Information Content of Higher-Order Galaxy Correlation Functions
Lado Samushia, Zachary Slepian, Francisco Villaescusa-Navarro
MMNRAS , 1–14 (2020) Preprint 4 February 2021 Compiled using MNRAS L A TEX style file v3.0
Information Content of Higher-Order Galaxy Correlation Functions
Lado Samushia, , ∗ Zachary Slepian, , † & Francisco Villaescusa-Navarro , ‡ Department of Physics, Kansas State University, 116 Cardwell Hall, 1228 N. 17 𝑡ℎ St., Manhattan, KS 66506, USA Abastumani Astrophysical Observatory, Tbilisi, GE-0179, Georgia Department of Astronomy, University of Florida, 211 Bryant Space Science Center, Gainesville, FL 32611 Physics Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94709 Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton NJ 08544, USA Center for Computational Astrophysics, Flatiron Institute, 162 5 𝑡ℎ Ave., New York, NY 10010, USA
ABSTRACT
The shapes of galaxy 𝑁 -point correlation functions can be used as standard rulers to constrain the distance-redshift relationshipand thence the expansion rate of the Universe. The cosmological density fields traced by late-time galaxy formation are initiallynearly Gaussian, and hence all the cosmological information can be extracted from their 2-Point Correlation Function (2PCF)or its Fourier-space analog the power spectrum. Subsequent nonlinear evolution under gravity, as well as halo and then galaxyformation, generate higher-order correlation functions. Since the mapping of the initial to the final density field is, on largescales, invertible, it is often claimed that the information content of the initial field’s power spectrum is equal to that of allthe higher-order functions of the final, nonlinear field. This claim implies that reconstruction of the initial density field fromthe nonlinear field renders analysis of higher-order correlation functions of the latter superfluous. We here show that this claimis false when the 𝑁 -point functions are used as standard rulers . Constraints available from joint analysis of the galaxy powerspectrum and bispectrum (Fourier-space analog of the 3-Point Correlation Function) can, in some cases, exceed those offered bythe initial power spectrum even when the reconstruction is perfect. We provide a mathematical justification for this claim and alsodemonstrate it using a large suite of 𝑁 -body simulations. In particular, we show that for the 𝑧 = 𝑘 max = . ℎ / Mpc, using the bispectrum alone offers a factor of two reductionin the variance on the cosmic distance scale relative to that available from the power spectrum. If these conclusions extend to theredshift-space galaxy density field, they suggest that future large-scale structure surveys may yet be able to significantly tightendark energy constraints by design focus on higher number densities.
Key words: cosmological parameters – dark energy – distance scale – large-scale structure of Universe – methods: statistical –cosmology: theory
In the consensus picture of cosmological structure formation, quan-tum fluctuations before inflation seeded cosmological density fieldsthat were then amplified by gravitational instability (e.g. Starobin-sky 1982; Bardeen et al. 1983). These amplified density fields aretraced by galaxies at lower redshifts, and their initial stochasticitypropagates to the late-time distribution of galaxies. The statisticalproperties of the late-time galaxy distribution can be described by aseries of 𝑁 -Point Correlation Functions (NPCFs), (cid:104) 𝛿 ( r ) 𝛿 ( r )(cid:105) ≡ 𝜉 ( r − r ) , (cid:104) 𝛿 ( r ) 𝛿 ( r ) 𝛿 ( r )(cid:105) ≡ 𝜁 ( r − r , r − r ) , (1) (cid:104) 𝛿 ( r ) . . . 𝛿 ( r )(cid:105) ≡ 𝜂 ( r − r , . . . , r − r ) ,. . . , ∗ E-mail: [email protected] † E-mail: zslepian@ufl.edu ‡ E-mail: [email protected] where 𝛿 ( r ) is the density fluctuation at position r and the anglebrackets denote ensemble averages over realizations (e.g. Peebles1980). Translational invariance forces the correlation functions todepend only on the difference between position vectors. For a ran-dom field that is sufficiently close to Gaussian, which is the case onlarge scales, a collection of all correlation functions contains all ofthe non-stochastic information about the field. It is often convenient In detail, the observed galaxy density field is subject to redshift-space dis-tortions (RSD) due to peculiar velocities, which are not translation-invariantbut rather spherically symmetric about the observer (e.g. Hamilton & Culhane1996). However, the induced breaking of translation symmetry is small. It has been noted that on small scales, knowing all of the zero-lag correlationfunctions (the moments) is not necessarily sufficient for reconstructing theone-point density PDF (Carron 2011; Carron & Neyrinck 2012; Carron &Szapudi 2017). However our goal here is not to reconstruct the one-point PDFfrom its cumulants, but rather to use the higher-order correlation functions asa standard ruler for the Universe’s expansion. © a r X i v : . [ a s t r o - ph . C O ] F e b Samushia et al. to analyze the Fourier Transforms (FTs) of the NPCFs, defined as 𝑃 ( k ) ≡ F { 𝜉 ( r )} ,𝐵 ( k , k (cid:48) ) ≡ F { 𝜁 ( r , r (cid:48) )} , (2) 𝑇 ( k , k (cid:48) , k (cid:48)(cid:48) ) ≡ F { 𝜂 ( r , r (cid:48) , r (cid:48)(cid:48) )} ,. . . , where F denotes a multivariate FT. The functions on the left-handside are referred to as polyspectra. The three lowest order polyspec-tra are the power spectrum, the bispectrum, and the trispectrum.For the full range of scales (and wave-vectors), and in the absenceof measurement systematics, binning or averaging, the informationcontent of the polyspectra is equivalent to that of the correlation func-tions (see e.g. Hoffmann et al. 2018 for discussion of this point forthe 3PCF/bispectrum). By information content we mean the Fisherinformation (Fisher 1922; Cramer 1999)—the precision to whichcosmological parameters can be inferred from measured NPCFs orpolyspectra. We now present two different ways in which the polyspectra canbe used for cosmology. First, at any given redshift, a polyspectrum’sshape as a function of the relevant wave-vectors will encode some in-formation about the cosmological parameters and the law of gravity.In particular, high-redshift physics such as Baryon Acoustic Oscilla-tions (BAO), the transition between radiation and matter-dominationsetting the equality scale , the contribution to the Hubble expan-sion by the effective energy density of relativistic species ( 𝑁 eff ), andSilk damping setup the initial shape of the polyspectra, especiallythe linear-theory power spectrum. As time goes on, the nonlinearinteraction of density perturbations with each other produces bispec-trum, trispectrum, etc. but these are simply set by integrals of simplekernels against the linear power spectrum. These kernels come fromperturbatively solving fluid equations governing the matter, and aresurprisingly insensitive to the cosmological parameters. Hence, the 𝑠ℎ𝑎𝑝𝑒 of the matter polyspectra is set primarily by high-redshiftphysics, and the “direct” information they contain on the cosmolog-ical parameters ultimately comes from how those parameters shapethe linear power spectrum at high redshift.However, the polyspectra can also be used indirectly, as standardrulers. In this use, they probe the overall expansion of the Universeover time, which in turn is sensitive to the cosmological parame-ters, as they dictate the behavior of the Friedmann equation. To usepolyspectra as standard rulers, one takes the measured shape at agiven redshift or redshifts and compares with the fiducial shape. Thisreveals both the distance to galaxies and the expansion rate of theUniverse at that time from how much the shape has dilated. Thislatter use of polyspectra is the focus of the present work.For completeness, we now briefly review previous use of thepolyspectra in both of the ways outlined above. We then separatelydiscuss density-field reconstruction. This produces a knee in the matter transfer function at 𝑘 eq ∼ . ℎ / Mpc . Indeed, we use the matter-dominated form for them even despite darkenergy, and it differs by less than a quarter of a percent from the matter-plus-dark energy form even at 𝑧 = We first discuss previous work on direct use of the shape to con-strain cosmology. The amplitude of the polyspectra at different wave-vectors is a function of the cosmological parameters. Consequently,unbiased and precise measurements of polyspectra can be used toinfer the values of the cosmological parameters. For instance, thewavelength of the BAO as viewed in Fourier space (a decaying os-cillatory feature in the polyspectra, e.g. Sunyaev & Zeldovich 1970;Peebles & Yu 1970; Eisenstein & Hu 1998a; Meiksin et al. 1999;Slepian & Eisenstein 2016) is set by the sound horizon at decoupling ( 𝑧 ∼ ) , which in turn is sensitive to the baryon and photondensities as well as the integrated expansion of the Universe (drivenprimarily by radiation and matter) of the Universe up to that point(e.g. Hu & Sugiyama 1995, 1996). As another example, the totalmass of neutrinos, 𝑚 𝜈 , changes the amplitude of the spectra at highwave-numbers (Bond et al. 1980; Hu et al. 1998; Lesgourgues &Pastor 2006; Abdalla & Rawlings 2007; Saito et al. 2009; Agarwal &Feldman 2011), and 𝑁 eff changes the phase (Baumann et al. 2017).As earlier mentioned, the polyspectras’ intrinsic shapes are setat early times and cosmological parameters that affect later-timeevolution of the Universe do not imprint characteristic scales on them.For instance, small amounts of spatial curvature, Ω K , parametersdescribing deviations from General Relativity (GR), and propertiesof dark energy, such as its energy density, Ω Λ , and the current valueof its equation of state, 𝑤 , do not strongly affect the polyspectras’average shape at a given redshift.Nonetheless, some “direct” constraints on these parameters canstill be derived from the dependence of the polyspectra on the angleof its wave-vectors to the line of