a user’s guide to extracting cosmological information from
TRANSCRIPT
A User’s Guide to Extracting Cosmological Information from Line-Intensity Maps
Jose Luis Bernal,1, 2, 3 Patrick C. Breysse,4 Hector Gil-Marın,1, 3 and Ely D. Kovetz5
1ICC, University of Barcelona, IEEC-UB, Martı i Franques 1, E08028 Barcelona, Spain2Dept. de Fısica Quantica i Astrofısica, Universitat de Barcelona, Martı i Franques 1, E08028 Barcelona, Spain
3Institut d’Estudis Espacials de Catalunya (IEEC), E08034 Barcelona, Spain4Canadian Institute for Theoretical Astrophysics, University of Toronto,
60 St. George Street, Toronto, ON, M5S 3H8, Canada5Department of Physics, Ben-Gurion University, Be’er Sheva 84105, Israel
Line-intensity mapping (LIM) provides a promising way to probe cosmology, reionization andgalaxy evolution. However, its sensitivity to cosmology and astrophysics at the same time is also anuisance. Here we develop a comprehensive framework for modelling the LIM power spectrum, whichincludes redshift space distortions and the Alcock-Paczynski effect. We then identify and isolatedegeneracies with astrophysics so that they can be marginalized over. We study the gains of using themultipole expansion of the anisotropic power spectrum, providing an accurate analytic expression fortheir covariance, and find a 10%-60% increase in the precision of the baryon acoustic oscillation scalemeasurements when including the hexadecapole in the analysis. We discuss different observationalstrategies when targeting other cosmological parameters, such as the sum of neutrino masses orprimordial non-Gaussianity, finding that fewer and wider bins are typically more optimal. Overall,our formalism facilitates an optimal extraction of cosmological constraints robust to astrophysics.
PACS numbers:
I. INTRODUCTION
Line-intensity mapping (LIM) [1] has recently arisenas a key technique to surpass and expand the remarkableachievements of precision observational cosmology overthe past decades. Impressive experimental efforts concen-trated mostly on two observables, namely the cosmic mi-crowave background (CMB) and galaxy number counts,have attained percent-level measurements of the stan-dard cosmological model, Λ-Cold Dark Matter (ΛCDM),and provided stringent constraints on possible deviationsfrom it [2, 3]. However, the former only allows limited(due to damping on small scales) access to a brief mo-ment in the early history of the Universe, while the latterloses effectiveness as it probes deeper into the Universe(as the discrete sources become too faint to detect).
LIM provides a way to bridge the gap between theCMB and galaxy surveys, and probe the huge swathsof the observable Universe that remain uncharted [4, 5].It measures the integrated emission from either atomicor molecular spectral lines originating from all thegalaxies—individually detectable or not—as well as fromthe diffuse intergalactic medium along the line of sight,and generates three-dimensional maps of the targetedvolume that in principle contain all the information thatcan be harvested from the incoming photons.
LIM’s experimental landscape is rapidly progressing.Probably the most studied line to date is the 21-cmspin-flip transition in neutral hydrogen, first detected byRef. [6]. A comprehensive suite of experiments is on itsway to map this line across the history of the observableUniverse, with several experiments targeting the epochsof cosmic dawn and reionization [7–10] and others fo-cusing on lower redshifts [11–16]. Meanwhile, growingattention has been given to other lines. Some exam-
ples include: carbon monoxide (CO) rotational lines [17–21], [CII] [22–24], Hα and Hβ [25, 26], oxygen lines [25]and Lyman-α [27, 28]. A few of these have alreadybeen (at least tentatively) detected at intermediate red-shifts [23, 29–32]. A significant effort is now being in-vested in LIM experiments targeting these lines, withsome instruments already observing and others to comeonline soon [33–42].
LIM holds unique promise for the study of both cos-mology and astrophysics [1, 5]. Since the line emissionoriginates from halos, the intensity of the spectral linesacts as a biased tracer of the underlying density dis-tribution (sourced by the cosmological primordial fluc-tuations), with the bias depending on the specific line.Meanwhile, the signal is intimately related to various as-trophysical processes that take place during reionizationand galaxy evolution [43–49]. A primary challenge istherefore to disentangle between the astrophysical andcosmological information contained in the maps.
Previous works have investigated the potential of us-ing LIM observations to study cosmology within the con-fines of ΛCDM and beyond it (see e.g., Refs. [15, 50–65]).However, existing studies are limited in the sense thatthey do not fully account for or properly model one ormore of the following: the observable signal, observinglimitations, intrument response, degeneracies between as-trophysics and cosmology, or the noise covariance. Ignor-ing these effects may result in a significant underestima-tion of the errors and bias of the best-fit parameters evenwhen dealing merely with forecasts (see e.g., Refs. [66, 67]for some examples related to galaxy surveys).
In this paper we aim to address this deficit by pro-viding a general and comprehensive formalism to opti-mally extract robust cosmological information from thepower spectrum of fluctuations in observed line-intensity
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maps. We focus especially on maximizing the precisionof baryon acoustic oscillation (BAO) measurements, butalso consider the potential of LIM to constrain exten-sions to ΛCDM, such as primordial non-Gaussianity orneutrinos with a sum of masses higher than 0.06 eV. Inparticular, we consistently account for the following:
• Degeneracies between cosmology and astrophysics.We identify and isolate the cosmological informa-tion encoded in the power spectrum that can beextracted given known astrophysical dependencies.
• LIM power spectrum anisotropies. We present amodel for the power spectrum which includes theanisotropic information from redshift-space distor-tions and the Alcock-Paczynski effect (see below).
• Multipole decomposition of the power spectrum.We show the optimal way to account for theseanisotropies, including the full covariance betweendifferent multipoles. We demonstrate genericallythe importance of going to higher multipoles.
• Effects of survey volume and instrument response.We show how to accurately model the suppressionof the power spectrum on large and small scales dueto survey and beam shapes. We discuss the effectsof redshift binning and experimental optimization.
In order to remain as generic as possible in our quan-titative estimates, we refrain from forecasting for spe-cific experiments and consider a generic LIM instrumentwhich can easily be replaced with any concrete experi-mental setup. This manuscript (hopefully) results in auseful manual for LIM enthusiasts who wish to preciselyquantify the cosmological information accessible in theirobserved maps.
To perform the analyses this work is based on, we mod-ify the public code LIM1. We analytically compute theanisotropic LIM power spectrum and the correspondingcovariance, using outputs from CAMB2 [68] to extract thematter power spectrum and other cosmological quanti-ties, and Pylians3 [69] to obtain the halo mass functionand halo bias. We plan to release an updated version ofthe LIM package upon publication.
Our calculations are based on several standard as-sumptions and parameter choices. Throughout this workwe adopt the halo mass function and halo bias pre-sented in Ref. [70]. We assume, unless otherwise stated,the Planck baseline ΛCDM model and take the best-fit parameter values from the combined analysis of thefull Planck 2018 and galaxy BAO measurements [2, 3]:baryon and dark matter physical densities at z = 0,Ωbh
2 = 0.0224 and Ωcdmh2 = 0.1193, respectively; spec-
tral index ns = 0.967; amplitude log(As1010
)= 3.047 of
1 https://github.com/pcbreysse/lim2 https://camb.info/3 https://github.com/franciscovillaescusa/Pylians
the primordial power spectrum of scalar perturbations atthe pivot scale; Hubble constant H0 = 67.67 km/s/Mpc;a sum of neutrino masses
∑mν = 0.06 eV; and Gaussian
primordial perturbations, i.e., fNL = 0.This paper is structured as follows. In Section II we
review the basics of LIM power spectrum modelling. Wehighlight the degeneracies between astrophysics and cos-mology and provide a formulation which clearly delin-eates them. Properties of the measured power spectrumare derived in Section III, including the Legendre mul-tipole expansion, and the effects of the window functionand the covariance matrix. A generalized LIM exper-iment is introduced in Section IV, along with the pre-scription for our Fisher-based forecasts. In Section Vwe investigate the extraction of cosmological informationfrom the LIM power spectrum, demonstrating the im-portance of higher multipoles when measuring BAO andRSD, and describing strategies to improve cosmologicalconstraints, including redshift binning and experimentaloptimization. We defer to an Appendix the adaptationof our methodology to cross-correlations between differ-ent lines or with galaxy surveys (Appendix A), as well asthe discussion of how to fold in the instrument responseand the signal suppression due to the finite volume sur-veyed, and the effects of foregrounds and line interlopers(Appendix B). We present our conclusions in Section VI.
II. MODELLING THE IM POWER SPECTRUM
A. From line luminosity to the power spectrum
The brightness temperature T and specific intensityI of a given radiation source are equivalent quantitiesthat depend on the expected luminosity density ρL of aspectral line with rest frame frequency ν at redshift z:
T (z) =c3(1 + z)2
8πkBν3H(z)ρL(z),
I(z) =c
4πνH(z)ρL(z),
(1)
where c is the speed of light, kB is the Boltzmann con-stant and H is the Hubble parameter [17]. Depending onthe frequency band of the experiment, either T or I areconventionally used. Hereinafter, we will use T in orderto homogenize the nomenclature, but all the expressionsare equally valid if intensity is used instead. Assuming aknown relation, L(M, z), between the luminosity of thespectral line and the mass M of the host halo, the ex-pected mean luminosity density can be computed usingthe halo mass function dn/dM(z):
〈ρL〉(z) =
∫dML(M, z)
dn
dM(M, z). (2)
The main statistic we will focus on in this work isthe LIM power spectrum, which is given by the Fouriertransform of the two-point correlation function of the
3
perturbations of the brightness temperature, denoted byδT ≡ T − 〈T 〉. Since spectral lines are sourced in halos,δT can be used to trace the halo distribution and, equiv-alently, the underlying matter density perturbations. Atlinear order, matter density and brightness tempera-ture perturbations are related by an effective linear bias,which is given in terms of the halo bias bh by
b(z) =
∫dML(M, z)bh(M, z) dn
dM (M, z)∫dML(M, z) dn
dM (M, z). (3)
In practice, T maps are obtained in redshift space,where the observed position along the line of sight isdisturbed with respect to real space due to peculiarvelocities, producing the so-called redshift-space distor-tions (RSD). RSD introduce anisotropies in the a-prioriisotropic real-space power spectrum. The relation be-tween real and redshift space perturbations (δr and δs,respectively) can be approximated by a linear factor, suchthat δs = FRSDδ
r, with FRSD in Fourier space given by:
FRSD(k, µ, z) =
(1 +
f(z)
b(z)µ2
)1
1 + 0.5 (kµσFoG)2 , (4)
where f(z) is the logarithmic derivative of the growthfactor (also known as the growth rate), k is the module
of the Fourier mode, and µ = k ·k‖ is the cosine of the an-
gle between the mode vector ~k and the line-of-sight com-ponent, k‖ (i.e. µ ∈ [−1, 1]). At linear scales, coherentpeculiar velocities boost the power spectrum in redshiftspace through the Kaiser effect (first term in Eq. (4))[71]. In turn, small-scale velocities suppress the cluster-ing on small scales, an effect known as the fingers of God(second term in Eq. (4)). We use a Lorentzian dampingfactor whose scale-dependence is driven by the parameterσFoG, the value of which is related to the halo velocitydispersion4. We assume a fiducial value of 7 Mpc. Werefer the interested reader to e.g., Ref. [72], for a reviewon RSD discussing these contributions.
On top of all these effects, there is a scale-independentshot noise contribution to the LIM power spectrum stem-ming from the discreteness of the source population,which would be present even in the absence of clustering.Thus we can express the anisotropic LIM power spectrumas the sum of clustering and shot noise contributions:
P (k, µ, z) = Pclust(k, µ, z) + Pshot(z);
Pclust(k, µ, z) = 〈T 〉2(z)b2(z)F 2RSD(k, µ, z)Pm(k, z);
Pshot(z) =
(c3(1 + z)2
8πkBν3H(z)
)2 ∫dML2(M, z)
dn
dM,
(5)
where Pm is the matter power spectrum.
4 The fingers-of-God damping can be also modelled with a Gaus-sian function, yielding similar results. We have checked that theconclusions of this work do not depend on this choice.
We note that the above expressions assume that eachhalo contains only a single point-source line emitter at itscenter. In many cases, there will be wider-scale emissionfrom diffuse gas within a halo, or there will be satellitegalaxies which contribute additional intensity to that ofthe central emitter. Either of these will lead to an ad-ditional “one-halo” contribution to the clustering termwhich traces the halo density profile. We direct the in-terested reader to Ref. [74] for a discussion of one-haloeffects in intensity mapping surveys. We also leave for fu-ture work the effect of anisotropic halo assembly bias [73].
B. Dependence on astrophysics and cosmology
The emission of any spectral line is intimately relatedto astrophysical processes. Therefore, the observed den-sity field is strongly dependent on astrophysics throughthe line luminosity function. Fortunately, according toEq. (5), the relation between astrophysics and the LIMpower spectrum at linear order is limited to the ampli-tude of the clustering and shot noise contributions5. Thismeans that the only accessible astrophysical informationpresent in the linear LIM power spectrum is contained inthe first two moments of L(M)6.
Since δT traces matter overdensities, LIM observationsalso carry cosmological information. This information ismostly encoded in Pm, FRSD and the BAO. For example,as seen in Eq. (4), FRSD depends on the growth factorf , and so can be efficiently used to constrain deviationsfrom general relativity.
In order to measure the power spectrum, redshifts needto be transformed into distances. This procedure intro-duces further anisotropies in the power spectrum if theassumed cosmology does not match the actual one. Thisis known as the Alcock-Paczynski effect [76]. Radial andtransverse distances are distorted in different ways, whichcan be modelled by introducing rescaling parameters toredefine distances:
α⊥ =DA(z)/rs
(DA(z)/rs)fid, α‖ =
(H(z)rs)fid
H(z)rs, (6)
where DA is the angular diameter distance, rs is thesound horizon at radiation drag, and the superscript ‘fid’denotes the corresponding values in the assumed (fidu-cial) cosmology. Due to this distortion, the true wavenumbers are related to the measured ones in the trans-verse and line-of-sight directions by ktrue
⊥ = kmeas⊥ /α⊥
and ktrue‖ = kmeas
‖ /α‖, respectively. These relations can
5 A detailed study of the nonlinearity and scale dependence of theastrophysical terms in the LIM power spectrum lies beyond thescope of this work.
6 Alternative probes such as the voxel intensity distribution [75]are more suitable to infer L(M), see Section VI.
4
then be expressed in terms of k and µ [77]:
ktrue =kmeas
α⊥
[1 + (µmeas)
2 (F−2
AP − 1)]1/2
,
µtrue =µmeas
FAP
[1 + (µmeas)
2 (F−2
AP − 1)]−1/2
,
(7)
where FAP ≡ α‖/α⊥. To correct for the modification ofthe volumes, the power spectrum is then multiplied by
the factor: H(z)/Hfid(z)×(DfidA (z)/DA(z)
)2. BAO pro-
vide useful cosmic rulers, as they allow high precisionmeasurements of α⊥ and α‖, which can effectively con-strain the expansion history of the Universe, see Ref. [78].Moreover, the Alcock-Paczynski effect has been proposedas a method to identify and remove line interlopers [79].
Other cosmological dependences in Eq. (5) arisethrough the halo bias and the halo mass function, andalso due to the halo velocity dispersion (via σFoG).However, the cosmological information encoded in thesequantities will be degenerate with the highly-uncertainL(M) modeling. Therefore, in our analysis below we willmostly focus on BAO and RSD.
C. Explicitly accounting for the degeneracies
As clearly evident, the quantities appearing in Eq. (5)present strong degeneracies. Therefore, after identifyingwhere the cosmological information is encoded, it is use-ful to reparametrize Eq. (5) to minimize the degeneracies.
The overall shape of Pm is not very sensitive to smallvariations of the cosmological parameters of ΛCDM.Thus, it is preferable to employ a template for Pm(k, z),computed based on the fiducial cosmology (parametriz-ing its amplitude with σ8, the root mean square of thedensity fluctuations within 8hMpc−1), and then measurethe terms that relate it to the LIM power spectrum.
Examining Eq. (5), we find that σ8, b and 〈T 〉 aredegenerate within a given single redshift, as well as σ8,fµ2, and 〈T 〉. These degeneracies can be broken with anexternal prior on 〈T 〉, or using tomography and interpret-ing the measurements under a given cosmological model,taking advantage of their different redshift evolutions.
Other ways to break these degeneracies, at least par-tially, include the combination of the LIM power spec-trum and higher-order statistics (see e.g., Ref. [80] in thecase of halo number counts, and Ref. [81] for the 21cmLIM case); multi-tracer techniques with several spectrallines and galaxy surveys [56]; or the exploitation of themildly non-linear regime of the LIM power spectrum bymodelling the halo clustering with Lagrangian perturba-tion theory [62].
Taking all this into account, we group all degenerateparameters and reparameterize the LIM power spectrum
from Eq. (5) as:
P (k, µ) =
(〈T 〉bσ8 + 〈T 〉fσ8µ
2
1 + 0.5 (kµσFoG)2
)2Pm(k, ~ς)
σ28
+ Pshot,
(8)where all quantities depend on z. From Eq. (8), the setof parameter combinations that can be directly measuredfrom the LIM power spectrum at each independent red-shift bin and observed patch of sky is:
~θ = α⊥, α‖, 〈T 〉fσ8, 〈T 〉bσ8, σFoG, Pshot, ~ς. (9)
In ~ς we group all parameters that modify the templateof the power spectrum used in the analysis, such as anyof the standard ΛCDM parameters, or extensions likeprimordial non-Gaussianity or neutrino masses.
As explained above, out of all the parameter combina-tions included in Eq. (9), only α⊥, α‖ and 〈T 〉fσ8 containuseful cosmological information (modulo 〈T 〉, in the caseof 〈T 〉fσ8). Given the difficulty in tracing cosmologicalinformation hidden in the other parameter combinations,we prefer to be conservative and consider them simply asnuisance parameters in our analysis.
Lastly, note that we choose this parameterization inorder to separate b and f . However, the LIM power spec-trum can be cast in different ways in order to target otherparameter combinations. For example, if a measurementof β = f/b is wanted, one could use:
P (k, µ) =
(〈T 〉bσ8
(1 + βµ2
)1 + 0.5 (kµσFoG)
2
)2Pm(k)
σ28
+ Pshot. (10)
In this case, the measured quantities would be β and〈T 〉bσ8, instead of 〈T 〉fσ8 and 〈T 〉bσ8.
III. MEASURING THE IM POWER SPECTRUM
A. The window function and multipole expansion
LIM experiments have a limited resolution and probea finite volume, which limits the minimum and maxi-mum accessible scales, respectively. This effectively ren-ders the observed brightness temperature fluctuationssmoothed with respect to the true ones. This smooth-ing can be modelled by convolving the true T map withwindow functions describing the instrument response andthe effects due to surveying a finite volume. This wouldyield an observer-space power spectrum, given by:
P (k, µ, z) = W (k, µ, z)P (k, µ, z) =
= Wvol(k, µ, z)Wres(k, µ, z)P (k, µ, z),(11)
where Wvol and Wres respectively are the survey-area andinstrument-response window functions in Fourier space.
The line-of-sight resolution of a LIM instrument isgiven by the width of the frequency channels. The res-olution in the plane of the sky depends on whether the
5
experiment uses only the auto-correlation of each of itsantennas (i.e., a single-dish approach) or employs inter-ferometric techniques. In the former case, the resolutionis determined by the antenna’s beam profile, while inthe latter, by the largest baseline of the interferometer,Dmax. The characteristic resolution limits in the radialand transverse directions are given by:
σ‖ =c(1 + z)δν
H(z)νobs;
σdish⊥ = χ(z)σbeam, σinterf
⊥ =cχ(z)
νobsDmax,
(12)
where δν is the frequency-channel width, νobs is the ob-served frequency, χ(z) is the radial comoving distance,and the width of the beam profile is given by σbeam =θFWHM/
√8 log 2 (where θFWHM is its full width at half
maximum). Then, the resolution window function Wres
that models the instrument response in Fourier space canbe computed as [20]:
Wres(k, µ) = exp−k2
[σ2⊥(1− µ2) + σ2
‖µ2]
. (13)
The beam profile, field of view, and frequency band win-dow might not be Gaussian; for instance, the frequencychannels are likely to be discrete bins (which would corre-spond to a top-hat window in real space). In practice, theexact instrument response will be accurately character-ized by each experiment. Investigation of more realisticwindow functions is beyond the scope of this work.
On the other hand, at a given redshift, observing apatch of the sky with solid angle Ωfield over a frequencyband ∆ν corresponds to a surveyed volume Vfield, whichin the absence of complex observation masks is given by:
Vfield =[χ2(z)Ωfield
] [c(1 + z)2∆ν
H(z)ν
]. (14)
The two factors in Eq. (14) are the transverse area andthe length of the radial side of the volume observed (i.e.,Vfield ∼ L2
⊥L‖). These are the largest scales that canbe probed by a single-dish like experiment, hence we de-
fine kmin,dish‖ ≡ 2π/L‖ and kmin,dish
⊥ ≡ 2π/L⊥. While an
interferometer shares this limitation in the radial direc-tion, the largest scales that an interferometer can mea-sure are determined by the shortest baselineDmin, so that
kmin,interf⊥ ≡ 2πνobsDmin/(cχ(z)). Note that kmin,interf
⊥will always be ultimately limited by L⊥. LIM powerspectrum measurements beyond these scales will be sup-pressed by the lack of observable modes.
To account for the loss of modes, we define a volumewindow function Wvol, which in Fourier space is given by:
Wvol(k, µ) =
(1− exp
−(
k
kmin⊥
)2 (1− µ2
))×
×
1− exp
−(
k
kmin‖
)2
µ2
.
(15)
We use a smoothed window in order to extend the lossof modes to smaller scales than those corresponding tokmin⊥,‖ . This mimics the effect of having residuals present
in the map after the removal of foregrounds (and line-interlopers), polluting the signal, since large scales arethe ones more likely to be affected7.
Though we have worked up to this point in (k, µ) coor-dinates, we must point out that while the Fourier trans-form of the observed map is performed using Cartesiancoordinates, the line of sight changes with different point-ings on the sky. It is therefore not parallel to any Carte-sian axis. Therefore, it is not possible to obtain a welldefined µ from the observations, meaning that we cannotdirectly measure P (k, µ). Nevertheless, it is possible todirectly measure the multipoles of the anisotropic powerspectrum using, e.g., the Yamamoto estimator [82].
Accounting for all the effects described in this section,the multipole expansion of the observed LIM power spec-trum is given by:
P`(kmeas) =
H(z)
Hfid(z)
(DfidA (z)
DA(z)
)22`+ 1
2×
×∫ 1
−1
dµmeasP (ktrue, µtrue)L`(µmeas),
(16)
where L` is the Legendre polynomial of degree `. In ouranalysis below, we will calculate the integral in Eq. (16)numerically. Analytic results (neglecting the fingers-of-God contribution to the RSD and not considering Wvol)can be found in Ref. [83]. We note that there is a non-vanishing contribution from the shot noise of the LIMpower spectrum to multipoles higher than the monopole,due to the effect of the window function.
Most previous works do account for the smearing ofthe brightness temperature maps, but choose to includethis effect in the covariance of the LIM power spectrumrather than in the observed signal. This is equivalentto deconvolving the observed LIM power spectrum byW (k, µ). As we show in Appendix B, this approach mayresult in a reduction of the significance of the measuredLIM power spectrum multipoles when their covariance isproperly modelled8, unless special treatment is applied toboth signal and errors. We explore this avenue and derivea nearly optimal estimator of the multipoles of the trueunsmoothed LIM power spectrum. However, given thecomplexity of applying this approach to observations, weadvocate for modelling the smoothed LIM power spec-trum rather than deconvolving the observed one.
7 One could assume a top-hat window in real space to perform amore aggressive analysis. Complex observational masks may en-tail more complicated Wvol, but these effects are very dependenton the particular circumstances of a given experimental setup.
8 The contribution to the covariance of the monopole is often mod-elled incorrectly, see Appendix B.
6
B. The power spectrum covariance matrix
There are three contributions to the LIM power spec-trum covariance: (i) sample variance; (ii) instrumentalnoise; and (iii) residual contamination after removal offoregrounds and line interlopers. Since the magnitudeof the effect of foregrounds and line interlopers greatlydepends on the spectral line and redshift of interest, weconsider only the first two contributions to the covari-ance and assume that the signal has been cleaned suchthat the effects included in Wvol are sufficient. We referthe reader interested in foreground and line-interloperremoval strategies to, e.g., Refs. [79, 84–89]. We alsoassume a Gaussian covariance without mode coupling.
The instrumental noise, set by the practical limitationsof the experiment, introduces an artificial floor bright-ness temperature in the observed map. The noise powerspectrum for a given single-dish like experiment or aninterferometer is given by (see e.g., Ref. [50]):
P dishn =
T 2sysVfield
∆νtobsNpolNfeedsNant;
P interfn =
T 2sysVfieldΩFOV
∆νtobsNpolNfeedsns,
(17)
where tobs is the total observing time, Nfeeds is the num-ber of detectors in each antenna, each of them able tomeasure Npol = 1, 2 polarizations, Tsys is the system tem-perature, ΩFOV = c2/(νobsDdish)2 is the field of view ofan antenna, and ns is the average number density of base-lines in the visibility space. Assuming a constant numberdensity of baselines, the latter is given by [50]:
ns =2πν2
obs(D2max −D2
min)
c2Nant(Nant − 1). (18)
Although the anisotropic power spectrum may not bemeasurable, it is useful to define the covariance per k andµ bin:
σ2(ki, kj , µ) =1√
Nmodes(ki, µ)
(P (ki, µ) + Pn
)×
× 1√Nmodes(kj , µ)
(P (kj , µ) + Pn
)δKij ,
(19)
where δK is the Kronecker delta (introduced because weneglect mode coupling). In Eq. (19), the first term inthe parentheses corresponds to the sample variance, andNmodes denotes the number of modes per bin in k and µin the observed field:
Nmodes(k, µ) =k2∆k∆µ
8π2Vfield, (20)
with ∆k and ∆µ referring to the width of the k and µbins, respectively.
The covariance matrix of the LIM power spectrummultipoles is comprised of the sub-covariance matrices of
each multipole, and those between different multipoles.The sub-covariance matrix for multipoles ` and `′ is givenby:
C``′(ki, kj) =(2`+ 1)(2`′ + 1)
2×
×∫ 1
−1
dµσ2(ki, kj , µ)L`(µ)L`′(µ),
(21)
where the Delta function we assumed in Eq. (19) makeseach sub-covariance matrix diagonal. However, modelingthe non-zero covariance between multipoles is essentiallfor an unbiased analysis. We refer the interested readerto Ref. [90] for a thorough analytic derivation of the co-variance of the multipoles of the galaxy power spectrumunder the Gaussian assumption.
IV. FORECASTING FOR IM EXPERIMENTS
A. Parameterizing a generic IM experiment
In order to illustrate the potential of our methodol-ogy, it is useful to consider a concrete example for aLIM experiment — without loss of generality — so thatwe can compute the window functions and the covari-ance. Without specifying the targeted emission line,we choose as our strawperson LIM experiment an am-bitious single dish-like instrument with a total frequencyband ∆ν/ν ≈ 0.2. We split the corresponding vol-ume into several redshift bins to study how the signal— and the extraction of cosmological information —changes with redshift. In our analysis below we use fivenon-overlapping, independent redshift bins centered atz = 2.7, 4.0, 5.3, 6.6, 7.9, such that log10 [∆(1 + z)] =log10 [∆(ν/νobs)] = 0.1.
In order to compute the signal, we need to assumesome numbers for the astrophysical quantities that arepresent in Eq. (5). Strictly as an example, we willassume a CO intensity mapping signal described bythe model from Ref. [20]. In the redshift bins definedabove, this model gives mean brightness temperature〈T 〉 = 2.9, 3.3, 3.5, 3.1, 2.4, luminosity-averaged biasb = 1.5, 2.1, 2.6, 3.1, 4.0, and shot noise contribution tothe LIM power spectrum Pshot = 354, 331, 214, 114, 59(Mpc/h)3µK2. Note that we are neglecting the evolu-tion of these quantities within each bin, simply using thevalue at the central redshift each time.
As an analogy with the futuristic experiment envi-sioned in Ref. [78], we consider an array of single dish an-tennas with total NpolNfeedsNanttobs/T
2sys = 10500 h/K2
and 18900 h/K2 for the first two redshift bins, respec-tively, and 25 × 103 h/K2 for the last three redshiftbins. The value of NpolNfeedsNanttobs/T
2sys changes with
redshift because we assume that Tsys depends on νobs
until it saturates: Tsys = max [20, νobs (K/GHz)] (seee.g., [91, 92]). Furthermore, we assume a spectral res-olution νobs/δν = 15450, 11500, 9150, 7600, 6500, and
7
θFWHM = 4 arcmin. Taking into account these exper-iment specifications, we choose Ωfield = 1000 deg2 tomaximize the significance of the measurement around thescales of the BAO. For this experiment, we would haveσ⊥ < σ‖ and L⊥ > L‖.
We emphasize that our choice of this particular COmodel is only an example. Our formalism is general andapplicable to any line and experiment, and the quantitieswe compute here can easily be calculated for any othermodel the reader might have in mind.
In Fig. 1 we demonstrate the effect of the LIM powerspectrum smoothing, discussed in Section III, for the spe-cific case of the first redshift bin of our strawperson exper-iment, comparing P` and P`. While for the monopole, Wjust suppresses the power spectrum at high and low k, theeffect on the quadrupole, and especially on the hexade-capole, is much more significant. The effect largely de-pends on the ratio between σ⊥ and σ‖, and between kmin
⊥and kmin
‖ . Therefore, the effect of W on the power spec-
trum is of course strongly dependent on the experiment.This was similarly demonstrated in Ref. [83], though thesurvey-volume window effect was not included there.
We also show how the LIM power spectrum multi-poles change when each of the parameters of Eq. (9) is5% larger than the fiducial values. We can see that thederivatives of the quadrupole and hexadecapole with re-spect to the parameters considerably change whether weconsider P or P . This figure provides an intuition of thedependence of the power spectrum on each of the consid-ered free parameters, both for the true and the measuredLIM power spectra, and may guide the design of futureexperiments depending on the parameter targeted.
B. Likelihood and Fisher Matrix
If we construct the vectors ~Θ(k) =[P0(k), P2(k), ...
],
and ~Θ =[P0(k0), P0(k1), ..., P2(k0), P2(k1), ...
], we can
compute S/N(k), the total signal-to-noise ratio per k bin,as well as the total S/N summed over all bins as:
[S/N( k)]2 = ~ΘT (k)C−1(k)~Θ(k),
[S/N]2
= ~ΘT C−1~Θ,(22)
where the superscript T denotes the transpose operatorand C(ki) is a N`×N` matrix (with N` being the numberof multipoles included in the analysis) made up of thecorresponding elements of C``′(ki, ki). Similarly, we cancompute the χ2 of the LIM power spectrum multipolesfor a given experiment, using:
χ2 = ∆~ΘT C−1∆~Θ, (23)
where ∆~Θ is the difference between the model predictionand the actual measurements.
We will use the Fisher matrix formalism to forecastconstraints from our generalized LIM experiment [93, 94].
The Fisher matrix is the average of the second par-tial derivatives of the logarithm of the likelihood, logL,around the best fit (or assumed fiducial model). Eq. (23)can be adapted to form the Fisher matrix by replacing
∆~Θ by the corresponding derivatives. The Fisher matrixelement corresponding to the parameters ϑa and ϑb is:
Fϑaϑb=
⟨∂2 logL∂ϑa∂ϑb
⟩=
(∂~ΘT
∂ϑaC−1 ∂
~Θ
∂ϑb
). (24)
V. EXTRACTING COSMOLOGY FROM LIM
Based on the previous sections, we are now equipped tocalculate quantitative estimates of the potential of LIMexperiments to set robust cosmological constraints. Thisis a good way to evaluate the methodology presentedabove. We will use our generic LIM experiment and fore-cast constraints on α⊥, α‖ and 〈T 〉fσ8, as well as on thesum of neutrino masses and primordial non-Gaussianity.
A. The importance of going to higher multipoles
In order to exploit the anisotropic BAO, it is neces-sary to measure at least the monopole and quadrupole ofthe LIM power spectrum, Eq. (16). This is equivalent toconstraining DA/rs and Hrs, rather than a combinationof them if only the isotropic signal is measured, with theobvious benefits that it entails [95]. However, while mea-suring the monopole and quadrupole is relatively easy,given their reasonably high S/N, detecting the hexade-capole is challenging, since the signal is typically wellbelow the noise. The situation is even more pessimisticfor ` > 4.
Neglecting the fingers-of-God effect and Wvol, Ref. [83]finds that including the hexadecapole (` = 4) doesnot significantly improve the estimated constraints onthe astrophysical parameters. Nevertheless, focusingon cosmology and taking into account that the Alcock-Paczynski effect is anisotropic, we find that including thehexadecapole might be useful. Even if the hexadecapoleis not detectable, it might help to break degeneracies be-tween the BAO and RSD parameters, since its amplitudeis mostly independent of the luminosity-averaged bias,but it does depend greatly on the α⊥ and α‖ configu-ration. For instance, including the hexadecapole in theanalysis of the quasar power spectrum in eBOSS resultedin smaller degeneracies between α‖ and α⊥, and betweenα‖ and fσ8, compared with the case without includingthe hexadecapole [96].
Moreover, the hexadecapole can be measured andestimated at the same time as the lower order mul-tipoles, without requiring additional computing time.However, robust measurements of the hexadecapole im-pose stronger requirements on experiments, given itsanisotropy. In order to assess the benefits of adding thehexadecapole to the LIM power spectrum analysis, we
8
Fid. T f 8 T b 8 FoG Pshot
102
103
104
105
P 0 [
K2 h3 M
pc3 ]
z = 2.73
10 2 10 1 100
k [hMpc 1]
0.90
0.95
1.00
1.05
P 0/P
fid 0
101
102
103
104
105
P 2 [
K2 h3 M
pc3 ]
10 2 10 1 100
k [hMpc 1]
100
101
102
103
104
(P2
Pfid 2)
101
102
103
104
P 4 [
K2 h3 M
pc3 ]
10 2 10 1 100
k [hMpc 1]
100
101
102
103
104
(P4
Pfid 4)
102
103
104
105
P 0 [
K2 h3 M
pc3 ]
z = 2.73
10 2 10 1 100
k [hMpc 1]
0.90
0.95
1.00
1.05
P 0/P
fid 0
101
102
103
104
105
P 2 [
K2 h3 M
pc3 ]
10 2 10 1 100
k [hMpc 1]
100
101
102
103
104
(P2
Pfid 2)
101
102
103
104
P 4 [
K2 h3 M
pc3 ]
10 2 10 1 100
k [h/Mpc]
100
101
102
103
104
(P4
Pfid 4)
FIG. 1: Comparison of the LIM power spectrum multipoles at z = 2.73 for the fiducial set of parameters (blue) and the caseswhere each of the parameters of Eq. (9) is increased by 5% (except for ~ς). The top panels show the true power spectrum,while the bottom panels show the smoothed power spectrum as measured by the generalized experiment considered. Fromleft to right, each column corresponds to the monopole, quadrupole and hexadecapole, respectively. The lower part of eachpanel shows the ratio or difference (when the multipoles cross zero) between the fiducial power spectrum and the ones with thevaried parameters. Dashed lines denote negative values. Note the effect of the window function on all multipoles, especiallythe quadrupole and hexadecapole: both the fiducial power spectra and their parameter dependence are considerably affectedby the smoothing of the map.
compare the projected results with and without its in-clusion.
We first evaluate the straightforward gain of includingmore multipoles than just the monopole: we comparethe S/N(k) of the LIM power spectrum using only themonopole and then consecutively adding the quadrupoleand hexadecapole. The results given our strawpersonexperiment in each of the five redshift bins are shownin Fig. 2. We can see that adding the quadrupole orthe hexadecapole does not significantly increase the totalS/N per k bin. Note that this does not mean that the S/N
of P2 or P4 is low, since the total S/N is not the sum of theindividual significances of each multipole; the covariancebetween multipoles must be included (see Eq. (22)).
It is also evident that the signal-to-noise decreases with
redshift. This is because the amplitude of the LIM powerspectrum decreases with redshift faster than the noiselevel over the range considered here. In addition, thewindow function suppresses the LIM power spectrum atwider k ranges. While the behavior in Fig. 2 dependson the specific experiment due to the covariance and theanisotropy of W , the qualitative result applies generally.
Nonetheless, adding multipoles beyond the monopolehelps to break degeneracies between parameters. Thecorrelation matrices for each redshift bin after marginal-izing over nuisance parameters are shown in Fig. 3. Wecompare the results with and without the hexadecapole.This figure shows a reduction of the correlation betweenthe parameters when the hexadecapole is included withrespect to the case in which it is not. Note that this re-
9
10 2 10 1 100
k [hMpc 1]10 3
10 2
10 1
100
101
102
S/N
(k)
= 0= 0, 2= 0, 2, 4
z = 2.73z = 4.01z = 5.30z = 6.58z = 7.87
FIG. 2: Signal-to-noise ratio per k bin of the measured LIMpower spectrum for each redshift bin of the generic experi-ment considered in this work (color coded) including only themonopole (dotted lines), the monopole and the quadrupole(dashed lines) and adding also the hexadecapole (solid lines).
FIG. 3: Correlation matrices of the parameters α⊥, α‖ and〈T 〉fσ8, marginalized over the nuisance parameters, for eachredshift bin, calculated for our generalized experiment. Thecorrelations in the lower triangular matrix correspond to thecase where the monopole and quadrupole are included. In theupper triangular matrix the hexadecapole is included as well.
duction is smaller for larger redshifts (where the globalS/N, and especially that of the hexadecapole, is smaller).
Lower absolute correlations have a positive impacton the final marginalized constraints. Table I reportsthe forecasted 68% confidence-level marginalized pre-cision of the measurements of the BAO rescaling pa-rameters and the parameter combination 〈T 〉fσ8, foreach of the redshift bins of our general experiment.
zσrel (α⊥) (%) σrel
(α‖)
(%) σrel (〈T 〉fσ8) (%)` ≤ 2 ` ≤ 4 ` ≤ 2 ` ≤ 4 ` ≤ 2 ` ≤ 4
2.73 1.5 1.1 2.0 1.3 3.4 2.14.01 1.4 1.1 2.0 1.4 3.5 2.55.30 1.7 1.5 2.4 1.9 3.8 3.06.58 3.8 3.4 4.8 4.1 5.9 5.17.87 10.6 9.5 12.2 11.0 9.5 8.6
TABLE I: Forecasted 68% confidence-level marginalized rela-tive constraints using our generalized experiment on the BAOparameters and 〈T 〉fσ8, expressed as percentages. We com-pare results for only the monopole and quadrupole to the caseof adding the hexadecapole, leading to marked improvement.
The fiducial values of α⊥ and α‖ are 1 for all red-shifts, the corresponding fiducial values of 〈T 〉fσ8 are0.78, 0.67, 0.56, 0.43, 0.29 µK for increasing z (assum-ing the 〈T 〉 values reported in Section IV A).
The worsening of the forecasted constraints with red-shift is expected, given the decreasing S/N (as shown inFig. 2). Interestingly, the marginalized constraints onthe BAO rescaling parameters are between 11% and 62%stronger when including the hexadecapole. The improve-ment obtained by including the hexadecapole decreaseswith redshift, as expected from the reduction of the dif-ference between correlations with and without the hex-adexapole shown in Fig. 3. Although measuring evenhigher multipoles might yield further gain, the improve-ment would be marginal, and the level of observationalsystematics would be too high to consider these measure-ments reliable. This is why we limit our study to ` ≤ 4.
B. Redshift binning and experimental optimization
When aiming to measure BAO and RSD, it is prefer-able to bin in redshift as much as possible, so that theevolution of the expansion of the Universe and growthof structure is better constrained. However, if the targetis a parameter that does not change with redshift, widerredshift bins are more optimal, since that reduces the co-variance of the LIM power spectrum (see Section III B).Therefore, the analysis should always be adapted to thetarget parameter. We illustrate this fact by forecast-ing constraints on the sum
∑mν of the neutrino masses
(when allowed to be larger than 0.06 eV), and on devia-tions from primordial Gaussian perturbations.
Any imprint caused by primordial deviations fromGaussian initial conditions would have been preservedin the ultra-large scales of the matter power spectrum,which have remained outside the horizon since inflation.In the local limit, primordial non-Gaussianity can beparametrized with fNL as the amplitude of the localquadratic contribution of a single Gaussian random field
10
φ to the Bardeen potential Φ 9:
Φ(x) = φ(x) + fNL
(φ2(x)− 〈φ2〉
). (25)
The skewness introduced in the density probability distri-bution by local primordial non-Gaussianity increases thenumber of massive objects, hence introducing a scale de-pendence on the halo bias [98–101]. Denoting the Gaus-sian halo bias with bGh , the total halo bias appearing inEq. (3) is given by bh(k, z) = bGh (z) + ∆bh(k, z), where:
∆bh(k, z) =[bGh (z)− 1
]fNLδec
3ΩmH20
c2k2Tm(k)D(z). (26)
Here Ωm is the matter density parameter at z = 0,δec = 1.68 is the critical value of the matter overdensityfor ellipsoidal collapse, D(z) is the linear growth factor(normalized to 1 at z = 0) and Tm(k) is the matter trans-fer function (which is approximately ∼ 1 at large scales).
The extensions we consider to ΛCDM therefore have∑mν and fNL as extra parameters, respectively, which
we include in ~ς in our Eq. (9), since they modify the powerspectrum template. Naturally, varying either
∑mν or
fNL would also slightly modify the halo mass functionand therefore alter the LIM power spectrum via modifi-cations of 〈T 〉, Pshot and b. We do not model this depen-dence, but our findings should not affected by this: thehalo mass function (minimally) affects the amplitudes ofthe Pclust and Pshot terms, which we marginalize over.
Moreover, we should note that more massive neutrinosinduce a slight scale dependence on the halo bias that canbe efficiently removed if the power spectrum of baryonsand cold dark matter (instead of the power spectrum ofall matter, including neutrinos) is used (see e.g., [102–104]). Since we do not aim here for detailed constraints,but for an illustration of the survey optimization, we ig-nore these effects and leave their study to future work.
We compare the performance using the redshift bin-ning proposed in Section IV A with those of a single red-shift bin centered at z = 3.80 and covering the whole fre-quency band of the experiment. In order to maximize thecoverage at large scales, we use logarithmic binning in kin both cases. All measured quantities present in Eq. (9),except for ~ς in some cases (e.g.,
∑mν or fNL), are differ-
ent in each redshift bin. Therefore, the final marginalizedconstraints on ~ς obtained using several redshift bins isgiven by the result of the combination of each individualmarginalized constraint obtained from each redshift bin:
σϑς=
∑z
[(F−1z
)ϑςϑς
]−1−1/2
, (27)
where Fz is the Fisher matrix of a given redshift bin.
9 Here we assume the convention of large scale structure ratherthan the one for CMB (fLSS
NL ≈ 1.3fCMBNL [97]).
Fig. 4 shows a comparison of the forecasted marginal-ized 68% confidence-level constraints on fNL and
∑mν
from our generalized experiment using one and five red-shift bins. It also shows the dependence of the constraintson Ωfield and tobsNfeeds/T
2sys. While the former affects
the volume probed (which determines Nmodes and Wvol),the latter only affects the amplitude of Pn. We restrictour investigation of the optimization of the survey to thevariation of these parameters (for a comprehensive studyof survey optimization to constrain fNL, see Ref. [60]).
Having five redshift bins reduces L‖ in each of them,
which suppresses the power spectrum up to higher kmin‖
values, through Wvol. This is the main reason that ourforecasted errors on fNL and
∑mν using a single red-
shift bin are approximately a factor of two better. Forhigh values of Ωfield, sample variance is smaller than theinstrumental noise, which is why the constraints improveso much when tobsNfeeds/T
2sys increases. The improve-
ment is smaller for low Ωfield since the sample variancecontribution to the error dominates and reducing the in-strumental noise does not significantly reduce the totalcovariance. Since the imprints of fNL lie on large scales,this effect is more critical than for
∑mν , which affects a
wider range of scales. Also, this effect is not very strongwith five redshift bins, since the P suppression limits theexploitation the very large line-of-sight scales.
C. Further improvements
Besides the strategies explored above, the extractionof cosmological information from the LIM power spec-trum can be further improved. One possibility would beto assign an arbitrary normalized weight, w, to each ofthe observed voxels in configuration space. Then, w(~r)can be chosen to maximize the S/N of the measured IMpower spectrum. With the introduction of these weights,a minimum-variance estimator can be derived, as pro-posed in Ref. [105] for galaxy surveys. An adaptation toLIM can be found in Ref. [106].
Different redshift binning strategies or overlapping binsmay be more optimal than the prescription used above.This will depend on the main objectives of the survey, asargued in Section V B. However, if overlapping bins areto be used, they would not be independent anymore, andthe corresponding covariance will have to be taken intoaccount. Moreover, the actual optimal frequency band ofthe experiment depends on the targeted parameters.
Finally, in this work we have enforced the redshift binsto be narrow enough such that the redshift evolution ofthe large-scale brightness temperature fluctuations doesnot change significantly (except in Section V B). How-ever, this can be incorporated into the analysis usingredshift-weighting techniques [107]: redshift-dependentweights are chosen in order to minimize the projected er-ror on the target cosmological or astrophysical parameterusing a Fisher matrix forecast, so that the constrainingpower is maximized. These techniques can become espe-
11
20
25
30
35
40
45
5010
3Npo
lNfe
edsN
antt o
bs
T2 sys
[h/K
2 ]
223038
1z bin
20
25
30
35
40
(f NL)
2000 4000 6000 8000 10000
field [deg2]
20
25
30
35
40
45
50
103N
polN
feed
sNan
tt obs
T2 sys
[h/K
2 ]
0.07
0.09
0.11 0.110.07
0.08
0.09
0.10
0.11
0.12
(m
) [eV
]
20
25
30
35
40
45
50
103N
polN
feed
sNan
tt obs
T2 sys
[h/K
2 ]
50
60
70
5z bins
40
50
60
70
80
(f NL)
2000 4000 6000 8000 10000
field [deg2]
20
25
30
35
40
45
50
103N
polN
feed
sNan
tt obs
T2 sys
[h/K
2 ]
0.12
0.15
0.18 0.10
0.12
0.14
0.16
0.18
0.20
(m
) [eV
]
FIG. 4: Forecasted 68% confidence-level marginalized constraints on fNL (top panels) and on∑mν (bottom panels) for our
general experiment using the monopole, quadrupole and hexadecapole, as function of Ωfield and NpolNfeedsNanttobs/T2sys, a
combination of instrumental parameters which determines the measurement sensitivity. We compare results using a singleredshift bin (Left panels) or five bins (Right panels). Note the change of scale in the color bars.
cially useful when wide redshift bins are needed, e.g. forprimordial non-Gaussianity, see Ref. [108].
VI. CONCLUSIONS
LIM techniques have attracted substantial attentiondue to their untapped potential to constrain astrophysicsand cosmology by probing huge volumes of the observableUniverse which are beyond the reach of other methods.Nonetheless, the versatility of LIM can also be consid-ered a nuisance, since cosmological information is alwaysintrinsically degenerate with astrophysics. These degen-eracies, as well as the effect of subtle contributions to theobserved LIM power spectrum and its covariance, as wellas various assumptions entailed in the analyses, are toocommonly ignored or only partially treated.
The aim of this work has been to provide a compre-hensive and general framework to optimally exploit thecosmological information encoded in the LIM power spec-trum. We have presented a reparameterization of it toidentify and isolate the degeneracies between cosmology
and astrophysics, including redshift-space distortions andthe Alcock-Paczynski effect. Furthermore, we have intro-duced and advocated for the use of the multipole expan-sion of the LIM power spectrum. We also derived anaccurate analytic covariance for the multipoles, and dis-cussed common errors in previous analyses.
Using a generalized experiment, and focusing onbaryon acoustic oscillations and redshift-space distor-tions measurements, we showed that adding the hexade-capole to the monopole and quadrupole of the LIM powerspectrum in the analysis returns a 10% − 60% improve-ment in the forecasted constraints. This is mainly dueto the reduction of the correlation between the param-eters, as the S/N of hexadecapole measurements is usu-ally low. However, redshift-space distortions as inferredfrom the LIM power spectrum are completely degeneratewith 〈T 〉. This degeneracy may be partially broken usingmildly non-linear scales and perturbation theory, as pro-posed by Ref. [62]. Other possible strategies include thecombination with other cosmological probes [56] or jointanalyses of two-point and higher order statistics measure-ments [80, 81].
12
We also investigated different survey strategies de-pending on the target of the experiment. As an illus-tration, we compared forecasts for constraints on
∑mν
and local primordial non-Gaussianity using one or multi-ple redshift bins over the same total volume, finding thatin both cases using one redshift bin returns stronger con-straints. We also explored the improvement of the con-straints by decreasing the instrumental noise and increas-ing the size of the patch of the sky observed. While theconstraints on
∑mν do not significantly improve when
varying Ωfield, the size of the patch of the sky observed iscritical to constrain primordial non-Gaussianity, since itssignatures are dominant at large scales (however, increas-ing Ωfield beyond ∼ 4000 − 6000 deg2 does not improvesignificantly the constraints, since in this case the instru-mental error dominates the error budget at large scales).
Finally, we briefly discussed further improvements tothe LIM power spectrum analyses to be studied in futureworks, especially to be applied to simulated and real ob-servations, rather than theoretical work. In the Appen-dices, we provide an adaptation of our proposed frame-work to cross-power spectra between spectral lines or be-tween LIM and galaxy surveys, and derive a close-to-optimal estimator for the true LIM power spectrum (i.e.,without smoothing), and compare different approaches.
Though we have focused here on power spectrum anal-yses, line-intensity maps contain a significant amount ofinformation beyond their power spectra. Line-intensityfluctuations are generated by highly complex gas dynam-ics on sub-galactic scales, which gives rise to a signif-icantly non-Gaussian intensity field. Alternative sum-mary statistics, such as higher order correlations or thevoxel intensity distribution [75], are then needed in orderto exhaust the information encoded in LIM observations.Ideally, these summary statistics will be measured andanalyzed with the power spectrum in tandem, account-ing for their correlation, as first explored in Ref. [109].
The methodology developed in this work should findample opportunities for implementation. As a com-pelling example, we demonstrate in a companion paper,Ref. [78], that LIM can be used to efficiently probe theexpansion history of the universe up to extremely highredshifts (z . 9), possibly weighing in on the growingHubble tension [110–112] and suggested models to ex-plain it (including models of evolving dark energy, exoticmodels of dark-matter, dark-matter–dark-energy interac-tion, modified gravity, etc., see e.g. Refs. [113–127]). Itcan also be used to constrain model-independent expan-sion histories of the Universe [128–130] at the few-percentlevel. Our formalism could also be adapted to measurethe velocity-induced acoustic oscillations [131], recentlyproposed in Ref. [132] as a standard ruler at cosmic dawn.
We are eager for the next generation of LIM observa-tions to be available for cosmological analyses, and hopethat the general framework reported in this manuscriptcan guide its precise and robust exploitation.
Acknowledgments
Funding for this work was partially provided bythe Spanish MINECO under projects AYA2014-58747-P AEI/FEDER UE and MDM-2014-0369 of ICCUB(Unidad de Excelencia Maria de Maeztu). JLBis supported by the Spanish MINECO under grantBES-2015-071307, co-funded by the ESF, and thanksJohns Hopkins University for hospitality during earlystages of this work. HGM acknowledges the supportfrom la Caixa Foundation (ID 100010434) which codeLCF/BQ/PI18/11630024.
Appendix A: Formalism for cross-correlations
1. The IM cross-power spectrum
Let us consider two generic brightness-temperaturemaps of two different spectral lines, denoted by X, Y ,respectively. In this case, the Kaiser effect present inthe RSD factor is different for each tracer. For instance,in FXRSD ∝
(1 + fµ2/bX
), with bX being the luminosity-
averaged bias for the line X. Then, the LIM cross-powerspectra of the X and Y lines is given by:
PXY = PXYclust + PXYshot ;
PXYclust = 〈TX〉〈TY 〉bXbY FXRSDFYRSDPm;
PXYshot =
(c3(1 + z)2
8πkBν3H(z)
)2 ∫dMLX(M)LY (M)
dn
dM,
(A1)
where we have dropped the explicit notation regardingdependence on k, µ and z (as will be done hereafter) forthe sake of readability (See Refs. [46, 49] for a deriva-tion). Following the same arguments as in Section II, weexpress the cross-power spectrum between two differentlines as:
PXY =Pm/σ
28(
1 + 0.5 [kµσFoG]2)2×
×(T
1/2XY bXσ8 + T
1/2XY fσ8µ
2)×
×(T
1/2XY bY σ8 + T
1/2XY fσ8µ
2)
+ PXYshot,
(A2)
where TXY = 〈TX〉〈TY 〉. Since we have two differentbiases and two different brightness temperatures (one perline), the measurable combinations of parameters are:
~θXY = α⊥, α‖, T1/2XY fσ8, T
1/2XY bXσ8, T
1/2XY bY σ8,
σFoG, PXYshot, ~ς .
(A3)
The covariance of the LIM cross-power spectrum is alsoslightly different. Considering, without loss of generality,the case in which each line is observed by a different
13
experiment, the covariance per µ and k bin of the cross-power spectra of two different lines is (neglecting modecoupling) given by:
σ2XY =
1
2
(P 2XY
Nmodes+ σX σY
), (A4)
where σX and σY are the square root of the correspond-ing covariances in Eq. (19).
Often, if the cross-power spectrum of two tracers canbe measured, the corresponding auto-power spectra canbe as well. In this case, when comparing the model toobservations, all this information needs to be taken intoaccount. In this case, Eqs. (22), (23), and (24) are stillcorrect, but both the data vector and the covariance ma-trix need to be changed. The former will include both
auto- and cross-power spectra, hence ~Θ would be the
concatenation of ~ΘXX , ~ΘXY and ~ΘY Y . In turn, the co-variance matrix will be formed by four square blocks ofthe same size, where the diagonal blocks would be CXXand CY Y , and the off-diagonal blocks, CXY .
2. Cross-correlation with galaxy surveys
LIM observations can also be cross-correlated withgalaxy number counts. Denoting the galaxy catalog andrelated quantities with subscript/superscript g, we havean RSD factor F gRSD ∝
(1 + fµ2/bg
). In this case, the
cross-power spectrum of galaxy number counts and LIMis:
PXg = PXgclust + PXgshot ;
PXgclust = 〈TX〉bXbgFXRSDFgRSDPm ;
PXgshot =c3(1 + z)2
8πkBν3H(z)
〈ρXL 〉gng
,
(A5)
where 〈ρXL 〉g is the expected luminosity density sourcedonly from the galaxies belonging to the galaxy catalogused, and ng is the number density of such galaxies[46, 49]. Note the difference in the shot noise terms inEq. (A5) with respect to Eqs. (5) and (A1). This is be-
cause contributions to PXgshot come only from the locationsoccupied by the galaxies targeted by the galaxy survey,and the shot noise between the galaxy distribution andthe luminosity sourced elsewhere vanishes. We also as-sume that the shot noise of the galaxy power spectrum isPoissonian (i.e., P ggshot = 1/ng). Nonetheless, clusteringand halo exclusion may introduce deviations from a Pois-sonian shot noise. This non-Poissonian contribution canchange the amplitude of the shot noise and even inducesa small scale dependence [133, 134].
Similarly to Eq. (A2), we prefer to express the cross-power spectrum of LIM and the galaxy number counts
as:
PXg =Pm/σ
28(
1 + 0.5 [kµσFoG]2)2×
×(〈TX〉1/2bXσ8 + 〈TX〉1/2fσ8µ
2)×
×(〈TX〉1/2bgσ8 + 〈TX〉1/2fσ8µ
2)
+ PXgshot.
(A6)
In this case, the parameter combinations measured wouldbe:
~θXg = α⊥, α‖, 〈TX〉1/2fσ8, 〈TX〉1/2bXσ8,
〈TX〉1/2bgσ8, σFoG, PXgshot, ~ς .
(A7)
This parameterization already accounts for the possi-
ble variation of the amplitude of PXgshot due to the non-Poissonian contributions mentioned above, since we have
marginalized over PXgshot. Note that if the goal is to mea-sure 〈ρXL 〉g, the non-Poissonian contribution to the shotnoise needs to be explicitly modelled. We neglect thepotential scale dependence introduced, which is a goodapproximation at this stage. As in the auto-spectrumcase, one may be able to access some of the degenerateastrophysical information in this cross-spectrum througha map’s one-point statistics, in this case by conditioningthe LIM statistics on those of the galaxy catalog [135].
The covariance per µ and k bin of the cross-power spec-tra of one line and galaxy number counts can be com-puted (neglecting mode coupling) as:
σ2Xg =
1
2
(P 2Xg
Nmodes+ σX
Pgg + 1ng√
Nmodes
), (A8)
where (Pgg + 1/ng) /√Nmodes is the square root of the
covariance of the galaxy power spectrum (when Poisso-nian shot noise is assumed).
Appendix B: True vs. observed power spectra
In most of the literature, the effect of the instrumentresponse, finite-volume surveyed and remaining residualsafter foreground and interlopers removal is included inthe error budget of the LIM power spectrum measure-ments, rather than incorporate it in the signal, as we dohere. This implies that the smearing of δT must be re-moved from the data (i.e., a deconvolution of the windowfunction is required), and therefore the true LIM powerspectrum would be estimated in the analysis.
The derivation of the observables, degeneracies and co-variance is equivalent to the one presented in Sections IIand III B, with two exceptions. On one hand, the sum-mary statistic would be P instead of P , which affectsthe power spectrum multipoles (Eq. (16)), as shown inFig. 1. On the other hand, since the noise power spec-trum Pn, depicted in Eq. (17), corresponds to the noise
14
of the observed temperature fluctuations, the inverse ofthe window needs to be applied to Pn (i.e., the sameoperation to obtain the true power spectrum from theobservations). In this case, the variance of P per k andµ bin (neglecting mode coupling) is:
σ2(k, µ) =1
Nmodes(k, µ)
(P (k, µ) +
PnW (k, µ)
)2
. (B1)
Eq. (B1) has an immediate consequence on the S/N ofthe power spectrum multipoles. When computing C``′(following Eq. (21), but using σ2 instead of σ2), the in-tegral over µ tends to asymptote to infinity beyond the
first resolution limit, i.e., k & min(kmax⊥ , kmax
‖
), due to
the presence of W in the denominator. The effect onC00 is shown in Fig. 5, considering only Wres. This be-havior was unnoticed in previous work because either theanistropic power spectrum was directly used (rather thanthe multipoles), or due to an incorrect computation ofthe covariance. In many previous works, the thermal-noise contribution to the monopole error is given as pro-portional to 1/
∫dµW (k, µ), when in fact it should be
proportional to∫dµ1/W (k, µ). Given the exponential
behavior of W , W−10 differs substantially from (W−1)0,
where the subindex 0 denotes the monopole.However, an optimal estimator of the multipoles of P
should contain the same information that is contained inP , if W is accurately modelled. In the following we dis-cuss a next-to-optimal estimator of the multipoles of thetrue power spectrum, P . Let us consider that we could
assign a weight, w(~k), to each mode of the brightnesstemperature perturbations in Fourier space, δkT , suchthat the temperature fluctuations, F , where P = 〈|F|2〉,becomes:
F =1
A1/2wδkT , (B2)
where A = V −1k
∫d3~kw2, Vk is the volume in Fourier
space, and all quantities depend on k and µ. Hereinafterwe will not show the explicit notation for the dependencefor the sake of simplicity and readability.
Applying the weights also to the noise power spectrum,we obtain P = w2P/A and Pn = w2Pn/A. If we use theseestimators for Eq. (B1), the covariance of the multipoles
of P is given by:
C``′ =(2`+ 1)(2`′ + 1)
2A2Nmodes×
×∫ 1
−1
dµw4PL`(µ)L`′(µ)δKij ,
(B3)
where P = (P + Pn/W )2. In order to obtain the mini-mum variance using this framework, we impose the sta-bility of the covariance under small changes in w. Sincemost of the S/N comes from the monopole (see Figure 2),let us consider the minimization of the average of the
10 2 10 1 100
k [hMpc 1]
100
101
102
103
104
105
106
P 0(k
), 00
[K2 h
3 Mpc
3 ]
P0
00
P0, 00
P0, 00
P0, 00
FIG. 5: Monopole of the LIM power spectrum (solid lines)and square root of its covariance (dashed lines) at z = 2.73for the generic experiment described in Section IV A. Differentcolors denote different modeling of the LIM power spectrumand covariance: the observed one, P (blue), the true one asin Eq. (B1), P (orange), and the true one with the weight
scheme discussed in this Appendix, P (green). For illustrationpurposes, here we only consider Wres and ignore Wvol.
covariance of the monopole in a shell in Fourier space.Imposing w = w0 + δw:
1
Vk
∫Vk
d3~kC00 ∝
∫d3~kw4
0
(1 + δw
w0
)4
P[∫d3~kw2
0
(1 + δw
w0
)2]2 ≈
≈
∫d3~kw4
0
(1 + 4 δww0
)P[∫
d3~kw20
(1 + 2 δww0
)]2 ≈≈∫
d3~kw40P[∫
d3~kw20
]2××
(1 + 4
∫d3~kw3
0δwP∫d3~kw4
0P− 4
∫d3~kw0δw∫
d3~kw20
),
(B4)
where we have expanded over δw/w0 up to second order.
The stability condition of C00 is fulfilled if:
w0 = P−1/2 =
(P +
PnW
)−1
. (B5)
We compare the three modellings of the LIM powerspectrum and the covariances discussed in this appendixin Fig. 5, only accounting for Wres and neglecting theeffects of Wvol for the sake of clarity in this illustration.We show the monopole of the measured LIM power spec-trum, P , of the true LIM power spectrum, P , and apply-ing the weights of Eq. (B5) to the true LIM power spec-
trum, P , as well as the square root of the corresponding
15
covariances. As can be seen, using P is very limited, sincethe S/N tends to zero at the first resolution limit. On
the contrary, P shows a slightly more optimal behaviourthan P . However, as discussed in Section III, measuring
δkT will be extremely difficult when the plane-parallelapproximation breaks down. Therefore, we advocate forthe use of P and C``′ , since their measurement is morestraightforward.
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