Constraints on Primordial non-Gaussianity from Future HI Intensity Mapping Experiments
aa r X i v : . [ a s t r o - ph . C O ] O c t Constraints on primordial non-Gaussianity from future H I intensity mapping experiments Y I -C HAO L I ∗ and Y IN -Z HE M A † School of Chemistry and Physics, University of KwaZulu-Natal,Westville Campus, Private Bag X54001, Durban, 4000, South Africa andNAOC-UKZN Computational Astrophysics Centre (NUCAC),University of KwaZulu-Natal, Durban, 4000, South Africa
The primordial non-Gaussianity induces scale-dependent bias of the H I with respect to the underlyingdark matter, which exhibits features on the very large scales of the 21-cm power spectrum potentiallyobservable with H I intensity mapping observations. We forecast the prospective constraints on the fourfundamental shapes of primordial non-Gaussianity (local, equilateral, orthogonal, and enfolded), with thecurrent and future H I intensity mapping experiments, BINGO, FAST, and SKA-I. With the current con-figuration of the experiments and assumed one-year observation time, we find that the SKA-I will pro-vide tighter constraints on the local shape of primoridal non-Gaussianity than Planck . The results are ( σ f localNL , σ f equilNL , σ f orthNL , σ f enfoldNL ) SKA − I = (0 . , , , , ( σ f localNL , σ f equilNL , σ f orthNL , σ f enfoldNL ) BINGO =(17 , , , , ( σ f localNL , σ f equilNL , σ f orthNL , σ f enfoldNL ) FAST = (9 . , , , . If the lower frequency bandof FAST can be used, the constraint on local-type primordial non-Gaussianity will be σ f NL ∼ . which isbetter than Planck . In addition, if the observation time for FAST could be extended to two years, the constrainton the equilateral shape of primordial non-Gaussianity would be improved to σ f NL ∼ . Similarly, if the ob-servational time of SKA-I could be extended to two years, the constraint on local and orthogonal shapes couldbe improved to . and , respectively, achieving better constraints than Planck . I. INTRODUCTION
The statistical properties of the primordial fluctuation offerrich insights into the physics of inflation and the early Uni-verse [1]. One of the widely discussed questions is whetheror not the primordial fluctuations deviated from the Gaus-sian distribution. The simple single-field slow-roll inflation-ary model predicts primordial fluctuation with almost Gaus-sian distribution [2–4]. However, many alternative modelsof single-field slow-roll inflation can produce different typesof primordial non-Gaussianity [5–14] (PNG), which leavesdistinctive features in the statistical properties of the cosmicmicrowave background (CMB) and the large-scale structure(LSS) of the Universe.If the primordial fluctuation is Gaussian, the two-point cor-relation function (i.e. the power spectrum in Fourier space)can describe all of the statistical properties of the PNG. There-fore, the most straightforward way to measure the PNG isthrough the higher-order correlation of CMB or LSS. Currentmeasurements of the temperature and polarization of CMBfrom the
Planck satellite provide state-of-the-art constraintson local, equilateral and orthogonal types of PNG [15] as f localNL = 0 . ± . , f equilNL = − ± , and f orthoNL = − ± at confidence level (C.L.).Besides the constraints from CMB, there have been manyefforts to measure f NL through large-scale structure surveys.This is because the PNG induces a scale-dependent bias ofthe galaxy with respect to the underlying dark matter distribu-tion tracer [16–22]. Reference [19] used spectroscopic andphotometric luminous red galaxy samples and quasar samples ∗ Electronic address: [email protected] † Electronic address: [email protected] from the SDSS survey to obtain the limit for local-type PNGas − − < f localNL < +70(+96) at ( . ) C.L.,which was comparable to the measurements at the time fromthe Wilkinson Microwave Anisotropy Probe ( WMAP ) five-yearresults. Reference [23] used radio sources from the NRAOVLA Sky Survey (NVSS), the quasar and MegaZ-LRG (DR7)catalogues of the SDSS, and the final SDSS II Luminous RedGalaxy (LRG) photometric redshift survey and found f localNL =48 ± ( σ C.L.). Reference [24] found f localNL = 90 ± at σ C.L. by using photometric SDSS data, but due to unac-counted systematics this result may be better interpreted as f localNL < at C.L. Reference [25] used the SDSS-IIIBaryon Oscillation Spectroscopic Survey (BOSS) data to con-strain the f localNL and found − < f localNL < at σ C.L. Inaddition, Ref. [26] used the correlation of the residual peculiarvelocities on different directions to constrain PNG and found | f localNL | < . at C.L. These limits are currently consis-tent with but weaker than the measurements from the
