Enhanced Detectability of Pre-reionization 21-cm Structure
SS UBMITTED TO A P J L
ETTERS J UNE
30, 2010
Preprint typeset using L A TEX style emulateapj v. 5/25/10
ENHANCED DETECTABILITY OF PRE-REIONIZATION 21-CM STRUCTURE M ARCELO
A. A
LVAREZ , U E -L I P EN , AND T ZU -C HING C HANG
Submitted to ApJ Letters June 30, 2010
ABSTRACTBefore the universe was reionized, it was likely that the spin temperature of intergalactic hydrogen was de-coupled from the CMB by UV radiation from the first stars through the Wouthuysen-Field effect. If the IGMhad not yet been heated above the CMB temperature by that time, then the gas would appear in absorption rela-tive to the CMB. Large, rare sources of X-rays could inject sufficient heat into the neutral IGM, so that δ T b > δ T b (cid:39)
250 mKon arcminute to degree angular scales, an order of magnitude larger in amplitude than that caused by ionizedbubbles during reionization, δ T b (cid:39)
25 mK. This signal could therefore be easier to detect and probe higherredshifts than that due to patchy reionization. For the case in which the first objects to heat the IGM are QSOshosting 10 − M (cid:12) black holes with an abundance exceeding ∼ − at z ∼
15, observations with either theArecibo Observatory or the Five Hundred Meter Aperture Spherical Telescope (FAST) could detect and imagetheir fluctuations at greater than 5- σ significance in about a month of dedicated survey time. Additionally,existing facilities such as MWA and LOFAR could detect the statistical fluctuations arising from a populationof 10 − M (cid:12) black holes with an abundance of ∼ Gpc − at z (cid:39) − Subject headings: cosmology: theory — dark ages, reionization, first stars — intergalactic medium INTRODUCTION
The 21 cm transition is among the most promising probes ofthe high-redshift universe. The observed differential bright-ness temperature from the fully neutral IGM at the mean den-sity with spin temperature T s ( z ) is given by δ T b ( z ) (cid:39)
29 mK (cid:18) + z (cid:19) / (cid:20) − T CMB ( z ) T s ( z ) (cid:21) (1)(e.g., Madau et al. 1997). In the absence of an external radi-ation field, only the hot, dense gas in minihalos would havebeen able to develop a spin temperature different from thatof the CMB at z ∼
20 (Shapiro et al. 2006). However, radi-ation emitted by the earliest generations of stars (e.g., Ciardi& Madau 2003; Barkana & Loeb 2005) and X-ray sources(Chuzhoy et al. 2006; Chen & Miralda-Escudé 2008) couldhave coupled the spin temperature to the IGM temperaturevia the “Wouthuysen-Field effect” (Wouthuysen 1952; Field1959) long before reionizing the universe, at redshifts as highas z = 20 −
30, causing the IGM to appear in absorption withrespect to the CMB, with δ T b (cid:39) −
250 mK.Only a modest amount of heating is necessary to raise thegas temperature above the CMB temperature of ∼ −
60 Kat z ∼ −
20. By the time reionization was well underway, itis generally believed that the neutral component of the IGMhad already been heated to T s (cid:29) T CMB , and therefore the ob-served brightness temperature will be an order of magnitudelower in amplitude, δ T b (cid:39)
25 mK, than when the IGM was inabsorption (e.g., Furlanetto et al. 2006).Clearly, there should be some transition epoch, duringwhich sources of X-ray radiation heated the neutral IGM,creating “holes” in the absorption. Natural candidates forthese early sources of heating are quasars (e.g., Chuzhoy et al.2006; Zaroubi et al. 2007; Thomas & Zaroubi 2008; Chen &Miralda-Escudé 2008). Unfortunately, little is known about
Electronic address: [email protected] Canadian Institute for Theoretical Astrophysics, University ofToronto, 60 St.George St., Toronto, ON M5S 3H8, Canada IAA, Academia Sinica, PO Box 23-141, Taipei, 10617, Taiwan the quasar population at z >
6, with the only constraints com-ing from the bright end at z ∼ ∼ > M (cid:12) , lumi-nosities ∼ > erg s − , and a comoving abundance ∼ > − .Quasars are not the only high-redshift objects able to pro-duce X-rays. High-redshift supernovae (Oh 2001) and X-raybinaries could have also been sources. Theoretical modelstypically parametrize X-ray production associated with starformation by f X , normalized so that f X = 1 corresponds to thatobserved for local starburst galaxies (e.g., Furlanetto 2006;Pritchard & Furlanetto 2007).Our aim in this paper is to explore the enhanced 21-cm sig-nature of early quasars. We will therefore assume that thefirst galaxies were efficient enough to couple the spin temper-ature to the kinetic temperature by z (cid:39)
20, but did not heat theIGM above the CMB temperature, i.e. f X (cid:28)
1. As shown byPritchard & Loeb (2010), even for values of f X > Ω m h , Ω b h , h , n s , σ ) =(0 . , . , . , . , . a r X i v : . [ a s t r o - ph . C O ] J un ALVAREZ, PEN & CHANG FORECAST
Our predictions will focus on accreting black holes withmasses greater than (cid:39) M (cid:12) . It is of course possible that amore abundant population of black holes with lower massesexisted at these early times, left behind as remnants of earlygenerations of massive Pop III stars (Heger et al. 2003), act-ing as “seeds” for the observed z ∼ (cid:39) M (cid:12) in the first halos with virial temperaturegreater than (cid:39) K (e.g., Bromm & Loeb 2003; Begelmanet al. 2006). Formation of black holes by this mechanism mayhave been quite a rare occurence (e.g., Dijkstra et al. 2008). Itis this scenario, in which most accreting black holes in the uni-verse were relatively rare and more massive than (cid:39) M (cid:12) ,that is most consistent with the predictions we make here.Because the heating is a time-dependent effect, we willparametrize the total energy radiated during accretion as E tot = Lt QSO = (cid:15) M BH c , where we take the radiative efficiency (cid:15) =0 .
1. In principle some of the rest mass energy, i.e. the ini-tial seed mass of the black hole, did not contribute to heatingthe surroundings, but in general the seed mass is expected tobe small compared to the mass of the black hole after it un-dergoes its first episode of radiatively-efficient accretion as aquasar, so we neglect it.
Quasar spectral energy distribution
We assume the spectral energy distribution of the quasaris given by a broken power-law, S ν ∝ ν − . at ν < ν b , and S ν ∝ ν − . at ν > ν b , with h ν b = 11 . L , then the spectral energy distribution is S ν = L ν b (cid:26) ( ν/ν b ) − . , ν < ν b , ( ν/ν b ) − . , ν > ν b . (2) The 21-cm profile around individual quasars
The temperature profile surrounding a quasar can be ob-tained by considering the fraction of the radiated energy ab-sorbed per atom at comoving distance r and redshift z , Γ HI ( r , z ) = (1 + z ) π Lr (cid:90) ∞ ν HI d ν S ν σ ν h ν ( h ν − h ν HI ) χ ν exp[ − τ ν ( r )] , (3)where χ ν is the fraction of photoelectron energy, h ν − h ν HI ,which goes into heat, with the rest being lost to secondaryionizations and excitations. In the limit in which the ionizedfraction of the gas is low, 0 . < χ ν < . h ν >
25 eV(Shull & van Steenberg 1985). In what follows we will makethe approximation that S ν = 0 . ν , and assume that theenergy radiated is Lt qso ≡ (cid:15) M BH c , with t qso short compared tothe Hubble time. In this case, the relative brightness tempera-ture is given by δ T b ( M BH , r , z ) = 29 mK (cid:20) − T CMB ( z ) T s ( M BH , r , z ) (cid:21) (cid:20) + z (cid:21) / , (4) F IG . 1.— Top: Profiles of the observed differential brightness temperatureversus comoving distance at three different redshifts around a quasar that hasradiated 10 per cent of the rest-mass energy of 10 , 10 , and 10 M (cid:12) , aslabeled. Bottom: Spin temperature of the IGM, obtained via equation (5). where T s ( M BH , r , z ) = T IGM ( z ) + (cid:15) M BH c k b Γ HI ( r , z ) . (5)In calculating the optical depth, we assume that the IGM iscompletely neutral at the mean density, n H ( z ), so that τ ν ( r , z ) = rn H ( z )(1 + z ) − σ ν . In reality, the quasar’s own H II region willreduce the opacity at small radii, but given that we are con-cerned with the small amount of heating happening at muchlarger radii, we neglect the H II region when calculating theoptical depth.Shown in Fig. 1 are profiles of the observed differentialbrightness temperature versus comoving distance at three dif-ferent redshifts. The total energy radiated by the quasar, Lt QSO in each curve corresponds to 10 per cent of the rest mass, aslabeled. Close to the quasar, the gas is heated above the CMBtemperature, and δ T b (cid:39) −
40 mK. Further away, the heatingfrom the quasar declines due to spherical dilution and attenu-ation of the lowest energy radiation, with δ T b finally reachingabout -220 to -340 mK at large distances. The FWHM size ofthe fluctuations are about 20, 80, and 400 Mpc for black holesof mass 10 , 10 , and 10 M (cid:12) , respectively.RE-REIONIZATION 21-CM STRUCTURE 3 Black hole abundance at high-z
How do the black hole densities and masses we assumehere compare to those observed at z ∼ M (cid:12) < M BH < × M (cid:12) at z = 6, finding it to be well-fitted by dn / d ln M BH (cid:39) φ ∗ ( M BH / M ∗ ) − exp( − M BH / M ∗ ), with φ ∗ = 5 .
