Reconstructing features in the primordial power spectrum
MMNRAS , 1–14 (2021) Preprint 19 February 2021 Compiled using MNRAS L A TEX style file v3.0
Reconstructing features in the primordial power spectrum
Yuhao Li, ★ Hong-Ming Zhu and Baojiu Li Astronomy Centre, Department of Physics and Astronomy, University of Sussex, Falmer, Brighton, BN1 9RH, UK Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 3H8, Canada Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Potential features in the primordial power spectrum, such as oscillatory patterns, have been searched for in galaxy surveys inrecent years, since these features can assist in understanding the nature of inflation and distinguishing between different scenariosof inflation. The null detection to date suggests that any such features should be fairly weak, and next-generation galaxy surveys,with their unprecedented sizes and precisions, are in a position to place stronger constraints than before. However, even ifsuch primordial features once existed in the early Universe, they would have been significantly weakened or even wiped out onsmall scales in the late Universe due to nonlinear structure formation, which makes them difficult to be directly detected in realobservations. A potential way to tackle this challenge for probing the features is to undo the cosmological evolution, i.e., usingreconstruction to obtain an approximate linear density field. By employing a suite of large N-body simulations, we show thata recently-proposed nonlinear reconstruction algorithm can effectively retrieve lost oscillatory features from the mock galaxycatalogues and improve the accuracy of the measurement of feature parameters (assuming such primordial features do exist). Wedo a Fisher analysis to forecast how reconstruction affects the constraining power, and find that it can lead to significantly morerobust constraints on the oscillation amplitude for a DESI-like survey. In particular, we compare the application of reconstructionwith other ways of improving constraints, such as increasing the survey volume and range of scales, and show that it can achievewhat the latter do, but at a much lower cost.
Key words: methods: numerical – large-scale structure of Universe
Inflation, the most successful theory to solve the problems of the hotBig Bang model and to explain the seeding of the observed large-scalestructures today, plays a crucial role in the development of moderncosmology. The simplest version of single-field inflation (Guth 1981;Linde 1982; Albrecht & Steinhardt 1982) predicts that primordialdensity fluctuations obey Gaussian statistics and the correspondingpower spectrum follows a simple power law, which is favoured by thecosmic microwave background (CMB) data released by the WMAP(Peiris et al. 2003; Spergel et al. 2007; Komatsu et al. 2009; Hinshawet al. 2013) and Planck (Ade et al. 2014a, 2016b; Akrami et al. 2020b)collaborations.However, the physical origin of the inflaton field that is believed tohave driven inflation is not fully understood yet, and the fact that thevery high energy in the early Universe makes it an ideal place to wit-ness the consequences of the laws of fundamental physics offers hopethat new physics could be revealed by cosmological observations ofthe large-scale structures. More sophisticated inflation models lead-ing to primordial non-Gaussianity have been developed across thelast decades (see Bartolo et al. 2004; Chen 2010; Celoria & Matar-rese 2020, for some reviews). These models predict unique featuresdeviating from those of the simple single-field inflation model, whichcan be classified into different types, each of which can be attributed ★ E-mail: [email protected] (YL) to different physical mechanisms (see Chen 2010; Chluba et al. 2015,for some reviews). Chen et al. (2016) briefly reviews several repre-sentative feature models of inflation, such as the sharp feature modelwhich oscillates in linear scale in the primordial power spectrum, canbe described by the template of the sinusoidal wiggle in the power-law primordial spectrum, which is also a good approximation forthe scalar power spectrum in the axion monodromy model (Flaugeret al. 2017). This oscillatory feature could be generated by a localisedsharp feature in inflationary potentials or internal field space, and thenature of the sharp feature could correspond to distinct mechanisms,such as a step or bump in the single-field inflationary potential (e.g.,Adams et al. 2001; Adshead et al. 2012; Bartolo et al. 2013; Hazraet al. 2014), or the embedding of new physics (Bozza et al. 2003).Other typical feature models include the resonance model in whichthe feature oscillates in logarithmic scale (see e.g. Wang et al. 2005;Bean et al. 2008; Flauger et al. 2010, for the mechanisms behind theresonance model), and the standard clock model which combines theform of the previous two feature models (e.g., Chen & Namjoo 2014;Chen et al. 2015).As a result, the (non)detection of primordial features can be used todistinguish among different scenarios of inflation. A variety of classesof well-motivated inflationary models, such as the so-called generalsingle-field inflation (e.g., Chen et al. 2007; Senatore et al. 2010),the multi-field inflation (see Byrnes & Choi 2010, for a review) andso on, are continuously tested with the updated release of data fromthe Planck mission (Ade et al. 2014b, 2016a; Akrami et al. 2020a), © a r X i v : . [ a s t r o - ph . C O ] F e b Y. Li, H.-M. Zhu & B. Li though none of them is preferable to the simplest single-field inflationmodel so far, which suggests that such features should be fairly weakif they do exist. Since the primordial features are not only imprintedin the CMB, and some can also leaves a signature in the matterpower spectrum, some future large-scale structure (LSS) surveyswith high sensitivity, such as Euclid (Racca et al. 2016), DESI (DESICollaboration et al. 2016) and SPHEREx (Doré et al. 2014) surveys,will provide the opportunity to search for the primordial features asa complement to CMB data (e.g., Chen et al. 2016; Ballardini et al.2016; Palma et al. 2018; L’Huillier et al. 2018; Zeng et al. 2019), orstrengthen the constrains on the strength of such features.However, any feature imprinted in the primordial density or cur-vature field by inflation is subject to the impact of cosmic evolutionthat leads to today. In particular, even if the primordial features onceexisted in the very early Universe, they would have been significantlyweakened or even wiped out on small scales in the late-time Universedue to nonlinear structure formation (Beutler et al. 2019; Ballardiniet al. 2020). Meanwhile, the available information on large scales,where the evolution can be described by linear theory, is limited dueto the cosmic variance. This can affect the confidence level at whichto measure or constrain these features. A potential method to addressthis issue is to undo the cosmological evolution in a process usuallycalled reconstruction, which can partially recover the initial densityfield, unlocking the information that once existed on small scales. Awell-known example is the reconstruction of baryonic acoustic os-cillation (BAO) features, which sharpens these features in the galaxycorrelation function which provides a standard ruler for distance mea-surements (e.g., Eisenstein et al. 2007; Kazin et al. 2014; Schmittfullet al. 2015; Zhu et al. 2017; Wang et al. 2017; Shi et al. 2018; Sarpaet al. 2019; Mao et al. 2021). Similarly, reconstructing the primor-dial power spectrum from the observed galaxy catalogues might bebeneficial for probing the primordial features by using future LSSsurveys, which is what we set out to check here.In this work, as a first step towards assessing the potential benefitof reconstruction, we assume additional simple oscillatory featuresin the power-law primordial power spectrum. By utilising a suite oflarge N-body simulations, we study the performance of a nonlin-ear reconstruction algorithm proposed recently by Shi et al. (2018);Birkin et al. (2019) to retrieve the lost primordial features from themock galaxy catalogue. In particular, we carry out parameter fitting tothe damped and reconstructed wiggles to assess whether reconstruc-tion can lead to more robust constraints on the feature parameters. Toinvestigate the impact of reconstruction in the real galaxy surveys, wealso forecast the constraints on the feature parameters for the DESIsurvey using the Fisher matrix approach, and compare the cases withand without reconstruction.This paper is organised as follows: in Section 2 we describe themodel of primordial features, the simulations used in this work, andthe methodology of assessing the performance of the reconstructionmethod to retrieve the lost primordial features due to structure for-mation. In Section 3 we give more details on the approach used toforecast the constraints on the feature parameters for the DESI survey.In Section 4 we show the results of reconstruction and forecast anddiscuss the implications of them. Finally, in Section 5 we concludeour findings and discuss potential future improvements.
We start with presenting the primordial power spectrum models withoscillatory features that we adopt in this paper for illustration purpose.We then describe the simulation runs for these models. It is followed by a brief review of the nonlinear reconstruction method which willbe used to recover the small-scale oscillation features from evolveddark matter and halo fields. Finally, we describe the analytic model toquantify the features measured in the power spectrum before givingthe details of the Fisher matrix forecast in the next section.
We take a powerlaw-type primordial power spectrum to be our fidu-cial no-wiggle model, which is given by 𝑃 nw ( 𝑘 ) = 𝐴 𝑠 (cid:18) 𝑘𝑘 ∗ (cid:19) 𝑛 𝑠 − , (1)where 𝑘 is the comoving wavenumber, 𝐴 𝑠 and 𝑛 𝑠 are respectively thescalar amplitude and spectral index with the pivot scale given by 𝑘 ∗ = .
05 Mpc − . Motivated by Ballardini et al. (2020), we consider threewiggled models which share the same form of oscillatory features,of which the featured primordial power spectrum is expressed as 𝑃 w ( 𝑘 ) = 𝑃 nw ( 𝑘 ) (cid:104) + 𝐴 cos (cid:0) 𝜔𝑘 𝑚 + 𝜙 (cid:1)(cid:105) , (2)where 𝐴 , 𝜔 and 𝜙 are respectively the amplitude, frequency and phaseof the oscillation, 𝑚 is the power of the comoving wavenumber 𝑘 .Note that even if the primordial features exist, they could be morecomplicated than any phenomenological models that we are currentlyusing. For now, we cannot determine the precise form of the features,thus we aim at something narrow, which is assuming that we knowthe functional form and verifying if reconstruction can improve theaccuracy of finding the feature parameters.The oscillation parameters of the four models are listed in Table 1.The initial oscillations of the three wiggled models are shown in thered dashed lines in the right panel of Fig. 1, where we have presentedthe difference between 𝑃 w and 𝑃 nw . Within our interested range ofscales, 𝑘 = ( . − . ) ℎ Mpc − , Model 1 has three peaks, Model 2and Model 3 have only two peaks. The first peak of Model 2 is chosento be on a smaller scale than that of Model 1. The two peaks of Model3 are at the same position as the first and third peaks of Model 1.By comparing the reconstructed wiggles of the three wiggled modelslater, we would be able to comprehend the effect of the reconstructionmethod on different scales. In the regime of linear perturbations, the primordial wiggles preservetheir shapes and amplitude 𝑃 w / 𝑃 nw . However, nonlinear large-scalestructure evolution will change this behaviour, leading to damping of 𝑃 w / 𝑃 nw at late times. This makes it harder to measure the propertiesof these primordial oscillations from an evolved density field, evenmore so for a late-time tracer (e.g., galaxy) field. In order to quantifysuch degrading effects, N-body cosmological simulations can proveto be a useful tool.We have run four simulation runs including the no-wiggle modeland three wiggled models. First we assume a flat universe and adoptPlanck 2018 cosmology, with ℎ = . Ω m = . Ω 𝑐 ℎ = . Ω 𝑏 ℎ = . Ω Λ = . 𝑛 𝑠 = .