sight thanks to Redshift Space Dis-tortions (RSD) imprinted by velocity fields (Hamilton 1998; Guzzoet al. 2008; Percival et al. 2011; Jennings et al. 2012; Linder 2013;Vlah & White 2019; Hernández-Aguayo et al. 2019; Wright et al.2019; García-Farieta et al. 2019; Gagrani & Samushia 2017a; Slepian& Eisenstein 2018; Sugiyama et al. 2019a; Sugiyama et al. 2020c;Garcia & Slepian 2020; Kamalinejad & Slepian 2020a).In addition to constraining the cosmological parameters, the in-trinsic shapes of galaxy polyspectra can be used to constrain biasparameters describing the connection between galaxies and theirhost halos (Sefusatti & Scoccimarro 2005; Gaztañaga et al. 2005;Sefusatti & Komatsu 2007; Guo & Jing 2009; Gaztañaga et al. 2009;Manera & Gaztañaga 2011; Marín 2011; McBride et al. 2011; Marínet al. 2013; Yuan et al. 2017, 2018). We note that it may also bepossible to extract these constraints without explicitly computing theNPCF by fitting the observed field directly (Jasche & Kitaura 2010;Jasche & Wandelt 2013; Seljak et al. 2017; Leclercq et al. 2019;Schmittfull et al. 2019; Horowitz et al. 2019; Villaescusa-Navarroet al. 2020a). Fortunately for us, the measured spectra can also be used as standardrulers (§2). We can compare apparent scales of different features inthe spectra to the ones expected from a model to infer how far awaythe galaxies at various redshifts are from us (Roukema & Mamon2000; Park & Kim 2010; Weinberg et al. 2013; Ntelis et al. 2018;Anselmi et al. 2019; Wagoner et al. 2020). This distance-redshiftrelationship is very sensitive to dark energy parameters that drivethe late-time time evolution of the Universe (Glazebrook & Blake2003; Samushia et al. 2011). This technique has become standardfor the power spectrum and 2PCF, and has offered much stronger
MNRAS , 1–14 (2020) igher-Order Clustering constraints on dark energy compared to what we would get fromthe direct dependence of the shape on the cosmological parameters(Aubourg et al. 2015; eBOSS Collaboration et al. 2020). Indeed, thedependence of the intrinsic shape of the power spectrum on darkenergy parameters is very mild, and BAO constraints on dark energyin fact come from the usage of the power spectrum and 2PCF asstandard rulers. Higher-order polyspectra are significantly more challenging to mea-sure and analyze. The difficulties range from computational (the algo-rithms are computationally expensive, though recent work by Slepian& Eisenstein (2015a,b); Portillo et al. (2018); Slepian & Eisenstein(2018); Pearson & Samushia (2019); Nuñez & Niz (2020); Garcia &Slepian (2020) has improved this for the 3PCF and anisotropic 3PCF,and augurs to do so for NPCFs as well (Cahn & Slepian 2020, Slepianet al. in prep. 2020). A graph database approach was developed bySabiu et al. (2019) and improves the speed of measuring e.g the 4PCF.Tomlinson et al. (2019) also provides a new fast scheme for measur-ing polyspectra, though it averages over the internal angles and tracksonly the side lengths. Algorithmic improvements have recently beenmade for the bispectrum Scoccimarro (2015); Pearson & Samushia(2018b); Sugiyama et al. (2019a). Computing theory predictions isalso a challenge of using polyspectra (perturbation theory does notconverge nearly as well for higher orders). Higher-order polyspec-tra have higher dimensionality (more wave-vectors to track), whichcreates additional practical difficulties in computing their covariancematrices.The power spectrum of galaxies has been used extensively (bothas a standard ruler and through its intrinsic shape) to derive high-precision cosmological constraints (Cole et al. 2005; Beutler et al.2011; Blake et al. 2012; Anderson et al. 2012; Howlett et al. 2015;Ross et al. 2015; Aubourg et al. 2015; Alam et al. 2017; Abbottet al. 2018, 2019; eBOSS Collaboration et al. 2020). A similar bis-pectrum analysis has been attempted in recent works (Scoccimarroet al. 2001; Gil-Marín et al. 2015a,b). Slepian et al. (2017) and Pear-son & Samushia (2018a) used the BAO feature in the 3PCF as astandard ruler to constrain the distance-redshift relationship. Slepian& Eisenstein (2018) proposed a convenient basis for the anisotropic3PCF, with improvements to the line of sight in Garcia & Slepian(2020), and Sugiyama et al. (2019b) proposed a similar basis for theanisotropic bispectrum and Sugiyama et al. (2020a) used it to mea-sure a BAO feature in BOSS-like mock catalogues. Slepian et al.(2015) measured the BOSS 3PCF in a compressed basis, and Slepianet al. (2018) used the BOSS 3PCF to put the tightest current con-straints on how high-redshift baryon-dark matter relative velocitiesbias galaxy formation. Gualdi et al. (2019a) and Gualdi et al. (2019b)used the compressed BOSS bispectrum to obtain cosmological con-straints.Sabiu et al. (2019) measured the BOSS 4PCF using a graph-database approach, and this is the only recent work of which we areaware measuring quadruplet correlations (thee Fourier space analogof which is the trispectrum). There has been theoretical work onthe trispectrum (e.g. Bertolini et al. (2016) but it is challenging to BOSS is an acronym for the Baryon Oscillation Spectroscopic Survey(Dawson et al. 2013). Our focus here is recent work, but there were a handful of papers measuringthe 3PCF or bispectrum and 4PCF from the 1960s-1980s, reviewed in Peebles(2001), beginning on page 6. pursue and hence compressions have recently been suggested (Mun-shi et al. 2011; Gualdi et al. 2020). Recent works have suggestedthat adding just the bispectrum to the standard cosmological anal-ysis has the potential to significantly tighten derived cosmologicalconstraints (Gagrani & Samushia 2017b; Karagiannis et al. 2018;Hahn et al. 2020; Hahn & Villaescusa-Navarro 2020; MoradinezhadDizgah et al. 2020; Gualdi & Verde 2020; Ravenni et al. 2020; Ka-malinejad & Slepian 2020b).At early times, for example, for cosmic microwave background(CMB) anisotropy data, the problem of lost information in higherorders is not as acute. Early cosmological fields are believed to be veryclose to being Gaussian (Bartolo et al. 2004; Planck Collaborationet al. 2014, 2016, 2020b; Gómez & Jimenez 2020). They are fullydescribed by their power spectrum; all higher orders are either zeroor are derivable from the two-point function via Isserlis’ Theorem(Isserlis 1918) (also know as Wick’s Theorem). We do note thatas the precision of the measurements increases, higher-order effectsdo become relevant even for these highly Gaussian fields, e.g. inthe analysis CMB lensing by the foreground structure (Blanchard &Schneider 1987; Lewis & Challinor 2006; Lewis et al. 2011; Böhmet al. 2016, 2018; Planck Collaboration et al. 2020a).
One approach to simplifying the analysis is to pursue “reconstruct-ing” the initial, nearly-Gaussian, version of the observed galaxy field(Monaco & Efstathiou 1999; Mohayaee et al. 2006; Padmanabhanet al. 2009b; Noh et al. 2009). This procedure has been successfullyperformed on galaxy samples and is part of the standard clusteringanalysis toolkit of galaxy surveys (Eisenstein et al. 2007b; Padman-abhan et al. 2009a; Padmanabhan et al. 2012; Xu et al. 2012; Kazinet al. 2014). It has been shown to make the BAO feature of the powerspectrum into a much sharper standard ruler (Eisenstein et al. 2007a).The reconstructed field is nearly Gaussian and there is very little lostby ignoring the leftover higher-order polyspectra. Reconstruction hasthe additional advantage of decorrelating the measured power spec-trum at each different wave-number, making the covariance matrixcalculations simpler (Ngan et al. 2012).The actual reconstruction algorithms are by no means trivial, andmuch recent work has focused on increasing their efficiency (Croft& Gaztanaga 1997; Tassev & Zaldarriaga 2012; Jasche & Wandelt2013; Wang et al. 2013; Burden et al. 2014; White 2015; Burdenet al. 2015; Achitouv & Blake 2015; Sherwin & White 2019a). In thiswork, we will not be concerned with the practicalities of the approach.We will assume that by some means the perfect reconstruction hasbeen achieved, i.e. the galaxies are at the same distance from us asthey were before the reconstruction, but the field was evolved backto its Gaussian version. Even in this case the field will have somenon-Gaussian features generated by non-linear biasing of galaxies(Bardeen et al. 1986; Fry & Gaztanaga 1993; Bernardeau 1996;Frieman & Gaztañaga 1999; Sheth & Tormen 1999), but we willassume that on large enough scales, where galaxy bias is almostlinear, these effects are subdominant.If nearly perfect reconstruction were achievable, one might askif there were any point in going through a complicated analysisof higher-order statistics. The bispectrum would still be useful inconstraining certain features of the cosmological model (e.g. neu-trino signatures at large wave-numbers, higher-order biasing, or non-Gaussian features in the initial conditions). But would it add anythingmeaningful to the dark energy constraints coming from large-scaleclustering (see e.g. discussion in Schmittfull et al. 2015)?After all, since the mapping from the final to the initial field is
MNRAS , 1–14 (2020)
Samushia et al. invertible on large scales, it seems obvious that no information canbe lost in the process of reconstruction. And, since the initial fieldis fully described by its power spectrum, it is often claimed that thepower spectrum of the perfectly reconstructed field would be as goodof a standard ruler as 𝑎𝑙𝑙 of the higher-order spectra of the originalnonlinear field.The main objective of this paper is to show that this latter claimis false. While the reconstruction conserves the information in somesense, the power spectrum of the Gaussian field is not always a betterstandard ruler than the combined polyspectra of the nonlinear field(§2). The series of nonlinear spectra can, in some circumstances,turn out to be more sensitive standard rulers and provide a strongerconstraint on dark energy parameters (§3).
