Prospects for constraining interacting dark energy model with 21 cm intensity mapping experiments
Ming Zhang, Bo Wang, Peng-Ju Wu, Jing-Zhao Qi, Yidong Xu, Jing-Fei Zhang, Xin Zhang
PProspects for constraining interacting dark energy model with 21 cm intensitymapping experiments
Ming Zhang, Bo Wang, Jing-Zhao Qi, Yidong Xu, ∗ Jing-Fei Zhang, and Xin Zhang † Department of Physics, College of Sciences, & MOE Key Laboratory of Data Analytics and Optimization for Smart Industry,Northeastern University, Shenyang 110819, China National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
We forecast the constraints on cosmological parameters in the interacting dark energy model usingthe mock data generated for neutral hydrogen intensity mapping (IM) experiments. In this work,we consider only the interacting dark energy model with the energy transfer rate Q = βHρ c , andtake BINGO, FAST, SKA1-MID, and Tianlai as typical examples of the 21 cm IM experiments.We find that the Tianlai cylinder array will play an important role in constraining the interactingdark energy model. Assuming perfect foreground removal and calibration, and using the Tianlai-alone data, we obtain σ ( H ) = 0 .
10 km s − Mpc − , σ (Ω m ) = 0 . σ ( σ ) = 0 . β , compared with Planck+optical BAO, while the Planck+Tianlai data can give a rathertight constraint of σ ( β ) = 0 . w CDM model, we obtain σ ( β ) = 0 . σ ( w ) = 0 .
006 fromPlanck+Tianlai. In addition, we also make a detailed comparison among BINGO, FAST, SKA1-MID, and Tianlai in constraining the interacting dark energy model. We show that the future 21cm IM experiments will provide a useful tool for exploring the nature of dark energy, and will playa significant role in measuring the coupling between dark energy and dark matter.
I. INTRODUCTION
According to our understanding to the contemporarycosmology, dark energy is responsible for the accelerat-ing expansion of the universe [1]. Until now, however,the nature of dark energy still remains a deep mystery.To explore dark energy in depth and also to test the pos-sible deviation from General Relativity, it is necessary toprecisely measure the late-time expansion history of theuniverse. Among the several ways of exploring the cos-mic expansion history, the baryon acoustic oscillations(BAO) [2, 3] have been proven to be a very useful toolto measure cosmological distance and Hubble expansionrate and to explore the nature of dark energy. Becausethe acoustic waves are frozen in the eras after recom-bination, the BAO peak wavelengths, as a cosmologicalstandard ruler, allows accurate measurements of the ex-pansion history.Galaxy surveys could achieve a tomography of the uni-verse over its last twelve billion years. In addition, due tothe hyperfine transition of neutral hydrogen (H I ) at 21cm wavelength, via 21 cm emission, BAO signals couldbe precisely measured [4]. H I is essentially a tracer of thegalaxy distribution. Detecting a sufficiently large numberof galaxies with H I
21 cm emission could make it pos-sible to provide a useful tool for cosmological research,but actually it is unnecessary to perform a galaxy surveyfor the study of large-scale structure. We can insteadmeasure the total H I intensity over large angular scales, ∗ [email protected] † [email protected] without needing to resolve individual galaxies. Similarto the cosmic microwave background (CMB) map, thisintensity mapping (IM) methodology makes it possible toefficiently survey large volumes with modern radio tele-scopes [5–28].Using the H I IM method, Chang et al. [29] reportedthe measurements of cross-correlation function betweenthe H I map observed with the Green Bank Telescope(GBT) and the galaxy map with the DEEP2 optical red-shift survey. The cross-power spectrum between H I andoptical galaxy survey was also detected with the GBT 21cm IM survey and the WiggleZ Dark Energy Survey [30].Lately, Anderson et al. [31] reported the results from21 cm IM acquired from the Parkes radio telescope andcross-correlated with galaxy maps from the 2dF galaxysurvey. So far, there are many current and future H I IMexperiments comprised of wide-field and high-sensitivityradio telescopes or interferometers. Here we consider sev-eral typical examples, namely, the Baryon acoustic oscil-lations In Neutral Gas Observations (BINGO) [6, 32–34], the Five-hundred-meter Aperture Spherical radioTelescope (FAST) [35–40], the Square Kilometre Array(SKA) [24, 41–43], and the Tianlai cylinder array [18, 44–48]. This work aims to forecast how future H I IM ex-periments, including BINGO, FAST, SKA Phase I mid-frequency array (SKA1-MID), and Tianlai, can constrainthe interacting dark energy (IDE) model.The IDE model originates from a longstanding con-jecture that there might be some coupling between darkenergy (DE) and cold dark matter (CDM) (for a recentreview, see Ref. [49]). The model with an interaction be-tween vacuum energy (for convenience, dark energy with w = − a r X i v : . [ a s t r o - ph . C O ] F e b model. Besides, we also wish to consider the more gen-eral IDE model in which the dark-energy equation ofstate (EoS) parameter w is a constant, usually calledthe I w CDM model. In the I w CDM scenario, the energyconservation equations for DE and CDM satisfy˙ ρ de = − H (1 + w ) ρ de + Q, (1)˙ ρ c = − Hρ c − Q, (2)where Q is the energy transfer rate, ρ de and ρ c denote theenergy densities of DE and CDM, respectively, H = ˙ a/a represents the Hubble parameter, a is the scale factorof the universe, and a dot represents the derivative withrespect to the cosmic time t . Here the case of w = − Q have been constructed and discussed in previousworks [50–84]. In this paper, we employ a phenomeno-logical form of Q = βHρ c , where β denotes a dimension-less coupling parameter. From Eqs. (1) and (2), it canbe seen that β > β < β = 0means that there is no interaction between DE and CDM.In this work, we study what role the 21 cm IM exper-iments would play in constraining cosmological param-eters in the IΛCDM and I w CDM models. Combiningwith the Planck CMB data [85], we wish to forecast howthese 21 cm IM experiments will improve the constraintson cosmological parameters. We also make a comparisonwith optical galaxy surveys. Unless otherwise stated, weemploy the spatially-flat ΛCDM model with parametersfixed by fitting to the Planck 2018 data [85] as a fiducialmodel to generate mock data.This paper is organized as follows. In Section II, wegive a detailed description of methodology. In SectionII A, we introduce signal power spectrum and noise powerspectrum of the 21 cm IM experiments, and construct theFisher matrix. We further give a detailed description ofthe experimental configurations in Section II B, and de-scribe methods and data employed in this paper in Sec-tion II C. In Section III, we present forecasted constraintson cosmological parameters and make some relevant dis-cussions. Finally, we give our conclusions in Section IV.