965 and 𝐴 𝑠 = × − (Aghanim et al. 2020). The value of 𝜎 is approximately 0.79 thoughit varies a little bit across different models. We then customise thefunction of the primordial power spectrum in the Einstein-Boltzmannsolver code camb (Lewis & Challinor 2011) to be Eq. (1) for the no-wiggle model and Eq. (2) for the wiggled models. We calculate thelinear theory matter power spectrum at 𝑧 =
49 using this version ofthe camb code, which is used as the input matter power spectrum for
MNRAS000
MNRAS000 , 1–14 (2021) econstruction of primordial features Table 1.
The oscillation parameters used for the no-wiggle model and threewiggled models. Columns respectively denote (1) the power of the comovingwavenumber; (2) the amplitude, (3) frequency and (4) phase of the oscillation. 𝑚 𝐴 𝜔 / 𝜋 𝜙 / 𝜋 [ ℎ − Mpc ] Fiducial 0Model 1 1 0 .
05 15 .
00 0Model 2 1 0 .
05 8 .
57 0Model 3 0 .
631 0 .
05 7 .
13 0 the publicly available code 2lptic (Crocce et al. 2006) to generate theinitial conditions used for the N-body simulations. In the left panel ofFig. 1 we compare the initial matter power spectrum given by camband the matter power spectrum measured from the initial conditionsgenerated using 2lptic; it can be seen that they are in good agreementfor all models within the range of scales of our interest (the blowingup at small scales is due to the finite particle resolution).To more conveniently describe the oscillatory features for the wig-gled models, as mentioned above, we define the relative wiggle pat-tern as 𝑃 rw ( 𝑘 ) = 𝑃 w ( 𝑘 ) 𝑃 nw ( 𝑘 ) − , (3)which are shown in the right panel of Fig. 1. This clearly shows thatthe oscillatory features are perfectly created in the initial conditionsof the simulations.Next, we run the simulations using the parallel N-body code ram-ses (Teyssier 2002) which is based on the adaptive mesh refinement(AMR) technique. Each simulation is performed with 𝑁 = dark matter particles in a box of size 1024 ℎ − Mpc, and we outputfour snapshots at different redshifts, respectively as 𝑧 =
0, 0 .
5, 1, and1 .
5. For each snapshot, we use the halo finder rockstar (Behrooziet al. 2013) to identify the haloes with the definition of the halo mass 𝑀 𝑐 , where 𝑀 𝑐 is the mass within a sphere whose average den-sity is 200 times the critical density. Since the low-mass haloes areunable to be fully probed due to the limited simulation resolution, wemeasure the cumulative halo mass functions (cHMFs) from the mainhaloes with more than 100 particles to check the validity of the sim-ulation, which show very good agreement with the analytic formulaein Tinker et al. (2008). For each snapshot we establish one dark mat-ter particle catalogue (hereafter DM) and two mock halo cataloguesrespectively with the number density of 1 × − ( ℎ − Mpc ) − (here-after H1) and 5 × − ( ℎ − Mpc ) − (hereafter H2). Both host haloesand subhaloes are included in the halo catalogues. We achieve thenumber density by applying a mass cutoff, i.e., neglecting the haloeswith smaller masses than the cutoff. By using the power spectrumestimator tool powmes (Colombi & Novikov 2011), we measure thenonlinear matter power spectrum from DM and nonlinear halo powerspectrum separately from H1 and H2. Finally, we take the ratio of thepower spectrum of the wiggled models to the corresponding powerspectrum of the no-wiggle model to obtain the quantity 𝑃 rw for allcases. In order to partially recover the primordial features lost in the struc-ture formation, we perform reconstruction of the initial density fieldfrom the late-time density field using the nonlinear reconstructionalgorithm described in Shi et al. (2018). This reconstruction methodis based on mass conservation. Without assuming any cosmological model or having free parameters except the size of the mesh used tocalculate the density field, it uses multigrid Gauss-Seidel relaxationto solve the nonlinear partial differential equation which governs themapping between the initial Lagrangian and final Eulerian coordi-nates of particles in evolved density fields. Previous tests show thatthe reconstructed density field is over ∼
80% correlated with the ini-tial density field for 𝑘 (cid:46) . ℎ Mpc − , if reconstruction is performedon the dark matter density field, which cover the scales of our interest,though the performance becomes poorer when the method is appliedon density fields calculated from sparse tracers (Birkin et al. 2019;Wang et al. 2020; Liu et al. 2020). This method is implemented in amodified version of the ecosmog code (Li et al. 2012, 2013), whichitself is based on ramses.We reconstruct the initial density field separately from the cata-logues DM, H1 and H2 for each snapshot. The halo catalogues, whichcontain both main and subhaloes, are assumed to be the same as mockgalaxy catalogues hereafter unless otherwise stated. The procedurefor the reconstruction from the halo catalogue is principally similarto that from the dark matter particle catalogue, apart from two thingsat the beginning. One is that we prepare the Gadget-format particledata for the ecosmog code in two ways. The halo catalogue is directlywritten into Gadget-format tracer particles due to its small numberdensity. However, the very large number of the simulation particles,along with their strongly non-uniform spatial distribution, in the darkmatter particle catalogues, leads to the requirement of large memoryfootprint when processing the data. To avoid this problem, we usethe publicly available dtfe code (Cautun & van de Weygaert 2011),based on Delaunay tessellation, to calculate the density field on aregular mesh with 512 cells employing the triangular shaped cloud(TSC) mass assignment scheme; then the mesh cells are regarded asuniformly-distributed fake particles with different masses, which aretransformed to Gadget format that can be directly read by ecosmog.The other particular thing is that we calculate the linear halo biasused for the reconstruction from the halo catalogue. The estimate ofthe halo bias is based on the relation 𝑏 ( 𝑟 ) = 𝜉 hh ( 𝑟 ) 𝜉 hm ( 𝑟 ) , (4)where 𝜉 hh ( 𝑟 ) is the auto-correlation function of haloes and 𝜉 hm ( 𝑟 ) isthe cross-correlation function between the haloes and the dark matterparticles. We use the publicly available cute code (Alonso 2012) tomeasure 𝜉 hh ( 𝑟 ) and 𝜉 hm ( 𝑟 ) from a given simulation snapshot, andtake the ratio between them to obtain the value of linear halo bias asa function of the distance 𝑟 . Since the linear halo bias is theoreticallya constant on large scales, we apply the method of least squares to thevalues on scales 𝑟 (cid:38) ℎ − Mpc to obtain an estimate of it. Note thatwhen dealing with observational data we do not necessarily have suchan accurate measurement of the linear halo or galaxy bias; however,Birkin et al. (2019) find that the exact value of linear bias is not veryimportant for this reconstruction method to recover the phases of theinitial density field.The following steps of reconstruction are then the same for bothdark matter particle catalogue and halo catalogues. First, ecosmogcalculates the density field in the Eulerian coordinates using the TSCmass assignment scheme, and solves the mapping between the Eu-lerian and Lagrangian coordinates, to get the displacement potentialas well as the displacement field on a regular mesh with 512 cells.We then use a Python code to transfer the output fields from the Eu-lerian coordinates to the Lagrangian coordinates. After that, becausethe Lagrangian coordinates are not uniform, we feed the dtfe codewith the Lagrangian coordinates and displacement field of the meshcells to calculate the reconstructed density field as the divergence of MNRAS , 1–14 (2021)
Y. Li, H.-M. Zhu & B. Li
Figure 1. [Colour Online] The left panel shows the comparison between the initial matter power spectra given by camb (red dashed lines) and the matter powerspectra measured from the initial conditions of the simulations generated using 2lptic (black lines), from the bottom up they are respectively the fiducial model,Model 1, Model 2 and Model 3, each model is shifted upwards by a factor of 10 successively to avoid the clutter of all curves. The right panel shows the 𝑃 rw results, cf., Eq. (3), obtained from the left panel for the three wiggled models, for instance, the bottom curves show the ratio of Model 1 to the fiducial model,followed by the ones for Model 2 and Model 3 upwards; each model is shifted upwards by a constant of 0 .
15 successively for the same reason as above. the displacement field w.r.t. the Lagrangian coordinates. Finally, wemeasure the reconstructed power spectrum from the reconstructeddensity field using a post-processing code.
