Issue 
A&A
Volume 615, July 2018



Article Number  A1  
Number of page(s)  35  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201732343  
Published online  04 July 2018 
Covariance of the galaxy angular power spectrum with the halo model
Département de Physique Théorique and Center for Astroparticle Physics, Université de Genève,
24 quai Ernest Ansermet,
1211
Geneva, Switzerland
email: fabien.lacasa@unige.ch
Received:
22
November
2017
Accepted:
3
January
2018
As the determination of density fluctuations becomes more precise with larger surveys, it becomes more important to account for the increased covariance due to the nonlinearity of the field. Here I have focussed on the galaxy density, with analytical prediction of the nonGaussianity using the halo model coupled with standard perturbation theory in real space. I carried out an exact and exhaustive derivation of all treelevel terms of the nonGaussian covariance of the galaxy C_{ℓ}, with the computation developed up to the third order in perturbation theory and local halo bias, including the nonlocal tidal tensor effect. A diagrammatic method was used to derive the involved galaxy 3D trispectra, including shotnoise contributions. The projection to the angular covariance was derived in all trispectra cases with and without Limber’s approximation, with the formulae being of potential interest for other observables than galaxies. The effect of subtracting shotnoise from the measured spectrum is also discussed, and does simplify the covariance, though some nonGaussian shotnoise terms still remain. I make the link between this complete derivation and partial terms which have been used previously in the literature, including supersample covariance (SSC). I uncover a wealth of additional terms which were not previously considered, including a whole new class which I dub braiding terms as it contains multipolemixing kernels. The importance of all these new terms is discussed with analytical arguments. I find that they become comparable to, if not bigger than, SSC if the survey is large or deep enough to probe scales comparable with the matterradiation equality k_{eq}. A short selfcontained summary of the equations is provided in Sect. 9 for the busy reader, ready to be implemented numerically for analysis of current and future galaxy surveys.
Key words: methods: analytical / galaxies: statistics / largescale structure of Universe
© ESO 2018
1 Introduction
Tracers of the large scale matter distribution in the Universe are one of the main probes of cosmology, with current galaxy surveys such as KiDS (Hildebrandt et al. 2017) and Dark Energy Survey (DES) already providing constraints competitive with Planck on some cosmological parameters (Abbott et al. 2017), and future surveys such as Euclid (Laureijs et al. 2011) and the Large Synoptic Sky Telescope (LSST; Abell et al. 2009) as scheduled to greatly improve our understanding of dark energy and the structuration of the Universe.
Late time tracers of the large scale structure (LSS) have, however, undergone significant nonlinear evolution, which makes their probability distribution function deviate from the Gaussianity of the initial density field. As such, twopoint statistics no longer retain all the information, and have their covariance increased by the presence of a nonvanishing trispectrum. The focus of the present article is on this increased covariance, in the case of galaxy clustering analysed with the angular power spectrum C_{ℓ}.
Covariances are central to the statistical inference process, being the only element besides signal prediction when using the popular Gaussian likelihood, though see Sellentin & Heavens (2018) for indications that current data may need a nonGaussian likelihood. In the past, covariances have been estimated through a variety of techniques. Jackknife or bootstrap methods allow estimates from the data itself, however it is known that the results from these methods are very noisy, and Lacasa & Kunz (2017) showed that they give biased estimates of the effect of supersample covariance (SSC), which I will introduce later. Numerical covariance estimation through simulations of the LSS remains costly, especially if one wants cosmologydependent covariance matrices, although newer techniques allowing fast mock creations (Klypin & Prada 2018) or data compression (Heavens et al. 2017) may help cut down that cost. Furthermore the induced numerical noise in the covariance must be propagated in the likelihood (Sellentin & Heavens 2016, 2017), enlarging our uncertainties. Finally Lacasa & Kunz (2017) showed that simulations also give biased estimates of SSC, unless the simulation is orders of magnitude larger than the survey, a difficult task for future surveys covering a large part of the sky up to high redshifts (Schneider et al. 2016).
Analytical modelling of the covariances is an approach which has the logical advantage of yielding a selfconsistent data analysis with complete analytical understanding of the physics. Conversely, not being able to predict the covariance analytically would question the confidence with which analytical prediction of the signal itself should be trusted.
Analytical covariances have been developed recently for LSS tracers including some nonGaussian effects (e.g. Lacasa & Rosenfeld 2016; Krause & Eifler 2017). They can provide both fast and noiseless covariance matrices, which could possibly be varied with model parameters. The current approaches are based on the halo model (Cooray & Sheth 2002) coupled with perturbation theory, and are state of the art applied to recent galaxy surveys (Hildebrandt et al. 2017; van Uitert et al. 2017; Krause et al. 2017; Abbott et al. 2017).
The Gaussian contribution to the covariance is normally simple enough to treat analytically: if C_{ℓ} is the total power spectrum (including shotnoise effects) of the signal considered, the fullsky Gaussian covariance is (1)
Supersample covariance (SSC; Hu & Kravtsov 2003; Takada & Hu 2013) is currently thought to be the dominant nonGaussian contributionto the covariance. The effect comes from the nonlinear modulation of local observables by long wavelength density fluctuations.In other words, the survey can be nonrepresentative of the universe by probing a region denser (or less dense) than average, as all observables react to such background density change. SSC has already had an impact on cosmological constraints from current surveys, with Hildebrandt et al. (2017) finding that failure to include it would lead to 1σ shift in their constraint on S_{8}.
Numerical investigation by the author in the case of photometric galaxy clustering have shown that indeed SSC is the dominant effect beyond the Gaussian covariance for specifications of ongoing survey such as DES (Lacasa 2017, slide 14). However when implementing specifications of future surveys, I found that other nonGaussian terms (1halo and 2halo1+3, which will be introduced in the article’s main text) have an impact on the covariance which is as important as SSC, sometimes even more important depending on redshift and scale (Lacasa 2017, slide 8). As these terms become significant, the question arises of the importance of all the other nonGaussian terms. It thus becomes timely to carry out an exhaustive derivation of all possible nonGaussian covariance terms, within the current modelling framework that is the halo model.
Here, I undertake the task of carrying out this exhaustive derivation in the case of the angular power spectrum of galaxies . The choice of the harmonic basis is the one underlying current covariance implementations, even when data are in the other popular basis: real space (e.g. Joachimi et al. 2008; Krause et al. 2017), indeed results can be converted straightforwardly through (e.g. Crocce et al. 2011) (2) (3)
The choice of signal is relevant, though most of the theoretical framework developed should adapt straightforwardly to another observable. I focus here on galaxy clustering using the halo model together with standard perturbation theory at treelevel. This is the most complex signal in the sense that galaxy discreteness (shotnoise) yields more covariance terms. I have left the application to other observables (clusters, weaklensing, secondary anisotropies of the CMB) to future works.
The methods used in the article are presented in Sect. 2. This includes in particular a diagrammatic approach to galaxy polyspectra with the halo model, and projection from 3D quantities to 2D observables or covariances. This should be of interest to theoreticians of largescale structure tracers. Sections 3–7 contain the main calculation of the article: all nonGaussian covariance terms are derived one by one; firstly in the most general casethen simplified with relevant approximations. An explanation of the origin of these terms and their ordering will be given in Sect. 2.5. A regrouping of terms, comparison with derivation of previous literature, and analytical discussion of the potential importance of these nonGaussian terms will be performed in Sect. 8. This should be of interest to give more physical interpretation of the derivation, and intuition on when and why it should be considered of importance. Finally, a selfconsistent summary of the results for the busy reader is given in Sect. 9, using simplifications of relevance to current data analysis. This is the section of reference for numerical implementations of the formulae, for inclusion in analysis of present and future surveys.
2 Methods
2.1 Conventions
2.1.1 Cosmological notations
I use the following notations for cosmological quantities: r(z) is the comoving distance to redshift z, G(z) is the growth factor and is the comoving volume element per unit solid angle. Unit vectors have an upper hat, for example, is a direction on the celestial sphere. In the Limber approximation, the peak of the spherical Bessel function j_{ℓ} (kr) is k_{ℓ} = (ℓ + 1∕2)∕r(z) and depends on an implicit redshift.
I also make use of quantities in the halo model (Cooray & Sheth 2002), such as the halo mass function , halo spherical profile u(kM, z) and halo bias b_{β}(M, z) where β = 1, 2, 3 for the local bias terms used here, and β = s2 for the quadratic tidal tensor bias (Chan et al. 2012; Baldauf et al. 2012).
2.1.2 Shortenings
I have used the following shortenings in order to keep long equations as readable as can be possible. Spherical harmonics indices are shortened through i ≡ (ℓ_{i}, m_{i}), including inthe case of indices in sums. The sum of Fourier wavevectors is shortened through k_{i+j} ≡k_{i} + k_{j}, implying in particular . When unambiguous, arguments of multivariate functions are shortened through f(z_{1234}) ≡ f(z_{1}, z_{2}, z_{3}, z_{4}). For example for polyspectra, (4)
Outside of function arguments, repetition of indices is used to note multiplication: X_{ij} ≡ X_{i} X_{j}, for example, in integration elements dM_{αβ} ≡dM_{α} dM_{β}.
The number of galaxy ntuples (pairs, triplets…) is shortened to (5)
2.1.3 Definitions
Inspired by the notations of Takada & Hu (2013), I defined the following integrals for galaxies: (6)
where μ is an integer (the galaxy tuplepower) and β is the bias type. I note that when k_{1}, …, k_{μ} → 0, as . becomes scaledependent only on small scales, of order of the halo sizes.
I also introduced the following integrals, useful later for angular quantities projected in a redshift bin (7)
and its generalisation to multiple Bessel functions (8)
In the following, when unambiguous I will leave redshift integration bounds implicit for simplicity of notation.
2.2 Diagrammatic
I modelled the galaxy density field using the halo model (Cooray & Sheth 2002) coupled with treelevel perturbation theory, allowing firstprinciple description of all galaxy statistics. In this context, the galaxy number density is written as (Lacasa et al. 2014): (9)
where the first sum runs over halos and the second over galaxies inside that halo.
In Fourier space, the (absolute) galaxy polyspectrum of order n is defined by^{1} (10)
Lacasa et al. (2014) introduced a diagrammatic method to compute the different terms of the galaxy polyspectrum with the halo model. This method was illustrated in more detail, including a trispectrum example, in Sect. 3.4.4 of Lacasa (2014).
For the polyspectrum of order n, the first step is to draw in diagrams all the possibilities of putting n galaxies in halo(s). Potentially two or more galaxies can lie at the same point (“contracted”) for the shotnoise terms. Then for each diagram, the galaxies should be labelled from 1 to n, as well as the halos for example, with α_{1} to α_{p}.
Each diagram produces a polyspectrum term which is an integral over the halo masses of several factors:

