Cosmology Lecture 24

Normalization of Power Spectrum

Recall we had the power spectrum:

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

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

• Use CMB data
• Use galaxy clustering strength

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

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

Finishing Inflation

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

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

Our equation of motion reads:

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

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

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

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

If ${\displaystyle T\ll T_{c}}$, we have the same value of ${\displaystyle \mu ^{4}}$ at ${\displaystyle \phi =0}$, but the potential decreases to a minimum at ${\displaystyle \phi =\mu }$, where the true vacuum is. After that point, ${\displaystyle V\propto \phi ^{4}}$; it increases again. Thus, the relationship between ${\displaystyle V}$ and ${\displaystyle \phi }$ is governed by the relationship between ${\displaystyle T}$ and ${\displaystyle T_{c}}$. Remember that in order to have inflation in the early universe, we need ${\displaystyle {\dot {\phi }}^{2}\ll V(\phi )}$. 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 (${\displaystyle {\dot {\phi }}^{2}\ll V(\phi )}$). As the universe makes the phase transition from the false vacuum to the true vacuum, ${\displaystyle \phi }$ will oscillate about the minimum vacuum energy at ${\displaystyle \phi =\mu }$. In our Lagrangian, ${\displaystyle \phi }$ is coupled to radiation, so this oscillation leads to a “reheating” of the radiation in the universe. This is effectively a frictional term on ${\displaystyle \phi }$, which allows ${\displaystyle \phi }$ to finally settle into the true vacuum. Adding this friction term to our equation for the evolution of ${\displaystyle \phi }$ (and setting ${\displaystyle \nabla \phi =0}$–we’ll assume the universe is spatially homogeneous) we have:

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

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

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

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 (${\displaystyle \delta t\approx {\delta \phi \over {\dot {\phi }}}}$). 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 ${\displaystyle \Phi }$, the potential fluctuations, is given by:

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

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

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

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

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

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