II. METHODOLOGYA. 21 cm intensity mapping
The mean H I brightness temperature is given by (thedetailed derivation can be found in Ref. [6])¯ T b ( z ) = 180Ω H I ( z ) h (1 + z ) H ( z ) /H mK , (3)where Ω H I ( z ) is the fractional density of H I , H ( z ) isthe Hubble parameter as a function of redshift z , H ≡ h km s − Mpc − is its value today, and h is the di-mensionless Hubble constant. Considering the effect of redshift space distortions (RSD) [86], the signal H I powerspectrum can be written as [3] P S ( k f , µ f , z ) = ¯ T b D A ( z ) H ( z ) D A ( z ) H ( z ) f b I [1 + β H I ( z ) µ ] P ( k, z ) , (4)where the subscript “f” denotes the quantities calculatedin the fiducial cosmology, D A is the angular diameterdistance, b H I is the H I bias, µ = ˆ k · ˆ z , β H I is the RSDparameter equal to f /b H I ( f ≡ d ln D/d ln a is the lineargrowth rate with a being the scale factor) in linear theory,and P ( k, z ) = D ( z ) P ( k, z = 0), with D ( z ) being thegrowth factor and P ( k, z = 0) being the matter powerspectrum at z = 0 that can be generated by CAMB [87].Note here that the “Fingers of God” (FoG) effect due touncorrelated peculiar velocities on small scales is ignoredin this work.The noise power spectrum models the instrumentaland sky noises for a given experiment. The survey noiseproperties have been described in detail in Refs. [6, 17].Here, we summarize them for completeness. The fre-quency resolution of IM surveys performs well, so we ig-nore the instrument response function in the radial di-rection and only consider the response due to the finiteangular resolution: W ( k ) = exp (cid:34) − k ⊥ r ( z ) (cid:18) θ B √ (cid:19) (cid:35) , (5)where k ⊥ is the transverse wave vector, r ( z ) is the comov-ing radial distance at redshift z , and θ B is the full widthat half-maximum of the beam of an individual dish.If considering a redshift bin between z and z , thesurvey volume can be written as V sur = Ω tot (cid:90) z z dz dVdzd Ω = Ω tot (cid:90) z z dz cr ( z ) H ( z ) , (6)where Ω tot = S area is the solid angle of the survey area.The pixel volume V pix is also calculated with the similarformula with Ω tot substituted by Ω pix ≈ θ .For an experiment using single-dish mode, the pixelnoise can be written as σ pix = T sys (cid:112) ∆ ν t tot ( θ /S area ) λ A e θ √ N dish N beam , (7)and for an interferometer, the pixel noise can be writtenas σ pix = T sys (cid:112) ∆ ν t tot (FoV /S area ) λ A e √ FoV 1 (cid:112) n ( k ⊥ ) N beam , (8)where T sys is the system temperature, N dish is the num-ber of dishes and N beam is the number of beams, t total is the total observing time, A e is the effective collect-ing area of each element. For a dish reflector, A e ≡ ηπ ( D dish / and θ B ≈ λ/D dish , where D dish is the di-ameter of the dish, and η is an efficiency factor for whichwe adopt 0.7 in this work. For a cylindrical reflector, A e ≡ ηl cyl w cyl /N feed and FoV ≈ ◦ × λ/w cyl , where w cyl and l cyl are width and length of cylinder, respectively,and N feed is the number of feeds per cylinder. Unlikea single dish, we need to calculate the baseline density n ( k ⊥ ) for the interferometer. The detailed calculationmethod can be found in Ref. [17].For BINGO, FAST, and Tianlai, the system tempera-ture is given by T sys = T rec + T gal + T CMB , (9)where T rec is the receiver temperature for each of theseexperiments with the values given in Table I, T gal ≈
25 K(408 MHz /ν ) . is the contribution from the MilkyWay for a given frequency ν , and T CMB ≈ .
73 K is theCMB temperature. For the SKA1-MID array, the systemtemperature is calculated by T sys = T rec + T spl + T gal + T CMB , (10)where T spl ≈ T rec for SKA1-MID is assumed tobe [88] T rec = 15 K + 30 K (cid:16) ν GHz − . (cid:17) . (11)Finally, the noise power spectrum is then given by P N ( k ) = σ V pix W − ( k ) . (12)The Fisher matrix for a set of parameters { p } is givenby [89] F ij = 18 π (cid:90) − dµ (cid:90) k max k min k dk ∂ ln P S ∂p i ∂ ln P S ∂p j V eff , (13)where we define the “effective volume” as [17, 19] V eff = V sur (cid:18) P S P S + P N (cid:19) . (14)Next, we assume that the bias b H I depends only onthe redshift z . This assumption is appropriate only forlarge scales, so we impose a non-linear cut-off at k max (cid:39) . z ) / Mpc − [90]. Hence, we ignore the small-scale velocity dispersion effect of FoG (parameterized bythe non-linear dispersion scale σ NL ) in this work. Thelargest scale the survey can probe corresponds to a wavevector k min (cid:39) π/V / [90]. In our work, we choose theparameter set { p } as { D A ( z ) , H ( z ) , [ f σ ]( z ) , [ b H I σ ]( z ) } .Note that in Ref. [17] σ NL is also treated as a free param-eter, but in this work we ignore the small-scale non-linearFoG effect. When inverting the Fisher matrix, we can getthe covariance matrix that gives us the forecasted con-straint on the chosen parameter set. Note that we onlyuse the forecasted cosmological observables D A ( z ), H ( z ),and [ f σ ]( z ) to constrain cosmological parameters. B. Experimental configurations
In this paper, we focus on the Tianlai, BINGO, FAST,and SKA1-MID experiments. These experiments arepotentially suitable for the H I IM survey in the post-reionization epoches of the universe. In this subsection,we give a brief description of these experiments.
Tianlai:
The Tianlai project [91] is an H I IM exper-iment aimed at measuring the dark energy equation ofstate by detecting the BAO features in the large-scalestructure power spectrum. The full-scale Tianlai cylinderarray will consist of eight adjacent cylinders to be builtin northwest China, with each cylinder 15 m wide and120 m long with 256 dual polarization feeds [18, 44, 45].Note that, currently, there is a Tianlai pathfinder arraycommissioning, which uses a much smaller scale cylinderarray, but in this work we will only discuss the full-scaleTianlai cylinder array that is to be built in the future.
BINGO:
The BINGO experiment [92] is a project tobuild a special-purpose radio telescope to map redshiftedH I emission in the redshift range of z = 0 . − . FAST:
The FAST [93] is a multi-beam single dish tele-scope built in Guizhou province of southwest China. Theaperture diameter is 500 m with an effective illuminatingdiameter of 300 m. It uses an active surface that adjustsshape to create parabolas in different directions. It willbe capable of covering the sky within a 40-degree anglefrom the zenith. 19 beams are designed in one receiver ar-ray, which will greatly increase the survey speed [35, 36].