for each halo α_{j} there is a corresponding:

halo mass function

average of the number of galaxy tuples in that halo. For example, for a single galaxy in that halo, for a pair, etc.;

as many halo profile as different points, where . For example, k = k_{i} for a noncontracted galaxy, while for a galaxy contracted twice;


the halo polyspectrum of order p, conditioned to the masses of the corresponding haloes:
where the sum runs over the indexes i of the galaxies inside the halo α_{j}.
Finally one should account for all the possible permutations of the galaxy labels 1 to n in the diagram.
Additionally, if one is interested in the polyspectrum of the relative density fluctuations , instead of the absolute fluctuations, one should add a prefactor to the preceding expression. This can prove useful for 3D observables; however for 2D projected observables, as the angular power spectrum studied in this article, it is the absolute fluctuations which naturally enter the equations.
2.3 Projection to 2D observables
The projected galaxy density in the direction in a redshift bin i_{z}, is the lineofsight integral: (11)
with dV = r^{2}dr being the comoving volume per unit solid angle. This projection neglects redshiftspace distortion and other general relativistic effects (for example, Bonvin & Durrer 2011), whose impacts are left for future studies.
In full sky, after spherical harmonic decomposition the harmonic coefficients are (for example, Lacasa & Rosenfeld 2016)
The galaxy power spectrum estimator is then
with the coming from the change k_{2} →−k_{2} and the parity of the Legendre polynomial P_{ℓ}.
The nonGaussian part of the galaxy spectrum covariance is thus: (19)
leaving redshift bins implicit hereafter.
As a power spectrum estimator, most of the contribution to Ĉ_{ℓ} Eq. (17) will come from k_{1} ≈ −k_{2}, that is, k_{1+2} ≪ k_{1} ≈ k_{2}. Thus, similarly to the case of the 3D power spectrum estimator (Takada & Hu 2013), the covariance Eq. (19) probes the trispectrum in the squeezed diagonal configuration represented in Fig. 1: k_{1+2} = k_{3+4} ≪ k_{1} ≈ k_{2}, k_{3} ≈ k_{4}. Contrary to Takada & Hu (2013) however, the present derivation does not rely on any approximation or Taylor expansion in terms of k_{1+2}.
The 3D trispectrum generally depends on six degrees of freedom (d.o.f.) that fix the shape of the quadrilateron k_{1} + k_{2} + k_{3} + k_{4} = 0. There is however no unique natural choice for these six d.o.f.^{2}: the choice will depend on the trispectrum term considered. For all the terms considered here (see Appendix A), four d.o.f. will be the quadrilateron sides k_{1}, k_{2}, k_{3}, k_{4}, then the trispectrum may also depend on the length of one of the diagonals k_{1+2}, k_{1+3}, k_{1+4} and/or on angles either between base wavevectors or between a diagonal and a base wavevector . Deriving the projection Eq. (19) analytically in all the necessary trispectrum cases proves a complex task, and is the subject of Appendices B and C. I list below the three less complex cases where the trispectrum does not depend on any angle, solely on lengths of base wavevectors or diagonals.
Firstly, the easiest case is of a diagonalindependent trispectrum, that is, T_{gal} (k_{1234}, z_{1234}) = T_{gal}(k_{1234}, z_{1234}). This case was treated by Lacasa et al. (2014); Lacasa (2014) for a general diagonalindependent polyspectrum. In the present case, one finds (see Appendix B.1 for a derivation): (20)
Using Limber’s approximation (see Appendix B.1), this simplifies to (21)
This case will be relevant for a large part of the covariance terms later on.
The second case of interest is of a trispectrum depending on the length of the squeezed diagonal, K = k_{1+2}, additionally to the length of the four sides k_{1234}. This case is treated in Appendix B.2, giving: (22)
Using Limber’s approximation on k_{1234} (but not on K, since this would be a poor approximation for j_{0} which has a large support and peaks at K = 0), this simplifies to (23)
This case will be relevant for SSC terms later on (Sect. 8.1).
The third case of interest is of a trispectrum depending on the length of one of the other diagonal, K = k_{1+3} (with K = k_{1+4} giving a symmetric result), additionally to the length of the four sides k_{1234}. This case is treated in Appendix B.3, giving: (24)
Using Limber’s approximation on k_{1234} but not on the diagonal, the covariance simplifies to (25)
This case will be relevant for braiding terms later on (Sect. 8.2).
There are furthermore eight cases of trispectra depending on angles between wavevectors: four cases where the trispectrum depends on one angle, tackled in Appendix C.1, and four cases where the trispectrum depends on two angles, tackled in Appendix C.2. Due to the complexity of these expressions, they are left to their respective appendices for the clarity of the main text. These formulae involve geometric coefficients which are shown in Appendix E to be related to Wigner 3nJ symbols: 6J, 9J and even the case of a reduced 12J symbol of the second kind. Reduction checks are performed in Appendix D to assure the robustness of the results.
These 2D projection formulae, although not the main aim of this article, can be viewed as standalone results that should be of interest for other analyses, for example, interested in the covariance of other signals or using a different modelling framework such as a different flavour of perturbation theory.
Fig. 1
3D trispectrum in the squeezed diagonal limit. 

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Fig. 2
Diagrams for the galaxy power spectrum. From left to right: twohalo (2h), onehalo (1h) and shotnoise (shot). 

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2.4 Example of the power spectrum
In this article, I am interested in the covariance of the galaxy power spectrum. Before coming to the covariance, the diagrammatic formalism and the 2D projection explained above can already be illustrated at the power spectrum level. This will already uncover technical details of later interest.
Figure 2 shows the power spectrum diagrams, recovering the well known fact that the spectrum decomposes in a twohalo term, a onehalo term and shotnoise. One immediate advantage, already underlined in Lacasa et al. (2014); Lacasa (2014), is that shotnoise is described consistently and does not need a separate formalism like counts in cells (e.g. Peebles 1980).
Applying the diagrammatic rules yields for example the following expression for the twohalo term of the power spectrum between twopossibly different redshifts: (26)
In the following, I will shorten the argument of mass and redshift to its indices:
At tree level the halo power spectrum takes the form (27)
so that one recovers the familiar equation
with the (scaledependent) galaxy first order bias (30)
I note however that things become more complex at 1loop order, where one would get additional contributions from higher order perturbation theory and halo biases, with a form more complex than Eq. (28). Since I will be working only at tree level, in the following I note P(k, z) instead of P_{lin}(k, z) for simplicity.
The 2halo part of the angular power spectrum is then given by (31)
In the following I note P(k) ≡ P(k, z = 0) for shortening.
Limber’s approximation simplifies to (32)
For the shotnoise power spectrum term, the diagrammatic rules give (33)
At this point I seem to face a slight incoherence: which redshift am I talking about, z_{1} or z_{2} ? This would be a real issue if one were computing the power spectra between slices of the universe at the same location but different times, in which case our correlation function could hit the same galaxy at two different redshifts. However real observables are located on the past light cone, so the two redshifts have to coincide. Whatever value is given to when z_{1} ≠ z_{2} should get canceled when the 2D projection of Sect. 2.3 is carried out. So I can take (34)
and it should give the same angular power spectrum.
Indeed one can check for instance with the first possibility: (35)
and one would get the same results with the two other possibilities given in Eq. (34). I note the appearance of a Kronecker between redshift bins, assuming they are not overlapping (see Appendix F.1 for the case of overlapping bins).
In the following, I adopt notation from the third possibility (i.e. , with being a Kronecker symbol) as it makes explicit that the redshift have to coincide. I also note that the Limber approximation is exact for this power spectrum term, as is constant with k.
Finally, the issue of Poissonian or nonPoissonian shotnoise is discussed in Appendix F.2.
For the onehalo power spectrum term, the diagrammatic rules give (36)
again I am faced with an apparent redshift incoherence. But since the two galaxies hit by the correlation function are in the same halo, and since observations are located on the past light cone, the two redshifts must be close, limited by δr < 2R(M, z) where R(M, z) is the virial radius of the halo. In this limited redshift interval there will be no appreciable evolution of the mass function, halo profiles, etc. So all redshifts can be taken to be equal, finding (37)
One can note that the Limber’s approximation is particularly well adapted to this power spectrum term. Indeed, at low ℓ/low k where the Limber’s approximation often gets wrong, goes to a constant so that Limber becomes exact, and starts to have a scaledependence only on halo size scales – small scales where Limber’s approximation works well. Thus one gets the angular power spectrum (38)
Again I note the presence of a Kronecker over redshift bins, and that other forms of coinciding for z_{1} = z_{2} would have given the same answer for the observable.
In general for higher order polyspectra, Limber’s approximation will be exact for wavevectors on which the polyspectrum does not depend, well justified for wavevectors for which the dependence is only through halo profiles u(k), justified only at high ℓ when there is a power spectrum dependence P(k), and unjustified when there is a dependence on the angles between wavevectors (e.g. through perturbation theory kernels as will be seen later).
In the following, I thus apply Limber’s approximation on wavevectors for which there is no dependence or only u(k) dependence, and will provide both the noLimber and Limber equations in the other cases.
2.5 Power spectrum covariance terms
Section 2.3 gave the projection equations from 3D to 2D. I now need the 3D trispectrum equations. Using the diagrammatic approach of Sect. 2.2, the involved diagrams are shown in Fig. 3.
This justifies the organisation of the next few sections of this article, as I derive the covariance terms in order of increasing complexity. I will start with the clustering terms: onehalo in Sect. 3, twohalo (both 2+2 and 1+3) in Sect. 4, threehalo in Sect. 5 and finally fourhalo in Sect. 6. I then move to shotnoise (all terms) in Sect. 7.
Fig. 3
Diagrams for the galaxy trispectrum. From left to right, top row: fourhalo (4h), threehalo (3h), threehalo shotnoise (3hshot3g), twohalo 1+3 (2h1+3). Middle row: twohalo 2+2 (2h2+2), twohalo threegalaxy shotnoise a (2hashot3g), twohalo threegalaxy shotnoise b (2hbshot3g), twohalo twogalaxy shotnoise a (2hashot2g), twohalo twogalaxy shotnoise b (2hbshot2g). Bottom row: onehalo (1h), onehalo threegalaxy shotnoise (1hshot3g), onehalo twogalaxy shotnoise a (1hashot2g), onehalo twogalaxy shotnoise b (1hbshot2g), onegalaxy shotnoise (shot1g). 

