Introduction of a factor Δℓ when summing equal distants 𝐶ℓ

[fab13]

TL;DR Summary

I would like to demonstrate the formula that defines the standard deviation of a 𝐶ℓ and mostly justify the presence of the factor Δℓ when we have a finite number of 𝐶ℓ's angular power spectrum and associated finite number of multipole ℓ (generated by a code ). I tried at the end of my post to calculate the mean Cℓ for a single bin but it is missing a factor 2. Any help would be really appreciated. This is a problem about the theorical/numerical frontier given the fact I have not Δℓ=1.

Hello,

In the context of Legendre expansion with $C_\ell$ quantities, below the following formula which is the error on a $C_\ell$ :

$\sigma_(C_{\ell})=\sqrt{\frac{2}{(2 \ell+1)\Delta\ell}}\,C_{\ell}\quad(1)$

where $\Delta\ell$ is the width of the multipoles bins used when computing the angular power spectra.

I would like to understand why this factor appears. Normally, we can infer the expression of $\sigma_(C_{\ell})$ and get :

$\sigma_(C_{\ell})=\sqrt{\frac{2}{(2 \ell+1)}}\,C_{\ell}$

That is to say, without having the $\Delta\ell$ factor under the root.

In my case, I have only a finite number of multipoles $\ell$ (60 exactly) in a range between 10 and 5000 and 60 corresponding $C_\ell$. From my tutor, to compute this standard-deviation, I have to take :

$\Delta\ell=(4990/60)$.

But I would like to understand why I have to use this factor when we have a finite number of equidistant multipoles with associated $C_\ell$ values.

For example, my tutor told me that, ideally, we would have a $C_\ell$ for each $\ell=1..N$, i.e with $\ell=1,2,3,4,...N$ with the formula :

$\sigma(C_{\ell})^2=\frac{2}{(2 \ell+1)}\,C_{\ell}^{2}$

But he mentionned that, like we have only a finite number of equidistant multipoles $\ell$, we are obliged to use this factor $\Delta\ell$ in our case (equation$(1)$).

My tutor told me it is like an average on the expression with multipole $\ell$, i.e on the factor $\sqrt{\dfrac{2}{(2\ell+1)}}$, but I didn't fully understand this meaning.

Could anyone help me to justify the using of this factor $\Delta\ell$ ?

UPDATE : Someone tried quickly to explain why we introduce this factor $\Delta\ell$ inside the square root, but I am still confused about this demonstration.

Here is his reasoning :

For the interval $b$: $b=[\ell, \ell+\Delta\ell]$, one has :

$C_{\ell,b}=\dfrac{1}{\Delta\ell}\,\sum_{\ell'=\ell}^{\ell'+\Delta\ell}\,C_{\ell'}$

If one takes the variance, then :

$\text{Var}(C_{\ell,b})=\dfrac{1}{\Delta\ell^2}\,\sum_{\ell'}\text{Var}(C_{\ell'})$

$\text{Var}(C_{\ell,b})=\dfrac{1}{\Delta\ell^2}\,\sum_{\ell'}\dfrac{2}{(2\ell'+1)}\,(C_{\ell'})\quad(2)$

Important step here, one considers the equality (which is actually an approximation) :

$\sum_{\ell'}\dfrac{2}{(2\ell'+1)}\,(C_{\ell'})=\dfrac{2}{(2\ell_{mean}+1)}\,(C_{\ell'_{mean}}^2)\,\Delta\ell\quad(3)$

This way, we have from equation $(2)$ :

$\text{Var}(C_{\ell,b})\simeq \dfrac{1}{\Delta\ell^2}\,\dfrac{2}{(2\ell_{mean}+1)}\,(C_{\ell'_{mean}}^2)\,\Delta\ell$

Finally the expression of the variance for the interval $b$:

$\text{Var}(C_{\ell,b})\simeq \dfrac{1}{\Delta\ell}\,\dfrac{2}{(2\ell_{mean}+1)}\,(C_{\ell'_{mean}}^2)$

I still didn't understand the step between $(2)$ and $(3)$.