Planck
CMB observation. In general, the scale-dependent bias signalcan be degenerated with the nonlinear bias between halo andunderlying dark matter, which is contributed from the nonlin-ear evolution of the matter fluctuations [27]. However, fore-casts indicate that the constraint errors could decrease or orders of the magnitude with the future LSS survey, especiallyfor the future radio survey. (see [28] and its references for re-view). Beside the constraint on the PNG amplitude, previousstudies also show that the scale-dependent bias introduced bythe PNG is sensitive mainly to the squeezed limit and, withthe future LSS surveys, it can be used to distinguish amongdifferent PNG shapes [29, 30].The scale-dependent bias not only affects the large-scalegalaxy bias, but also affects the H I distribution. A moreefficient method of the radio survey is to map out a largevolume of the Universe through the intensity mapping tech-nique, which measures the combined H I emission of the unre-solved galaxies. Therefore, in principle one can obtain a three-dimensional H I distribution that can provide more modes offluctuation than the CMB two-dimensional sphere. Therehave been several works to forecast the detectability of PNGthrough the H I intensity mapping technique [31–33], but thoseforecasts are exclusively only for the local and equilateral typeof PNG and limited experimental cases (SKA and Tianlai). Inthis work, we will calculate the scale-dependent bias of allfour typical types of PNG by using the halo model and cal-culate their imprints on the power spectrum of H I . Thenwe forecast the detectability of all three ongoing H I imag-ing surveys, i.e. BAO as Integrated Neutral Gas Observation(BINGO) [34], Five-Hundred-Metre Aperical Spherical Tele-scope (FAST) [35, 36] and Square Kilometre Array Phase-I(SKA-I) [37].This paper is organized as follows. In Sec. II, we summa-rize the primordial bispectrum and discuss different types ofPNG to be forecasted in this work. In Sec. III, we calculatethe scale-dependent bias of the LSS induced by the PNG, andthen the power spectrum of H I . In Sec. IV, we introduce theFisher matrix forecast method that used in our analysis. InSec.V, the detailed experiment parameters are discussed. InSec. VI, we present our results and some discussion. Conclu-sion will be in the last section.Besides the PNG parameters, we will adopt a spatially flatUniverse with cosmological parameters fixed as Planck Ω m = 0 . ; Ω Λ = 0 . ; σ =0 . ; and h = 0 . , where the Hubble constant is H =100 h km s − Mpc − . The amplitude and tilt of scalar powerspectrum are A s ( k ) = 2 . × − and n s = 0 . , wherepivot scale is k = 0 .
002 Mpc − . II. PRIMORDIAL BISPECTRUM
The inflationary models predict the primordial curvaturefluctuations with the deviation from Gaussian distribution [4,39–41]. The deviation is particularly described by writing thegauge-invariant Bardeen’s potential φ as the sum of a Gaus-sian random field and a quadratic correlation [40, 42], φ = φ G + f NL ( φ − h φ i ) , (1)in which f NL is a dimensionless, phenomenological parame-ter describing the magnitude of the PNG.To extract more information of the non-Gaussian primor-dial fluctuations, we need to go beyond the statistics of thepower spectrum. The lowest-order statistics sensitive to thePNG is the three-point function or bispectrum B φ ( k , k , k ) ,in which φ is the primordial Bardeen potential which is di-rectly related to the curvature perturbation [43]. The potentialof the primordial curvature perturbation is related to the New-tonian potential during the matter domination via the transferfunction T ( k ) which satisfies T ( k →
0) = 1 . By applyingthe Poisson equation, φ is related to the matter density field δ m ( k ) by δ m ( k ) = M ( k ) φ ( k ) , where M ( k ) = 23 k T ( k )Ω m H . (2) The configuration shape of B φ ( k , k , k ) is related to thephysical mechanisms during the inflation. In our analysis, weconsider four classes of bispectrum shape characterizing thelocal, equilateral, enfolded and orthogonal types of PNG. A. Local shape
The local-type PNG can be produced in different inflation-ary models, such as the multifield model [5, 44], curvatonmodel [6], inhomogeneous reheating [45] or new Ekpyroticmodels [46]. In these cases, f localNL can be substantially differ-ent from zero.The potential bispectrum of the local-type PNG has the sim-ple form, B φ ( k , k , k ) = 2 f localNL [ P φ ( k ) P φ ( k ) + (cyc . )] , (3)in which, P φ ( k ) = 2 π A s ( k )( k/k ) n s − is the power spec-trum of the Gaussian Bardeen potential. B. Equilateral shape