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3 Onehalo term
The onehalo term is the tenth diagram of Fig. 3 (bottom row, first from the left), while the other diagrams with a single halo (remainder of the bottom row in Fig. 3) are shotnoise terms which will be treated in Sect. 7. Applying the diagrammatic rules from Sect. 2.2 and the notes in Sect. 2.4 about coincident redshifts, the corresponding trispectrum part is:
where , i.e. it is equal to 1 when all redshifts are equal, and 0 otherwise.
This trispectrum term is diagonalindependent, according to the nomenclature of Sect. 2.3.
When projecting onto the angular covariance, as argued in Sect. 2.4, Limber’s approximation is justified on all wavevectors, since no P(k) factors are present. One thus obtains:
4 Twohalo terms
4.1 Twohalo 1+3 term
This term isthe fourth diagram of Fig. 3 (upper right corner). Applying the diagrammatic rules from Sect. 2.2, the corresponding trispectrum part is:
This trispectrum term is diagonalindependent following the nomenclature of Sect. 2.3.
For the permutation presented above, Limber’s approximation is justified on k_{2}, k_{3}, k_{4}, but may not be justified on k_{1} for low ℓ. One finds the covariance term (45)
Using the Limber’s approximation also on k_{1}, one finds:
4.2 Twohalo 2+2 term
This term isthe fifth diagram of Fig. 3: middle row, first from the left. The other diagrams in the middle row with two halos are shotnoise terms which will be treated in Sect. 7. Applying the diagrammatic rules from Sect. 2.2, the corresponding trispectrum part is:
The three permutations can be viewed respectively as flat rhyme (aabb), alternate rhyme (abab) and enclosed rhyme (abba).
For the first permutation (flat rhyme, Eq. (49)), Limber’s approximation is justified on k_{1}, k_{2}, k_{3}, k_{4} but not on the squeezed diagonal k_{1+2}, especially since the later aliases into the monopole. Equation (23) must then be used for the projection. One finds: (52)
where explicitly z_{a} ∈ i_{z} and z_{b} ∈ k_{z}. There are two possible ways to compute this equation numerically, depending whether one first goes with the wavevector integral or the redshift integrals. Respectively
is the matter angular power spectrum (that would be measure if one could directly see total matter instead of galaxies), that is involved in several other equations below.
For the two other permutations (alternate and enclosed rhymes, Eqs. (50) and (51)), Limber’s approximation is justified on k_{1}, k_{2}, k_{3}, k_{4} but may not be justified on the alternate diagonal k_{1+3} (resp. k_{1+4}). Equation (25) must then be used for the projection.
Again, thereare two possible ways to compute this equation numerically, depending whether one first goes for the wavevector integral or the redshift integrals. Respectively
where z_{a} ∈ i_{z} and z_{b} ∈ j_{z}.
5 Threehalo term
This section is longer than the previous ones due to the increased complexity of the term, and is thus split into smaller subsections for clarity.
5.1 Trispectrum
This term isthe second diagram of Fig. 3. The other diagram containing three different halos (third diagram of Fig. 3) is a shotnoise term which will be treated in Sect. 7. Applying the diagrammatic rules from Sect. 2.2, the corresponding trispectrum part is: (59)
The halo bispectrum splits into three terms (b2, s2 and 2PT, see Appendix A), and thus the 3h galaxy trispectrum too. Using notations from Appendix A.2, one finds: (60)
This can be rewritten to regroup terms and explicit the 18 permutations, grouping first the six terms involving angles between base wavevectors (), and then the twelve terms involving angles with a diagonal k_{α+β}: (61)
After Legendre decomposition of the angle dependence, the 2PT term yields three subterms (n = 0, 1, 2), while the b2 and s2 terms yield one subterm each (n = 0 and n = 2 respectively). Accounting for all permutations I thus have a total of 90 subterms.
5.2 Covariance
Let us first compute the contribution coming from the six terms with an angle between base wavevectors k_{α} ⋅ k_{β}. Using results from Appendix C.1.1, for X ∈{b2, s2, 2PT}, one finds
where the first term (Eq. (62)) comes from the pairs (1, 2) and (3, 4), while the second term (Eq. (63)) comes from the pairs (1, 3), (1, 4), (2, 3) and (2, 4).
Limber’s approximation can be applied on the first term only if ℓ_{a} = ℓ, which is the sole contributiononly for n = 0. Similarly, Limber’s approximation can be applied to the second term only for n = 0. I return to the n = 0 case later.
I tackle here the contribution coming from the twelve terms with an angle with a diagonal k_{α+β} ⋅k_{γ}. Using results from Appendix C.1.2, for X ∈{b2, s2, 2PT}, one finds
where the first term (64) comes from the permutations involving the squeezed diagonal (123, 124, 341, 342), while the second term (65) comes from the permutations involving an alternate diagonal (132, 134, 142, 143, 231, 234, 241, 243)^{3}.
5.3 Simplifications
In the n = 0 case, the covariance gets simpler, as it corresponds to an angleindependent trispectrum. The covariance becomes:
Limber’s approximation can also be used (except on the squeezed diagonal), if one first performs the wavenumber integrals before the redshift integrals (instead of the opposite, that was used in the previous noLimber equations). The resulting covariance is (70)
where in Eq. (71) z_{a} ∈ i_{z} and z_{b} ∈ k_{z}, while in Eq. (72) z_{a} ∈ i_{z} and z_{b} ∈ j_{z}.
One can perform the summation over X ∈{b2, s2, 2PT} by introducing the notation (73)
6 Fourhalo terms
6.1 Trispectrum
This term is the first diagram of Fig. 3. Applying the diagrammatic rules from Sect. 2.2, the corresponding trispectrum part is:
Following Appendix A, the halo trispectrum splits into three terms, and thus the 4h galaxy trispectrum too:
where the permutations are explicited for example in Appendix A.3.
After Legendre decomposition of the angles and accounting for all permutations, the b3 term splits into four subterms, the 3PT term into 9 × 3 × 4 = 108 subterms, and the 2 × 2 term into 25 × 12 = 300 subterms. I thus have a total of 412 subterms to compute.
6.2 Covariance
First, the term from third order halo bias is (80)
This term is the simplest as the trispectrum does not have an angle dependence. As such, if valid, the Limber approximation may easily be applied, and gives (81)
Next, the term from third order perturbation theory splits in two parts: one coming from trispectrum permutations involving the squeezed diagonal k_{1+2}, and one coming from permutations involving an alternate diagonal k_{1+3} or k_{1+4}.
I note that the last integral (over K and x_{a}) is purely analytic, however I did not find a closed form expression for it, except in the case n′ = 0 which will be tackled later.
The second part of the 3PT term is
where the symbol is related to a contraction of a 12J symbol of the second kind in Appendix E.4. Again the last integral (over K and x_{a}) is purely analytic with no known closed form expression, except in the case n′ = 0 which will be tackled later.
Finally the 2 × 2 trispectrum term also splits in two parts: one coming from trispectrum permutations involving the squeezed diagonal k_{1+2}, and one coming from permutations involving an alternate diagonal k_{1+3} or k_{1+4}.
The second part of the 2 × 2 term is (86)
6.3 Simplifications
For some cases of the Legendre decomposition, the covariance equations get simpler; I tackle these cases here.
First, beginning with 3PT terms, when n′ = 0, the 3PTsqueezed term simplifies to (87)
If I further set n = 0, I have (88)
and Limber’s approximation can be used on all wavevectors to yield: (89)
where I usedthe fact that F_{3;0,0} is independent of its arguments and superscripts (see Appendix A.3).
If n′ = 0, the 3PTalternate term also simplifies: (90)
and Limber’s approximation can be used on all wavevectors to yield: (92)
This result is exactly twice that of the squeezed term^{4}: . As such, the two terms can be grouped together: (93)
Another straighter way to get to this equation is that taking (n, n′) = (0, 0) is equivalent to inputting F_{3}(k_{1}, k_{2}, k_{3}) = 3 F_{3;0,0} in Eq. (78) and realising the analogy with the 4hb3 case. Thus Eq. (80) can be used with the replacements . This remarks allows us to unify the b3 and 3PT terms into a single equation: (94)
with .
Second, let us look now at 2 × 2 terms which give symmetric roles to n and n′ and thus significantly simplify only when both are zero. When n = n′ = 0, the 2 × 2squeezed term simplifies to (96)
Using Limber’s approximation on k_{1}, k_{3}, and definition (73) of , one finds: (97)
where z_{a} ∈ i_{z}, z_{b} ∈ k_{z}.
The 2 × 2alternate term also simplifies when n = n′ = 0 (98)
Using Limber’s approximation on k_{1}, k_{2}, and definition (73) of , one finds: (99)
where z_{a} ∈ i_{z}, z_{b} ∈ j_{z}.
7 Shotnoise
7.1 Onegalaxy shotnoise
This term is the last diagram of Fig. 3. Applying the diagrammatic rules from Sect. 2.2, the corresponding trispectrum part is: (100)
since k_{1} + ⋯ + k_{4} = 0 and the halo profile is normalised to u(0M, z) = 1.
The corresponding covariance term is (101)
7.2 Twogalaxy shotnoise
These termsare the 8th, 9th, 12th, and 13th diagrams of Fig. 3. One could apply the diagrammatic rules from Sect. 2.2 to write down the corresponding trispectrum part; however it was realised by Lacasa et al. (2014) that these diagrams are identical to ones of lower order polyspectra. For instance for the 8th diagram (2hashot2g) (102)
These diagrams can then be resumed to reveal the clustering part of the lower order polyspectrum (here the power spectrum). Noting and writing down explicitly all involved permutations of (1234), one finds: (103)
where the first two lines in Eq. (103) come from “1+3” diagrams (2hbshot2g and 1hbshot2g, respectively 9th and 13th diagrams in Fig. 3), and the last three lines come from “2+2” diagrams (2hashot2g and 1hashot2g, respectively 8th and 12th diagrams in Fig. 3).
The corresponding covariance is
where is the clustering part (i.e. without shotnoise) of the galaxy angular power spectrum.
7.3 Threegalaxy shotnoise
These terms are the 3rd, 6th and 11th diagrams of Fig. 3 (3hshot3g, 2hshot3g and 1hshot3g). As in the previous subsection, I remark that these diagrams are identical to ones of the galaxy bispectrum and can be resumed. Noting and writing explicitly all involved permutations of (1234), one finds: (106)
The corresponding covariance is (first with terms in the same order) (107) (108)
where is the clustering part (i.e. without shot noise) of the galaxy angular bispectrum.
7.4 Shotnoise subtraction
The shotnoise contribution to the power spectrum is where is the actual number of galaxies in the survey (in a redshift bin, and per steradian). This number is perfectly known, and thus it can be subtracted from the measurement in order to reveal power up to smaller scales. This subtraction is indeed actually applied in power spectrum measurement by past and current surveys working with the relative fluctuation δ_{gal}: the corrected spectrum is .
Naively this subtraction shouldn’t affect the covariance, since covariances are invariant under addition of a constant. However it does affect it, because one is subtracting the actual number of galaxies, not the modelpredicted number. The actual number of galaxies is itself a random variable, so subtracting it will add covariance terms. This random number is, in fact, positively correlated with the galaxy power spectrum measurement, thus the spectrum covariance will be reduced by the shotnoise subtraction.
Explicitly, the shotnoise subtraction removes several of the power spectrum covariance terms. There are two equivalent ways to see which terms are going to be canceled. The first way uses the fact that the shotnoise contribution to the power spectrum corresponds to the diagram with coinciding galaxies (right most diagram in Fig. 2), thus subtracting shotnoise corresponds to only taking the two other spectrum diagrams, that is, forbidding galaxies 1 and 2 to coincide. At the covariance level, galaxies 1 and 2 correspond to C_{ℓ} and galaxies 3 and 4 correspond to ; shotnoise subtraction (for both C_{ℓ} and ) thus corresponds to forbidding diagrams with a coincidence 1 = 2 and/or a coincidence 3 = 4. This removes many of the terms presented in the previous Sects. 7.1–7.3, however there are still terms remaining, corresponding to the coincidences 1 = 3 or instance. The second equivalent way goes through realspace: in w(θ), shotnoise corresponds to a dirac at θ = 0. So shotnoise subtraction will kill all terms that yield a dirac at θ = 0 and/or θ′ = 0 in the realspace covariance . Harmonic transforming this back to , this corresponds to all terms which have no dependence on at least one of the multipole. For example, in the shot3g term (Sect. 7.3) has no dependence on ℓ, and thus will be canceled by the shotnoise subtraction.
Through either equivalent way, one finds that the following shotnoise covariance terms are canceled by shotnoise subtraction:

;

the “1+3” part of : and the symmetric term in ℓ′;

the squeezed part of : ;

the squeezed part of : and the symmetric term in ℓ′.
The following covariance terms are however still present:

the alternate part of :

the alternate part of :
In summary, most shotnoise effects are canceled out. Most, but not all: a small group of terms, later participating in the braiding covariance (see Sect. 8.2), resist because one cannot erase the discreteness of the galaxy density field. Inference from a discretely sampled field cannot be the same as from the underlying continuous field.
8 Discussion of the results
8.1 Supersample covariance
SSC has been studied in the past literature, mostly for 3D surveys (e.g. Takada & Hu 2013) but then also in spherical harmonics: Lacasa & Rosenfeld (2016) derived its impact on the crosscovariance between cluster counts and the galaxy angular power spectrum. However its impact on the autocovariance of an angular power spectrum has never been derived rigorously, but was surmised using the 3D results or the crosscovariance result. Here I show that SSC does emerge naturally from the halo model derivation, and does recover the postulation from Lacasa & Rosenfeld (2016) based on the crosscovariance result. I use only equations after Limber’s approximation, leaving the issue of noLimber SSC to be tackled in future works. Also, I first tackle in Sect. 8.1.1 the simpler case of SSC terms coming from angleindependent trispectra (n = 0), which will be the terms going into the summary Sect. 9. Then in Sect. 8.1.2, I will discuss the case of angledependent terms (n = 1, 2) and how to generalise SSC to other modelisation and partial sky coverage.
8.1.1 Angleindependent terms
We see SSC emerging when grouping all covariance terms where the trispectrum has a dependence on the squeezed diagonal through a P(k_{1+2}), making a appear in the covariance. Specifically, there is such dependence in the 2h2+2sqz term, Eq. (53), in the 3hdiagsqz0 term, Eq. (71), and in the 4h2X2sqz00 term, Eq. (97). There are also shotnoise contributions which need a bit more work to yield a form unifiable with the clustering terms.
The first shotnoise contribution comes from the twohalo part of Eq. (105) (109)
where I assumed that k_{1+2} is sufficiently small to neglect the scale dependence of , meaning that galaxy bias can be considered constant for supersurvey modes, a reasonable approximation. The second shotnoise contribution comes from the two and threehalo parts of Eq. (107).
Combining all these terms (2h2+2sqz, 3hdiagsqz, 4h2X2sqz, shot2gsqz2h, shot3gsqz23h), one finds: (112)
More familiar equations can be found by expliciting the sum over X ∈{b2, s2, 2PT}, using that (114)
introducing the effective galaxy bias (115)
and the variance of the background matter density (116)
With all these notations, one finds (117)
I thus nicely recover the same SSC equation as the one derived in Lacasa & Rosenfeld (2016) in the crosscovariance case.
In the literature, for example Takada & Hu (2013), the first term in Eq. (118) is called beatcoupling (BC) and the third term is called halo sample variance (HSV); the second and fourth terms were discovered by Lacasa & Rosenfeld (2016), they come respectively from the nonlinear response of halos to matter density and from shot noise.
However the SSC shotnoise terms will be canceled by the shotnoise subtraction described in Sect. 7.4, the shotnoise corrected SSC then becomes: (119)
8.1.2 Angledependent terms
Two subleading SSC effects found in the recent literature are not present in the previous subsection. Here I show that is because they come from the n = 1 and n = 2 terms.
The first effect is the socalled dilation effect found by Li et al. (2014) in the 3D P(k) case. From Li et al. (2014), one sees that this term comes from the (121)
part of the 2PT kernel F_{2}(k_{1}, k_{2}).
In my derivation this part of F_{2} yields the 2PT n = 1 Legendre term, cf. Appendix A. Hence the dilation effect is present here if one considers n = 1.
The second effect is from supersurvey tidal fields. This has been first uncovered by Li et al. (2018), Akitsu & Takada (2018) for the redshift space power spectrum of galaxies. Then Barreira et al. (2018), which appeared the same day as this article v1 on arXiv, showed that it also affects the isotropic power spectrum of weaklensing. To discus this issue, I consider matter only, not galaxies, and adopt the same notations as Barreira et al. (2018). The central notion is that of the LSS response to long wavelength (= soft) modes. The first order response , for one such soft mode, is defined through the squeezed limit of the bispectrum (122)
and Barreira et al. (2018) decompose this response into its isotropic and tidal field part (123)
where P_{2} is the second order Legendre polynomial.
In this article, I use standard perturbation theory at treelevel. Then from Appendix A.2 giving the 2PT bispectrum, one sees that the resulting power spectrum response is (124)
From this one sees easily that the n = 2 Legendre term in F_{2} will source the tidal response. Taking appropriate limits when p → 0, one can further see that the n = 1 Legendre term in F_{2} will source both the isotropic and the tidal responses. The n = 0 term considered in the previous subsection sources the socalled growthonly part of the isotropic response. Hence the decomposition of F_{2} in n = 0, 1, 2 Legendre terms is equivalent to including growthonly, dilation and tidal effects in the response approach to SSC.
Although the derivation presented here uses standard perturbation theory at treelevel, it is possible to generalise the SSC equations to another modelling of matter, simply by adapting the power spectrum response. For instance, this response can be fitted to numerical simulations and then fed in the equations (see e.g. Barreira & Schmidt 2017; Barreira et al. 2018). This appears a solution for observables directly sensitive to the matter power spectrum (e.g. weaklensing), but it may not be feasible for galaxies. Indeed for the galaxy spectrum, the onehalo response is a critical element at intermediate to small scales, and it is fully nonperturbative and heavily dependent on the galaxy selection function. As such it necessitates a halo modelisation to be predicted correctly.
The advantage of the present derivation is that it does not rely on any particular soft mode limit p → 0 or any Taylor expansion as in Takada & Hu (2013). Instead, the present derivation is exact within the modelling assumption. Hence it remains valid even on large scales comparable to the survey size, while the response approach is limited to k ≪ p, in other words, at small scales.
A side note is that I developed here all the covariance equations in the fullsky limit. However this is not a practical limitation, as Lacasa et al. (2018) has recently developed the formalism to predict analytically SSC in the more realistic case of partial sky coverage with an arbitrary survey mask. One basically needs to change σ^{2} in Eq. (119) to account for the effect of the mask power spectrum.
Finally, there has been a lot of emphasis on SSC in literature, however the systematic derivation presented in this article finds a wealth of other nonGaussian covariance terms which have never been considered before. The following subsections are devoted to these new terms and their potential importance.
8.2 Braiding terms
Braiding terms are those that arise when the trispectrum has a dependence on one of the alternate diagonal through a P(k_{1+3}) (or P(k_{1+4})), making a appear in the covariance as well as a 3J symbol mixing the multipoles ℓ and ℓ′. For any SSC term (studied in Sect. 8.1) there is a corresponding braiding term. Namely the clustering contributions are the 2h2+2alt term, Eq. (57), the 3hdiagalt term, Eq. (76), and the 4h2X2alt term, Eq. (99). Defining the braiding kernel (125)
these three clustering terms can be unified into: (126)
which is relatively similar to defined in Eq. (113), except for the shotnoise part and the multipolecoupling structure. In fact, one has the identity (128)
Shotnoise contributions to the braiding covariance are given by Eq. (104) (129)
and Eq. (108) (130)
In order to include these terms in a unified formula similar to Eq. (126), one has to add some approximations: (131)
which neglects several bispectrum terms^{5}, and uses Limber’s approximation on ℓ and ℓ′.
With these, one gets the unified formula (133)
8.3 Importance of terms
Among the nonGaussian covariance terms, SSC is the main reference against which to compare the new terms discovered in this article. Indeed it has already been well studied in the literature, including for the galaxy angular power spectrum, for example, in combination with cluster number counts (Lacasa & Rosenfeld 2016) or with weak lensing (Krause & Eifler 2017). Its importance is already well recognised, and it is indeed included in the analysis of current galaxy surveys (e.g. van Uitert et al. 2017; Krause et al. 2017; Abbott et al. 2017) having an impact both on the cosmological error bars as well as central values (Hildebrandt et al. 2017).
As already mentioned in the introduction, numerical investigations I performed (Lacasa 2017) show that the 1h and 2h1+3 terms have an impact comparable to SSC on the signaltonoise ratio of , when using survey specifications representative of future missions like Euclid. As these terms become important, there is a priori no reason for others not to be, so I now turn to analytical arguments comparing all the other terms to SSC in adequate regimes where they can be compared.
The braiding terms (Sect. 8.2) are the easiest ones to be compared with SSC, as it has already been noted that they have some similarity to it^{6}. Indeed, using Eq. (128), in the case ℓ = ℓ′ and i_{z} = j_{z} = k_{z} = l_{z} one has the identity (135)
which can be compared with the corresponding SSC case (136)
in particularone sees that the term ℓ_{a} = 0 in the sum of Eq. (135) implies (137)
so at low multipoles, braiding must be nonnegligible. To go further, assumptions need to be made. If one can assume that the matter power spectrum is slowly varying over the range of multipoles of interest, then (138)
