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Inflation.html
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<!DOCTYPE html>
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<title>CMB physics</title>
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<h2>Problems for Inflation</h2>
What happened before Inflation?
when inflation began — if it happened at all — and how long it lasted.
<p>Two important features of Inflation: Gaussianity and B modes. See <a href = "https://www.nature.com/articles/458820a">Cosmology: The test of inflation</a></p>
<h2>Alternative theories to Inflation</h2>
In 2001, the two physicists proposed a radical alternative to inflation called ekpyrosis, from the Greek for 'out of fire'.
It grew out of discussions with string theorists, who see the visible world as inhabiting lower-dimensional membranes, or branes, in a universe made up of at least 10 dimensions.
Steinhardt and Turok proposed two universes on separate three-dimensional branes that would oscillate back and forth along a mutually perpendicular dimension, like sheets hung out to dry on parallel washing lines.
Every trillion years or so, after each universe had dissipated into darkness during an expansive phase, the two branes would approach one another and collide,
releasing a fireball of energy to start each universe afresh. "It would mean that the Big Bang wouldn't be a beginning but a collision," says Steinhardt.
<p> See <a href = "brandenberger2008.pdf">Alternative to Inflation</a> and <a href="https://arxiv.org/abs/2202.12897"> Alternative ideas in cosmology </a> </p>
<ol>
<li>Topological Defects and varying light speed </li>
<li>Bouncing cosmologies:pre–Big Bang scenario;ekpyrotic scenario;cyclic universe </li>
<li>String gas</li>
</ol>
Inflationary cosmology, although extremely successful phenomenologically, is faced with serious conceptual problems
that have motivated alternative models. At present, though,none of the proposed alternatives is as well developed as inflationary cosmology; all of them lack a formulation in terms
of an effective field theory that can describe the entire period
of cosmological evolution from when the cosmological perturbations are produced until when they are measured. Nor
do any of the proposed alternatives solve the famous problems of SBB cosmology as elegantly as inflation does.
Nevertheless, it is now clear that alternative cosmological scenarios exist that can generate a nearly scale-invariant
spectrum of cosmological perturbations to explain the details of the CMB anisotropy spectrum.
Most important, the alternative models can be distinguished from inflation by observational means. And cosmologists expect a lot of new observational data over the next few years.
<h2> Inflation and Big Bang -- Which one comes first?</h2>
See Section 7.9 of Hannu's lecture note.
<h2>comoving horizon</h2>
\begin{equation}
\eta(t) \equiv \int_{0}^{t} \frac{d t^{\prime}}{a\left(t^{\prime}\right)} = \int_{0}^{a} d \ln a^{\prime} \frac{1}{a^{\prime} H\left(a^{\prime}\right)}
\end{equation}
Thus, the comoving horizon $\eta$ is the logarithmic integral of the comoving Hubble radius, $1 / a H$.
The comoving Hubble radius is the approximate distance over which light can travel in the course of one expansion time, i.e., the time in which the scale factor increases by a factor of $e$.
It provides a yardstick to assess <font color="red">whether particles can, at the given epoch, communicate within one $e$ -fold of expansion</font>.
