Bulk viscosity of a gas of neutrinos and coupled scalar particles, in the era of recombination

0.15 MB·English

by  R. F. Sawyer

Bulk viscosity of a gas of neutrinos and coupled scalar particles, in the era of
recombination
R. F. Sawyer1
1Department of Physics, University of California at Santa Barbara, Santa Barbara, California 93106
Bulk viscosity may serve to damp sound waves in a system of neutrinos coupled to very light
scalarparticles,intheeraafternormalneutrinodecouplingbutbeforerecombination. Wecalculate
the bulk viscosity parameter in a minimal scheme involving the coupling of the two systems. We
add some remarks on thebulk viscosity of a system of fully ionized hydrogen plus photons.
PACSnumbers: 14.60.Lm,98.80.-k,95.35+d
6 INTRODUCTION can remove most of the neutrinos from the universe as
0 the temperature declines into the region in which the
0
Ithas longbeenknownthata simple formofbulk vis- neutrinos become non-relativistic, a region that roughly
2
cosity is a potential source of dissipation in mixtures of corresponds to the recombination region itself. However
n relativistic and nonrelativistic fluids. For the cosmolog- Hannestad[15] pointed outthat the acoustic oscillations
a
J ically interesting case of photons and matter in the era ofthefluidofinteractingneutrinosandφ’sinthispicture
just before recombination,where shear viscositydoes in- could alter the CMB angular power spectrum in ways
3
2 deed strongly damp waves for wave-lengths up to some that conflict with the data 1.
critical length, Weinberg [1] has given the expression for Beginning from this observation, Raffelt and Hannes-

1 the bulk viscosity parameter, tad [17] have given coupling constant limitations that a
v
true free-streaming requirement would put on the ν φ
5 1 ∂P 2 −
2 ζ =4aT4τγ . (1) coupling. But more detailed calculations of the effect of
5 h3−(cid:16)∂ρ(cid:17)ni ν scattering, annihilation or decay on the CMB and the
1 matter power spectrum appear to show that the land-
Here τ is a mean free time for photon in the medium
0 γ scape of coupling constant limitations is considerably
and the electron (and proton) number density n is held
6
more complicated that dictated by the “free-streaming”
0 fixed in the derivative. Since the derivative in (1) is of
/ the order (1/3 10−10) in the early universe, in view of criterion, and, indeed, that in some coupling schemes,
h − parameters that give a “tightly coupled” ν φ fluid at
p the large photon/electron ratio, the effect is absolutely −
the recombination temperature are compatible with the
- negligible in this system.
o data [18].
For what comes later in this paper, we comment on
r
t the parameter, τ . The development that led to (1) was The first purpose of the present paper is to point out
s γ
a basedinpartonthesolutionbyThomas[2]ofarelativis- a possible role for bulk viscosity in controlling the size
: tic transport equation for photons. However, Thomas’ of neutrino density fluctuations under some conditions.
v
i treatmentincludedonlyphotonabsorptionandemission. Clearlytheregionofapplicationwouldbeinbetweenthe
X When Comptonscatteringis the interactionmechanism, tightly coupled case and the free streaming case. The
r itappearsthatτ shouldbethemeanfreetimeforComp- origin of this dissipation is qualitatively similar to that
a γ
tonscatteringweightedbyanefficiencyfactorforenergy which underlies (1). We have two gases, one completely
transfer, that is to say, multiplied by a number of order relativistic, the other in the relativistic-non-relativistic
ofthenumberofcollisionsrequiredfortemperatureequi- transitional region. Thus if we apply an adiabatic com-
librationofasystemwhichinitiallyhasdifferenttemper- pression,they change temperature by different amounts.
atures for the matter and for the radiation. When the There is a time lag in the equilibration, and consequent
scatteringmechanismisComptonscatteringandthesys- creationofentropy. Now,however,theanalogofthe fac-
temisneartherecombinationtemperaturethisisalarge torthatproducedthesquareoftheelectron-photonratio
number indeed. in(1)neednotgiveanextremelysmallnumber,sincethe
The motivation for the present paper is the possible numbers of ν’s and φ’s are comparable.
