Constraints on Primordial non-Gaussianity from Future HI Intensity Mapping Experiments

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by  Yi-Chao Li

Constraints onPrimordialnon-Gaussianity from Future HI Intensity Mapping Experiments
YI-CHAO LI∗ and YIN-ZHE MA†
School of Chemistry and Physics, University of KwaZulu-Natal,
Westville Campus, Private Bag X54001, Durban, 4000, South Africa and
NAOC-UKZN Computational Astrophysics Centre (NUCAC),
University of KwaZulu-Natal, Durban, 4000, South Africa
The primordial non-Gaussianity induces scale-dependent bias of the HI with respect to the underlying
dark matter, which exhibits features on the very large scale of 21-cm power spectrum potentially observ-
able with HI intensity mapping observations. We forecast the prospective constraints on the four funda-
mental shapes of primordial non-Gaussianity (local, equilateral, orthogonal and enfolded), with the cur-
7
rent and future HI intensity mapping experiments, BINGO, FAST and SKA-I. With the current config-
1
uration of the experiments and assumed one-year observation time, we find that the SKA-I will pro-
0
vide tighter constraints on local shape of primoridal non-Gaussianity than Planck. The results are
2
n (σfNloLcal,σfNeqLuil,σfNorLth,σfNenLfold)SKA−I = (0.54,86,25,43), (σfNloLcal,σfNeqLuil,σfNorLth,σfNenLfold)BINGO =
a (17,100,128,164), (σfNloLcal,σfNeqLuil,σfNorLth,σfNenLfold)FAST = (9.5,44,75,94). If thelower frequency band
J ofFASTcanbeused, theconstraintonlocal-typeprimordialnon-GaussianitywillbeσfNL ∼ 1.62whichis
betterthanPlanck. Inaddition,iftheobservationtimeforFASTcouldbeextendedtotwoyears,theconstraint
1
on equilateral shape of primordial non-Gaussianity will be improved to be σ ∼ 32. Similarly, if obser-
fNL
vationaltimeofSKA-Icouldbeextendedtotwoyears, theconstraintonlocalandorthogonal shapescanbe
]
O improvedto0.43and20respectively,achievingbetterconstraintsthanPlanck.
C
.
h I. INTRODUCTION from SDSS survey and obtain the limit for local type PNG
p as 31( 96) < flocal < +70(+96) at 95% (99.7%) CL,
- wh−ichwa−scomparaNblLetothethenmeasurementsfromWilkin-
o Thestatisticalpropertiesoftheprimordialfluctuationoffer
son Microwave Anisotropy Probe (WMAP) five-year results.
r richinsightsintothephysicsofinflationandtheearlyuniverse
t Ref. [19] used radio sourcesfromthe NRAO VLA Sky Sur-
s [1]. One of the wildly discussed questionsis whetheror not
a vey(NVSS),thequasarandMegaZ-LRG(DR7)cataloguesof
theprimordialfluctuationsaredeviatedfromGaussiandistri-
[ theSDSS,andthefinalSDSSIILuminousRedGalaxy(LRG)
bution. The simple single-field slow-roll inflationary model
photometricredshiftsurveyandfoundflocal = 48 20(1σ
v1 tpiroendic[t2s]p.riHmoowrdeivaelrfl,umcatunaytiaolntewrnitahtivaelmmosotdGelasuossfiasinndgilset-rfiibeuld- CL). Ref. [20] found fNloLcal = 90 ± 30NaLt 1σ CL±by using
1 photometricSDSS data, but due to unaccountedsystematics
slow-roll inflation can produce different types of primordial
2 thisresultmaybebetterinterpretedasflocal < 120at84per
2 non-Gaussianity [3–10] (hereafter PNG), which leaves dis- centCL.Ref.[21]usedSDSS-IIIBaryonNOLscillationSpectro-
0 tinctivefeaturesinstatisticalpropertiesofcosmicmicrowave scopicSurvey(BOSS)datatoconstraintheflocal,andfound
0 background(CMB)andthelarge-scalestructure(LSS)ofthe NL
45 < flocal < 195 at 2σ CL. In addition, Ref. [22] used
1. Universe. −thecorrelaNtLionoftheresidualpeculiarvelocitiesondifferent
0 IftheprimordialfluctuationisGaussian,thetwo-pointcor- directionstoconstrainPNGandfound flocal <25.7at68%
7 relation function (i.e. power spectrum in Fourier space) can CL.Theselimitsarecurrentlyconsisten|twNLith|butweakerthan
1
describe all of the statistical properties of PNG. Therefore, the measurementsfromthe PlanckCMB observation. How-
:
v themoststraightforwardwaytomeasurethePNGisthrough ever, forecasts indicate that the constraint errors could de-
Xi the higher-order correlation of CMB or LSS. Current mea- creaseoneortwoordersofthemagnitudewiththefutureLSS
surementsof the temperatureand polarizationof CMB from survey,especiallyforthefutureradiosurvey.(see[23]andits
r
a Plancksatelliteprovidesthestate-of-the-artconstraintsonlo- referencesforreview).