As we discussed above, cosmic structure formation leads to dampingof the primordial wiggles. Reconstruction is expected to revert someof this damping, but cannot completely undo it. So we need a modelfor the wiggles of the reconstructed matter or halo power spectrum.Ideally this should be an analytical model since it can be more easilyused in the Fisher analysis later. In this subsection, we describe howthis is achieved by using a fitting function.Instead of fitting the absolute matter and halo power spectra, wepropose an analytic formula to directly fit the quantity 𝑃 rw obtainedfrom the simulations and reconstructions, cf. Eq. (3), which combinesthe oscillatory feature model and a Gaussian damping function, givenby 𝑃 rw ( 𝑘, 𝑧 ) = 𝐴 cos (cid:0) 𝜔𝑘 𝑚 + 𝜙 (cid:1) exp (cid:20) − 𝑘 𝜁 ( 𝑧 ) (cid:21) , (5)where 𝜁 ( 𝑧 ) is the damping parameter that depends on the redshift 𝑧 .For the fitting of each measured 𝑃 rw ( 𝑘 ) , we let 𝜔 , 𝜙 and 𝜁 be the freeparameters because 𝜔 and 𝜙 play an essential role in determiningthe position of the peaks, and 𝜁 quantifies the extent of the dampingeffect. The parameters 𝐴 and 𝑚 are taken to be their theoretical valuesin Table 1. We apply the least-squares estimator to obtain the best-fitparameters by minimising 𝜒 = 𝑁 ∑︁ 𝑖 = (cid:2) 𝑃 rw , i ( 𝑧 ) − 𝑃 rw ( 𝑘 i , 𝑧 ; 𝜔, 𝜙, 𝜁 ) (cid:3) , (6)where 𝑃 rw , i ( 𝑧 ) are the data points of wiggle spectrum in the 𝑖 th 𝑘 binat reshift 𝑧 . Since there is only one realisation of simulation for eachmodel, we assume that the uncertainties of all data points 𝑃 rw , i ( 𝑧 ) arethe same and follow the same Gaussian distribution. Note that, as thequantity we fit is 𝑃 rw = 𝑃 w / 𝑃 nw −
1, this is equivalent to doing thefitting of 𝑃 w with √ 𝑃 nw as uncertainty (e.g., Feldman et al. 1994).We calculate the uncertainties of the best-fit parameters based on95 % confidence interval, as a rough estimate of the size of the errors.To minimise the influence of the cosmic variance on very large scales,we fit the data within the interval of 𝑘 = ( . − . ) ℎ Mpc − , whichcovers our intended range of scales. In order to investigate the impact of reconstruction, we will forecastthe constraints on the feature parameters for the DESI survey usingthe Fisher information matrix, and compare with the case of doing noreconstruction. For this purpose, we first model the observed broad-band galaxy power spectrum. Then we describe how to calculate theFisher information matrix, followed by its analytic marginalisation.Finally, we give the specifications of the DESI survey.
By combining the Eqs. (3) and (5), the featured galaxy power spec-trum in real space can be modelled as, 𝑃 mod ( 𝑘, 𝑧 ) = 𝑃 nl ( 𝑘, 𝑧 ) (cid:20) + 𝐴 cos (cid:0) 𝜔𝑘 𝑚 + 𝜙 (cid:1) exp (cid:18) − 𝑘 𝜁 ( 𝑧 ) (cid:19)(cid:21) , (7)where 𝑃 nl ( 𝑘, 𝑧 ) is the nonlinear matter power spectrum without theprimordial oscillatory features at 𝑧 , which includes the BAO wigglesand is equivalent to the nonlinear matter power spectrum of the no-wiggle model. However, since there is only one simulation realisationfor a single no-wiggle model, which cannot provide a smooth non-linear matter power spectrum, and since a fast method to get 𝑃 mod ismore convenient in the Fisher analysis, we use the halofit model inthe camb code to calculate 𝑃 nl ( 𝑘, 𝑧 ) instead.The broadband galaxy power spectrum in real space is not a directobservable due to the measurement in the angular and redshift coor-dinates instead of the 3D comoving coordinates. In order to relate theobserved galaxy power spectrum 𝑃 obs ( 𝒌 , 𝑧 ) to the modelled galaxypower spectrum 𝑃 mod ( 𝑘, 𝑧 ) , the standard practice is to project thegalaxies to their comoving positions assuming some reference cos-mology via the coordinate transformation based on the relations 𝑘 ref ⊥ = 𝐷 A ( 𝑧 ) 𝐷 refA ( 𝑧 ) 𝑘 ⊥ , 𝑘 ref (cid:107) = 𝐻 ref ( 𝑧 ) 𝐻 ( 𝑧 ) 𝑘 (cid:107) , (8)where 𝑘 (cid:107) and 𝑘 ⊥ are respectively the light-of-sight and transversecomponents of the wavevector 𝒌 , i.e., 𝑘 = | 𝒌 | = 𝑘 ⊥ + 𝑘 (cid:107) , the su-perscript ref denotes the reference cosmology, note that the referencecosmology hereafter is the same one used in the simulations unlessotherwise stated; 𝐷 A ( 𝑧 ) = 𝑟 ( 𝑧 )/( + 𝑧 ) is the angular diameter dis-tance at 𝑧 with the comoving distance 𝑟 ( 𝑧 ) : under the assumption of MNRAS000
By combining the Eqs. (3) and (5), the featured galaxy power spec-trum in real space can be modelled as, 𝑃 mod ( 𝑘, 𝑧 ) = 𝑃 nl ( 𝑘, 𝑧 ) (cid:20) + 𝐴 cos (cid:0) 𝜔𝑘 𝑚 + 𝜙 (cid:1) exp (cid:18) − 𝑘 𝜁 ( 𝑧 ) (cid:19)(cid:21) , (7)where 𝑃 nl ( 𝑘, 𝑧 ) is the nonlinear matter power spectrum without theprimordial oscillatory features at 𝑧 , which includes the BAO wigglesand is equivalent to the nonlinear matter power spectrum of the no-wiggle model. However, since there is only one simulation realisationfor a single no-wiggle model, which cannot provide a smooth non-linear matter power spectrum, and since a fast method to get 𝑃 mod ismore convenient in the Fisher analysis, we use the halofit model inthe camb code to calculate 𝑃 nl ( 𝑘, 𝑧 ) instead.The broadband galaxy power spectrum in real space is not a directobservable due to the measurement in the angular and redshift coor-dinates instead of the 3D comoving coordinates. In order to relate theobserved galaxy power spectrum 𝑃 obs ( 𝒌 , 𝑧 ) to the modelled galaxypower spectrum 𝑃 mod ( 𝑘, 𝑧 ) , the standard practice is to project thegalaxies to their comoving positions assuming some reference cos-mology via the coordinate transformation based on the relations 𝑘 ref ⊥ = 𝐷 A ( 𝑧 ) 𝐷 refA ( 𝑧 ) 𝑘 ⊥ , 𝑘 ref (cid:107) = 𝐻 ref ( 𝑧 ) 𝐻 ( 𝑧 ) 𝑘 (cid:107) , (8)where 𝑘 (cid:107) and 𝑘 ⊥ are respectively the light-of-sight and transversecomponents of the wavevector 𝒌 , i.e., 𝑘 = | 𝒌 | = 𝑘 ⊥ + 𝑘 (cid:107) , the su-perscript ref denotes the reference cosmology, note that the referencecosmology hereafter is the same one used in the simulations unlessotherwise stated; 𝐷 A ( 𝑧 ) = 𝑟 ( 𝑧 )/( + 𝑧 ) is the angular diameter dis-tance at 𝑧 with the comoving distance 𝑟 ( 𝑧 ) : under the assumption of MNRAS000 , 1–14 (2021) econstruction of primordial features flat universe it is given by 𝑟 ( 𝑧 ) = 𝑐𝐻 ∫ 𝑧 d 𝑧 (cid:48) (cid:104) Ω m ( + 𝑧 ) + Ω Λ (cid:105) − , (9)where Ω Λ = − Ω m is the current density parameter of the cosmo-logical constant, and the Hubble