We will denote the ordered hierarchy of polyspectra by
S ≡ (cid:104) 𝑃 (cid:16) 𝑘 𝑃 (cid:17) , 𝐵 (cid:16) 𝑘 𝐵 , 𝑘 𝐵 (cid:17) , 𝑇 ( 𝑘 𝑇 , 𝑘 𝑇 , 𝑘 𝑇 ) , . . . (cid:105) , (3)where the superscripts 𝑘 explicitly identify the parent functions ofeach argument (P for power spectrum, B for bispectrum, T for trispec-trum). We will use a superscript 𝐼 for the initial (Gaussian) field and 𝐹 for the final (nonlinear) field. S is a function of series of wave-numbers (a single wave-number for the power spectrum, pairs ofwave-numbers for the bispectrum, etc.). We will denote a collectionof these wave-numbers as k = (cid:104) 𝑘 𝑃 , 𝑘 𝐵 , 𝑘 𝐵 , 𝑘 𝑇 , 𝑘 𝑇 , 𝑘 𝑇 , . . . (cid:105) . (4) 𝑆 𝐼 is then just the initial power spectrum, but 𝑆 𝐹 goes through allhigher orders. We will assume that on large scales they are bothinvertible as functions of the cosmological parameters 𝜽 . If that isthe case, they can be uniquely mapped to each other by S 𝐹 ( k , 𝜽 ) = S 𝐹 (cid:16) S 𝐼 ( k (cid:48) , 𝜽 ) , k , 𝜽 (cid:17) . (5)The above equation states that given for all cosmological parametervalues, all polyspectra of the nonlinear field can be computed giventhe initial power spectrum. We put k (cid:48) as an argument of S 𝐼 to high-light the fact that gravitational evolution mixes modes and the finalpolyspectra at a specific wave-number depending on the initial powerspectrum in a wide range of wave-numbers. We primed the argumentto ensure that it does not get confused with a similar argument of S 𝐹 from which it is distinct (e.g. it does not contribute to the derivativesof S 𝐹 with respect to the wave-number). The 𝜽 , on the other hand,have the same values in both places they appear.The functional dependence (5) is highly nonlinear and is impossi-ble to write down in a closed analytic form. Good approximations toit can be achieved by means of perturbation theory series or 𝑁 -bodysimulations. In Eulerian Standard Perturbation Theory (SPT) up to If we require a superscript for another designation, in some contexts 𝐼 and 𝐹 will appear in the subscript. second order the equivalent of equation (5) is 𝑃 𝐹 ( k ) = 𝐷 𝑃 𝐼 ( k )+ 𝐷 ∫ (cid:2) 𝐹 ( k − k (cid:48) , k (cid:48) ) (cid:3) 𝑃 𝐼 ( k − k (cid:48) ) 𝑃 𝐼 ( k (cid:48) ) d k (cid:48) + 𝐷 ∫ 𝐹 ( k , k (cid:48) , − k (cid:48) ) 𝑃 𝐼 ( k ) 𝑃 𝐼 ( k (cid:48) ) d k (cid:48) ,𝐵 𝐹 ( k , k ) = 𝐷 𝐹 ( k , k ) 𝑃 𝐼 ( k ) 𝑃 𝐼 ( k ) + cyclic .𝐷 is the linear growth factor; the form of the 𝐹 𝑛 kernels is given ine.g. Bernardeau et al. (2002).In real surveys, polyspectra cannot be measured with arbitrarilyfine binning in k . The finite volume only lets us measure them atdiscrete sets of modes. From now on, we will write them as vectors S 𝑖 ( 𝜽 ) where the subscript index runs over wave-number bins (orsets of wave-number bins for higher-order polyspectra) with somearbitrary ordering.We take it that the binned final polyspectra are measured withinsome uncertainty described by a covariance matrix, (cid:104) 𝛿 S 𝐹𝑖 𝛿 S 𝐹𝑗 (cid:105) = (cid:104) 𝐶 𝑆𝐹 (cid:105) 𝑖 𝑗 . (6)The subscript 𝐹 denotes “final” field and the superscript S (on therighthand side) that it is the covariance of a polyspectrum S . Thebest possible uncertainty within which the initial power spectrum canbe reconstructed, and its covariance, are then given respectively by 𝛿 S 𝐼𝑖 ∼ ∑︁ 𝑗 𝜕 S 𝐼𝑖 𝜕 S 𝐹𝑗 𝛿 S 𝐹𝑗 , (7) (cid:104) 𝐶 𝑆𝐼 (cid:105) 𝑖 𝑗 ≡ (cid:104) 𝛿 S 𝐼𝑖 𝛿 S 𝐼𝑗 (cid:105) ∼ ∑︁ 𝑘ℓ 𝜕 S 𝐼𝑖 𝜕 S 𝐹𝑘 𝜕 S 𝐼𝑗 𝜕 S 𝐹ℓ (cid:104) 𝐶 𝑆𝐹 (cid:105) 𝑘ℓ , (8)where C 𝑆𝐼 is a close to diagonal matrix, since the power spectrumof the initial Gaussian field is not correlated across wave-numbers.We will assume that the reconstruction procedure is perfect and theactual covariance of the reconstructed polyspectra is not significantlydifferent from the one in equation (8). We will also assume that thereconstruction is free of biases (e.g. the bias from assuming a wrongcosmological model, survey window effects, etc.). This issue in thecontext of BAO reconstruction is discussed in Sherwin & White(2019b).The best precisions one can get for the cosmological parametersfrom, respectively, the initial and final polyspectra are (cid:104) 𝐶 𝜃𝐼 (cid:105) − 𝑘ℓ = ∑︁ 𝑖 𝑗 𝜕 S 𝐼𝑖 𝜕𝜃 𝑘 𝜕 S 𝐼𝑗 𝜕𝜃 ℓ (cid:104) 𝐶 𝑆𝐼 (cid:105) − 𝑖 𝑗 , (9) (cid:104) 𝐶 𝜃𝐹 (cid:105) − 𝑘ℓ = ∑︁ 𝑖 𝑗 𝜕 S 𝐹𝑖 𝜕𝜃 𝑘 𝜕 S 𝐹𝑗 𝜕𝜃 ℓ (cid:104) 𝐶 𝑆𝐹 (cid:105) − 𝑖 𝑗 . (10)To compare these precisions we need to relate the derivatives of theinitial and the final polyspectra with respect to cosmological param-eters (i.e. the sensitivity to cosmology) bin by bin. From equation (5)this relationship is 𝑑 S 𝐹𝑖 𝑑 𝜽 = ∑︁ 𝑗 𝜕 S 𝐹𝑖 𝜕 S 𝐼𝑗 𝜕 S 𝐼𝑗 𝜕 𝜽 + 𝜕 S 𝐹𝑖 𝜕 𝜽 . (11) We will denote all covariance matrices by C . The superscript will be usedto indicate what quantity this covariance matrix is of, and the subscript willindicate the origin of the measurement. E.g. C 𝑎𝑏 indicates a covariance of 𝑎 when it is inferred from 𝑏 .MNRAS000
S ≡ (cid:104) 𝑃 (cid:16) 𝑘 𝑃 (cid:17) , 𝐵 (cid:16) 𝑘 𝐵 , 𝑘 𝐵 (cid:17) , 𝑇 ( 𝑘 𝑇 , 𝑘 𝑇 , 𝑘 𝑇 ) , . . . (cid:105) , (3)where the superscripts 𝑘 explicitly identify the parent functions ofeach argument (P for power spectrum, B for bispectrum, T for trispec-trum). We will use a superscript 𝐼 for the initial (Gaussian) field and 𝐹 for the final (nonlinear) field. S is a function of series of wave-numbers (a single wave-number for the power spectrum, pairs ofwave-numbers for the bispectrum, etc.). We will denote a collectionof these wave-numbers as k = (cid:104) 𝑘 𝑃 , 𝑘 𝐵 , 𝑘 𝐵 , 𝑘 𝑇 , 𝑘 𝑇 , 𝑘 𝑇 , . . . (cid:105) . (4) 𝑆 𝐼 is then just the initial power spectrum, but 𝑆 𝐹 goes through allhigher orders. We will assume that on large scales they are bothinvertible as functions of the cosmological parameters 𝜽 . If that isthe case, they can be uniquely mapped to each other by S 𝐹 ( k , 𝜽 ) = S 𝐹 (cid:16) S 𝐼 ( k (cid:48) , 𝜽 ) , k , 𝜽 (cid:17) . (5)The above equation states that given for all cosmological parametervalues, all polyspectra of the nonlinear field can be computed giventhe initial power spectrum. We put k (cid:48) as an argument of S 𝐼 to high-light the fact that gravitational evolution mixes modes and the finalpolyspectra at a specific wave-number depending on the initial powerspectrum in a wide range of wave-numbers. We primed the argumentto ensure that it does not get confused with a similar argument of S 𝐹 from which it is distinct (e.g. it does not contribute to the derivativesof S 𝐹 with respect to the wave-number). The 𝜽 , on the other hand,have the same values in both places they appear.The functional dependence (5) is highly nonlinear and is impossi-ble to write down in a closed analytic form. Good approximations toit can be achieved by means of perturbation theory series or 𝑁 -bodysimulations. In Eulerian Standard Perturbation Theory (SPT) up to If we require a superscript for another designation, in some contexts 𝐼 and 𝐹 will appear in the subscript. second order the equivalent of equation (5) is 𝑃 𝐹 ( k ) = 𝐷 𝑃 𝐼 ( k )+ 𝐷 ∫ (cid:2) 𝐹 ( k − k (cid:48) , k (cid:48) ) (cid:3) 𝑃 𝐼 ( k − k (cid:48) ) 𝑃 𝐼 ( k (cid:48) ) d k (cid:48) + 𝐷 ∫ 𝐹 ( k , k (cid:48) , − k (cid:48) ) 𝑃 𝐼 ( k ) 𝑃 𝐼 ( k (cid:48) ) d k (cid:48) ,𝐵 𝐹 ( k , k ) = 𝐷 𝐹 ( k , k ) 𝑃 𝐼 ( k ) 𝑃 𝐼 ( k ) + cyclic .