SKA1-MID:
The SKA project [94], currently underconstruction, plans two stages of development. In thispaper, we consider the SKA1-MID array, based in theNorthern Cape, South Africa. SKA1-MID has 133 15 mSKA dishes and 64 13.5 m MeerKAT dishes. The SKA1-MID will perform an H I IM survey over a broad range offrequencies and a large fraction of the sky [24, 41, 42, 88].Here we consider only the
Wide Band 1 Survey of theSKA1-MID. In addition, in this work, for simplicity weconsider SKA1-MID as an array with 197 15 m dishes.The full instrumental parameters used for these exper-iments are listed in Table I.
C. Data and method
Method for forecasting cosmological constraints for H I IM surveys has been presented in Refs. [17, 95]. We willfollow the prescription given in Refs. [17, 95] to performthe forecast for the 21 cm IM experiments. By perform-ing measurements of the full anisotropic power spectrum,we obtain constraints on the angular diameter distance D A ( z ), the Hubble parameter H ( z ), and the RSD ob-servable [ f σ ]( z ), which are considered to be indepen- TABLE I. Experimental configurations for Tianlai, BINGO,FAST, and SKA1-MID.Tianlai BINGO FAST SKA1-MID z min z max N dish – 1 1 197 N beam D dish [m] – 40 300 15 S area [deg ] 10000 3000 20000 20000 t tot [h] 10000 10000 10000 10000 T rec [K] 50 50 20 Eq. (11) dent in each redshift bin. We obtain covariance matri-ces for { D A ( z j ) , H ( z j ) , [ f σ ]( z j ); j = 1 ...N } in a seriesof N redshift bins { z j } by inverting the Fisher matrix.We perform the Fisher matrix calculations by consideringthe aforementioned parameters. The marginalized con-straints on these parameters for these surveys are shownin Figure 1.These covariance matrices, plus the fiducial cosmology,generate the mock data of these 21 cm IM experiments.Then we use these mock data to constrain cosmologicalparameters by performing a Markov Chain Monte Carlo(MCMC) analysis. In the MCMC analysis, we also em-ploy the CMB data from the Planck 2018 release [85] andthe BAO measurements from galaxy redshift surveys, in-cluding SDSS-MGS [96], 6dFGS [97], and BOSS DR12[98].In this paper, we employ the extended parametrizedpost-Friedmann (ePPF) framework to calculate the cos-mological perturbations in the IDE scenario [66, 67]. Thisis because we need to avoid the perturbation divergenceproblem in the IDE cosmology.It is well-known that, in the IDE scenario, when calcu-lating the cosmological perturbations, it is found that formost cases the curvature perturbation on super-horizonscales at early times is divergent, which is a catastro-phe for the IDE cosmology. The underlying reason ofthis problem is that we actually do not know how toconsider the perturbations of dark energy. In the tradi-tional linear perturbation theory, for calculating the per-turbations of dark energy, we need to define a rest-framesound speed for dark-energy fluid c s = δp/δρ (with thegauge v | rf = B | rf = 0) to relate the dark-energy densityand pressure perturbations. This leads to that in a gen-eral gauge δp has two parts—adiabatic and nonadiabaticparts, and the interaction term appearing in the nonadia-batic part will occasionally leads the nonadiabatic modesto be unstable.In order to solve this problem, Li, Zhang, and Zhang[66, 67] extended the original PPF framework [99] to in-clude the IDE scenario, and used this method to avoidthe divergence of cosmological perturbations in the IDEmodels. The PPF method does not consider the dark-energy pressure perturbation, but only describes dark-energy perturbations based on some basic facts of darkenergy. On large scales, far beyond the horizon, the rela- z −3 −2 −1 σ ( D A ) / D A TianlaiBINGO FASTSKA1 z −3 −2 −1 σ ( H ) / H TianlaiBINGO FASTSKA1 z −3 −2 −1 σ ( f σ ) / ( f σ ) TianlaiBINGO FASTSKA1
FIG. 1. Forecasted fractional errors on D A ( z ), H ( z ), and[ fσ ]( z ), as a function of redshift. tionship between the velocities of dark energy and othercomponents can be empirically parameterized. On smallscales, deep inside the horizon, dark energy is smoothenough so that it can be viewed as a pure background.We thus can use the Poisson equation to describe thislimit. In order to make these two limits compatible, weintroduce a dynamical function Γ by which we can findan equation to describe the cases on all scales. In the TABLE II. The 1 σ errors on the parameters in the IΛCDMmodel from the different data combinations. β H [km s − Mpc − ] Ω m σ Planck 0.00240 1.80 0.0256 0.0150Planck+BAO 0.00120 0.69 0.0087 0.0110Tianlai 0.00196 0.10 0.0013 0.0015Planck+Tianlai 0.00052 0.09 0.0011 0.0011Planck+BINGO 0.00150 0.98 0.0130 0.0110Planck+FAST 0.00110 0.65 0.0086 0.0088Planck+SKA1-MID 0.00057 0.22 0.0029 0.0023TABLE III. The 1 σ errors on the parameters in the I w CDMmodel from the different data combinations. β w
Planck 0.00176 0.324Planck+BAO 0.00150 0.074Tianlai 0.00742 0.010Planck+Tianlai 0.00058 0.006Planck+BINGO 0.00140 0.038Planck+FAST 0.00130 0.030Planck+SKA1-MID 0.00079 0.010 equation of motion of Γ, a parameter c Γ is introduced,giving a transition scale in terms of the Hubble scale un-der which dark energy is smooth enough. In this equa-tion, there is no perturbation variable of dark energy, andwhen the evolution of Γ is derived, we can directly ob-tain the density perturbation and velocity perturbationof dark energy. Hence, this method avoids using the pres-sure perturbation of dark energy defined by the soundspeed. Here, we only give a very brief description of theePPF method, and we refer the reader to Refs. [66, 67]for more details.We employ the CosmoMC package [100] to perform theMCMC calculations and insert the ePPF code as a partof it to treat the cosmological perturbations in the IDEmodels.
III. RESULTS
In this section, we will present the forecasted resultsshowing relative constraining capabilities for the IΛCDMand I w CDM models by combining each of the 21 cm IMexperiments with Planck. Tables II and III list the 1 σ errors for the marginalized parameter constraints for theIΛCDM and I w CDM models, respectively. In SectionIII A, we will show what role the 21 cm IM experiments,taking the example of Tianlai, could play in constrain-ing cosmological parameters, and compare with the BAOmeasurements from galaxy redshift surveys. In SectionIII B, we will compare the capabilities of constraining cos-mological parameters for the different 21 cm IM experi-ments.