existenceofascalarfield,φ,withmasssmallerthanneu- In principle, a formalism for the evolution of inhomo-
trino masses and coupled to neutrinos. Over time, the-
orists have found a number of reasons to entertain such
a possibility [3]- [8]. The most stringent bounds on the
couplingstrength,atleastuntilrecently,havecomefrom 1 In ref. [16] from 1987 it had already been noted that an inter-
supernovae [9]- [13]. Recently Beacom, Bell and Dodle- actingseaofneutrinosandmajoronscouldformstructureinthe
universe; of course the details of the interesting application of
son [14] demonstrated the rather surprising result that,
these ideasarenow different, inviewofallofthe datathat has
staying within these bounds, processes ν +ν φ+φ comeinsincethen.
→
2
geneities that followed the detailed evolution of the po- The formalism that we shall present is completely ap-
sition and momentum distributions of each kind of par- plicable to the case of photons and ordinary matter and
ticle would have no need at all to mention viscosity ex- this may possibly be of interest in other astronomical
plicitly. Indeed, shear viscosity is subsumed within the contexts. We shall apply our method to the ionized-
standard evolution codes for the baryon, electron, pho- hydrogen system, with a result related to (1)
tonsystems,enteringatthe quadrupolelevelwherenon-
isotropic stresses are taken into account. Bulk viscosity
CALCULATION
is not included, nor need it be, in the system of matter
plus photons; but interacting neutrinos can be another
story. We consider an adiabatic compression specified by a
There are two important ways in which our ν φ sys- value of δV/V, where we do not allow energy transfer
tem requires computational mechanics somewha−t differ- between−the two species, leading to respective tempera-
entfromthosenecessarytoderivetheWeinbergformula: turechangesforthespeciesofδTφ,ν. Definingtheindices
γ for the respective species as,
1) In contrast to the case with matter and photons, φ,ν
where there is a conserved number of protons and elec-
δT δV
φ,ν
trons,the neutrino systemwillbe takento be symmetri- = γφ,ν 1 , (3)
Tφ,ν − V (cid:16) − (cid:17)
calbetweenparticlesandantiparticles,andthe annihila-
tion process into φ’s is an integral part of the evolution. we take
This makesthe determinationofthe analogofthe factor
δV
in (1), (1 [∂p ), different than in the photon-matter =beiωt, (4)
case. 3 − ∂ρin V
whereitistobeunderstoodthatweconsistentlytakereal
2) Looking at the parametersfor the problem at hand
parts to get linear perturbations in the physical quanti-
it appears that the frequency dependence of the bulk
ties. Thetimerateofchangeofthetemperaturesisthus,
viscosity will play a role in the application. The con-
sequencesofthis dependencehavebeenimportantinthe d
role of bulk viscosity in damping the gravitational radi- Tφ,ν =ib(γφ,ν 1)ωTφ,ν. (5)
dt −
ation instabilities in rapidly rotating neutron stars [19]-
[22]. This dependence, being essentially a non-local ef- The rate of heat transfer per unit volume between the
fect, does not rigorously fit into the framework that is two baths is of the order of,
best suited to the analysis of a relativistic fluid, viz, ad-
(T T )
dition of an additional term in the energy-momentum Q˙ =Γρ φ− ν , (6)
φ
T
tensor in the comoving system, 0
where T is the mean temperature of both baths, ρ the
0 φ
δTi,j = δi,jζ(ω) U, (2) energy density of the φ particle bath, and Γ an effective
− ∇·
collision rate for φ’s in the medium. By “effective” we
whereUisthespacecomponentofthefour-velocity. We mean the rate for collisions of a φ with the ν cloud, sup-
believe that it is legitimate, however, to crank the for- plemented by rate for φ+φ 2ν, and multiplied by
→
malism forward using (2), and then, at the point when an energy transfer efficiency factor which is of the order
Fourier transforms are introduced in position space, to of, but less than, unity, (and model dependent as well).
