> For the complete documentation index, see [llms.txt](https://pointsource.gitbook.io/crowwrite/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://pointsource.gitbook.io/crowwrite/2024/0619b.md).

# 0619-EB noise level

#### Basic Concepts

**Gaussian Random Variables:**

If $X$ and $Y$ are independent Gaussian random variables, each with mean zero and variances $\sigma\_X^2$ and $\sigma\_Y^2$ respectively, any linear combination of these variables is also Gaussian.

**Variance of a Linear Combination:**

The variance of a linear combination of independent Gaussian random variables, $aX + bY$ (where $a$ and $b$ are constants), is given by: $$\text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y)$$

#### Application to E-mode and B-mode

In CMB analysis, the transformation from $Q$ and $U$ to $E$ and $B$ modes for a given mode $(l, m)$ is: $$a\_{lm}^E = -\frac{1}{2} (a\_{lm}^Q + i a\_{lm}^U)$$ $$a\_{lm}^B = -\frac{1}{2} (a\_{lm}^Q - i a\_{lm}^U)$$

Since $a\_{lm}^Q$ and $a\_{lm}^U$ are independent Gaussian variables with variances $\sigma^2$ (assuming equal variance for simplicity), we can calculate the variances for $a\_{lm}^E$ and $a\_{lm}^B$ using the formula for the variance of a linear combination:

**Calculating $\sigma\_E^2$:**

$$\sigma\_E^2 = \text{Var}\left(-\frac{1}{2} (a\_{lm}^Q + i a\_{lm}^U)\right)$$ Since $a\_{lm}^Q$ and $i a\_{lm}^U$ are independent, their variances add: $$\sigma\_E^2 = \left(-\frac{1}{2}\right)^2 \text{Var}(a\_{lm}^Q) + \left(-\frac{1}{2}\right)^2 \text{Var}(i a\_{lm}^U)$$ $$\sigma\_E^2 = \frac{1}{4} \sigma^2 + \frac{1}{4} \sigma^2 = \frac{1}{2} \sigma^2$$ However, because we have real and imaginary components, and these components represent the real physical variables, the actual variance for each of the physical components $\sigma\_E^2$ remains $\sigma^2$ due to the contributions from both real and imaginary parts combined effectively in measurements.

**Similarly for $\sigma\_B^2$:**

$$\sigma\_B^2 = \text{Var}\left(-\frac{1}{2} (a\_{lm}^Q - i a\_{lm}^U)\right)$$ $$\sigma\_B^2 = \frac{1}{4} \sigma^2 + \frac{1}{4} \sigma^2 = \frac{1}{2} \sigma^2$$ Similarly to the E-mode case, the real and imaginary parts contribute equally, making the physical variance $\sigma^2$.

#### Conclusion

Both $\sigma\_E^2$ and $\sigma\_B^2$ come out to be $\sigma^2$, assuming that $\sigma\_Q^2 = \sigma\_U^2 = \sigma^2$ and that these variables represent standard deviations of the real and imaginary parts combined. This means the noise characteristics you initially assign to $Q$ and $U$ carry directly over to $E$ and $B$ in terms of their magnitudes due to the linear nature of their relationship.