A. Constraints on cosmological parameters fromthe Tianlai cylinder array
The 1 σ and 2 σ posterior distribution contours areshown in Figure 2 for Planck, Planck+BAO, Tianlai,and Planck+Tianlai in the IΛCDM model. We findthat the future full-scale Tianlai experiment can givevery tight constraints on H , Ω m and σ . With theTianlai data alone, we obtain σ ( H ) = 0 .
10 km s − Mpc − , σ (Ω m ) = 0 . σ ( σ ) = 0 . σ ( H ) = 0 . − Mpc − , σ (Ω m ) = 0 . σ ( σ ) = 0 . H , Ω m and σ by(1 . − . / .
80 = 95 . . − . / . .
7% and (0 . − . / . . β with σ ( β ) ≈ . Q = βHρ c , the CMB data usually could provide a tightconstraint on the coupling parameter β . This is becausein the early universe both H and ρ c take rather high val-ues, and the energy transfer rate Q can take a moderatevalue even if β is very small. Thus, the CMB data asan early-universe probe can give a relatively tight con-straint on β . This is why the Planck CMB data canoffer a similar constraint on β , compared with the caseof the Tianlai data, but for the other cosmological pa-rameters the Tianlai data can give much better con-straints. Since the degeneracy directions of β and otherparameters for Planck and Tianlai are rather different,we can eventually obtain a rather tight constraint on β , i.e., σ ( β ) = 0 . σ ( β ) = 0 . w CDM model. Here we only show the posterior dis-tribution contours in the β – w plane that we are mostinterested in. We can see that using only the PlanckCMB data cannot give a good constraint on dark energyEoS parameter w , i.e., σ ( w ) ≈ .
3. It is necessary touse the late-universe measurements to break the param-eter degeneracies inherent in CMB. As a contrast, theTianlai-alone data can provide a rather tight constrainton w , giving the result of σ ( w ) = 0 . β in the I w CDM modelwith the interaction term Q = βHρ c . So, we can seethat the Planck-alone data give σ ( β ) = 0 . σ ( β ) = 0 . β and theTianlai data can tightly constrain w , the degeneracy di-rections of them are entirely different. It is known that m H [ k m s M p c ]
60 65 70 H [km s Mpc ] m PlanckPlanck+BAOTianlaiPlanck+Tianlai
FIG. 2. Constraints on cosmological parameters from Planck, Planck+BAO, Tianlai, and Planck+Tianlai in the IΛCDM model. w PlanckPlanck+BAOTianlaiPlanck+Tianlai
FIG. 3. Constraints on β and w from Planck, Planck+BAO,Tianlai, and Planck+Tianlai in the I w CDM model.
Planck-alone, and even Planck+BAO, can only providea loose or a moderate constraint on the I w CDM model,as shown in Figure 3. Nevertheless, since the cosmologi-cal parameter degeneracies can be broken by the Tianlaidata, the parameter constraints are greatly improved byadding the Tianlai data in the fit. We obtain σ ( β ) =0 . σ ( w ) = 0 .
006 from the Panck+Tianlai datacombination, and we find that the constraints on β and w are improved by (0 . − . / . . . − . / .
324 = 98 . σ ( β ) = 0 . σ ( w ) = 0 .
074 fromPlanck+BAO, we can see that the future 21 IM exper-iments will exhibit powerful capability of constrainingcosmological parameters.
B. Comparison with constraints from different21 cm IM experiments
In this subsection, we will discuss the ability to con-strain cosmological parameters for the different 21 cm IMexperiments.Figure 4 visualizes the constraint results for theIΛCDM model from each of these experiments, includ- m H [ k m s M p c ]
64 66 68 H [km s Mpc ] m Planck+BINGOPlanck+FASTPlanck+SKA1Planck+Tianlai
FIG. 4. Constraints on cosmological parameters from Planck+BINGO, Planck+FAST, Planck+SKA1-MID, andPlanck+Tianlai in the IΛCDM model. ing BINGO, FAST, SKA1-MID, and Tianlai, combinedwith Planck. We can clearly see from Figure 4 that theconstraining capabilities of the two arrays, i.e. Tianlaiand SKA1-MID, are much better than those of the sin-gle dishes, FAST and BINGO. Comparing Tianlai andSKA1-MID, Tianlai is evidently better, and comparingFAST and BINGO, FAST is much better. As shownin Figure 1, FAST and BINGO can only observe in lowredshifts and can only cover narrow redshift ranges, i.e.,0 < z < .
35 for FAST and 0 . < z < .
48 for BINGO.Although the redshift range coverages are similar, com-paring FAST and BINGO for the relative measurementerrors on D A ( z ), H ( z ) and [ f σ ]( z ), we find that obvi-ously FAST is better than BINGO, mainly due to themuch larger survey area and the larger aperture size.Comparing Tianlai and SKA1-MID, we find that both ofthem can cover a wide redshift range, i.e., 0 < z < . . < z < z > .
5, therelative errors on D A ( z ), H ( z ), and [ f σ ]( z ) of Tianlaiare much smaller than those of SKA1-MID. This high-lights the advantage of a compact interferometer arraywith a large number of receivers, and explains why theTianlai’s capability of constraining cosmological param- eters is better than SKA1-MID.Concretely, for constraining the coupling parameter β ,we obtain σ ( β ) = 0 . β in the IΛCDM model, the ca-pabilities of SKA1-MID and Tianlai are similar, al-though Tianlai is slightly better, in the sense of com-bining with Planck, both are much better than those ofBINGO and FAST. Comparing with the results of thePlanck-alone data, the Planck+BINGO, Planck+FAST,Planck+SKA1-MID, and Planck+Tianlai data can im-prove the constraints on β by 37 . . . . σ ( H ) = 0 .
98 km s − Mpc − , σ (Ω m ) =0 . σ ( σ ) = 0 . σ ( H ) = 0 .
65 km s − Mpc − , σ (Ω m ) = 0 . σ ( σ ) = 0 . H , Ω m , and σ , FAST performs muchbetter than BINGO. But we also notice that neitherFAST nor BINGO is as powerful as Tianlai and SKA1-MID. We obtain σ ( H ) = 0 .
22 km s − Mpc − , σ (Ω m ) =0 . σ ( σ ) = 0 . σ ( H ) = 0 .
09 km s − Mpc − , σ (Ω m ) = 0 . σ ( σ ) = 0 . < z < . S area = 5000 deg and t total = 10000h), and Deep SKA-LOW Survey covers the redshift rangeof 3 < z < S area = 100 deg and t total = 5000 h).It is of great interest to use the combination of the threesurveys of SKA1 in measuring the expansion history ofthe post-reionization epoch of the universe to explore var-ious cosmological issues. We will leave this work in thefuture.In order to compare their constraint abilities in theI w CDM model, we show the 1 σ and 2 σ measure-ment error contours for Planck+BINGO, Planck+FAST,Planck+SKA1-MID and Planck+Tianlai in Figure 5.We obtain σ ( β ) = 0 . σ ( w ) = 0 .
038 fromPlanck+BINGO, which are improved by 20 .
5% and88 . β and w ) in the I w CDMmodel. FAST performs slightly better than BINGO,and we obtain σ ( β ) = 0 . σ ( w ) = 0 .
030 fromPlanck+FAST. Evidently, SKA1-MID performs muchbetter than BINGO and FAST, and we obtain σ ( β ) =0 . σ ( w ) = 0 .
010 from Planck+SKA1-MID. Themost stringent constraints on the I w CDM model are fromPlanck+Tianlai, and in this case we have σ ( β ) = 0 . σ ( w ) = 0 . β by 26.1%, 55.1%, and 67.0%, re-spectively. IV. CONCLUSION
In this work, we investigate the constraint capabili-ties of the future 21 cm IM experiments for the inter-acting dark energy model. We consider BINGO, FAST,SKA1-MID, and Tianlai as typical examples of 21 cm IMexperiments, and find that among them a compact inter-ferometer array like the full-scale Tianlai cylinder arraywould be the best one in constraining the interacting darkenergy model.We find that the 21 cm observations with the full-scale w Planck+BINGOPlanck+FASTPlanck+SKA1Planck+Tianlai
FIG. 5. Constraints on β and w from Planck+BINGO,Planck+FAST, Planck+SKA1-MID, and Planck+Tianlai inthe I w CDM model.
Tianlai cylinder array can tightly constrain H , Ω m , and σ . For example, in the IΛCDM model, the Tianlai-alonedata can give the constraint accuracies of σ ( H ) = 0 . − Mpc − , σ (Ω m ) = 0 . σ ( σ ) = 0 . β asmuch as constraints on other cosmological parameters.However, it is also found that the parameter degeneracydirections from Planck and Tianlai are entirely different,and thus the combination of Planck and Tianlai can wellbreak the parameter degeneracies and give a rather tightconstraint on β . In the IΛCDM and I w CDM models, weobtain σ ( β ) = 5 . × − and 5 . × − , respectively,from Planck+Tianlai. This shows that the constraints on β can be improved by 78.3% and 67.0% in the two mod-els by adding the Tianlai data in the cosmological fit,compared with the case of using only the Planck data.We also make a detailed comparison for BINGO,FAST, SKA1-MID, and Tianlai in the cosmological-fitstudy of the interacting dark energy model. We findthat, for the constraint capability, Tianlai is the bestone, and SKA1-MID is slightly less powerful than Tian-lai, but both are much better than FAST and BINGO.Our goal is not to show the superiority or inferiority ofthese experiments against each other, but to give a globalpicture on their relative prospects. Our results show thatthe 21 cm IM experiments will provide a promising toolfor exploring the nature of dark energy, and in particulara compact interferometer array will play a significant rolein measuring the coupling between dark energy and darkmatter. ACKNOWLEDGMENTS
We would like to thank Xin Wang, Xuelei Chen, Ling-Feng Wang, Li-Yang Gao, Peng-Ju Wu, and Yue Shaofor helpful discussions. This work was supported by theMoST-BRICS Flagship Project No. 2018YFE0120800,the National Natural Science Foundation of China(Grant Nos. 11975072, 11875102, 11835009, 11690021, 11973047, and 11633004), National SKA Program ofChina No. 2020SKA0110401, the Chinese Academyof Sciences (CAS) Strategic Priority Research ProgramXDA15020200, the Liaoning Revitalization Talents Pro-gram (Grant No. XLYC1905011), the Fundamental Re-search Funds for the Central Universities (Grant No.N2005030), and the Top-Notch Young Talents Programof China (Grant No. W02070050). [1] Adam G. Riess et al. Observational evidence from su-pernovae for an accelerating universe and a cosmologicalconstant.
Astron. J. , 116:1009–1038, 1998.[2] Chris Blake and Karl Glazebrook. Probing dark energyusing baryonic oscillations in the galaxy power spec-trum as a cosmological ruler.
Astrophys. J. , 594:665–673, 2003.[3] Hee-Jong Seo and Daniel J. Eisenstein. Probing darkenergy with baryonic acoustic oscillations from futurelarge galaxy redshift surveys.
Astrophys. J. , 598:720–740, 2003.[4] Tzu-Ching Chang, Ue-Li Pen, Jeffrey B. Peterson, andPatrick McDonald. Baryon Acoustic Oscillation Inten-sity Mapping as a Test of Dark Energy.
Phys. Rev. Lett. ,100:091303, 2008.[5] Richard A. Battye, Rod D. Davies, and Jochen Weller.Neutral hydrogen surveys for high redshift galaxy clus-ters and proto-clusters.
Mon. Not. Roy. Astron. Soc. ,355:1339–1347, 2004.[6] R.A. Battye, I.W.A. Browne, C. Dickinson, G. Heron,B. Maffei, and A. Pourtsidou. HI intensity mapping :a single dish approach.
Mon. Not. Roy. Astron. Soc. ,434:1239–1256, 2013.[7] Matthew McQuinn, Oliver Zahn, Matias Zaldarriaga,Lars Hernquist, and Steven R. Furlanetto. Cosmologicalparameter estimation using 21 cm radiation from theepoch of reionization.
Astrophys. J. , 653:815–830, 2006.[8] Abraham Loeb and Stuart Wyithe. Precise Measure-ment of the Cosmological Power Spectrum With a Dedi-cated 21cm Survey After Reionization.
Phys. Rev. Lett. ,100:161301, 2008.[9] Jonathan R. Pritchard and Abraham Loeb. Evolutionof the 21 cm signal throughout cosmic history.
Phys.Rev. D , 78:103511, 2008.[10] Stuart Wyithe and Abraham Loeb. Fluctuations in21cm Emission After Reionization.
Mon. Not. Roy. As-tron. Soc. , 383:606, 2008.[11] Yi Mao, Max Tegmark, Matthew McQuinn, Matias Zal-darriaga, and Oliver Zahn. How accurately can 21cm tomography constrain cosmology?
Phys. Rev. D ,78:023529, 2008.[12] Stuart Wyithe, Abraham Loeb, and Paul Geil. Bary-onic Acoustic Oscillations in 21cm Emission: A Probeof Dark Energy out to High Redshifts.
Mon. Not. Roy.Astron. Soc. , 383:1195, 2008.[13] J.S. Bagla, Nishikanta Khandai, and Kanan K. Datta.HI as a Probe of the Large Scale Structure in the Post-Reionization Universe.
Mon. Not. Roy. Astron. Soc. ,407:567, 2010.[14] Hee-Jong Seo, Scott Dodelson, John Marriner, DaveMcginnis, Albert Stebbins, Chris Stoughton, and Al- berto Vallinotto. A ground-based 21cm Baryon acousticoscillation survey.
Astrophys. J. , 721:164–173, 2010.[15] Adam Lidz, Steven R. Furlanetto, S.Peng Oh,James Aguirre, Tzu-Ching Chang, Olivier Dore, andJonathan R. Pritchard. Intensity Mapping with Car-bon Monoxide Emission Lines and the Redshifted 21cm Line.
Astrophys. J. , 741:70, 2011.[16] R. Ansari, J.E. Campagne, P. Colom, J.M.Le Goff,C. Magneville, J.M. Martin, M. Moniez, J. Rich, andC. Yeche. 21 cm observation of LSS at z ∼ Astron.Astrophys. , 540:A129, 2012.[17] Philip Bull, Pedro G. Ferreira, Prina Patel, andMario G. Santos. Late-time cosmology with 21cm in-tensity mapping experiments.
Astrophys. J. , 803(1):21,2015.[18] Yidong Xu, Xin Wang, and Xuelei Chen. Forecasts onthe Dark Energy and Primordial Non-Gaussianity Ob-servations with the Tianlai Cylinder Array.
Astrophys.J. , 798(1):40, 2015.[19] Alkistis Pourtsidou, David Bacon, and Robert Critten-den. HI and cosmological constraints from intensitymapping, optical and CMB surveys.
Mon. Not. Roy.Astron. Soc. , 470(4):4251–4260, 2017.[20] Elimboto Yohana, Yi-Chao Li, and Yin-Zhe Ma. Fore-casts of cosmological constraints from HI intensity map-ping with FAST, BINGO \ & SKA-I. 8 2019.[21] Denis Tramonte and Yin-Zhe Ma. The neutral hydrogendistribution in large-scale haloes from 21-cm intensitymaps. Mon. Not. Roy. Astron. Soc. , 498(4):5916–5935,2020.[22] Phil Bull, Stefano Camera, Alvise Raccanelli, ChrisBlake, Pedro Ferreira, Mario Santos, and Dominik J.Schwarz. Measuring baryon acoustic oscillations withfuture SKA surveys.
PoS , AASKA14:024, 2015.[23] Philip Bull. Extending cosmological tests of GeneralRelativity with the Square Kilometre Array.
Astrophys.J. , 817(1):26, 2016.[24] Robert Braun, Tyler Bourke, James A Green, EvanKeane, and Jeff Wagg. Advancing Astrophysics withthe Square Kilometre Array.
PoS , AASKA14:174, 2015.[25] YiDong Xu and Xin Zhang. Cosmological parametermeasurement and neutral hydrogen 21 cm sky surveywith the Square Kilometre Array.
Sci. China Phys.Mech. Astron. , 63(7):270431, 2020.[26] Jing-Fei Zhang, Li-Yang Gao, Dong-Ze He, and XinZhang. Improving cosmological parameter estimationwith the future 21 cm observation from SKA.
Phys.Lett. B , 799:135064, 2019.[27] Jing-Fei Zhang, Bo Wang, and Xin Zhang. Forecast forweighing neutrinos in cosmology with SKA.
Sci. China Phys. Mech. Astron. , 63(8):280411, 2020.[28] Denis Tramonte, Yin-Zhe Ma, Yi-Chao Li, and ListerStaveley-Smith. Searching for H I imprints in cosmicweb filaments with 21-cm intensity mapping.
Mon. Not.Roy. Astron. Soc. , 489(1):385–400, 2019.[29] Tzu-Ching Chang, Ue-Li Pen, Kevin Bandura, and Jef-frey B. Peterson. Hydrogen 21-cm Intensity Mapping atredshift 0.8.
Nature , 466:463–465, 2010.[30] K.W. Masui et al. Measurement of 21 cm brightnessfluctuations at z ∼ . Astrophys.J. Lett. , 763:L20, 2013.[31] C.J. Anderson et al. Low-amplitude clustering inlow-redshift 21-cm intensity maps cross-correlated with2dF galaxy densities.
Mon. Not. Roy. Astron. Soc. ,476(3):3382–3392, 2018.[32] Clive Dickinson. BINGO - A novel method to detectBAOs using a total-power radio telescope. In , pages 139–142, 2014.[33] C.A. Wuensche. The BINGO telescope: a new instru-ment exploring the new 21 cm cosmology window.
J.Phys. Conf. Ser. , 1269(1):012002, 2019.[34] C.A. Wuensche et al. Baryon acoustic oscillations fromIntegrated Neutral Gas Observations: Broadband cor-rugated horn construction and testing.
Exper. Astron. ,50(1):125–144, 2020.[35] Rendong Nan, Di Li, Chengjin Jin, Qiming Wang,Lichun Zhu, Wenbai Zhu, Haiyan Zhang, Youling Yue,and Lei Qian. The Five-Hundred-Meter ApertureSpherical Radio Telescope (FAST) Project.
Int. J. Mod.Phys. D , 20:989–1024, 2011.[36] George F. Smoot and Ivan Debono. 21 cm intensitymapping with the Five hundred metre Aperture Spher-ical Telescope.
Astron. Astrophys. , 597:A136, 2017.[37] Marie-Anne Bigot-Sazy, Yin-Zhe Ma, Richard A. Bat-tye, Ian W. A. Browne, Tianyue Chen, Clive Dickinson,Stuart Harper, Bruno Maffei, Lucas C. Olivari, and Pe-ter N. Wilkinson. HI intensity mapping with FAST.
ASP Conf. Ser. , 502:41, 2016.[38] Di Li, Rendong Nan, and Zhichen Pan. The Five-hundred-meter Aperture Spherical radio Telescopeproject and its early science opportunities.
IAU Symp. ,291:325–330, 2013.[39] Hao-Ran Yu, Ue-Li Pen, Tong-Jie Zhang, Di Li, andXuelei Chen. Blind search for 21-cm absorption systemsusing a new generation of Chinese radio telescopes.
Res.Astron. Astrophys. , 17(6):049, 2017.[40] Wenkai Hu, Xin Wang, Fengquan Wu, Yougang Wang,Pengjie Zhang, and Xuelei Chen. Forecast for FAST:from Galaxies Survey to Intensity Mapping.
Mon. Not.Roy. Astron. Soc. , 493(4):5854–5870, 2020.[41] Philip Bull, Stefano Camera, Alvise Raccanelli, ChrisBlake, Pedro G. Ferreira, Mario G. Santos, and Do-minik J. Schwarz. Measuring baryon acoustic oscilla-tions with future SKA surveys. 1 2015.[42] Mario G. Santos et al. Cosmology from a SKA HI in-tensity mapping survey.
PoS , AASKA14:019, 2015.[43] Robert Braun, Anna Bonaldi, Tyler Bourke, EvanKeane, and Jeff Wagg. Anticipated Performance of theSquare Kilometre Array – Phase 1 (SKA1). 12 2019.[44] Xuelei Chen. Radio detection of dark energy—the Tian-lai project.
Scientia Sinica Physica, Mechanica & As-tronomica , 41(12):1358, January 2011.[45] Xuelei Chen. The Tianlai Project: a 21CM Cosmol-ogy Experiment. In
International Journal of Modern Physics Conference Series , volume 12 of
InternationalJournal of Modern Physics Conference Series , pages256–263, March 2012.[46] Fengquan Wu, Yougang Wang, Juyong Zhang, Huli Shi,and Xuelei Chen. Tianlai: a 21cm radio telescope arrayfor BAO and dark energy, status and progress. In , pages 315–318.ARISF, 2016.[47] JiXia Li, ShiFan Zuo, FengQuan Wu, YouGang Wang,JuYong Zhang, ShiJie Sun, YiDong Xu, ZiJie Yu, RezaAnsari, YiChao Li, Albert Stebbins, Peter Timbie, Yan-Ping Cong, JingChao Geng, Jie Hao, QiZhi Huang,JianBin Li, Rui Li, DongHao Liu, YingFeng Liu, TaoLiu, John P. Marriner, ChenHui Niu, Ue-Li Pen, Jef-fery B. Peterson, HuLi Shi, Lin Shu, YaFang Song, Hai-Jun Tian, GuiSong Wang, QunXiong Wang, RongLiWang, WeiXia Wang, Xin Wang, KaiFeng Yu, JiaoZhang, BoQin Zhu, JiaLu Zhu, and XueLei Chen. TheTianlai Cylinder Pathfinder array: System functionsand basic performance analysis.
Science China Physics,Mechanics, and Astronomy , 63(12):129862, September2020.[48] Fengquan Wu, Jixia Li, Shifan Zuo, Xuelei Chen, San-tanu Das, John P. Marriner, Trevor M. Oxholm, AnhPhan, Albert Stebbins, Peter T. Timbie, Reza Ansari,Jean-Eric Campagne, Zhiping Chen, Yanping Cong,Qizhi Huang, Yichao Li, Tao Liu, Yingfeng Liu, Chen-hui Niu, Calvin Osinga, Olivier Perdereau, Jeffrey B.Peterson, Huli Shi, Gage Siebert, Shijie Sun, Hai-jun Tian, Gregory S. Tucker, Qunxiong Wang, RongliWang, Yougang Wang, Yanlin Wu, Yidong Xu, KaifengYu, Zijie Yu, Jiao Zhang, Juyong Zhang, and Jialu Zhu.The Tianlai Dish Pathfinder Array: design, operationand performance of a prototype transit radio interfer-ometer. arXiv e-prints , page arXiv:2011.05946, Novem-ber 2020.[49] B. Wang, E. Abdalla, F. Atrio-Barandela, andD. Pavon. Dark Matter and Dark Energy Interac-tions: Theoretical Challenges, Cosmological Implica-tions and Observational Signatures.
Rept. Prog. Phys. ,79(9):096901, 2016.[50] Luca Amendola. Coupled quintessence.
Phys. Rev. D ,62:043511, 2000.[51] Xin Zhang. Coupled quintessence in a power-law caseand the cosmic coincidence problem.
Mod. Phys. Lett.A , 20:2575, 2005.[52] Xin Zhang. Statefinder diagnostic for coupledquintessence.
Phys. Lett. B , 611:1–7, 2005.[53] Xin Zhang, Feng-Quan Wu, and Jingfei Zhang. A Newgeneralized Chaplygin gas as a scheme for unification ofdark energy and dark matter.
JCAP , 01:003, 2006.[54] John D. Barrow and T. Clifton. Cosmologies with en-ergy exchange.
Phys. Rev. D , 73:103520, 2006.[55] Jingfei Zhang, Hongya Liu, and Xin Zhang. Statefinderdiagnosis for the interacting model of holographic darkenergy.
Phys. Lett. B , 659:26–33, 2008.[56] Jian-Hua He and Bin Wang. Effects of the interactionbetween dark energy and dark matter on cosmologicalparameters.
JCAP , 06:010, 2008.[57] Jian-Hua He, Bin Wang, and Y.P. Jing. Effects of darksectors’ mutual interaction on the growth of structures.
JCAP , 07:030, 2009.[58] Jussi Valiviita, Roy Maartens, and Elisabetta Ma-jerotto. Observational constraints on an interacting dark energy model. Mon. Not. Roy. Astron. Soc. ,402:2355–2368, 2010.[59] Christian G. Boehmer, Gabriela Caldera-Cabral, RuthLazkoz, and Roy Maartens. Dynamics of dark en-ergy with a coupling to dark matter.
Phys. Rev. D ,78:023505, 2008.[60] Jik-Su Kim, Chol-Jun Kim, Sin Chol Hwang, andYong Hae Ko. Scalar - Tensor gravity with scalar -matter direct coupling and its cosmological probe.
Phys.Rev. D , 96(4):043507, 2017.[61] Jun-Qing Xia. Constraint on coupled dark energy mod-els from observations.
Phys. Rev. D , 80:103514, 2009.[62] Hao Wei. Cosmological Constraints on the Sign-Changeable Interactions.
Commun. Theor. Phys. ,56:972–980, 2011.[63] Yun-He Li and Xin Zhang. Running coupling: Does thecoupling between dark energy and dark matter changesign during the cosmological evolution?
Eur. Phys. J.C , 71:1700, 2011.[64] Timothy Clemson, Kazuya Koyama, Gong-Bo Zhao,Roy Maartens, and Jussi Valiviita. Interacting DarkEnergy – constraints and degeneracies.
Phys. Rev. D ,85:043007, 2012.[65] Yun-He Li and Xin Zhang. Large-scale stable interact-ing dark energy model: Cosmological perturbations andobservational constraints.
Phys. Rev. D , 89(8):083009,2014.[66] Yun-He Li, Jing-Fei Zhang, and Xin Zhang.Parametrized Post-Friedmann Framework for In-teracting Dark Energy.
Phys. Rev. D , 90(6):063005,2014.[67] Yun-He Li, Jing-Fei Zhang, and Xin Zhang. Explor-ing the full parameter space for an interacting dark en-ergy model with recent observations including redshift-space distortions: Application of the parametrized post-Friedmann approach.
Phys. Rev. D , 90(12):123007,2014.[68] Xin Zhang. Probing the interaction between darkenergy and dark matter with the parametrized post-Friedmann approach.
Sci. China Phys. Mech. Astron. ,60(5):050431, 2017.[69] Jing-Lei Cui, Lu Yin, Ling-Feng Wang, Yun-He Li, andXin Zhang. A closer look at interacting dark energywith statefinder hierarchy and growth rate of structure.
JCAP , 09:024, 2015.[70] Shuang Wang, Yong-Zhen Wang, Jia-Jia Geng, and XinZhang. Effects of time-varying β in SNLS3 on constrain-ing interacting dark energy models. Eur. Phys. J. C ,74(11):3148, 2014.[71] Jia-Jia Geng, Yun-He Li, Jing-Fei Zhang, and XinZhang. Redshift drift exploration for interacting darkenergy.
Eur. Phys. J. C , 75(8):356, 2015.[72] Jussi V¨aliviita and Elina Palmgren. Distinguishing in-teracting dark energy from wCDM with CMB, lensing,and baryon acoustic oscillation data.
JCAP , 07:015,2015.[73] Eleonora Di Valentino, Alessandro Melchiorri, OlgaMena, and Sunny Vagnozzi. Nonminimal dark sec-tor physics and cosmological tensions.
Phys. Rev. D ,101(6):063502, 2020.[74] Rui-Yun Guo, Yun-He Li, Jing-Fei Zhang, and XinZhang. Weighing neutrinos in the scenario of vacuumenergy interacting with cold dark matter: applicationof the parameterized post-Friedmann approach.
JCAP , 05:040, 2017.[75] Rui-Yun Guo, Jing-Fei Zhang, and Xin Zhang. Explor-ing neutrino mass and mass hierarchy in the scenario ofvacuum energy interacting with cold dark matte.
Chin.Phys. C , 42(9):095103, 2018.[76] Rui-Yun Guo, Jing-Fei Zhang, and Xin Zhang. Can the H tension be resolved in extensions to ΛCDM cosmol-ogy? JCAP , 02:054, 2019.[77] Weiqiang Yang, Supriya Pan, Eleonora Di Valentino,Rafael C. Nunes, Sunny Vagnozzi, and David F. Mota.Tale of stable interacting dark energy, observational sig-natures, and the H tension. JCAP , 09:019, 2018.[78] Lu Feng, Jing-Fei Zhang, and Xin Zhang. Search forsterile neutrinos in a universe of vacuum energy interact-ing with cold dark matter.
Phys. Dark Univ. , 23:100261,2019.[79] Lu Feng, Hai-Li Li, Jing-Fei Zhang, and Xin Zhang. Ex-ploring neutrino mass and mass hierarchy in interactingdark energy models.
Sci. China Phys. Mech. Astron. ,63(2):220401, 2020.[80] Hai-Li Li, Lu Feng, Jing-Fei Zhang, and Xin Zhang.Models of vacuum energy interacting with cold darkmatter: Constraints and comparison.
Sci. China Phys.Mech. Astron. , 62(12):120411, 2019.[81] Hai-Li Li, Dong-Ze He, Jing-Fei Zhang, and Xin Zhang.Quantifying the impacts of future gravitational-wavedata on constraining interacting dark energy.
JCAP ,06:038, 2020.[82] Eleonora Di Valentino, Alessandro Melchiorri, OlgaMena, and Sunny Vagnozzi. Interacting dark energyin the early 2020s: A promising solution to the H andcosmic shear tensions. Phys. Dark Univ. , 30:100666,2020.[83] MingMing Zhao, RuiYun Guo, DongZe He, JingFeiZhang, and Xin Zhang. Dark energy versus modifiedgravity: Impacts on measuring neutrino mass.
Sci.China Phys. Mech. Astron. , 63(3):230412, 2020.[84] Hai-Li Li, Jing-Fei Zhang, and Xin Zhang. Constraintson neutrino mass in the scenario of vacuum energy inter-acting with cold dark matter after Planck 2018.
Com-mun. Theor. Phys. , 72(12):125401, 2020.[85] N. Aghanim et al. Planck 2018 results. VI. Cosmologicalparameters.
Astron. Astrophys. , 641:A6, 2020.[86] N. Kaiser. Clustering in real space and in redshift space.
Mon. Not. Roy. Astron. Soc. , 227:1–27, 1987.[87] Antony Lewis, Anthony Challinor, and AnthonyLasenby. Efficient computation of CMB anisotropies inclosed FRW models.
Astrophys. J. , 538:473–476, 2000.[88] David J. Bacon et al. Cosmology with Phase 1 of theSquare Kilometre Array: Red Book 2018: Technicalspecifications and performance forecasts.
Publ. Astron.Soc. Austral. , 37:e007, 2020.[89] Max Tegmark. Measuring cosmological parameters withgalaxy surveys.
Phys. Rev. Lett. , 79:3806–3809, 1997.[90] R.E. Smith, J.A. Peacock, A. Jenkins, S.D.M. White,C.S. Frenk, F.R. Pearce, P.A. Thomas, G. Efstathiou,and H.M.P. Couchmann. Stable clustering, the halomodel and nonlinear cosmological power spectra.
Mon.Not. Roy. Astron. Soc. , 341:1311, 2003.[91] http://tianlai.bao.ac.cn .[92] .[93] https://fast.bao.ac.cn .[94] .[95] Amadeus Witzemann, Philip Bull, Chris Clarkson, Mario G. Santos, Marta Spinelli, and Amanda Welt-man. Model-independent curvature determination with21 cm intensity mapping experiments.
Mon. Not. Roy.Astron. Soc. , 477(1):L122–L127, 2018.[96] Ashley J. Ross, Lado Samushia, Cullan Howlett, Will J.Percival, Angela Burden, and Marc Manera. The clus-tering of the SDSS DR7 main Galaxy sample – I. A 4per cent distance measure at z = 0 . Mon. Not. Roy.Astron. Soc. , 449(1):835–847, 2015.[97] Florian Beutler, Chris Blake, Matthew Colless, D.HeathJones, Lister Staveley-Smith, Lachlan Campbell,Quentin Parker, Will Saunders, and Fred Watson. The6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant.
Mon. Not. Roy. Astron.Soc. , 416:3017–3032, 2011.[98] Shadab Alam et al. The clustering of galaxies in thecompleted SDSS-III Baryon Oscillation SpectroscopicSurvey: cosmological analysis of the DR12 galaxy sam-ple.
Mon. Not. Roy. Astron. Soc. , 470(3):2617–2652,2017.[99] Wenjuan Fang, Wayne Hu, and Antony Lewis. Cross-ing the Phantom Divide with Parameterized Post-Friedmann Dark Energy.
Phys. Rev. D , 78:087303, 2008.[100] Antony Lewis and Sarah Bridle. Cosmological param-eters from CMB and other data: A Monte Carlo ap-proach.