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Initial conditions

So far we have derived all the equations we need to evolve perturbations from the early Universe and till today, but we haven't talked about how what set up the initial fluctuations and how to set the initial conditions needed to solve these equations.


See Chapter 2 of Baumann for a thorough review of inflation that goes into more detail on all the technical things. We will here just briefly go through some key things for us: why we need inflation, roughly how it works, its predictions and how this relates to setting the initial conditions of our perturbations.

Why do we need inflation and how does it work?

For us its easy to explain why we need (something like) inflation because we need to know what the hell we should put as the initial conditions when solving the perturbation system. And this is one important role of inflation, to provide the seeds for structures in our Universe. However even before inflation was invented people "knew" what they "should" be: the "natural" form of the initial power-spectrum is a scale-invariant spectrum. Any preferred scale need to come from some physics and we did not have any preferred mechanism for generating it so scale-invariance was a good guess. Inflation provided such a mechanism and says that this guess was almost spot on: there is some small correction to perfect scale-invariance that provides one way of testing the idea of inflation. There is also some additional important reasons to have inflation.

Horizon problem

Measurements of the cosmic microwave radiation have revealed that the temperature is the same in any direction to high precision. Light coming from the "north pole" (any direction) of our Universe wasn't in causal contact with light coming from the "south pole" (the opposite direction) at the time of recombination so how is this possible? The main mechanism we know for making the temperature of two different things equal is by having these things in contact for a longer period of time so that they can obtaim thermal equilibrium. We can compute how far a sound wave in the radiation field starting from the big-bang have traveled until recombination. This is called the sound-horizon: $$r_s = \int_0^{\eta_{\rm rec}} c_sd\eta$$ where $c_s$ is the sound speed which in the absence of baryons is just $\frac{c}{\sqrt{3}}$. If we compare this to the angular diameter distance at the last scattering surface $d_A = (\eta_0 - \eta_{\rm rec})a_{\rm rec} \sim 14$ Mpc we find that only photons coming from direction separated by $\sim 2$ degrees on the sky could have been in thermal contact. Thus the full CMB contains $\sim$ 10000 independent patches for which there are no reason for why they should have the same temperature to one part in ten thousand (the full sky is $4\pi$ which is $4\pi(360/2\pi)^2 \sim 42000$ square degrees). This, the so-called horizon problem, represents a serious problem in need of an explaination. And inflation provides such an explanation. The regions we today observe at different directions was indeed in causal contact in the very early Universe and could thermalize before a very rapid expansion kicked in and separated these regions.

Flatness problem

There are also some additional problems inflation solves like the flatness problem. If the Universe turns out to not be perfectly flat then for the Universe to close to flat as we observe it today it had to be extremely close (see problem set) to flat in the early Universe. With the exponential expansion in inflation the curvature evolves as $\Omega_k(a) \propto e^{-2Ht}$ providing a natural explanation for why the Universe is flat no matter what $\Omega_k$ was at the start of inflation.

Exotic relic problem

In several theoretical extensions of the standard model (like grand unified theories) its expected that exotic particles like magnetic monopoles are produced in large quantities in the early Universe when the temperature was extremely hot. Why haven't we observed any? This is the exotic relic problem. Either they don't exist and these theories predicting them are not realized in nature or if they are indeed correct then inflation saves the day by massively diluting the density of these during the exponential expansion. There are many good reasons to take these theories seriously, but on the other hand there is still no observational basis for them though. All we have are some hints. Thus if this is a real problem depends a bit on your theoretical priors. It's anyway nice that inflation is able to expain this (though it almost does a too good job at hiding such particles - it would be much better if we were able to observe them!).

    Figure 1: Inflationary solution to the horizon problem. The comoving Hubble sphere with radius $\frac{1}{\mathcal{H}}$ shrinks during inflation and expands during the conventional Big Bang evolution (at least until dark energy takes over at late times). Conformal time during inflation is negative. The spacelike singularity of the standard Big Bang is replaced by the reheating surface, i.e. rather than marking the beginning of time it now corresponds simply to the transition from inflation to the standard Big Bang evolution. All points in the CMB have overlapping past light cones and therefore originated from a causally connected region of space. Figure taken from Baumann's lecture notes.