The equilateral-type of PNG can be produced in the infla-tionary models with higher-derivative interactions. Usuallythere are two dominant interaction terms of the inflation fieldgiving rise to the PNG peaking in the equilateral limit, whichcan be represented by a unique template with the equilateralshape.The primordial bispectrum of the equilateral type takes theform [8], B φ ( k , k , k ) = 6 f equilNL γ ( k , k , k ) × h − (cid:16) P φ ( k ) P φ ( k ) + (cyc . ) (cid:17) − (cid:16) P φ ( k ) P φ ( k ) P φ ( k ) (cid:17) / + (cid:16) P / φ ( k ) P / φ ( k ) P φ ( k ) + (cyc . ) (cid:17) i , (4)in which function γ ( k , k , k ) takes into account the runningof f equilNL and reads [47], γ ( k , k , k ) = (cid:20) k + k + k k CMB (cid:21) − κ , (5)where k CMB = 0 . h Mpc − , roughly corresponding to thelargest ℓ used to estimate the non-Gaussianity with WMAP data [48]. The free parameter κ is assumed to be constant.Following the discussion in the works of [47, 49], we usesmall negative κ = − . to enhance the non-Gaussianity onsmall scales. In the rest of this paper, the equilateral-type bis-pectrum always take the form of Eq. (4) with κ = − . . C. Orthogonal shape
The shapes of PNG caused by the two dominant terms ofhigher-derivative interactions, as we introduced above, areslightly different around flattened triangles k + k ≃ k .By taking an appropriate linear combination, the resulting or-thogonal shape of the PNG can minimize the similarities andmaximize the differences. The orthogonal shape is well ap-proximated by the following template [41, 50]: B φ ( k , k , k ) = 6 f orthNL h − (cid:16) P φ ( k ) P φ ( k ) + (cyc . ) (cid:17) − (cid:16) P φ ( k ) P φ ( k ) P φ ( k ) (cid:17) / + 3 (cid:16) P / φ ( k ) P / φ ( k ) P φ ( k ) + (cyc . ) (cid:17) i , (6) D. Enfolded shape
It is well studied that if the initial vacuum state for the in-flation deviates from the standard Bunch-Davies vacuum, theresulting bispectrum takes the enfolded shape [9–12], whichcan be approximated by B φ ( k , k , k ) = 6 f enfoldNL h (cid:16) P φ ( k ) P φ ( k ) + (cyc . ) (cid:17) + 3 (cid:16) P φ ( k ) P φ ( k ) P φ ( k ) (cid:17) / − (cid:16) P / φ ( k ) P / φ ( k ) P φ ( k ) + (cyc . ) (cid:17) i . (7)Note that as pointed out in Appendix C of [51], the squeezedlimit of this type of non-Gaussianity will result in a negligiblescale-dependent bias. Reference [51] suggested a new factor-izable template with correct squeezed limit. III. HI BIAS AND POWER SPECTRA OF 21-CM
The H I bias is the bias of H I distribution with respect to theunderlying dark matter distribution and the H I bias function, b H I , can be obtained by assuming a model for the amount ofH I mass in a dark matter halo of mass M , M H I ( M ) , and inte-grating over the halo mass function d n/ d M . Here we use theSheth-Tormen halo mass function [52] with mass range [ , ] M ⊙ b H I ( z ) = 1 ρ H I ( z ) Z M max M min d M d n d M ( M, z ) M H I ( M ) b ( M, z ) , (8)in which b ( M, z ) is the real-space halo bias and ρ H I ( z ) is, ρ H I ( z ) = Z M max M min d M d n d M ( M, z ) M H I ( M ) . (9) For the H I intensity mapping experiments, we follow the as-sumption discussed in [53] and consider a simple power lawmodel for the amount of H I mass, M H I ( M ) = AM α , α ≃ . , (10)which is a redshift independent function. The prefactor A willbe canceled with the normalization of ρ H I . A. The Lagrangian bias
FIG. 1: Three models of Lagrangian bias b L ( z ) , i.e., Matarrese andVerde [17], Mo and White [54], and Mo and White [55]. The Lagrangian bias describes the statistical bias of the halodistribution to the primordial dark matter fields. The PNG af-fects the initial conditions of the primordial density fields, so itis more convenient to study such effects in Lagrangian space.On the other hand, it is also necessary to study the statisticsof the evolved halo field at low redshifts in Eulerian space,which is conveniently related to the observation. The bias inLagrangian space, b L , relates to the Eulerian space bias, b E ,via b E = b L + 1 [54]. The extra unity factor of b E reflects themotions of primordial peaks at later times [26]. The uniformlydistributed halos in the initial epoch, which have b L = 0 , willlead to unbiased distribution to the dark matter field at a latertime. The