# Power Spectrum Multipoles of Survey Data (ConvolvedFFTPower)¶

The ConvolvedFFTPower class computes the power spectrum multipoles $$P_\ell(k)$$ for data from an observational survey that includes non-trivial selection effects. The input data is expected to be in the form of angular coordinates (right ascension and declination) and redshift. The measured power spectrum multipoles represent the true multipoles convolved with the window function. Here, the window function refers to the Fourier transform of the survey volume. In this section, we provide an overview of the FFT-based algorithm used to compute the multipoles and detail important things to know for the user to get up and running quickly.

Note

To jump right into the ConvolvedFFTPower algorithm, see this cookbook recipe for a detailed walk-through of the algorithm.

## Some Background¶

We begin by defining the estimator for the multipole power spectrum, often referred to as the “Yamamoto estimator” in the literature (Yamamoto et al 2006):

(1)$P_\ell = \frac{2\ell+1}{A} \int \frac{\d\Omega_k}{4\pi} \left [ \int \d\vrone \ F(\vrone) e^{i \vk \cdot \vrone} \int \d\vrtwo \ F(\vrtwo) e^{-i \vk \cdot \vrtwo} \L_\ell(\vkhat \cdot \vrhat_2) - P_\ell^{\rm noise}(\vk) \right ].$

where $$\Omega_k$$ represents the solid angle in Fourier space, and $$\L_\ell$$ is the Legendre polynomial of order $$\ell$$. The weighted density field $$F(\vr)$$ is defined as

(2)$F(\vr) = \wfkp(\vr) \left[ n_g'(\vr) - \alpha' \ n'_s(\vr)\right],$

where $$n_g'$$ and $$n_s'$$ are the number densities for the galaxy catalog and synthetic catalog of randoms object, respectively, and $$\alpha'$$ is the ratio of the number of real galaxies to random galaxies. Often, the catalog of random objects has a much higher number density than the galaxy catalog, and the factor of $$\alpha'$$ re-normalizes to the proper number density. The normalization $$A$$ is given by

(3)$A = \int \d\vr [n'_g(\vr) \wfkp(\vr)]^2.$

The shot noise $$P_\ell^{\rm noise}$$ in equation (1) is

(4)$P_\ell^{\rm noise}(\vk) = (1 + \alpha') \int \d\vr \ n'_g(\vr) \wfkp^2(\vr) \L_\ell (\vkhat \cdot \vrhat).$

The FKP weights, first derived in Feldman, Kaiser, and Peacock 1994, minimize the variance of the estimator at a desired power spectrum value. Denoted as $$w_\mathrm{FKP}$$, these weights are given by

(5)$w_\mathrm{FKP}(\vr) = \frac{1}{1 + n_g'(\vr) P_0},$

where $$P_0$$ is the power spectrum amplitude in units of $$h^{-3} \mathrm{Mpc}^3$$ where the estimator is optimized. For typically galaxy survey analyses, a value of order $$P_0 = 10^{4} \ h^{-3} \mathrm{Mpc}^3$$ is usually assumed.

In our notation, quantities marked with a prime ($$'$$) include completeness weights. These weights, denoted as $$w_c$$, help account for systematic variations in the number density fields. Typically the weights for the random catalog are unity, but for full generality we allow for the possibility of non-unity weights for $$n_s$$ as well. For example, $$\alpha' = N'_\mathrm{gal} / N'_\mathrm{ran}$$, where $$N'_\mathrm{gal} = \sum_i^{N_\mathrm{gal}} w_c^i$$.

## The Algorithm¶

We use the FFT-based algorithm of Hand et al 2017 to compute the power spectrum multipoles. This work improves upon the previous FFT-based estimators of $$P_\ell(k)$$ presented in Bianchi et al 2015 and Scoccimarro 2015 by only requiring the use of $$2\ell+1$$ FFTs to compute a multipole of order $$\ell$$. The algorithms uses the spherical harmonic addition theorem to express equation (1) as

(6)$P_\ell(k) = \frac{2\ell+1}{A} \int \frac{\d\Omega_k}{4\pi} F_0(\vk) F_\ell(-\vk),$

where

(7)\begin{align}\begin{aligned}F_\ell(\vk) &= \int \d\vr \ F(\vr) e^{i \vk \cdot \vr} \L_\ell(\vkhat \cdot \vrhat),\\ &= \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{\ell} \Ylm(\vkhat) \int \d\vr \ F(\vr) \Ylm^*(\vrhat) e^{i \vk \cdot \vr}.\end{aligned}\end{align}

The sum over $$m$$ in equation (7) contains $$2 \ell + 1$$ terms, each of which can be computed using a FFT. Thus, the multipole moments can be expressed as a sum of Fourier transforms of the weighted density field, with weights given by the appropriate spherical harmonic. We evaluate equation (7) using the real-to-complex FFT functionality of the pmesh package and use the real form of the spherical harmonics $$\Ylm$$.

