$\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$T$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$($}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$)$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\propto $}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\{$}\begin{array}{cc}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}}^{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$2$}}\hfill & \colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\ll $}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\hfill \\ \colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{ln}$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$$}\frac{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}}{{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}}\hfill & \colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\gg $}{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$k$}}_{\colorbox[rgb]{0.901960784313726,0.901960784313726,0.901960784313726}{$\mathrm{eq}$}}\hfill \end{array}$$ | (424) |
The power spectrum is constructed out of density squared, so the observed matter power spectrum is
$$P(k)\propto T{(k)}^{2}{P}_{\mathcal{R}}(k)\text{.}$$ | (425) |
Figure 18 shows ${k}^{-3}P(k)$. (The ${k}^{-3}$ renormalisation is just an alternative convention used for the power spectrum, effectively giving the “power per mode”, rather than the “total power from a shell in $\mathrm{ln}k$” which has been our convention thus far.)
On large scales (small $k$), ${k}^{-3}P(k)$ grows as $k$.
The power spectrum turns over around $k\sim 0.01h{\mathrm{Mpc}}^{-1}$ corresponding to the horizon size at mattter-radiation equality.
Beyond the peak, the power ${k}^{-3}P(k)$ falls as ${k}^{-3}$ (or more precisely, ${k}^{-3}{\mathrm{ln}}^{2}(k/{k}_{\mathrm{eq}})$ once the Meszaros logarithmic growth is included), where ${k}_{\mathrm{eq}}$ is the wavenumber of a mode that enters the horizon at the matter-radiation transition.
There are small amplitude baryon acoustic oscillations in the spectrum. This arises from the oscillations in the baryon-photon plasma discussed in Section 7; we will discuss this further in the context of the CMB next.
Linear theory applies on scales ${k}^{-1}>10\mathrm{Mpc}$ at $z=0$, but the dashed line shows estimates for non-linear corrections that apply at smaller scales. These corrections can be calculated using numerical simulations, higher-order perturbation theory, or some combination of the two.
The shape of the matter power spectrum as inferred from galaxy clustering agrees well with the theoretical prediction in Fig. 18.
The inflationary proposal requires a huge extrapolation of the known laws of physics. In the absence of a complete theory, a phenomenological approach has been employed, where an effective potential $V(\varphi )$ is postulated. Ultimately, $V(\varphi )$ has to be derived from a fundamental theory, and significant progress in implementing inflation in string theory has been made in recent years. While it is challenging to understand the origin of inflation from a particle physics point of view, it is also a great opportunity to learn about ultra-high-energy physics from cosmological observations.
We have made these predictions by piecing together the course so far, but in this chapter we added a number of ingredients:
Density and gravitational wave fluctuations are assumed to be generated by quantum random noise.
To determine the amplitude of this noise, one needs an action for the degree of freedom under consideration; from this we derive a canonical momentum which can then be used in the famous uncertainty relationship $\mathrm{\Delta}x\mathrm{\Delta}p\ge 1/2$ to get the minimum level of fluctuations.
The level of fluctuations is set in the sub-horizon regime, i.e. on very small scales.
The fluctuations are then stretched outside the horizon by inflation, and their amplitudes are tracked by using the classical equations of motion.
Both scalar field fluctuations and gravitational wave fluctuations have a solution which tends to a constant as the mode leaves the horizon. This non-zero residual fluctuation is what makes it into the observable Universe.
The above physics determines the primordial power spectrum.
But the observed power spectrum is not the primordial one: for a start, density fluctuations have a scale-dependent relationship with the curvature perturbations
As the Universe continues to evolve, we keep tracking the amplitude of the modes. In particular, when the density fluctuations re-enter the horizon, their behaviour depends on whether the Universe is radiation or matter dominated. The relationship between primordial and observed power is summarised in the transfer function $T(k)$.
As a result, small scale modes (which entered during radiation domination) are strongly suppressed in amplitude relative to large scale modes (which enter later, during matter domination).
We are now in a position to summarise the observational characteristics of inflationary theories:
Flat geometry, i.e. the observable universe should have no spatial curvature. Flatness has been verified at the 1% level by the location, or, better, separation, of the CMB acoustic peaks combined with some low-redshift distance information.
Gaussianity, i.e. the primordial perturbations should correspond to Gaussian random variables to a very high precision.
Scale-invariance, i.e. to a first approximation, there should be equal power at all length-scales in the primordial spectrum, without being skewed towards high or low wavenumbers. In terms of the parameterisation above, this corresponds to ${n}_{s}=1$ and ${n}_{t}=0$. However, small deviations from scale-invariance are also a typical signature of inflationary models and tell us about the dynamics of inflation. Observationally, these small deviations must be disentangled from the vastly larger transfer function effects discussed above.
Super-Hubble fluctuations, i.e. there exist correlations between anisotropies on scales larger than the apparent causal horizon, beyond which two points could not have exchanged information at light-speed during the history of a non-inflationary universe. This corresponds to angular separations on the sky larger than $\sim {2}^{\circ}$.
Primordial gravitational waves, which give rise to temperature and polarization anisotropies. These tensor modes must exist; however, their predicted amplitude can vary by many orders of magnitude depending on the energy scale at which inflation took place.
Adiabaticity. We have not discussed this point, but after reheating, there are no perturbations in the relative number densities of different species on super-Hubble scales. This follows from the assumption that only a single field is important during inflation.