evaluate ζ(ω) at ω =v k where v is the speed of sound Adding the effect of heat transfer to the right hand side
s s
in the medium. of (5) gives,
In one respect, however, our ν φ system is simpler
d T (t) T (t)
tinhaanstinhgelemcaotltleisri-opnhoistornelsaytisvteemly;etffih−ecieennte.rgTyheiqsuisilibbercaatuiosne dtTφ(t)=i(γφ−1)bωT0eiωt−Γρφh φ T−0 ν i[cV(φ)]−1,
d T (t) T (t)
we have the direct transformation, ν+ν φ+φ.2 T (t)=i(γ 1)bωT eiωt+Γρ φ − ν [c(ν)]−1.
→ dt ν ν − 0 φh T0 i V
(7)
2 In models that couple light scalars to neutrinos, the neutrinos Subtracting, defining δT(t)=Tφ(t)−Tν(t), and solving,
mightbetakeneitherasDiracneutrinos,withpossiblecouplings we obtain,
that preserve lepton number, leading to ν+ν¯ → φ+φ, or as
dwMieffaejhoraervanentaν(inn+eauνntrg→iunloaφsr+tdheφap.tenvTdihoelenactaeem,lpfeolpirttuoenxdaensmufpmolreb)tehrb,euittnwwowehcciaacsnhestcaaaksreee δT(t)=Rehiω+Γρiφω([(cγφVφ]−−1γ+ν)[TcνV0b]−1)T0−1eiωti. (8)
anestimateoftotal rate,asafunctionofcoupling,G,tobethe
In the calculation of work, P dV, below, the volume
sameinbothcases. Thedistinctionwillnotenterfurtherinthis
paper,except hereandthereinterminology. differential will be given by [Rd/dt(δV)]dt = bVωsin(ωt).
3
We will therefore retain only the sin(ωt) terms in the asΓ=G4Tg(m /T),where Gis the ν χ coupling con-
ν
−
δT’s, since the cos(ωt) terms do not contribute to the stant, and the dimensionless function g(m /T) is given
ν
dissipation. We denote these parts and the associated in the appendix. Introducing k 3ω, we write ζ in the
≈
pressure changes by δTs, δPs. form,
From (7) we then find that the sin(ωt) term in the os-
cillationofthetemperaturedifferenceisdividedbetween 104G−4ρ T−1H mν
the two baths as, ζ = φ 0 (cid:16)T0 (cid:17), (16)
1+k2/k2
0
[c(φ)]−1
δTs(t)=δTs(t) V , where
φ [c(φ)]−1+[c(ν)]−1
V V 10−4(γ 1/3) αν 1/3 T2
[c(ν)]−1 H = ν − (cid:16)cV(ν) − (cid:17) 0 . (17)
δTνs(t)=−δTs(t)[c(φ)]−1V+[c(ν)]−1 . (9) g(mν/T0) [c(Vφ)]−1+[cV(ν)]−1 2ρ2φ
V V (cid:16) (cid:17)
Next we calculate the induced pressure changes of the The “cutoff” wave number k is given as,
0
two clouds,
m
k =10−3G4T F ν , (18)
δPs(t)=δTs(t) dPφ α δTs(t), 0 0 (cid:16)T0 (cid:17)
φ φ h dT i≡ φ φ where
δPνs(t)=δTνs(t)hddPTνi≡ανδTνs(t). (10) F(cid:16)mT0ν(cid:17)=103[g(mν/T0)]1/23−1/2
Taking the φ mass to be zero, we list the three relevant ρ T−1 [c(φ)]−1+[c(ν)]−1 .
parameters for the φ cloud, × φ 0 (cid:16) V V (cid:17)
(19)
2π2T3
αφ = 450 , c(Vφ) =3αφ , γφ =4/3, (11) F and H are dimensionless functions of mν/T0, plotted
in figs. 1 and 2 for the range of values in which we are
where we use units ¯h = c = k = 1. For the neutrino
B most interested. They are both of order unity in the
cloud in the domain in which the mass plays a role, nu-
semirelativistic region in which there may be significant
mericalcalculationsofthe correspondingparametersare
effects of bulk viscosity. The calculation of the compo-
described in the appendix.
nents of the above functions, α , γ , and g is discussed
ν ν
We write the net pressure oscillation amplitude as,
in the appendix.