b L for halos is defined as positive. But for otherdark matter tracers, it can be negative. The tracers anticor-related with the initial dark matter fields will lead to the lessclustered distribution than the dark matter field at later time.It the past years, people have been developing differentanalytical, semianalytical and parametric models of the biasfunction. Below, we list the three most typical and commonlyused ones.Based on the Press and Schechter (PS) halo mass function[56] and its extensions, Mo and White (1996) [54] give thebias factor for the halo of mass M , b L ( M, z ) = 1 δ c (cid:2) ν ( M, z ) − (cid:3) , (11)where ν ( M, z ) = δ c ( z ) /σ R . δ c ( z ) = δ c /D ( z ) , where D ( z ) is the linear growth function and we use Eq. (10) in [55] tocompute it. δ c ≃ . is the critical density contrast forspherical collapse. With the approximation of high-peak, theabove bias factor can be expressed as b L ( M, z ) = δ c ( z ) /σ R (Matarrese and Verde 2008 [17]). With the ellipsoidal col-lapse model [57], Mo and White (2002) [55] give another ex-pression, b L ( M, z ) = 1 δ c ( z ) h ν ′ + bν ′ − c ) − ν ′ c / √ aν ′ c + b (1 − c )(1 − c/ (cid:21) , (12)in which, ν ′ = √ aν and a = 0 . , b = 0 . , c = 0 . .Figure 1 shows the three models of Lagrangian bias we dis-cussed above. B. The scale-dependent bias
As we analyzed before, PNG affects the distribution of thepeaks at the initial stage of matter fluctuations; therefore, it iscorrelated with the Lagrangian bias. In the presence of PNG,the halo bias can be written as the combination of a usualscale-invariant bias, b ( M, z ) , and a scale-dependent modifi-cation, ∆ b ( M, z, k ) , b NG ( M, z, k ) = b ( M, z ) + ∆ b ( M, z, k ) . (13)By substituting Eq. (13) into Eq. (8), we can obtain the scale-dependent H I bias, which can be expressed as, b NG H I ( z, k ) = b H I ( z ) + ∆ b H I ( z, k ) , (14)in which b NG H I ( z, k ) is the total bias, b H I ( z ) is the scale-independent term, and ∆ b H I ( z, k ) is the scale-dependentterm, which is obtained by integrating ∆ b ( M, z, k ) over thehalo mass function and the H I mass model, ∆ b H I ( z, k ) = 1 ρ H I ( z ) Z M max M min d M × d n d M ( M, z ) M H I ( M )∆ b ( M, z, k ) , (15)where ρ H I ( z ) is calculated in Eq. (9).Dalal et al. [16] firstly derived the expression of scale-dependent correction to the bias of galaxies and halos forlocal-shape bispectrum, ∆ b D ( z, k ) = 2( b E − f NL δ c m a ( z ) g ( z ) r H k , (16)in which, δ c is the critical density, a ( z ) g ( z ) = D ( z ) is thelinear growth factor and r H = 1 /H . Equation (16) is derivedby only considering the high peaks of the density contrast,which means that the expression only works at the large scaleswith k → .More accurate analytical expressions for the scale-dependent bias have been studied [17–22]. A widely usedexpression is derived by Matarrese and Verde [17], ∆ b MV ( M, z, k ) = 2 f NL (cid:18) δ ( z ) σ R (cid:19) F ( k ) M R ( k ) , (17) in which, δ c ( z ) = δ c /D ( z ) and M R ( k ) is Eq. (2) smoothedwith window function W R ( k ) , M R ( k ) = 23 T ( k ) k H Ω m W R ( k ) , (18)where R denotes a smoothing radius which defines the halomass M by M = 3 H Ω m πG πR . (19)So ∆ b MV is also a function of halo mass, M . F ( k ) is relatedto the bispectrum of primordial potential field B φ ( k , k , k ) ,and the power spectrum P φ ( k ) , F ( k ) = 116 π σ R Z d k k M R ( k ) × Z − d µ M R ( k ) B φ ( k , k , k ) P φ ( k ) , (20)where k = k + k + 2 kk µ and σ R is the rms of the under-lying dark matter fluctuation fields smoothed on scale R givenin Eq. (19).If we substitute the local-shape bispectrum into Eq. (20),and take the limit of k → , then the dependence of ∆ b MV ( M, z, k ) on the halo mass automatically drops of, F ( k → → T ( k → → M R ( k → → (2 / k / ( H Ω m ) , and, ∆ b MV ( z, k → → b E − f NL δ c a ( z ) g ( z ) 32 H Ω m k = ∆ b D ( z, k ) ∼ k − , , (21)i.e. the general expression of scale-dependent bias in Eq. (17)recovers the bias proposed in Dalal et al. [16]. The advantageof using Eq. (17) is that it can be used to calculate any shapeof PNG, provided that the bispectrum B φ function is given.The scale-dependent bias for equilateral, orthogonal andenfolded shapes of PNG can be obtained by substitutingEqs. (4), (6) and (7) into Eq. (20). In Fig. 2, we show theabsolute value of the scale-dependent part of the bias, i.e.Eq. (15) for the four shapes of PNG at z = 0 (left panel)and z = 2 (right panel). One can see that the local shape hasthe most