𝐷 is the linear growth factor; the form of the 𝐹 𝑛 kernels is given ine.g. Bernardeau et al. (2002).In real surveys, polyspectra cannot be measured with arbitrarilyfine binning in k . The finite volume only lets us measure them atdiscrete sets of modes. From now on, we will write them as vectors S 𝑖 ( 𝜽 ) where the subscript index runs over wave-number bins (orsets of wave-number bins for higher-order polyspectra) with somearbitrary ordering.We take it that the binned final polyspectra are measured withinsome uncertainty described by a covariance matrix, (cid:104) 𝛿 S 𝐹𝑖 𝛿 S 𝐹𝑗 (cid:105) = (cid:104) 𝐶 𝑆𝐹 (cid:105) 𝑖 𝑗 . (6)The subscript 𝐹 denotes “final” field and the superscript S (on therighthand side) that it is the covariance of a polyspectrum S . Thebest possible uncertainty within which the initial power spectrum canbe reconstructed, and its covariance, are then given respectively by 𝛿 S 𝐼𝑖 ∼ ∑︁ 𝑗 𝜕 S 𝐼𝑖 𝜕 S 𝐹𝑗 𝛿 S 𝐹𝑗 , (7) (cid:104) 𝐶 𝑆𝐼 (cid:105) 𝑖 𝑗 ≡ (cid:104) 𝛿 S 𝐼𝑖 𝛿 S 𝐼𝑗 (cid:105) ∼ ∑︁ 𝑘ℓ 𝜕 S 𝐼𝑖 𝜕 S 𝐹𝑘 𝜕 S 𝐼𝑗 𝜕 S 𝐹ℓ (cid:104) 𝐶 𝑆𝐹 (cid:105) 𝑘ℓ , (8)where C 𝑆𝐼 is a close to diagonal matrix, since the power spectrumof the initial Gaussian field is not correlated across wave-numbers.We will assume that the reconstruction procedure is perfect and theactual covariance of the reconstructed polyspectra is not significantlydifferent from the one in equation (8). We will also assume that thereconstruction is free of biases (e.g. the bias from assuming a wrongcosmological model, survey window effects, etc.). This issue in thecontext of BAO reconstruction is discussed in Sherwin & White(2019b).The best precisions one can get for the cosmological parametersfrom, respectively, the initial and final polyspectra are (cid:104) 𝐶 𝜃𝐼 (cid:105) − 𝑘ℓ = ∑︁ 𝑖 𝑗 𝜕 S 𝐼𝑖 𝜕𝜃 𝑘 𝜕 S 𝐼𝑗 𝜕𝜃 ℓ (cid:104) 𝐶 𝑆𝐼 (cid:105) − 𝑖 𝑗 , (9) (cid:104) 𝐶 𝜃𝐹 (cid:105) − 𝑘ℓ = ∑︁ 𝑖 𝑗 𝜕 S 𝐹𝑖 𝜕𝜃 𝑘 𝜕 S 𝐹𝑗 𝜕𝜃 ℓ (cid:104) 𝐶 𝑆𝐹 (cid:105) − 𝑖 𝑗 . (10)To compare these precisions we need to relate the derivatives of theinitial and the final polyspectra with respect to cosmological param-eters (i.e. the sensitivity to cosmology) bin by bin. From equation (5)this relationship is 𝑑 S 𝐹𝑖 𝑑 𝜽 = ∑︁ 𝑗 𝜕 S 𝐹𝑖 𝜕 S 𝐼𝑗 𝜕 S 𝐼𝑗 𝜕 𝜽 + 𝜕 S 𝐹𝑖 𝜕 𝜽 . (11) We will denote all covariance matrices by C . The superscript will be usedto indicate what quantity this covariance matrix is of, and the subscript willindicate the origin of the measurement. E.g. C 𝑎𝑏 indicates a covariance of 𝑎 when it is inferred from 𝑏 .MNRAS000 , 1–14 (2020) igher-Order Clustering The second term in this expression describes how much the mappingbetween initial and final spectra itself depends on cosmological pa-rameters. In cosmological models without dark energy this derivativeis negligible; to compute evolved polyspectra at all wave-numbers,one only needs to know the linear power spectrum but not the value of Ω m (Martel & Freudling 1991; Bernardeau 1994). In the presence ofa time-independent cosmological constant this is not formally true,but the actual dependence of the mapping on Ω m and Ω Λ is ex-tremely weak (Scoccimarro et al. 1998; Fosalba & Gaztanaga 1998).This also seems to be the case for time-dependent dark energy modelson large scales (McDonald et al. 2006).Hence, on large scales and for conventional cosmological models,the above equation simplifies to 𝑑 S 𝐹𝑖 𝑑 𝜽 = ∑︁ 𝑗 𝜕 S 𝐹𝑖 𝜕 S 𝐼𝑗 𝜕 S 𝐼𝑗 𝜕 𝜽 . (12)Combining this result with equations (7) and (8) ensures that the cos-mological constraints derivable from the initial and final polyspectragiven by 𝐶 𝜽 𝐼 and 𝐶 𝜽 𝐹 are exactly equal. This is the sense in whichthe information is conserved by the reconstruction. The origin ofthis conservation is also clear. The covariance and the sensitivityof the polyspectra scale as inverses of each other under reconstruc-tion. This conforms with our intuition that invertible transformationsshould conserve information.It is worth reflecting on what equations (9) and (10) actually mean.They both tell us that if we were given a box with a galaxy field,from which we measure polyspectra with some accuracy, the cos-mological constraints derivable from those polyspectra depend ontwo things: how well one measures each spectrum at different wave-numbers, and how sensitive that wave-number is to the cosmologicalparameters (i.e. by how much would it change if we changed 𝜽 bya small amount). The procedure is that we are fitting to each modeindependently (except that the cross-correlation in measurements isaccounted for in the covariance matrices) and are summing up theconstraint coming from all the modes of all the polyspectra. Theseequations are appropriate when the fits are performed on the intrin-sic shape of the polyspectra (e.g. constraining the amplitude andamplitude-like signals such as that of the neutrino mass). However,they are not appropriate when the polyspectra are used as standardrulers.To see why this is the case, imagine a hypothetical universe inwhich the polyspectra do not depend on cosmological parameters,perhaps due to some accidental cancellation of linear and nonlinearevolution effects. No matter what set of cosmological parameters weconsider, we always have the same initial conditions and the samenonlinear evolution. Equations (7) and (8) would then tell us that wecannot extract constraints on 𝜽 from measured polyspectra, since thederivatives are zero.This result makes perfect sense. Given a box of galaxies, we cannotguess what the input cosmological parameters were if all cosmolog-ical parameters result in the polyspectra of identical shapes. It is alsoclear that for real observations, we could still use these polyspectra asstandard rulers, by comparing their known sizes in physical units totheir apparent sizes on the sky. We can then use these measurementsof distance to constrain dark energy (as long as the expansion historyof the Universe still depends on dark energy). In a certain sense, if theintrinsic shapes of the polyspectra did not depend on cosmologicalparameters they would make even better standard rulers, because wewould not have to worry about cosmology dependence of the tem-plate. This toy model suggests that when computing the constrainingpower of a standard ruler we need to modify our formalism. The ge- ometrical probes contain additional information to the one that canbe retrieved just from looking at the intrinsic shape of NPCF. We will start this section with a brief recap of how the standardruler measurements with galaxy polyspectra are made. Appendix Bprovides a more detailed review. Since we observe galaxies on thesky and do not know how to convert distances and angles to physi-cal separations 𝑎 𝑝𝑟𝑖𝑜𝑟𝑖 , we initially compute polyspectra in somefiducial cosmology. Unless we are lucky, the fiducial cosmology willnot be identical to the real cosmology of the Universe. The measuredpower spectrum 𝑃 m ( 𝑘 ) will have the same intrinsic shape as the truepower spectrum, 𝑃 t ( 𝑘 ) , but since we did not put galaxies at correctdistances, it will be uniformly dilated with respect to the real powerspectrum in physical units as 𝑃 m ( 𝑘 ) = 𝑃 t ( 𝛼𝑘 ) . (13)The dilation parameter 𝛼 is the ratio of the fiducial distance weassumed to the true distance to the galaxies: 𝛼 = 𝐷 t ( 𝑧 )/ 𝐷 fid ( 𝑧 ) ,where 𝐷 is the comoving distance (see Appendix B for details). Bymeasuring it from the power spectrum, we can estimate the distanceto a galaxy sample at redshift 𝑧 . This distance-redshift relationshipdepends on cosmological parameters (including dark energy parame-ters) and can be used to obtain very accurate cosmological constraintseven if the intrinsic shape of the power spectrum does not have sucha dependence.Small uncertainties in the measured power spectrum propagate tosmall uncertainties in the recovered value of 𝛼 via 𝛿𝑃 m ( 𝑘 ) ∼ 𝜕𝑃 m ( 𝑘 ) 𝜕𝛼 𝛿𝛼 = 𝜕𝑃 t ( 𝛼𝑘 ) 𝜕 ln 𝑘 𝛿𝛼. (14)This equation tells us that the power spectrum makes a better standardruler if its intrinsic shape changes rapidly in wave-number. If thepower were constant across wave-numbers, we would not be able touse it as a standard ruler. The presence of characteristic scales andoscillations (e. g. the BAO) where the slope rapidly changes withthe wave-number makes the power spectrum a better standard rulerthan it would have been if it were a smoother shape. How sensitivethe intrinsic shape is to the variations in cosmological parameters is,in this case, irrelevant. The intrinsic shape could have a very weakor no dependence on cosmological parameters for all we care. Theimportant thing is how sensitive is the spectrum to uniform dilation(given by equation 14) and how sensitive is the distance-redshiftrelationship to the change in cosmology ( 𝜕𝛼 / 𝜕𝜃 ).The actual standard ruler analysis is much more complicated thanthis. There are two 𝛼 parameters describing the line-of-sight andacross-the-line-of-sight dilation and we have to properly accountfor the fact that the cosmology dependence is both in the dilation(standard ruler effect) and in the template (intrinsic dependence of theshape). None of this affects the essence of our main argument, whichis that the goodness of the field as a standard ruler is not conservedunder gravitational evolution. For simplicity, we will assume that thestandard ruler is used to constrain a single scale.Use of the bispectrum as a standard ruler is similar. We relatemeasured bispectra in the fiducial cosmology to a template via 𝐵 m ( 𝑘 , 𝑘 ) = 𝐵 t ( 𝛼𝑘 , 𝛼𝑘 ) , (15)and how good of a standard ruler the bispectrum is will be determinedby 𝛿𝐵 m ( 𝑘 , 𝑘 ) ∼ (cid:20) 𝜕𝐵 t ( 𝑘 , 𝑘 ) 𝜕 ln 𝑘 + 𝜕𝐵 t ( 𝑘 , 𝑘 ) 𝜕 ln 𝑘 (cid:21) 𝛿𝛼. (16) MNRAS , 1–14 (2020)
Samushia et al.
