Heating and cooling of the intergalactic medium by resonance photons

0.15 MB·

by  Leonid Chuzhoy

Draftversion February5,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
HEATING AND COOLING OF THE EARLY INTERGALACTIC MEDIUM BY RESONANCE PHOTONS
Leonid Chuzhoy 1 and Paul R. Shapiro 1
Draft versionFebruary 5, 2008
ABSTRACT
During the epoch of reionization a large number of photons were produced with frequencies below
the hydrogen Lyman limit. After redshifting into the closest resonance, these photons underwent
multiple scatterings with atoms. We examine the effect of these scatterings on the temperature of
the neutral intergalactic medium (IGM). Continuum photons, emitted between the Lyα and Lyγ
frequencies, heat the gas after being redshifted into the H Lyα or D Lyβ resonance. By contrast,
7
photonsemittedbetweentheLyγ andLy-limitfrequencies,produceeffectivecoolingofthe gas. Prior
0
0 toreionization,theequilibriumtemperatureof 100Kforhydrogenandheliumatomsissetbythese
∼
2 two competing processes. At the same time, Lyβ resonance photons thermally decouple deuterium
fromotherspecies,raisingitstemperatureashighas104 K.Ourresultshaveimportantconsequences
n
forthecosmic21-cmbackgroundandtheentropyflooroftheearlyIGMwhichcanaffectstarformation
a
and reionization.
J
Subject headings: intergalactic medium – diffuse radiation – radiation mechanisms: general
7
2

1. INTRODUCTION some other mechanism such as photoelectric heating by
4
a strong X-ray background, can raise the IGM tempera-
v Duringtheepochofreionization,UVphotonsbetween
ture above T without fully ionizing it.
3 the Lyα and Ly-limit frequencies are produced in abun- CMB
8 dance. Unlike the ionizing photons above the Ly-limit, However, two important physical mechanisms were
4 whose mean free path in the neutral IGM is quite short, missing in previous calculations. The first is the forbid-
4 photons between Lyα and the Ly-limit scatter only af- den transition from the 2s to 1s level in hydrogen. The
0 cascade which follows absorptionof most photons which
ter redshifting into one of the atomic resonances, which
6 redshiftintohighatomicresonances(Lyβ,Lyγ,Lyδetc.)
allows them to move freely through cosmological dis-
0 proceedspreferentiallynotviathe 2plevel(whose decay
tances. Eventually scattering these photons exchange
/ to 1s produces a Lyα “injected photon”) but via the 2s
h energy with atoms, thereby providing a mechanism for
level, instead. This significantly reduces the number of
p raising or lowering the gas temperature far from any ra-
- diation source. injected photons, raisingthe equilibrium temperature to
o about 100 K. The second is the impact of deuterium.
Several authors previously considered this process.
r Despiteitsverylowabundance,deuteriummakesanim-
t Madau, Meiksin & Rees (1997) estimated that photons
s portantcontributiontothegasheatingbyitsinteraction
loseaboutonepercentoftheirenergytothegasasthey
a
with Lyβ photons. At high radiation intensities, the D
: redshiftthroughLyαresonance. Theyfoundaccordingly
v Lyβ resonance becomes more important for heating the
that, if Lyα scattering controls the HI hyperfine level
i gasthan HLyα. In addition,scatterings by D atoms re-
X population, as required to produce a 21-cm background
duce the number of injected photons for hydrogen, thus
distinctfromtheCosmicMicrowaveBackground(CMB),
r decreasing the cooling rate.
a recoil heating sufficed to raise T above T and
IGM CMB
In 2 and 3 we calculate the effect of resonant scat-
to move the 21-cm background into emission. Later,
§ §
tering on H and D atoms, respectively. In 4 and 5
however, Chen & Miralda-Escude (2004) showed that § §
we discuss the impact of this scattering on the thermal
the inclusion of atomic thermal motion (neglected by
evolution, 21-cm signal and reionization history of the
Madauetal.) intheanalysisreducestheheatingrateby
IGM.