cal,equilateralandorthogonaltypesofPNG[11]asfNloLcal = The scale-dependent bias not only affects the large-scale
0.8 5.0,fequil = 4 43andfortho = 26 21at68% galaxy bias, but also affects the HI distribution. A more
± NL − ± NL − ±
confidencelevel(CL). efficient method of the radio survey is to map out a large
volume of the Universe through the intensity mapping tech-
Besides the constraints from CMB, there have been a lot
ofeffortsofmeasuringf throughlarge-scalestructuresur- nique,whichmeasuresthecombinedHIemissionoftheunre-
NL
solvedgalaxies.Therefore,inprincipleonecanobtainathree-
veys.ThisisbecausethePNGinducesascale-dependentbias
ofthegalaxywithrespecttotheunderlyingdarkmatterdistri- dimensional HI distribution that can provide more modes of
fluctuations than the CMB two-dimensional sphere. There
butiontracer [12–18]. Ref.[15]usedspectroscopicandpho-
havebeenseveralworkstoforecastthedetectabilityofPNG
tometric luminous red galaxy samples, and quasars samples
through HI intensity mapping technique [24–26], but those
forecastsareexclusivelyonlyforlocalandequilateraltypeof
PNG and limited experimental cases (SKA and Tianlai). In
∗Electronicaddress:[email protected] this work, we will calculate the scale-dependent bias of all
†Electronicaddress:[email protected] fourtypicaltypesof PNG by using the halo model, and cal-
2
culatetheirimprintsonthepowerspectrumof HI . Thenwe A. Localshape
forecastthedetectabilityofallofthethreeon-goingHIimag-
ing surveys, i.e. BAO as Integratd Neutral Gas Observation The standard single-field inflation predicts the local-type
(BINGO)[27],Five-Hundred-MetreApericalSphericalTele- PNG [37]. But the parameter flocal is expected to be very
NL
scope (FAST) [28, 29] and Square Kilometre Array Phase-I closetozero[2]. However,largeamountofnon-Gaussianity
(SKA-I)[30]. can be produced with different inflationary model, such as
Thispaperisorganizedasfollows. InSect.IIwesumma- multifieldsmodel[3]orcurvatonmodels[4]. Inthesecases,
rize the primordialbispectrumand discuss differenttypesof flocalcanbesubstantiallydifferentfromzero.
NL
PNG to be forecasted in this work. In Sect. III we calculate Thepotentialbispectrumofthelocal-typePNGhasthesim-
thescale-dependentbiasoftheLSSinducedbythePNG,and pleform,
thenthepowerspectrumofHI. InSect.IVweintroducethe
Fisher matrix forecast method that used in our analysis. In B (k ,k ,k )=2flocal[P (k )P (k )+(cyc.)], (3)
φ 1 2 3 NL φ 1 φ 2
Sect.V,the detailedexperimentparametersarediscussed. In
Sect.VIwepresentourresultsandsomediscussion. Conclu- inwhich,Pφ(k)=2π2As(k0)(k/k0)ns−4isthepowerspec-
sionwillbeinthelastsection. trumoftheGaussianBardeenpotential.
BesidesthePNGparameters,wewilladoptaspatiallyflat
UniversewithcosmologicalparametersfixedasPlanck2015
B. Equilateralshape
best-fittingvalues[31],i.e. Ω = 0.309;Ω = 0.691;σ =
m Λ 8
0.809; and h = 0.68, where the Hubble constant is H =
0
100hkms−1Mpc−1. Theamplitudeandtiltofscalarpower Theequilateral-typeofPNGcanbeproducedbytheinfla-
spectrumareA (k )=2.141 10−9andn =0.961,where tionary models with higher-derivative interactions. Usually
s 0 s
pivotscaleisk =0.002Mpc×−1. therearetwodominantinteractiontermsoftheinflationfield
0
givingrisetothePNG.Oneofthedominanttermsleadstothe
equilateral-typeofPNG,andanotherleadstotheorthogonal-
type.
II. PRIMORDIALBISPECTRUM Theprimordialbispectrumoftheequilateraltypetakesthe
form[6],
Mostofthesingle-fieldinflationarymodelspredictthepri- B (k ,k ,k )=6fequilγ(k ,k ,k )
mordialcurvaturefluctuationswiththedeviationfromGaus- φ 1 2 3 NL 1 2 3
sian distribution [32–34]. The deviation is particularly de- P (k )P (k )+(cyc.)
φ 1 φ 2
scribed by writing the gauge-invariantBardeen’s potential φ ×h−(cid:16) (cid:17)
2/3
asthesumofaGaussianrandomfieldandaquadraticcorre- 2 P (k )P (k )P (k )
φ 1 φ 2 φ 3
lation[33,35], − (cid:16) (cid:17)
+ P1/3(k )P2/3(k )P (k )+(cyc.) ,
φ 1 φ 2 φ 3
φ=φG+fNL(φ2G−hφ2Gi), (1) (cid:16) (cid:17)i(4)
inwhich,fNL isadimensionless,phenomenologicalparame- inwhich,functionγ(k1,k2,k3)takesintoaccountoftherun-
terdescribingthemagnitudeofthePNG. ningoffequilandreads [38],
NL
To extract more information of the non-Gaussian primor-
dialfluctuations,weneedtogobeyondthestatisticsofpower γ(k ,k ,k )= k1+k2+k3 −2κ, (5)
1 2 3
spectrum. The lowest-order statistics sensitive to the PNG (cid:20) kCMB (cid:21)
is the three-point function or bispectrum B (k ,k ,k ), in
which,φistheprimordialBardeenpotentialwφhic1hi2sdi3rectly wherekCMB =0.086hMpc−1andκ= 0.2.
