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\pdf_title "Temperature fluctuations on Pixel in FRLW"
\pdf_author "Renan Alves de Oliveira"
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\begin_body

\begin_layout Title
Temperature fluctuations on Pixel in FRLW
\end_layout

\begin_layout Author
Renan Alves de Oliveira
\begin_inset Newline newline
\end_inset

Thiago dos Santos Pereira
\end_layout

\begin_layout Date

\end_layout

\begin_layout Standard

\end_layout

\begin_layout Standard
For this work, we are interested in large scale effects which is dominated
 by the Sachs Wolfe effect: 
\begin_inset Formula 
\begin{eqnarray*}
\frac{\Delta\mathrm{T}}{\mathrm{T}}\left(\mathbf{r}\right) & = & \frac{1}{3}\Phi\left(\mathbf{r}\right),
\end{eqnarray*}

\end_inset

where 
\begin_inset Formula $\frac{\Delta\mathrm{T}}{\mathrm{T}}$
\end_inset

 is the temperature fluctuations evaluated at 
\begin_inset Formula $\mathbf{r}$
\end_inset

 and 
\begin_inset Formula $\Phi\left(\mathbf{r}\right)$
\end_inset

 is the gravitational potential.
 Using
\begin_inset Formula 
\begin{eqnarray*}
\frac{\Delta\mathrm{T}}{\mathrm{T}}\left(\mathbf{r}\right) & = & \frac{1}{3\left(2\pi\right)^{3}}\int\mathrm{d^{3}}\mathbf{k}e^{i\mathbf{k}\cdot\mathbf{r}}\Phi\left(\mathbf{k}\right),
\end{eqnarray*}

\end_inset

 let us suppose that this system is embedded in a box, with volume 
\begin_inset Formula $V$
\end_inset

, side 
\begin_inset Formula $L$
\end_inset

, and periodicity 
\begin_inset Formula 
\begin{eqnarray*}
\Phi\left(\mathbf{r}\right) & = & \Phi\left(\mathbf{r}+\mathbf{L}\right)=\Phi\left(\mathbf{r}+L\hat{\mathbf{x}}\right)=\Phi\left(\mathbf{r}+L\hat{\mathbf{y}}\right)=\Phi\left(\mathbf{r}+L\hat{\mathbf{z}}\right).
\end{eqnarray*}

\end_inset

Plugging this in the Fourier transform and comparing terms:
\begin_inset Formula 
\begin{eqnarray*}
e^{i\mathbf{k}\cdot\mathbf{r}} & = & e^{i\mathbf{k}\cdot\left(\mathbf{r}+L\hat{\mathbf{x}}\right)}\Rightarrow e^{iL_{x}k_{x}}=1,
\end{eqnarray*}

\end_inset

and doing the same for 
\begin_inset Formula $k_{y}$
\end_inset

 and 
\begin_inset Formula $k_{z}$
\end_inset

, we have
\begin_inset Formula 
\begin{eqnarray*}
\mathbf{k} & = & \frac{2\pi}{L}\left(\mathbf{n}_{x}+\mathbf{n}_{y}+\mathbf{n}_{z}\right).
\end{eqnarray*}

\end_inset

We will use a discretization of the Fourier space.
 Let us rewrite the Fourier transform as
\begin_inset Formula 
\begin{eqnarray*}
\frac{\Delta\mathrm{T}}{\mathrm{T}}\left(\mathbf{r}\right) & = & \frac{1}{3V}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}}\Phi\left(\mathbf{k}\right)\Rightarrow\frac{\Delta\tilde{\mathrm{T}}}{\tilde{\mathrm{T}}}\left(\mathbf{r}\right)=\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}}\Phi\left(\mathbf{k}\right).
\end{eqnarray*}

\end_inset

For a homogeneous distribution,
\begin_inset Formula 
\begin{eqnarray*}
\left\langle \Phi\left(\mathbf{k}\right)\Phi^{*}\left(\mathbf{q}\right)\right\rangle  & = & P\left(\mathbf{k}\right)\delta\left(\mathbf{k}-\mathbf{q}\right),
\end{eqnarray*}

\end_inset

however, for a Gaussian variables 
\begin_inset Formula $\phi\left(\mathbf{k}\right)$
\end_inset


\begin_inset Formula 
\begin{eqnarray*}
\left\langle \phi\left(\mathbf{k}\right)\phi^{*}\left(\mathbf{q}\right)\right\rangle  & = & \delta\left(\mathbf{k}-\mathbf{q}\right).
\end{eqnarray*}

\end_inset

Let us assume that
\begin_inset Formula 
\begin{eqnarray*}
\phi\left(\mathbf{k}\right) & \equiv & \frac{\Phi\left(\mathbf{k}\right)}{\sqrt{P\left(\mathbf{k}\right)}}\Rightarrow\Phi\left(\mathbf{k}\right)=\phi\left(\mathbf{k}\right)\sqrt{P\left(\mathbf{k}\right)}.
\end{eqnarray*}

