Geometrical constraints on curvature from galaxy-lensing cross-correlations
GGeometrical constraints on curvature from galaxy-lensing cross-correlations
Yufei Zhang
1, 2, ∗ and Wenjuan Fang
1, 2, † CAS Key Laboratory for Research in Galaxies and Cosmology,Department of Astronomy, University of Science and Technology of China,Hefei, Anhui, 230026, People’s Republic of China School of Astronomy and Space Science, University of Science and Technology of China,Hefei, Anhui, 230026, People’s Republic of China
Accurate constraints on curvature provide a powerful probe of inflation. However, curvature con-straints based on specific assumptions of dark energy may lead to unreliable conclusions when usedto test inflation models. To avoid this, it is important to obtain constraints that are independent onassumptions for dark energy. In this paper, we investigate such constraints on curvature from thegeometrical probe constructed from galaxy-lensing cross-correlations. We study comprehensivelythe cross-correlations of galaxy with magnification, measured from type Ia supernovae’s bright-nesses (“ gκ SN ”), with shear (“ gκ g ”), and with CMB lensing (“ gκ CMB ”). We find for the LSST andStage IV CMB surveys, “ gκ SN ” , “ gκ g ” and “ gκ CMB ” can be detected with signal-to-noise ratio
S/N = 104 , , K of0 .
723 from “SN + gκ SN ”, 0 . gκ g ”, and 0 .
04 from “SN + gκ g + gκ CMB ” for theLSST and Stage IV CMB surveys. The last one is more competitive than a Stage IV BAO survey(“BAO”). When galaxy-lensing cross-correlations are added to the combined probe of “SN + BAO+ CMB”, where “CMB” stands for Planck measurement for the CMB acoustic scale, we obtainconstraint on Ω K of 0 . I. INTRODUCTION
The Universe’s curvature is one of its fundamentalproperties, and deserves to be accurately measured.More importantly, stringent constraints on curvature alsoprovide powerful probes of early Universe physics suchas inflation [1–3]. Though predictions of the inflationaryscenario have been shown to be consistent with accu-rate measurements of the cosmic microwave background(CMB) anisotropies [4], details of the scenario remain tobe uncovered by further measurement results includingmeasurement of the curvature.In inflation models with a large number of e-foldings,curvature is predicted to be undetectable, i.e., the magni-tude of Ω K —the curvature density parameter—is below10 − , the measurement limit from local fluctuations inthe spatial curvature within our Hubble volume. Whileif ordinary slow-roll inflation is preceded by false vacuumdecay, potentially observable open curvature can be pro-duced [5–9]. Specifically, the analyses by [10] and [11]find that future detection of closed curvature at the levelof | Ω K | ∼ > − will exclude eternal inflation and posechallenges to the inflationary scenario, while detection ofopen curvature at the same level will suggest that falsevacuum decay happened before the observable inflation.Therefore, accurate measurement of curvature providesa powerful tool to probe inflation.Current constraints on curvature come mainly from ∗ [email protected] † [email protected]. Corresponding author. measurements of the Universe’s geometry. Due to severedegeneracy between curvature and dark energy, most ofthese constraints adopt simple assumptions for dark en-ergy. For example, by assuming dark energy to be thecosmological constant, the Planck collaboration obtainsΩ K (cid:39) . ± . a r X i v : . [ a s t r o - ph . C O ] F e b of background galaxies, and the remapping of the CMBfields.The galaxy-weak lensing cross-correlation itself cer-tainly involves information on structure growth. How-ever, by comparing the signals for the same lensing galax-ies but sources at different redshifts, pure geometric in-formation can be extracted [19–21]. With more and moreambitious cosmological surveys starting or to start oper-ation, especially the planned stage IV dark energy ex-periments such as the LSST [22], CSST [23, 24], thestage IV CMB experiment [25], this geometrical probeis drawing more attention these days. For example, ra-tios of distances have recently been measured from realsurveys with high significance using the cross-correlationbetween galaxy and cosmic shear (the so-called galaxy-galaxy lensing), and the cross-correlation between galaxyand CMB lensing [26, 27]. At the same time, the cross-correlation between galaxy and supernovae magnificationitself has not been conclusively detected with currentdata yet. For recent trials, see [28–30]. The opportu-nities with future surveys such as the LSST should bewide-ranging, see [31] and our investigations in Sec. III Abelow.Gravitational lensing uniquely probes the angular di-ameter distance from the lens (rather than from observersat z = 0) to the source, in addition to the distancesto the lens and to the source. It is known that thethree distances altogether provide a pure metric probefor curvature [32, 33], see Fig. 1. In particular, [32] pro-posed to obtain model-independent constraints on cur-vature by using the geometrical probe constructed fromgalaxy-galaxy lensing. However, their analysis done inreal space adopts an oversimplified assumption that ob-servables along different line of sights are completelyindependent. In this paper, without directly dealingwith the correlations between observables along differ-ent line of sights, we perform an analysis in Fourierspace that automatically takes into account the corre-lations. In addition, we extend the analysis to includegalaxy-supernovae magnification and galaxy-CMB lens-ing cross-correlations. We notice that cosmological con-straints from the cross-correlation between galaxy andsupernovae magnification have been barely explored, ifnot completely none. We forecast the dark energy inde-pendent constraints on curvature from galaxy-weak lens-ing cross-correlations for Stage IV dark energy experi-ments, and see whether the desired accuracy level of 10 − can be reached when they are combined with other pop-ular geometrical probes.The rest of this paper is organized as follows. InSec. II, we present the formulas we use to forecast thepure geometrical constraints from galaxy-lensing cross-correlations. In Sec. III, we forecast the constraintson curvature with different assumptions for dark en-ergy from the geometrical probe of galaxy-lensing cross-correlations and the ultimate combination with super-novae, BAO and CMB. We discuss our results in Sec. IVand summarize in Sec. V. Ω K r L + r LS r S FIG. 1. The comoving angular diameter distances to thelens r L , to the source r S , and from the lens to the source r LS provide a probe of curvature altogether. Shown in this plot isthe dependence on Ω K for the ratio of ( r L + r LS ) to r S , while r L , r S fixed at 1500 and 3000 h − Mpc respectively. Here, wehave used the approximation for r LS when | Ω K | (cid:28) II. THEORETICAL CALCULATIONS
In this section, we present the theoretical calculationsto forecast the pure geometrical constraints from galaxy-lensing cross-correlations. To break the parameter de-generacies, we will combine the constraints with thosefrom the supernovae Hubble diagram, which is avail-able from the same supernovae survey used to measurethe galaxy-supernovae cross-correlation, and the calcu-lation of which is given in Sec II A. There are greatsimilarity among the three types of galaxy-lensing cross-correlations. The differences are mainly in source red-shifts and noises for measuring the lensing signals. Thus,we elaborate the calculations for the galaxy-supernovaecross-correlation, which are much less presented in theliterature, while briefly mention those for the other two.We note our calculations in Sec.II B 1 applies to othertypes of standard candles as well.