parameter 𝐻 ( 𝑧 ) is given by 𝐻 ( 𝑧 ) = 𝐻 (cid:104) Ω m ( + 𝑧 ) + Ω Λ (cid:105) . (10)Along with several main factors being considered, i.e., the redshift-space distortions (RSD) and shot noise, we can model the observedgalaxy power spectrum as 𝑃 obs ( 𝑘, 𝜇, 𝑧 ) = (cid:34) 𝐷 refA ( 𝑧 ) 𝐷 A ( 𝑧 ) (cid:35) 𝐻 ( 𝑧 ) 𝐻 ref ( 𝑧 ) 𝐹 FoG ( 𝑘, 𝜇, 𝑧 ) 𝜎 ( 𝑧 ) 𝑃 mod ( 𝑘, 𝑧 )+ 𝑁 gal ( 𝑧 ) , (11)where 𝜎 ( 𝑧 ) is the R.M.S. linear density fluctuations on the scale of8 ℎ − Mpc, 𝑁 gal ( 𝑧 ) = / 𝑛 g ( 𝑧 ) is the shot noise with 𝑛 g ( 𝑧 ) being thegalaxy number density, and the Finger-of-God factor 𝐹 FoG ( 𝑘, 𝜇, 𝑧 ) describing the effect of RSD is modelled as Ballardini et al. (2020) 𝐹 FoG ( 𝑘, 𝜇, 𝑧 ) = (cid:2) 𝑏 ( 𝑧 ) 𝜎 ( 𝑧 ) + 𝑓 ( 𝑧 ) 𝜎 ( 𝑧 ) 𝜇 (cid:3) + 𝑘 𝜇 𝜎 𝑟, 𝑝 / (cid:0) − 𝑘 𝜇 𝜎 𝑟,𝑧 (cid:1) , (12)where 𝑏 ( 𝑧 ) is the linear halo bias at 𝑧 , 𝑓 ( 𝑧 ) = d ln 𝐷 ( 𝑎 ) d ln 𝑎 , (13)is the linear growth rate at 𝑧 with 𝐷 ( 𝑎 ) and 𝑎 respectively beingthe linear growth factor and the scale factor (note that we normalise 𝐷 ( 𝑎 ) so that 𝐷 ( 𝑎 = ) = 𝜇 = cos 𝜃 with 𝜃 being theangle between the wavevector 𝒌 and the line of sight, i.e., 𝜇 = 𝑘 (cid:107) / 𝑘 , 𝜎 𝑟, 𝑝 = 𝜎 𝑝 /[ 𝐻 ( 𝑧 ) 𝑎 ] is the distance dispersion corresponding to thephysical velocity dispersion 𝜎 𝑝 whose fiducial value is taken to be290 km s − , and the last exponential damping factor accounts for theredshift error 𝜎 ( 𝑧 ) with 𝜎 𝑟,𝑧 = 𝑐𝜎 ( 𝑧 )/ 𝐻 ( 𝑧 ) . The Fisher matrix approach provides a method to propagate the un-certainties of the observable to the constraints on the cosmologicalparameters. Our calculation of the Fisher matrix is based on Tegmark(1997) and Seo & Eisenstein (2003), assuming that the power spec-trum of a given 𝑘 mode satisfies a Gaussian distribution which has avariance equal to the power spectrum itself, and that different bins of 𝑘 are independent of each other for large surveys, the Fisher matrixfor each redshift bin, with bin centre at 𝑧 = 𝑧 c , can be approximatedas 𝐹 𝑖 𝑗 ( 𝑧 c ) = 𝑉 eff ( 𝑧 c ) 𝜋 ∫ d 𝜇 × ∫ 𝑘 max 𝑘 min d 𝑘 𝑘 𝜕 ln 𝑃 obs ( 𝑘, 𝜇, 𝑧 c ) 𝜕𝜃 𝑖 𝜕 ln 𝑃 obs ( 𝑘, 𝜇, 𝑧 c ) 𝜕𝜃 𝑗 , (14)where 𝑘 min , 𝑘 max are respectively the minimum and maximum valuesof 𝑘 used for the forecast. We set 𝑘 min = . ℎ Mpc − and adopttwo values of 𝑘 max , respectively 0 . ℎ Mpc − and 0 . ℎ Mpc − , tocompare the constraints for different range of scales. The effectivevolume of the redshift bin 𝑉 eff ( 𝑧 c ) is expressed as 𝑉 eff ( 𝑧 c ) = (cid:20) + 𝑛 g ( 𝑧 ) 𝑃 obs ( 𝑘, 𝜇, 𝑧 ) (cid:21) − 𝑉 surv ( 𝑧 c ) , (15) where 𝑛 g ( 𝑧 ) 𝑃 obs ( 𝑘, 𝜇, 𝑧 ) is the signal-to-noise, the comoving surveyvolume 𝑉 surv ( 𝑧 c ) with the redshift bin width Δ 𝑧 is given by 𝑉 surv ( 𝑧 c ) = 𝜋 (cid:20) 𝑟 (cid:16) 𝑧 c + Δ 𝑧 (cid:17) − 𝑟 (cid:16) 𝑧 c − Δ 𝑧 (cid:17) (cid:21) Ω surv Ω sky , (16)where Ω surv and Ω sky are respectively the survey area and the areaof the full sky. Additionally, 𝜃 is the 8-dimensional parameter vectorwhich consists of five cosmological parameters and three oscillationparameters, 𝜔 c = Ω c ℎ , 𝜔 b = Ω b ℎ , ℎ, 𝑛 s , 𝐴 s , 𝐴, 𝜔, 𝜙. (17)The partial derivatives of 𝑃 obs ( 𝑘, 𝜇, 𝑧 c ) w.r.t. the cosmological pa-rameters are calculated numerically using the finite difference, 𝜕𝑃 obs ( 𝑘, 𝜇, 𝑧 c ) 𝜕𝜃 𝑖 = 𝑃 obs ( 𝜃 fid 𝑖 + Δ 𝜃 𝑖 ) − 𝑃 obs ( 𝜃 fid 𝑖 − Δ 𝜃 𝑖 ) Δ 𝜃 𝑖 , (18)where Δ 𝜃 𝑖 is taken to be 10% of the fiducial value of 𝜃 fid 𝑖 , though wehave explicitly checked that the partial derivative is insensitive to thesize of Δ 𝜃 𝑖 . By contrast, the partial derivatives w.r.t. the oscillationparameters can be calculated analytically due to the analytic form ofthe oscillations.The Fisher matrices of all redshift bins are then summed up toreturn a 8 × 𝐹 M 𝛼𝛽 = 𝐹 𝛼𝛽 − 𝐹 𝛼𝑚 𝐹 − 𝑚𝑛 𝐹 𝑛𝛽 , (19)where the subscripts 𝛼 and 𝛽 denote the target parameters, while 𝑚 and 𝑛 denote the nuisance parameters. Finally, we get the uncertain-ties of the oscillation parameters from the covariance matrix, i.e., theinverse of the marginalised Fisher matrix. The parameters used in the Fisher analysis, including those associatedwith the specifications of the DESI survey (DESI Collaboration et al.2016) are discussed here.We start with the most crucial parameter, the damping parameter 𝜁 displayed in Table 2, which depends not only on the redshifts but alsoon the halo number densities and – more importantly – whether thereconstruction is applied. We only have values of 𝜁 for four redshifts,i.e., 𝑧 =
0, 0 .
5, 1, 1 .
5, and two different halo number densities, i.e., 𝑛 halo = × − ( ℎ − Mpc ) − and 5 × − ( ℎ − Mpc ) − , but the fore-casted number density achievable in the DESI survey varies over theredshift range, so the values of 𝜁 may not apply to the entire redshiftrange. As a result, we cut off some high redshift bins which have thenumber density much smaller than 5 × − ( ℎ − Mpc ) − . We use abilinear interpolation between the redshift and the number densityto estimate an appropriate value of 𝜁 for a given combination of theredshift and number density. For those the number density is largerthan 1 × − ( ℎ − Mpc ) − or smaller than 5 × − ( ℎ − Mpc ) − , wesimply adopt the values of 𝜁 for 𝑛 halo = × − ( ℎ − Mpc ) − or 𝑛 halo = × − ( ℎ − Mpc ) − instead. In this work, we use differ-ent values of 𝜁 for the different models as obtained using the fittingmethod described in Section 2.4, and we will comment on this pointagain later.As we consider both emission line galaxies (ELGs) and luminous MNRAS , 1–14 (2021)
Y. Li, H.-M. Zhu & B. Li red galaxies (LRGs) in the DESI survey, which have different numberdensities and redshift distributions, different range of redshift binsis chosen for ELGs and LRGs in the Fisher analysis. After throwingaway the redshift bins with very small number densities, we take therange of 𝑧 = ( . − . ) for ELGs and 𝑧 = ( . − . ) for LRGs,and the redshift bin width is by default Δ 𝑧 = .
1. In addition to thecalculation of effective survey volume, by following the DESI survey,the fixed values of 𝑛 g ( 𝑧 ) 𝑃 obs ( . , . , 𝑧 ) are used for the signal-to-noise, two survey areas are considered including the expected surveyarea of 14,000 deg and 9,000 deg as the pessimistic case (DESICollaboration et al. 2016). As for the Finger-of-God factor, the linearhalo bias for ELGs and LRGs is simply defined in terms of the growthfactor via (DESI Collaboration et al. 2016) 𝑏 ELG ( 𝑧 ) 𝐷 ( 𝑧 ) = .