\end_inset

The function 
\begin_inset Formula $\Delta\mathrm{T}/\mathrm{T}$
\end_inset

 is a real function, for our sum be a real function as well we should impose
 that
\begin_inset Formula 
\begin{eqnarray*}
\Phi\left(\mathbf{k}\right) & = & \Phi^{*}\left(-\mathbf{k}\right).
\end{eqnarray*}

\end_inset

Let us break our sum in two hemispheres using the parity condition above
 using 
\begin_inset Formula $\left(\theta_{k},\phi_{k}\right)\rightarrow\left(\pi-\theta_{k},\pi+\phi_{k}\right)$
\end_inset

 as
\begin_inset Formula 
\begin{eqnarray*}
\frac{\Delta\tilde{\mathrm{T}}}{\tilde{\mathrm{T}}}\left(\mathbf{r}\right) & = & \sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}}\Phi\left(\mathbf{k}\right)+\sum_{-\mathbf{k}}e^{-i\mathbf{k}\cdot\mathbf{r}}\Phi\left(-\mathbf{k}\right),\\
 & = & \sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}}\Phi\left(\mathbf{k}\right)+\sum_{-\mathbf{k}}e^{-i\mathbf{k}\cdot\mathbf{r}}\Phi^{*}\left(\mathbf{k}\right),\\
 & = & \sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}}\Phi\left(\mathbf{k}\right)+e^{-i\mathbf{k}\cdot\mathbf{r}}\Phi^{*}\left(\mathbf{k}\right).
\end{eqnarray*}

\end_inset

Using the redefinition of the Power Spectrum, and specifically for this
 case, 
\begin_inset Formula $P\left(\mathbf{k}\right)=P\left(k\right)$
\end_inset

 leading that 
\begin_inset Formula $\sqrt{P\left(\mathbf{k}\right)}$
\end_inset

 is already a real function, however
\begin_inset Formula 
\begin{eqnarray*}
\phi\left(\mathbf{k}\right) & = & \phi^{R}\left(\mathbf{k}\right)+i\phi^{I}\left(\mathbf{k}\right).
\end{eqnarray*}

\end_inset

Plugging the results above in the discretized Fourier transform:
\begin_inset Formula 
\begin{eqnarray*}
\frac{\Delta\tilde{\mathrm{T}}}{\tilde{\mathrm{T}}}\left(\mathbf{r}\right) & = & \sum_{\mathbf{k}}\sqrt{P\left(\mathbf{k}\right)}\left\{ e^{i\mathbf{k}\cdot\mathbf{r}}\left[\phi^{R}\left(\mathbf{k}\right)+i\phi^{I}\left(\mathbf{k}\right)\right]+e^{-i\mathbf{k}\cdot\mathbf{r}}\left[\phi^{R}\left(\mathbf{k}\right)-i\phi^{I}\left(\mathbf{k}\right)\right]\right\} ,
\end{eqnarray*}

\end_inset

leading to
\begin_inset Formula 
\[
\boxed{\frac{\Delta\tilde{\mathrm{T}}}{\tilde{\mathrm{T}}}\left(\mathbf{r}\right)=2\sum_{\mathbf{k}}\sqrt{P\left(\mathbf{k}\right)}\left[\cos\left(\mathbf{k}\cdot\mathbf{r}\right)\phi^{R}\left(\mathbf{k}\right)-\sin\left(\mathbf{k}\cdot\mathbf{r}\right)\phi^{I}\left(\mathbf{k}\right)\right].}
\]

\end_inset

Using 
\begin_inset Formula ${\bf k}$
\end_inset

, and defining
\begin_inset Formula 
\begin{eqnarray*}
\mathbf{n} & \equiv & \mathbf{n}_{x}+\mathbf{n}_{y}+\mathbf{n}_{z},
\end{eqnarray*}

\end_inset

where 
\begin_inset Formula $\mathbf{n}\in\mathbb{Z}$
\end_inset

, and
\begin_inset Formula 
\begin{eqnarray*}
\mathbf{k} & = & \frac{2\pi}{L}\mathbf{n}\Rightarrow k=\frac{2\pi}{L}n.
\end{eqnarray*}

\end_inset

Let
\begin_inset Formula 
\begin{eqnarray*}
\mathbf{r} & = & R\left(\sin\theta\cos\phi\hat{\mathbf{x}}+\sin\theta\sin\phi\hat{\mathbf{y}}+\cos\theta\hat{\mathbf{z}}\right),
\end{eqnarray*}