A. Type Ia Supernovae
The apparent magnitude of a supernova at redshift z i is given by m i = 5 log [ H d L ( z i )] + M + (cid:15) i , (1)We introduce M i as a quantity involving the supernova’sintrinsic luminosity and the Hubble constant H , andseparate M i into its mean M and statistical variation( M i − M ). The latter is included in (cid:15) i , which we useto represent the total variation in m i . The luminositydistance d L is related to the comoving angular diameterdistance r by d L = (1 + z ) r , while r is given as a functionof the comoving radial distance χ by r ( χ ) = 1 √− Ω K H sin (cid:104)(cid:112) − Ω K H χ (cid:105) , (2)with χ calculated by χ = (cid:82) z dz (cid:48) /H ( z (cid:48) ).We assume the total number of supernovae discov-ered from a supernovae survey is N tot , and their redshiftdistribution is dn/dz , which we normalize to be 1. Toforecast parameter constraints from a supernovae-aloneprobe — the supernovae Hubble diagram, following [22],we neglect all possible correlations between the appar-ent magnitudes of different supernovae, and for the vari-ance in an individual supernova’s apparent magnitude σ m , we assume it is totally due to statistical variationof the supernova’s intrinsic luminosity, and take it tobe a constant. We note that correlations between dif-ferent supernovae induced by various systematic effectsand other types of statistical variations should be takeninto account for a more accurate forecast. We then usethe Fisher matrix technique to forecast the anticipatedconstraints, which is constructed as F αβ = N tot (cid:90) dz dndz σ m ∂ ¯ m ( z ) ∂p α ∂ ¯ m ( z ) ∂p β . (3)Here, besides the cosmological parameters, our param-eter set also includes M which involves both the meanof the supernovae’s intrinsic brightness and H . In ob-taining constraints on the cosmological parameters, wemarginalize over M . It can be seen that the parameterconstraints will be inversely proportional to N / . B. Galaxy-lensing cross-correlation
1. Galaxy-supernovae cross-correlation
In the previous section, we have considered the to-tal variation in an individual supernova’s brightness ata given redshift comes only from its intrinsic luminosity.In reality, an additional type of variation will be intro-duced by the magnification effect of gravitational lensingby matter distribution in the foreground [35–38]. Somesupernovae are magnified, and some demagnified. In thelimit of weak lensing, (cid:15) i in Eq. (1) will have an addi-tional term of − / (ln 10) κ i , besides ( M i − M ). Here, κ i is the lensing convergence for a source at the supernova’slocation. This magnification effect can be statisticallymeasured by cross-correlating the supernovae’s bright-nesses with the distribution of large-scale structure trac-ers in their foreground such as galaxies (see e.g., [39–41]),which we study in this section.Specifically, we consider the cross-correlation between the following two observables, δ Dn ( (cid:126)θ ) ≡ (cid:90) dzW L ( z ) δ n ( z, (cid:126)θ ) , (4) M ( (cid:126)θ ) ≡ (cid:90) dzW S ( z ) m ( z, (cid:126)θ ) , (5)where δ Dn and δ n are the two and three-dimensionalgalaxy overdensities; W L and W S are the redshift selec-tion functions for the galaxies (“Lens”) and supernovae(“Source”) respectively, both of which have been nor-malized to be 1, i.e., (cid:82) W ( z ) dz = 1; M ( (cid:126)θ ) representsthe average apparent magnitude for the selected super-novae whose angular positions are within a solid angle d θ ( d θ →
0) around (cid:126)θ . Intrinsic fluctuations in su-pernovae’s brightnesses drop out in the cross-correlation,and by a Fourier transform, we get the following cross-correlation power spectrum, C LS(cid:96) = −
152 ln(10) Ω m H (cid:90) dzW L ( z ) g S ( z ) ar P gm ( k = (cid:96)r , z ) , (6)where Ω m is the density parameter for matter, a is thescale factor, P gm is the galaxy-matter power spectrum,and g S ( z ) is given by, g S ( z ) = (cid:90) dz (cid:48) W S ( z (cid:48) ) r ( χ ) r ( χ (cid:48) − χ ) r ( χ (cid:48) ) Θ( χ (cid:48) − χ ) , (7)where Θ is the Heaviside step function. In our derivationfor Eq (6), we have used the Limber approximation [42,43].Galaxy-supernovae cross-correlation potentially pro-vides a pure geometrical probe of the Universe, as isthe case for galaxy-galaxy lensing [19, 21]. A directway to see this is by taking the limit that the fore-ground galaxies are all selected to be at a single redshift,i.e., W L ( z ) → δ ( z − z L ), then we have C LS(cid:96) /C LS (cid:48) (cid:96) → g S ( z L ) /g S (cid:48) ( z L ), i.e., the ratio of the cross-correlationsbetween these galaxies and supernovae selected with dif-ferent selection functions, W S ( z ) and W S (cid:48) ( z ), probes