84 and 𝑏 LRG ( 𝑧 ) 𝐷 ( 𝑧 ) = . . (20)The DESI survey defines the redshift error as 𝜎 ( 𝑧 ) = . /( + 𝑧 ) ,in this case, the exponential damping factor is very close to 1 for ourintended range of scales, so we neglect it in the calculation. In this section, we will first compare the linear, nonlinear and recon-structed 𝑃 rw measured for all models and redshifts. Then we presentthe results of the analytic fit to more quantitatively demonstrate theimprovement by the reconstruction. Finally we show the results ofthe constraints on the oscillation parameters and give forecast for theDESI survey. In Fig. 2 we compare the results of the linear, nonlinear and recon-structed 𝑃 rw ( 𝑘 ) obtained from DM, H1 and H2 at the four redshiftsfor the three wiggled models. The black solid lines represent thelinear 𝑃 rw ( 𝑘 ) obtained from the initial conditions of the simulations,which are equivalent to the primordial oscillatory features. The bluedashed lines represent the nonlinear 𝑃 rw ( 𝑘 ) obtained from the outputsnapshots of the simulations, which are also referred to as the unre-constructed 𝑃 rw ( 𝑘 ) for convenience. It can be seen that the wiggleson small scales are gradually damped as the redshift decreases. Thered dash-dotted lines represent the reconstructed 𝑃 rw ( 𝑘 ) obtainedfrom the reconstructed density field, which helps to partially retrievethe lost wiggles.The 𝑃 rw ( 𝑘 ) results shown in the first column are obtained fromDM, which exhibit some common characteristics for all three wiggledmodels. By comparing the unreconstructed results with the linear-theory predictions, it seems that the scale at which the wiggles start tobe weakened becomes larger as time progresses, though the specificrange differs a bit for different models due to the discrepancies in theiroriginal shape of the oscillations. For instance, deviation from lineartheory at 𝑧 = 𝑘 (cid:38) . ℎ Mpc − for Model 1, 𝑘 (cid:38) . ℎ Mpc − for Model 2 and 𝑘 (cid:38) . ℎ Mpc − for Model 3. Furthermore, thewiggles on scales 𝑘 (cid:38) . ℎ Mpc − are almost totally lost at 𝑧 = . (cid:46) 𝑘 (cid:46) . ℎ Mpc − would be an important objective of reconstruction. By comparingthe reconstructed 𝑃 rw results with the linear-theory prediction, it canbe seen that, despite some imperfection, the reconstruction still to alarge extent achieves this by retrieving the initial oscillations on ourinterested scales, i.e., 0 . (cid:46) 𝑘 (cid:46) . ℎ Mpc − .The success of the reconstruction from the dark matter particlesis largely thanks to their high number density, which allows the late-time nonlinear density field to be accurately produced: in this sense, reconstruction from DM can be considered as an idealised case or anupper limit, which is difficult to achieve in real observations. For arough comparison, we have shown, in the middle and right columnsof Fig. 2, the 𝑃 rw results obtained from the two halo catalogues, H1and H2, which have number densities similar to typical real galaxycatalogues. These results are less impressive than those for the darkmatter particles because of the much smaller halo number densities.Also due to the small halo number densities, these results are noisier,which in theory can be made smoother by having more realisationsof simulations, or equivalently a larger volume.By comparing the results of H1 and H2 for the same model, we findthat there is no significant difference in the unreconstructed 𝑃 rw ( 𝑘 ) at the same redshift because the number densities of these two halocatalogues only differ by a factor of 2. In most cases the reconstructed 𝑃 rw results of H1 seem slightly better compared to those of H2, asa result of the slightly larger number densities in H1, though thedifference is again insignificant visually; we will revisit this pointwhen discussing the analytic fit in the next subsection. Comparingthe results with and without reconstruction, it is clear that the formerdoes give less damped and sharper oscillation features, confirmingthat reconstruction can help to partially recover the lost wiggles. Thisrecovery is more substantial at lower redshifts than at higher redshifts,since at higher redshifts there is little damping to start with. At lowerredshifts, on the other hand, reconstruction can even recover someof the wiggles at 𝑘 ∼ . ℎ Mpc − where there is literally nothing inthe unreconstructed case. We expect that this will greatly help in theaccurate measurements of wiggle parameters, especially in modelswith few wiggles at 𝑘 (cid:46) . ℎ Mpc − – we will discuss this in theparameter fittings next. Figs. 3, 4 and 5 show, respectively, the results of the analytic fit to theunreconstructed and reconstructed 𝑃 rw results for the three modelsstudied in this work. It can be seen that, in most cases, the analyticmodel Eq. (5), with a Gaussian damping function characterised bythe parameter 𝜁 ( 𝑧 ) , fits the data very well.The corresponding best-fit parameters of 𝜔 , 𝜙 and 𝜁 ( 𝑧 ) , as well astheir uncertainties, are displayed in Table 2, which assist the under-standing from a quantitative perspective. As mentioned before, wemainly focus on the results of H1 and H2, and thus the results of DMwould be taken as a reference and not be discussed in detail. Thethree parameters are mainly determined by the remaining peaks inthe wiggles. We shall first discuss the results of the damping param-eter, followed by the oscillation parameters, and then combine themto clarify the improvement given by reconstruction.The damping parameter 𝜁 effectively describes the extent of thenonlinear effects in structure formation and characterises the sup-pression of the primordial oscillations. It is zero in the linear regime,such as at 𝑧 =
49, and gradually increases as the redshift decreasesbecause the structures become progressively more nonlinear. Thusreconstruction aims to reduce 𝜁 and retrieve the primordial oscilla-tions. In Table 2 it shows that, despite the reconstructed values of 𝜁 are not reduced to zero due to the imperfection of reconstruction, theyare evidently smaller than the unreconstructed values in all cases, andthe uncertainties of 𝜁 are also reduced after reconstruction in mostlow-redshift ( 𝑧 <
1) cases, which confirms that the reconstructionsucceeds in retrieving the lost wiggles to a great extent. Specifically,by comparing the cases among different models but same cataloguesand redshifts, the corresponding values after reconstruction seem tobe nearly independent of the model, which implies that the improve-ment on the recovery of the wiggles does not depend on the shape of
MNRAS , 1–14 (2021) econstruction of primordial features Figure 2. [Colour Online] Comparisons among the linear (black solid line), nonlinear (blue dash-dotted line) and reconstructed (red dashed line) 𝑃 rw . Thelinear 𝑃 rw is measured from the initial conditions generated using 2LPTic, the nonlinear 𝑃 rw is measured from the output snapshots of the simulations, andthe reconstructed 𝑃 rw is obtained from the reconstructed density field. Each row represents one redshift 𝑧 which is shown on the right side. The three columnsdenote, respectively, the results from the dark matter particle catalogue DM and the halo catalogues H1 and H2. Every four rows from the bottom up respectivelybelong to Model 1, Model 2 and Model 3. the primordial oscillations . This is similar to the unreconstructed 𝜁 ,which supports the qualitative inference in the previous subsectionthat the nonlinear regime is similar at the same redshift for differentmodels, although the unreconstructed values of 𝜁 in Model 1 arecommonly a bit larger than those in Model 2 and Model 3. Addition-ally, for each model, the reconstructed values of 𝜁 in H1 are smallerthan those in H2 at the same redshift, and the same trend can beseen in the unreconstructed values as well, which is attributed tothat H1 has twice the halo number density as H2. Furthermore, foreach catalogue, it appears that the reconstructed 𝜁 is only reducedwith increasing redshift at low redshift ( 𝑧 < 𝑧 > 𝜁 decreases withincreasing redshift, thus the difference between the reconstructed and This makes sense given that the amplitude of the primordial oscillations issmall in this work, and so the wiggles can be considered as small perturbationsto the primordial density field. Reconstruction, on the other hand, is sensitiveto the overall distribution of matter. unreconstructed 𝜁 seems to be large at low redshift and small at highredshift. In other words, the improvement given by the reconstructionis effective at low redshift but relatively limited at high redshift.Next, we shall focus on whether the improved wiggles after recon-struction can lead to more accurate measurements of the oscillationparameters 𝜔 and 𝜙 . Regarding the oscillation frequency 𝜔 , the re-constructed values of 𝜔 are much closer to the theoretical values thanthe unreconstructed values in all cases, which is especially evident atlow redshifts. Except for a few high-redshift cases, the improvementon the uncertainties after reconstruction is evident in most cases aswell. Specifically, when comparing amongst different models, it ap-pears that the reconstructed 𝜔 values of Model 1 show slightly betterperformances over those of Model 2 and Model 3, which is becausethe reconstructed wiggles of Model 1 have four evident peaks withinthe fitting range of scales at all redshifts, while Model 2 and Model 3only have two; clearly more peaks can enhance the accuracy of the fit.Similarly, the unreconstructed values of Model 1 seem to be closerto the theoretical values than those of Model 2 and Model 3; moredetails will be discussed later. For each model, the reconstructed val-ues of H1 are a little bit better than those of H2 at the same redshiftin most low-redshift cases, which can be explained by the larger halonumber density of H1. By contrast, there is no evident distinction of MNRAS , 1–14 (2021)
Y. Li, H.-M. Zhu & B. Li
Figure 3. [Colour Online] The analytic fit to the unreconstructed and reconstructed 𝑃 rw for Model 1. The black solid lines represent the measured 𝑃 rw and thered dashed lines represent the fitting curves given by our analytic model, Eq. (5). The thin lines are for the unreconstructed cases and the thick lines are for thereconstructed cases. The three columns from left to right respectively denote the dark matter particle catalogue DM, and the halo catalogues H1 and H2. Everytwo rows from the bottom up represent the same redshift shown on the right side. In each group of two rows, the upper one is for the unreconstructed, and thelower one for the reconstructed, case. Figure 4. [Colour Online] The same as Fig. 3 but for Model 2.MNRAS000
Figure 3. [Colour Online] The analytic fit to the unreconstructed and reconstructed 𝑃 rw for Model 1. The black solid lines represent the measured 𝑃 rw and thered dashed lines represent the fitting curves given by our analytic model, Eq. (5). The thin lines are for the unreconstructed cases and the thick lines are for thereconstructed cases. The three columns from left to right respectively denote the dark matter particle catalogue DM, and the halo catalogues H1 and H2. Everytwo rows from the bottom up represent the same redshift shown on the right side. In each group of two rows, the upper one is for the unreconstructed, and thelower one for the reconstructed, case. Figure 4. [Colour Online] The same as Fig. 3 but for Model 2.MNRAS000 , 1–14 (2021) econstruction of primordial features Figure 5. [Colour Online] The same as Fig. 3 but for Model 3.
Table 2.
The best-fit parameters of 𝜔 , 𝜙 and 𝜁 and their 95% uncertainties for the three wiggled models studied in this work. The values of 𝜔 and 𝜙 in the tableare respectively in the units of 𝜋 ℎ − Mpc and 𝜋 , and their corresponding theoretical values are shown below the title of each model on the top of the table. DMdenotes the dark matter particle catalogue, H1 the halo catalogue with 𝑛 halo = × − ( ℎ − Mpc ) − and H2 the halo catalogue with 𝑛 halo = × − ( ℎ − Mpc ) − .Each group of six rows includes the unreconstructed and reconstructed cases for the same redshift.Model 1 Model 2 Model 3 𝜔 = 𝜙 = 𝜔 = . 𝜙 = 𝜔 = . 𝜙 = 𝑧 parasDM H1 H2 DM H1 H2 DM H1 H20 . 𝜔 . ± .
24 14 . ± .
36 14 . ± .
37 7 . ± .
74 7 . ± .
34 7 . ± .
41 6 . ± .
16 6 . ± .
25 6 . ± . 𝜙 . ± .
03 0 . ± .
05 0 . ± .
04 0 . ± .
08 0 . ± .
04 0 . ± .
05 0 . ± .
04 0 . ± .
06 0 . ± . 𝜁 . ± .
26 6 . ± .
40 7 . ± .
41 7 . ± .
95 6 . ± .
44 6 . ± .
53 6 . ± .
23 6 . ± .
35 6 . ± . 𝜔 . ± .
02 15 . ± .
21 14 . ± .
20 8 . ± .
03 8 . ± .
09 8 . ± .
23 7 . ± .
02 7 . ± .
09 7 . ± . 𝜙 − . ± .
01 0 . ± .
05 0 . ± .
04 0 . ± . − . ± . − . ± . − . ± . − . ± .
03 0 . ± . 𝜁 . ± .
03 3 . ± .