at least three orders of magnitude. Furthermore, they
showed, while photons emitted between Lyα and Lyβ 2. HYDROGENRESONANCES
frequencies (called by them the “continuum photons”)
When photons redshift through the Lyα resonance,
heat the gas when redshifted into Lyα resonance, those
multiple scatterings affect their spectrum. The in-
injected directly into Lyα by the cascade from higher
tensity J(ν) varies with frequency in the neighbor-
resonances (called the “injected photons”), cool the gas.
hood of the resonance at ν according to the following
With increasing gas temperature, the efficiency of “con- α
(Chuzhoy & Shapiro 2005)
tinuum photons” as heaters declines, while that of “in-
jected photons” as coolers rises. Therefore, if, as these J(x)=J(0)e−2π3γax3−2ηx, (1)
authors claimed, the numbers of these two types of pho-
tonsweresimilar,thencoolingwouldprevailattempera- for injected photons and x>0, otherwise
tbuyreLsyaαbpovuem1p0inKg.wIfousold, thhaev2e1b-cemenbianckagbrsoourpntdiopnr,oudnulceesds J(x)=2πJ0γa−1e−2π3γax3−2ηx x e2π3γaz3+2ηzz2dz, (2)
−∞
Z
1McDonald Observatory and Department of Astronomy, The where a = A (2kT /mc2)−1/2/4πν , T is the hy-
21 H α H
University of Texas at Austin, RLM 16.206, Austin, TX 78712, drogen kinetic temperature, A is the Einstein spon-
USA;[email protected]; [email protected] 21
taneous emission coefficient of the Lyα transition, x =
2
(ν/ν 1)/(2kT /mc2)1/2 isthefrequencydistancefrom atoms, produce a cascade to the 2s level, from which
α H
−
line center divided by the thermal width of the reso- an electron decays to the ground level by emitting
nance, γ−1(1 + 0.4/T )−1 = τ 1 is the Gunn- two photons below Lyα (as also noted by Hirata 2006;
s GP
≫
Peterson optical depth, J is the UV intensity far away Pritchard& Furlanetto 2006). Another fraction are ab-
0
from the resonance. The recoil parameter, η, equals sorbed by D atoms and either destroyed or converted to
[hν /(2kT mc2)1/2][(1+0.4/T )/(1+0.4/T )],whereT D Lyα photons, as previously explained. The rest, after
α H s H s
is the spin temperature of the 21 cm hyperfine transi- redshifting into the closest H resonance and producing
tion, which equals T (T ) at high (low) radiation a cascade to the 2p level, become the injected photons.
H CMB
intensities. The intensity at x=0, J(0), is given by Therefore, the ratio of injected and continuum photons,
is
JJ(00) = πζ(cid:16)J31(ζ)√−3J−13(ζ)(cid:17) + 1F2(cid:18)1;31,23;−ζ42(cid:19)(,3) JJci =n∞=3Zνnνn+1Js(ν)Pn(1−fD,n)dν/
X
whereζ =(16η3a/9πγ)1/2,1F2isahypergeometricfunc- ν3 ∞ νn+1
J (ν)dν + J (ν)P f dν , (9)
tionandJ1 andJ−1 arethe Besselfunctions ofthe first s s n D,n
kind. 3 3 "Zν2 nX=3Zνn #
Since photons to the red (blue) of the resonance scat- where νn is the frequency of np 1s transition (i.e.,
ter preferentially off atoms moving towards (away) from νn = νlim(1 n−2), where νlim =→13.6 h−1 eV is the
−
them,theaverageenergyanatomgainsfromascattering Ly-limit frequency), Pn is the fraction of all cascades
photon depends on its frequency: from level np that go through 2p (0, 0.26 for n = 3 and
4, respectively, and roughly 1/3 for n > 4), J (ν) is the
∆E(x)= (hν)2 1 kTHφ′(x) , (4) spectralprofileoftheradiationsource,andfD,ns isthera-
mc2 − h φ(x) tio of the cascades from level np occurring in deuterium
(cid:18) (cid:19)
andhydrogen. Itcanbeshownthat,sincethedeuterium
where φ(x) is the normalized scattering cross-section
optical depth, τ , in the np 1s transition, was low
D,n
(gCashufrzohmoyea&chShpahpoitroon2a0s05it).paTshseestothtaroluenghertghyegraeisnonofanthcee during reionization, fD,n ≈ 1→− e−τD,n(1−frec,n), where
f is the probability for the electron in level np to
is determined by the radiation intensity, the probability rec,n
decay directly to 1s. For a hot source with surface tem-
of scattering and the average gain at each frequency:
perature T 105 K, J (ν) ν2 between Lyα and Ly-
bb s
≫ ∝
J(x) limit, and J/J 0.17 (the exact value depends on the
i c
∆E = ∆E(x)φ(x)dx, (5) ≈
tot J opticaldepth ofdeuterium whichis redshift dependent).