prominent feastures of scale-dependent bias at largescales, which can be constrained with 21-cm intensity map-ping observation on large angular scales. The orthogonal and This is consistent with Eq. 13 in [17]. The “ ∆ c ” defined in [17] is equal to δ c in this paper. In Ref. [17], b E − b L = δ c /σ R FIG. 2: The absolute value of scale-dependent bias | ∆ b ( z, k ) | [Eq. (15)] for different PNG shapes at z = 0 (left panel) and z = 2 (rightpanel) with assumed f NL = 1 . The four shapes of PNG are shown in different colors and dashed lines listed in the legend. The reason to plotthe absolute value is because the orthogonal shape of ∆ b is negative (see also Fig. 1 in [58]). The approximation of the local shape of PNGby Dalal et al. [16] [Eq. (16)] is shown in the brown dashed line, which is consistent and almost completely overlapped with the computationfrom the halo model [Eqs. (3) and (17)] shown with the red solid line. enfolded shapes have less prominent features but are possiblydetectable at small k . The scale-dependent bias induced byequilateral shape is too small on large scales so it will be hardto detect. The results shown in Fig. 2 are consistent with theanalysis in [22] and Fig. 1 in [58].We can see the asymptotic behavior of scale-dependent bias[Eq. (15)] on large scales by taking the limit of k → , then ∆ b → ( F / M R ) . Therefore, ∆ b (Local) ∼ k − ∆ b (Equilateral) ∼ const∆ b (Enfolded) ∼ k − ∆ b (Orthogonal) ∼ k − . (22)These asymptotic behaviors of ∆ b are consistent with thecomputation of halo models in Fig. 2. C. Power spectrum
We employ the H I tomographic angular power spectrum asthe observable in our analysis, The expression of the angularpower spectrum of the i th and the j th redshift bins is C ijℓ = 4 πT ij b Z d ln k W iℓ ( k ) W jℓ ( k )∆ ζ ( k ) , (23)in which, ∆ ζ ( k ) is the dimensionless power spectrum of pri-mordial curvature perturbation and T ij b = T b ( z i ) T b ( z j ) isthe multiplication of H I mean brightness temperature of the i th and j th redshift bins. We use the expression of T b ( z ) inChang et al.(2008) [59], T b ( z ) = 0 . (cid:18) Ω H I − (cid:19) (cid:18) z . (cid:19) . × (cid:18) Ω m + (1 + z ) − Ω Λ . (cid:19) − . mK , (24) -2 -1 ℓ ( ℓ + ) C i j ℓ / ( π ) [ µ K ] ℓ−1.0−0.50.00.51.0 C i j ℓ / C ii ℓ R e d s h i f t W i n d o w ( j ) Redshift Window(i) = 3.06
FIG. 3: Upper panel: Cross-correlated angular power spectrum be-tween redshift z i = 3 . and z j , which ranges from . to . shown with different colors. Lower panel: The radio of tomographicangular cross-power spectrum between z i and z j to the auto-powerspectrum of z i . where Ω H I is the fractional H I density assumed to be . × − [60]. The window function W ℓ ( k ) is, W ℓ ( k ) = Z d χ d N g ( χ )d χ j ℓ ( kχ ) b NG H I ( χ ( z ) , k ) T δ ( χ, k ) , (25)where j ℓ is a spherical Bessel function, d N g ( χ ) / d χ is the red-shift distribution of galaxy number, T δ ( χ, k ) is the transferfunction for the galaxy number over-density, and b NG H I is thetotal bias of H I (Eq. (14)). To calculate the angular powerspectrum, we use the C AMB S OURCES package [61].Figure 3 shows the tomographic angular power spectrum.The upper panel shows the cross-power spectrum betweenredshift z i = 3 . and z j , which ranges from . to . shown with different colors. The lower panel shows the ra-tio of the cross-power spectrum of different redshift bins tothe auto-power spectrum of the same redshift bin. We can seethat the cross-power spectrum decreases as the redshift devi-ates from z i = 3 . . This is what we expected, since thecross-correlated signal should drop if the frequency windowsmove away from each other. IV. FISHER MATRIX FORECAST
To forecast the potential for constraining f NL , we performthe Fisher matrix analysis. If we assume that the model like-lihood surface in parameter space can be well approximatedby a multivariant Gaussian, the Fisher matrix F is then agood approximation for the inverse of the parameter covari-ance. In the 21-cm tomography, each frequency band willprovide a map of 21-cm intensities, so we need to sum overthe Fisher matrix in both ℓ -space and frequency space. Since ν = 1420MHz / (1 + z ) , each frequency corresponds to aunique redshift slice. The Fisher matrix is F αβ = f sky ℓ max X ℓ min (cid:18) ℓ + 12 (cid:19) tr[ C ℓ,α Σ ℓ C ℓ,β Σ ℓ ] , (26)in which C ℓ is an n z × n z matrix, in which each element is theH I cross angular power spectrum between the two frequencybins. Σ ℓ = ( C ℓ + N ℓ ) − is the total noise inverse matrix, inwhich