<h2>Statistical properties of Gaussian perturbations </h2>
Going back to coordinate space, we find
$$
\langle g(\mathbf{x})\rangle=\left\langle\sum_{\mathbf{k}} g_{\mathbf{k}} e^{i \mathbf{k} \cdot \mathbf{x}}\right\rangle=\sum_{\mathbf{k}}\left\langle g_{\mathbf{k}}\right\rangle e^{i \mathbf{k} \cdot \mathbf{x}}=0
$$
The square of the perturbation can be written as
$$
g(\mathbf{x})^{2}=\sum_{\mathbf{k}} g_{\mathbf{k}}^{*} e^{-i \mathbf{k} \cdot \mathbf{x}} \sum_{\mathbf{k}^{\prime}} g_{\mathbf{k}^{\prime}} e^{i \mathbf{k}^{\prime} \cdot \mathbf{x}}
$$
since $g(\mathrm{x})$ is real. The typical amplitude of the perturbation is described by the variance, the expectation value of this square,
\begin{aligned}
\left\langle g(\mathbf{x})^{2}\right\rangle &=\sum_{\mathbf{k} \mathbf{k}^{\prime}}\left\langle g_{\mathbf{k}}^{*} g_{\mathbf{k}^{\prime}}\right\rangle e^{i\left(\mathbf{k}^{\prime}-\mathbf{k}\right) \cdot \mathbf{x}}=\sum_{\mathbf{k}}\left\langle\left|g_{\mathbf{k}}\right|^{2}\right\rangle=2 \sum_{\mathbf{k}} s_{\mathbf{k}}^{2} .=\left(\frac{2 \pi}{L}\right)^{3} \sum_{\mathbf{k}} \frac{1}{4 \pi k^{3}} \mathcal{P}(k) \\
& \rightarrow \frac{1}{4 \pi} \int \frac{d^{3} k}{k^{3}} \mathcal{P}(k)=\int_{0}^{\infty} \frac{d k}{k} \mathcal{P}(k)=\int_{-\infty}^{\infty} \mathcal{P}(k) d \ln k
\end{aligned}
Note, an alternative definition for the power spectrum is
\begin{equation}
P(k) \equiv V\left\langle\left|g_{\mathbf{k}}\right|^{2}\right\rangle
\end{equation}
This $\mathcal{P}(k)$ is the common <font color="red"><i>dimensionless power spectrum</i> </font> $\Delta(k)$
\begin{equation}
P(k)=\frac{2 \pi^{2}}{k^{3}} \mathcal{P}(k)
\end{equation}
<h2> Tensor-to-scalar ratio</h2>
Referring to Chapter 8.7 of Hannu's lecture.
inflation generates primordial scalar perturbations $\mathcal{R}_{k}$ with the power spectrum
\begin{equation}
\mathcal{P}_{\mathcal{R}}(k)=\left[\left(\frac{H}{\dot{\varphi}}\right)\left(\frac{H}{2 \pi}\right)\right]_{\mathcal{H}=a k}^{2}=\frac{1}{4 \pi^{2}}\left(\frac{H^{2}}{\dot{\varphi}}\right)_{t=t_{k}}^{2}
\end{equation}
and primordial tensor perturbations with the power spectrum
\begin{equation}
\mathcal{P}_{h}(k)=\frac{8}{M_{\mathrm{Pl}}^{2}}\left(\frac{H}{2 \pi}\right)_{t=t_{k}}^{2}
\end{equation}
The <font color="red"><i>tensor-to-scalar ratio </i> </font> is the ratio of the two primordial spectra
\begin{equation}
r \equiv \frac{\mathcal{P}_{h}(k)}{\mathcal{P}_{\mathcal{R}}(k)}=\frac{8}{M_{\mathrm{Pl}}^{2}}\left(\frac{\dot{\bar{\varphi}}}{H}\right)_{\mathcal{H}=k}^{2}
\end{equation}
Applying the slow-roll equations
$
H^{2}=\frac{V}{3 M_{\mathrm{Pl}}^{2}} \quad \text { and } \quad 3 H \dot{\varphi}=-V^{\prime} \quad \Rightarrow \quad \frac{\dot{\varphi}}{H}=-M_{\mathrm{Pl}}^{2} \frac{V^{\prime}}{V}
$
they become
\begin{equation}
\begin{aligned}
\mathcal{P}_{\mathcal{R}}(k) &=\frac{1}{24 \pi^{2}} \frac{1}{M_{\mathrm{Pl}}^{4}} \frac{V}{\varepsilon} \\
\mathcal{P}_{h}(k) &=\frac{2}{3 \pi^{2}} \frac{V}{M_{\mathrm{Pl}}^{4}}
\end{aligned}
\end{equation}
where $\varepsilon$ is the slow-roll parameter. The tensor-to-scalar ratio is thus
\begin{equation}
r \equiv \frac{\mathcal{P}_{h}(k)}{\mathcal{P}_{\mathcal{R}}(k)}=16 \varepsilon
\end{equation}
<h2>Harrison-Zeldovich spectrum</h2>
Since during inflation, $V$ and $V^{\prime}$ change slowly while a wide range of scales $k$ exit the horizon. $\mathcal{P}_{\mathcal{R}}(k)$ and $\mathcal{P}_{h}(k)$ should be slowly varying functions of $k$.
We define the $\bf{spectral\ indices}$ $n_{s}$ and $n_{t}$ of the primordial spectra as
\begin{equation}
\begin{aligned}
n_{s}(k)-1 & \equiv \frac{d \ln \mathcal{P}_{\mathcal{R}}}{d \ln k} \\
n_{t}(k) & \equiv \frac{d \ln \mathcal{P}_{h}}{d \ln k} .
\end{aligned}
\end{equation}
if the spectral index is independent of $k$, we say that the spectrum is $\bf{scale \ free}$. In this case the primordial spectra have the power-law form
\begin{equation}
\mathcal{P}_{\mathcal{R}}(k)=A_{s}^{2}\left(\frac{k}{k_{p}}\right)^{n_{s}-1} \quad \text { and } \quad \mathcal{P}_{h}(k)=A_{t}^{2}\left(\frac{k}{k_{p}}\right)^{n_{t}}
\end{equation}
where $k_{p}$ is some chosen reference scale, "pivot scale", and $A_{s}$ and $A_{t}$ are the amplitudes at this pivot scale.
If the power spectrum is constant,
$$
\mathcal{P}=\text { const. }
$$
corresponding to $n_{s}=1$ and $n_{t}=0$, we say that the spectrum is scale invariant. A scaleinvariant scalar spectrum is also called the Harrison-Zeldovich spectrum.
<h2>Consistency condition</h2>
\begin{equation}
n_{t}=-\frac{r}{8}
\end{equation}
<h2> comoving curvature perturbation $\mathcal{R}$ and density perturbation </h2>
For small scales, the gravitational potential and density perturbation are related to the
curvature perturbation as
\begin{equation}
\begin{aligned}
\Phi_{\mathbf{k}} &=-\frac{3}{5} \mathcal{R}_{\mathbf{k}} \quad(\text { mat.dom, subhorizon}) \\
\delta_{\mathbf{k}} &=-\frac{2}{3}\left(\frac{k}{\mathcal{H}}\right)^{2} \Phi_{\mathbf{k}}=\frac{2}{5}\left(\frac{k}{\mathcal{H}}\right)^{2} \mathcal{R}_{\mathbf{k}}
\end{aligned}
\end{equation}
For the scale-invariant primordial power spectrum
\begin{equation}
\mathcal{P}_{\mathcal{R}}(k)=A_{s}^{2}=\text { const. }
\end{equation}
giving
\begin{equation}
\begin{aligned}
\mathcal{P}_{\Phi}(k) &=\frac{9}{25} \mathcal{P}_{\mathcal{R}}(k)=\frac{9}{25} A_{s}^{2}=\text { const } \\
\mathcal{P}_{\delta}(t, k) &=\frac{4}{9}\left(\frac{k}{\mathcal{H}}\right)^{4} \mathcal{P}_{\Phi}(k)=\frac{4}{25}\left(\frac{k}{\mathcal{H}}\right)^{4} \mathcal{P}_{\mathcal{R}}(k) \\
&=\frac{4}{25}\left(\frac{k}{\mathcal{H}}\right)^{4} A_{s}^{2} \propto t^{4 / 3} k^{4}
\end{aligned}
\end{equation}
Thus perturbations in the gravitational potential are scale invariant, but perturbations in density
are not. Instead the density perturbation spectrum is steeply rising, meaning that there is much
more structure at small scales than at large scales. Thus the scale invariance refers to the
gravitational aspect of perturbations.
The relation between density and gravitational potential perturbations reflects the nature of
gravity: A 1% overdense region 100 Mpc across generates a much deeper potential well than a
1% overdense region 10 Mpc across, since the former has 1000 times more mass. Therefore we
need much stronger density perturbations at smaller scales to have an equal contribution to $\Phi$.
However, if we extrapolate above equation back to horizon entry, $k = \mathcal{H}$, we get
\begin{equation}
\delta_{H}^{2}(k) \equiv " \mathcal{P}_{\delta}\left(k, t_{k}\right) " \equiv \frac{4}{25} \mathcal{P}_{\mathcal{R}}(k)=\left(\frac{2}{5} A_{s}\right)^{2}=\mathrm{const}
\end{equation}
Thus for scale-invariant primordial perturbations, <font color="red"><i>density perturbations of all scales enter the horizon with the same amplitude</i> </font>(Same aspect with Carlo's lecture.).
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