28 4 . ± .
25 2 . ± .
05 3 . ± .
13 4 . ± .
32 2 . ± .
04 3 . ± .
14 3 . ± . . 𝜔 . ± .
09 14 . ± .
20 14 . ± .
26 8 . ± .
29 8 . ± .
23 8 . ± .
22 6 . ± .
06 6 . ± .
15 6 . ± . 𝜙 . ± .
01 0 . ± .
03 0 . ± .
04 0 . ± .
04 0 . ± .
03 0 . ± .
03 0 . ± .
02 0 . ± .
05 0 . ± . 𝜁 . ± .
10 5 . ± .
23 6 . ± .
30 5 . ± .
40 5 . ± .
32 5 . ± .
29 5 . ± .
08 4 . ± .
23 5 . ± . 𝜔 . ± .
02 15 . ± .
08 14 . ± .
13 8 . ± .
02 8 . ± .
07 8 . ± .
15 7 . ± .
02 7 . ± .
07 7 . ± . 𝜙 − . ± .
01 0 . ± .
02 0 . ± .
03 0 . ± . − . ± . − . ± . − . ± . − . ± . − . ± . 𝜁 . ± .
04 3 . ± .
10 3 . ± .
17 1 . ± .
04 3 . ± .
10 3 . ± .
20 1 . ± .
04 3 . ± .
11 3 . ± . . 𝜔 . ± .
04 14 . ± .
17 14 . ± .
17 8 . ± .
13 8 . ± .
16 8 . ± .
19 6 . ± .
02 6 . ± .
13 6 . ± . 𝜙 . ± .
01 0 . ± .
03 0 . ± .
03 0 . ± .
02 0 . ± .
03 0 . ± .
03 0 . ± .
01 0 . ± .
04 0 . ± . 𝜁 . ± .
05 4 . ± .
21 5 . ± .
20 4 . ± .
18 4 . ± .
22 5 . ± .
26 4 . ± .
03 4 . ± .
20 4 . ± . 𝜔 . ± .
01 15 . ± .
10 15 . ± .
15 8 . ± .
01 8 . ± .
07 8 . ± .
14 7 . ± .
01 7 . ± .
09 7 . ± . 𝜙 . ± .
01 0 . ± . − . ± .
03 0 . ± . − . ± .
02 0 . ± . − . ± .
01 0 . ± .
03 0 . ± . 𝜁 . ± .
03 3 . ± .
14 3 . ± .
20 1 . ± .
04 3 . ± .
11 3 . ± .
20 1 . ± .
03 3 . ± .
13 3 . ± . . 𝜔 . ± .
02 14 . ± .
17 14 . ± .
22 8 . ± .
07 8 . ± .
14 7 . ± .
18 6 . ± .
01 6 . ± .
15 6 . ± . 𝜙 . ± .
01 0 . ± .
03 0 . ± .
03 0 . ± .
01 0 . ± .
03 0 . ± .
03 0 . ± .
01 0 . ± .
05 0 . ± . 𝜁 . ± .
02 4 . ± .
21 5 . ± .
26 3 . ± .
10 4 . ± .
20 4 . ± .
25 3 . ± .
02 3 . ± .
22 4 . ± . 𝜔 . ± .
01 15 . ± .
09 14 . ± .
23 8 . ± .
01 8 . ± .
10 8 . ± .
23 7 . ± .
01 7 . ± .
11 7 . ± . 𝜙 . ± . − . ± .
02 0 . ± .
05 0 . ± .
01 0 . ± . − . ± .
05 0 . ± . − . ± . − . ± . 𝜁 . ± .
03 3 . ± .
12 3 . ± .
29 0 . ± .
03 3 . ± .
15 3 . ± .
32 0 . ± .
02 3 . ± .
16 3 . ± . , 1–14 (2021) Y. Li, H.-M. Zhu & B. Li the unreconstructed values at the same redshift between H1 and H2,which could result from the non-negligible noise in the wiggle spec-tra. It can be checked by having more realisations of the simulationsin the future. Moreover, for each catalogue, the reconstructed valuesbecome closer to the theoretical values as the redshift increases in thelow-redshift range, but it cannot be further improved at high redshift,which is similar to how the redshift affects the reconstructed 𝜁 as wefound above. By contrast, it is shown that the unreconstructed valuesof DM become more accurate with increasing redshift, but those ofH1 and H2 do not show the same trend, which could be caused by theevident noise in the wiggle spectra due to the wide gap between thenumber densities of dark matter particles and haloes. Therefore, thereconstruction mainly improves the prediction of 𝜔 at low redshift,and the predictions for Model 2 and Model 3 are improved more thanthose for Model 1.The situation is quite different in the case of the oscillation phase 𝜙 .The unreconstructed values of Model 1 and Model 2 are determinedvery well in most cases, so the reconstructed values only show a littlebit improvement on the unreconstructed 𝜙 even for low-redshift cases.However, for Model 3 the unreconstructed values largely deviate fromthe theoretical value in all cases, and the unreconstructed values of H2deviate even further than those of H1 at the same redshift. Thereforethe reconstruction once again shows its advantage of allowing moreaccurate measurement of 𝜙 , especially for H2 at low redshift.When combining the results of all three parameters, it seems thatthe reconstruction is most useful at low redshifts, 𝑧 <
1, and Model2 and Model 3 benefit more from it than Model 1. Although the re-maining peaks of Model 1 are kept very well so that its reconstructedresults are better than those of the other two models, the improve-ment is relatively limited for it. Thus the improvement depends lesson how great the reconstructed wiggles are, and more on how poorthe primordial wiggles are kept before reconstruction; in other words,the reconstruction would be more useful if the primordial wiggles arelost to a greater extent. As we mentioned before, the wiggles on scales 𝑘 (cid:38) . ℎ Mpc − are totally lost at 𝑧 =
0, Model 1 has exactly the firsttwo original peaks outside this range of scales, so these two peaks areeffectively preserved at low redshift. By contrast, Model 3 has oneoriginal peak at the same position of the first peak of Model 1 whichis effectively preserved, and its second peak is at the same position ofthe third peak of Model 1, which is almost lost. Hence the primordialwiggles of Model 3 are preserved less well than those of Model 1,and Model 3 would benefit more from the reconstruction. Similarly,Model 2 has two original peaks in the range 𝑘 (cid:46) . ℎ Mpc − : thefirst is on a smaller scale compared with the first peak of other twomodels, and so it is not preserved as well as the first peak of the othertwo models due to the larger damping effect, while the second peakis totally wiped out. Thus the primordial wiggles of Model 2 are kepteven worse than those of Model 3. However, since the fitting range ofscales includes the first trough to the left of the first peak for Model 2but not for Model 3, this partially balances the accuracy of the fit forModel 2. Therefore the improvement by the reconstruction is simi-lar for Model 2 and Model 3, and both benefit from reconstructionsubstantially more than Model 1. Since the results of the constraints on the oscillation parameters aresimilar between Model 2 and Model 3, we take Model 1 and Model2 as two examples to illustrate and discuss how the reconstructioncould improve the constraints in a real galaxy survey. We shall firsttalk about some common features exhibited in both models, and thenclarify the distinctions between them. Finally, we also forecast how much the uncertainties of the feature amplitude can be improved afterreconstruction for three models.Figs. 6 and 7 show, respectively, the constraints on the oscillationparameters for a DESI-like survey with a survey area of 14,000 deg ,based on the primordial oscillations of Model 1 and Model 2. Themarginalised likelihoods of the oscillation parameters shown in theupper panels of both figures indicate that the unreconstructed caseswith 𝑘 max = . ℎ Mpc − (red lines) give better constraints than theunreconstructed cases with 𝑘 max = . ℎ Mpc − (grey), because inthe former case more 𝑘 modes are included in the Fisher matrix whichincrease the accuracy of the constraints. Additionally, by comparingthe likelihoods of the same 𝑘 max , we find that reconstruction leadsto more robust constraints on the parameters, because it successfullyrecovers some of the lost peaks in the nonlinear regime and providesmore effective 𝑘 modes. Furthermore, stronger constraints are shownfor ELGs (right panels) compared with LRGs (left panels), sincethe former has more available redshift bins and larger halo numberdensity for the same redshift bins.The above trends are also shown in the marginalised 2D confidencecontours in the lower part of the corner plots in Fig. 6 and 7. Inparticular, it can be seen in most contours that the cases for 𝑘 max = . ℎ Mpc − are largely improved by reconstruction as the boundaryof 68 % limits shrinks to be around the boundary of 95 % limits afterreconstruction. It is because the peaks on scales 𝑘 (cid:38) . ℎ Mpc − are heavily damped at low redshift, and the recovered wiggles at 𝑘 = ( . − . ) ℎ Mpc − significantly contribute to the constraints.By contrast, since the peaks on scales 𝑘 (cid:46) . ℎ Mpc − are preservedvery well, the recovery of wiggles for 𝑘 max = . ℎ Mpc − is not asimportant as in the case for 𝑘 max = . ℎ Mpc − . Besides, the 𝐴 - 𝜔 and 𝐴 - 𝜙 contours for 𝑘 max = . ℎ Mpc − show that these parametersare degenerate with each other, and these degeneracies are brokenand replaced with stronger constraints when 𝑘 max = . ℎ Mpc − ,which include more 𝑘 modes.However, in all cases the 𝜔 - 𝜙 contours show that the two param-eters are strongly degenerate. It is understood from the 𝑃 rw resultsthat the region 𝑘 = ( . − . ) ℎ Mpc − dominates the results ofconstraints, because on larger scales ( 𝑘 (cid:46) . ℎ Mpc − ) the un-certainty is large due to cosmic variance, while on smaller scales( 𝑘 (cid:38) . ℎ Mpc − ) the wiggles are damped which have less influenceon the constraints. We have checked explicitly that combinations of 𝜔 and 𝜙 along the degeneracy direction shown in Figs. 6 and 7 lead tolittle change to the 𝑃 rw curves in the range of 𝑘 = ( . − . ) ℎ Mpc − .This is a feature of the oscillation model itself, which is why a strongdegeneracy is still present even in the cases of reconstruction (albeitsignificantly less strong than the unreconstructed cases).When comparing the constraints between Model 1 and Model 2,it appears that the choice of 𝑘 max has a different influence on theconstraints. For both models, the constraints are similar for 𝑘 max = . ℎ Mpc − but significantly different for 𝑘 max = . ℎ Mpc − . ForModel 2 there is a wide gap between the marginalised 1D distri-butions of both the unreconstructed and reconstructed cases, whenincreasing 𝑘 max from 0 . ℎ Mpc − to 0 . ℎ Mpc − , which could beattributed to the fact that Model 2 has few peaks at 𝑘 (cid:46) . ℎ Mpc − .Thus for the primordial oscillations with relatively low frequency, in-creasing 𝑘 max would be very beneficial to improve the constraints.Furthermore, the improvement on the constraints by the reconstruc-tion shows that Model 2 benefits more from the reconstruction thanModel 1, which is consistent with the analysis in Section 4.2.Lastly, as Ballardini et al. (2020), we show the marginalised un-certainties of 𝐴 as a function of 𝜔 for the three models in Fig. 8 anddiscuss the implications of the results.First, we focus on Model 1 and Model 2 which show almost iden- MNRAS , 1–14 (2021) econstruction of primordial features Figure 6. [Colour Online] Forecasts of constraints on the oscillatory feature parameters for a DESI-like survey with a survey area of 14 ,
000 deg , for theprimordial oscillations of Model 1. The left side is for LRGs and the right side is for ELGs. The upper panels show the 1D marginalised likelihoods. The middleand lower panels show the marginalised 68% and 95% confidence contours for every two out of three feature parameters. The green and grey colours represent,respectively, the cases for 𝑘 max = . ℎ Mpc − with and without reconstruction, while the blue and red colours represent the cases for 𝑘 max = . ℎ Mpc − withand without reconstruction. Figure 7. [Colour Online] The same as Fig. 6 but for Model 2. MNRAS , 1–14 (2021) Y. Li, H.-M. Zhu & B. Li
Figure 8. [Colour Online] Forecasts of the marginalised uncertainties of the oscillation amplitude 𝐴 as a function of the frequency 𝜔 , for the three models. Thefirst column is for LRGs and the second column is for ELGs. The bottom panels are for Model 1, followed by Model 2 and Model 3 upwards. The dotted blacklines mark the theoretical amplitudes of the oscillations, 𝐴 = .