apure geometrical quantity. Furthermore, in the limitof W S ( z ) → δ ( z − z S ) and W S (cid:48) ( z ) → δ ( z − z S (cid:48) ), C LS(cid:96) /C LS (cid:48) (cid:96) → r ( χ S − χ L ) r ( χ S (cid:48) ) /r ( χ S ) r ( χ S (cid:48) − χ L ), whichdirectly probes the ratio of the angular diameter dis-tances.In this section, we investigate the pure geometri-cal probe constructed from the galaxy-supernovae cross-correlation. We consider a survey of both supernovaeand galaxies, and assume the former to be observed toa maximum redshift z max , and the latter to a redshiftbeyond that. For measurements of the cross-correlation,galaxies at z > z max would be of no use, since their cross-correlation with the supernovae would vanish. We divideboth the supernovae and galaxies from z = 0 to z = z max into redshift bins of equal size ∆ z . Hereafter, we use “S”as the index for the supernova (source) redshift bins, and“L” for the galaxy (lens) redshift bins. The measuredcross power spectra C LS(cid:96) and C L (cid:48) S (cid:48) (cid:96) (cid:48) have the followingcovariance,Cov( C LS(cid:96) , C L (cid:48) S (cid:48) (cid:96) (cid:48) ) = δ K(cid:96)(cid:96) (cid:48) (cid:96) ∆ (cid:96)f sky (cid:16) C SS (cid:48) (cid:96) C LL (cid:48) (cid:96) + C LS (cid:48) (cid:96) C L (cid:48) S(cid:96) (cid:17) , (8)where δ K is the Kronecker delta, ∆ (cid:96) is the size of themultipole bin used to measure C LS(cid:96) , and f sky is the frac-tion of sky covered by the survey. Both the auto powerspectra for the supernovae brightnesses C SS (cid:48) (cid:96) and for thegalaxy distribution C LL (cid:48) (cid:96) have two contributions: onefrom large-scale structure (“LSS”), and the other fromshot noise (“shot”), given by C SS (cid:48) (cid:96), shot = δ KSS (cid:48) σ m / ¯ n DS , (9) C SS (cid:48) (cid:96), LSS = ( 152 ln(10) Ω m H ) (cid:90) dz dχdz × g S ( z ) g S (cid:48) ( z ) a r P mm ( k = (cid:96)r , z ) , (10) C LL (cid:48) (cid:96), shot = δ KLL (cid:48) / ¯ n DL , (11) C LL (cid:48) (cid:96), LSS = (cid:90) dzW L ( z ) W L (cid:48) ( z )( dχdz ) − r − × P gg ( k = (cid:96)r , z ) , (12)where σ m , as in Sec. II A, is the variance of an individ-ual supernova’s brightness due to its intrinsic luminosity,¯ n DS , ¯ n DL are the mean angular number densities for su-pernovae in the “S”th redshift bin and galaxies in the“L”th redshift bin respectively, while P mm and P gg arethe matter and galaxy power spectra in turn. Note C SS (cid:48) (cid:96), LSS is the power spectrum of the E-mode shear except for aconstant factor of (5 / ln(10)) , see e.g., [21, 44].Same as before, we use the Fisher matrix techniqueto forecast parameter constraints from galaxy-supernovaecross-correlation. Since statistical isotropy implies thatdifferent multipoles are uncorrelated, the Fisher matrixcan be written as a sum of contributions from differentmultipoles, F αβ = (cid:88) (cid:96) (cid:88) ( LS ) , ( L (cid:48) S (cid:48) ) ∂C LS(cid:96) ∂p α (Cov (cid:96) ) − ∂C L (cid:48) S (cid:48) (cid:96) ∂p β , (13)where ( LS ) or ( L (cid:48) S (cid:48) ) labels distinct cross power spectra.Cov (cid:96) is the subblock of the full covariance matrix (Eq. 8)for all the cross power spectra with multipole (cid:96) . We noteits inverse is proportional to 2 (cid:96) ∆ (cid:96)f sky , hence F αβ ∝ f sky ,and the parameter constraints will be proportional to f − / .To extract the pure geometrical constraints, we chooseour redshift bins to be narrow enough such that the fol-lowing approximation (under the limit of ∆ z →
0) to the cross power spectrum holds to a good accuracy, C LS(cid:96) ≈ −
152 ln(10) Ω m H r ( χ S − χ L ) r ( χ S ) r ( χ L ) 1 a ( z L ) × P gm ( k = (cid:96)r ( χ L ) , z L )Θ( χ S − χ L ) , (14)where χ S = χ ( z S ), χ L = χ ( z L ), with z S , z L representingthe mean redshifts of the narrow supernova and galaxybins respectively. With this approximation, we can eas-ily separate geometrical information ( r ( χ S − χ L ) /r ( χ S ))from what remains whose prediction typically involvesuncertainties in galaxy bias and matter power spectrumin the nonlinear regime, which we hereafter denote as C gm(cid:96) ( z L ). Next, we take the C gm s at different multipolesand redshifts also as parameter entries for the Fisher ma-trix, and marginalize over them for the final geometricalconstraints on the cosmological parameters. Includingthese extra parameters significantly increases the dimen-sion of the Fisher matrix, hence increases the difficultyfor its inversion. However, from Eq. (13), we find that F αβ = 0, when p α and p β correspond to C gm at differentmultipoles. This feature greatly simplifies the inversionof the Fisher matrix with the method of “inversion bypartitioning” [45].
2. Galaxy-galaxy lensing
Compared to galaxy-supernovae cross-correlation,galaxy-galaxy lensing [46] can be detected with a strongersignificance and to a higher redshift, for it is much easierto observe a large number of galaxies to a high redshiftthan to observe supernovae. In this section, we considerthe pure geometrical probe from galaxy-galaxy lensing.While galaxy-supernovae cross-correlation is the cor-relation between the distribution of a foreground galaxypopulation and magnifications in the background super-novae’s brightnesses, galaxy-galaxy lensing is the corre-lation between the former and distortions in the back-ground galaxies’ images caused by weak lensing - the cos-mic shear field. To be explicit, galaxy-galaxy lensing isthe cross-correlation between δ Dn , given by Eq. (4), andΓ i , given by the following,Γ i ( (cid:126)θ ) = (cid:90) dzW S ( z ) γ i ( z, (cid:126)θ ) , (15)where γ i is the shear field estimated through measure-ments of background galaxies’ ellipticities, and “ i ” labelsthe two shear components. Note, W S ( z ) here is the selec-tion function for the “sources”, i.e., background galaxies.Similar to M ( (cid:126)θ ), Γ i ( (cid:126)θ ) represents the average of the shearestimated from background galaxies whose angular posi-tions are within a solid angle d θ ( d θ →
0) around (cid:126)θ .The shear field can be decomposed into a curl-freeE-mode component and a divergence-free B-mode com-ponent. With scalar perturbations alone, only the E-mode shear exists, which is equivalent to the lens-ing convergence. Therefore, we only need to studythe cross-correlation between galaxy and the E-modeshear, which we recognize to be exactly the same as thegalaxy-supernovae cross-correlation, except for a factorof 5 / ln(10).In this section, we consider a weak lensing survey. Asbefore, we divide the galaxies into redshift bins of equalsize ∆ z , and denote the galaxy-galaxy lensing powerspectrum as C LS(cid:96) . Different from before, “S” here labelsthe redshift bin for the “source” galaxies. The expressionfor the covariance of C LS(cid:96) and C L (cid:48) S (cid:48) (cid:96) (cid:48) remains the same asbefore, except the following differences: (1) C SS (cid:48) (cid:96), LSS hasnot a factor of (5 / ln(10)) ; (2) C SS (cid:48) (cid:96), shot is now given by C SS (cid:48) (cid:96), shot = δ KSS (cid:48) ¯ n DS (cid:90) dzW S ( z ) γ ( z ) , , (16)where ¯ n DS is the mean angular number density for sourcegalaxies in the “S”th redshift bin, and γ rms is the rms ofshear in each component from galaxies’ intrinsic elliptic-ities. In the end, we use the same method as in Sec II B 1to forecast the pure geometrical constraints on cosmolog-ical parameters from galaxy-galaxy lensing.
3. Galaxy-CMB lensing cross-correlation
The Universe’s large-scale structure gravitationally de-flects the photons of CMB as well, and thus perturbs theCMB power spectra. The weak lensing convergence forthe CMB can be reconstructed from the various lensedCMB power spectra using the minimum variance estima-tor, which minimizes the reconstruction noise [47]. Forcosmic shear, the source galaxies are typically distributedacross a relative broad range of redshift, and the sig-nals are then averaged over this distribution. However,the CMB photons originate from a very narrow rangeof comoving distance, thus the source redshift distribu-tion can be approximated as a Dirac δ -function with thevalue of redshift known to a very precise level. For thecross-correlation between galaxy and CMB lensing, theexpressions for the cross power C LS(cid:96) and the covariancebetween C LS(cid:96) and C L (cid:48) S(cid:96) (cid:48) (“S” here labels the redshift binfor the “source” of CMB) remain the same as for thegalaxy-galaxy lensing, except C SS(cid:96), shot is now replaced bythe CMB lensing reconstruction noise, whose expressionis given by, e.g. Eq.(42) in [47].
III. CURVATURE CONSTRAINTS
In this section, we present the constraints on curva-ture from the geometrical probe constructed from galaxy-lensing cross-correlations. In Sec III A, we forecast theconstraints from the galaxy-lensing cross-correlations incombination with the supernovae Hubble diagram. Wemake our forecast for fiducial surveys mimicking theLSST and Stage IV CMB experiment. In addition to de-tecting a large number of type Ia supernovae, the LSST will measure both the galaxy distribution and cosmicshear field at the same time. Hence, three of the probesdiscussed in the last section, i.e., the supernovae Hub-ble diagram, galaxy-supernovae cross-correlation, andgalaxy-galaxy lensing, will be available from the LSST,while the galaxy-CMB lensing cross-correlation can bemeasured from the overlapping area between the LSSTand Stage IV CMB experiment. (We assume the StageIV CMB survey overlaps completely with the LSST.) InSec III B, we add in BAO and CMB to further tighten theconstraints. In Sec III C, we study improvements in theconstraints from increasing the high redshift extension ofsupernovae while keeping their total number fixed.
A. Combination of galaxy-lensing cross-correlationwith supernovae Hubble diagram
The supernovae Hubble diagram probes the angulardiameter distance from a given redshift to an observer onthe earth, i.e., at z = 0, while the galaxy-lensing cross-correlation can additionally probe the angular diameterdistances from a given redshift (the sources’ redshift) toobservers at all intermediate redshifts with z (cid:54) = 0 (thelenses’ redshifts). Hence, the latter provides importantcomplementary information.The LSST is about to survey a sky area of approxi-mately 20,000 deg for a duration of 10 years startingby 2022 [48]. It will detect about half a million type Iasupernovae to a redshift slightly beyond z = 1, see Fig. 2for the supernovae’s redshift distribution [22]. In the fol-lowing, we assume the total number of supernovae N tot to be 4 × [48], and the rms of their intrinsic bright-nesses σ m to be 0 . − , which is the so-called “gold” sampleof the LSST galaxies, the redshift distribution of which isshown as the solid line in Fig. 2 [22]. Among these galax-ies, we assume about 60% of them can be used for shearmeasurement [48], hence, for galaxy-galaxy lensing, thenumber density of source galaxies is 30 arcmin − , and weassume γ rms has a redshift-independent value of 0 .
28 [22].For the Stage IV CMB experiment, we assume a 1 arcminbeam and 1 µ K arcmin noise [25], and assume it coversthe LSST survey area.For the galaxy-lensing cross-correlations, we choose aredshift bin size of ∆ z = 0 . z = 0 .
15 for thelens galaxies, and finds the correction from the narrowtracer bin approximation to be negligible compared totheir measurement errors. For galaxy-supernovae cross-correlation, we have 11 supernovae bins from z = 0 to z max = 1 . × /
2) distinct cross power spectra. Forgalaxy-galaxy lensing, we cut off the galaxy distributionat z max = 4 in our numerical calculation, which includes ∼
99% of all the galaxies, and we have 780 distinct cross d n / d z FIG. 2. Normalized redshift distributions of type Ia super-novae (histogram) and galaxies (solid line) for the LSST. power spectra. This also gives us 40 galaxy-CMB lens-ing cross power spectra. In Fig. 3, we explicitly show thepower spectra for the galaxy-supernovae cross-correlation(“ gκ SN ”, black) and galaxy-galaxy lensing (“ gκ g ”, red)for the same redshift bins of lenses and sources. Note, theformer has been divided by a factor of 5 / ln(10), hencethe two curves agree. The cross power spectrum for thegalaxy and CMB lensing for the same lens galaxies isalso shown as the green curve. The error bands includ-ing both sample variance and shot noise are forecastedaccording to Eq. (8), with appropriate adaptions for gκ g and gκ CMB as discussed in Sec. II. For the auto powerspectrum of CMB lensing and the reconstructed noisesfor the Stage IV CMB experiment as considered here, werefer the readers to [49].For CMB lensing, we use the public available package quicklens [50] to do lensing reconstruction from the T,E and B modes of CMB, and we cut off the multipolesat (cid:96) = 3000, due to the difficulty of cleaning temperatureforegrounds at (cid:96) > gκ SN and gκ g , since we marginalize overparameters directly describing the galaxy-matter powerspectrum, there is no need to worry about nonlinear ef-fects and baryonic effects on small scales, we use infor-mation on angular scales up to (cid:96) ∼ [21]. Of course,not much information can be obtained from modes withhigh enough multipoles due to significant noise.If we define the total S/N square for the galaxy-lensingcross-correlation as (cid:18) SN (cid:19) ≡ (cid:88) (cid:96) (cid:88) ( LS ) , ( L (cid:48) S (cid:48) ) C LS(cid:96) (Cov (cid:96) ) − C L (cid:48) S (cid:48) (cid:96) , (17)we find for the LSST, the galaxy-supernovae cross-correlation has S/N = 103 .
8, while the galaxy-galaxylensing has
S/N = 2291 .
2, about 22 times larger due to ‘ -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 C L S ‘ g SN g g g CMB
FIG. 3. The power spectra for the galaxy-supernovae cross-correlation (“ gκ SN ”, black), galaxy-galaxy lensing (“ gκ g ”,red), and galaxy-CMB lensing cross-correlation (“ gκ CMB ”,green). Error bands are forecasted for the LSST and StageIV CMB experiment. We have divided the galaxy-supernovaecross power and its errors by a factor of 5 / ln(10). The lensgalaxies are in the redshift bin of z:[0.4, 0.5], while the su-pernovae and source galaxies are in z:[0.5, 0.6]. The errorbands for the galaxy-CMB lensing cross power are plotted upto (cid:96) = 3000.TABLE I. Total S/N for detecting the galaxy-lensing cross-correlation from the LSST and Stage IV CMB experiments.S/N gκ SN gκ g gκ CMB its wider redshift coverage and smaller shot noise (seeTable I). While for the LSST and Stage IV CMB ex-periment, the galaxy-CMB lensing cross-correlation has
S/N = 1842 . Λ =0 .
689 [12]; when calculating P mm , we further choose { Ω b h , Ω m h , σ , n s } = { . , . , . , . } [12],and adopt the Smith et al prescription [51] to accountfor the effects of non-linear evolution; when calculating P gg and P gm , we assume a simple linear bias of b = 1.Degeneracy between dark energy parameters and cur-vature is expected, since they both affect the Universe’sexpansion rate, hence the comoving radial distance,though curvature has an additional effect on the comov-ing angular diameter distance. Therefore, the constraintson curvature will depend on specific assumptions aboutdark energy, characterized by assumptions for its equa-tion of state w which determines the evolution of its en-ergy density.For comparison, we forecast the constraints on curva-ture for a wide range of choices of w . We first considerthe following common choices: (1) w is fixed to be (-1),i.e., dark energy is the cosmological constant; (2) w isa constant whose value needs to be determined; (3) w varies with time as w = w + w a (1 − a ), with the valuesof w , w a to be determined. From the first to the thirdchoice, the degrees of freedom in w is increasing. Next,we consider a fourth more general parametrization where w is a binned function parametrized by its values in N equal-sized a bins from a = 0 to a = 1, which we denoteas w i , with i = 1 , ..., N . Each w i is allowed to vary inde-pendently. This parametrization potentially can encloseall possible models for w , hence the constraints on cur-vature obtained with this choice after marginalizing overall dark energy parameters can be considered as indepen-dent of models of dark energy. In the following, we willstudy the constraints on curvature with N =10, 20, and50, and the results when N = 50 will be our most darkenergy independent constraints on curvature.The constraints on curvature with the above assump-tions for dark energy forecasted for the LSST and StageIV CMB experiments are shown in Table II. In differentcolumns, we show the constraints from different probes,with the second column showing the constraints from thesupernovae Hubble diagram alone (“SN”), and the thirdand fourth columns giving the constraints from the com-binations of “SN” with “ gκ SN ” and “ gκ g ” respectively.The last column gives the constraints from the combina-tions of “SN” with “ gκ g ” and “ gκ CMB ”. We do not showthe constraints from either “ gκ SN ”, “ gκ g ” or “ gκ CMB ”alone, because they are less interesting due to severe pa-rameter degeneracy [32]. In different sections, we showthe constraints with different assumptions for dark en-ergy. From the first to the third sections, we show inturn the constraints with our first three choices of w ,while from the fourth to the last, we give the constraintsfor the fourth choice with N = 10 , ,
50 respectively.Note, from the top to the bottom sections, the uncer-tainty in our knowledge about w is increasing.By comparing the constraints in different sections foreach probe, we find that when there are more degreesof freedom in dark energy’s equation of state, the con-straints on curvature get weaker, consistent with one’s ex-pectation. Our strongest constraints are hence obtainedwhen w = −
1, and the weakest ones obtained when w isa binned function with 50 bins in a . It can also be eas-ily seen that the constraints from the combined probesof either “SN + gκ SN ” or “SN + gκ g ” are better than“SN” alone, due to the important complimentary infor-mation provided by the galaxy-lensing cross-correlationwhich breaks the degeneracy between curvature and darkenergy parameters.It is interesting to find that the improvements in thecurvature constraints by adding in “ gκ SN ” or “ gκ g ” getmore significant when dark energy has more degrees of TABLE II. Parameter constraints forecasted for the LSSTand Stage IV CMB experiments. From left to right, thecolumns are constraints from the supernovae Hubble dia-gram (“SN”), its combination with galaxy-supernovae cross-correlation (“SN + gκ SN ”), and combination with galaxy-galaxy lensing (“SN + gκ g ”), and combination with galaxy-galaxy lensing and galaxy-CMB lensing(“SN + gκ g + gκ CMB ”). Note, our constraints from “ gκ SN ” , “ gκ g ” and” gκ CMB ”are pure geometrical. From top to bottom, theconstraints in different sections are based on dark energyparametrizations with more degrees of freedom in its equa-tion of state w : the first three sections are for (1) w = − w =const, (3) w = w + w a (1 − a ); while the last threesections assume w is a binned function parametrized by itsvalues in N bins from a = 0 to a = 1, with N = 10 , , gκ SN SN+ gκ g SN+ gκ g + gκ CMB Ω K Λ K Λ w K Λ w w a K Λ K Λ K Λ freedom, with the improvements being minimal whendark energy is known to be the cosmological constant andmaximal when w is parametrized by 50 w i s. This high-lights the importance of galaxy-lensing cross-correlationin obtaining dark energy independent constraints on cur-vature. Moreover, the constraints from the combinationof “SN + gκ g ” are better than from “SN + gκ SN ”. For N = 50, the improvement factor is 233 by adding in“ gκ g ” to “SN”, while 13 when adding in “ gκ SN ”. Thisis expected as compared to “ gκ SN ”, “ gκ g ” has a widerredshift coverage and lower shot noise. To tighten theconstraints more, we add in “ gκ CMB ” to “SN + gκ g ” inthe last column, we do not add “ gκ SN ” here to avoid thestrong correlations between “ gκ SN ” and “ gκ g ” . We no-tice, the improvements by adding in “ gκ CMB ” to “SN + gκ g ” get milder when dark energy has more degrees offreedom, ranging from 4 when dark energy is known tobe the cosmological constant to 1.04 when N = 50.Finally, we find that though the curvature constraintskeep getting weaker when w is allowed to have moredegrees of freedom, the constraints from the combinedprobes of “SN + gκ g + gκ CMB ” do not degrade muchwhen we take w to be a binned function and increasethe number of bins from 10 to 20 to 50. As discussed in[52], a limited number of bins that is equally-spaced inscale factor is enough for parameter constraints to con-verge. Therefore, we hereafter quote the constraints with w a binned function with 50 bins in a as our final “darkenergy independent” constraints.To summarize, we obtain dark energy independent con-straints on Ω K of 0 .
723 from the combination of “SN+ gκ SN ” , 0 . gκ g ” for the LSST, and0 .
04 from “SN + gκ g + gκ CMB ” for the LSST and StageIV CMB experiments. We find the galaxy-lensing cross-correlation plays a significant role in obtaining these re-sults. It improves the curvature constraints by breakingthe degeneracy between curvature and the dark energyparameters from “SN” alone. If we know dark energy tobe the cosmological constant, we are able to get muchtighter constraints of 0 . gκ SN ”, 0 . gκ g ”, and 0 . gκ g + gκ CMB ”.We note that in obtaining these results, we do not applyany priors. These results do not reach the desired ac-curacy level of 10 − , so below we add other geometricalprobes to improve the constraints on curvature further. B. Adding in BAO and CMB
In this section, we are interested in tightening the con-straints on curvature further by combining with othergeometrical probes. Specifically, we include the probesutilizing the standard ruler of sound horizon at recom-bination s . Measurements of the CMB anisotropies canprobe the angular size extended by the sound horizon atrecombination θ ∗ , hence the angular diameter distanceto recombination r ∗ (= s/θ ∗ ). Late-time BAO measure-ments can also probe the sound horizon through its im-prints on matter distribution. Its extensions in the trans-verse direction ( δθ = s/r ) and the line-of-sight direction( δz = sH ) probe the late-time angular diameter distance r and Hubble expansion rate H respectively. In this sec-tion, we include these two probes in our forecast.For the CMB constraints, we simply incorporate thePlanck measurement for θ ∗ , which is at an accuracy levelof ∼ . m h and Ω b h as priors forour calculation. Strictly speaking, late-time BAO measurements probe the soundhorizon at the end of the baryon drag epoch. We here neglectthe small difference following [16].
TABLE III. Constraints from “BAO + CMB”, “SN + BAO +CMB” and “SN + gκ g + gκ CMB + BAO + CMB”(denoted as“All”). The upper section assumes dark energy is the cosmo-logical constant, while the lower section assumes w is a binnedfunction parametrized by 50 w i s - its values in 50 bins from a = 0 to a = 1. For each w i , a prior of ∆ w i = 10 √ N ( N = 50)is applied.parameters BAO+CMB SN+BAO+CMB AllΩ K Λ K Λ For the BAO constraints, we follow [16] and considera Stage IV BAO experiment that maps 25% of the fullsky from z = 0 to z = 3 with errors ∼
80% larger thanthe linear theory sample variance errors (to account fornon-negligible shot noise and non-linear degradation ofthe BAO signal). Such an experiments can be collec-tively achieved by the BAO programs that are currentlyongoing or under design, such as the programs from Eu-clid [54], WFIRST [55], and DESI [56]. We adopt theforecasted covariance matrix for the measured quantitiesof r/s and sH from [16], and then use the Fisher matrixto derive constraints on the cosmological parameters.The constraints we obtained from BAO and CMB areshown in Table III. By adding BAO and CMB to our pre-vious calculations we get stronger constraints, also shownin Table III. Here we only show the constraints obtainedwith the assumption that dark energy is the cosmologi-cal constant (top section), and with our most uncertainassumption for dark energy (i.e. w is a binned functionwith 50 equal-sized bins from a = 0 to a = 1, bottomsection). Note, whenever CMB is included, we apply aweak Gaussian prior with width ∆ w i = 10 √ N on all the w i s [16].We focus on discussions about the “dark energy-independent constraints on curvature”, which we simplyrefer to as “constraints on curvature” unless otherwiseexplicitly stated in the following. From Table III, it canbe seen that our constraint on curvature from “BAO +CMB” is better than that from “SN + gκ g + gκ CMB ”.Therefore, “BAO + CMB” is more promising in con-straining curvature in a dark-energy independent waythan the combination of “SN” and galaxy-lensing crosscorrelations.At the same time, we also look at the constraint oncurvature from BAO alone , which we find to be 0 . gκ g ” and “SN + gκ g + gκ CMB ”. Therefore, for StageIV dark energy surveys, the combination of “SN” and Planck priors on Ω m h and Ω b h are still included here. galaxy-lensing cross correlations provides a slightly bet-ter probe of curvature than BAO.To see the importance of galaxy-lensing cross correla-tions when they are added to the combined probe of “SN+ BAO + CMB”, we compare the 3rd and 4th columns ofTable III, and find they are only mildly helpful when darkenergy is the cosmological constant, but can tighten theconstraint on curvature by approximately a factor of 7when dark energy has a general parametrization. There-fore galaxy-lensing cross-correlations play an importantrole in extracting information on the Universe’s curvaturein a dark energy independent way.Our ultimate constraint on curvature from the com-bination of the five geometrical probes “SN + gκ g + gκ CMB + BAO + CMB” now reaches 0 . gκ g + gκ CMB ” or “BAO+ CMB”, which give curvature constraints only at thelevel of 10 − , reflecting the strong complementarity ofthe two combined probes.To conclude, when BAO and CMB are included, theimprovements on the curvature constraints can be oneorder of magnitude. We find the dark energy indepen-dent constraints on curvature now can be as tight as ∼ × − , but still one order of magnitude away fromthe desired level of 10 − . Moreover, even if dark energyis the cosmological constant, the constraint on curvaturefrom the combination of the five geometrical probes of“SN + gκ g + gκ CMB + BAO + CMB” is only ∼ . × − . In thefollowing we explore how much we can gain on the cur-vature constraint by broadening the redshift coverage ofsupernovae. C. Increasing z max of Supernovae It is known that for a supernovae alone probe, abroader redshift coverage can give better parameter con-straints provided the total number of supernovae is keptfixed [57]. This is because supernovae at different red-shifts usually lead to different degeneracy directionsamong the parameter space, and a wider redshift cov-erage results in better complementarity. In this sectionwe investigate the possibility of tightening the curvatureconstraints more with a wider redshift coverage of super-novae or standard candles in general.Specifically, we keep the total number of supernovae tobe fixed at 4 × , and assume their redshift distributionfollows that of the LSST galaxies but cuts off at z max . Inthe following, we will forecast the constraints on curva-ture for two choices of z max : z max = 2 and z max = 3.Supernovae at such high redshifts may be challenging tobe surveyed with ground-based telescopes like the LSST,but may be easier to observe with future space-basedones. For example, the WFIRST mission is about to findsupernovae to z max = 1 .
7, but with N tot only ∼ × . Thus, we note the forecasts we makein this section may be too optimistic for type Ia super- novae surveys currently in plan. However, they may bemore realistic for other types of standard candles such asquasars and gamma-ray bursts, which can be observedto much higher redshifts [59, 60].The dark energy independent constraints on curvatureobtained by extending z max from z max = 1 . z max = 2and z max = 3 are plotted in Fig. 4. It can be easilyseen that the curvature constraints from either “SN”,“SN+ gκ SN ”, “SN + gκ g + gκ CMB ”, or “SN + gκ g + gκ CMB + BAO + CMB” all get better when the super-novae’s redshift distribution can reach a higher z max . Wealso find that the improvements in the constraints by in-creasing z max from z max = 1 . z max = 2are more significant than increasing it from z max = 2 to z max = 3. For the dark energy independent constraintson curvature from “SN ”, we obtain 1 .
12 with z max = 2(improved by a factor of ∼ . z max = 1 . .
728 with z max = 3 (improved by a factor of ∼ . z max = 2); while for the dark energy independent con-straints from “SN + gκ g + gκ CMB ”, we get 0 . z max = 2 (improved by a factor of ∼ . z max = 1 . . z max = 3 (improved by a factor of ∼ . z max = 2). Even if we can increase the supernovae’sredshift distribution only to z max = 2, the efforts are re-warding enough judging from the improvements on cur-vature constraints from “SN” and its combination withgalaxy-lensing cross-correlations. Again, we find as be-fore that combining galaxy-lensing cross-correlations and“SN” resulting in significant improvement, even for thecase with gκ SN , which has a relative lower S/N .However, for the combination of all five probes “SN + gκ g + gκ CMB + BAO + CMB”, the improvement fromincreasing z max is much milder, especially when increas-ing from z max = 2 to z max = 3, reflecting the subdomi-nant role of “SN” in the combined probes (probably be-cause BAO already provides the high- z information upto z = 3), and the rapid decrease with redshift of galaxydistribution at high z . We find the constraints from thecombination of the five probes is still ∼ − . However,significant improvement on curvature constraint can bepossible with standard candles whose redshift distribu-tion has a larger fraction at high redshift, say z > IV. DISCUSSION
Our calculations above have adopted several simplifi-cations. We have neglected a few systematic effects suchas photometric redshift errors, shear calibration errors,galaxy intrinsic alignments etc. For photometric red-shift errors, the LSST galaxies’ photometry will have highenough quality to provide a rms accuracy σ/ (1+ z ) of 0.02[48]. This is in general much smaller compared to ourbin width of ∆ z = 0 .
1. Therefore, we expect photomet-ric redshift errors would not change at least the order ofmagnitude of our results. For shear calibration errors, re-cent analysis by [27] found that if the multiplicative shear0 z max -3 -2 -1 σ Ω K SNSN+ g SN SN+ g g + g CMB
All
FIG. 4. Forecast 1 σ constraints on curvature as a function ofthe maximal redshift of supernovae for “SN”(black) , “SN + gκ SN ”(red), “SN + gκ g + gκ CMB ”(green) and “SN + gκ g + gκ CMB + BAO + CMB”(blue, denoted as “All”), respectively.The total number of supernovae is kept fixed at 4 × . Thedark energy equation of state parameter w is assumed to bea binned function parametrized by its values in 50 bins from a = 0 to a = 1. bias m from LSST can be calibrated to σ ( m ) = 0 . − ,one order of magnitude larger than the desired level of10 − , we do not analyze the effect of intrinsic alignmenttogether with other systematic effects (including thosewe have not mentioned in the above, such as the nar-row lens bin approximation, lensing dilution and galaxylensing boost factors) quantitatively here, which will notchange our primary finding.In this work, we obtain our dark energy-independentconstraints on curvature by assuming the equation ofstate of dark energy is parametrized by its values in 50 a bins from a = 0 to a = 1 and marginalizing over allthese parameters. Compared to previous work on thistopic by [61] and [62], this approach is more model-independent: the method proposed by [61] depends onthe assumption that dark energy is completely negligiblein “matter-dominated” regime, while ours allows dark en-ergy to be non-negligible even at early times (early darkenergy); [62] parametrizes w of dark energy at low red- shift ( z ∼ < .
7) with 15 principle components (PCs), butcalculations of the PCs typically depend on the specificsof both the data set and cosmological model used to ob-tain the Fisher matrix from which they are derived.However, compared to previous work by [32], our ap-proach is not as model-independent. [32] probes curva-ture purely from the relationship between r L , r LS and r S . By marginalizing over the distances of r L , r S whichare integrals of functions of the Hubble expansion rate,the obtained constraints do not depend on any energycomponent of the Universe or dynamics that governs itsexpansion, but only on the validity of the FRW metric.Our approach is less general in the sense that we in ad-dition assume energy-momentum conservation and thevalidity of the Friedmann equation if cosmic accelerationis due to dark energy, or if cosmic acceleration is due tomodified gravity, its effect on the Universe’s expansioncan be viewed equivalently as an effective dark energy,which holds for most interesting modified gravity models(see e.g., [63, 64]). Therefore, though in this paper weuse the term of “dark energy”-independent constraintson curvature, our constraints are actually independenton the unknown mechanism for cosmic acceleration in-cluding both dark energy and modified gravity.Since we obtain our curvature constraints by marginal-izing over the contribution of dark energy or “effective”dark energy to the Universe’s expansion, while [32] ob-tain theirs by marginalizing over the distances, our con-straints will be tighter than theirs. We conclude that [32]provide a pure metric probe of curvature which does notdepend on how the Universe expands, while we probe thecurvature in a way that is independent on how (“effec-tive”) dark energy affects the Universe’s expansion.There are also many works in the literature that uti-lize measurements of the angular diameter distances andHubble expansion rates to obtain model-independentconstraints on curvature, e.g. [65–70]. These works typ-ically need to estimate derivatives of the angular diam-eter distances or to reconstruct the Hubble expansionhistory using some model-independent smoothing tech-niques such as the Gaussian process. Thus, it may behard to achieve an accuracy as tight as σ Ω K ∼ − ro-bustly using these methods.In this work, we have focused on constraining curva-ture using probes of the Universe’s geometry. One cansurely add in probes of the Universe’s growth of struc-ture to tighten the constraints. Actually, [62] has studiedmodel independent constraints on curvature from com-bining geometry probes with growth probes through mea-suring the abundance of X-ray clusters. However, theirwork is done within the “smooth” dark energy paradigm[71] and assumes the validity of general relativity, hence itdoes not apply to dark energy with nontrivial clusteringproperties or modified gravity. Also, probes of structuregrowth are usually subject to systematics from theoret-ical modeling of baryonic physics and growth of struc-ture on nonlinear scales. Future work on using probes ofstructure growth to constrain curvature in a dark energy-1independent way should take into account of all theseproblems, which may be challenging. V. SUMMARY
Accurate constraints on curvature provide a powerfulprobe of inflation. However, current accurate constraintson curvature are almost all derived upon simple assump-tions of dark energy such as assuming it is the cosmolog-ical constant. Considering the large uncertainties in ourtheoretical understanding about dark energy, constraintswith these assumptions may lead to unreliable conclu-sions when they are used to test inflation models. Hence,for a robust test of inflation models, it is important toobtain constraints on curvature that are independent onuncertainties in our knowledge about dark energy. In thispaper, we have investigated such constraints on curva-ture from the geometrical probe constructed from galaxy-lensing cross-correlations and its combination with othercommon geometrical probes.We study the galaxy-magnification, galaxy-shear, andgalaxy-CMB lensing cross-correlations, with magnifica-tion measured from the type Ia supernovae’s bright-nesses. We find for the Stage IV dark energy surveyof LSST and the Stage IV CMB survey, the galaxy-magnification cross-correlation (“ gκ SN ”) can be detectedwith signal-to-noise ratio S/N = 104, the galaxy-shearcross-correlation (“ gκ g ”) with S/N = 2291, and thegalaxy-CMB lensing cross-correlation (“ gκ CMB ”) with
S/N = 1842. We include the supernovae Hubble dia-gram (“SN”) to break parameter degeneracy, which isavailable with the same supernovae data set used to mea-sure “ gκ SN ”. We obtain dark energy independent con-straints on Ω K of 0 .
723 from “SN + gκ SN ”, 0 . gκ g ”, and 0 .
04 from “SN + gκ g + gκ CMB ” for theLSST and Stage IV CMB experiment. We find that thegalaxy-lensing cross-correlation plays a significant role intightening the curvature constraint by breaking the de-generacy between curvature and the dark energy parame-ters, especially when dark energy is completely unknown. We find the constraint from “SN + gκ g + gκ CMB ” is bet-ter than that from a Stage IV BAO experiment, but notas good when BAO is combined with the Planck mea-surement for the acoustic scale in the CMB. Adding thegalaxy-lensing cross-correlations to the combined probeof “SN + BAO + CMB” results in a factor of 7 improve-ment in the dark energy independent constraints on cur-vature, but much milder improvement when dark energyis known to be the cosmological constant. We obtain ourultimate constraint on Ω K of 0 . gκ g + gκ CMB + BAO + CMB”. Our analysis also shows thattighter constraints can be obtained with better knowl-edge about dark energy.We investigate the possibility of tightening the curva-ture constraints further by increasing the redshift exten-sion of supernovae or standard candles in general, whilekeeping the total number fixed at the same value. We findthough the “SN” alone and its combination with galaxy-lensing cross correlations have significant improvementson curvature constraints, the combined probes of “SN + gκ g + gκ CMB + BAO + CMB” does not. However, im-provements can still be achievable with a larger fractionof standard candles at high redshift, larger than that forthe LSST galaxies which we have assumed for the su-pernovae in our analysis. While this is hard to realizewith supernovae, it can be easier to achieve with othertypes of standard candles such as quasars [60]. We planto investigate more about this in a future paper.
ACKNOWLEDGEMENTS
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