δPs+δPs, obtaining,
φ ν
H
δPs δPs+δPs = ωbζsinωt,
≡ φ ν −
2.5
(12)
2
where
1.5
(γ γ )Γρ α 1
ν φ φ ν
ζ = − . 1
ω2+Γ2ρ2([c(φ)]−1+[c(ν)]−1)2T−2 hc(ν) − 3i
φ V V 0 V
0.5
(13)
m T-1
3 4 5 6 7 8 9
The amount of mechanical energy converted to thermal
energy in one period is given by
2π/ω
∆E = dt2ζωsin(ωt)dV(t)=2πωb2ζ, (14) FIG. 1: The function H(mν/T) which appears in the bulk
Z
0 viscosity result, (16).
giving the time averageddissipation per unit volume,
Thedampingrateforaplanesoundwaveinamedium
with bulk viscosity, ζ, is given by eq.(2.56) of ref.[1],
E˙/V =ζω2b2/2, (15)
k2ζ
identifying ζ as the bulk viscosity. Γdamp = . (20)
2(ρ+p)
Using eq.(5)of ref.[18] and stayingin the temperature
regionthat is (nearly) non-relativistic for the ν’s we cal- We divide by the frequency for a sound wave of wave-
culate the reactionrate forφ+φ ν+ν in the mixture numberk,takev 1/√3,andshifttoappropriateunits,
s
→ ≈
4
F
and absorption (not scattering) provides the mechanism
2.5 for energy transfer, and in which the photon absorp-
tionrateisindependentofenergy,bybeginningfromthe
2
Boltzmannequationforthe photondistributionfunction
f(ω),
1.5
∂
m T-1 ∂tf =τγ−1[(1−e−ω/Tm)f −e−ω/Tm]. (24)
3 4 5 6 7 8 9
WeestimatetherateofenergytransferQ˙ andfromitthe
coefficient Γ in (6) by inserting f = (exp[ω/T 1)−1,
γ
−
expanding to first order in (T T ), multiplying by ω
γ m
−
and integrating d3k. We obtain,
FIG. 2: The function F(mν/T) which appears in the result
for the wave-numbercutoff k0 (18). ∞
1 1
Γ= dωω4 =3.83τ−1,
τ W(T )Z [exp(ω/T ) 1] γ
γ 0 0 0 −
obtaining, (25)
k 10−6 4 .2eV where,
Γ /(kv ) .28
damp s ≈ (cid:16)[1Mpc]−1(cid:17)(cid:16) G (cid:17) (cid:16) T0 (cid:17)
1
×H(cid:16)mT0ν(cid:17)(1+k2/k02)−1. (21) W(T0)=Z dωω3exp(ω/T0)−1. (26)
We can directly apply (23) to determine bulk viscos-
where
ity in regions in which it is dominated by the “free-free”
k0 G 4 T0 mν photoemissionandphotoabsorptionprocess,rather than
=30.3 F . (22)
[1Mpc]−1 (cid:16)10−6(cid:17) (cid:16).2eV (cid:17) (cid:16)T0 (cid:17) by Compton scattering. This process is completely ef-
ficient in energy transfer, so that our previous caveats
From(21)and(22)weseethattherecanbestrongdamp-
with respect to an efficiency factor do not apply. The
ing in several oscillations when the system is at the re-
results will be most interesting in a system in which we
combination temperature, for a neutrino mass that is of
have neither n >> n , nor n << n . We take the
the order of 1 eV and coupling G of the order 10−6. e γ e γ
photo-absorption rate, Γ from the Kramers formula,
abs
modified by the Gaunt factor correction [23];
MATTER PLUS PHOTON SYSTEMS.
16πe6n2 2π ω
Γ (ω)= e eω/(2T)K , (27)
abs 3m ω3 rm T 0 2T
e e (cid:16) (cid:17)
All of our development is applicable to the system of
ionized hydrogen considered by Weinberg, and we com- where K is the modified Bessel function.
