Cosmology Lecture 24

Normalization of Power Spectrum

Recall we had the power spectrum:

$P(k)=Ak^{n}T^{2}(k)\,\!$

$Ak^{n}$ is set in the primordial universe, and we have no way of theoretically predicting $A$, the normalization constant. We have two choices for getting $A$:

• Use CMB data
• Use galaxy clustering strength

The variance of linear perturbation on the scale of $R$ (or $M={4\pi \over 3}{\bar {\rho }}R^{3}$) is given by the equation:

${\begin{aligned}\sigma ^{2}(R)&\equiv \int {{d^{3}k \over (2\pi )^{3}}P(k){\tilde {W}}^{2}(kR)}\\&=\int {d\ln k{4\pi k^{3}P(k) \over (2\pi )^{3}}{\tilde {W}}^{2}(kR)}\\&=\int {d\ln k\Delta {\tilde {W}}^{2}}\\\end{aligned}}\,\!$

where $W$ is some window function, and ${\tilde {W}}$ is the Fourier transform of that window. In general, people have decided to normalize this to an average over the scale of $8H^{-1}Mpc$, and to denote this as $\sigma _{8}$.

Finishing Inflation

Recall from last time that $\phi$ is the inflaton field. The energy density and pressure from $\phi$ are given by:

${\begin{aligned}\rho _{\phi }&=T^{00}={\frac {1}{2}}{\dot {\phi }}^{2}+{\frac {1}{2}}(\nabla \phi )^{2}+V(\phi )\\P_{\phi }&=T^{ii}=\phi {\dot {\phi }}^{2}-{1 \over 6}(\nabla \phi )^{2}-V(\phi )\\\end{aligned}}\,\!$

For this pair of equations, if ${\dot {\phi }}^{2}\ll V(\phi )$ and $(\nabla \phi )^{2}\ll V(\phi )$, then $\rho _{\phi }=-P_{\phi }$. We found that this means $w=-1$, and we have ${\ddot {a}}>0$, so we have an exponentially expanding universe.

Our equation of motion reads:

${\ddot {\phi }}-\nabla ^{2}+3{{\dot {a}} \over a}{\dot {\phi }}=-{dV(\phi ) \over d\phi }\,\!$

So what is $V(\phi )$? Typically $V(\phi ,T=0)\propto (\phi ^{2}-\mu ^{2})^{2}$, where $T$ is the temperature. At finite temperature, The effective potential is related to the $T=0$ potential by:

$V_{eff}(\phi ,T)=V(\phi ,T=0)+\mu ^{2}\left({T \over T_{c}}\right)^{2}\phi ^{2}\,\!$

For $T\gg T_{c}$, $V_{eff}\propto \phi ^{2}$, and $V_{eff}=\mu ^{4}$ when $\phi =0$. This non-zero potential at $\phi =0$ is called the false vacuum.

If $T\ll T_{c}$, we have the same value of $\mu ^{4}$ at $\phi =0$, but the potential decreases to a minimum at $\phi =\mu$, where the true vacuum is. After that point, $V\propto \phi ^{4}$; it increases again. Thus, the relationship between $V$ and $\phi$ is governed by the relationship between $T$ and $T_{c}$. Remember that in order to have inflation in the early universe, we need ${\dot {\phi }}^{2}\ll V(\phi )$. Therefore, as we have a gradual roll-off from the false vacuum into the true vacuum, we have inflation.

The End of Inflation

Inflation stops when the slow roll-off doesn’t hold anymore (${\dot {\phi }}^{2}\ll V(\phi )$). As the universe makes the phase transition from the false vacuum to the true vacuum, $\phi$ will oscillate about the minimum vacuum energy at $\phi =\mu$. In our Lagrangian, $\phi$ is coupled to radiation, so this oscillation leads to a “reheating” of the radiation in the universe. This is effectively a frictional term on $\phi$, which allows $\phi$ to finally settle into the true vacuum. Adding this friction term to our equation for the evolution of $\phi$ (and setting $\nabla \phi =0$–we’ll assume the universe is spatially homogeneous) we have:

${\ddot {\phi }}+3H{\dot {\phi }}=-{dV \over d\phi }-\Gamma _{\phi }{\dot {\phi }}\,\!$

Similarly the evolution of the radiation density is modified by this interaction:

${d\rho _{r} \over dt}+4H\rho _{r}=\Gamma _{\phi }{\dot {\phi }}^{2}\rho _{\phi }\,\!$

Origin of Fluctuations

Quantum fluctuations in the vacuum field can cause different portions of the universe to undergo the phase transition into the true vacuum state at slightly different times ($\delta t\approx {\delta \phi \over {\dot {\phi }}}$). This causes different portions of the universe to undergo heating at slightly redshifts. Areas which undergo heating at earlier redshifts will thus end up a little cooler at the time that inflation ends. We can show that $\Phi$, the potential fluctuations, is given by:

$\nabla ^{2}\Phi =4\pi G{\bar {\rho }}a\delta \,\!$

where $\delta$ is the density fluctuation. Thus, $k^{2}\Phi _{k}\propto \delta _{k}$, so:

$\Delta _{\Phi }^{2}\propto k^{3}P_{\Phi }(k)\propto k^{3}\Phi ^{2}\Rightarrow P_{\Phi }\propto k^{-4}P_{\delta }\,\!$

From messy QFT calculations, we can say $\langle \delta \phi \rangle ={H \over 2\pi }$:

$\Delta _{\Phi }\approx H\delta t\approx H{\delta \phi \over {\dot {\phi }}}\approx {H^{2} \over 2\pi {\dot {\phi }}}\,\!$

So $\Delta _{\Phi }^{2}\approx \left({H^{2} \over 2\pi {\dot {\phi }}}\right)^{2}\approx$constant. This means $P_{\Phi }(k)\propto k^{-4}$, and so $P_{\delta }(k)\propto k^{+1}$. However, it should be noted that although $H$ is constant, ${\dot {\phi }}$ is not perfectly constant, and so $P_{\delta }(k)$ doesn’t scale perfectly as $k$.