\end_inset

and 
\begin_inset Formula $\mathbf{n}=n\left(\sin\theta_{n}\cos\phi_{n}\hat{\mathbf{x}}+\sin\theta_{n}\sin\phi_{n}\hat{\mathbf{y}}+\cos\theta_{n}\hat{\mathbf{z}}\right)$
\end_inset

,
\begin_inset Formula 
\begin{eqnarray*}
\mathbf{k}\cdot\mathbf{r} & = & 2\pi n\frac{R}{L}\cos\gamma,
\end{eqnarray*}

\end_inset

where
\begin_inset Formula 
\begin{eqnarray*}
\cos\gamma & = & \cos\theta\cos\theta_{n}+\sin\theta\sin\theta_{n}\cos\left(\phi-\phi_{n}\right),
\end{eqnarray*}

\end_inset

 and
\begin_inset Formula 
\begin{eqnarray*}
\theta_{n} & = & \arccos\left(\frac{n_{z}}{n}\right),\qquad\phi_{n}=\arctan\left(\frac{n_{y}}{n_{x}}\right).
\end{eqnarray*}

\end_inset

We can rewrite the equation for the variation of the temperature as
\begin_inset Formula 
\begin{eqnarray*}
\frac{\Delta\tilde{\mathrm{T}}}{\tilde{\mathrm{T}}}\left(\mathbf{r}\right) & = & 2\sum_{\mathbf{k}}\sqrt{P\left(\mathbf{k}\right)}\left[\cos\left(2\pi n\frac{R}{L}\cos\gamma\right)\phi^{R}\left(\mathbf{k}\right)-\sin\left(2\pi n\frac{R}{L}\cos\gamma\right)\phi^{I}\left(\mathbf{k}\right)\right].
\end{eqnarray*}

\end_inset

To speed up this numerical code, we can evaluate all possible norms in terms
 of the first octant, and we do not take care of the terms of the south
 hemisphere below 
\begin_inset Formula $k_{z}$
\end_inset

 since we'd used the parity relation.
 Also, we can map all the other points using only the first octant:
\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Ocatant
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Relationship
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\cos\gamma$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\left(\theta,\phi\right)$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\cos\theta\cos\theta_{k}+\sin\theta\sin\theta_{k}\cos\left(\phi-\phi_{k}\right)}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
II
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\left(\theta,\pi-\phi\right)}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\cos\theta\cos\theta_{k}-\sin\theta\sin\theta_{k}\cos\left(\phi+\phi_{k}\right)}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
III
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\left(\theta,\pi+\phi\right)}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\cos\theta\cos\theta_{k}-\sin\theta\sin\theta_{k}\cos\left(\phi-\phi_{k}\right)}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
IV
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\left(\theta,-\phi\right)}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\ensuremath{\cos\theta\cos\theta_{k}+\sin\theta\sin\theta_{k}\cos\left(\phi+\phi_{k}\right)}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
This implies that
\begin_inset Formula 
\begin{eqnarray*}
\frac{\Delta\tilde{\mathrm{T}}}{\tilde{\mathrm{T}}}\left(\mathbf{r}\right) & = & 2\sum_{\mathbf{k}}\sqrt{P\left(\mathbf{k}\right)}\left\{ \phi_{1}^{R}\left(\mathbf{k}\right)\cos\left(2\pi n\frac{R}{L}\cos\gamma_{I}\right)-\phi_{1}^{I}\left(\mathbf{k}\right)\sin\left(2\pi n\frac{R}{L}\cos\gamma_{I}\right)\right.\\
 &  & +\phi_{2}^{R}\left(\mathbf{k}\right)\cos\left(2\pi n\frac{R}{L}\cos\gamma_{II}\right)-\phi_{2}^{I}\left(\mathbf{k}\right)\sin\left(2\pi n\frac{R}{L}\cos\gamma_{II}\right)\\
 &  & +\phi_{3}^{R}\left(\mathbf{k}\right)\cos\left(2\pi n\frac{R}{L}\cos\gamma_{III}\right)-\phi_{3}^{I}\left(\mathbf{k}\right)\sin\left(2\pi n\frac{R}{L}\cos\gamma_{III}\right)\\
 &  & \left.+\phi_{4}^{R}\left(\mathbf{k}\right)\cos\left(2\pi n\frac{R}{L}\cos\gamma_{IV}\right)-\phi_{4}^{I}\left(\mathbf{k}\right)\sin\left(2\pi n\frac{R}{L}\cos\gamma_{IV}\right)\right\} .
\end{eqnarray*}

\end_inset

We will start first with the invariant scale Harrison-Zel'dovich power spectrum:
\begin_inset Formula 
\begin{eqnarray*}
\mathcal{P}(k) & = & Ak^{-3},
\end{eqnarray*}

\end_inset

 and later, for other geometries.
\end_layout

\end_body
\end_document