---
# System prepended metadata

title: 'LIM math:'
tags: [LIM-Lytomo]

---

# LIM math:
###### tags: `LIM-Lytomo`
#### P(k) and $\sigma_{P(k)}$:

$\sigma_{P_{A}}(k, \mu) = \frac{P_{noiseless}(k,\mu) + P_{A, noise}(k,\mu)/W^2_{A}(k,\mu)}{\sqrt{N_{modes}(k,\mu)}}$


Where :

$P_{noiseless}(k,\mu)$ : The 2D power spectrum of the signal on a fine-grid of $(1 cMcp/h)^3$

$\sigma^{-2}_{P_{A}}(k) = \sum_{\mu} \sigma^{-2}_{P_{A}}(k,\mu)$
### CO:

- $P_{noise, CO} = \sigma^2_n V_{vox, COMAP}$ 
with   :
$V_{vox, COMAP} = 4.7 \times 4.7 \times 2.6 \ (cMcp/h)^3$ and $\sigma_n = 17.8 \mu K$

- $W^2_{CO}(k, \mu) = e^{- k ^2 \sigma^2_{\perp}} e^{-\mu^2 k^2 (\sigma^2_{||} - \sigma^2_{\perp})}$ 
with : ($\sigma_{||}, \sigma_{\perp}) = (2.6,4.7) \ cMpc/h$


### Lya:

- $P_{noise,Lya}= P_{los}(k_{||} = k\mu)/n_{2D}$ with $n_{2D} = 1/d^2_{\perp}$

- $W^2_{Lya}(k, \mu) = 1$


### Galaxies:

- $P_{noise, gal}(k,\mu) = 1/n_{3D, Gals}$

 - The noise above (which we are currently using) is ~ 1/2 of the this :
   
   $P_{noise, gal}(k,\mu) = <\tilde{d} \tilde{d}^*>$ with $d(\vec{r}) = \mathcal{G} \left( noiseless \ map, \sigma_{||} = c \sigma_z / H(z) \right) - noiseless \ map$
$\mathcal{G}()$ is a 1D gaussian kernel



- $W^2_{gal}(k, \mu) = e^{- \mu^2 k^2 \sigma_{||}}$, again : $\sigma_{||} = c \sigma_z / H(z)$

$\sigma_z$ is the redshift uncertainty : $R_z = \sigma_z / (1+z)$


### Cross power:


$\sigma_{P_{\times}} ^2 = \frac{\sigma_{P_{f_1}}\times \sigma_{P_{f_2}}}{2} + \frac{P_{\times}^2}{2 N_{modes}}$