We use the symbolic manipulation functionality available in the SymPy Python package to compute the spherical harmonic expressions in terms of Cartesian vectors. This allows the user to specify the desired multipoles at runtime, enabling the code to be used to compute multipoles of arbitrary $$\ell$$.

## Getting Started¶

Here, we outline the necessary steps for users to get started using the ConvolvedFFTPower algorithm to compute the power spectrum multipoles from an input data catalog.

### The FKPCatalog Class¶

The ConvolvedFFTPower algorithm requires a galaxy catalog and a synthetic catalog of random objects without any clustering signal, which we refer to as the “data” and “randoms” catalogs, respectively. The CatalogSource object responsible for handling these two types of catalogs is the FKPCatalog class.

The FKPCatalog determines the size of the Cartesian box that the “data” and “randoms” are placed in, which is then also used during the FFT operation. By default, the box size is determined automatically from the maximum extent of the “randoms” positions. In this automatic case, the size of the box can be artificially extended and padded with zeros via the BoxPad keyword. Users can also specify a desired box size by passing in the BoxSize keyword.

The FKPCatalog object can be converted to a mesh object, FKPCatalogMesh, via the to_mesh() function. This mesh object knows how to paint the FKP weighted density field, given by equation (2), to the mesh using the “data” and “randoms” catalogs. With the FKP density field painted to the mesh, the ConvolvedFFTPower algorithm uses equations (6) and (7) to compute the multipoles specified by the user.

### From Sky to Cartesian Coordinates¶

The Position column, holding the Cartesian coordinates, is required for both the “data” and “randoms” catalogs. We provide the function nbodykit.transform.SkyToCartesian() for converting sky coordinates, in the form of right ascension, declination, and redshift, to Cartesian coordinates. The conversion from redshift to comoving distance requires a cosmology instance, which can be specified via the Cosmology class.

For more details, see Converting from Sky to Cartesian Coordinates.

### Specifying $$n'_g(z)$$¶

The number density of the “data” catalog as a function of redshift, in units of $$h^{3} \mathrm{Mpc}^{-3}$$, is required to properly normalize the power spectrum using equation (3) and to compute the shot noise via equation (4). The “data” and “randoms” catalog should contain a column that gives this quantity, evaluated at the redshift of each object in the catalogs.

When converting from a FKPCatalog to a FKPCatalogMesh, the name of the $$n'_g(z)$$ column should be passed as the nbar keyword to the to_mesh() function. The $$n'_g(z)$$ column should have the same name in the “data” and “randoms” catalogs.

By default, the name of the nbar column is set to NZ. If this column is missing from the “data” or “randoms” catalog, an exception will be raised.

Note that the RedshiftHistogram algorithm can compute a weighted $$n(z)$$ for an input catalog and may be useful if the user needs to compute $$n_g(z)$$.

Important

By construction, the objects in the “randoms” catalog should follow the same redshift distribution as the “data” catalog. Often, the “randoms” will have an overall number density that is 10, 50, or 100 times the number density of the “data” catalog. Even if the “randoms” catalog has a higher number density, the nbar column in both the “data” and “randoms” catalogs should hold number density values on the same scale, corresponding to the value of $$n_g(z)$$ at the redshift of the objects in the catalogs.

### Using Completeness Weights¶

The ConvolvedFFTPower algorithm supports the use of completeness weights to account for systematic variations in the number density of the “data” and “randoms” objects. In our notation in the earlier background section, quantities marked with a prime ($$'$$) include completeness weights. The weighted and unweighted number densities of the “data” and “randoms” fields are then related by:

\begin{align}\begin{aligned}n'_g(\vr) &= w_{c,g}(\vr) n_g(\vr),\\n'_s(\vr) &= w_{c,s}(\vr) n_s(\vr),\end{aligned}\end{align}

Typically the weights for the random catalog are unity, but for full generality, we allow for the possibility of non-unity weights for $$n_s$$ as well. For example, $$\alpha' = N'_\mathrm{gal} / N'_\mathrm{ran}$$, where $$N'_\mathrm{gal} = \sum_i^{N_\mathrm{gal}} w_{c,g}^i$$ and $$N'_\mathrm{ran} = \sum_i^{N_\mathrm{ran}} w_{c,s}^i$$.

When converting from a FKPCatalog to a FKPCatalogMesh, the name of the completeness weight column should be passed as the comp_weight keyword to the to_mesh() function.

By default, the name of the comp_weight column is set to the default column Weight, which has a value of unity for all objects. If specifying a different name, the column should have the same name in both the “data” and “randoms” catalogs.

### Using FKP Weights¶

Users can also specify the name of a column in the “data” and “randoms” catalogs that represents a FKP weight for each object, as given by equation (5). The FKP weights do not weight the individual number density fields as the completeness weights do, but rather they weight the combined field, $$n'_g(\vr) - \alpha' n'_s(\vr)$$ (see equation (2)).