More explicitly, if is an increasing function of ℓ over [0, 2ℓ], then one finds (139)
whereas if is decreasing (140)
The first situation occurs when the survey probes scales larger than the matterradiation equality where P(k) has a maximum, i.e. ℓ ≲ ℓ_{eq} = k_{eq}r(z), which will be the case for future surveys covering large portions of the sky. The second situation occurs at smaller scales. From current constraints in the cosmologically interesting domain; then from power counting argument one gets that the covariance ratio is . So in summary
Another regime where braiding is important is crossredshifts: from Eq. (112) one sees that SSC vanishes for crossspectra i_{z} ≠ j_{z} as a consequenceof Limber’s approximation; however from Eq. (126) one sees that braiding covariance does not vanish in this regime. Hence braiding will be of importance for effects producing nonvanishing crossspectra, for example when dropping Limber’s approximation or accounting for general relativistic effects (e.g. Cardona et al. 2016).
The next terms which may be important are the third order ones involved in the 4halo term, either the third order bias term (4hb3) or the term from third perturbation theory (4h3PT). For these terms, it is simpler to argue at the level of the 3D trispectrum:both these terms give a trispectrum
whereas the SSC from 4h2X2sqz (beat coupling BC–BC in the literature) comes from a trispectrum
where the prefactor of both trispectra are of the same order.
Thus these terms will be more important than BC–BC in the range of multipoles where P(k_{ℓ}z) ≳ P(k_{1+2}z). In this full sky derivation, k_{1+2} becomes aliased in the monopole ℓ = 0, but in general k_{1+2} is a supersurvey mode, so for a general survey covering a fraction f_{SKY} of the sky, one gets the rule of thumb that these terms are going to be important for multipoles where (144)
This certainly happens for ℓ ≲ ℓ_{eq} if the surveyis large enough to see the matterradiation equality scale. Interestingly, Eq. (144) is basically equivalent to the condition for braiding to be important with respect to SSC, although in the former case this was argued only on the diagonal ℓ = ℓ′ whereas here it suffices that one of the multipoles fullfills the condition, either ℓ or ℓ′.
The penultimate terms are coming from the threehalo term, explicitly 3hbase which has two contributions whose trispectra follow
whereas the SSC from 3hdiagsqz (BCHSV in the literature) comes from a trispectrum
where the prefactor of both trispectra are of the same order.
Thus these terms will be more important than BCHSV in the range of multipoles where P(k_{ℓ}z) ≳ P(k_{1+2}z). Hence the condition Eq. (144) is again the one that rules the importance of these terms.
Finally, shotnoise terms impact the measurements in a manner inversely proportional to the number of observed galaxies. More precisely the impact on the signaltonoise of is of order , where N_{ℓ} is the number of multipoles considered. For future surveys, this effect can thus be expect to be well below the percent level, unless targeting high multipoles with thin redshift bins at the lowest and highest redshifts, where galaxy numbers decrease. In summary, most terms have a chance to be of importance if the survey considered probes scales comparable to or larger than the matterradiation equality k_{eq}, and shotnoise can be of importance for a spectroscopic survey targeting information on small scales.
9 Summary
This section summarises covariance terms that should be considered in the simplest case where one uses Limber’s approximation and shotnoise subtraction (Sect. 7.4), as is usually done in current galaxy surveys, and further considering only n = 0 for angledependent kernels. The more general equations can be found in the main body of the text. This section can thus be considered by the busy reader as the reference summary containing the first order equations to be implemented numerically.
9.1 Notations and remarks
In order for this section to be selfcontained, I recapitulate here the particular notations which are used in the covariance terms.
To begin with, as discussed in Sect. 2, I consider the power spectrum of the absolute fluctuations δn_{gal}(x) and not the relative fluctuations . One can convert my absolute power spectrum into a relative one by dividing by the factor N_{gal}(i_{z}) N_{gal}(j_{z}), where i_{z} and j_{z} are the indices of the two redshift bins considered.
The power spectrum covariance is noted for simplicity
and needs to be divided by a factor
if one wants relative fluctuations instead of absolute ones.
Most importantly, I use the following definitions: k_{ℓ} = (ℓ + 1∕2)∕r(z) is the comoving wavenumber given by Limber’s approximation at multipole ℓ and redshift z, (145)
is an integral that will appear frequently, (146)
is the sum of second order contributions, and (147)
is the sum of third order contributions.
The angular power spectrum of matter is (148)
and in fullsky the SSC and braiding kernels are respectively
9.2 Covariance terms
This subsection simply lists the different covariance contributions, ordered in term of simplicity.
The first contribution is the onehalo term (Sect. 3) (151)
then there is the twohalo 1+3 term (Sect. 4.1) (152)
the threehalo base term (Sect. 5) (153)
and the fourhalo term from third order contributions (Sect. 6) (154)
Then there are groups of terms unifying contributions with different numbers of halos.
First is the supersample covariance (Sect. 8.1)
where z_{a} ∈ i_{z}, z_{b} ∈ k_{z}, and (155)
Second is the braiding covariance (Sect. 8.2) (156)
where z_{a} ∈ i_{z}, z_{b} ∈ j_{z}, and (157)
Finally, shotnoise terms, where the only surviving shotnoise subtraction (Sect. 7.4) are braiding ones (Sect. 8.2) (158)
The importance of all these covariance terms is partially discussed in Sect. 8.3 with analytical arguments. The actual importance for a galaxy survey will strongly depend on the survey specifications, galaxy selection, choice of data vector (e.g. redshift and scale cuts) and so on, and as such cannot be forecast easily analytically. Instead numerical analysis must be carried out, which will be the subject of future works. I expect that at least several of these terms, if not most of them, will be of importance for cosmological constraints from future surveys such as Euclid and LSST.
10 Conclusion
I have carried out an exhaustive analytic derivation of all nonGaussian covariance terms of the galaxy angular power spectrum when using the halo model at tree level. The calculation of the involved trispectrum is developed up to third order both in halo bias and standard perturbation theory, including nonlocal halo bias and all shotnoise terms.
The projection of the trispectrum into the angular covariance has been derived in all trispectra cases, including complex cases with several dependence on angle between wavenumbers, in two appendices (Appendices B and C) together with robustness checks of the formulae performed (Appendix D). These derivations, though not the original aim of the article, are standalone results that can be used in order to model the angular covariance of other signals or alter the modelling framework, for example, using a different flavour of perturbation theory.
A wealth of nonGaussian covariance terms has been found, providing a rigorous derivation of the already known SSC in the angular case, and more importantly discovering several new terms. A whole new class of terms, which I dub braiding covariance, stems from the same physical effects that lead to SSC but leads to different couplings between multipoles and redshift bins. Other terms (3hbase, third order contributions) furthermore exist, and I provide a unified treatment of shotnoise terms, including how they are affected by the popular habit of subtracting from the observed power spectrum.
A clean executive summary is provided in Sect. 9 in the simplest case where one uses Limber’s approximation, shotnoise subtraction (Sect. 7.4), and retains only n = 0 terms for angledependent kernels. This section can serve as a reference for the minimal number of nonGaussian terms to quantify numerically for galaxy surveys.
The potential importance of the new nonGaussian terms has been discussed with analytical arguments in Sect. 8.3, in particular in comparison with SSC, as the latter has already shown to have an impact on constraint from current surveys. It was found that some terms (braiding, 3hbase, 4h3) should become comparable to, if not bigger than, SSC on scales comparable to the matterradiation equality. Other terms (1h, 2h1+3) can become important for deep surveys with a high portion of satellite galaxies, with only numerical calculation which can decide precisely on their actual impact. Finally shotnoise terms can become relevant when analysing small scales with spectroscopic (more sparse) surveys, such as when constraining neutrinos or noncold dark matter.
Numerical codes computing SSC and the onehalo term already exist and have been used. The one by the author, for instance, can compute covariance on a standard laptop in a matter of seconds or tens of seconds depending on the number of multipoles. Including the new nonGaussian terms presented here shall prove very feasible and will be studied in future works. It is expected that this inclusion will not alter the order of magnitude of the speed of the calculation. Hence the analytical approach to covariances will remain feasible, and in fact the most competitive, for current and future surveys.
Acknowledgements
I thank Ruth Durrer, Vittorio Tansella and Alexandre Barreira for helpful discussions, Pierre Fleury for help with 9J symbols, and Elena Sellentin and Martin Kunz for proofreading and suggestions that improved this article. I acknowledgesupport by the Swiss National Science Foundation.
Appendix A 3D halo polyspectra
The density field of halos at a given mass is composed of a local bias hierarchy (Fry & Gaztanaga 1993) and a nonlocal bias term from tidal forces (Chan et al. 2012; Baldauf et al. 2012) (A.1)
where I cutthe local hierarchy at third order, and the quadratic tidal tensor field s^{2} is related to the matter density field via (Baldauf et al. 2012) (A.2)