−
relatedtothecurvatureperturbation[36].Thepotentialofpri-
mordialcurvatureperturbationisrelatedtotheNewtonianpo-
C. Orthogonalshape
tentialduringthematterdominationviathetransferfunction
T(k)which satisfies T(k 0) = 1. By applyingthe Pois-
sonequation,φisrelatedto→thematterdensityfieldδ (k)by TheorthogonalshapeofPNGbispectrumisnearlyorthog-
m
δ (k)= (k)φ(k),where, onaltoboththelocalandequailateralformsofPNG,whichis
m
M wellapproximatedbythefollowingtemplate[39]
2k2T(k)
(k)= . (2) B (k ,k ,k )=6forth 3 P (k )P (k )+(cyc.)
M 3 ΩmH02 φ 1 2 3 NL h− (cid:16) φ 1 φ 2 (cid:17)
2/3
8 P (k )P (k )P (k )
The configuration shape of Bφ(k1,k2,k3) is related to the − (cid:16) φ 1 φ 2 φ 3 (cid:17)
physicalmechanismsduringtheinflation. Inouranalysis,we +3 P1/3(k )P2/3(k )P (k )+(cyc.) ,
consider four classes of bispectrum shape characterizing the (cid:16) φ 1 φ 2 φ 3 (cid:17)i
(6)
local,equilateral,enfoldedandorthogonaltypesofPNG.
3
D. Enfoldedshape
It is well studied that if the inital vacuum state for the in-
flationdeviatesfromthestandardBunch-Daviesvacuum,the
resultingbispectrumtakestheenfolded-shape [7–10],which
canbeapproximateby
B (k ,k ,k )=6fenfold P (k )P (k )+(cyc.)
φ 1 2 3 NL φ 1 φ 2
h(cid:16) (cid:17)
2/3
+3 P (k )P (k )P (k )
φ 1 φ 2 φ 3
(cid:16) (cid:17)
P1/3(k )P2/3(k )P (k )+(cyc.) .
−(cid:16) φ 1 φ 2 φ 3 (cid:17)i
(7)
FIG.1: ThreemodelsofLagrangianbiasbL(z),i.e.Matarreseand
Verde[13],MoandWhite[42],andMoandWhite[43].
III. HIBIASANDPOWERSPECTRAOF21-CM
TheHIbiasisthebiasofHIdistributionwithrespecttothe otherdarkmattertracers,itcanbenegative. Thetracersanti-
underlyingdarkmatterdistributionandthe HI biasfunction, correlated with the initial dark matter fields will lead to the
bHI,canbeobtainedbyassumingamodelfortheamountof less clustered distribution than the dark matter field at later
HImassinadarkmatterhaloofmassM,MHI(M),andinte- time.
gratingoverthehalomassfunctiondn/dM.Hereweusethe Itthepastthirtyyears,peoplehavebeendevelopingdiffer-
Sheth-Tormenhalomassfunction[40]withmassrange[108, ent analytical, semi-analytical and parametric models of the
1016]M⊙ bias function. In below, we list the three most typical and
commonly-usedones.
1 Mmax dn
Basingon thePress& Schechter(hereafterPS) halomass
b (z)= dM (M,z)M (M)b(M,z),
HI ρHI(z)ZMmin dM HI function [44] and its extensions, MoandWhite (1996) [42]
(8) givesthebiasfactorforhalosofmassM,
inwhich,b(M,z)isthereal-spacehalobiasandρ (z)is,
HI
1
Mmax dn bL(M,z)= δ ν2(M,z)−1 , (11)
ρHI(z)=Z dMdM(M,z)MHI(M). (9) c (cid:2) (cid:3)
Mmin whereν(M,z) = δ (z)/σ . δ (z) = δ /D(z),whereD(z)
c R c c
Forthe HI intensitymappingexperiments,we followthe as- is the linear growth function which we use eq. (10) in [43]
sumptiondiscussed in [41] andconsidera simple powerlaw to computeit. δ 1.686is the critical density contrastfor
c
≃
modelfortheamountofHImass, sphericalcollapse. Withtheapproximationofhigh-peak,the
abovebiasfactorcanbeexpressedas b (M,z) = δ (z)/σ2
MHI(M)=AMα, α≃0.6, (10) (MatarreseandVerde 2008 [13]). WithL the ellipsoicdal coRl-
lapsemodel[45],MoandWhite(2002)[43]givesanotherex-
which is a redshift independent function. The pre-factor A
pression,
willbecanceledwiththenormalizationofρ .
HI
1
b (M,z) = ν′2+bν′2(1−c)
L
δ (z)
A. TheLagrangianbias c h
ν′2c/√a
, (12)
TheLagrangianbiasdescribesthestatisticalbiasofthehalo − ν′2c+b(1 c)(1 c/2)(cid:21)
− −
distributiontotheprimordialdarkmatterfields. ThePNGaf-
inwhich,ν′ =√aν anda=0.707, b=0.5, c=0.6.
fectstheinitialconditionsoftheprimordialdensityfields,soit
Figure1showsthethreemodelsofLagrangianbiaswedis-
ismoreconvenienttostudysucheffectsinLagrangianspace.
cussedabove.