The precision to which 𝛼 can be measured from the initial and thefinal polyspectra is (cid:2) 𝐶 𝛼𝐼 (cid:3) − = ∑︁ 𝑖 𝑗 𝜕 S 𝐼𝑖 𝜕𝛼 𝜕 S 𝐼𝑗 𝜕𝛼 (cid:104) 𝐶 𝑆𝐼 (cid:105) − 𝑖 𝑗 = ∑︁ 𝑖 𝑗 𝜕 S 𝐼𝑖 𝜕 ln k 𝜕 S 𝐼𝑗 𝜕 ln k (cid:104) 𝐶 𝑆𝐼 (cid:105) − 𝑖 𝑗 , (17) (cid:2) 𝐶 𝛼𝐹 (cid:3) − = ∑︁ 𝑖 𝑗 𝜕 S 𝐹𝑖 𝜕𝛼 𝜕 S 𝐹𝑗 𝜕𝛼 (cid:104) 𝐶 𝑆𝐹 (cid:105) − 𝑖 𝑗 = ∑︁ 𝑖 𝑗 𝜕 S 𝐹𝑖 𝜕 ln k 𝜕 S 𝐹𝑗 𝜕 ln k (cid:104) 𝐶 𝑆𝐹 (cid:105) − 𝑖 𝑗 , where the derivatives with respect to k denote a combined derivativeof the polyspecrtra with respect to all wave-numbers as in 𝜕 S 𝜕 ln k ≡ ∑︁ 𝑖 𝜕 S 𝜕 ln k 𝑖 , (18)where the components of k are given by equation (4). This sumwill contain the terms given by equations (14) and (16) and similarderivatives for the higher-order NPCFs.The precision to which cosmological parameters can be derivedfrom 𝛼 is (cid:104) 𝐶 𝜽 𝛼 (cid:105) − 𝑖 𝑗 = 𝜕𝛼𝜕 𝜽 𝑖 𝜕𝛼𝜕 𝜽 𝑗 (cid:2) 𝐶 𝛼 (cid:3) − , (19) C 𝛼 is the covariance matrix of 𝛼 and can be either C 𝛼𝐼 or C 𝛼𝐹 andsince 𝛼 is a single number this is just its variance. The equation abovedoes not change between the initial and the final fields. How the 𝛼 errors propagate to the cosmological parameter errors depends on thevariance of 𝛼 but does not depend on the origin of the measurement.To determine which standard ruler (initial or final) results in betterconstraints we have to compare the first and second equations of (17).The covariance matrices scale by equation (8) as before. The shapederivatives of the final polyspectra, on the other hand, are 𝜕 S 𝐹 𝜕 ln k = 𝜕𝜕 ln k (cid:110) S 𝐹 (cid:16) S 𝐼 ( k (cid:48) , 𝜽 ) , k , 𝜽 (cid:17)(cid:111) . (20)Unlike equation (12), where we were able to claim a simple relation-ship between the two derivatives, there is no general relationship thatrelates the above to 𝜕 S 𝐼 / 𝜕 ln k in this case. The difference in the twocovariance matrices in equation (17) does not get canceled by thedifference in the derivatives.This would be the case even if the mapping between initial andfinal polyspectra were linear, for instance as in S 𝐹 ( k , 𝜽 ) = ∫ 𝐾 ( k , k (cid:48) , 𝜽 )S 𝐼 ( k (cid:48) , 𝜽 ) d k (cid:48) . (21)The only kernel 𝐾 that can enforce the equality C 𝛼𝐼 = C 𝛼𝐹 is a wave-number independent scaling 𝐾 ( k , k (cid:48) , 𝜽 ) ∝ 𝛿 D ( k − k (cid:48) ) where 𝛿 D is aDirac delta function.On reflection, this is not surprising. The cosmological evolutiondoes not know that we will be using polyspectra as standard rulers,so barring a coincidence there is no reason why it should conservethe strength of the standard rulers over time.To summarize, even though the evolution from the initial to thefinal field is invertible on large scales, it does not conserve the field’scumulative capacity as a standard ruler. How good of a standardruler the polyspectra are is determined by how sensitive they arewith respect to wave-number dilation, or how rapidly they changeacross wave-numbers. This property is not conserved by gravitationalevolution simply because there is no underlying mathematical reasonfor it to be conserved. In the previous section, we established that the power spectrum of theinitial Gaussian field does not have to provide the same constraintson the scale parameter 𝛼 as the hierarchy of polyspectra of the finalnonlinear field. Which one is a better standard ruler is difficult toguess without performing actual calculations. The explicit analyticcalculations, unfortunately, are difficult to make. Doing so wouldrequire explicitly writing down the mapping given by equation (5)between the polyspectra of the initial and the final fields.The power spectrum of the nonlinear field is a worse standard rulerthan the power spectrum of the Gaussian field for two reasons. Grav-itational evolution partially erases the BAO feature making the non-linear power spectrum smoother (smaller derivative in equation 14).The nonlinear power spectrum is also more correlated between dif-ferent wave-numbers, making it a less powerful measurement overall.On the other hand, the nonlinear field gains additional polyspectra,like a bispectrum, that can be used as standard rulers. The real ques-tion is whether the bispectrum as a standard ruler is good enough tomake up for the lost constraining power due to partially erased BAOwiggles in the nonlinear power spectrum.There is no universal answer to this question. For specific galaxysamples the answer depends on many factors such as the numberdensity, galaxy bias, range of wave-numbers used in fitting, fittingprocedure, etc. Our goals is not to argue that the nonlinear polyspec-tra are better standard rulers but to to demonstrate that there is nofundamental reason why they cannot be.To make our results as transparent as possible, we take a real-spacematter field at redshift zero. Performing the same analysis in redshift-space would significantly complicate it (introduction of two dilationparameters, measuring anisotropic bispectrum), and for what we aretrying to show it would not matter. If anything, we expect our resultsto be even stronger in redshift-space, where the bispectrum has amore non-trivial shape.One of the difficulties of analyzing higher-order polyspectra is theneed for accurate covariance matrices. Measuring the bispectrum upto 𝑘 max = . ℎ Mpc − in bins of Δ 𝑘 = . ℎ Mpc − results in 925bispectrum measurements. An accurate estimate of the 925 × Quijote simulations (Villaescusa-Navarro et al. 2020b) of 1 ℎ − Gpc volume each. We compute theangle-averaged power spectrum and bispectrum in bins of Δ 𝑘 = . ℎ Mpc − . We estimate the covariance of the measurements fromthe sample variance between 8,000 simulations (see Appendix A forthe details of these computations). We use equations (9) and (10) toestimate the information on the parameters coming from the intrinsicshape, and equations (17) to estimate the information coming fromtheir usage as standard rulers. Appendix A explains the details ofhow these derivatives are computed.Fig. 1 shows the cumulative signal-to-noise ratios of the powerspectrum and bispectrum measured from Quijote boxes,S / N = ∑︁ 𝑖 𝑗 [ PB ] 𝑖 (cid:104) 𝐶 − (cid:105) 𝑖 𝑗 [ PB ] T 𝑗 , (22)where PB is a row vector constructed from the power spectrum andbispectrum bins and C is their covariance matrix. For each pointon the horizontal axis, we used all the power spectrum bins up tothat value of 𝑘 max and all the bispectrum bins that have all theirwave-numbers below that 𝑘 max .We compute the cumulative signal-to-noise because it seems to bea very popular proxy for the information content of the spectra in therecent literature. In terms of parameter fits, it corresponds to the case MNRAS , 1–14 (2020) igher-Order Clustering .
05 0 .
10 0 .
15 0 . k max in h/ Mpc C u m u l a t i v e S i g n a l − t o − N o i s e Nonlinear Power SpectrumBispectrumLinear Power Spectrum
Figure 1.