This bias model has been shown to reproduce adequately the halo clustering by Hoffmann et al. (2018). The bias parameters can be either taken as free parameters to be fit together with cosmology in observations, or theoretical inputs can be used. Indeed the peakbackground split prescribes the local bias as (e.g. Mo et al. 1997) (A.4)
where ν is the peak height and f(ν) the mass function. Alternatively a nearuniversal relation between b_{2} and b_{1} was found by Hoffmann et al. (2017).
For the tidal tensor bias, the local lagrangian model gives (Baldauf et al. 2012) (A.5)
A.1 Power spectrum
At tree level, the only contribution is that from first order bias and first order (linear) perturbation theory. Thus the halo power spectrum is given by (A.6)
A.2 Bispectrum
For the bispectrum, the first order contribution is zero, so one needs to go at second order in both perturbation theory and local and nonlocal bias. The halo bispectrum thus splits into three terms (A.7)
where the second order local halo bias term is (A.8)
and finally the second order perturbation theory term is (A.11)
with (Fry 1984) (A.12)
These can be summarised in a general equation (A.13)
with X ∈{b2, s2, 2PT}, the kernels are (A.14)
and the biases are the expected ones (A.15)
These notations will become useful in the next subsection.
A.3 Trispectrum
For the trispectrum, at treelevel one needs either to go at second other in two density contrasts simultaneously, or go to third order in one density contrast. The halo trispectrum thus splits into three terms
where, using the notations of Appendix A.2 (A.16)
Because the involved permutations are not completely trivial, here is another writing of this equation: (A.17)
where the sums runs over all six possible pairs {α, β}, and where {γ, δ} are the remaining two indices in {1, 2, 3, 4}.
The trispectrum from third order halo bias is (A.18)
and finally the trispectrum from third order perturbation theory is (A.19)
For the third order perturbation theory kernel, Bernardeau et al. (2002, Eq. 394344) gives the recurrence equations which can be used to derive all F_{n} kernel (A.20)
where and .
Using this recurrence equation, after symmetrisation of the kernel I find (A.21)
This result agrees with TrØst Nielsen & Durrer (2017).
A.4 Legendre decomposition
It will be necessary for Appendix C to project the dependence of trispectra on wavevectors angles onto Legendre polynomials P_{n}. To this end, I give here the decomposition of the halo bispectrum and trispectrum in Legendre series.
Using results from Appendix A.2, the halo bispectrum writes (A.23)
Using results from Appendix A.3, the halo trispectrum writes
The third order bias bispectrum has no dependence on wavevectors angles (equivalently it decomposes with just a (n, n′) = (0, 0) term).
For the 3PT contribution, noting that
thus the trispectrum term from third order perturbation is (A.33)
Appendix B Projection of the 3D trispectrum to the covariance of the angular power spectrum I: angleindependent case
In both this section and Appendix C, I derive the projection of the 3D trispectrum into the angular covariance , examining the various cases of functional dependence of the trispectrum. In all these cases, the underlying strategy is basically the same. Starting from Eq. (19), the Legendre polynomials are expanded in spherical harmonics, and the wavevector dirac(s) is (are) Fourier transformed introducing one (diagonalindependent case) or two (diagonaldependent case) auxiliary comoving volume integrand. Then integration of spherical harmonics over unit vectors is performed, yielding either Kroneckers of harmonic indices or Gaunt coefficients. Next, integration over unit vectors of the auxiliary comoving volume(s) is performed, yielding more Kroneckers or Gaunt coefficients, and any appearing simplification is carried out. Finally, sums over azimuthal parameters m_{i} are performed and other simplifications may appear. I end up with a sum over potential auxiliary multipoles of a geometric coefficient multiplying an angular trispectrum.
That trispectrum is an integral over comoving volumes and wavenumbers, without dependence on any angle or unit wavevector. In the coming calculations, three type of functional form for these trispectra are going to appear.
The first type happens when there is a dependence on the squeezed diagonal K = k_{1+2}: (B.1)
The second type happens when there is a dependence on the alternate diagonal K = k_{1+3}: (B.2)
(another type would appear for the last diagonal K = k_{1+4}, but this is just a symmetric version of the 1+3 type).
In the case ℓ_{a} = ℓ_{b} and f(K, k_{1}, k_{2}, k_{3}, k_{4}) independentof K, both trispectra reduce to the third type (B.3)
These notations prove useful in unifying the results of all the next subsections, both in this section and Appendix C.
I now turn to examining each possible 3D trispectrum functional form and derive the corresponding covariance.
B.1 Diagonalindependent trispectrum
The trispectrum is said diagonalindependent if T_{gal}(k_{1234}, z_{1234}) = T_{gal}(k_{1234}, z_{1234}), that is, it depends only on the length of the four Fourier wavevectors and not on their relative orientations. This is for example the case of the onehalo term, the twohalo 1+3 term and some shotnoise terms. In that case, the power spectrum covariance is
where the angular trispectrum is (B.10)
Using Limber’s approximation (Limber 1953; Loverde & Afshordi 2008) on all wavevectors, the angular trispectrum simplifies to
with (where ℓ_{1} = ℓ_{2} = ℓ and ℓ_{3} = ℓ_{4} = ℓ′) and z ∈ i_{z}.
B.2 Trispectrum depending on the squeezed diagonal
In this casethe trispectrum also depends on the length of the squeezed diagonal k_{1+2} This case proceeds in a manner similar to that of Appendix B.1, except that I force the diagonal to appear in the wavenumberdirac through the identity (B.11)
where the angular trispectrum is (B.15)
with Limber’s approximation on k_{1234} (but not on K, since it aliases into the monopole) simplifying it to (B.16)
where x_{a∕b} = r(z_{a∕b}), z_{a} ∈ i_{z}, z_{b} ∈ k_{z}, and (B.17)
B.3 Trispectrum depending on one of the other diagonals
In this case the trispectrum depends on K = k_{1+3} (or k_{1+4}, which is a symmetric case), additionally to k_{1234}. The case proceeds similarly to Appendix B.2, except that the diagonal will now produce a mixing between ℓ and ℓ′
where the sum runs over multipoles following the triangular inequality ℓ − ℓ′≤ ℓ_{a} ≤ ℓ + ℓ′ and the parity condition ℓ + ℓ′ + ℓ_{a} even, and the angular trispectrum is (B.23)
Using Limber’s approximation on k_{1234}, the angular trispectrum simplifies to (B.24)
where x_{a∕b} = r(z_{a∕b}) and (B.25)
If Limber’s approximation is also applied on ℓ_{a}, one gets (B.26)
where z_{a} = z_{b} = z and .
Appendix C Projection of the 3D trispectrum to the covariance of the angular power spectrum II: angledependent case
The threehalo and fourhalo terms of the covariance involve respectively the halo bispectrum and trispectrum. For both these polyspectra,the terms coming from the tidal tensor and from perturbation theory contain kernels depending on angles between some wavevectors. Using AlKashi’s theorem (law of cosines) one could express all these angles in term of just wavenumbers k_{i} and k_{ij}, and thus reduce to the cases studied in the previous section Appendix B. This is however not adequate as it yields nonseparable expressions, and will be incorrect when using the Limber’s approximation, due to angles being potentially fastvarying function of wavenumbers, particularly in the squeezed limit. The next subsection give the correct equations for projecting these angledependent trispectra into angular covariances.
C.1 Trispectrum depending on one angle
The following properties of Gaunt coefficients will be useful (C.1)
C.1.1 Angle between base wavevectors
A first useful remark is that a trispectrum dependence on an angle never appears together with a dependence on one diagonal (k_{1+2}, k_{1+3} or k_{1+4}). So, accounting for symmetries, the two case I have to consider are (C.4)
where P_{n} is the nth order Legendre polynomial.
C.1.1.1 The case of
In the case involving , the covariance reads
where the sum runs over multipoles following the triangular inequality ℓ − n≤ ℓ_{a} ≤ ℓ + n and the parity condition ℓ + n + ℓ_{a} even, and the angular trispectrum is (C.12)
Limber’s approximation, if valid, can be used on k_{3} and k_{4}, but not on k_{1} and k_{2}, because a closed form expression for the integral of two distinct Bessel function^{7} is unknown to the author. One finds: (C.13)
with .
C.1.1.2 The case of
In the case involving , the covariance reads
where the sum runs over multipoles following the triangular inequalities ℓ − n≤ ℓ_{1} ≤ ℓ + n and ℓ′ − n≤ ℓ_{3} ≤ ℓ′ + n, and the parity conditions ℓ + n + ℓ_{1} and ℓ′ + n + ℓ_{3} even^{8}, and the angular trispectrum is (C.21)
Limber’s approximation, if valid, can be used on k_{2} and k_{4}, but not on k_{1} and k_{3}. This yields: (C.22)
with k_{ℓ} = (ℓ + 1∕2)∕x and .
C.1.2 Angle with a diagonal or
A useful remark is that there is never a dependence on an angle between two diagonals, for example, , the angles involved will always be between a diagonal and a base wavevector . Moreover, using that for example, k_{1+2} = −k_{3+4}, the case can always reduced to that of an angle where l≠i, j.
Finally,a dependence on an angle with one diagonal never appears together with a dependence on a wavenumber of another diagonal. For instance if there is a dependence on , there can be a dependence on k_{1+2} (and k_{1}, k_{2}, k_{3}, k_{4}) but there will be no dependence on k_{1+3} or k_{1+4}.
Armed with these considerations and accounting for symmetries, there are two cases I have to consider: (C.23)
where P_{n} is the nth order Legendre polynomial.
C.1.2.1 Angle with the squeezed diagonal
where the sum runs over multipoles following the triangular inequality ℓ′ − n≤ ℓ_{3} ≤ ℓ′ + n, and the parity condition ℓ′ + n + ℓ_{3} even, and the angular trispectrum is (C.31)
Limber’s approximation, if valid, can be used on k_{1}, k_{2} and k_{4}, but not on k_{3} and K. It yields: (C.32)
with k_{ℓ} = (ℓ + 1∕2)∕r_{1} and .
C.1.2.2 Angle with an alternate diagonal
Then the covariance reads (C.34)
where I defined the geometric coefficient (C.39)