For me, an average is under the form :

$C_{\ell'_{mean}}=\dfrac{1}{2}\,(C_{\ell'}+C_{\ell'+\Delta\ell})$ but I can't see this factor $\dfrac{1}{2}$ in this reasoning.

If someone could explain this critical step between $(2)$ and $(3)$.

 

[Orodruin]

The average $\bar x$ of $N$ different quantities $x_i$ satisfies $\sum x_i = \bar x N$. This is what is being used.

 

[fab13]

@Orodruin : thanks for your quick answer.

Could you be please more explicit with the notations I took ? your formula is trivial but I have difficulties to apply it with my notations.

Moreover, do you give a reasoning on a single bin of width $\Delta\ell$ or on the total interval $[\ell_{min},\ell_{max}]$ ?

In the formula $\sum x_i = \bar x N$, does $N$ correspond to $\Delta\ell$ in my formula ?

Thanks in advance, Regards

 

[fab13]

Isn't really there nobody who could explain me my problem of understanding about my last message above ?

 

[hutchphd]

Summary:: I would like to demonstrate the formula that defines the standard deviation of a 𝐶ℓ and mostly justify the presence of the factor Δℓ when we have a finite number of 𝐶ℓ's angular power spectrum and associated finite number of multipole ℓ (generated by a code ). I tried at the end of my post to calculate the mean Cℓ for a single bin but it is missing a factor 2. Any help would be really appreciated. This is a problem about the theorical/numerical frontier given the fact I have not Δℓ=1.

For me, an average is under the form :

Cℓmean′=12(Cℓ′+Cℓ′+Δℓ) but I can't see this factor 12 in this reasoning.

If someone could explain this critical step between (2) and (3).

Can you not work it "backwards" to see what N must be in this situation? You have the answer provided to you, It is worrisome that this step eludes you.

 

[fab13]

Hi everyone,

It seems that I may have a "demonstration" about my issue of understanding :

If i take the expression for the bin "$b$" $[\ell_b,\ell_b+\Delta\ell_b]$ :

$C_{\ell,b}=\dfrac{1}{\Delta\ell_b}\,\sum_{\ell=\ell_b}^{\ell_b+\Delta\ell_{b}}\,(C_{\ell})$

Then :

$\text{Var}(C_{\ell,b})=\dfrac{1}{\Delta\ell_b^2}\,\sum_{\ell=\ell_b}^{\ell_b+\Delta\ell_{b}}\,\text{Var}(C_{\ell})\quad(2)$

By taking $\Delta\ell=1$, we can have :

$\text{Var}(C_{\ell_b})=\dfrac{1}{\Delta\ell_{b}^2}\,\sum_{\ell=\ell_b}^{\ell+\Delta\ell}\dfrac{2}{(2\ell+1)}\,(C_{\ell}^2)\Delta\ell \simeq \int_{\ell=\ell_{b}}^{\ell_{b}+\Delta\ell_b} \dfrac{1}{\Delta\ell_{b}^2} \dfrac{2}{2 \ell+1}\,C_\ell^2\text{d}\ell \quad(3)$

and this integral can be approximated by rectangular integration method :

$\int_{\ell=\ell_{b}}^{\ell_{b}+\Delta\ell_b} \dfrac{1}{\Delta\ell_{b}^2} \dfrac{2}{2 \ell+1}\,C_\ell^2\text{d}\ell \simeq \dfrac{1}{\Delta\ell_{b}^{2}} \dfrac{2}{(2\ell_{b}+1)} \,C_{\ell_{b}}^{2} \Delta{\ell_{b}} =\quad(4)$

where $\Delta{\ell_{b}}$ is the delta width of bin "$b$" :

then, I could write :

$\text{Var}(C_{\ell,b})=\dfrac{1}{\Delta\ell_{b}} \dfrac{2}{(2 \ell_{b}+1)}\,C_{\ell_b}^{2} \quad(5)$

Do you think my reasoning is right ?