05, used in the forecasts. The meanings of the different colours and line styles are indicated in thelegends. The same colours represent the cases with same 𝑘 max and same situation of reconstruction but different survey areas; the thick lines are for the surveyarea of 14 ,
000 deg and the thin lines are for 9 ,
000 deg . tical variations of the uncertainties w.r.t. 𝜔 , due to their identicalform of the oscillations — we note that these are the same oscillationmodel with different choices of 𝜔 , and thus we should have expected exactly identical constraints in Fig. 8. However, as we have discussedabove, the best-fit values of 𝜁 are slightly different (cf. Table 2), eventhough we expect any model dependence of 𝜁 to be weak (this sug-gests that more realisations of simulations are needed to measure 𝜁 as functions of halo number density, redshift, and model (or modelparameter), etc., more accurately, which will be left for future work). As expected, ELGs place slightly tighter constraints than LRGs dueto their larger number densities and redshift range. The sharp peaksappeared at 𝜔 (cid:39) ℎ − Mpc are due to the degeneracy between theoscillatory features and the BAO wiggles over the signal-dominatedrange of scales. For 𝜔 (cid:38) ℎ − Mpc the uncertainties do not varywith 𝜔 , but for smaller 𝜔 things are complicated for different 𝑘 max .For 𝑘 max = . ℎ Mpc − we can see an increase in the uncertaintiesat 𝜔 (cid:46) ℎ − Mpc, while a similar increase starts to appear at evensmaller 𝜔 – 20 ℎ − Mpc – for 𝑘 max = . ℎ Mpc − . Thus larger 𝑘 max MNRAS000
000 deg . tical variations of the uncertainties w.r.t. 𝜔 , due to their identicalform of the oscillations — we note that these are the same oscillationmodel with different choices of 𝜔 , and thus we should have expected exactly identical constraints in Fig. 8. However, as we have discussedabove, the best-fit values of 𝜁 are slightly different (cf. Table 2), eventhough we expect any model dependence of 𝜁 to be weak (this sug-gests that more realisations of simulations are needed to measure 𝜁 as functions of halo number density, redshift, and model (or modelparameter), etc., more accurately, which will be left for future work). As expected, ELGs place slightly tighter constraints than LRGs dueto their larger number densities and redshift range. The sharp peaksappeared at 𝜔 (cid:39) ℎ − Mpc are due to the degeneracy between theoscillatory features and the BAO wiggles over the signal-dominatedrange of scales. For 𝜔 (cid:38) ℎ − Mpc the uncertainties do not varywith 𝜔 , but for smaller 𝜔 things are complicated for different 𝑘 max .For 𝑘 max = . ℎ Mpc − we can see an increase in the uncertaintiesat 𝜔 (cid:46) ℎ − Mpc, while a similar increase starts to appear at evensmaller 𝜔 – 20 ℎ − Mpc – for 𝑘 max = . ℎ Mpc − . Thus larger 𝑘 max MNRAS000 , 1–14 (2021) econstruction of primordial features has an extra advantage of significantly reducing the uncertainties forsmall 𝜔 , in addition to giving more stringent constraints (everythingelse the same) for all 𝜔 overall. By comparing the pairs of curves withthe same colours, i.e., the same cases ( 𝑘 max and reconstructed vs. un-reconstructed) but different survey areas, we find that, as expected, alarger survey area always gives better constraints.Most interestingly, everything else equal, doing reconstruction cansignificantly reduce the uncertainties of 𝐴 . As an example, for large 𝜔 values, in the case of 𝑘 max = . ℎ Mpc − and survey area equal to14 ,
000 deg , reconstruction reduces 𝜎 ( 𝐴 ) from ∼ .
003 to ∼ . ,
000 to 14 ,
000 deg with 𝑘 max fixed to 0 . . ℎ Mpc − , or increasing 𝑘 max from 0 .
25 to 0 . ℎ Mpc − keepingthe survey area fixed to either 9 ,
000 or 14 ,
000 deg . A similarly goodimprovement can be seen with 𝑘 max = . ℎ Mpc − or survey areaequal to 9 ,
000 deg , when doing reconstruction. In certain cases,e.g., the large- 𝜔 regime of the lower panels of Fig. 8, reconstructionwith 𝑘 max = . ℎ Mpc − and a survey area equal to 9 ,
000 deg (the thin green dashed line) can lead to comparable constraints to notdoing reconstruction but with 𝑘 max = . ℎ Mpc − and a survey areaequal to 14 ,
000 deg (the thick orange dot-dashed line). Given thatincreasing survey area is usually not possible, and increasing 𝑘 max isalso not straightforward given the effort required to model the non-linear regime of matter clustering, especially with RSD, performingreconstruction seems to be a cheap way to maximise the exploitationand scientific return of survey data.The behaviours of Model 3 are broadly similar to those of Model1 and Model 2, e.g., both the absolute and the relative heights of thedifferent curves, as well as their shapes are the same as before. Thereare, however, some notable differences, e.g., the main peaks in 𝜎 ( 𝐴 ) in Model 3 are at slightly different values of 𝜔 from the other models,and the curves are also less smooth. As mentioned above, the bump(which has the structure of a double peak) of 𝜎 ( 𝐴 ) for Model 1 andModel 2 is related to the BAO peak in the matter/galaxy correlationfunction, which is at (cid:39) ℎ − Mpc. The primordial wiggles for thosetwo models, in configuration space, correspond to a spike at matter orhalo separation 𝑟 = 𝜔 . In those two models, when 𝜔 (cid:29) ℎ − Mpc,the BAO and primordial peaks are separated afar and thus the formerdoes not affect the accuracy of the measurement for the latter. As 𝜔 approaches 100 ℎ − Mpc from above, the BAO and primordial peaksstart to ‘interfere’, leading to changes of both the amplitude and shapeof the latter, making it harder to measure its parameters accurately.We speculate that the dip — which causes the double-peak structurein 𝜎 ( 𝐴 ) for Model 1 and Model 2 — is due to the fact that, when theprimordial peak does not coincide well with the centre of the (ratherwide) BAO peak, its shape can be affected in an asymmetric manner,making the measurement of its parameters even more inaccurate. Incontrast, the structure of the primordial wiggles in Model 3 is muchmore complicated in configuration space, because 𝑚 ≠ 𝜎 ( 𝐴 ) between thisand the other models. In this paper, we have investigated the effect of density reconstructionon retrieving hypothetical oscillatory features in the primordial powerspectrum which are erased on small scales in the late-time Universedue to the nonlinear cosmological evolution.We considered three different oscillatory features that are addedto a simple power-law primordial power spectrum, for which we ranN-body simulations and identified dark matter halo catalogues from the output snapshots at a number of redshifts. We reconstructed theinitial density field from the particle data, and halo catalogues withdifferent number densities. Finally, we compared the fitted feature pa-rameters from the unreconstructed and reconstructed density fields,to identify the improvement by reconstruction. We showed that re-construction can be very effective in helping retrieve the lost wiggles— with the finite volume of our simulations, not only does it lead tomuch less biased best-fit values of the feature parameters, but it alsosubstantially shrinks the measurement uncertainty. The improvementwas especially strong where the primordial features have been moreseverely erased to start with, such as at 𝑧 < 𝑘 range.While reconstruction is commonly used in improving the measure-ment of the BAO scale, and hence the determination of the expansionrate and properties of dark energy, this work has demonstrated thatsimilar applications are possible in other cases where certain featuresin matter clustering are present. This is particularly true if these fea-tures are in the mildly nonlinear regime, 0 . (cid:46) 𝑘 /( ℎ Mpc − ) (cid:46) . a priori — this is howwe forecasted constraints on the oscillation amplitude 𝐴 . However,the reconstruction step is completely independent of any assumptionof a particular primordial feature, and hence any method developedfor detecting general features from the matter clustering should applyto and benefit from the reconstructed density field.As a first step, the present study is based on various simplifications,and we discuss a couple here which can be improved in the future. Thefirst is related to the damping parameter 𝜁 post-reconstruction. Aswe have seen, 𝜁 controls the improvement over the unreconstructedcase. The simulations carried out for this work — due to their limitedresolution, box size, output snapshots numbers, coverage of modelsand number of realisations — did not allow us to more accuratelyquantify how 𝜁 depends on the oscillation model (though we suspectthat any dependence on parameters 𝐴, 𝜔 should be weak as long as 𝐴 is small), redshift, or the tracer type or number density. It certainlywould be great if better simulations will become available, allowingfurther improvements on these aspects.The second is related to the modelling of redshift-space distortions(RSD), for which we have adopted a simplistic prescription and wellpushed beyond the limit (e.g., 𝑘 (cid:39) . ℎ Mpc − ) where it is expectedto work. This is not an issue for a forecast work, but for constraintsusing real data it should be treated more carefully. The reconstructionmethod here has been extended to remove RSD from observed galaxycatalogues (Wang et al. 2020), though that is unlikely to work reliablyat 𝑘 as large as (cid:39) . ℎ Mpc − . Of course, we can always cut 𝑘 max to something that we are comfortable with. However, as mentionedabove, if we would like to take maximum benefit from reconstruction, MNRAS , 1–14 (2021) Y. Li, H.-M. Zhu & B. Li it is likely that we need to go substantially beyond 𝑘 (cid:39) . ℎ Mpc − .This can be achieved, for example, by using emulators of redshift-space galaxy or halo clustering (see, e.g., Zhai et al. 2019; Kobayashiet al. 2020); actually, as long as the primordial oscillations are weak(as implied by current null detections), one might assume that theirpresence has little or negligible impact on RSD.The ultimate objective, of course, is to apply this method to realobservation data from future galaxy surveys such as Euclid and DESI.For this, the above-mentioned improvements, amongst many others,would need to be done properly. These will be left for future works,in which we plan to carry out updated forecasts for these surveys andeventually real constraints. ACKNOWLEDGEMENTS
DATA AVAILABILITY
Simulation data used in this work can be made available upon requestto the authors.