where the parity conditions respected by the Gaunt coefficients ensure that is real^{9}. This coefficient will be shown to be, in fact, a 6J symbol in Appendix E.2.
The angular trispectrum is given by (C.40)
Limber’s approximation, if valid, can be used on k_{1}, k_{3} and k_{4}, but not on k_{2} and K. It yields: (C.41)
with k_{ℓ} = (ℓ + 1∕2)∕x_{a} and .
C.2 Trispectrum depending on two angles
From Appendix A.3, one sees that the halo trispectrum brings terms of the form (C.42)
for some function of the wavevector moduli.
Furthermore the 3PT halo trispectrum term brings subterms of the form (C.43)
for another function of the moduli. There will also be similar terms in k_{2+3} = k_{1+4} which will be deducible from the k_{1+3} case by symmetry, and terms in k_{1+2} = k_{3+4} (squeezed diagonal). In the following I will first tackle the two cases needed for the X × Y terms, then the cases needed for the 3PT term.
C.2.1 X × Y terms
C.2.1.1 Squeezed diagonal case
Then the covariance reads (C.47)
where the sum runs over multipoles following the triangular inequalities ℓ − n≤ ℓ_{1} ≤ ℓ + n, ℓ′ − n′≤ ℓ_{3} ≤ ℓ′ + n′ and n − n′≤ ℓ_{c} ≤ n + n′, and the parity conditions ℓ + n + ℓ_{1}, ℓ′ + n′ + ℓ_{3} and n + n′ + ℓ_{c} even (which ensure that is real). The angular trispectrum is (C.51)
Limber’s approximation, if valid, can be used on k_{2} and k_{4}, but not on k_{1}, k_{3} and K. It yields: (C.52)
with k_{ℓ} = (ℓ + 1∕2)∕x_{a} and .
C.2.1.2 Alternate diagonal case
where I had to define the geometric coefficient^{10} (C.58)
which will be shown in Appendix E.3 to be, in fact, a 9J symbol.
The angular trispectrum is (C.59)
Limber’s approximation, if valid, can be used on k_{3} and k_{4}, but not on k_{1}, k_{2} and K. It yields (C.60)
with .
C.2.2 3PT terms
Following Eq. (A.21) for the 3PT kernel, and accounting for symmetries, the two cases I have to consider are (C.61)
C.2.2.1 Squeezed diagonal case
Then the covariance reads (C.64) (C.67) (C.68) (C.69)
where the sum runs over multipoles following the triangular inequalities ℓ − n≤ ℓ_{1} ≤ ℓ + n and ℓ′ − n′≤ ℓ_{3} ≤ ℓ′ + n′, and the parity conditions ℓ + n + ℓ_{1} and ℓ′ + n′ + ℓ_{3} even (the latter ensuring that is real). The angular trispectrum is (C.70)
Limber’s approximation, if valid, can be used on k_{4}, but not on k_{1}, k_{2}, k_{3} and K. It yields: (C.71)
with .
C.2.2.2 Alternate diagonal case
where I had to define the geometric coefficient^{11} (C.78)
and the angular trispectrum is (C.79)
Limber’s approximation, if valid, can be used on k_{4}, but not on k_{1}, k_{2}, k_{3} and K. It yields: (C.80)
with .
Appendix D Reduction of covariance cases
In this section, I show that the various complex trispectrum cases computed in Appendices B and C, reduce to simpler trispectrum cases when appropriate. This is a consistency (sanity) check of the derivations presented above.
D.1 Diagonalindependence
In this subsection, I show that the two trispectrum cases with a diagonal dependence, presented in Appendix B, do reduce to the diagonal independent case when appropriate.
The following properties will be useful here: (D.1) (D.2)
D.1.1 Squeezed diagonal
In this case, the 3D trispectrum is (D.3)
and the resulting covariance, derived in Appendix B.2, is (D.4)
where the angular trispectrum is (D.5)
If in fact T_{gal}(k_{1+2}, k_{1234}, z_{1234}) does not depend on k_{1+2}, using Eq. (D.1), one sees that the angular trispectrum reduces to the diagonalindependent case: (D.6)
And the covariance equation is already equal to the diagonalindependent case.
D.1.2 Alternate diagonal
In this case,the 3D trispectrum is (D.7)
and the resulting covariance, derived in Appendix B.3, is (D.8)
where the angular trispectrum is (D.9)
If in fact T_{gal}(k_{1+3}, k_{1234}, z_{1234}) does not depend on k_{1+3}, as in previous Appendix D.1.1, one sees that the angular trispectrum reduces to the diagonalindependent case, becoming independentof ℓ_{a}. Then using Eq. (D.2), one sees that the covariance equation also reduces to the diagonalindependent case.
D.2 Angle independence
In this subsection, I show that the trispectrum cases with an angle dependence, presented in Appendix C.1, do reduce to the angle independent cases of Appendix B when appropriate. I then show that the most complex cases of a trispectrum depending on two angles, presented in Appendix C.2, do reduce to the cases depending on a single angle when appropriate. This ends the reduction tree from the most complex to the most simple cases.
All thesereductions will correspond to take the case n = 0 for some Legendre polynomial. The following properties will thus be useful:
As a side note, Eq. (D.10) can be interpreted diagrammatically with the diagrams introduced in Appendix E: it corresponds to cutting a line, and on each extremity vertex letting the two concurring lines reconnect into a single one.
D.2.1 One angle
D.2.1.1 case
and the resulting covariance, derived in Appendix C.1.1, is (D.13)
where the angular trispectrum is (D.14)
For n = 0, Eq. (D.11) shows that the covariance equation reduces to the simplest trispectrum case (angle and diagonal independent) and forces ℓ_{a} = ℓ, implying that the angular trispectrum also reduces to the standard case.
D.2.1.2 The case of
and the resulting covariance, derived in Appendix C.1.1, is (D.16)
where the angular trispectrum is (D.17)
For n = 0, Eq. (D.11) shows that the covariance equation reduces to the simplest trispectrum case (angle and diagonal independent) and forces ℓ_{1} = ℓ and ℓ_{3} = ℓ′, implying that the angular trispectrum also reduces to the standard case.
D.2.1.3 The case of
and the resulting covariance, derived in Appendix C.1.2, is
where the angular trispectrum is (D.19)
For n = 0, Eq. (D.11) shows that the covariance equation reduces to the angle independent trispectrum case (depending on the squeezed diagonal) and forces ℓ_{3} = ℓ′, implying that the angular trispectrum also reduces to the appropriate case.
D.2.1.4 The case of
and the resulting covariance, derived in Appendix C.1.2, is (D.21)
where the geometric coefficient is (D.22)
and the angular trispectrum is given by (D.23)
For n = 0, Eq. (D.10) reduces the H coefficient to (D.24)
Diagrammatically, this reduction can be seen as the transformation of the tetrahedron of Fig. E.2 into an oat grain diagram (Fig. E.1) when the side n is cut. Inputting this to the covariance equation, one sees that the angular trispectrum reduces to the appropriate angle independent case (depending on the alternate diagonal), and the covariance is
as required.
D.2.2 Two angles
D.2.2.1 X × Y squeezed diagonal case
and the resulting covariance, derived in Appendix C.2.1, is (D.26)
where the angular trispectrum is (D.27)
For n = 0, Eq. (D.11) shows that the covariance equation reduces to the trispectrum case, and forces ℓ_{1} = ℓ, implying that the angular trispectrum also reduces to the appropriate case.
The manifest symmetry between n and n′ means that one does not have to check the case n′ = 0.
D.2.2.2 X × Y alternate diagonal case
and the resulting covariance, derived in Appendix C.2.1, is (D.29)
where the geometric coefficient is (D.30)
and the angular trispectrum is (D.31)
For n = 0, Eq. (D.10) reduces the J coefficient to (D.32)
Diagrammatically, this reduction can be seen as the transformation of the hexagon of Fig. E.3 into a tetrahedron diagram (Fig. E.2) when the side n is cut.
i.e. reducing to the case as required.
The manifest symmetry between n and n′ means that one does not have to check the case n′ = 0.
D.2.2.3 3PT squeezed diagonal case
and the resulting covariance, derived in Appendix C.2.2, is (D.34)
where the angular trispectrum is (D.35)
For n = 0, Eq. (D.11) shows that the covariance equation reduces to the trispectrum case, and forces ℓ_{1} = ℓ, implying that the angular trispectrum also reduces to the appropriate case.
For n′ = 0, Eq. (D.11), and using that ℓ + ℓ_{1} + n is even, shows that the covariance equation reduces to the trispectrum case. Furthermore it forces ℓ_{3} = ℓ′, together with Eq. (D.1), this implies that the angular trispectrum also reduces to the appropriate case.
D.2.2.4 3PT alternate diagonal case
and the resulting covariance, derived in Appendix C.2.2, is (D.37)
where the geometric coefficient is (D.38)
and the angular trispectrum is (D.39)
For n = 0, Eq. (D.10) reduces the K coefficient to (D.40)
Diagrammatically, this reduction can be seen as the transformation of the prism of Fig. E.4 into a tetrahedron diagram (Fig. E.2) when the side n is cut. Hence the covariance is
i.e. reducing to the case as required.
For n′ = 0, Eq. (D.10) reduces the K coefficient to (D.41)
Diagrammatically, this reduction can be seen as the transformation of the prism of Fig. E.4 into a tetrahedron diagram (Fig. E.2) when the side n is cut.
Then further reductions occur: the Kronecker symbols reduce the angular trispectrum to . But given that the 3D trispectrum does not depend on k_{1+3}, I can employ Eq. (D.1) to show that the angular trispectrum further reduces to , independent of ℓ_{a} and corresponding to the angular trispectrum required for the case. Now I can perform the sum over ℓ_{a} and use the identity (D.42)
So that, when performing the partial sum over (ℓ_{a}, ℓ_{b}) in the covariance equation, one finds
Hence the covariance equation becomes (D.43)
in other words, finally, reducing to the case as required.
With this final reduction, I have completed the reduction tree, that is, proved that all equations for the complex bispectrum case reduce to the simpler case when appropriate. Hence the covariance derivations presented in all the preceding appendices have successfully passed this sanity/consistency check.
Appendix E Relation of geometric coefficients with 3nJ symbols
The following relation between Gaunt coefficients and 3J symbols will be useful in this section: (E.1)
E.1 Diagrammatic of 3J symbols
Geometric coefficients found in the covariance derivation of the previous appendices can be represented diagrammatically, by representinga 3J symbol as a vertex from which start 3 lines labelled with the corresponding multipoles. This representation was devised by Yutsis et al. (1962), and in the following I am going to call these Yutsis diagrams.
Fig. E.1
Oatgrain diagram. 