    How do we get an inflation period?

    As we know from dark energy to have exponential expansion $a = e^{Ht}$ the equation of state $w$ should be close to $-1$. We could do this with a cosmological constant, however then inflation would never end! A simple way of generating this is to add a scalar field (in the standard mode the Higgs is an example of a scalar field) $\phi$ with some self-interaction potential $V$. The equation of state of such a model is given by $$w = \frac{\frac{1}{2}\dot{\phi}^2 - V}{\frac{1}{2}\dot{\phi}^2 + V}$$ so if the kinetic energy of the scalar field is much less than the potential energy (called slow-roll - the field slowly rolls down its potential) we have $w\simeq -1$. Once the field reaches the bottom of the potential we have $V \approx 0 \to w \approx 1$ and inflation stops. At the bottom it will oscillate and the energy in the scalar field is converted into standard model particles in a process (that we don't know the details off, but know of several physical mechanisms to achieve this) called reheating. After this (presumably complicated and highly out of equiibrium process) we have the normal Big Bang picture we discussed when going through the thermal history of the Universe. I should also mention that the scenario disussed here is only one possible way of realizing an inflationary epoch and that there are other proposals out there. There are also other scenarios that have been proposed to solve the problems we have talked about here, but that is beyond the scope of this course. Our presentation here has been very brief and schematic and we will not get into much more detail about this. What is important for us is to know the basic problems with the standard Big Bang model and the predictions for the initial conditions models like inflation introduce that are relevant for our goal of computing the predictions for the CMB. I refer you to Baumann for more technical details regarding inflation if you want to learn more.

      Figure 2: Scalar-field rolling down its potential. In the shaded region we have (slow-roll) inflation. The field eventually ends up at the bottom of the potential where reheating happens (the energy in the scalar field gets converted to standard model particles). Figure taken from Baumann's lecture notes.

      Predictions of inflation: primordial power-spectrum

      The scalar field does not just roll down its potential. There are also quantum fluctuations (recall the uncertainity principle) on top of the classical trajectory. These tiny fluctuations means that some regions will expand faster or slower than others so inflation will end at different times at different places. Since the expansion is really fast during inflation these tiny fluctuations gets streched to macroscopic scales. The effect of this is that the local densities after inflation will vary across space and this will leave us with perturbations in the metric. One of inflations key preditions is that the so-called curvature perturbation $\mathcal{R}$ is set up as a near gaussian random field with a power-spectrum $$\left<\mathcal{R}(\vec{k})\mathcal{R}(\vec{k}')^*\right> = (2\pi)^3\delta(\vec{k}-\vec{k}')P(k)$$ that is (for the simplest single field inflation models) on the form $P(k) = \frac{2\pi^2}{k^3}\mathcal{P}_\mathcal{R}(k)$ where $\mathcal{P}_\mathcal{R}(k) = A_s(k/k_{\rm pivot})^{n_s-1}$. The parameter $A_s$, the primoridal amplitude, is related to the energy scale (the value of $H$) for when inflation happens. The parameter $n_s$, the spectral index, depends on how long inflation lasted (the longer the closer $n_s$ is to the scale-invariant value $n_s = 1$). To solve the horizon problem we typically need $\sim 60$ e-folds of inflation (i.e. the universe expands by a factor $e^{60}$) so we expect $n_s$ to be at max a few percent away from $1$. Our best knowledge today is that $n_s\sim 0.96$.

      Connecting inflation to initial conditions: Conserved curvature perturbation

        Figure 3: The Curvature perturbations during and after inflation: The comoving horizon $\frac{1}{\mathcal{H}}$ shrinks during inflation and grows in the subsequent evolution. This implies that comoving scales exit the horizon at early times and re-enter the horizon at late times. While the curvature perturbations are outside of the horizon they don't evolve, so the primordial power-spectrum we can predict from inflation (and which are comptued at horizon exit during inflation) can be related directly to observables at late times and for our purposes. Figure taken from Baumann's lecture notes.