When converting from a FKPCatalog to a FKPCatalogMesh, the name of the FKP weight column should be passed as the fkp_weight keyword to the to_mesh() function.

By default, the name of the fkp_weight column is set to FKPWeight, which has a value of unity for all objects. If specifying a different name, the column should have the same name in both the “data” and “randoms” catalogs.

The ConvolvedFFTPower algorithm can also automatically generate and use FKP weights from the input nbar column if the user specifies the use_fkp_weights keyword of the algorithm to be True. In this case, the user must also specify the P0_FKP keyword, which gives the desired $$P_0$$ value to use in equation (5).

## The Results¶

### The Multipoles¶

The multipole results are stored as the poles attribute of the initialized ConvolvedFFTPower object. This attribute is a BinnedStatistic object, which behaves like a structured numpy array and stores the measured results on a coordinate grid defined by the wavenumber bins specified by the user. See Analyzing your Results for a full tutorial on using the BinnedStatistic class.

The poles attribute stores the following variables:

• k :

the mean value for each k bin

• power_L :

complex array storing the real and imaginary components for the $$\ell=L$$ multipole

• modes :

the number of Fourier modes averaged together in each bin

Note that measured power results for bins where modes is zero (no data points to average over) are set to NaN.

Note

The shot noise is not subtracted from any measured results. Users can access the shot noise value for monopole, computed according to equation (4) in the meta-data attrs dictionary. Shot noise for multipoles with $$\ell>0$$ is assumed to be zero.

### The Meta-data¶

Several important meta-data calculations are also performed during the algorithm’s execution. This meta-data is stored in both the ConvolvedFFTPower.attrs attribute and the ConvolvedFFTPower.poles.attrs atrribute.

1. data.N, randoms.N :

the unweighted number of data and randoms objects

2. data.W, randoms.W :

the weighted number of data and randoms objects, using the column specified as the completeness weights. This is given by:

\begin{align}\begin{aligned}W_\mathrm{data} &= \sum_\mathrm{data} w_c\\W_\mathrm{ran} &= \sum_\mathrm{randoms} w_c\end{aligned}\end{align}
3. alpha :

the ratio of data.W to randoms.W

4. data.norm, randoms.norm :

the normalization $$A$$ of the power spectrum, computed from either the “data” or “randoms” catalog (they should be similar). They are given by:

(8)\begin{align}\begin{aligned}A_\mathrm{data} &= \sum_\mathrm{data} n'_g w_c \wfkp^2\\A_\mathrm{ran} &= \alpha \sum_\mathrm{randoms} n'_g w_c \wfkp^2\end{aligned}\end{align}
5. data.shotnoise, randoms.shotnoise :

the contributions to the monopole shot noise from the “data” and “random” catalogs. These values are given by:

\begin{align}\begin{aligned}P^\mathrm{shot}_\mathrm{data} &= A_\mathrm{ran}^{-1} \sum_\mathrm{data} (w_c \wfkp)^2\\P^\mathrm{shot}_\mathrm{ran} &= \alpha^2 A_\mathrm{ran}^{-1} \sum_\mathrm{randoms} (w_c \wfkp)^2\end{aligned}\end{align}
6. shotnoise :

the total shot noise for the monopole power spectrum, which should be subtracted from the monopole measurement in ConvolvedFFTPower.poles. This is computed as:

$P^\mathrm{shot} = P^\mathrm{shot}_\mathrm{data} + P^\mathrm{shot}_\mathrm{ran}$
7. BoxSize :

the size of the Cartesian box used to grid the “data” and “randoms” objects on the Cartesian mesh.

### From $$P_\ell(k)$$ to $$P(k,\mu)$$¶

The ConvolvedFFTPower.to_pkmu() allows users to rotate the measured multipoles, stored as the ConvolvedFFTPower.poles attribute, into $$P(k,\mu)$$ wedges (bins in $$\mu$$). The function returns a BinnedStatistic holding the binned $$P(k,\mu)$$ data.

Results can easily be saved and loaded from disk in a reproducible manner using the ConvolvedFFTPower.save() and ConvolvedFFTPower.load() functions. The save function stores the state of the algorithm, including the meta-data in the ConvolvedFFTPower.attrs dictionary, in a JSON plaintext format.
• The Position column is missing from the input “data” and “randoms” catalogs. The input position columns for this algorithm are assumed to be in terms of the right ascension, declination, and redshifts, and users must add the Position column holding the Cartesian coordinates explicitly to both the “data” and “randoms” catalogs.
• Normalization issues may occur if the number density columns in the “data” and “randoms” catalogs are on different scales. Similar issues may arise if the FKP weight column uses number density values on different scales. In all cases, the number density to be used should be that of the data, denoted as $$n'_g(z)$$. The algorithm will compute the power spectrum normalization from both the “data” and “randoms” catalogs (as given by equation (8)). If the values do not agree, there is likely an issue with varying number density scales, and the algorithm will raise an exception.