Ontheotherhand,itisalsonecessarilytostudythestatistics
of the evolved halo field at low redshifts in Eulerian space,
whichisconvenientlyrelatedtotheobservation. Thebiasin
Lagrangianspace, b , relates to the Eulerian space bias, b , B. Thescale-dependentbias
L E
via b = b +1 [42]. The extra unity factor of b reflects
E L E
themotionsofprimordialpeaksatlatertimes[22]. Theuni- Asweanalyzedbefore,PNGaffectsthedistributionofthe
formly distributed halos in the initial epoch, which have the peaksattheinitialstageofmatterfluctuations,thereforeitis
b = 0, will lead to unbiased distributionto the darkmatter correlatedwiththeLagrangianbias. InthepresenceofPNG,
L
fieldatlatertime.Theb forhalosispositivedefined.Butfor the halo bias can be written as the combination of a usual
L
4
FIG.2: Theabsolutevalue of scale-dependent bias|∆b(z,k)|(Eq.(15))for differentPNGshapes atz = 0(leftpanel) andz = 2(right
panel)withassumedfNL =1.ThefourshapesofPNGareshownindifferentcolouranddashedlineslistedinthelegend. Thereasontoplot
absolutevalueisbecauseorthogonalshapeof∆bisnegative(seealsofig.1in[46]).TheapproximationoflocalshapeofPNGbyDalaletal.
[12](Eq.(16))isshowninthebrowndashedline,whichisconsistentandalmostcompletelyoverlappedwiththecomputationfromthehalo
model(Eqs.(3)and(17))inredsolidline.
scale-invariant bias, b(M,z), and a scale-dependent modifi- in which, δ (z) = δ /D(z) 1 and (k) is the Eq. (2)
c c R
M
cation,∆b(M,z,k), smoothedwithwindowfunctionW (k),
R
bNG(M,z,k)=b(M,z)+∆b(M,z,k). (13) 2T(k)k2
(k)= W (k), (18)
MR 3 H2Ω R
BysubstitutingEq.(13)intoEq.(8),wecanobtainthescale- 0 m
dependentHIbias,whichcanbeexpressedas, where R denotes a smoothing radius which defines the halo
bNG(z,k)=b (z)+∆b (z,k), (14) massM by,
HI HI HI
in which bNHIG(z,k) is the total bias, bHI(z) is the scale- M = 3H02Ωm4πR3. (19)
independent term, and ∆b (z,k) is the scale-dependent 8πG 3
HI
term, which is obtained by integrating ∆b(M,z,k) over the
So∆bMV isalsoafunctionofhalomass,M. (k)isrelated
halomassfunctionandHImassmodel, F
to thebispectrumof primordialpotentialfield B (k ,k ,k),
φ 1 2
1 Mmax andthepowerspectrumPφ(k),
∆b (z,k)= dM
HI ρHI(z)ZMmin (15) 1
dn (M,z)M (M)∆b(M,z,k), F(k)= 16π2σ2 Z dk1k12MR(k1)
× dM HI R (20)
1 B (k ,k ,k)
φ 1 2
whereρHI(z)iscalculatedinEq.(9). ×Z dµMR(k2) P (k) ,
Dalal et al. [12] firstly derived the expression of scale- −1 φ
dependent correction to the bias of galaxies and halos for wherek2 =k2+k2+2kk µandσ isthermsoftheunder-
2 1 1 R
local-shapebispectrum,
lyingdarkmatterfluctuationfieldssmoothedonscaleRgiven
inEq.(19).
3Ω
∆bD(z,k)=2(b 1)f δ m , (16) If we substitute the local-shape bispectrum into Eq. (20),
E− NL c2a(z)g(z)r2 k2
H and take the limit of k 0, then2 the dependence of
in which, δ is the critical density, a(z)g(z) = D(z) is the ∆bMV(M,z,k)onhaloma→ssautomaticallydropsoff
c
lineargrowthfactorandr =1/H . Equation(16)isderived
H 0
(k 0) 1
by only considering the high peaks of the density contrast,
F → →
whichmeansthattheexpressiononlyworksatthelargescales T(k 0) 1
→ →
withk 0. (k 0) (2/3)k2/(H2Ω ),
More→accurate analytical expressions for the scale- MR → → 0 m
dependentbiashavebeenstudied[13–18]. Awildlyusedex-
pressionisderivedbyMatarreseandVerde[13],
δ2(z) (k) 1Thisisconsistentwitheq.13in[13].The“∆c”definedin[13]isequalto
∆bMV(M,z,k)=2f c F , (17) δcinthispaper.
NL(cid:18) σR2 (cid:19)MR(k) 2InRef.[13]bE−1=bL=δc/σR2
5
and,
δ 3H2Ω Redshift Window(i) = 3.06
∆bMV(z,k→0)→2(bE −1)fNLa(z)gc(z)2 0k2m 2K] 102 3.0
, µ
=∆bD(z,k) )[ 101
π 2.7
2
k−2, /(
∼ ijCℓ 100 2.4
(21) )
i.e. thegeneralexpressionofscale-dependentbiasinEq.(17) ℓ(ℓ+1 10-1 2.1ow(j)
roefcuosvienrgsEthqe.b(1ia7s)pisrotphoasteidtcinanDbaelaulseetdatlo.[c1a2l]c.uTlahteeaadnvyasnhtaapgee 10-2 1.8Wind
ofPNG,providedthatthebispectrumBφ functionisgiven. 1.0 1.5hift
The scale-dependent bias for equilateral, orthogonal and s
d
enfolded shapes of PNG can be obtained by substituting 0.5 1.2Re
Eqs.(4),(6)and(7)intoEq.(20).InFig.2,weshowtheabso- iiCℓ
lutevalueofthescale-dependentpartofthebias,i.e. Eq.(15) ij/Cℓ 0.0 0.9
for the four shapes of PNG at z = 0 (left panel) and z = 2
−0.5 0.6
(rightpanel).Onecanseethatthelocalshapehasmostpromi-
nent feasture of scale-dependent bias at large scales, which
−1.0
canbeconstrainedwith21-cmintensitymappingobservation
101 102
onlargeangularscales. Theorthogonalandenfoldedshapes
ℓ
haveless prominentfeaturesbutpossiblydetectableatsmall
k. The scale-dependentbias induced by equilateral shape is
FIG.3: Upperpanel: Cross-correlatedangularpowerspectrumbe-
toosmallonlargescalessoitwillbehardtobedetected.The
tween redshift zi = 3.06 and zj, which ranges from0.37 to 3.06
resultsshowninFig.2areconsistentwiththeanalysisin[18] shownwithdifferentcolors.Lowerpanel:Theradiooftomographic
andfig.1in[46]. angularcross-powerspectrumbetweenzi andzj totheauto-power
Wecanseetheasymptoticbehaviorofscale-dependentbias spectrumofzi.
(Eq.(15)) onlargescalesbytakingthelimitofk 0, then
→
∆b ( / ). Therefore,
R
→ F M
∆b(Local)∼k−2 1w0h−e3re[4Ω8H].I TishtehweifnradcotwionfuanlcHtiIodnensℓi(tyk)asiss,umedtobe0.62×
W
∆b(Equilateral) const
∼ (22)
∆b(Enfolded) k−1 (k)= dχdNg(χ)j (kχ)bNG(χ(z),k)T (χ,k), (25)
∼ Wℓ Z dχ ℓ HI δ
∆b(Orthogonal) k−1.
∼
wherej isasphericalBesselfunction,dN (χ)/dχisthered-
These asymptotic behaviors of ∆b is consistent with the ℓ g
shift distribution of galaxy number, T (χ,k) is the transfer
computationofhalomodelsinFig.2. δ
function for the galaxy number over-density, and bNG is the
HI
total bias of HI (Eq. (14)). To calculate the angular power
C. Powerspectrum spectrum,weusetheCAMB SOURCESpackage[49].
Figure 3 shows the tomographic angular power spectrum.
The Upper panel shows the cross-power spectrum between
WeemploytheHItomographicangularpowerspectrumas
redshift z = 3.06 and z , which ranges from 0.37 to 3.06
theobservableinouranalysis, Theexpressionoftheangular i j
shownwithdifferentcolors. TheLowerpanelshowstheratio
powerspectrumofthei-thandthej-thredshiftbinsis,
of the cross-power spectrum of different redshift bins to the
auto-powerspectrumofthesameredshiftbin.Wecanseethat
Cij =4πTij d lnk i(k) j(k)∆2(k), (23)
ℓ b Z Wℓ Wℓ ζ the cross-power spectrum decreases as the redshift deviates
from z = 3.06. This is what we expected, since the cross-
inwhich,∆2(k)isthedimensionlesspowerspectrumofpri- i
ζ correlatedsignalshoulddropifthefrequencywindowsmove
mordial curvature perturbation and Tij = T (z )T (z ) is awayfromeachother.
b b i b j
the multiplication of HI mean brightness temperature of the
i-thandj-thredshiftbins. WeusetheexpressionofT (z)in
b
Changetal.(2008)[47], IV. FISHERMATRIXFORECAST
Ω 1+z 0.5
Tb(z) = 0.39(cid:18)10H−I3(cid:19)(cid:18) 2.5 (cid:19) ToforecastthepotentialforconstrainingfNL,weperform
theFishermatrixanalysis. Ifweassumethatthemodellike-
Ω +(1+z)−3Ω −0.5 lihood surface in parameter space can be well approximated
m Λ mK, (24)
× (cid:18) 0.29 (cid:19) by a multivariant Gaussian, the Fisher matrix F is then a
6
good approximationfor the inverse of the parameter covari- a. BINGO The BINGO experiment is a single-dish
ance. In the 21-cm tomography, each frequency band will HI intensity mapping experiment, which aims at map-
providea map of 21-cm intensities, so we need to sum over ping the HI emission at frequencies between 960MHz and
theFishermatrixinbothℓ-spaceandfrequencyspace. Since 1260MHz [27, 53]. The telescope of BINGO experiment
ν = 1420MHz/(1 + z), each frequency corresponds to a hasnomovingpartsanditconductsadrift-scanstrategy. To
uniqueredshiftslice. TheFishermatrixis, achieveenoughsurveyarea,awideinstantaneousfieldofview
(FOV)withmultiplefeedsisrequired.Atotalof60feedslaid
ℓmax 2ℓ+1 outinarectangleof16m 15matthefocalplane. Thiswill
Fαβ =fsky (cid:18) 2 (cid:19)tr[Cℓ,αΣℓCℓ,βΣℓ], (26) form a FOV of about 10×◦(in Declination direction) 9◦(in
ℓXmin Right Ascension direction). With the 10◦ wide stri×p cen-
inwhich,C isann n matrix,inwhicheachelementisthe tering at Declination of 45◦, the total survey area is about
ℓ z× z 2500deg2. −
HI crossangularpowerspectrumbetweenthetwofrequency
bins. Σ = (C +N )−1 isthetotalnoiseinversematrix,in b. FAST FAST is the largest single-dish telescope,
ℓ ℓ ℓ
whichN isthen n experimentalnoisepowerspectrum. which also has the multi-beam system of 19 feed-horns ar-
ℓ z z
× ray [28, 29]. The multi-beamsystem is proposedto work at
Herewemakeasimpleassumptionthatthenoisesindifferent
frequency(redshift)binsareuncorrelated,thereforetheN is frequenciesfrom1.05GHzto1.45GHzwithsystemtempera-
ℓ
adiagonalmatrix. Inreality,21-cmintensitymapsarehighly tureof25K. Inouranalysis,weonlyincludethefrequencies
contaminatedbytheforeground,suchasGalacticsynchrotron up to 1.35GHz. With the 300m illuminated aperture, each
ofthefeed-hornhasthebeamsize(FullWidthatHalfMaxi-
emission, extragalactic point sources, and atmospheric sig-
nal. One needs to apply foreground removale technique to mum)of2.9′,andforma26′FOVwith19beams. Duetothe
longslewing time, FAST can only workon drift-scanobser-
reducetheforegroundcontamination[50–52].However,there
vationmode. SimilartotheBINGOexperiment,FASTscans
always be some level of residual Galactic foreground after
applying such techniques to the maps. Therefore the cross- a26′ widestripalongtheRightAscensiondirectionforeach
sidereal day. But the zenith angle of FAST can be adjusted
correlationof noisesbetweendifferentfrequencybandsmay
notcompletelybezero. fromDec: 14◦12′ to Dec:65◦48′. Withoutoverlappingbe-
Underoursimplifiedassumption,theelementofN matrix tweenscan−ningstrips,ittakesabouthalfyeartocoverall80◦
ℓ
is Declination range . With one-year observation (3.15 107
second),themaximumsurveyareaisabout24000deg2×.
Nij = δijNHI c. SKA-I The SKA Phase I (SKA-I) plans to construct
ℓ ℓ
= δijT2 S /(N N t ∆ν). (27) 190movable15mdishes[26]. Themaximumsurveyareais
sys survey ant feed TOT about 25000deg2. A efficient survey area is need to be ex-
T = T + T is the system temperature, which plored to minimal the constraint errors. In our analysis, we
sys rec sky
is contributed from the sky temperature, T = 60 onlyconsidertheautocorrelationofeachdishes,whichmeans
sky
(300MHz/ν)2.55, andreceivertemperatureT foreachex×- that the SKA-I works as 190 single dishes. Without the in-
rec
periment. N andN arethenumberofantennaandthe terferometry, the SKA-I has very low resolution and is only
ant feed
number of feed horn in each antenna respectively. The de- sensitivetothelow-ℓmodes.
tailedexperimentalparametersforFAST,SLA-IandBINGO Figure 4 shows the noise power spectra of different ex-
arelistinTableI. periments in at redshift bin z = 0.37 (left upper panel) and
z = 3.06 (right upper panel). The black solid line in the
upperpanelofeachfigureshowsthestandardangularpower
V. EXPERIMENTPARAMETERS spectrum of 21-cm (f = 0); The black dash-dotted, dot-
NL
tedanddashedlinesshowthenoisepowerspectraofSKA-I,
FAST and BINGO experiments. One-year observation time
and 2500deg2 survey area are assumed for all the experi-
TABLEI: TheexperimentparametersforFAST,SKA-IandBINGO. ments.ThepartialderivativesofCiiwithrespecttoparameter
D istheilluminatedaperture. ℓ
dish f areshowninthelowerpanel. Thedifferentcolorscorre-
NL
FAST SKA-I BINGO spondtodifferenttypesofPNG.
νmin[MHz] 1050 350 960 Comparing to the BINBO experiment, FAST and SKA-I
νmax[MHz] 1350 1050 1260 canhaveverylargesurveyarea. However,withthelimitinte-
grationtime,largesurveyareamaynotbeabletobeatdown
∆ν[MHz] 10 10 10
theconstrainterror.WewilldiscussthedetailsinSect.VI.
nν(nz) 30 70 30
D [m] 300 15 25
dish
Nant×Nfeed 1×19 190×1 1×60
VI. RESULTSANDDISCUSSION
tTOT[yr] 1 1 1
Trec[K] 25 28 50
Figure 5 shows the σ contoursfor local-shape PNG in
Ssurvey[deg2] <24000 <25000 2500 the plane of the surveyfaNreLa and total observationtime. The
leftandmiddlepanelsofFig.5showthecontoursforSKA-I
7
104 104
Redshift Window(i) = 0.37 Redshift Window(i) = 3.06
103 103
2] 2]
K K
)[µ 102 )[µ 102
π π
2 2
/( 101 /( 101
iiCℓ iiCℓ
1) 100 1) 100
+ fNL=0 + fNL=0
ℓ(ℓ 10-1 Noise Level SKAI ℓ(ℓ 10-1 Noise Level SKAI
Noise Level FAST Noise Level FAST
Noise Level BINGO Noise Level BINGO
10-2 10-2
2] 2]
µK 10-1 µK 10-1
[L [L
fN 10-2 fN 10-2
∂ ∂
ii/Cℓ10-3 iiC/ℓ10-3
π)∂10-4 π)∂10-4
)/(2 10-5 LEoqcuaillateral )/(2 10-5 LEoqcuaillateral
1 1
+ 10-6 Orthogonal + 10-6 Orthogonal
ℓ Enfolded ℓ Enfolded
( (
ℓ ℓ
101 102 101 102
ℓ ℓ
FIG. 4: Upper panels–Comparison between the noise power spectra of different experiments and the 21-cm power spectrum in standard
model (fNL = 0) for thetwo representative redshift bins (left and right panels). Inboth panels, one-year observation time(equivalent to
3.15×107sec)and2500deg2surveyareaareassumedforalltheexperiments. Lowerpanels–ThepartialderivativesofCiiwithrespectto
ℓ
parameterfNLforfourshapesofPNG.
3 SKA-I Local FAST Local
104 .00 104 101 SKA-I Local FAST Local
0
SKA-I Equilateral FAST Equilateral
SKA-I Orthogonal FAST Orthogonal
××Nt[year]feedTOT110023 1.500.5000 1100NL01fσ ××Nt[year]feedTOT110023 16..500000 3.000 10NL1fσ /σσ)(minffNLNL SKA-I Enfolded FAST Enfolded
Nant101 Nant101
15.000 100
6.000 15.00030.000 30.000 100
102 103 104 105 102 103 104 105 102 103 104 105
Ssurvey[deg2] Ssurvey[deg2] Ssurvey[deg2]
FIG.5: Theleft(forSKA-I)andmiddle(forFAST)panelsshowtheσ contoursforlocal-shapePNGintheparameterspaceofthesurvey
fNL
areaandtotalobservationtime. ThedashedcontouristheerrorofconstraintwithPlancktemperatureandpolarizationdata[11]. Theright
panelshowstheσfNL/(σfNL)minasafunctionofsurveyareaforvariousPNGtypes.ThesolidlinesshowtheresultsforSKA-Iwithone-year
observationaltimeand190dishes;thedashedlinesshowtheresultsforFASTwithone-yearobservationaltimeand19beams.
and FAST experiments respectively. The colour going from optimization is about 6000deg2 for FAST experiment. For
red to blue meansthat the constraintsbecomestronger. Dif- other shapes, the optimized survey areas are approaching to
ferent black solid lines are the contours of the same error themaximumskycoverageofSKA-IorFAST.Thelargesur-
of flocal. Therefore, the error tends to become smaller if veyareacanhelptobeatthecosmicvarianceonlargescales,
NL
N N t becomesbigger.Thusthemostefficient buttheintegrationtimeperpixelbecomessmaller,leadingto
ant feed TOT
× ×
way toreducetheconstrainterroristo increasethe observa- largerpixelnoise.
tiontimeorthenumberofdishes(feeds). Assumingone-year One can see from the right panel of Fig. 5 that generally
observation time and the maximum dish(feeds) number for speaking the larger survey area is, the smaller the error of
SKA-I and FAST experiments, the constraint errors of vari- f is, except for measuring equilateral shape of PNG us-
NL
ous PNG types as function of survey area are shown in the ing FAST survey. This is different from the situation of us-
rightpanelof Fig. 5. In orderto have a clear view, the con- ing21-cmintensitymappingtomeasuretheangularscale of
straint errors, σ , are divided by the their minimal values. BAOacousticoscillation,whichhavetheoptimalsurveyarea
fNL
Itistruethattheoptimalsurveyareamaynotbethemaximal around6000deg2(ForBINGOseefig.7in[27],andforFAST
surveyarea.Forexample,inthecaseofEquilateralshape,the seefig.1in[54]).Thereasonisbecausescale-dependentbias
8
fromPNGisalwaysprominentonverylargescales,sobeat- tween350MHzand1050MHz, whichisthesameasthefre-
ing down cosmic variance is more important than lowering quency range of SKA-I experiment, the constraint for local
downthepixelnoise. However,BAOscaleissub-horizonfor shapePNG willbeσ 1.62withtheoptimizedsurvey
which one needs to find a balance between lowering down areaof6000deg2.ThfeNlocLcoalns∼trainterrors(σ )fororthogonal
pixel noise and beating down cosmic variance. We use dif- fNL
andenfoldedshapesbecome39and64respectively,whichare
ferent optimized survey areas for different cases in the later
allhighlyreduced.
analysis.
Figure 6 shows the σ as a function of ℓ if we fix
fNL min
ℓ = 600. Different PNG shapes are shown in different
max
panels.Ineachpanel,differentcolorsindicatedifferentexper-
imentsas shown in the legends. The optimizedsurveyareas
areappliedto theanalysis. Theconstrainterrorsof different VII. CONCLUSION
PNG shapes from Planck satellite are shown with the black
dashed lines [11]. The σ of different PNG shapes fore-
fNL In this work, we explored the constraining power on the
castedwithdifferentexperimentsarelistinTableII.
primordial non-Gaussianity (PNG), with the future single-
Wecanseethat,forthelocalshapePNG,theSKA-Iexper-
dish HI intensity mappingobservationswith BINGO, FAST
iment is potentially able to constrain f better than Planck
NL and SKA-I. Four fundamentalshapes of PNG are studied in
experiment. But we should realize that it is only the most
our analysis, including local, equilateral, orthogonaland en-
idealcase. Itiswellknownthat,oneofthebigchallengesfor
folded. Wefocusontheeffectofscale-dependentbiastothe
observationsof HI intensity mapping is the foregroundsub- underlyingdarkmattertracer,inducedbytheprimordialnon-
traction, and the low-ℓ modes may not be detectable due to
Gaussinaity. The properties of such scale-dependentbias at
the foregroundcontamination. Our results show that, to ob-
large-scale limit are discussed in our analysis. The forecast
tainaremarkableconstraintonf withtheSKA-Iintensity
NL resultsarelistinTableII.
mappinginthefuture,weneedtorecovertheangularpower
spectrumof HIwiththeminimalℓmax 50. Thisistheaim Our forecasts show that with the current configuration of
ofseveralrecenteffortsofrestoringlarg≃eangularpowerwith the experimentsone-yearobservationtime, the constrainton
cross-correlationwithweakgravitationallensing[55,56].We local shape of PNG from SKA-I intensity mapping experi-
also find that the constraint error for orthogonalshape PNG ment can be better than the current Planck experiment. The
with SKA-I is 25, which is at the same level of current optimized survey area of 25000deg2 is applied in the anal-
Planck limit. If∼the observation can be extended to 2 years, ysis of SKA-I, but the results are more sensitive to the total
theerrorwillbereducedto 20. observationtimethanthesurveyarea.However,theHIinten-
Theconstrainterrorfore∼quilateralshapePNG withFAST sity mapping experimentsmay be contaminatedby the fore-
is 44,whichisbetterthantheresultsofSKA-IandBINGO groundandthelow-ℓmodesmaybebedetectable.Ouranaly-
ex∼periments. The FAST error on fequil is close to the cur- sisshowsthattheSKA-Iexperimentcanstillhavetheremark-
rent limit of Planck experiment. ThNiLs is because the scale- ableconstraintwithoutthemodesof ℓ < 50 With two-years
dependent bias induced by the equilateral shape PNG has observation,theconstraintonorthogona∼lshapePNGis 20,
∼
higher signal-to-noise ratio at small scales and the FAST whichisalsobetterthantheconstraintfromPlanckmeasure-
experiment is more sensitive to the small-scale modes than ment.
SKA-IsingledishmodeandBINGO. The FAST experimenthasits advantageof higherangular
We also test the possible extensionsof the currentconfig- resolution,andmoresensitivetothesmall-scalemodes,which
uration by adding more integration time. If the observation is good for constraining the equilateral shape of PNG. With
time for SKA-I and FAST could be extendedto 2 years, the thecurrentconfigurationandtwo-yearsobservation,thecon-
constraintsonfNL canbeimprovedquantitatively. Thefore- strainterrorforEquilateralshapeofPNGwillbeσfNL = 32,
castedconstraintondifferentshapesofPNGarelistedinTa- whichisbetterthanthecurrentlimitofPlanckobservation.
bleII.Itisworthnoticingthattheconstrainterroronorthog-
Similar constraint on the local shape of PNG can be
onalshapePNGwithSKA-IandequilateralshapePNGwith
FASTbecomesmallerthanlimitsofPlanckwithextendedob- achievedbytheFAST HI intensitymapping,ifitsfrequency
bandwidthcouldbeextendedtothelowerfrequencies(ultra-
servationaltime.
wideband).Assumingthesameworkingfrequencyrange,the
A good extension for FAST experiment is to extend its
bestconstraintfromFASTonlocalshapeofprimordialnon-
bandwidth to the lower frequencies, which are correspond-
Gaussianiyisσ 1.62.
ingtothehigherredshifts.SofartheFASTtelescopehasone fNL ∼
ultra-wide band receiver working on 270MHz 1.62GHz. The studieswe conducthere are the standardpowerspec-
∼
Unfortunately, the ultra-wide band receiver has only one tra analysis of 21-cm. There have been efforts on using the
beam. Itwilltakequitelongtimetoachievethesameobser- multi-tracertechniquetobeatthecosmicvarianceandobtain
vationtime asthemulti-beamreceiver. Nowthe multi-beam tighter constraintson f [57–59]. In addition, using three-
NL
system of the FAST telescope is designed to work on fre- point correlation function is another way to measure PNG.
quencies between 1050MHz and 1350MHz. Assuming that Thesemethodswillbeexploredtomeasureallshapesoff
NL
the FAST multi-beam system works on the frequencies be- inthefuturework.
9
Local Shape Equilateral Shape
104 Sum over ℓ to ℓmax=600 104 Sum over ℓ to ℓmax=600
SKA-I 25000deg2 SKA-I 25000deg2
FAST 24000deg2 FAST 6000deg2
103 BINGO 2500deg2 103 BINGO 2500deg2
Planck Planck
NL NL
σf 102 σf 102
101 101
100 100
101 102 101 102
ℓ ℓ
Enfolded Shape Orthogonal Shape
104 Sum over ℓ to ℓmax=600 104 Sum over ℓ to ℓmax=600
SKA-I 25000deg2 SKA-I 25000deg2
FAST 24000deg2 FAST 24000deg2
103 BINGO 2500deg2 103 BINGO 2500deg2
Planck
NL NL
σf 102 σf 102
101 101
100 100
101 102 101 102
ℓ ℓ
FIG.6: TheσfNL asafunctionofℓminforvariousexperimentsandPNGshapes. TheblackdashedlineisthecurrentconstraintwithPlanck
temperatureandpolarizationdata[11].
TABLEII: σ ofdifferentPNGshapesforecastedwithdifferentexperiments. Theoptimizedsurveyareasareappliedintheanalysis. The
fNL
“Planck2015”columnshowstheconstrainterrorwithPlanktemperatureandpolarizationdata[11]. Thenumbersinboldcharacterarethe
constraintsbetterthanPlanck.
CurrentConfiguration Extentions
Planck2015 FAST SKA-I BINGO SKA-I2yr† FAST2yr†† FASTlow‡
Local 5 9.5 0.54 17 0.43 7.4 1.6
Equilateral 43 44 86 100 66 32 53
Orthogonal 21 75 25 128 20 59 39
Enfolded – 94 43 164 36 70 64
†SKA-Iwithtwo-yearobservation;††FASTwithtwo-yearobservation;‡FASTwithlowfrequenciesrange
from350MHzto1050MHz
Acknowledgments on CAMB SOURCES . This work is supported by National
Research Foundation of South Africa with grant no.105925
WethankNealDalal,DiLi,RoyMaartensandXiao-Dong andUniversityofKwaZulu-Natalstaffstart-upgrant.
Xu for helpful discussions and Stefano Camera for his help
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