Cumulative signal-to-noise ratio of respectively the Gaussian powerspectrum (black line), the nonlinear power spectrum (blue line), and thenonlinear bispectrum (orange line), as a function of maximum wave-numberconsidered in the analysis. This comes from the direct shape and therefore thelinear power spectrum contains all the information. Derived for the 1 ℎ − Gpc box of matter distribution (negligible shot-noise) at 𝑧 = where the power spectrum (bispectrum) intrinsic shape is known ex-actly and the amplitude needs to be determined from data, i.e. fittingto unknown 𝐴 P and 𝐴 B in 𝐴 P 𝑃 ( 𝑘 ) and 𝐴 B 𝐵 ( 𝑘, 𝑘 (cid:48) , 𝑘 (cid:48)(cid:48) ) . On largescales, the measurement of 𝐴 P then equivalent to the measurementof ( 𝑏 𝜎 ) and 𝐴 B to that of ( 𝑏 𝜎 ) , where 𝑏 is the linear bias and 𝜎 (or 𝜎 ) is one of the possible parametrizations of the amplitudeof matter clustering (Feldman 2010; Sánchez 2020).The lines on Fig. 1 show signal-to-noise as a function of themaximum wave-number considered in the analysis for the nonlinearpower spectrum, nonlinear bispectrum, and linear power spectrumperfectly reconstructed and placed at redshift zero. As expected, thebispectrum contains significantly less information on the amplitudeand this information grows with the maximum wave-number. Yet wenotice that the information extracted from the joint fit is always belowthe information extractable from the linear power spectrum. Thisresult is not surprising. The amplitude is measured in the intrinsicshape of the polyspectra bin by bin, and the results are in line withour expectation from equations (9) and (10).Fig. 2 shows a similar plot of the information on the dilation pa-rameter 𝛼 . These constraints come from the usage of the polyspectraas standard rulers (not the intrinsic shape) and are described by equa-tion (17). These equations do not force the cumulative informationin the final and initial fields to be equal, and it is indeed the casethat at about 𝑘 = . ℎ Mpc − the bispectrum becomes a betterstandard ruler for constraining 𝛼 then the initial power spectrum. By 𝑘 max = . ℎ Mpc − the improvement reaches a factor of two.The step-like structure with increasing 𝑘 max in the two lines cor-responding to the power spectrum information is not a numericalartifact. These steps appear because the main feature in the powerspectrum—the BAO wiggles—are stronger standard rulers at theedges of a given wave-form (more sensitive to dilation) than at theirminima and maxima. The slope of the information curve is thereforeflattened as we pass over peaks in the BAO wiggles. The low-redshiftpower spectrum line is slightly above the Gaussian power spectrum atvery low wave-numbers for a similar reason: the nonlinear evolutionmoves the crest of the BAO to slightly higher wave-numbers. Techni-cal details behind these computations are presented in Appendix A.One could argue that comparing the Gaussian power spectrum and the bispectrum at a fixed 𝑘 max is not fair since the bispectrum“siphons the information” from smaller scales in the initial field.This is certainly true, but our findings suggest that the bispectrumis a much better standard ruler at 𝑘 max = . ℎ Mpc − even whencompared to the Gaussian power spectrum up to 𝑘 max = . ℎ Mpc − . In the previous sections we showed that, in general, there is norelationship between how good standard rulers from the nonlinearfield are compared to their Gaussian field counterpart. Whether thereconstructed power spectrum or the nonlinear bispectrum is a betterstandard ruler will depend on the specifics of a galaxy sample suchas its redshift, number density and bias (Gagrani & Samushia 2017b;Chan & Blot 2017; Yankelevich & Porciani 2019; Colavincenzoet al. 2019; Philcox & Eisenstein 2019; Hahn et al. 2020; Gualdi& Verde 2020; Sugiyama et al. 2020b; Leicht et al. 2020; Hahn &Villaescusa-Navarro 2020). It is undeniable that the linear powerspectrum is significantly easier to analyze due to the size of the data,and the ease of modeling and computing covariance matrices. Evenif the bispectrum is a better standard ruler in principle, we may notbe able to reliably extract this information in practice.On the other hand, our tests on
Quijote simulations suggest thatat redshift zero the real-space nonlinear matter field is a better stan-dard ruler by a factor of two! There is no reason why going to theredshift-space will reverse this order of precedence. If anything, thebispectrum analysis should benefit more from the addition RSD. Wepresented our main results for the bare-bones case of the real-spaceunbiased tracers to keep the physical picture simple, but we checkedthat adding extra nuisance parameters accounting for e.g. bias param-eters and non-Poissonian shot-noise does not affect our conclusions.One may wonder why it is that for the BOSS (Dawson et al. 2013)and eBOSS (Dawson et al. 2016) samples the constraints comingfrom the joint analysis of the power spectrum and the bispectrumare slightly lower than the constraints from the reconstructed powerspectrum. We think this is very likely due to the effect of shot-noise inthose samples. The shot-noise affects the variance of the high wave-number modes more than it affects that of the low wave-numbermodes. The bispectrum starts overtaking the linear power spectrumas a standard ruler at wave-numbers of 𝑘 ∼ . ℎ Mpc − . As theshot-noise increases, the contribution of these wave-number binsgets down-weighted. This and the fact that the nonlinear bias termsare expected to affect the bispectrum more at higher wave-numbersmay be the reason behind the apparent “conservation of information”between nonlinear and linear fields in the BOSS and eBOSS samples.The Bright Galaxy Survey (BGS) sample at low redshifts and theEmission Line Galaxy (ELG) sample at around 𝑧 ∼ 𝑛 = − ℎ Mpc − ,and the forecasts for the higher-order analysis look very promisingWang et al. (2019b). 21-cm intensity mapping surveys (Pritchard &Loeb 2012; Villaescusa-Navarro et al. 2014; Bull et al. 2015), whichhave very low shot-noise, are another potentially rich candidate foruse of these methods (Saiyad Ali et al. 2006; Yoshiura et al. 2015;Majumdar et al. 2018; Bharadwaj et al. 2020). Detailed forecasts forthese surveys are very complicated and fall outside of the scope ofthis paper. MNRAS , 1–14 (2020)
Samushia et al. .
05 0 .
10 0 .
15 0 . k max in h/ Mpc C u m u l a t i v e F i s h e r I n f o r m a t i o n o n α Nonlinear Power SpectrumBispectrumLinear Power Spectrum
Figure 2.
The information on the isotropic dilation parameter obtainable from the Gaussian power spectrum (black line), the nonlinear power spectrum (blueline), and the nonlinear bispectrum (orange line), as a function of maximum wave-number considered in the analysis. The constraints from the nonlinearfield (bispectrum) exceed the ones obtainable from the initial field (linear power spectrum). Derived for the 1 ℎ − Gpc box of matter distribution (negligibleshot-noise) at 𝑧 =
0. The Fisher information of the power spectrum flattens around the peaks of the BAO features. A region remains flat until you go over theBAO peak. Three flat regions correspond to the three BAO peaks traversed (see Fig. A1). This is not as apparent in the bispectrum since at each 𝑘 we sum overmultiple 𝑘 and 𝑘 . Another interesting question is what happens for the NPCFs oforders four and higher. This question is very difficult to answer with-out performing specific calculations. We were unable to derive re-liable estimates for the trispectrum from the
Quijote simulations athigh enough wave-numbers. The number of simulations, even thoughlarge for other purposes, was not large enough to reliably computecovariance matrices for the large number of trispectrum configura-tions. In general, the higher-order polyspectra do provide additionalstandard rulers but are also significantly noisier. It may well be thatthe bispectrum provides a sweet spot where the increased noise iscompensated by the additional sensitivity to dilation, and for higherorders the noise scale too steeply for them to make a reasonablecontribution. There is no reason why if the bispectrum is a betterstandard ruler than the power spectrum, the trispectrum has to bean even better standard ruler. Such an intuition would be borrowedfrom perturbation theory, where if a certain infinite series diverges Gualdi et al. (2020) presented integrated trispectrum and its covarianceestimates from 5,000 simulations, but their measurements extended only upto 𝑘 max = . ℎ Mpc − . The number of distinct trispectra scales as 𝑘 andgoing to 𝑘 max = . ℎ Mpc − with roughly the same accuracy would requireup to an order of magnitude more simulations. for lower orders it must also diverge for higher orders. But the prob-lem at hand has nothing to do with the perturbative expansion inthe linear field, and so we do not think this intuition is necessarilyapplicable here.There are a few reasons why the conclusions of this paper may atfirst seem counter-intuitive. One of them is due to the coincidencenoted at the beginning of the section—the BOSS and eBOSS samplesjust happened to have number densities and biases that resulted inapparent conservation of information under reconstruction. Anothersource of this uneasiness is the basic intuition from statistics telling usthat invertible transformations cannot create or destroy information.If one made certain measurements and recorded proper covariances,one can multiply these measurements by some numbers, raise them toa power, or apply a wide range of nonlinear transformations. As longas those transformations are invertible and properly accounted forin the covariance matrices, they are not going to affect the amount ofinformation one can extract from the fields. Current literature tends touse the signal-to-noise as a universal proxy for the information con- Data reduction techniques such as binning are not always invertible.MNRAS000