        One particular important combination of metric petrubations is the so-called curvature perturbation $\mathcal{R}$ defined (in the Newtonian gauge) as $$\mathcal{R} = \Phi + \frac{\mathcal{H}^2(\frac{d\Phi}{d\log a} - \Psi)}{4\pi G a^2 (\overline{\rho} + \overline{P})}$$ Why is this so important? First of all its a gauge invariant quantity, secondly it has the property that $$\frac{d\mathcal{R}}{dt} \approx 0$$ for any mode that is outside the horizon $k \ll \mathcal{H}$. This is what makes it possible to connect the initial conditions in the early Universe (but long after inflation) to what was generated in inflation. A lot of complicated physics happens (like reheating) between inflation and the time we start solving our equations, but we don't have to know this to know the initial conditions which is great. All the modes we observe today corresponds to fluctuations that were outside the horizon in the early Universe. Inflation set up these initial fluctuations and as inflation goes on the co-moving horizon decreases with time so a given mode will eventually leave the horizon and get frozen in there. Since $\mathcal{R}$ is conserved outside the horizon we can easily connect the predictions for the flutuations put up by inflation to what we should put as the initial conditions for $\Phi$. The next step is how to relate fluctuations in the metric to density perturbations in the different matter components.

        Adiabatic initial conditions

        We now know how to set the initial conditions for the metric potential, but how do we related that to the perturbations in the energy density of the different species? This is where we need some assumptions to proceed (that importantly can be tested with observations). Density fluctuations could be present initially or they can be generated from stresses in the matter which causally push matter around. The former assumption is that of adiabatic initial conditions while the latter represents so-called isocurvature perturbations. Adiabatic perturbations have a constant matter-to-radiation ratio everywhere so $$\frac{n_i}{n_\gamma} = \frac{\overline{n}_i}{\overline{n}_\gamma}$$ where $i$ is baryons, cold dark matter, neutrinos etc. Another way of phrasing this is to say that the energy density of all matter matter components is the same as in the background at some slightly different time that varies from place to place. Since $\overline{\rho}_i(a) \propto 1/a^{3(1+w_i)}$ we have $\rho_i(x,a) = \overline{\rho}_i(a+\delta a(x)) \simeq \overline{\rho}_i(a)(1 - 3(1+w_i)\delta a(x)/a)$ so $\delta_i(x,a) = -3(1+w_i)\delta a(x) / a$ and adiabatic perturbations therefore have $$\frac{\delta_i}{1+w_i} = \frac{\delta_j}{1+w_j}$$ for any two species $i,j$. In particular this implies that $$4\Theta_0 = 4\mathcal{N}_0 = \delta_b = \delta_{\rm CDM}$$ Now that we have all this what remains is to relate this to $\Phi,\Psi$. We'll give a simplified treatment of this here. A proper analysis would do this systematically by expanding every term in a power-series of $\frac{ck}{\mathcal{H}} \simeq k \eta \ll 1$ and using the evolution equations to match order by order. Anyway, lets study the equations. First of all remember that initially all modes of interest today is outside the horizon so $\frac{ck}{\mathcal{H}} \ll 1$. Secondly the metric perturbations evolves very slowly outside the horizon so we roughly have $\Phi^\prime \approx 0$ and $\Psi^\prime \approx 0$. Looking at the Poisson equation we then see that $$2\Theta_0 = 2\mathcal{N}_0 \simeq -\Psi$$ which together with $4\Theta_0 = 4\mathcal{N}_0 = \delta_b = \delta_{\rm CDM}$ gives us the initial condition for all the density perturbations. Then lets consider the other Einstein equation. Since the photons are tightly coupled to baryons (remember this drives $\Theta \to \Theta_0$ so the higher order multipoles are very small) so $\Theta_2 \ll \mathcal{N}_2$ which gives us $$\mathcal{N}_2 \simeq -\frac{(\Phi + \Psi)c^2k^2a^2}{12 f_\nu H_0^2}$$ where $f_\nu = \frac{\Omega_\nu}{\Omega_\gamma + \Omega_\nu}$. Next looking at the velocity equations. The $v_{\rm CDM}$ equation $v_{\rm CDM}^\prime = - v_{\rm CDM} - \frac{ck}{\mathcal{H}}\Psi$ can be written $$(v_{\rm CDM}a)^\prime = -\frac{cka}{\mathcal{H}}\Psi$$ The right hand side is $\propto a^2$ since $\mathcal{H} \propto 1/a$ and $\Psi$ is constant. Integrating gives us $$v_{\rm CDM} = \frac{C}{a} - \frac{ck}{2\mathcal{H}}\Psi$$ The first term is a decaying mode so can be ignored and we see that the relevant initial conditions is $v_{\rm CDM} = -\frac{ck}{2\mathcal{H}}\Psi$. The same analysis for the other species gives $$v_b \simeq -\frac{ck}{2\mathcal{H}}\Psi$$ $$\Theta_1 \simeq \frac{ck}{6\mathcal{H}}\Psi$$ $$\mathcal{N}_1 \simeq \frac{ck}{6\mathcal{H}}\Psi$$ Note that the velocity of all the species are the same (remember that $v_\gamma = -3\Theta_1$) as expected - they move as one fluid. From the equation for the neutrino moment $\mathcal{N}_\ell$ (skipping the details) it follows $$\mathcal{N}_\ell \simeq \frac{ck}{(2\ell+1)\mathcal{H}} \mathcal{N}_{\ell-1}\,\,\,\text{for}\,\,\, \ell \gt 2$$ and a similar analysis for the photon equation (again skipping the details) gives us $$\Theta_\ell \simeq -\frac{\ell}{2\ell+1}\frac{ck}{\mathcal{H}\tau^\prime}\Theta_{\ell-1}\,\,\,\text{for}\,\,\, \ell \gt 2$$ Finally we need to set the initial value of $\Phi,\Psi$. Since the equation-set is linear we can freely choose the normalization of the system before solving and then plug back in the correct normalization afterwards (note that the full expression for a mode $f(k) = A e^{i\theta}$ will not just have an amplitude $A$, but also a random phase $\theta$. For a gaussian random field $\theta$ is a uniform random number in $[0,2\pi)$, but this will not be relevant for us since this always cancel out in the power-spectrum $|f|^2 = A^2$. Any given Universe corresponds to different realisations of these phases and amplitudes from the underlying random field and if we could have predicted these numbers we would have been able to predict exacly where galaxy clusters would form and what the temperature of the CMB is in any given direction. We obviously cannot do this and have to settle for the statistical observables). What we will do is to choose the normalization such that the curvature perturbations at the initial time is unity. For a mode outside the horizon in the radiation era we have $$\mathcal{R} \simeq \Phi - \frac{1}{2}\Psi$$ The relation between the two potentials is given by (see Exercise 2 in Chapter 6 of Dodelson) $$\Phi + \Psi \simeq -\frac{2f_\nu}{5}\Psi$$ so $\mathcal{R} = 1$ gives us $$\Psi \simeq \frac{1}{\frac{3}{2} + \frac{2f_\nu}{5}}$$ $$\Phi \simeq -(1 + \frac{2f_\nu}{5})\Psi$$ Note that if you don't include massless neutrinos then we simply have $\Phi = -\Psi = 1$. This completes the discussion regarding initial conditions. We skipped a lot of details here so see Chapter 6 in Dodelson for a more thorough treatment. The important thing to remember here is what assumptions you are can make when massaging the equations to extract the initial conditions we need 1) small derivatives outside the horizon 2) $k\eta \ll 1$ 3) tight coupling which implies tiny multipoles for photons compared to the monopole and dipole 4) in the radiation era $\mathcal{H} \propto 1/a = e^{-x}$.

        What are the other options for the initial conditions? Well if we don't have adiabatic initial conditions then this would imply that $$ \frac{\delta_i}{1+w_i} - \frac{\delta_j}{1+w_j} \not = 0$$ This is the so-called iso-curvature perturbations (see e.g. this and this). However observations show no evidence for this and supports the assumption that the initial perturbations are (very close to) adiabatic so we will not go into more details about this posibilliy.

        Truncating the Boltzmann hierarchy

        There is a final thing to deal with. The photon multipoles form an infinite hierarchy of equations and this is not something we can deal with numerically. Luckily the higher order multipoles are generally small so we only need to include a finite amount of them $0,1,2,\ldots,\ell_{\rm max}$. What do we do with the equation for $\ell_{\rm max}$ with depends on the $\ell_{\rm max}+1$'st multipole? One possible solution is to simply take $\Theta_{\ell} = 0$ for $\ell \gt \ell_{\rm max}$, however from practice this turns out to not be the best idea as it requires (you can try this in your code) a really large $\ell_{\rm max}$ in order for this approximation to not spoil the solution. The error, even though small, will over time propagate down to the lowest order multipoles we want to compute accurately and over time this will produce inaccuracies (the error propagates down on a timescale $\eta \sim \ell_{\rm max}/k$ due to the way the system is coupled so $\ell_{\rm max}$ needs to be very large for the largest $k$). There is a better way. We will try to figure out roughly how $\Theta_{\ell_{\rm max}}$ evolves and use this to propose a slightly modified last equation. Consider the equation for $\ell \gt 2$: $$\Theta^\prime_\ell = \frac{\ell ck}{(2\ell+1)\mathcal{H}}\Theta_{\ell-1} - \frac{(\ell+1)ck}{(2\ell+1)\mathcal{H}}\Theta_{\ell+1} + \tau^\prime \Theta_\ell$$ If we can ignore the last term (which is not a terrible approximation after tight coupling has ended and photons decouple) then the recursion relation is the same as that of the so-called spherical Bessel function $\Theta_\ell = j_\ell(k\eta)$ since this function satisfy $$j_\ell'(x) = \frac{\ell}{2\ell+1}j_{\ell-1}(x) - \frac{(\ell+1)}{(2\ell+1)}j_{\ell+1}(x)$$ This function also satisfy the recursion relation $$j_{\ell+1}(x) = \frac{2\ell+1}{x}j_\ell(x) - j_{\ell-1}(x)$$ so if we simply assume that $\Theta_\ell$ satisfy the relation above, i.e. $$\Theta_{\ell+1}(x) \approx \frac{2\ell+1}{k\eta}\Theta_\ell - \Theta_{\ell-1}$$ and use this to rewrite the $\Theta_{\ell+1}$ term we get $$\Theta^\prime_\ell \approx \frac{ck}{\mathcal{H}}\Theta_{\ell-1} - \frac{(\ell+1)}{\eta\mathcal{H}}\Theta_\ell + \tau^\prime \Theta_\ell$$ This equation no longer depends on higher order multipoles than $\ell$ and can be used to truncate the hierarchy by using this for the largest multipole we include $\ell = \ell_{\rm max}$. This turns out to work really well and allows us to use a really low $\ell_{\rm max}\sim 6-8$ and still get an accurate solution for the three lowest multipoles (the only ones we need to compute the CMB power-spectrum using the line of sight integration technique we will go through in the next chapter). The same procedure as here also applies for the neutrino and polarization multipoles.


        • Inflation is one possible solution to the flatness problem, the horizon problem, the exotic relic problem (if you consider this a problem) and provides the seed for structures in our Universe.
        • Inflation generates a slightly non-scale invariant spectrum $P(k) = \frac{2\pi^2}{k^3} A_s (k/k_{\rm pivot})^{n_s-1}$ characterized by a spectral index $n_s$ slightly less than $1$. This is the strongest evidence we have today as this was not something we expected, but a prediction from inflation that was later confirmed by observations.
        • The fluctuations are close to gaussian. This we could "expect" otherwise so not a strong evidence, however a key predictions of inflation models is that they predict slight non-gaussianity that could be measured. This would be strong evidence. The backside is that the landscape of inflationary models predict anything from almost no to a lot of non-gaussianity so any observation can probably be accounted for. Nevertheless if we stick to the simplest single field slow-roll we discussed here there is a clear prediction which would provide strong evidence for that scenario.
        • Inflation generates gravitational waves which leads to a B-mode polarization signal. This is the main goal of current and future CMB experiments and would provide even stronger evidence for inflation (some alternatives to inflation predicts zero gravitational waves).

