Calculate variance on the ratio of 2 angular power spectra

[fab13]

TL;DR Summary

I am looking for a calculation about the variance on the ratio between 2 angular power spectra.

In the context of Survey of Dark energy stage IV, I need to evaluate the error on a new observable called "O" which is equal to :

\begin{equation}
O=\left(\frac{C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}}{C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}}\right)=\left(\frac{b_{s p}}{b_{p h}}\right)^{2}
\end{equation}

with spectroscopic $C_{\ell, \text { gal,sp }}^{\prime}$ defined by the integral :
$$
C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}=\int_{l_{m i n}}^{l_{\max }} C_{\ell, \mathrm{gal}, \mathrm{ph}}(\ell) \mathrm{d} \ell=b_{s p}^{2} \int_{l_{\min }}^{l_{\max }} C_{\ell, \mathrm{DM}}(\ell) \mathrm{d} \ell=b_{s p}^{2} C_{\ell, \mathrm{DM}}^{\prime}
$$
Same for photometric $C_{\ell, \text { gal,ph }}^{\prime}$ :
$$
C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}=\int_{l_{m i n}}^{l_{\max }} C_{\ell, \mathrm{gal}, \mathrm{ph}}(\ell) \mathrm{d} \ell=b_{p h}^{2} \int_{l_{m i n}}^{l_{\max }} C_{\ell, \mathrm{DM}} \mathrm{d} \ell=b_{p h}^{2} C_{\ell, \mathrm{DM}}^{\prime}
$$
We can infer this ratio of bias from the following definitions :
$$
\begin{array}{l}
C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}(k, z)=b_{s p}^{2} C_{\ell, \mathrm{DM}}^{\prime}(k, z) \\
C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}(k, z)=b_{p h}^{2} C_{\ell, \mathrm{DM}}^{\prime}(k, z)
\end{array}
$$

I need to estimate the variance $\sigma_o^{2}$ of this new observable. How could I begin a such task ?

For convenience, I am going to begin with ony one bin, so the integrals would disappear.

The $C_l^{GG}$ is defined by (by taking $\gamma = G$

$$
C_{i j}^{\gamma \gamma}(\ell) \simeq \frac{c}{H_{0}} \int \mathrm{d} z \frac{W_{i}^{\gamma}(z) W_{j}^{\gamma}(z)}{E(z) r^{2}(z)} P_{\delta \delta}\left[\frac{\ell+1 / 2}{r(z)}, z\right]
$$
where $i$ and $j$ identify pairs of redshift bins, $E(z)$ is the dimensionless Hubble parameter of Eq. (11), $r(z)$ is the comoving distance, $P_{\delta \delta}(k, z)$ is the matter power spectrum evaluated at $k=k_{\ell}(z) \equiv(\ell+1 / 2) / r(z)$ due to the Limber approximation; we define the weight function $W^{\gamma}(z)$ as
$$

\begin{aligned}
W_{i}^{\gamma}(z) &=\frac{3}{2} \frac{H_{0}}{c} \Omega_{\mathrm{m}, 0}(1+z) \tilde{r}(z) \int_{z}^{z_{\max }} \mathrm{d} z^{\prime} n_{i}\left(z^{\prime}\right)\left[1-\frac{\tilde{r}(z)}{\tilde{r}\left(z^{\prime}\right)}\right] \\
&=\frac{3}{2} \frac{H_{0}}{c} \Omega_{\mathrm{m}, 0}(1+z) \tilde{r}(z) \widetilde{W}_{i}(z)
\end{aligned}
$$
with $z_{\max }$ the maximum redshift of the source redshift distribution. With respect to the standard formalism, we replace the comoving distance $r(z)$ with its dimensionless scaled version $\tilde{r}(z)=r(z) /\left(c / H_{0}\right)$ to highlight that the dependence of $W_{i}^{\gamma}(z)$ on the cosmological.

So, as you can see, this is not going to be easy to compute the variance on the ration between $C\ell^{GG}$.

In the paper https://arxiv.org/abs/1910.09273, they simply put the following variance without demonstration :

\begin{equation}
\Delta C_{i j}^{\epsilon \epsilon}(\ell)=\sqrt{\frac{2}{(2 \ell+1) \Delta \ell f_{\mathrm{sky}}}} C_{i j}^{\epsilon \epsilon}(\ell)
\end{equation}

That's why any suggestion to start is welcome.

Thanks in advance for your remark.

 

[AndyW]

Thank you for your question. I understand your need to evaluate the error on a new observable in the context of the Survey of Dark Energy Stage IV. It is important to accurately estimate the variance of this observable in order to make meaningful and reliable conclusions.

To begin with, it is helpful to understand the definition of the new observable "O" that you have provided. This observable is defined as the ratio of the spectroscopic and photometric bias terms, which are themselves defined as integrals of the galaxy power spectrum. The goal is to estimate the variance of this observable, denoted by σo2.

As you have mentioned, it may be convenient to start with just one bin, in which case the integrals will disappear. However, it is important to keep in mind that the variance of the observable will depend on the number of bins and other parameters such as the maximum redshift of the source distribution.

One approach to estimating the variance of this observable is to use the standard formalism for the galaxy power spectrum, which involves the Limber approximation and the weight function Wγ(z). In this case, the variance of the observable can be expressed as the square root of the sum of the variances of the spectroscopic and photometric bias terms, multiplied by their respective weights. This approach may be computationally intensive, but it can provide a more accurate estimate of the variance.

Alternatively, as mentioned in the paper you referenced, a simpler approach may be to use the formula given in equation (2), which assumes a Gaussian distribution for the galaxy power spectrum. This approach may be less computationally intensive, but it may also be less accurate.

In conclusion, there are different approaches to estimating the variance of the new observable "O". Depending on the specific details of your study, one approach may be more suitable than the other. I hope this response has provided some useful insights and suggestions to help you begin your task. Good luck with your research!