Quijote simulations athigh enough wave-numbers. The number of simulations, even thoughlarge for other purposes, was not large enough to reliably computecovariance matrices for the large number of trispectrum configura-tions. In general, the higher-order polyspectra do provide additionalstandard rulers but are also significantly noisier. It may well be thatthe bispectrum provides a sweet spot where the increased noise iscompensated by the additional sensitivity to dilation, and for higherorders the noise scale too steeply for them to make a reasonablecontribution. There is no reason why if the bispectrum is a betterstandard ruler than the power spectrum, the trispectrum has to bean even better standard ruler. Such an intuition would be borrowedfrom perturbation theory, where if a certain infinite series diverges Gualdi et al. (2020) presented integrated trispectrum and its covarianceestimates from 5,000 simulations, but their measurements extended only upto 𝑘 max = . ℎ Mpc − . The number of distinct trispectra scales as 𝑘 andgoing to 𝑘 max = . ℎ Mpc − with roughly the same accuracy would requireup to an order of magnitude more simulations. for lower orders it must also diverge for higher orders. But the prob-lem at hand has nothing to do with the perturbative expansion inthe linear field, and so we do not think this intuition is necessarilyapplicable here.There are a few reasons why the conclusions of this paper may atfirst seem counter-intuitive. One of them is due to the coincidencenoted at the beginning of the section—the BOSS and eBOSS samplesjust happened to have number densities and biases that resulted inapparent conservation of information under reconstruction. Anothersource of this uneasiness is the basic intuition from statistics telling usthat invertible transformations cannot create or destroy information.If one made certain measurements and recorded proper covariances,one can multiply these measurements by some numbers, raise them toa power, or apply a wide range of nonlinear transformations. As longas those transformations are invertible and properly accounted forin the covariance matrices, they are not going to affect the amount ofinformation one can extract from the fields. Current literature tends touse the signal-to-noise as a universal proxy for the information con- Data reduction techniques such as binning are not always invertible.MNRAS000 , 1–14 (2020) igher-Order Clustering tent, and for the signal-to-noise and other amplitude-like parametersthe information is indeed conserved.In §2 we explained why this intuition fails with standard rulers. Thestandard ruler tests rely on apparent effects that do not really exist innature as such. One would struggle to identify the parts referring orrelated to a standard ruler test in the Euler or the Poisson equations.Therefore there is no fundamental reason why the laws of natureshould conserve the efficiency of standard rulers as cosmologicalprobes over time. In fact, the sensitivity of polyspectra to cosmologi-cal parameters sometimes is not conserved even when the constraintsare coming from the analysis of their intrinsic shape (as opposed totheir usage as standard rulers). To derive “the conservation of in-formation” we had to assume that the mapping between the initialand the final polyspectra depends on the cosmological parametersweakly in equation (11). This is not the case e.g. when constrainingthe mass of neutrinos. In this case the mapping depends on the valueof the parameter (given exactly the same initial power spectrum, thedamping of the amplitude at high wave-numbers depends on the totalmass of neutrinos). The evolution itself imprints a useful feature intothe polyspectra and the reconstructed version would obviously havemuch lower sensitivity to the parameter of interest.We can think of a few toy models to put our intuition at ease. Onehypothetical example is of a universe that has a flat power spectruminitially. Let us suppose that the laws of gravity in this universe aresuch that they start imprinting a “hump” in this initially flat powerspectrum with time, so that at later times we have a power spectrumwith a feature. At late times then we have a standard ruler that canbe used to measure the distance-redshift relationship and derive darkenergy constraints. If we reconstruct the field we will go back tothe featureless power spectrum that cannot be used as a standardruler. In this hypothetical universe the gravitational evolution actuallycreates “information” out of nowhere, the information is clearly notconserved, and reconstruction would erase this information.An even simpler example is an initial field that has only one wave-number and a single phase, and creates a sinusoidal pattern across thesky. The initial power spectrum is in this case a Dirac delta functionof the special frequency. This special frequency would be much moreeasily detected in the nonlinearly-evolved field.A less hypothetical example is that of standard candles. The stan-dard candles are in many respects similar to standard rulers. Theyhave a known intrinsic luminosity (or a luminosity that 𝑐𝑎𝑛 be stan-dardized) and by measuring their apparent luminosity we can putvery stringent constraints on the distance-redshift relationship andconsequently on dark energy. These standard candles are used asindividual objects. In standard analyses we do not care how they arearranged in space with respect to each other. These objects were alsocreated from initial Gaussian fields with tiny fluctuations that werefully described by their power spectrum. One could ask a similarquestion of them: where does the information on the dark energythey provide at late times come from? Which part of the initial Gaus-sian power spectrum is it encoded in? It is clear that this informationdoes not really come from anywhere. It does not reside in the smallpatch of the Universe that later collapsed to make a supernova, andit clearly does not reside in the primordial power spectrum. We werelucky that the standard candles happened to exist in the Universe andcosmologists had sufficient ingenuity to realise they could be used tomeasure distances. We were also lucky that the distance as a functionof redshift happened to depend very strongly on dark energy.The situation with polyspectra and distance measurements is sim-ilar. Low-redshift polyspectra can be used as standard rulers. Theyalso happened to be generated from the seed Gaussian fields. Buttheir origin is largely irrelevant for their usage as standard rulers. When a carpenter uses a meter stick to measure the width of a win-dow, the information about the width of that window does not comefrom the wood that made the meter stick. The information comes thecarpenter’s use of the meter stick in a certain way. The carpenter canmeasure as many lengths and widths as he or she wishes. There isno limit on how much information he or she can collect, at least nolimit imposed by the physical origin of the meter stick.Another reason why we feel the information must come fromthe Gaussian power spectrum is that we often choose to representthe information content of a random fields by the signal-to-noiseratio. For the signal-to-noise and other amplitude-like parameters,equations (9) and (10) hold. Fig. 1 offers a simple and familiarexample of the amplitude-like parameters for which the informationis conserved, although it is not clear what kind of information itis exactly. For uncorrelated data, signal-to-noise is the sum of allmeasurements divided by their errors.If one makes independent observations of the same thing manytimes over, the signal-to-noise is a measure of how well that quantitycan be constrained cumulatively by all the measurements. When thedata are measurements of different things, the signal-to-noise doesnot really have a clear meaning. The power spectrum measurementsat two different wave-numbers are a measurement of two differentphysical quantities. The dependence of the two on the underlyingparameters is different. It is therefore not entirely clear what thesignal-to-noise represents in this case.The signal-to-noise rises as the quality of our measurements in-creases so it can be used as a reasonably good proxy for the relativegoodness of two pieces of data, but in some cases this kind of com-parison can be misleading. E.g. while samples with higher signal-to-noise result in better BAO constraints, they also result in worse RSDconstraints (Pearson et al. 2016). Computing the signal-to-noise ofthe power spectrum and the bispectrum bin by bin and summing itup only makes sense if one is interested in how well the amplitudeof the polyspectra would be measured if their shapes were perfectlyknown. These kind of fits are rarely performed in practice. ACKNOWLEDGEMENTS
MNRAS , 1–14 (2020) Samushia et al.
DATA AVAILABILITY
Computer codes used in producing the results presented in thismanuscript and instructions about
Quijote access are avail-able online. REFERENCES
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APPENDIX A: COMPUTING THE BISPECTRUM AND ITSDERIVATIVES
We start by dividing each
Quijote simulation box into a uniformgrid of 𝑁 = cells. We then go over all particles in thesimulation and assign each cell a number depending on how far awaythe cell center is from the particle using the formula 𝑊 = ( − 𝑠 + 𝑠 )/ ≤ 𝑠 < ( − 𝑠 )/ ≤ 𝑠 <
20 for 2 ≤ 𝑠 , (A1)where 𝑠 is the distance between the particle and the cell center in unitsof the cell size. This assignment is based on a piecewise cubic splineprescription (Chaniotis & Poulikakos 2004) and ensures that thegridding effects on the scales of interest (up to 𝑘 ∼ . ℎ Mpc − ) arenegligible (Sefusatti et al. 2016). Contributions from all particles areadded up. This results in a 512 array of numbers - 𝑛 𝑖 𝑗ℓ . We performa discrete real-to-complex Fourier transform of 𝑛 𝑖 𝑗ℓ , to obtain a512 × ×
257 grid of complex numbers - (cid:101) 𝑛 𝑖 𝑗ℓ , using the FFTWalgorithm (Frigo & Johnson 2005). The cells of this grid correspondto the wave-numbers at 𝑘 𝑥 = 𝑘 f (cid:18) 𝑖 − 𝑁 grid (cid:19) , 𝑖 = , . . . , 𝑁 grid ,𝑘 𝑦 = 𝑘 f (cid:18) 𝑗 − 𝑁 grid (cid:19) , 𝑗 = , . . . , 𝑁 grid , (A2) 𝑘 𝑧 = 𝑘 f ℓ, ℓ = , . . . , 𝑁 grid / ,𝑘 f = 𝜋𝐿 (A3)From this grid, we estimate the power spectrum 𝑃 ( 𝑘 ) in bins of Δ 𝑘 = . ℎ Mpc − starting with 𝑘 =
0, and the bispectrum 𝐵 ( 𝑘 , 𝑘 , 𝑘 ) in bins of Δ 𝑘 𝑖 = . ℎ Mpc − starting with 𝑘 𝑖 = (cid:101) 𝑛 𝑖 𝑗ℓ , compute thewave-number 𝑘 = √︃ 𝑘 𝑥 + 𝑘 𝑦 + 𝑘 𝑧 , and compute the absolute valueof the entry, | (cid:101) 𝑛 𝑖 𝑗𝑘 | . For each 𝑃 ( 𝑘 ) bin we simply average overcontributions from all cells that fall into that bin.For the bispectrum, we go over all pairs of cells in (cid:101) 𝑛 𝑖 𝑗ℓ , find a thirdunique cell such that the “triangular condition” k + k = k holds forthe three associated wave-numbers, and take a product (cid:101) 𝑛 (cid:101) 𝑛 (cid:101) 𝑛 ★ . Theasterix in this equation denotes complex conjugation and subscriptsrefer to the triplet of integer indeces. For each 𝐵 ( 𝑘 , 𝑘 , 𝑘 ) bin wetake the average over contributions from all pairs of cells (and thethird associated cell that is uniquely determined by the triangularcondition) that fall into that bin. We do not look at the bispectrumbins unless at least some combination of the wave-numbers in that binsatisfies a triangular condition. To avoid counting equivalent tripletsmultiple times, we impose a condition 𝑘 > 𝑘 > 𝑘 . . . . . . k in h /Mpc P ( k ) k i n M p c / h Linear Power SpectrumNonlinear Power Spectrum
Figure A1.
Nonlinear power spectrum at 𝑧 = Quijote matter particles. The linear power spectrum ismeasured from particles at 𝑧 =
99 and rescaled to have a matching amplitudeon large scales. . . . . . k in h /Mpc01234 B ( k , k , k ) k k k i n ( M p c / h ) Bispectrum
Figure A2.
Bispectrum of
Quijote matter particles at 𝑧 = 𝑘 . Multiple points at the same abscissa showall bispectrum bins that share that value of 𝑘 but have different values of 𝑘 and 𝑘 . We multiply the bispectrum by a factor of 𝑘 𝑘 𝑘 . This factor isproportional to the number of fundamental triangles in the bin and separatesdifferent points for better visibility. Fig. A1 shows the average nonlinear and linear power spectraof matter particles from
Quijote simulations. We rescaled the lin-ear power spectrum to have a matching amplitude at small wave-numbers. The two are reasonably close for large-scale modes (upto 𝑘 ∼ . ℎ Mpc − after which the additional power in the non-linear power spectrum becomes visible. The partial erasure of theBAO signal is also visible by eye. Fig. A2 shows the average bis-pectrum of matter particles from Quijote simulations at 𝑧 = C , by comput- MNRAS000
Quijote simulations. We rescaled the lin-ear power spectrum to have a matching amplitude at small wave-numbers. The two are reasonably close for large-scale modes (upto 𝑘 ∼ . ℎ Mpc − after which the additional power in the non-linear power spectrum becomes visible. The partial erasure of theBAO signal is also visible by eye. Fig. A2 shows the average bis-pectrum of matter particles from Quijote simulations at 𝑧 = C , by comput- MNRAS000 , 1–14 (2020) igher-Order Clustering B × BP × BP × P − . − . − . − . . . . . . Figure A3.