N ℓ is the n z × n z experimental noise power spectrum.Here we make a simple assumption that the noises in differentfrequency (redshift) bins are uncorrelated, therefore the N ℓ isa diagonal matrix. In reality, 21-cm intensity maps are highlycontaminated by the foreground, such as Galactic synchrotronemission, extragalactic point sources, and atmospheric sig-nal. One needs to apply foreground removal technique to re-duce the foreground contamination [62–64]. However, therealways be some level of residual Galactic foreground afterapplying such techniques to the maps. Therefore the cross-correlation of noises between different frequency bands maynot completely be zero.Under our simplified assumption, the element of N ℓ matrixis N ijℓ = δ ij N H I ℓ = δ ij T S survey / ( N ant N feed t TOT ∆ ν ) . (27) T sys = T rec + T sky is the system temperature, whichis contributed from the sky temperature, T sky = 60 × (300MHz /ν ) . , and receiver temperature T rec for each ex-periment. N ant and N feed are the number of antenna and thenumber of feed horn in each antenna respectively. The de-tailed experimental parameters for FAST, SKA-I and BINGOare listed in Table I. V. EXPERIMENT PARAMETERS
TABLE I: The experiment parameters for FAST, SKA-I and BINGO. D dish is the illuminated aperture.FAST SKA-I BINGO ν min [MHz] 1050 350 960 ν max [MHz] 1350 1050 1260∆ ν [MHz] 10 10 10 n ν ( n z ) 30 70 30 D dish [m] 300 15 25 N ant × N feed ×
19 190 × × t TOT [yr] 1 1 1 T rec [K] 25 28 50 S survey [deg ] < < a. BINGO The BINGO experiment is a single-dishH I intensity mapping experiment, which aims at map-ping the H I emission at frequencies between and [34, 65]. The telescope of the BINGO experi-ment has no moving parts and it conducts a drift-scan strat-egy. To achieve enough survey area, a wide instantaneousfield of view (FOV) with multiple feeds is required. A totalof 60 feeds laid out in a rectangle of × at the fo-cal plane. This will form a FOV of about ◦ (in Declinationdirection) × ◦ (in Right Ascension direction). With the ◦ wide strip centering at Declination of − ◦ , the total surveyarea is about . b. FAST FAST is the largest single-dish telescope,which also has the multibeam system of 19 feed-horns ar-ray [35, 36]. The multibeam system is proposed to work atfrequencies from . to . with system temperatureof . In our analysis, we only include the frequencies upto . . With the illuminated aperture, each of thefeed-horn has the beam size (Full Width at Half Maximum)of . ′ , and form a ′ FOV with beams. Due to the longslewing time, FAST can only work on drift-scan observationmode. Similar to the BINGO experiment, FAST scans a ′ wide strip along the Right Ascension direction for each side-real day. But the zenith angle of FAST can be adjusted fromDec: − ◦ ′ to Dec: ◦ ′ . Without over lapping betweenscanning strips, it takes about half year to cover all ◦ Decli-nation range . With one-year observation ( . × second),the maximum survey area is about . c. SKA-I The SKA Phase I (SKA-I) plans to construct movable dishes [33]. The maximum survey area isabout . A efficient survey area is need to be ex-plored to minimal the constraint errors. In our analysis, weonly consider the autocorrelation of each dishes, which meansthat the SKA-I works as single dishes. Without the in-terferometry, the SKA-I has very low resolution and is onlysensitive to the low- ℓ modes.Figure 4 shows the noise power spectra of different ex-periments in at redshift bin z = 0 . (left upper panel) and z = 3 . (right upper panel). The black solid line in theupper panel of each figure shows the standard angular power -2 -1 ℓ ( ℓ + ) C ii ℓ / ( π ) [ µ K ] f NL =0Noise Level SKAINoise Level FASTNoise Level BINGO ℓ10 -6 -5 -4 -3 -2 -1 ℓ ( ℓ + ) / ( π ) ∂ C ii ℓ / ∂ f N L [ µ K ] LocalEquilateralOrthogonalEnfolded
Redshift Window(i) = 0.37 -2 -1 ℓ ( ℓ + ) C ii ℓ / ( π ) [ µ K ] f NL =0Noise Level SKAINoise Level FASTNoise Level BINGO ℓ10 -6 -5 -4 -3 -2 -1 ℓ ( ℓ + ) / ( π ) ∂ C ii ℓ / ∂ f N L [ µ K ] LocalEquilateralOrthogonalEnfolded
Redshift Window(i) = 3.06
FIG. 4: Upper panels: Comparison between the noise power spectra of different experiments and the 21-cm power spectrum in standardmodel ( f NL = 0 ) for the two representative redshift bins (left and right panels). In both panels, one-year observation time (equivalent to . × sec) and survey area are assumed for all the experiments. Lower panels: The partial derivatives of C iiℓ with respect toparameter f NL for four shapes of PNG. spectrum of 21-cm ( f NL = 0 ); The black dash-dotted, dot-ted and dashed lines show the noise power spectra of SKA-I,FAST and BINGO experiments. One-year observation timeand survey area are assumed for all the experi-ments. The partial derivatives of C iiℓ with respect to parameter f NL are shown in the lower panel. The different colors corre-spond to different types of PNG.Comparing to the BINBO experiment, FAST and SKA-Ican have very large survey area. However, with the limitintegration time, the large survey area may not be able tobeat down the constraint error. We will discuss the detailsin Sec. VI. VI. RESULTS AND DISCUSSION
Figure 5 shows the σ f NL contours for local-shape PNG inthe plane of the survey area and total observation time. Theleft and middle panels of Fig. 5 show the contours for SKA-I and FAST experiments respectively. The color going fromred to blue means that the constraints become stronger. Dif-ferent black solid lines are the contours of the same errorof f localNL . Therefore, the error tends to become smaller if N ant × N feed × t TOT becomes bigger. Thus the most efficientway to reduce the constraint error is to increase the observa-tion time or the number of dishes(feeds). Assuming one-yearobservation time and the maximum dish(feeds) number forSKA-I and FAST experiments, the constraint errors of vari-ous PNG types as a function of survey area are shown in theright panel of Fig. 5. In order to have a clear view, the con-straint errors, σ f NL , are divided by the their minimal values. It is true that the optimal survey area may not be the maximalsurvey area. For example, in the case of equilateral shape,the optimization is about for the FAST experiment.For other shapes, the optimized survey areas are approachingthe maximum sky coverage of SKA-I or FAST. The large sur-vey area can help to beat the cosmic variance on large scales,but the integration time per pixel becomes smaller, leading tolarger pixel noise.One can see from the right panel of Fig. 5 that, generallyspeaking, the larger the survey area is, the smaller the errorof f NL , except for measuring equilateral shape of PNG us-ing the FAST survey. This is different from the situation ofusing 21-cm intensity mapping to measure the angular scaleof BAO acoustic oscillation, which have the optimal surveyarea around deg (For BINGO, see Fig. 7 in [34], andfor FAST, see Fig. 1 in [66]). The reason is because scale-dependent bias from PNG is always prominent on very largescales, so beating down cosmic variance is more importantthan lowering down the pixel noise. However, BAO scale issubhorizon for which there is always a trade-off between low-ering down pixel noise and beating down cosmic variance. Weuse different optimized survey areas for different cases in thelater analysis.Figure 6 shows the σ f NL as a function of ℓ min if we fix ℓ max = 600 . Different PNG shapes are shown in differentpanels. In each panel, different colors indicate different exper-iments as shown in the legends. The optimized survey areasare applied to the analysis. The constraint errors of differentPNG shapes from Planck satellite are shown with the blackdashed lines [15]. The σ f NL of different PNG shapes fore-casted with different experiments are listed in Table II. S survey [deg ]10 σ f N L / ( σ f N L ) m i n SKA-I LocalSKA-I EquilateralSKA-I OrthogonalSKA-I Enfolded FAST LocalFAST EquilateralFAST OrthogonalFAST Enfolded
FIG. 5: The left (for SKA-I) and middle (for FAST) panels show the σ f NL contours for local-shape PNG in the parameter space of the surveyarea and total observation time. The dashed contour is the error of constraint with Planck temperature and polarization data [15]. The rightpanel shows the σ f NL / ( σ f NL ) min as a function of survey area for various PNG types. The solid lines show the results for SKA-I with one-yearobservational time and dishes; the dashed lines show the results for FAST with one-year observational time and beams. ℓ10 σ f N L Local ShapeSum over ℓ to ℓ max = 600
SKA-I 25000deg FAST 24000deg BINGO 2500deg Planck ℓ10 σ f N L Equilateral ShapeSum over ℓ to ℓ max = 600
SKA-I 25000deg FAST 6000deg BINGO 2500deg Planck ℓ10 σ f N L Enfolded ShapeSum over ℓ to ℓ max = 600
SKA-I 25000deg FAST 24000deg BINGO 2500deg ℓ10 σ f N L Orthogonal ShapeSum over ℓ to ℓ max = 600
SKA-I 25000deg FAST 24000deg BINGO 2500deg Planck
FIG. 6: The σ f NL as a function of ℓ min for various experiments and PNG shapes. The black dashed line is the current constraint with Planck temperature and polarization data [15].
