Cosmology Lecture 21

Power Spectrum Shape

Recall that we defined the power spectrum $P$ by:

$\langle \delta ({\vec {k}})\delta ^{*}({\vec {k}}^{\prime })\rangle =P(k)(2\pi )^{3}\delta _{D}({\vec {k}}-{\vec {k}}^{\prime })\,\!$

We also defined a dimensionless power spectrum $\Delta$:

$\Delta ^{2}(k)\equiv {4\pi k^{3}P(k) \over (2\pi )^{3}}\,\!$

Now let’s consider the effects of matter fluctuation on $P(k)$. For this, we will consider $\delta =\delta _{matter}=\delta _{CDM}$ (since CDM is primarily responsible for clustering).

The primordial shape of the power spectrum, as predicted by the standard model, is:

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

where $n$ is the spectral index, which most inflationary models predict to be $\approx 1$. The $n=1$ model is called the Harrison-Zeldovich spectrum. In this model, we have nearly a nearly scale-invariant spectrum for fluctuations in a gravitational potential $\Phi$. Working backward, we can show that scale invariance implies a spectral index of $n=1$:

$\Delta _{\Phi }^{2}\propto k^{3}P_{\Phi }(k)\propto constant\,\!$

Using the Poisson Equation to relate $\Phi$ and $\delta$ in comoving coordinates:

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

Taking the Fourier Transform, we have:

$k^{2}\Phi (k)\propto \delta (k)\,\!$

Then by definition of $P(k)$, we know $\Phi ^{2}\propto P_{\Phi }$ and $\delta ^{2}\propto P_{\delta }$. Therefore, squaring both sides of the above gives us:

${\begin{aligned}k^{4}P_{\Phi }(k)&\propto P_{\delta }(k)\\P_{\delta }(k)&\propto k\Delta _{\Phi }^{2}(k)\propto k^{1}\\\end{aligned}}\,\!$

Thus, $n=1$. Now, we have the shape of the Power Spectrum as it was set by the early universe, but we need to find how it evolves as the universe develops. Thus we define the shape of the Processed Power Spectrum with some transfer function $T(k)$:

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

To solve for $T(k)$, we need to find the time-dependence of interactions on a for a given length scale. Recall that on PS#9 we introduced the physical horizon size:

$d_{H}\propto ct\propto {\begin{cases}a^{2}\ radiation-dominated\\a^{3 \over 2}\ matter-dominated\end{cases}}\,\!$

We showed that for a matter-dominated universe at redshift $(1+z)\gg \Omega _{0,M}^{-1}$, the horizon size as a function of redshift was:

$d_{H}(z)=2c(\Omega _{0,M}H_{0}^{2})^{-{\frac {1}{2}}}(1+z)^{-{3 \over 2}}\,\!$

Features at some length scale $\ell$ grow with the universe as $\ell \propto a$. However, since the horizon grows as $d_{H}\propto a^{3 \over 2}$, longer features come into communication with each other at later times. Thus, we must have a $k$ dependence (and $\Omega _{0,M}$ dependence) in our transfer function.

In PS#7 we found that for $y\equiv {\rho _{m} \over \rho _{r}}={a \over a_{eq}}$:

$\delta _{m}\propto y+{2 \over 3}\,\!$

Thus, $\delta _{m}\propto constant$ for $a\ll a_{eq}$ (radiation-dominated), and $\delta _{m}\propto a$ for $a\gg a_{eq}$ (matter-dominated). In the radiation-dominated era, perturbation modes with $\ell <d_{H}(z_{eq})$ enter the horizon, but $\delta$ is constant. On the other hand, in the matter-dominated era, length scales $\ell >d_{H}(z_{eq})$ enter the horizon, and $\delta \propto a$, and therefore, $\delta$ grows. Thus, the Power Spectrum must have a break at the critical length scale of the horizon size at matter-radiation equality. On PS#5 we found that $z_{eq}\approx 24000(\Omega _{0,M}h^{2})$ for $(1+z_{eq})\gg \Omega _{0,M}^{-1}$. Therefore:

$d_{H}(z_{eq})={2c \over {\sqrt {\Omega _{0,M}}}}h_{0}(1+z_{eq})^{-{3 \over 2}}=6000(24000)^{-{3 \over 2}}(\Omega _{0,M}h^{2})^{-2}\,\!$

In comoving coordinates, this comes to:

$d_{H}(z_{eq})=6000(24000)^{-{\frac {1}{2}}}(\Omega _{0,M}h^{2})^{-1}=39(\Omega _{0,M}h^{2})^{-1}Mpc\,\!$

If we’d been more accurate and included radiation, we’d have gotten the answer:

${d_{H}(z_{eq})\approx 16(\Omega _{0,M}h^{2})^{-1}Mpc}\,\!$

Using $k_{eq}\sim {1 \over d_{H}(z_{eq})}$, we can find the corresponding wavenumber:

${k_{eq}\sim 0.06(\Omega _{0,M}h^{2})Mpc^{-1}}\,\!$

For $k<k_{eq}$ (large $\ell$), features entered the horizon in the matter-dominated era, and features grow as $a$, preserving the initial power law. Therefore $P(k)\propto k^{n}\propto k^{1}$. For $k>k_{eq}$ (small $\ell$), features entered the horizon in the radiation-dominated era, when they couldn’t grow. Thus $P(k)\propto k^{-3}$. So $P(k)$ had its shape imprinted at equality time, after which the whole curve grows as $P(k)\propto \delta ^{2}\propto a^{2}$.

Note that the dependence of $k_{eq}$ on $\Omega _{0,M}$ means we can infer $\Omega _{0,M}$ from measuring $P(k)$. Defining the “shape parameter” $\Gamma \equiv \Omega _{0,M}h$, then $T(k)$ is only a function of $q\equiv {k \over \Gamma }$.

Finally, we find that for small $k$, $\Delta ^{2}(k)\propto k^{4}$, and for large $k$, $\Delta ^{2}(k)\propto k^{0}$. Thus, we have hierarchical structure formation, where smaller objects (large $k$) go non-linear and collapse earlier.