Z 0 Colder sources would produce a lower J/J ratio, but
i c
Figure1showstheheatingandthecoolingratesobtained unless the spectrum color temperature is below 5 104
by numerical integration of equation (5).2 For TH &100 K, Ji/Jc >0.1. ∼ ·
K and the range of optical depths corresponding to the Combining equations(6) and(7), we find that, for the
mean density IGM at 10 < z < 30, ∆Etot can be well mean IGM, the equilibrium temperature at which cool-
approximated by the asymptotes ing by injected photons balances heating by continuum
∆E /k 0.37T−1/3(1+z)K (6) photons is
c ∼ H 1/2 −3/2
1+z J/J
for continuum photons and T 130K i c (10)
eq
≈ 10 0.15
∆Ei/k∼−0.3TH1/3(1+z)1/2 K (7) This greatly exce(cid:18)eds the(cid:19)me(cid:18)an tem(cid:19)perature prior to
reionization, if only adiabatic cooling occurs once the
for injected photons. The total heating/cooling rate is
IGM decouples from the CMB.
J
H =N˙ (∆E + i∆E), (8) 3. DEUTERIUMRESONANCES
α α c i
J
c
Unlike hydrogen, the most important res-
whereN˙ isthenumberofthephotonsthatpassthrough onance for deuterium is not Lyα, but Lyβ
α
the Lyα resonance per H atom per unit time. (Chuzhoy & Shapiro 2005). As photons redshift
The continuum photons (i.e., those that are red- through the D Lyβ resonance, a significant fraction of
shifted into the Lyα resonance) can be produced in them aredestroyedvia absorptionandcascade. Priorto
two ways. Most originate as photons emitted be- reionization(z &10),thephotondestructionprobability
tween Lyα and Lyβ, which eventually redshift into the is above 0.2. Therefore the blue wing of the resonance
Lyα resonance. The rest come from higher-frequency is significantly higher than the red wing, so that the
photons that are absorbed by D atoms and cascade radiation color temperature around the resonance is
into the D Lyα resonance, from which they then red- negative. Because of their negative color temperature,
shift into the H Lyα resonance. The injected pho- the D Lyβ photons are much more efficient than the
tons (i.e., those injected directly into the Lyα reso- Lyα photons at heating the gas.
nance)comefromthe photonsemittedbetweenLyγ and If the velocity distribution of deuterium atoms is
the Ly-limit. Most of the latter photons, when red- Maxwellian,thenthe radiationintensityaroundtheLyβ
shifted into the closest resonance and absorbed by H resonance varies according to
2 Results are virtually insensitive to the choice of Lorentz vs e−x2J(x) γ J′(x)=
D
Voigtprofileforφ(x). √π −
3
∞
′ ′ ′
0.5f Erfc(Max[x , x])J(x)dx, (11)
rec,3 3.5
−∞ | | | |
Z
where γ−1 = τ 3 103(1 + z)3/2(n /n ) on) 3
D D,3 ≈ · D H ot
(Chuzhoy & Shapiro 2005). Solving equations (5) and h
p 2.5
(11) numerically, we find that for neutral IGM between α
y
z =10and20theaverageenergyeachLyβ photontrans- L
fers to the gas is er 2
p
n
∆E /k 0.012T1/2 1+z 3/2 K, (12) aryo 1.5
tot ≈ D (cid:18) 15 (cid:19) er b 1
p
where T is the deuterium kinetic temperature. K
PhysicDal insight into the above result can be gained H ( 0.5
from the following argument. Photons are scattered by
0
atoms in whose rest frame they are close to resonance 0 1 2 3 4
10 10 10 10 10
(i.e., ν(1 v/c) νLyβ, where v is the velocity of an T (K)
− ≈
atom). Thus photons in the red (blue) wing of the res-
Fig. 1.— The heating/cooling rate due to continuum (solid
onance are scattered preferentially by atoms moving to-
and dash-dotted lines for Ts =TH and Ts =TCMB respectively)
wards (away) from them. When the spectrum around and injected (dashed line) H Lyα photons, and D Lyβ photons
the resonance is flat, then to first approximation, the (dottedline)forflatspectrumsource(Ji/Jc≈0.15,Jβ/Jc≈0.35)
energy gain of atoms from photons in the blue wing is and z =12. For Lyα and Lyβ photons the results are plotted vs
temperatureofHandDatoms,respectively.
compensated by energy loss to photons on the red wing.
However, when the spectrum is tilted, as in our case, balance between heating and cooling rates, L =H ,
D β
each extra photon on the blue wing adds to the gas
r∆aEblteott/okw∼ha(htνwLyeβg/okt)vitnheerqmuaal/tcion≈(01.20)5.TD1W/2eKno,tceomthpaat-, TD ≈5·103 K(cid:18)2n·D1/0n−H5(cid:19)−1 10N−˙β14!×
unlike H Lyα photons, D Lyβ photons become more ef-
ficient heaters when the temperature rises. Ωbh2 −1 1+z −3/2 σHD −1
. (15)
Since radiation affects H and D atoms differently, we 0.02 10 10−15cm2
now have to make separate estimates of their respective (cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17)
temperatures 3. From equation (12) we find that the 4. THERMALEVOLUTIONOFIGM
heating rate per D atom is When resonancephotons arethe only importantheat-
H 4 10−28 erg s−1 ing source, the temperature of H and D atoms evolves
β ≈ · · × according to
n /n −1 N˙ 1+z 3/2
(cid:18)2·D10−H5(cid:19) 10−β14!(cid:18) 10 (cid:19) TD1/2, (13) ddTtH =32k(Hα+LDnnDH)− 43TtH, (16)
dT 2 4T
where N˙β is the number of photons passing through the D = (Hβ LD) D (17)
Lyβ resonance per H atom per second. Elastic collisions dt 3k − − 3t
with H atoms and to a smaller extent, with He atoms, wherethe term4T/3taccountsforadiabaticcoolingdue
make D atoms lose energy at a rate to Hubble expansion. The hydrogentemperature at any
redshift,T (z),doesnotdependverystronglyonthehis-
L k(T T ) 3kTD + 3kTH 1/2n σ = toryofphoHtonproduction(see Fig. 2). ThusTH(z) may
D ≈ D− H 2m 2m H HD beexpressedasafunctionofthetotalnumberofphotons
(cid:18) D H(cid:19)
betweenLyαandtheLy-limitperbaryonemittedbyred-
1/2
T
1.3 10−31 erg s−1(TD TH) D +TH shiftz,Nγ,H(Fig. 3). IfPopIIIstarsarethemainradia-
· · − 2 × tionsources,then,bytheendofreionizationN 200
(cid:18) (cid:19) γ,H ∼
Ω0b.0h22 11+0z 3 10−σ1H5Dcm2 , (14) (∼C1ia0r0diK&,wMhaildeaTuD2i0s0r3a)i.sedThtois∼im1p04li(eσsHtDh/a1t0T−H15rcimse2s)−to1
(cid:18) (cid:19)(cid:18) (cid:19) (cid:16) (cid:17) K.However,bothmuchhigherandmuchlowermaximum
values for N are not completely excluded.
where σ is the momentum transfer cross-section for γ,H
HD
collisionsbetweenHandDatoms4. Whentheradiation
5. IMPLICATIONS
intensity is high, so that T T , T is set by the
D H D
≫ We have shown that, during the reionization epoch,
3 In principle the same is alsotrue for other species (He, p, e), resonance photons can heat the neutral IGM to about
but inpractice only the temperature of deuterium is significantly 100 K, but not much higher. When gas collapses out of
different.
theIGM,theentropycannotdecrease(withoutradiative
4 Some element of uncertainty is present inthe above equation
cooling), so its overdensity inside virialized halos cannot
due to the absence of experimental measurements of σHD. The
collisional cross sections of atoms belonging to different species exceed(T /T )3/2,whereT (T )arethetemperatures
vir 0 0 vir
can be roughly determined by the size of the atom, which for D before (after) collapse, respectively. Consequently, for
and H atoms gives σHD = 0.35·10−15cm2 (Clementietal. 1963;
minihalos with T . 4000 K that form out of gas pre-
Slater1964). However, since D and H atoms energy levels are vir
nearlydegenerateitispossiblethatσHD issignificantlylarger. heatedto100K,thepost-collapseoverdensityislessthan
4
minihalos. Minihalos would also be more easily photoe-
4
10
vaporated, thereby reducing their consumption of ioniz-
ing photons during reionization (Shapiro, Iliev & Raga
2004).
103 This preheating also has a strong impact on the red-
shifted 21-cm signal from the beginning of reionization.
UV resonance photons may raise T above T , thus
H CMB
K) 2 movingthe gasfromabsorptionto emission(seeFig. 4).
T (10 In fact, even relatively low intensities (.10 photons per
baryon) can drastically affect the signal.
Ifthe firstradiationsourcesdepositasubstantialfrac-
1
10 tionoftheirenergyinX-rays,theaveragetemperatureof
theIGMmayclimbevenhigher(Venkatesan et al. 2001;
50
0
10
15 20 25 30 35 40
z 0
Fig. 2.— Thermal evolution of H (solid lines) and D (dashed −50
lines)atoms forσHD =10−15 cm2 andNγ,H(z =11)=200. The
comovingradiationintensityisassumedtogrowase−z/∆z,where K) −100
∆z=1,2,3correspondtolower,middleandupperlines. m
(
b
T −150
3 δ
10
−200
−250
2
10 −300
15 20 25 30 35 40
)
K z
(
H
T Fig. 4.— Theevolution of H 21-cm differential brightness tem-
101 peratureforNγ,H(z=11)=10with(solid)andwithout(dotted)
taking the gas heating into account, and for Nγ,H(z =11)=200
with (dashed) and without (dashed-dotted) gas heating. The
comoving radiation intensity is assumed to grow as e−z/2 and
σHD=10−15 cm2.
0
10
−1 0 1 2 3
10 10 10 10 10 Oh & Haiman 2003). However, unless the X-ray spec-
N trumisveryhard,mostoftheenergyisdepositedwithin
γ,H
1 comoving Mpc of the source. By contrast, the UV
Fig. 3.— Hydrogen kinetic temperature at z = 15 (solid and ∼photons considered here, which travel cosmological dis-
dotted lines)and z =11 (dashed and dashed-dotted lines) versus
Nγ,H(z), for σHD = 10−15 cm2. The dotted and dashed-dotted thaenacteinsgbmefeocrheasnciasmtte.riTngh,erperfoordeu,ceinarevgeiroynshtohmaotgdenoenouots
linesaretheresultswhentheheatingofdeuteriumisneglected.
have a radiation source nearby, heating by resonance
200. Minihalos with such lowered central densities
photons can be more important even if, on average, X-
∼
form H molecules more slowly and are more vulnera-
2 ray heating is stronger.
ble to H photodissociation (Oh & Haiman 2003). The
2
photodissociation is caused by UV photons in the range
from11.6to13.6eV,whichoverlapsthe rangefrom10.2 This work was supported by the W.J. McDonald Fel-
to12.7eV(fromLyαtoLyγ)responsibleforpreheating. lowship for LC and NASA Astrophysical Theory Pro-
ThismaysuppressH coolingandstarformationinthese gram grants NAG5-10825 and NNG04G177G.
2
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