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MNRAS 000, 1–11 (0000) Preprint 3 August 2016 Compiled using MNRAS LATEX style file v3.0

Constraining the EoR model parameters with the 21cm

bispectrum

Hayato Shimabukuro1,2,3, Shintaro Yoshiura2, Keitaro Takahashi 2, Shuichiro Yokoyama5

and Kiyotomo Ichiki1,41Department of Physics, Graduate School of Science, Nagoya University, Aichi, 464-8602, Japan2Department of Physics, Kumamoto University, Kumamoto, Japan3Observatoire de Paris, LERMA, Paris, France4Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya, 464-8602, Japan5Department of Physics, Rikkyo University, Tokyo, Japan

3 August 2016

ABSTRACT

We perform a Fisher analysis to estimate expected constraints on the Epoch of Reion-ization (EoR) model parameters (minimum virial temperature, ionizing efficiency,mean free path of ionizing photons) considering with thermal noise of ongoing tele-scopes, MWA and LOFAR. We consider how the inclusion of the 21cm bispectrumimproves the constraints compared with the power spectrum alone. With assumptionthat we perfectly remove foreground, we found that the bispectrum, which is calcu-lated by 21cmFAST, can constrain the EoR model parameters more tightly than thepower spectrum since the bispectrum is more sensitive to the EoR model parame-ters than the power spectrum. We also found that degeneracy among the EoR modelparameters can be broken by combining the bispectrum with the power spectrum.

Key words: cosmology: theory — intergalactic medium — Epoch of Reionization— 21cm line

RUP-16-23

1 INTRODUCTION

Since the cosmic recombination, there had been no lu-minous objects. This epoch is often called “Dark Ages(DA)”. The Dark Ages end by the formation of first lumi-nous objects because these objects illuminate the Universe(Yoshida et al. 2006; Fialkov et al. 2013; Visbal et al. 2012;Santos et al. 2011) and this epoch is called “Cosmic Dawn(CD)”. According to the standard hierarchical structure for-mation model based on ΛCDM cosmology, massive objectssuch as galaxies formed after the formation of smaller ob-jects. These first generation objects play an important roleon both thermal and ionization histories of the Intergalacticmedium (IGM) (Mesinger et al. 2012; Christian et al. 2013;Fialkov et al. 2014; Yoshiura et al. 2016b). As structure for-mation proceeds more, ionizing photons emitted by galax-ies result in ionization of IGM. This transition of stateof hydrogen in IGM is known as cosmic reionization. Theepoch when cosmic reioization occurred in the universe ev-erywhere is called “Epoch of Reionization”. Some recentobservations provide us with the fruitful information onthe EoR. For example, high-z QSO absorption lines im-

printed in their spectra tell us that reionization finishedby z ∼ 6(e.g. (Fan et al. 2006)), and rapid evolution of Ly-α luminosity function with redshift constrains the neutralhydrogen fraction at z > 6 (Konno et al. 2014). Further-more, the optical depth of Thomson scattering is measuredand the value obtained by PLANCK is τe = 0.066 ± 0.016(Planck Collaboration 2015).

The redshifted 21cm line signal, which is emissiondue to the hyperfine structure in a neutral hydrogenatom, is expected to be a promising tool to investi-gate matter density fluctuations, ionization state andthe spin temperature at the EoR (Furlanetto et al. 2006;Pritchard & Loeb 2012) directly. Although the cosmologi-cal 21cm signal has not been observed yet, improvementsof current instruments and foreground removal methodswill push observational 21cm cosmology into a new era innear future. In addition, on-going projects such as MWA(Tingay et al. 2012), LOFAR (Rottgering 2003) and PA-PER (Jacobs et al.2015) have potential to detect the 21cmsignal statistically. Furthermore, future instrument SKA isthought to be able to detect the 21cm power spectrum athigher redshift beyond the EoR and to map the brightnesstemperature (Mesinger et al. 2013; Pritchard et al. 2015;Mesinger et al. 2015; Hasegawa et al. 2016).

One of the fundamental statistical values for the

c© 0000 The Authors

2 H.Shimabukuro et al.

21cm signal is the 21cm power spectrum. If the 21cmsignal follows gaussian distribution, we can extractall of the statistical information from the power spec-trum alone. However, we expect that 21cm signal hasnon-gaussian features on its distribution because ofastrophysical effects such as X-ray heating and reion-ization (Wyithe & Morales 2007; Barkana & Loeb 2008;Ichikawa et al. 2010; Shimabukuro et al. 2016). In orderto estimate the non-gaussian features of the 21cm signal,various statistical quantitities are suggested such as the bis-pectrum, one-point statistics, and the Minkowski functionals(Watkinson & Pritchard 2013; Shimabukuro et al. 2015;Watkinson & Pritchard 2015; Shimabukuro et al. 2016;Yoshiura et al. 2016a). In our previous works, we studiedthe properties of the 21cm bispectrum and its detectability(Yoshiura et al. 2015; Shimabukuro et al. 2016). Yoshiuraet al calculated the sensitivity of the 21cm bispectrum by es-timating the contribution from thermal noise. They showedthat the 21cm bispectrum at the EoR can be detectable byMWA and LOFAR at large scale k . 0.3Mpc−1 and alsodetectable at k . 0.7 with SKA (Yoshiura et al. 2015). Asobservational techniques are developed, we expect to getmuch more information on the EoR from both combinationof the 21cm power spectrum and bispectrum. It is impera-tive to estimate the parameters of EoR models if we succeedto detect cosmic 21cm signal. Some previous works performforecast for the EoR parameters by using Markov ChainMonte Carlo (MCMC) method or Fisher forecast applying tothe 21cm power spectrum and on the global 21cm signal byFisher forecast (Pober et al 2014; Greig & Mesinger 2015;Liu et al. 2015; Mirocha et al. 2015; Harker et al. 2015).Furthermore, our previous work performed Fisher analysisfor the variance and skewness of the brightness temperature(Kubota et al. 2016).

In this paper, we consider the forecast of the param-eter constraint by using Fisher analysis for the on-goingobservations of the 21cm signal. We focus on MWA andLOFAR as on-going observations. Previous work performsFisher analysis for the 21cm power spectrum to constrainEoR model parameters with MWA and LOFAR as supposedtelescopes(Pober et al 2014). It obtains 10−20% of 1σ errorsfor fiducial value of the EoR model parameters. In our work,we estimate how these constraints for the EoR parametersare improved by including the 21cm bispectrum.

In this paper, we employ the best fit valuesof the standard cosmological parameters obtained in(Komatsu et al. 2010).

2 FORMULATION AND SET UP

2.1 Formulation for the 21cm bispectrum

A fundamental quantity of 21cm signal is the brightness tem-perature, which is described as the spin temperature offset-ting from CMB temperature, which is given by (see, e.g,

(Furlanetto et al. 2006))

δTb(ν) =TS − Tγ

1 + z(1− e−τν0 )

∼ 27xH(1 + δm)

(

H

dvr/dr +H

)(

1− Tγ

TS

)

×(

1 + z

10

0.15

Ωmh2

)1/2(Ωbh

2

0.023

)

[mK]. (1)

Here, TS and Tγ respectively represent gas spin temperatureand CMB temperature, τν0 is the optical depth at the 21cmrest frame frequency ν0 = 1420.4 MHz, xH is neutral fractionof the hydrogen gas, δm(x, z) ≡ ρ/ρ−1 is the evolved matteroverdensity, H(z) is the Hubble parameter and dvr/dr is thecomoving gradient of the gas velocity along the ling of sight.All quantities are evaluated at redshift z = ν0/ν − 1.

Let us focus on the spatial distribution of the bright-ness temperature. The spatial fluctuation of the brightnesstemperature can be defined as

δ21(x) ≡ (δTb(x)− 〈δTb〉)/〈δTb〉, (2)

where 〈δTb〉 is the mean brightness temperature obtainedfrom brightness temperature map and 〈...〉 expresses theensemble average. From this definition, we have the powerspectrum of δ21 defined as

〈δ21(k)δ21(k′

)〉 = (2π)3δ(k+ k′

)P21(k), (3)

If the statistics of the brightness temperature fluctuationsis a pure Gaussian, the statistical information of the bright-ness temperature should be completely characterized by thepower spectrum. The statistics of the brightness tempera-ture fluctuations completely follows that of the density fluc-tuations δm if both of the spin temperature and the neutralfraction are completely homogeneous. However, in the eraof CD and EOR, it is expected that the spin temperatureand the neutral fraction should be spatially inhomogeneousand the statistics of the spatial fluctuations of those quan-tities would be highly non-Gaussian due to the various as-trophysical effects. Accordingly, the statistics of the bright-ness temperature fluctuations would deviate from the pureGaussian and it should be important to investigate the non-Gaussian feature of the brightness temperature fluctuations.Although such a non-Gaussian feature can be investigatedthrough the skewness of the one-point distribution functionsas done in our work (Shimabukuro et al. 2015), the scale-dependent feature has been integrated out in the skewness.On the other hand, the higher order correlation functionsin Fourier space such as a bispectrum and a trispectrumcharacterize the non-Gaussian features and also have thescale-dependent information. Here, in order to see the non-Gaussian feature of the brightness temperature fluctuationsδ21, we focus on the bispectrum of δ21 which is given by

〈δ21(k1)δ21(k2)δ21(k3)〉 = (2π)3δ(k1+k2+k3)B(k1,k2,k3).

(4)

In order to calculate the bispectrum, we have to characterizethe shape of the bispectrum in k-space. In this work, wechoose the equilateral type bispectrum (k1 = k2 = k3 = k)because the equilateral type of bispectrum normalized bywavenumber shows stronger signal than other configurations[see (Shimabukuro et al. 2016)].

MNRAS 000, 1–11 (0000)

Constraining the EoR model parameters with the 21cm bispectrum 3

2.2 Calculation of the 21cm bispectrum

We calculate the bispectrum of the brightness tem-perature fluctuations by making use of 21cmFAST(Mesinger & Furnaletto 2007; Mesinger et al. 2011). Thiscode is based on a semi-numerical model of reionization andthermal history of the IGM, and generates maps of mat-ter density, velocity, spin temperature, ionized fraction andbrightness temperature at the designated redshifts.

We perform simulations in a 200Mpc3 comoving boxwith 3003 grids, which corresponds to 0.66 comoving Mpcresolution or ∼ 14.1 arcsec and 1.19deg2 field of viewat 127 MHz (z = 10), from z = 200 to z = 5adopting the following parameter set, (ζ, ζX , Tvir, Rmfp) =(15, 1056/M⊙, 10

4 K, 30 Mpc). Here, ζ is the ionizing effi-ciency, ζX is the number of X-ray photons emitted by sourceper solar mass, Tvir is the minimum virial temperature of ha-los which produce ionizing photons, and Rmfp is the meanfree path of ionizing photons through the IGM. In our cal-culation, we ignore, for simplicity, the gradient of peculiarvelocity whose contribution to the brightness temperature isrelatively small (a few %) (Chara et al. 2014).

2.3 Parameter dependence of the 21cm

bispectrum

We study the parameter dependence of the 21 cm bispec-trum in order to prepare for the Fisher forecast. We choosethree key parameters as the EoR model parameters. Webriefly summarize the key parameters we choose as follows:

1. ζ, ionizing efficiency; ζ is composed by a numberof parameters related to ionizing photons escaping fromhigh redshift galaxies and given as ζ = fescf∗Nγ/(1 + nrec)(Furlanetto et al. 2006). Here, fesc is the fraction of ionizingphotons escaping from galaxies into the IGM, f∗ is thefraction converted from baryons to stars, Nγ is the numberof ionizing photons per baryon in stars and nrec is therecombination rate per baryon. In our calculation, we adoptζ = 15 as a fiducial value to satisfy observed constraints onthe ionization history.

2. Tvir, minimum virial temperature of halos producing

ionizing photons; Tvir parameterizes the minimum mass ofhalos producing ionizing photons at the EoR. Typically,Tvir is chosen to be 104K corresponding to the temperatureabove which atomic cooling becomes effective. Tvir includesphysics of the high redshift galaxy formation. If there isno radiative feedback, atomic cooling is thought to becomeeffective at Tvir=104K. Hydrogen molecule cooling becomeseffective below this temperature. If stars or star forminggalaxies begin to form in a halo and radiative feedbackby such objects exists, minimum virial temperature isexpected to become higher since radiative feedback suchas the photodissociation of H2 prevents the gas fromcooling (Sobacchi & Mesinger 2013). On the other hand,positive feedback such as enhancement of H2 moleculesdue to increase of electrons pushes the minimum virialtemperature to lower values because cooling becomes moreeffective. We parameterize Tvir as a responsible parameterfor uncertainties in radiative feedback effects discussed

above.

3. Rmfp, maximum mean free path of ionizing photons

; This parameter determines maximum HII bubble size.Physically, the mean free path of ionizing photons aredetermined by the number density and the optical depthof Lyman-limit systems. In our calculation, we chooseRmfp=30[comoving Mpc] as a fiducial value.

We show the parameter dependence of ionization his-tory in Fig.1. We adopt ζ = 15, 20, 25, Tvir = 104, 3×104, 5×104[K] and Rmfp=15,30,60. Fig.1 shows that larger ζ andsmaller Tvir cause reionization earlier. Larger ζ means thatmuch more photons can contribute to ionization of the neu-tral hydrogen gas. This leads to the progress of the EoR.Since lower Tvir corresponds to smaller mass of halo, the for-mation epoch of halos capable to produce ionizing photonsbecomes earlier. These are because larger ζ and smaller Tvir

cause reionization earlier. Furthermore, larger Rmfp causesreionization efficiently. This is because large mean free pathof ionizing photons is capable to make large ionized bub-bles. However, the ionization history does not depend onRmfp at higher redshift before reionization does not progressefficiently. This is because Rmfp affects the epoch after ion-ized bubbles grows to some extent. In Fig.2, we show scaledependence of the bispectrum at each redshift for variousζ, Tvir, Rmap. From 1st column of this figure, we first cansee that larger ζ suppresses the bispectrum as redshift de-creases. This is because higher ζ drives reionization earlierand neutral fraction becomes smaller at whole scale. From2nd column, we can see that the power of the bispectrumis suppressed in the case of smaller Tvir, especially at largerscales. Smaller Tvir easily can form large number of haloeswhich is capable of producing ionizing photons. These haloesproduce large number of ionized bubbles. Therefore, theselarge number of ionized bubbles grow as reionization pro-ceeds and then the 21cm signal coming from these bubblesbecomes small. Thus, the power of the bispectrum corre-sponding to bubble size is suppressed. That’s why the powerof the bispectrum is suppressed at larger scales at z = 7 withsmall Tvir. On the other hand, the effect of Rmfp becomesremarkable at z = 7 although there are slight differenceamong different values of Rmap at z = 8, 9. As we can seefrom Fig.1, the effect of Rmfp appears in the epoch afterneutral hydrogen fraction decreases less than ∼ 0.5(lowerredshift). Therefore, the effect of Rmfp on the bispectrum isalso slight at z = 9, 8 and it appears at z = 7.

3 FISHER INFORMATION MATRIX &

ESTIMATION OF THERMAL NOISE FOR

THE BISPECTRUM

In order to forecast constraints on the EoR model param-eters, we use the Fisher information matrix Fij . Given ob-servational data, the maximum likelihood analysis gives aset of parameters which maximize the likelihood functionL (the probability distribution function for measured dataset as a function of model parameters). The Fisher for-malism assumes that the likelihood function L follows amulti-dimensional Gaussian form in given parameters. Bythe Fisher analysis (Coe. 2009; Verde 2010), we can estimate

MNRAS 000, 1–11 (0000)

4 H.Shimabukuro et al.

0.0

0.2

0.4

0.6

0.8

1.0

7 8 9 10 11 12

x H

z

ζ=15ζ=20ζ=25

0.0

0.2

0.4

0.6

0.8

1.0

7 8 9 10 11 12

x H

z

Tvir=104[K]Tvir=3*104[K]Tvir=5*104[K]

0.0

0.2

0.4

0.6

0.8

1.0

7 8 9 10 11 12

x H

z

Rmfp=15Rmfp=30Rmfp=60

Figure 1. Ionization histories with varying ζ(left), Tvir(middle) and Rmfp(right). We adopt ζ = 15, 20, 25 and Tvir = 104, 3 × 104, 5 ×

104[K], Rmfp = 15, 30, 60 [Mpc].

the expected constraints on the model parameters with sup-posed instruments.

The Fisher matrices for the 21 cm power spectrum andthe 21 cm bispectrum are respectively given by

Fij,PS =N∑

l

(

1

δPN (kl)

)2∂P (kl; ~p)

∂pi

∂P (kl; ~p)

∂pj

∣

∣

∣

∣

~p=~pfid

(5)

Fij,BS =N∑

l

(

1

δBN (kl)

)2∂B(kl; ~p)

∂pi

∂B(kl; ~p)

∂pj

∣

∣

∣

∣

~p=~pfid

(6)

where, ~p is the model parameter vector, ~p = (p1, p2, · · ·),and ~pfid is a set of fiducial model parameters, ~pfid =(p1,fid, p2,fid, · · ·). l expresses the l-th bin of wavenumber.δPN , δBN are thermal noise of power spectrum and bispec-trum, respectively. Note that we need to take the error co-variance into account for precise evaluations of Fisher ma-trix. However previous work shows that the correlation be-tween different modes is weak at large scales (k . 0.6Mpc−1)in the power spectrum(Mondal et al. 2016). Therefore weassume that the correlation between difference scales arenegligible in our study because we are focusing on rel-atively large scales accessible by the MWA and LOFAR(k . 0.3Mpc−1). Likewise, we also assume that the errorcovariance of the bispectrum is small. As we show later,dominant contribution for Fisher matrix comes from largerscales where error covariance becomes small. Therefore, wecan justify the assumption that we neglect the effect of errorcovariance at larger scales. Given the Fisher matrix, we canestimate the expected 1-σ error of i-th parameter:

σpi =√

F−1ii. (7)

Next, we estimate the thermal noises of the power spec-trum and bispectrum. Different from the power spectrumof thermal noise, the ensemble average of the bispectrumof thermal noise is actually zero if thermal noise followsGaussian distribution. However, the variance of the thermal

noise bispectrum is non zero and this variance contributes to21cm bispectrum signal as thermal noise. The thermal noisedue to the variance of the 21cm bispectrum is derived by(Yoshiura et al. 2015). We use the formula of power spec-trum of thermal noise and the variance of thermal noisebispectrum shown in eqs.8, 9 respectively, which are derivedby (McQuinn et al. 2006; Yoshiura et al. 2015).

δPN(k) ≈[

k3

∫ arcsin[min( k∗k

,1)]

arccos[min( yk2π

,1)]

dθ sin θǫ(n(k sin θ))2A3

eB2t20

(2π)2x2yλ6T 4sys

]−1/2

,(8)

δBN (k) =(2π)

5

2

√∆θ2k5/2ǫ

(

x2yλ2

Ae

)(

T 2sysλ

2

AeBt0

)

3

2

×[ ∫

dθ1

∫

dα sin θ1 sin θ2 sin γ(θ1, α)

n(k1)n(k2)n(k3)

]−1

2

, (9)

where k∗ is the longest transverse wavenumber vector corre-sponding to the maximum baseline length. The lower limitof the integral is determined by pixel size. The other quan-tities are wavenumber λ, system temperature Tsys, effectivearea Ae, bandwidth B, integral time t0 and the number den-sity of baselines n. x, y , determined by cosmology, are thequantities which convert uv space to Fourier space ~k. Otherquantities (θ, α) are angles in Fourier space, which are re-quired to calculate spherical averaged noise power spectrumand variance of the thermal noise bispectrum. Please referto (McQuinn et al. 2006; Yoshiura et al. 2015) for detail ex-planation. Here, we use telescope parameters used in table1 of (Yoshiura et al. 2015) (Although they assume a MWA-512T, we assume MWA-256T. Thus, we reduce the numberof antennae to half in this work).

In Figs.3 and 4, we show the scale dependence of thepower spectrum and bispectrum with thermal noise esti-mated with MWA and LOFAR. We assume 1000 hours forthe total observing time. Note that we multiply the square

MNRAS 000, 1–11 (0000)

Constraining the EoR model parameters with the 21cm bispectrum 5

Figure 2. The 21cm bispectrum as function of wavenumber at z = 9 (1st row), 8(2nd row), 7(3rd row) with varying ζ(left), Tvir(middle)and Rmfp(right). We adopt ζ = 15, 20, 25 and Tvir = 104, 3× 104, 5× 104[K], Rmfp = 15, 30, 60 [Mpc].

(cube) of the average brightness temperature for the powerspectrum (bispectrum). From Figs.3 and 4, we can see thatthe noise increases at smaller scales in both bispectrum andpower spectrum. This is because the number of longer base-lines corresponding to smaller scales is deficient. On theother hand, the sensitivity at large scales is limited by thefield of view.

For both bispectrum and power spectrum noises, wecannot calculate the sensitivity at k . 0.03Mpc−1 for theLOFAR telescope. Since we set antenna size as minimumbaseline length, we cannot calculate larger scales correspond-ing to that minimum baseline length. Comparing the sensi-tivity for the power spectrum with that for the bispectrum,the signal to noise ratio in the case of the power spectrumis slightly larger that in the case of bispectrum. If we focusjust on the sensitivity, the 21 cm power spectrum is moredetectable than the bispectrum. However, estimation of theexpected constraint does not depend on only the sensitivitybut also the parameter dependences in the power spectrumor bispectrum.

4 RESULT

We show the result of the Fisher analysis applying to thepower spectrum and the bispectrum. Here, we focus on equi-lateral type of the bispectrum. As we referred before, weconstrain the EoR model parameters assuming ongoing tele-scopes, MWA and LOFAR, to study how the bispectrum im-proves constraint on the EoR parameters. Note that we useboth power spectrum and bispectrum for the Fisher analysisat k=0.03 -1.0 Mpc−1 divided into 9 bins.

We will show the confidence regions of the EoR modelparameters. Note that the confidence regions obtained bythe Fisher analysis include physically meaningless regionssuch as Tvir < 0. Thus, we put on physically meaningfulboundary condition on the parameter space and exclude neg-ative value regions.

First, we show constraints on the EoR model parame-ters obtained by the bispectrum at z=7, 8, 9 in Fig.5. Wecan see that the constraints at z =8 is stronger than otherredshifts. This is because the 21cm bispectrum as function

MNRAS 000, 1–11 (0000)

6 H.Shimabukuro et al.

10-2

10-1

100

101

102

103

104

0.01 0.1 1

k3 P(k

)/2π

2 [mK

2 ]

k[Mpc-1]

z=7

MWALOFAR

signal

10-2

10-1

100

101

102

103

104

0.01 0.1 1

k3 P(k

)/2π

2 [mK

2 ]

k[Mpc-1]

z=8

MWALOFAR

signal

10-1

100

101

102

103

104

0.01 0.1 1

k3 P(k

)/2π

2 [mK

2 ]

k[Mpc-1]

z=9

MWALOFAR

signal

Figure 3. The comparison of 21cm power spectrum signal (short-dashed line) with thermal noise for various telescopes at z=7,8,9. Astelescopes, we choose the MWA(solid line), LOFAR(long-dashed line).

10-2

100

102

104

106

108

1010

0.01 0.1 1

k6 B(k

)/2π

2 [mK

3 ]

k[Mpc-1]

z=7

MWALOFAR

signal

10-2

100

102

104

106

108

1010

1012

0.01 0.1 1

k6 B(k

)/2π

2 [mK

3 ]

k[Mpc-1]

z=8

MWALOFAR

signal

10-2100102104106108

101010121014

0.01 0.1 1

k6 B(k

)/2π

2 [mK

3 ]

k[Mpc-1]

z=9

MWALOFAR

signal

Figure 4. The comparison of 21cm bispectrum signal(short-dashed line) with thermal noise for various telescopes at z=7,8,9. Astelescopes, we choose the MWA(solid line), LOFAR(long-dashed line).

of redshift has peak at z ∼ 8 in our model and thus thebispectrum is sensitive to EoR parameters at z ∼ 8. We canalso see that the constraint assuming LOFAR is tighter thanthat obtained by MWA. This comes from that the sensitiv-ity and resolution of LOFAR are better than those of MWA.

The physical meaning of inclination of ellipse is referredas follows. If ζ and Rmfp become larger, neutral hydrogenatoms are ionized more efficiently. Similarly, decreasing Tvir

also drives progress of reionization. Both increasing ζ, Rmfp

and decreasing Tvir play same role on reionization. Thus,

MNRAS 000, 1–11 (0000)

Constraining the EoR model parameters with the 21cm bispectrum 7

there is a degeneracy between these parameters. However,the direction of degeneracy is different among redshifts. Thisdifference of direction among redshifts enables us to breakdegeneracy.

Next, we compare error constraints obtained by thepower spectrum with that obtained by the bispectrum. Weshow the result in Fig.6 and table. 1. Here, we combine thebispectrum and the power spectrum at z=7, 8, 9. We findthat the constraints by the bispectrum are tighter than thatby the power spectrum. As you can see table. 1, if we usethe bispectrum, each parameter can be determined with afew %− ∼ 20% accuracy for MWA and with ∼ 0.1 − 0.5%accuracy for LOFAR. This accuracy is 1-2 order of magni-tude better than constraints obtained from the power spec-trum. We also find that LOFAR telescope puts on tighterconstraints than MWA.

We can find that the bispectrum can constrain the EoRparameters tighter than the power spectrum although sig-nal to noise ratio of the power spectrum is better than thatof the bispectrum. The reason why the bispectrum can givetight constraint is that derivative of the bispectrum with re-spect to the EoR parameters is much larger than that in thepower spectrum. However, we know that the Fisher matrixcan be determined by not only derivative of signal but alsoby the thermal noise. In order to study the balance betweenthe derivative of the signal and the thermal noise, we showa ratio of square of derivative with respect to virial tem-perature and ionizing efficiency for the bispectrum and the

power spectrum, r =(

∂B∂pi

/ ∂P∂pi

)2

, and a ratio of square of

the thermal noise error for MWA and LOFAR, (δPN/δBN )2,in Fig.7. Both of them are function of wave number. Top ofFig.7 implies that the derivative of the bispectrum with re-spect to both parameters is larger than that of the powerspectrum (r > 1) except for the derivative with respect tovirial temperature at z=7. Consequently, the derivative ofthe bispectrum contributes to the Fisher matrix more thanthat of the power spectrum (large Fisher matrix can puton tighter constraint). The difference among redshifts is re-markable for a ratio of the derivative. In particular, a ratio ofthe derivative is large at z=8,9 although there are slight dif-ferences for a ratio of the thermal noise among redshifts. Onthe other hand, we can see how much the thermal error con-tributes to the Fisher matrix in bottom of Fig.7. Althoughthe signal of the bispectrum is larger than that of the powerspectrum, the thermal noise of the bispectrum is also largerthan that of the power spectrum. Therefore, an advantageof the large value of derivative of the bispectrum is offset bythe thermal noise and the Fisher matrix for the bispectrumbecomes larger than that for the power spectrum by takingboth derivative and thermal noise into account.

5 SUMMARY

In order to explore the EoR parameter region with MWAand LOFAR observations, we estimated expected 1-σ errorsand constrain parameter region by the Fisher analysis withthe 21 cm power spectrum and the bispectrum. First, wefound that we can put tighter constraints on the EoR pa-rameters with LOFAR than with MWA. LOFAR can give1-2 order of magnitude better constraints on the parame-

∆ζfid/ζfid ∆Tvir/Tvir,fid ∆Rmfp/Rmfp,fid

PS, MWA 0.507 0.765 0.305

PS, LOFAR 0.180 0.223 0.0487

BS, MWA 0.035 0.209 0.014

BS, LOFAR 1.57× 10−3 5.03×10−3 9.23×10−4

Table 1. Constraints on ζ, Tvir and Rmfp estimated by the Fisherforecast, these are estimated by the power spectrum and the bis-pectrum at z=7, 8, 9.

ters than MWA. This comes from that the thermal noisefor LOFAR is lower than that for MWA. This means thatthe sensitivity of LOFAR is better than that of MWA. Thedifference of specification between MWA and LOFAR comesfrom effective area and maximum baseline length. Althoughthe number density of antennae in core region for MWAis larger, larger effective area of LOFAR compensates defi-ciency of the less number density of antennae.

Next, we found that expected errors obtained by thebispectrum improve that obtained by power spectrum. Thebispectrum can give constraints on each parameter with∼ 0.1−a few % accuracy although the power spectrum con-strain each parameter with a few× 10 %. This is because thebispectrum is more sensitive to the EoR parameters thanpower spectrum and the thermal noise for the bispectrumis smaller than that for the power spectrum. We also foundthat the combination of the power spectrum with the bispec-trum breaks degeneracy between EoR parameters as shownin Fig.6. In particular, degeneracies in the case of LOFARare broken remarkably. Therefore, we expect that we can ob-tain tight constraints on the EoR parameters by combiningboth bispectrum and power spectrum.

Some recent works forecast not only expectedconstraints on the EoR model parameters butalso the parameters at the epoch of the X-rayheating (Pober et al 2014; Greig & Mesinger 2015;Liu et al. 2015; Mirocha et al. 2015; Harker et al. 2015;Ewall-Wice et al. 2015) based on the Fisher analysis orthe Markov Chain Monte Carlo (MCMC) method for thepower spectrum assuming PAPER. In this work, we showthat expected constraints on the EoR model parameters areimproved by combining power spectrum with bispectrum.However, it should be emphasized that we ignore thesample variance and effect of foreground removal in ouranalysis for simplicity. In both case of the bispectrum andthe power spectrum, the sample variance is thought to beeffective at larger scales and foreground also affects largerscales (Pober et al 2014). Hence, we need to estimate theseeffects adequately to obtain more realistic results. This isour future work.

ACKNOWLEDGEMENT

This work is supported by Grant-in-Aid from the Min-istry of Education, Culture, Sports, Science and Technol-ogy (MEXT) of Japan, Nos. 24340048(K.T. and K.I.),26610048, 15H05896, 16H05999. (K.T.), No. 25-3015(H.S.),16J01585(Yoshiura) and 15K17659,16H01103(S.Y.).

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8 H.Shimabukuro et al.

0

5

10

15

20

25

30

0 10000 20000

ζ

Tvir

z=7,MWAz=8,MWAz=9,MWA

14.6

14.7

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15

15.1

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15.3

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9500 10000 10500

ζ

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22

24

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34

36

5 10 15 20 25

Rm

fp

ζ

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29.8

29.85

29.9

29.95

30

30.05

30.1

30.15

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Rm

fp

ζ

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20

25

30

35

40

0 5000 10000 15000 20000

Rm

fp

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z=7,MWAz=8,MWAz=9,MWA

29.6

29.8

30

30.2

30.4

9500 10000 10500

Rm

fp

Tvir

z=7,LOFARz=8,LOFARz=9,LOFAR

Figure 5. 1-σ contours of the EoR model parameters obtained by the bispectrum assuming MWA(left) and LOFAR(right).

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Constraining the EoR model parameters with the 21cm bispectrum 9

0

5

10

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10 15 20

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fp

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45

5000 10000 15000

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fp

Tvir

MWA,BSMWA,PS

25

30

35

9000 9500 10000 10500 11000

Rm

fp

Tvir

LOFAR,BSLOFAR,PS

Figure 6. 1-σ contours of the EoR model parameters obtained by bispectrum (redshift) and power spectrum(green) assuming MWA(left)and LOFAR(right). Here we use the bispectrum and the power spectrum at z=7, 8, 9.

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10 H.Shimabukuro et al.

10-1100101102103104105106107108

0.1 1

r[m

K2 ]

k[h-1Mpc-1]

Tvirz=7z=8z=9

100101102103104105106107108

0.1 1

r[m

K2 ]

k[h-1Mpc-1]

ζz=7z=8z=9

104

105

106

107

108

109

1010

0.1 1

r[m

K2 ]

k[h-1Mpc-1]

Rmfp

z=7z=8z=9

10-14

10-12

10-10

10-8

10-6

10-4

10-2

0.1 1

(δP N

/δB

N)2 [m

K-2

]

k[h-1Mpc-1]

MWA

z=7z=8z=9

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

0.1 1

(δP N

/δB

N)2 [m

K-2

]

k[h-1Mpc-1]

LOFAR

z=7z=8z=9

Figure 7. (Top) Ratio of square of the derivative of the bispectrum and the power spectrum with respect to virial temperature(left),

ionizing efficiency (middle) and maximum mean free path(left). r =(

∂B∂pi

/ ∂P∂pi

)2. (Bottom) The ratio of square of the thermal noise

error for MWA(left) and LOFAR(right), (δPN/δBN )2

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