REFERENCES
Adams J. A., Cresswell B., Easther R., 2001, Phys. Rev. D, 64, 123514Ade P. A. R., et al., 2014a, Astron. Astrophys., 571, A22Ade P. A. R., et al., 2014b, Astron. Astrophys., 571, A24Ade P. A. R., et al., 2016a, Astron. Astrophys., 594, A17Ade P. A. R., et al., 2016b, Astron. Astrophys., 594, A20Adshead P., Dvorkin C., Hu W., Lim E. A., 2012, Phys. Rev. D, 85, 023531Aghanim N., et al., 2020, Astron. Astrophys., 641, A6Akrami Y., et al., 2020a, Astron. Astrophys., 641, A9Akrami Y., et al., 2020b, Astron. Astrophys., 641, A10Albrecht A., Steinhardt P. J., 1982, Phys. Rev. Lett., 48, 1220Alonso D., 2012, arXiv e-prints, p. arXiv:1210.1833Ballardini M., Finelli F., Fedeli C., Moscardini L., 2016, JCAP, 10, 041Ballardini M., Murgia R., Baldi M., Finelli F., Viel M., 2020, JCAP, 04, 030Bartolo N., Komatsu E., Matarrese S., Riotto A., 2004, Phys. Rept., 402, 103Bartolo N., Cannone D., Matarrese S., 2013, JCAP, 10, 038Bean R., Chen X., Hailu G., Tye S. H. H., Xu J., 2008, JCAP, 03, 026Behroozi P. S., Wechsler R. H., Wu H.-Y., 2013, Astrophys. J., 762, 109Beutler F., Biagetti M., Green D., Slosar A., Wallisch B., 2019, Phys. Rev.Res., 1, 033209Birkin J., Li B., Cautun M., Shi Y., 2019, Mon. Not. Roy. Astron. Soc., 483,5267Bozza V., Giovannini M., Veneziano G., 2003, JCAP, 05, 001Byrnes C. T., Choi K.-Y., 2010, Adv. Astron., 2010, 724525Cautun M. C., van de Weygaert R., 2011, arXiv e-prints, p. arXiv:1105.0370 Celoria M., Matarrese S., 2020, Proc. Int. Sch. Phys. Fermi, 200, 179Chen X., 2010, Adv. Astron., 2010, 638979Chen X., Namjoo M. H., 2014, Phys. Lett. B, 739, 285Chen X., Huang M.-x., Kachru S., Shiu G., 2007, JCAP, 01, 002Chen X., Namjoo M. H., Wang Y., 2015, JCAP, 02, 027Chen X., Dvorkin C., Huang Z., Namjoo M. H., Verde L., 2016, JCAP, 11,014Chluba J., Hamann J., Patil S. P., 2015, Int. J. Mod. Phys. D, 24, 1530023Colombi S., Novikov D., 2011, POWMES: Measuring the Power Spectrumin an N-body Simulation (ascl:1110.017)Crocce M., Pueblas S., Scoccimarro R., 2006, Mon. Not. Roy. Astron. Soc.,373, 369DESI Collaboration et al., 2016, arXiv e-prints, p. arXiv:1611.00036Doré O., et al., 2014, arXiv e-prints, p. arXiv:1412.4872Eisenstein D. J., Seo H.-j., Sirko E., Spergel D., 2007, Astrophys. J., 664, 675Feldman H. A., Kaiser N., Peacock J. A., 1994, Astrophys. J., 426, 23Flauger R., McAllister L., Pajer E., Westphal A., Xu G., 2010, JCAP, 06, 009Flauger R., Mirbabayi M., Senatore L., Silverstein E., 2017, JCAP, 10, 058Guth A. H., 1981, Phys. Rev. D, 23, 347Hazra D. K., Shafieloo A., Smoot G. F., Starobinsky A. A., 2014, JCAP, 08,048Hinshaw G., et al., 2013, Astrophys. J. Suppl., 208, 19Kazin E. A., et al., 2014, Mon. Not. Roy. Astron. Soc., 441, 3524Kobayashi Y., Nishimichi T., Takada M., Takahashi R., Osato K., 2020, Phys.Rev. D, 102, 063504Komatsu E., et al., 2009, Astrophys. J. Suppl., 180, 330L’Huillier B., Shafieloo A., Hazra D. K., Smoot G. F., Starobinsky A. A.,2018, Mon. Not. Roy. Astron. Soc., 477, 2503Lewis A., Challinor A., 2011, CAMB: Code for Anisotropies in the Mi-crowave Background (ascl:1102.026)Li B., Zhao G.-B., Teyssier R., Koyama K., 2012, JCAP, 01, 051Li B., Barreira A., Baugh C. M., Hellwing W. A., Koyama K., Pascoli S.,Zhao G.-B., 2013, JCAP, 11, 012Linde A. D., 1982, Phys. Lett. B, 108, 389Liu Y., Yu Y., Li B., 2020, arXiv e-prints, p. arXiv:2012.11251Mao T.-X., Wang J., Li B., Cai Y.-C., Falck B., Neyrinck M., Szalay A., 2021,Mon. Not. Roy. Astron. Soc., 501, 1499Palma G. A., Sapone D., Sypsas S., 2018, JCAP, 06, 004Peiris H. V., et al., 2003, Astrophys. J. Suppl., 148, 213Racca G. D., et al., 2016, Proc. SPIE Int. Soc. Opt. Eng., 9904, 0OSarpa E., Schimd C., Branchini E., Matarrese S., 2019, Mon. Not. Roy. Astron.Soc., 484, 3818Schmittfull M., Feng Y., Beutler F., Sherwin B., Chu M. Y., 2015, Phys. Rev.D, 92, 123522Senatore L., Smith K. M., Zaldarriaga M., 2010, JCAP, 01, 028Seo H.-J., Eisenstein D. J., 2003, Astrophys. J., 598, 720Shi Y., Cautun M., Li B., 2018, Phys. Rev. D, 97, 023505Spergel D. N., et al., 2007, Astrophys. J. Suppl., 170, 377Taylor A. N., Kitching T. D., 2010, Mon. Not. Roy. Astron. Soc., 408, 865Tegmark M., 1997, Phys. Rev. Lett., 79, 3806Teyssier R., 2002, Astron. Astrophys., 385, 337Tinker J. L., Kravtsov A. V., Klypin A., Abazajian K., Warren M. S., YepesG., Gottlober S., Holz D. E., 2008, Astrophys. J., 688, 709Wang X., Feng B., Li M., Chen X.-L., Zhang X., 2005, Int. J. Mod. Phys. D,14, 1347Wang X., Yu H.-R., Zhu H.-M., Yu Y., Pan Q., Pen U.-L., 2017, Astrophys.J. Lett., 841, L29Wang Y., Li B., Cautun M., 2020, Mon. Not. Roy. Astron. Soc., 497, 3451Zeng C., Kovetz E. D., Chen X., Gong Y., Muñoz J. B., Kamionkowski M.,2019, Phys. Rev. D, 99, 043517Zhai Z., et al., 2019, Astrophys. J., 874, 95Zhu H.-M., Yu Y., Pen U.-L., Chen X., Yu H.-R., 2017, Phys. Rev. D, 96,123502This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS000