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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T=0} potential by:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{eff}(\phi ,T)=V(\phi ,T=0)+\mu ^{2}\left({T \over T_{c}}\right)^{2}\phi ^{2}\,\!}

For Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T\gg T_{c}} , $V_{eff}\propto \phi ^{2}$, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{eff}=\mu ^{4}} when Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi =0} . This non-zero potential at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi =0} is called the false vacuum.

If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T\ll T_{c}} , we have the same value of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mu ^{4}} at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi =0} , but the potential decreases to a minimum at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi =\mu } , where the true vacuum is. After that point, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \nabla \phi =0} –we’ll assume the universe is spatially homogeneous) we have:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 $\Phi$, the potential fluctuations, is given by:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle \delta \phi \rangle ={H \over 2\pi }} :

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta _{\Phi }\approx H\delta t\approx H{\delta \phi \over {\dot {\phi }}}\approx {H^{2} \over 2\pi {\dot {\phi }}}\,\!}

So Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta _{\Phi }^{2}\approx \left({H^{2} \over 2\pi {\dot {\phi }}}\right)^{2}\approx } constant. This means Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P_{\Phi }(k)\propto k^{-4}} , and so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P_{\delta }(k)\propto k^{+1}} . However, it should be noted that although $H$ is constant, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\dot {\phi }}} is not perfectly constant, and so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P_{\delta }(k)} doesn’t scale perfectly as $k$.