Correlation coefficient of the power spectrum (first 50 bins, goesup to 𝑘 max = . ℎ Mpc − ) and the bispectrum (last 150 bins, goes up to 𝑘 , max = . ℎ Mpc − ) measurements. Colormap is adjusted in such a waythat red, blue, and white cells correspond to the positive, negative, and zerocorrelations respectively. ing the sample covariance over 8,000 Quijote simulations, 𝑋 𝑖 = 𝑁 sim 𝑁 sim ∑︁ ℓ = 𝑋 𝑖ℓ , (A4) 𝐶 𝑖 𝑗 = 𝑁 sim 𝑁 sim ∑︁ ℓ = ( 𝑋 𝑖ℓ − 𝑋 𝑖 )( 𝑋 𝑗ℓ − 𝑋 𝑗 ) . (A5) 𝑋 𝑖ℓ are the power spectrum and bispectrum measurements, wherethe first index goes over the bins and the second index goes oversimulations. 𝑁 sim is the number of simulations.Fig. A3 shows the correlation matrix of these measurements de-fined as 𝑅 𝑖 𝑗 = 𝐶 𝑖 𝑗 √︁ 𝐶 𝑖𝑖 𝐶 𝑗 𝑗 . (A6)We ordered the measurements so that the first 50 elements are binned 𝑃 ( 𝑘 ) up to 𝑘 max = . ℎ Mpc − . They are followed by binned 𝐵 ( 𝑘 , 𝑘 , 𝑘 ) arranged in such a way that the bispectrum bin withsmaller 𝑘 goes first (smaller 𝑘 when 𝑘 are equal, and smaller 𝑘 when both 𝑘 and 𝑘 are equal). The first 150 bins arranged in thisway accommodate all bispectra up to 𝑘 , max = . ℎ Mpc − . Thecorrelation between power spectra at high wave-numbers is clearlyvisible in this plot. The bispectrum measurements are also correlatedbetween themselves and the power spectra but to a lesser extent.We apply a correction factor of H = 𝑁 sim 𝑁 sim − 𝑁 bins − 𝑁 bins is the total numberof power spectrum bin and bispectrum bin combinations. This factorcorrects for the fact that the raw inverse of the sample covariancematrix tends to underestimate the errors (Hartlap et al. 2007; Do-delson & Schneider 2013). This correction factor is the main reasonwe are unable to extend results presented on Figs. 1 and 2 to higherwave-numbers. The correction factor grows very rapidly with the 𝑘 max , it is negligible for the most of the 𝑘 max but grows to 13 percent at 𝑘 max = . ℎ Mpc − for our choice of binning.We compute the derivatives of power spectrum with respect to 𝛼 by constructing a cubic spline interpolation over power spectrum bins and then taking a backward numeric derivative,d 𝑃 d 𝛼 = 𝑃 ( 𝑘 ) − 𝑃 ( 𝑘 ( − 𝜖 )) 𝜖 , (A8)with 𝜖 = × − . To use a forward numeric derivative,d 𝑃 d 𝛼 = 𝑃 ( 𝑘 ( + 𝜖 )) − 𝑃 ( 𝑘 )) 𝜖 . (A9)We use forward numerical differentiation for the lowest 𝑘 value forwhich the backward differentiation would require an extrapolation.The bispectrum derivatives are computed similarly. We first con-struct a three-dimensional cubic spline and then compute a numericderivative,d 𝐵 d 𝛼 = 𝐵 ( 𝑘 , 𝑘 , 𝑘 ) − 𝐵 ( 𝑘 ( − 𝜖 ) , 𝑘 ( − 𝜖 ) , 𝑘 ( − 𝜖 )) 𝜖 , (A10)except for some bispectra bin combinations with small values of 𝑘 where backwards numerical differentiation would require extrapolat-ing our spline fit. APPENDIX B: STANDARD RULERS
We need to assume a fiducial cosmological model to convert galaxyredshifts to comoving distances 𝑑 c , 𝑑 c = 𝑑 c ( 𝑧, 𝜽 ) . (B1)The measured distances between galaxies will then be different fromthe real distances by | r − r | = | r − r | measured 𝑑 c ( 𝑧, 𝜽 fid ) 𝑑 c ( 𝑧, 𝜽 ) (B2) ≡ | r − r | measured 𝛼 ( 𝑧, 𝜽 fid , 𝜽 ) , where 𝑑 c is the physical distance to those galaxies. In conventionalcosmological models this distance is given by 𝑑 c ( 𝑧 ) = 𝑐 ∫ 𝑧 𝑑𝑧 (cid:48) 𝐻 ( 𝑧 (cid:48) ) , (B3)where 𝑐 is the speed of light and 𝐻 ( 𝑧 ) is the Hubble expansionparameter as a function of redshift. When fitting models of the cor-relation function 𝜉 to the measurements we have to account for thisscale-independent dilation, 𝜉 ( 𝑟, 𝜽 ) = 𝜉 ( 𝑟𝛼 ( 𝜽 ) , 𝜽 ) . (B4)The dilation effect is similar for the higher-order NPCFs, but we willkeep our discussion to the 2PCF for simplicity. The power spectrum,being the Fourier transforms of the 2PCF, scales by the inverse factor, 𝑃 ( 𝑘, 𝜽 ) = 𝑃 ( 𝑘 / 𝛼 ( 𝜽 ) , 𝜽 ) . (B5)Rather then being a nuisance, the presence of this scaling providesadditional opportunities for constraining cosmological parameters.Small changes in dark energy parameters, for example, affect theintrinsic shape of the power spectrum (the second argument) weakly,but affect the distance-redshift relationship and consequently the di-lation of the shape strongly. Recent works suggested that it may bemore natural to make NPCF measurements directly in the observablespace of redshifts and angles (Tadros et al. 1999; Bonvin & Durrer2011; Nicola et al. 2014; Yoo et al. 2018; Jalilvand et al. 2020). In thisapproach, the dilation is absent from the measurements (which aremade in redshift and angular coordinates) and the distance-redshiftrelationship becomes part of the model. We present our argumentsin a more familiar setting, because then, it makes the usage of the MNRAS , 1–14 (2020) Samushia et al. polyspectra as standard rulers more explicit. Our results would notchange if we instead formulated our arguments in terms of observa-tional coordinates (redshifts and angles).Small uncertainties in the measured power spectrum propagate toinferred cosmological parameters as 𝛿𝑃 ( 𝑘 ) ∼ − 𝜕𝑃 ( 𝑘 ) 𝜕𝑘 𝜕𝛼𝜕 𝜽 𝑘𝛼 𝛿 𝜽 + 𝜕𝑃 ( 𝑘 ) 𝜕 𝜽 𝛿 𝜽 . (B6)Fitting only the BAO feature in the power spectrum makes the sep-aration between the standard ruler and intrinsic shape constraintsclearer. The power spectrum can be divided into BAO and smoothcomponents, 𝑃 ( 𝑘, 𝜽 ) = 𝑃 BAO ( 𝑘, 𝜽 , 𝝈 ) 𝑃 smooth ( 𝑘, 𝜽 , 𝝂 ) , (B7)where by 𝝈 and 𝝂 we denoted nuisance parameters needed in realanalyses to compensate for observational effects and inaccuracies oftheoretical modeling (see e.g. Eisenstein & Hu 1998b, for details ofhow this split is performed in practice). 𝝈 will include parametersthat describe the nonlinear damping of the BAO feature, which isdifficult to link to the cosmological parameters 𝜽 directly, and 𝝂 will include smooth polynomials in 𝑘 that describe the effect ofgalaxy bias and nonlinear evolution (also difficult to compute from 𝜽 based on first principles). Both components will dilate by 𝛼 butsince the second component is “smooth” the effects of dilation will bedegenerate with the nuisance parameters. The BAO part of the powerspectrum has a decaying oscillatory feature the dilation of whichcannot be mimicked by nuisance parameters. For most conventionalcosmological models, the 𝑃 BAO between two models can be matchedby rescaling the power spectrum by a factor of 𝑟 d ( 𝜽 ) and adjustingnuisance parameters 𝝈 as in 𝑃 , BAO (cid:2) 𝑟 , d ( 𝜽 ) 𝑘, 𝝈 (cid:3) = 𝑃 , BAO (cid:2) 𝑟 , d ( 𝜽 ) 𝑘, 𝝈 (cid:3) , (B8)where subscripts 1 and 2 denote two different cosmologies. 𝑟 d canbe accurately computed in each cosmology. For these BAO onlyfits then measuring a power spectrum in a fiducial cosmology andcomparing it to a fiducial shape through dilation allows us to measurethe combination 𝑟 d / 𝛼 through 𝑃 ( 𝑘𝑟 d , fid ) = 𝑃 ( 𝛼𝑘𝑟 d ) . (B9)The measurement of 𝛼 can then be interpreted as a measurement of 𝛼 ( 𝜽 ) = 𝑑 c ( 𝑧, 𝜽 ) 𝑟 d , fid 𝑟 d ( 𝜽 ) 𝑑 c , fid . (B10)Sensitivity of measured power-spectrum to 𝛼 is determined by itslog-derivative with respect to the wave-number, 𝛿𝛼 ∼ (cid:20) 𝜕𝑃 ( 𝑘 ) 𝜕 ln 𝑘 (cid:21) − 𝛿𝑃 ( 𝑘 ) . (B11)All other things equal a feature at high wave-numbers would providea better constraint of 𝛼 . The power spectrum (or higher order spectra)can be used as standard rulers as long as they are not flat in 𝑘 in whichcase they have no sensitivity to dilation. Parts of the power spectrumthat scale as pure power law also do not contribute to the 𝛼 constraintssince they dilation in this case is fully degenerate with the linearbias parameter. Linear bias cannot be determined from cosmologicalparameters and would have to be measured from the power spectrumalong with 𝛼 . To see why this occurs, consider measuring the powerspectrum at 𝑁 different wave-number bands centered around 𝑘 𝑖 ,where 𝑖 = . . . 𝑁 , and in that in this wave-number range the powerspectrum can be modeled as 𝑃 𝑖 = 𝑏 𝑘 𝑛𝑖 . The derivatives of the powerspectrum with respect to 𝛼 and unknown bias will then be, 𝜕𝑃 𝑖 𝜕𝛼 = 𝑛𝑏 𝑘 𝑛𝑖 , 𝜕𝑃 𝑖 𝜕𝑏 = 𝑏𝑘 𝑛𝑖 . (B12) These derivative vectors are linearly dependent; therefore for thepure power-law power spectrum, the constraints on 𝛼 will be fullydegenerate with the constraints on the amplitude. This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS000