Precise Measurement of the Reionization Optical Depth from The Global 21-cm Signal Accounting for Cosmic Heating

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by  Anastasia Fialkov

Draftversion March1,2016
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
PRECISE MEASUREMENT OF THE REIONIZATION OPTICAL DEPTH FROM THE GLOBAL 21-CM
SIGNAL ACCOUNTING FOR COSMIC HEATING
Anastasia Fialkov
Department ofAstronomy,HarvardUniversity,60GardenStreet,MS−51,Cambridge,MA,02138U.S.A.
and
Abraham Loeb
6
Department ofAstronomy,HarvardUniversity,60GardenStreet,MS−51,Cambridge,MA,02138U.S.A.
1
Draft version March 1, 2016
0
2 ABSTRACT

b As a result of our limited data on reionization, the total optical depth for electron scattering, τ,
e limits precision measurements of cosmological parameters from the Cosmic Microwave Background
F
(CMB). It was recently shown that the predicted 21-cm signal of neutral hydrogen contains enough
9 information to reconstruct τ with sub-percent accuracy, assuming that the neutral gas was much
2 hotter thanthe CMB throughoutthe entire epochofreionization. Here we relaxthis assumptionand
usetheglobal21-cmsignalalonetoextractτ forrealisticX-rayheatingscenarios. Wetestourmodel-

] independent approachusing mock data for a wide range of ionization and heating histories and show
O
that an accurate measurement of the reionization optical depth at a sub-percent level is possible in
C mostof the consideredscenariosevenwhen heating is not saturatedduring the epochof reionization,
assuming that the foregrounds are mitigated. However, we find that in cases where heating sources
.
h had hard X-ray spectra and their luminosity was close to or lower than what is predicted based on
p low-redshiftobservations,theglobal21-cmsignalaloneisnotagoodtracerofthereionizationhistory.
-
o Subject headings: cosmology: cosmological parameters, dark ages, reionization, first stars; X-rays:
r binaries, galaxies, general
t
s
a
[ 1. INTRODUCTION One of the most promising tools to constrain reion-
The reionization of the intergalactic medium (IGM), ization is the predicted 21-cm signal of neutral hy-
2 drogen, HI, e.g., see Furlanetto et al. (2006) and
between redshifts z ∼ 13 and 6 (Zahn et al. 2012;
v
Pritchard& Loeb (2012). The brightness temperature
8 George 2015; Ade et al. 2015; Becker et al. 2015), is
ofthissignal,δT (z),dependsonthefractionalamountof
5 one ofthe leaststudiedepochs in the historyofthe Uni- b
hydrogenatomsintheIGMwhichareneutral,x ,and,
0 verse and is a research frontier in present-day cosmol- HI
thus,isexpectedtoprovideexclusiveinformationonthe
3 ogy(Loeb & Furlanetto 2013). During this process,the
reionizationhistoryoftheUniverse. Recently,Liu et al.
0 neutralintergalacticgaswaslikelyionizedbyultra-violet
(2015) advocated that the sky-averaged (global) 21-cm
. (UV) photons emitted by stars. In addition, sources of
1
0 X-rayphotons, such as X-ray binaries,mini-quasarsand signal, δTb(z), alone has enough information to fully re-
hot gas in galaxies, also had an effect on the epoch of construct the reionization history and measure the opti-
6
reionization (EoR) by pre-heating and mildly ionizing cal depth to reionization with great precision. To alle-
1
: the gas far from the sources (Oh 2001; Mesinger et al. viatethe computationalcosts,the authorsassumedthat
v 2013; Fialkov et al. 2015). the 21-cm signal tracks the ionization history, which is
Xi The EoR also affects the Cosmic Microwave Back- true only when X-ray sources heat up the cosmic gas to
ground (CMB) through its scattering off free electrons. atemperature abovethe CMB wellbefore the beginning
ar Thisscatteringdegradestheaccuracywithwhichcosmo- of reionization. In this case, the dependence of δTb(z)
logicalparameterscanbeextractedfromtheCMBdata. on the gas temperature is saturated (the so-called satu-
In particular, measurements of the amplitude of primor- ratedheatingregime),δTb(z)isproportionaltoxHI,and
dialfluctuations, A ,is degeneratewiththe totaloptical the reionization history can be fully extracted from the
s
depth, τ, sincethe totalamplitudeis estimatedfromthe global 21-cm signal measurements despite the presence
temperature power spectrum of the CMB as A e2τ. Be- of strong foregrounds. In particular, Liu et al. (2015)
s
cause the precision with which τ can be measured using showed that assuming saturated heating, the 21-cm sig-
the CMB is very poor (e.g., the 68% confidence level in nal allows to determine τ with much higher accuracy
τ corresponds to a relative error of ∼ 24% when mea- than it is possible from the CMB measurements.
suredfrom the temperature power spectrum), the errors However, the assumption that heating is saturated all
inA arehigh. As aresult, τ is sometimesreferredto as the way through reionization is debated (Fialkov et al.
s
a nuisance parameter for CMB cosmology. Luckily, al- 2014), and the nature and efficiency of early X-ray
ternativeprobesofreionizationcanprovideindependent sources could have a significant impact on the inten-
constraints on τ and remove the related uncertainty. sity of the redshifted 21-cm signal even at the end of
theEoR(Pritchard & Loeb 2012;Mesinger et al. 2013;
Fialkov et al. 2014; Pacucci et al. 2014). The nature
anastasia.fi[email protected] of the first X-ray sources is still unknown and possi-
[email protected]
2 Fialkov & Loeb
ble candidates include X-ray binaries (Mirabel et al. upandstopsaccretingintohalosbelow∼108−109 M⊙,
2011;Fragos et al. 2013;Fialkov et al. 2014)andmini- thus suppressing star formation in low-mass halos.
quasars(Madau et al. 2004;Fialkov et al. 2015)which In our simulation, ionization by UV photons is com-
emithardX-rayswithspectralenergydistribution(SED) puted following the excursion-set formalism, by com-
peakingatfewkeV,softX-raysourcessuchashotgasin paring the time-integrated number of ionizing pho-
galaxieswhichcanbewelldescribedbyapower-lawspec- tons to the number of neutral atoms in each region
tralshape(Furlanetto 2006),aswellasmoreexoticcan- (Furlanetto et al. 2004). Specifically, a simulation cell
didates such as annihilating dark matter (Cirelli et al. isionizedifζ f ≥(1−x ), whereζ is the ioniza-
UV coll e UV
2009). The efficiency of high-redshift sources, f , tionefficiencynormalizedtoyieldτ,f isthecollapsed
X coll
defined through the relation between their luminos- fraction, and x is the fraction of free electrons. In ad-
e
ity and the star formation rate is another unknown dition, we account for partial ionization of the neutral
(Fialkov et al. (2015)andreferencestherein),calibrated IGM by X-rays, which boost the free electron fraction
sothatthevalueoff =1correspondstotheluminosity far from the sources and have a non-negligible effect on
X
ofobservedlow-redshiftsourcesboostedbyametallicity- the topology of reionization.
dependent factor (Fragos et al. 2013). Thereionizationhistoryisstronglylinkedtothemech-
Here we consider a completely model-independent anism of star formation and its timing depends on the
method to reconstruct τ from the global 21-cm signal minimal mass of halos that can form stars, M . The
min
measurements after relaxing the saturated heating as- smaller is M , the earlier reionization starts and the
min
sumption and examining realistic X-ray sources with more gradual is the grows of the ionized fraction. Be-
hard and soft spectra varying their efficiency. Our re- cause star formation at high redshifts is very uncon-
sultsaretimelysincemanyoftheexperimentssuchasthe strained and is biased by multiple feedback mechanisms
Experiment to Detect the Global Epoch of Reionization (Greif 2015; Bromm 2013), we consider three different
Signature (EDGES, Bowman & Rogers (2010)), Large- scenariosvaryingthe low-masscutoffofstarformingha-
Aperture Experiment to Detect the Dark Ages (LEDA, los:
Greenhill & Bernardi (2012), Bernardi et al. (2015)),
Dark Ages Radio Explorer (DARE, Burns et al. • “Massivehalos”: StarsforminhalosofMmin &109
(2012)), and New extension in Nancay upgrading LO- M⊙ (circular velocity ≥35.5 km/s).
FAR (NenuFAR, Zarka et al. (2012)) are on their way
• “Atomic cooling”: Stars form through the atomic
to detect this signal for the first time while next gener-
ation telescopes, such as the Hydrogen Epoch of Reion- cooling channel in halos of Mmin &107 M⊙ (circu-
larvelocity≥16.5km/s)with activephotoheating
izationArray(HERA1)andthe SquareKilometerArray
feedback.
(SKA, Koopmans et al. (2015)), are expected to exten-
sively explore the EoR. • “Molecularcooling”: starformationhappens inall
In Section 2 we set up the stage describing simulation halos with circular velocity ≥ 4.2 km/s (M &
min
methods andmodel parameters. In Section3 we explore 105 M⊙). In this case we include the photoheat-
towhichextenttheglobal21-cmsignaltrackstheneutral
ing feedback, account for the effect of v , but ex-
bc
fraction in each case and propose a model-independent
clude the effect of Lyman-Werner (LW) photons
way to reconstruct the heating and ionization history
which are expected to destroy molecular hydro-
from the global 21-cm signal. In Section 4 we calcu- gen acting as negative feedback to star formation.
latetheopticaldepthfromthereconstructedreionization
The degree to which the LW feedback is efficient
history and discuss the accuracy with which it can be is a topic of active research (Schauer et al. 2015;
detected by global 21-cm experiments. Finally, we con- Visbal et al. 2014); therefore, we ignore the effect
cludeinSection5. Throughoutthispaperweassumethe
of this feedback here to optimize the contribution
standard Planck satellite cosmology (Ade et al. 2015).
of the molecular cooling halos and increase the di-
2. THEMOCKUNIVERSE versity of ionization histories. The case of molec-
ular cooling with LW and v included is close to
bc
We simulate the mock global 21-cm signal from the
the atomic cooling scenario (Fialkov et al. 2013)
redshift range z = 6 − 40 using a hybrid simulation,
which we considerseparately. Althoughthe role of
first introduced by Visbal et al. (2012) and described
molecular cooling halos in reionization is expected
inmore detailby Fialkov et al. (2014). This simulation
tobesmallbasedonthelowopticaldepthfoundby
allows to estimate the non-local impact of X-ray, Ly-α
Planck satellite, their contributionis not ruled out
and UV sources on the redshifted 21-cm signal of neu-
considering large uncertainty in τ measurements.
tral hydrogen as well as on the ionization history of the
IGM,andincludesthe effectofsupersonicflowsbetween In all the above cases we assume a star formation effi-
darkmatterandgas,v (Tseliakhovich & Hirata 2010), ciency of f = 5%. We consider the contribution of hy-
bc ⋆
which has an impact on high-redshift star formation in drogen and first helium reionizatio to τ, assuming that
105−107M⊙halos(Stacy et al. 2011;Maio et al. 2011) singly ionized helium and hydrogen are ionized to the
and, consequently, on the 21-cm signal (Dalal et al. same fraction, x (Wyithe et al. 2003), and normalize
e
2010; Tseliakhovichet al. 2011; Fialkov et al. 2012; ourmodelstoyieldτ consistentwithPlanck(Ade et al.
McQuinn & O’Leary 2012; Fialkov 2014). In addition, 2015) while also requiring reionization to end by z = 6
weaccountfor the photoheatingfeedback(Cohen et al. or earlier. For atomic and massive halos we choose
2015) which happens when the intergalactic gas heats τ = 0.082 which gives the redshift of full reionization,
z , being z ∼ 6.5 and z ∼ 8 respectively. This value
r r r
1 http://reionization.org/ ofτ isbetweenthe PlanckandWMAPmeasurementsof
Precise Measurement of Reionization Optical Depth from Global 21-cm with Realistic Heating 3
the optical depth and is 1σ away from the Planck’s best quencies corresponding to redshifts where the IGM was
fit value of 0.066. In the case of molecular cooling the colder than the CMB (the signal is seen in absorption).
process of reionization is very gradual, and we need to The minimal value of δT is reached at the beginning of
b
takeτ =0.114(3σ awayfromthePlanck’sbestfitvalue) heating era at redshift z when the first population of
min
to have reionization end by zr ∼6. X-raysourcesturnedon. Atthispointalsothetempera-
Finally, we consider two types of heating sources: (i) tureofthegas,whichwasadiabaticallycooledbycosmic
X-raybinaries with hardSED, and(ii) soft sources with expansion, reaches its minimum (right column of Figure
power-law SED (as described by Fialkov et al. (2014)). 1). X-ray sources inject energy into the IGM heating it
In addition, we consider three different values of heat- up and above the temperature of the CMB, if heating is
ing efficiency for each type of sources: fX = 0.3 (low), sufficiently strong. In this case the 21-cm signal is seen
fX = 1 (standard) and fX = 30 (high). The choice of in emission against the CMB at redshifts lower than z0
low and high heating efficiencies is motivated by rather where T = T . The emission signal peaks at z
K CMB max
poor observational constrains on the temperature of the and its amplitude declines at lower redshifts as reioniza-
IGM before the end of reionization. The unresolvedsoft tion progresses. If heating is not strong enough, pockets
cosmic X-ray background, which amounts to ∼ 25% of of neutral gas remain colder than the CMB throughout
thefluxinthe0.5−2keVChandraband(Lehmer 2012), the EoR,markedby a grey bandin eachpanel of Figure
sets an upper limit on fX when attributed to the high 1, and the 21-cm signalis seen in absorptionall the way
redshift sources (Dijkstra et al. 2012; Mesinger et al. downtoz . Welistz ,z andz inTable1forevery
r min 0 max
2013; Fialkov et al. 2015). Depending on the details considered model.
of star formation and for EoR ending at zr ∼ 6 this As Figure 1 suggests (and as was recently reported by
measurement yields an upper bound of fX ∼ 16− 36 Fialkov et al. (2014)),thesaturatedheatingassumption
(fX ∼ 45−75) in the case of hard (soft) X-rays; while may be justified only in the case of high fX (green lines
for zr ∼ 8.5 the efficiencies should be ∼ 5 times higher in the Figure) where the IGM is indeed hotter than the
(Fialkov et al. 2015). Here wechoosefX =30asa rep- CMB at the beginning of the EoR. In other cases the
resentativevalueofthe highheatingefficiencyforallthe gas is colder than the CMB at the beginning of reion-
considered models. The lower limit on fX comes from ization and undergoes the heating transition during the
thedatacollectedbythePrecisionArrayforProbingthe EoR. The most interesting case is that of massive halos,
EoR (PAPER, Pober et al. (2015), Ali et al. (2015)) which is also well-motivated by the low optical depth
whichrulesout21-cmfluctuationsofpowergreaterthan measurements. For this star formation scenario heating
∼500mK2atz =8.4inthek=0.15-0.5hMpc−1range, is slower than reionization and the neutral gas is always
where h is the Hubble constant in units of 100 km s−1 colderthanthe CMB intwo outofsix cases,namely the
Mpc−1. This constraintranslatesinto fX &0.01(0.001) caseofhardX-raysourceswithstandardandlowheating
for hard (soft) X-ray sources in the atomic cooling case. efficiency.
However, for such low efficiency, the gas appears to be
colderthantheCMBbytheendofEoR,andthemethod 3. EXTRACTINGTHENEUTRALFRACTIONFROMTHE
whichwepresentinthispaperdoesnotapply. Therefore GLOBAL21-CMSIGNAL
we choose f =0.3 as our low heating efficiency value. We would now like to mimic a global 21-cm exper-
X
Foreverymodelweoutputglobalneutralfraction,x¯ iment, assuming the foregrounds fully under control,
HI
(which we refer to as the true reionziation history), av- wherewerelyonLiu et al. (2015)whoshowedthatfore-
erage kinetic gas temperature T and the 21-cm signal ground contamination from Galactic synchrotron emis-
K
which depends on the ionization and thermal history in sion (de Oliveira-Costa et al. 2008), can be mitigated,
the following way, allowingprecisereconstructionofthe opticaldepthfrom
the global 21-cm signal. We first examine to which ex-
δT ≈δT (1+z)1/2x (1+δ) 1− TCMB , (1) tenttheglobal21-cmsignalcanbeusedtoconstrainthe
b b,0 HI T ionization history and reconstruct the neutral fraction,
(cid:18) S (cid:19)
and then (in the next Section) use this information to
whereδTb,0 isaconstantthatdependsonatomicphysics extract the total CMB optical depth.
andcosmologicalparameters,δ isthebaryonoverdensity We start by adopting the saturated heating assump-
which is statistically known from cosmology, and TCMB tion. Given the data, δTb, we estimate the neutral frac-
istheCMBtemperature. Hereweignorethepeculiarve- tion from Eq. (1) excluding temperature effects
locity term, which adds a small correction to the global
21-cm signal (Bharadwaj & Ali 2004; Barkana & Loeb δT
xsat ≡ b (2)
2005). Finally, TS is the spin temperature of the 21-cm HI δT (1+z)1/2
b,0
transitionwhich depends on environment. In particular,
when Ly-α coupling is saturated, which is usually true and check up to which values of x¯ (listed in Table 1)
HI
forz <25,wecanequatethespintemperaturetogaski- thetrueneutralfractionisfollowedbytheestimatedone.
netic temperature, T ≈T (Madau et al. 1997); while (FollowingLiu et al. (2015),weincludethefactor(1+δ)
S K
T → 1 when the IGM is much hotter than the CMB into the definition of x¯ , thus the quantity x is, in
S HI HI
(the saturated heating case). In the latter case Eq. (1) reality,xsat(1+δ). However,theeffectofdensityfluctu-
HI
can be further simplified, δT ∝ (1+z)1/2x (1 +δ), ationsonthe globalsignalisnotverylargeandomitting
b HI
and the 21-cm signal can be used as a tracer of neutral this contribution would not alter our conclusions.)
fraction weighted by the density fluctuations. As can be seen from the Table, the saturated heating
Typical global spectrum of the 21-cm signal (left col- assumption is not accurate even in the case of high f ,
X
umn of Figure 1) features a prominent trough at fre- and, although the gas is hotter than CMB by the begin-
4 Fialkov & Loeb
ν [MHz] ν [MHz]
142.8 71.4 47.6 35.7 142.8 71.4 47.6 35.7
50 4
0
3
K] −50 )K
m T
δ T [b−100 log(102
−150
1
−200
5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40
1+z 1+z
ν [MHz] ν [MHz]
142.8 71.4 47.6 35.7 142.8 71.4 47.6 35.7
50 4
0
3
K] −50 )K
m T
δ T [b−100 log(102
−150
1
−200
5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40
1+z 1+z
ν [MHz] ν [MHz]
142.8 71.4 47.6 35.7 142.8 71.4 47.6 35.7
50 4
0
3
K] −50 )K
m T
δ T [b−100 log(102
−150
1
−200
5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40
1+z 1+z
Fig.1.—Left: Global21-cmsignalforalltheconsideredmodels: massivehalos(toppanel),atomiccooling(middlepanel)andmolecular
cooling(bottom panel)areshownforthecasesofhardSED(solid)andsoftSED(dashed)forfX =0.3(blue),fX =1(red)andfX =30
(green). ThegreybandmarkstheEoRfromx¯HI =0.95tox¯HI =0.05andtheverticallinemarksthemiddlepointoftheEoR(x¯HI =0.5).
Right: KineticgastemperatureoftheIGMforthemodelsshownontheleft(samecolorcode). Thedottedlineisthetemperatureofthe
CMB.
Precise Measurement of Reionization Optical Depth from Global 21-cm with Realistic Heating 5
TABLE 1
Summaryoftheresultsforeachstructure formation(column1)andheating(column2)model. First,we
summarizethecriticalpointsoftheglobal21-cmsignal: the redshiftatwhichthe signalisminimal(zmin,
column 3),vanishes(z0, column4) andis maximal(zmax,column 5). Next, we note the valueof xHI atthe
pointin time when x¯sHaIt and x¯THI deviate by 5%fromthis value. We define the deviation (in %) as
∆xsHaIt≡|x¯HI−xsHaIt|/xHI =5% (column 6) and∆xTHI ≡|xHI−xTHI|/xHI =5% (column 7). Next, we list the
valuesof x¯THI atthe point atwhich dTb/dz ismaximal(x¯∗HI,column 8). Finally,we list zi (column 9)for
which ∆τ/τ takes itsminimalvalue (∆τmin/τ, column10).
Model Heating zmin z0 zmax xHI(∆xsHaIt=5%) xHI(∆xTHI =5%) x¯∗HI zi ∆τmin/τ
Massive Hard,fX =0.3 12.2 8.3 none 0% 0% none - >1%
Soft,fX =0.3 13.4 9.0 8.6 0% 0% 25% - >1%
Hard,fX =1 13.1 8.1 none 0% 0% none - >1%
Soft,fX =1 14.2 10.5 9.2 0% 0% 31.4% - >1%
Hard,fX =30 15.5 12.7 11.0 25.7% 54.9% 54.9% 15.3 0.007%
Soft,fX =30 16.9 14.6 13.0 58.8% 60.8% 64.2% 14.0 0.03%
Atomic Hard,fX =0.3 15.8 8.7 7.4 0% 0% 16.3% - >1%
Soft,fX =0.3 17.3 11.5 9.8 0% 0% 30.6% 16.3 0.01%
Hard,fX =1 16.9 10.9 9.2 0% 22.6% 24.1% - >1%
Soft,fX =1 18.3 13.5 11.0 40.1% 61.0% 36.7% 15.9 0.09%
Hard,fX =30 20.1 16.5 13.2 75.9% 93.5% 56.6% 14.8 0.04%
Soft,fX =30 21.4 18.6 15.0 87.4% 90.0% 71.1% 15.3 0.01%
Molecular Hard,fX =0.3 21.0 11.3 9.2 1.8% 2.2% 14.9% - >1%
Soft,fX =0.3 22.4 15.0 12.2 1.7% 1.7% 25.6% - >1%
Hard,fX =1 22.4 14.5 12.0 1.8% 26.6% 25.8% - >1%
Soft,fX =1 23.9 17.6 14.5 41.0% 46.5% 34.9% 26.8 0.08%
Hard,fX =30 26.6 21.9 17.5 78.8% 82.9% 48.8% 24.0 0.1%
Soft,fX =30 27.8 24.0 19.2 89.0% 89.0% 60.0% 24.4 0.03%
6 Fialkov & Loeb
reconstruct the thermal history at z < z assuming
min
1 adiabatic cooling at higher redshifts. The true tempera-
ture found in our simulation and the reconstructed one
areshowninFigure3forthe caseofatomiccoolingwith
0.8
f = 1 and f = 30. In the same figure we also show
X X
thefactor(1−T /T )foundfromourmockdataand
CMB S
compare it to the reconstructed value (1−T /Trec)
0.6 CMB K
which is always equal to 1 within the saturated heating
HI regime. Despite being a very crude approximation, Trec
x K
follows the general trend of T , and the reconstructed
0.4 K
factor(1−T /Trec)correctlyreproducesthefeatures
CMB K
of the true value of (1−T /T ). Undoubtedly, this
CMB S
0.2 is a much better approximationthat the saturatedheat-
ing assumption; however, a better guess of the thermal
history during the EoR would be very beneficial for the
0 xHI extraction.
7 10 15 20 We use the reconstructed factor (1−T /Trec) to
1+z CMB K
estimate the ionizationhistory accordingto Eq. (3). An
Fig.2.— The reionization history for the atomic cooling model example of xT is shown in Figure 2, and we list the
with hard X-rays, fX = 1 (red) and fX = 30 (black). We show HI
the true, x¯HI (solid), and the estimated, x¯sHaIt (dotted) and x¯THI vnaeluutersaloffrxacHtIiofnorfrwomhichthtehetrudeevoianteioniso5f%thienesTtaimblaete1d.
(dashed),neutralfractions. Thesquaresindicateuptowhichred-
shiftwe trustthe reconstructed historyxTHI whenfitting the ion- With the temperature information added, xTHI follows
izationhistoryinSection4. the true neutral fraction up to x¯HI ∼ 23% in the case
of f = 1, hard SED and atomic cooling shown in the
X
ning of EoR, thermal effects continue to play a role. In Figure(redlines), while this case wascompletely missed
particular, for soft (hard) X-rays xsat succeeds to track by x¯sat. Moreover,for the rest of the considered models
HI HI
the true reionization history from the end of EoR all the redhsiftat whichthe deviationreaches5%is pushed
the way up to x¯ ∼ 89% (x¯ ∼ 79%) for molecular deeper into the first half of reionization with the excep-
HI HI
cooling, x¯HI ∼ 87% (x¯HI ∼ 76%) in the case of atomic tion of all the cases with fX =0.3 for which the neutral
cooling and x¯HI ∼ 59% (x¯HI ∼ 26%) for massive halos. IGM is barely (or not at all) heated to TCMB by the
Ontheotherhand,forlowandstandardheatingefficien- end of reionization as well as the case of massive halos
cies, xsHaIt is a very poor approximationwith a fractional with fX = 1 and hard SED. This inability to track x¯HI
error ∆x¯ /x¯ being greater than 5% for all models is explained by the fact that our Trec is a too poor ap-
HI HI K
exceptformolecularandatomiccoolingwithsoftX-rays proximation and lacks precision to serve in the regime
andstandardheatingefficiencyinwhichcasexsHaIt follows TK .TCMB.
the true neutral fraction up to x¯ ∼40%. We show an
HI
example of the true ionizationhistory and the saturated 3.1. Complete Ionization History
heating approximation in Figure 2.
Next, we would like to develop a model-independent
The situation can be alleviated with information on
method to reconstruct the entire reionization history
the thermal state of the IGM used. Assuming that the
basedontheglobal21-cmsignal. Tothisend,wechoose
gaskinetictemperature,Trec,canbereconstructedfrom
K to use xT (and notx¯sat)as a tracer ofthe neutralfrac-
the global 21-cm spectrum, we can estimate the neutral HI HI
tion. FromouranalysisinthepreviousSection,weknow
fraction as
thatthisapproximationworkswellduringthelatestages
δT T −1 ofthe EoR;however,we do nothave agoodmeasurefor
xT = b 1− CMB, , (3)
HI δT (1+z)1/2 Trec thecriticalredshift(orthevalueoftheneutralfraction),
b,0 (cid:18) K (cid:19) z∗ (x∗HI),uptowhichthisapproximationholds. Herewe
where we also adopted saturated Ly-α coupling approx- adoptaratherconservativeapproach,outlinedbelow,to
imation. Eq. (3) improves over the saturated heating define this instant and to reconstruct the full ionization
assumption and promises to be a better tracer of the history, x¯rec.
HI
true neutral fraction than xsat. First, we keep all the measured data points for which
HI
As a proof of concept, we use a very simple method xT is guaranteed to follow the true neutral fraction
HI
to extract TKrec from δTb. Two critical points of the starting from the end of reionizaion at zr and up to z∗.
global spectrum can inform us about the heating his- We adopt the next model-independent criterion to find
tory: (i) the redshift of the heating transition, z0, where z∗: ifthesignalisseeninemissionattheadvancedstages
the gas temperature equates that of the CMB, TCMB = of the EoR, we searchfor a redshift (z∗) between zr and
2.725(1+ z ), and (ii) the trough of the 21-cm signal theemissionpeakatwhichthederivativedT /dz ismax-
0 b
at z which represents the beginning of the heating imal. Intuitively, this instant marks the change in the
min
era. We know that the gas cooled down adiabatically behavior of the global signal when it transits between
from z ∼ 200 to z ∼ z , and, given the values of cos- ionization-driven to heating-driven evolution. Clearly,
min
mological parameters, we can estimate the gas kinetic this approach does not apply to the cases with no emis-
temperature at z using publicly available codes such sion feature. This definition is rather conservative, and
min
as RECFAST (Seager et al. 2000). We interpolate be- in the cases with high degree of heating we lose some
tween these two values of redshift and temperature to information. In particular, x∗ is typically lower than
HI
Precise Measurement of Reionization Optical Depth from Global 21-cm with Realistic Heating 7
1000 1.05
1
300
0.95
100 S 0.9
T
K] /B
T [K TCM0.85
−
30 1 0.8
0.75
10
0.7
3 0.65
7 10 15 20 25 30 8 10 12 14 16 18 20
1+z 1+z
Fig.3.— Left: Heating history. Solidlines show the kinetic gas temperature drawnfromthe simulations inthe case of atomic cooling
for hard SED with fX = 1 (red) and fX = 30 (black) and soft SED for fX = 1 (green) and fX = 30 (blue). Dashed lines show the
correspondingreconstructedtemperaturedependence(onlythelog-loginterpolatedpiece)basedineachcaseontwopointsextractedfrom
the global 21-cm signal: zmin (open circles) and z0 (filled circles). The temperature of the gas which is cooled adiabatically is shown
with the solid grey curve, the temperature of the CMB is shown with the dotted line. Right: For each case from the left panel we plot
(1−TCMB/TS)(solid)and(1−TCMB/TKrec)(dashed). Inthecaseofsaturatedheating, thisfactorisalwaysequalto1(dotted line).
the value of xT where it deviates from the true neu- the low-redshift neutral fraction does no longer follow
HI
tral fraction by more than 5% (Table 1); moreover, in the collapsed fraction (as it does in the case of massive
these cases x¯sat works as well as xT at redshifts below halos which are immune to the photoheating feedback).
HI HI
z∗. However, this definition of z∗ works very well in the For X-ray binaries with fX = 1 (red curve in Figure 2)
cases when heating is weak and extracting the reioniza- formed in atomic cooling halos the true neutral fraction
tion information from the global signal is difficult, e.g., followsthecollapsedfractionathighvaluesofx¯HI,while
in the cases of fX =1 with hard SED for molecular and changing its behavior at x¯HI ∼30% due to the presence
atomic cooling. We find that in these cases x∗ is very of a feature (a bump) introduced by the photoheating
HI
close to the marginal value of x¯ at which xT ceases feedback. In this particular case, the information which
HI HI wecanextractfromx¯T isdominatedbythephotoheat-
to be a good approximation. In other words, when us- HI
ingeffectsanddoesnotgiveusanyinsightontheprocess
ing this model-independent criterion we do succeed to
of reionization at higher redshifts which we try to fit.
retain all the useful information in the “difficult” cases
In total, our reconstructed neutral fraction, which we
with weak heating; while we do lose some information
use in the next Section to find τ, is
in the “easy”caseswith enoughheating (however,as we
seeinthe nextsection,this lossdoes notaffectourmain
results). xTHI, z <z∗
Second, we assume that EoR starts at zi with the xrHeIc = F(z), z∗ <z <zi. (4)
Universe being neutral at higher redshifts. This ”an- ( 1, z ≥zi
chor” point can be determined from independent exper-
iments, e.g., using the kinetic Sunyaev-Zeldovich effect We find that our method works well for the majority
(Zahn et al. 2012); therefore, we do not include zi in of cases with x∗HI & 30% and F(z) does a decent job
thelistofourfreeparameterswhenfittingtheionization reconstructing x¯ when the starting point of reioniza-
HI
history. tion, z ,is chosenclosetothe truevalue. Figure4shows
i
Third, in the intermediate redshift range (z∗ < z < twoexamplesofxrHeIc: (i)acasewherethereconstruction
zi) x¯rHeIc is completed using a fitting function F(z). We workswell(atomic cooling with hardSED andfX =30,
tried several options and found that the best results in x∗ ∼57%,shownwithblackcurvesintheFigure),and
HI
terms of the final optical depth estimate are achieved (ii) where it fails (atomic cooling with hard SED and
withathree-parameterfunction whichappearsto fitthe f = 0.3, x∗ ∼ 16%, red curves). Here we clearly see
X HI
reionizationhistoryreasonablywellforalltheconsidered that in the case of the low heating efficiency the photo-
casesfor whichour approachcan be applied (i.e., all the heatingfeatureis verymisleadinganddoesnotallowfor
cases whichundergo the heating transitionuntil the end a more accurate fitting.
of the EoR). In particular, here we choose cumulative Asimplerfit,suchasthecommonlyusedtanh(x)func-
distribution function of Gamma distribution tion, works well for a subset of models which we con-
sider here, but with only two free parameters it does
1 z−c
F(z)= ta−1e−t/bdt, notcapturethedifferentshapesoftheionizationhistory.
baΓ(a) In our case, this fit worked sufficiently well to describe
Z0
the atomiccoolingcasewithstrongheatingbutfailedto
whereaistheshapeparameter,bisthe scaleparameter,
match the cases of molecular cooling and massive halos.
and c marks the end of reionization. It is worth noting
that in addition to the temperature effects, photoheat-
ingfeedbackcomplicatesthefittingprocedureforatomic 4. RECONSTRUCTINGTHEREIONIZATION OPTICAL
and molecular cooling. In the presence of this feedback, DEPTH
8 Fialkov & Loeb
5. CONCLUSIONS
2
The total CMB optical depth is a long-standing nui-
sance for CMB cosmology. Here we have examined to
1 which extent the global 21-cm signal can be used to
probe the total CMB optical depth in realistic cases
of IGM heating, including hard and soft X-ray sources
withlow,standardandhighheatingefficiency. Following
0.4
Fialkov et al. (2014), we have shown that the intensity
HI
x of the 21-cmsignalproduced during the EoRis strongly
0.2 affectedbythethermalstateoftheIGMinadditiontoits
ionization,whichmakesithardertoextractthereioniza-
tion history from the global 21-cm signal compared to a
0.1
scenarioin whichheatingis saturated(Liu et al. 2015).
We have developed a simple and model independent
approach to reconstruct the neutral fraction from a re-
0.047 10 15 20 alistic global 21-cm signal and used it to estimate the
1+z optical depth for a large variety of models with differ-
entionizationandheating histories. The method can be
Fig. 4.—Anexampleofreconstructedneutralfraction(dashed)
summarized as follows: (i) at low redshifts we extract
compared to x¯HI (solid) for atomic cooling with fX = 0.3 (red)
and fX =30 (black). The squares show z∗ and x∗ for each case. the neutral fraction from the global 21-cm signal going
Hereweusedzi=17atwhichthetrueneutralfractionis98%. beyond the saturated heating assumption and using in-
TheCMBopticaldepthisdependentontheionization formation on the thermal state of the IGM extracted
history directly from the mock global 21-cm signal; (ii) we as-
sume that the redshift at which reionization starts, z ,
i
τ = (1−x¯HI)n¯eσTdl, (5) is known with the Universe neutral at that epoch; (iii)
Z we complement the neutral fraction in the intermediate
where n¯ is the average number density of free electrons redshift range using a three-parameter fitting function
e
inionizedregionsaccountingforhydrogenionizationand which works well for the different types of reionization
first helium ionization, σ is the Thomson cross-section histories which we have explored.
T
anddl istheline-of-sightproperdistanceelement. Thus, One of the main conclusions we reach is that with the
knowingtheionizationhistoryfromtheglobal21-cmsig- thermal history added a better estimation of the reion-
nal should allow estimating the optical depth. ization history is possible, and the neutral fraction can
Although the reconstruction x¯rec does not work per- be reconstructed even when the 21-cm signal is affected
HI
fectly well to reproduce x¯ as can be seen from Figure bythermalhistoryallthewaythroughouttheEoR.Asa
HI
4, the error in τ is expected to be much smaller than proofofconcept,weadoptaverysimple methodto esti-
theerrorinx¯ itselfbecause: (i)thelargestpartofthe mate the temperature of neutral IGM using two critical
HI
opticaldepthiscontributedbyredshiftsz <z whenthe points of the global signal, namely (i) the heating tran-
r
Universe was fully ionized (in our case of massive halos sition at which the gas kinetic temperature equates that
with reionization ending at z ∼ 8 only 30% of the op- of the CMB, and (ii) the beginning of the heating era
r
tical depth is sourced by the ionized patches during the when X-ray sources turn on. Even this simple method
EoR); and (ii) the fit over- and under-predicts x¯ at improves over the saturated heating approximation.
HI
different redshifts which results is partial cancellation of Finally, we calculate the optical depth using the ex-
the error. tracted reionization history and show that an accurate
Usingx¯rec wecomputetheopticaldepthτrec andcom- measurement of τ, with fractional error below 1% over
HI
pare it to the true value, τ, found directly from the sim- a wide range of z , is possible even when the IGM heat-
i
ulationdata. Theaccuracywithwhichtheopticaldepth ing is not saturated all the way throughout the EoR.
can be extracted from the global signal depends on the We have blindly tested our method on a large variety of
value of z , as can be seen from Figure 5 where the frac- ionization histories for different star formation scenarios
i
tional error in the optical depth, ∆τ/τ = |τrec −τ|/τ, varying the low-mass cutoff of star-forming halos.
is shown as a function of z for all the cases where the Ourresultsaretimelyconsideringtheplethoraofexist-
i
fitting procedure converged. In most of our cases ∆τ/τ ing and planned global 21-cm experiments which might
features a broad minimum (of ∆z ∼ 2) within which remove the optical depth nuisance from the CMB cos-
i
the fractional error in τ is below 1%. The location of mology in near future, allowing for a much more precise
this feature is very close to the true beginning of EoR, determination of the cosmological parameters.
markedbygreybarsinFigure5whichcorrespondtothe
0.5−2%valuesofionizedfraction. Theminimalvalueof
the fractional error, which we quote in Table 1 together
with the corresponding z is below 0.1%, which is much
i
better thanthe current1σ confidencelevelofthePlanck We thank R. Barkana and A. Cohen for their contri-
satellite(∼24%). Incaseswherethereconstructiondoes bution to preceding works which provided a solid basis
not workwell andthe fractionalerrordoes not feature a for this paper. We thank R. Barkana for his valuable
minimum, ∆τ/τ remains below ∼ 10−20% level in the comments on the draft of this paper. This work was
0.5−2% range of the ionized fraction. supported in part NSF grant AST-1312034(for A.L.).
Precise Measurement of Reionization Optical Depth from Global 21-cm with Realistic Heating 9
100 100 100
10−1 10−1 10−1
10−2 10−2 10−2
∆ττ /10−3 ∆ττ /10−3 ∆ττ /10−3
10−4 10−4 10−4
13 14 15 16 17 18 13 14 15 16 17 18 21 22 23 24 25 26 27
z z z
i i i
Fig.5.— Total relativeerror ∆τ/τ is shown as a function of zi formassive halos (left), atomic cooling(middle) and molecular cooling
(right)scenarioswithhard(solid)andsoft(dashed)X-raysourcesofheatingefficiencyfX =0.3(blue),fX =1(red)andfX =30(green).
Thehorizontalblackdottedlinemarksthe∆τ/τ =0.01threshold. Wealsoshowthetruebeginningofreionizationinourmodels(shown
forhardSED withfX =1ineach case): thethick greybar marks 0.5%-2%range inionizedfractionand the crossmarks 1% ionization.
Hereweusearesolutionof∆zi=0.1,i.e.,ourerrorcurvesaresmoothedonthisscales.
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