We can see that, for the local shape PNG, the SKA-I exper-iment is potentially able to constrain f NL better than Planck experiment. But we should realize that it is only the mostideal case. It is well known that, one of the big challenges forobservations of H I intensity mapping is the foreground sub- traction, and the low- ℓ modes may not be detectable due tothe foreground contamination. Our results show that, to ob-tain a remarkable constraint on f NL with the SKA-I intensitymapping in the future, we need to recover the angular powerspectrum of H I with the minimal ℓ max ≃ . This is the aim TABLE II: σ f NL of different PNG shapes forecasted with different experiments. The optimized survey areas are applied in the analysis. The“ Planck
Planck . Current Configuration Extentions
Planck † FAST 2yr †† FAST low ‡ Local 5 9.5 Equilateral 43 44 86 100 66
59 39Enfolded – 94 43 164 36 70 64 † SKA-I with two-year observation; †† FAST with two-year observation; ‡ FAST with low frequencies rangefrom to of several recent efforts of restoring large angular power withcross-correlation with weak gravitational lensing [67, 68]. Wealso find that the constraint error for orthogonal shape PNGwith SKA-I is ∼ , which is at the same level of current Planck limit. If the observation can be extended to years,the error will be reduced to ∼ .The constraint error for equilateral shape PNG with FASTis ∼ , which is better than the results of SKA-I and BINGOexperiments. The FAST error on f equilNL is close to the cur-rent limit of Planck experiment. This is because the scale-dependent bias induced by the equilateral shape PNG hashigher signal-to-noise ratio at small scales and the FASTexperiment is more sensitive to the small-scale modes thanSKA-I single dish mode and BINGO. So far, in our analy-sis, we assume perfect knowledge of the power spectrum anddo not include the theoretical error. However, it has beenshown that the higher derivative terms contribute to the scale-dependent bias on small scales [69, 70]. Such contributionsinduce extra uncertainties to the scale-dependent bias mea-surements and reduce the detectability of equilateral PNG.We also test the possible extensions of the current con-figuration by adding more integration time. If the observa-tion time for SKA-I and FAST could be extended to years,the constraints on f NL can be improved quantitatively. Theforecasted constraint on different shapes of PNG are listedin Table II. It is worth noticing that the constraint error onthe orthogonal-shaped PNG with SKA-I and the equilateral-shaped PNG with FAST becomes smaller than the limits of Planck with extended observational time.A good extension for FAST experiment is to extend itsbandwidth to the lower frequencies, which are correspondingto the higher redshifts. So far the FAST telescope has one ul-trawide band receiver working on ∼ . . Un-fortunately, the ultrawide band receiver has only one beam.It will take quite a long time to achieve the same observa-tion time as the multibeam receiver. Now the multibeam sys-tem of the FAST telescope is designed to work on frequen-cies between and . Assuming that theFAST multibeam system works on the frequencies between and , which is the same as the frequencyrange of SKA-I experiment, the constraint for local shapePNG will be σ f localNL ∼ . with the optimized survey areaof . The constraint errors ( σ f NL ) for orthogonal and enfolded shapes become and respectively, which are allhighly reduced. VII. CONCLUSION
In this work, we explored the constraining power on theprimordial non-Gaussianity (PNG), with the future single-dish H I intensity mapping observations with BINGO, FASTand SKA-I. Four fundamental shapes of PNG are studied inour analysis, including local, equilateral, orthogonal and en-folded. We focus on the effect of scale-dependent bias to theunderlying dark matter tracer, induced by the primordial non-Gaussinaity. The properties of such scale-dependent bias atlarge-scale limit are discussed in our analysis. The forecastresults are listed in Table II.Our forecasts show that with the current configuration ofthe experiments one-year observation time, the constraint onlocal shape of PNG from SKA-I intensity mapping experi-ment can be better than the current Planck experiment. Theoptimized survey area of is applied in the anal-ysis of SKA-I, but the results are more sensitive to the totalobservation time than the survey area. However, the H I inten-sity mapping experiments may be contaminated by the fore-ground and the low- ℓ modes may be be detectable. Our analy-sis shows that the SKA-I experiment can still have the remark-able constraint without the modes of ℓ < ∼ With two-yearsobservation, the constraint on orthogonal shape PNG is ∼ ,which is also better than the constraint from Planck measure-ment.The FAST experiment has the advantage of higher angu-lar resolution and is more sensitive to the small-scale modes,which is good for constraining the equilateral shape of PNG.With the current configuration and two years observation,the constraint error for the equilateral shape of PNG will be σ f NL = 32 , which is better than the current limit of the Planck observation. However, such a limit is achieved by ignoringthe extra uncertainties caused by the higher derivative terms.Previous studies show that such extra uncertainties may notbe negligible. The detailed limit for the equilateral-type PNGneeds to be investigated in the future analysis.Similar constraint on the local shape of PNG can beachieved by the FAST H I intensity mapping, if its frequencybandwidth can be extended to the lower frequencies (ultraw-0ide band). Assuming the same working frequency range, thebest constraint from FAST on the local shape of primordialnon-Gaussianiy is σ f NL ∼ . .The studies we conduct here are the standard power spec-tra analysis of 21 cm. There have been efforts on using themultitracer technique to beat the cosmic variance and obtaintighter constraints on f NL [71–73]. In addition, using three-point correlation function is another way to measure PNG.These methods will be explored to measure all shapes of f NL in the future work. Acknowledgments
We thank Neal Dalal, Di Li, Roy Maartens, JeromeGleyzes, Yi Wang and Xiao-Dong Xu for helpful discussionsand Stefano Camera for his help on C
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