        The (adiabatic) initial conditions are given by: $$ \begin{align} \Psi &= \frac{1}{\frac{3}{2} + \frac{2f_\nu}{5}}\\ \Phi &= -(1+\frac{2f_\nu}{5})\Psi \\ \delta_{\rm CDM} &= \delta_b = -\frac{3}{2} \Psi \\ v_{\rm CDM} &= v_b = -\frac{ck}{2\mathcal{H}} \Psi\\ \Theta_0 &= -\frac{1}{2} \Psi \\ \Theta_1 &= +\frac{ck}{6\mathcal{H}}\Psi \\ \Theta_2 &= \left\{ \begin{array}{l} -\frac{8ck}{15\mathcal{H}\tau^\prime} \Theta_1, \quad\quad \textrm{(with polarization)} \\ -\frac{20ck}{45\mathcal{H}\tau^\prime} \Theta_1, \quad\quad \textrm{(without polarization)} \end{array}\right. \\ \Theta_\ell &= -\frac{\ell}{2\ell+1} \frac{ck}{\mathcal{H}\tau^\prime} \Theta_{\ell-1}\\ \Theta_0^P &= \frac{5}{4} \Theta_2 \\ \Theta_1^P &= -\frac{ck}{4\mathcal{H}\tau'} \Theta_2 \\ \Theta_2^P &= \frac{1}{4}\Theta_2 \\ \Theta_\ell^P &= -\frac{\ell}{2\ell+1} \frac{ck}{\mathcal{H}\tau^\prime} \Theta_{\ell-1}^P \\ \mathcal{N}_0 &= -\frac{1}{2} \Psi \\ \mathcal{N}_1 &= +\frac{ck}{6\mathcal{H}}\Psi \\ \mathcal{N}_2 &= -\frac{c^2k^2 a^2 (\Phi+\Psi)}{12H_0^2\Omega_{\nu 0}}\\ \mathcal{N}_\ell &= \frac{ck}{(2\ell+1)\mathcal{H}} \mathcal{N}_{\ell-1}, \quad\quad \ell \ge 3 \end{align} $$ where $f_{\nu} = \frac{\Omega_{\nu 0}}{\Omega_{\gamma 0} + \Omega_{\nu 0}}$. If you don't include neutrinos then set $f_\nu = 0$. Since the equation system is linear we are free to choose the normalization of $\Psi$ as we want when we solve it (the normalization can be done in the end). The particular normalization we use here is such that the curvature perturbations $\mathcal{R} = 1$.

        The Boltzmann hierarchy can be truncated by using a modified equation for the last multipole we include: $$\Theta^\prime_\ell = \frac{ck}{\mathcal{H}}\Theta_{\ell-1} - \frac{(\ell+1)}{\eta\mathcal{H}}\Theta_\ell + \tau^\prime \Theta_\ell$$ $$\Theta^\prime_{P\, \ell} = \frac{ck}{\mathcal{H}}\Theta^P_{\ell-1} - \frac{(\ell+1)}{\eta\mathcal{H}}\Theta^P_\ell + \tau^\prime \Theta^P_\ell$$ $$\mathcal{N}^\prime_\ell = \frac{ck}{\mathcal{H}}\mathcal{N}_{\ell-1} - \frac{(\ell+1)}{\eta\mathcal{H}}\mathcal{N}_\ell$$ We only need to inlcude $6-8$ multipoles for photons and polarization and $10-12$ for neutrinos (a bit more are needed as the higher multipoles don't have the additional $1/\tau'$ suppression as photons have in the early Universe) to get a resonably accurate result.