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For example the identity (E.2)
appearing for example, in the covariance case derived in Appendix B.3, can be represented with the oat grain diagram shown in Fig. E.1.
E.2 and 6J symbols
6J symbols are defined as a sum over products of four 3J symbols^{12} (see also Eq. (34.4.1) of DLMF (2017) for an equivalent definition) (E.4)
Recall the definition of (E.5)
Fig. E.2
Tetrahedron corresponding to the 6J symbol involved in this article. 

Open with DEXTER 
If one uses Eq. (E.1) to relate Gaunt coefficients to 3J symbols, up to change of silent variables m →−m, one sees that H can be put in the form of a 6J symbol, namely (E.6)
The Yutsis diagram of this symbol is a tetrahedron, visible in Fig. E.2 where each vertex represents a 3J symbol.
E.3 and 9J symbols
9J symbols are defined as a sum over products of six 3J symbols (see Eq. (34.6.1) of DLMF (2017)^{13}) (E.7)
Recall the definition of (E.8)
Fig. E.3
Hexagon corresponding to the 9J symbol involved in this article. 

Open with DEXTER 
If one uses Eq. (E.1) to relate Gaunt coefficients to 3J symbols, up to change of silent variables m →−m and using the parity condition on azimuthal parameters (e.g. m_{1} + m_{2} + m_{3} = 0), one sees that J is in fact a 9J symbol, namely (E.9)
The Yutsis diagram of this symbol is an hexagon, visible in Fig. E.3.
E.4 and 12J symbols
Recall the definition of (E.10)
If one uses Eq. (E.1) to relate Gaunt coefficients to 3J symbols and the parity condition on azimuthal parameters, one sees that K can be rewritten as a sum of six 3J symbol, namely (E.11)
The corresponding diagram is a triangular prism visible in Fig. E.4.
Fig. E.4
Triangular prism corresponding to the K symbol involved in this article. 

Open with DEXTER 
This coefficient can not be written in terms of a 9J symbol nor a simple sum of a few 6J symbols. It can however be written as reduced 12J symbol of the second kind (without braiding), that is the symbol whose Yutsis diagram is a cube, reduced to the case where one multipole is zero. See, for instance, Ališauskas (2000) for a definition and a representation of this 12J symbol. Explicitly, one finds: (E.12)
Appendix F Shot noise
F.1 Case of overlapping redshift bins
Equation (35) for the shotnoise angular power spectrum assumed that the redshift bins were not overlapping. In the case of overlapping bins one would have instead (F.1)
where i_{z} ∩ j_{z} denotes the range of redshift overlap.
F.2 Poissonianity?
In Sect. 2.4, the power spectrum shotnoise term is given by Eq. (35), restated here for convenience (F.2)
This power spectrum is the same as the one of a random map with galaxies drawn independently with a Poisson distribution. Thus an underlying Poissonian assumption seems hidden in the modelling.
I clarifyhere that there is no such explicit assumption: the power spectrum shotnoise term simply results from its definition asthe part of the correlation function with galaxy coincidence (third diagram in Fig. 2).
In a map with N galaxies, the twopoint correlation function hits twice the same galaxy exactly N times. Equations (33) and (35) are simply a restatement of this fact, which has no underlying assumption.
In the modelling used in this article, the total power spectrum asymptotes to the shotnoise value as ℓ →∞. In practice in a given survey analysis, the power spectrum may not asymptote to the shotnoise value due either to data analysis or physical effects, see for example, Paech et al. (2017). One such physical effect is nonlinear clustering, whose modelisation is absent from Paech et al. (2017). In the present article, nonlinear clustering is given by the onehalo term of the power spectrum. This term is constant on scales larger than the typical halo radius, so that a survey limited to large scale would indeed see a constant component in the power spectrum with value larger than the shotnoise value: “superPoissonian noise”. Another such physical effect is halo and/or galaxy exclusion, which is not modelled in this article and would lead to an anticlustering on halo and/or galaxy scales: “subPoissonian noise”.
Hence in a survey the high frequency limit of the power spectrum may not be given by the shotnoise value. However I refrain from calling this a nonPoissonian shotnoise or similar, as this would just create confusion in my opinion. Instead I reserve the word shot noise for the discreteness effect given by Eqs. (33) and (35), and call other high frequency terms by the physical effect they originate from (nonlinear clustering, halo and/or galaxy exclusion…). Finally, I note that as in Sect. 7.4, the shotnoise value can be subtracted from the measured power spectrum, and thus the measurement can be used to constraint these high frequency effects and the underlying physics.
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All Figures
Fig. 1
3D trispectrum in the squeezed diagonal limit. 

Open with DEXTER  
In the text 
Fig. 2
Diagrams for the galaxy power spectrum. From left to right: twohalo (2h), onehalo (1h) and shotnoise (shot). 

Open with DEXTER  
In the text 
Fig. 3
Diagrams for the galaxy trispectrum. From left to right, top row: fourhalo (4h), threehalo (3h), threehalo shotnoise (3hshot3g), twohalo 1+3 (2h1+3). Middle row: twohalo 2+2 (2h2+2), twohalo threegalaxy shotnoise a (2hashot3g), twohalo threegalaxy shotnoise b (2hbshot3g), twohalo twogalaxy shotnoise a (2hashot2g), twohalo twogalaxy shotnoise b (2hbshot2g). Bottom row: onehalo (1h), onehalo threegalaxy shotnoise (1hshot3g), onehalo twogalaxy shotnoise a (1hashot2g), onehalo twogalaxy shotnoise b (1hbshot2g), onegalaxy shotnoise (shot1g). 

Open with DEXTER  
In the text 
Fig. E.1
Oatgrain diagram. 

Open with DEXTER  
In the text 
Fig. E.2
Tetrahedron corresponding to the 6J symbol involved in this article. 

Open with DEXTER  
In the text 
Fig. E.3
Hexagon corresponding to the 9J symbol involved in this article. 

Open with DEXTER  
In the text 
Fig. E.4
Triangular prism corresponding to the K symbol involved in this article. 

Open with DEXTER  
In the text 
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