34 Gpc − and M ∗ = 2 . × M (cid:12) . Integrating the black hole mass function,one finds ρ BH ( > M BH ) (cid:39) × and 3 × M (cid:12) Gpc − , for M BH = 10 , and 10 M (cid:12) , respectively, somewhat larger thanthe black hole mass density we find which maximizes the 21-cm power spectrum (§3.2). Matching the abundance of blackholes greater than a given mass to the dark matter halo massfunction of Warren et al. (2006) at z = 6 (“abundance match-ing” – e.g., Kravtsov et al. 2004), we obtain M halo = 2 . × and 4 . × M (cid:12) for M BH = 10 and 10 M (cid:12) , respectively,implying a value of M BH / M halo (cid:39) × − to 2 × − overthe same range.More detailed predictions would require extrapolating the M BH − M halo relationship to lower masses and higher redshifts,or making highly uncertain assumptions about the formationmechanism of the high-redshift seeds and their accretion his-tory. For example, the ratio M BH / M halo (cid:39) − we determinehere by abundance matching at z = 6 and M BH = 10 − M (cid:12) would be a significant underestimate in atomic cooling haloswhere black hole formation by direct collapse took place, inwhich it is possible that M BH / M halo could approach the limit-ing value of Ω b / Ω m (cid:39) .
17. Clearly much more work is re-quired in understanding the high-redshift quasar population,and for this reason the constraints provided by either detec-tion or non-detection of the signal we predict here would bevery valuable. DETECTABILITY
In this section, we estimate the detectability of individ-ual sources as well the statistical detection of their powerspectrum. In the case of individual objects, we focus on anovel approach, using single-dish filled aperture telescopeslike Arecibo and FAST , while for the power spectrum wewill simply refer to existing sensitivity estimates for facilitiessuch as LOFAR , MWA , and SKA . Individual quasars
In order to determine the necessary integration time, weconvolve the profiles shown in Fig. 1 with a half-power beamwidth of θ b = 26 (cid:48) (cid:18) + z (cid:19) (cid:18) d dish
300 m (cid:19) − , (6)where d dish is the effective dish diameter. Converting angle onthe sky to comoving distance, we obtain the comoving reso-lution, D (cid:39)
70 Mpc (cid:18) d dish
300 m (cid:19) − (cid:18) + z (cid:19) . . (7)This implies that the fluctuation created by a 10 M (cid:12) -blackhole would be just resolvable with a single 300-m dish like http://fast.bao.ac.cn IG . 2.— Bottom: Peak fluctuation amplitude, measured with respect tothe mean background absorption (cid:104) ∆ T b (cid:105) ( z ), as a function of black hole mass,for three different redshift. Top: Signal-to-noise ratio at the same redshiftsfor a bandwidth corresponding to the beam width and an integration time of10 minutes. The detectability declines rapidly in the interval 10 < z < Arecibo (see Fig. 1), lower mass black holes would be un-resolved point sources, and the profile of higher mass blackholes could actually be measured. For an integration time of ∆ t , the sensitivity is given by δ T err = 22 mK (cid:18) ∆ t
60 s (cid:19) − / (cid:18) d dish
300 m (cid:19) / (cid:18) + z (cid:19) . (8)where we have used ∆ ν = 4 MHz (cid:18) D
70 Mpc (cid:19) (cid:18) + z (cid:19) − / (9)for the bandwidth corresponding to a comoving distance D , δ T err = T sys / √ ∆ ν ∆ t , and T sys (cid:39) × mK[(1 + z ) / . (e.g., Furlanetto et al. 2006).We calculate the maximum fluctuation amplitude as a func-tion of quasar black hole mass and redshift, δ T b , max ( M bh , z ),which is a convolution of the beam with the individual pro-files plotted in Fig. 1, δ T b , max = 2 θ D (cid:90) ∞ d θθ e − θ θ (cid:90) D / dl δ T b ( r l θ ) , (10)where we use a Gaussian profile with a FWHM of θ b ( z ), such ALVAREZ, PEN & CHANG F IG . 3.— Shown is the spherically-averaged three-dimensional power spec-trum of 21-cm fluctuations for a black hole density 1 M (cid:12) / Mpc , for threedifferent redshifts, as labeled. Large, relatively rare 10 M (cid:12) black holes havea power spectrum which peaks at k ∼ .
03 Mpc − , while for M BH = 10 M (cid:12) the power spectrum which peaks at k ∼ . − . Such a signal at z = 10should be easily detectable by LOFAR, MWA, or SKA. that θ b ( z ) = 2 √ θ g ( z ), and r l θ = r θ + l , where r θ is theprojected comoving distance perpendicular to the line of sightcorresponding to the angle θ .Shown in Fig. 2 are the resulting fluctuation ampli-tudes with respect to the mean back ground absorption, δ T b , max ( M bh , z ) − (cid:104) δ T b (cid:105) ( z ), as well as the signal-to-noise ratio,[ δ T b , max ( M bh , z ) − (cid:104) δ T b (cid:105) ( z )] /δ T b , err ( z ) for an integration time of10 minutes. As can be seen from the figure both the signal,and to a greater extent the signal-to-noise, decline rapidly withincreasing redshift. This is due to several reasons. First, theintrinsic signal declines with increasing redshift because ab-sorption is relatively weaker at higher redshifts when the IGMhas not had as much time to cool (see Fig. 1). In addition, thebeamwidth gets larger, in proportion to 1 + z , while the angu-lar size of the fluctuations at fixed black hole mass actuallydecline with redshift, as seen from Fig. 1. Power spectrum from Poisson fluctuations
To estimate the fluctuating background from a superposi-tion of sources, we assume all black holes at a given red-shift have the same mass, M BH , and comoving number den-sity, n BH . We then generate a realization of random blackhole positions within a periodic box, and calculate the heat-ing from each source using equation (5). Because the heatingrate is proportional to mass and we assume a random spa-tial distribution, the two length scales in the system are themean separation of sources, r sep ∝ n − / , and the distance outto which an individual black hole can effectively heat the IGMabove the CMB temperature, r heat ∝ M / . For this simplifiedcase, in which the spatial distribution of black holes is ran-dom and all black holes have the same mass, the shape of thepower spectrum is determined only by the ratio r heat / r sep ∝ ( n BH M BH ) / = ρ / . Our choice of ρ BH = 2 × M (cid:12) Gpc − is roughly that density at which the signal is maximized; a lowerblack hole mass density results in a lower overall signal, whilea higher density leads to overlap of the individual regions, andthe fluctuations become saturated.Shown in Fig. 3 is our predicted power spectrum for Pois-son fluctuations of black holes of mass M BH = 10 and 10 M (cid:12) at z =10, 15, and 20. Because the black hole mass densityis the same in both cases, the curves have the same shape,but are shifted in wavenumber such that k − ∝ M BH . Be-cause the individual regions are only weakly overlapping for ρ BH = 2 × M (cid:12) Gpc − , the shape of the curve is quite closeto the Fourier transform of an individual region, so that P ( k ) / ∝ (cid:90) ∞ r dr δ T b ( r ) sin krkr , (11)with δ T b ( r ) given by eqs. (4) and (5).The amplitude of this signal at k ∼ . − ([ k P ( k ) / (2 π )] / (cid:39)
50 mK) is almost an order of magni-tude greater than that expected from ionized bubbles when T S (cid:29) T CMB ( (cid:39) z = 12) δ T , err ( k ∼ . − ) (cid:39) for LOFARand MWA, and T , err ( k ∼ . − ) (cid:39) . for SKA, fortheir adopted array configurations and 1000 hr of integrationtime (Fig. 6 and Table 1 of McQuinn et al. 2006). Thus, thepower spectrum shown in Fig. 4 would be easily detectableby either of these three experiments at z ∼
12 for the surveyparameters used by McQuinn et al. (2006). SURVEY STRATEGIES
Since the signal comes from a large angular scale on thesky, a filled aperture maximizes the sensitivity. The twolargest collecting area telescopes are Arecibo and, under con-struction, FAST. They are at latitudes δ = 18 (26) degrees, re-spectively. To stabilize the baselines, a drift scan at 80 MHzmaps a strip of width 40 (90) arc minutes by length 360 de-grees cos δ every day. With Arecibo, a custom feedline withpairwise correlations of dipoles would allow a frequency de-pendent illumination of the mirror, allowing the frequency in-dependent removal of foregrounds. It would also allow op-eration as an interferometer, increasing stability and rejectionof interference, and enabling arbitrary apodization of the sur-face. Should grating sidelobes from support structures be-come a problem, one could also remove the carriage house,and mount the feedline on a pole from the center of the dish.This would result in a clean, unblocked aperture with fre-quency independent beam. Even with the carriage supportblocking, the side lobes would still be frequency independentfor an appropriately scaled illumination pattern.For FAST, a focal plane array also enables a frequency de-pendent illumination of the primary, resulting in a frequencyindependent beam on the sky. It also increases the surveyspeed by the number of receivers used. Only one receiver isneeded every half wave length, roughly two meters. Hundredpixel surveys seem conceivable at low cost, since only smallbandwidths would be needed, and the system temperature issky limited, even with cheap TV amplifiers.RE-REIONIZATION 21-CM STRUCTURE 5 F IG . 4.— Number of black holes expected within a 1-month (upper curves)and 1-day (lower curves) survey in drift-scan mode that scans each pointin the sky nine times, as a function of comoving black hole density. Theintegration time per pixel in each case is about 7, 14, and 25 minutes for z = 10, 15, and 20. With a 14-minute integration time, a 10 − M (cid:12) black holewould be detectable at greater than 5- σ significance (see Fig. 2) We note that the ratio of signal to foreground in this regimeis comparable to that during reionization. With a filled aper-ture, it may be easier to achieve foreground subtraction. Thishas been demonstrated for intensity mapping at z ∼ . z ∼
15) with a 300-m dish like Arecibo, which scans each point in the sky ninetimes would have a pixel size of about 0.4 deg during whicheach pixel was integrated for about 15 minutes and 1140 deg were surveyed. If 10 − M (cid:12) black holes had a number densityof 1 Gpc − at that time, one would expect to discover aboutthree or four of them at greater than 5- σ significance sinceeach pixel would have been integrated for about 15 minutesat z = 15 (see dotted lines in Figs. 2 and 4). Black holes withmasses greater than 10 M (cid:12) and similar abundances would beeasily detectable, allowing for followup with longer baselinefacilities such as LOFAR to determine the detailed shape oftheir 21-cm profile. Detecting a 10 − M (cid:12) black hole at 5- σ significance would require a significantly longer integrationtime on each pixel, about 4 hours, so that only about 70 deg could be surveyed in a month. Thus, the minimum spatialdensity of 10 − M (cid:12) black holes would be about 5 Gpc − at z = 15. Smaller black holes would mean even smaller fields ofview and longer integration times per pixel, so in those casesdetecting their statistical signature in the spherically-averagedpower spectrum discussed in §3.2 may be a better approach.Regardless of whether high-redshift quasar X-ray sourceswill be detected directly by the means we propose here, itis clear that the pre-reionization universe offers a wealth ofinformation that can be uniquely probed with the 21-cm tran-sition, and it is therefore vital that theoretical models for thesignal originating from before the epoch of reionization con-tinue to be developed.We wish to thank J. Peterson and R. M. Thomas for helpfuldiscussions on Arecibo and 21-cm signatures of early QSOs,resepectively, and J. R. Pritchard for comments on an earlierdraft of the paper. M. A. A. and T. C. are grateful for thehospitality of the Aspen Center for Physics, where this workwas completed. We acknowledge financial support by CIfARand NSERC.= 15. Smaller black holes would mean even smaller fields ofview and longer integration times per pixel, so in those casesdetecting their statistical signature in the spherically-averagedpower spectrum discussed in §3.2 may be a better approach.Regardless of whether high-redshift quasar X-ray sourceswill be detected directly by the means we propose here, itis clear that the pre-reionization universe offers a wealth ofinformation that can be uniquely probed with the 21-cm tran-sition, and it is therefore vital that theoretical models for thesignal originating from before the epoch of reionization con-tinue to be developed.We wish to thank J. Peterson and R. M. Thomas for helpfuldiscussions on Arecibo and 21-cm signatures of early QSOs,resepectively, and J. R. Pritchard for comments on an earlierdraft of the paper. M. A. A. and T. C. are grateful for thehospitality of the Aspen Center for Physics, where this workwas completed. We acknowledge financial support by CIfARand NSERC.