0
pare our result for this case to (1) in the low-frequency Defining the parameterto be used inthis casefor Γ in
limit, ζ(ω = 0). This will provide both a check on our (6) by Γ , we have, in place of (25),
ff
pedestrianformalism,andpossibleapplicationtoionized
hydrogen systems at much higher densities and temper- 1 ∞ ω4
atures than prevail in the recombination region in cos- Γ = dωΓ (ω)
ff abs
W(T )Z exp(ω/T ) 1
mology. In (13), replacing the subscript φ by γ and 0 0 0 −
ν by “mat”, we use the adiabatic indices for the mat- =114.e6m−3/2T−7/2n2. (28)
e 0 e
ter, γ = 5/3, α = 2n and the specific heat,
mat mat e
c(mat) = 3n where n is the density of the protons and TakingT/meastheefficiencyfactorforenergytransfer
V e e
in the Compton process, we find that the bulk viscosity
the electrons. For the photons we use the numbers from
is determined by Γ , rather than Compton scattering
(11) but with α and C doubled, since there are two ff
V
in all regions except those in which n << T3. In the
polarization states. We obtain, e
latter regions,however it is much smaller than the shear
16aT4n2 viscosity by virtue of the n dependence of (23).
ζ(0)= e , (23) e
Γ(4aT3+3n )2
e
with a = π2/15, the black-body constant in our units. DISCUSSION
The Weinberg formula (1) yields the same result if we
choose Γ = 4τ−1. We can check the relation between The principal result of this paper is the expression for
γ
these parameters, in a system in which photon emission thebulkviscosityoftheinteractinggasofν’sandφ’s. We
5
took one neutrino flavor, and an initial state with van- neutrino (and anti-neutrino, if we use Dirac ν’s) number
ishing chemical potential, i. e. equal numbers of ν’s and in our periodic changes we need to introduce an oscillat-
ν¯’s (or equal numbers of right-handed and left-handed ing chemicalpotential perturbationδµ as wellasanos-
ν
neutrinos in the Majorana variation). cillatingvalue forδV/V. Forthe Dirac neutrinocasethe
For the tightly coupled case with, say, G 10−5, we δµ are the same for neutrino and antineutrino (in con-
s
≈
wouldfindverylittle dissipationinour models. Forcou- trast to an equilibrium case, where they would be equal
plings of G 10−6, and a neutrino mass of order of an and opposite if there were zero net lepton number.) We
≈
eV we find relatively strong damping of a plane sound now have an energy equation,
wave in the medium. We note the following about this
δV ∂ρ ∂ρ
regime: (P +ρ) = νδT νδµ , (29)
ν
V −∂T − ∂µ
1) In the simplest coupling schemes, at least, there
would be a negligible number of φ’s produced prior to and a number conservation equation,
neutrino-matter decoupling. But for G 10−6 there
would be production at a rate that serves≈to create an δV 1 ∂nν ∂nν
= δT + δµ . (30)
equilibrium distribution during the period in which the V −nνh ∂T ∂µν i
neutrinos are still completely relativistic and prior to
where µ is to be evaluated at zero after differentiation.
photon decoupling. ν
Defining,
2) In this domain, the shear viscosity and bulk vis-
cosity will make comparable contributions to the damp- 1 ∂ρ 1 ∂n 1∂n 1 ∂ρ −1
ν ν
A= ,
ing of a plane wave. However, when the shift is made hρν +pν ∂T − n∂Tihn∂µ − p+ρν ∂µνi
to a proper cosmological calculation, in which the shear
(31)
viscosity is implicit in the way the scattering rate en-
ters the quadrupole terms, and the bulk velocity damps we have
monopoles, it is less clear to us what the relative contri-
butions of bulk viscosity to the actual signal will be. δµν =AδT , (32)
As we move into the region G < 10−6 the dissipation
becomesgreater,anditappearsthatoureffectswouldbe ∂ρ ∂ρ
an essentialelement of the analysis of the perturbations. γν 1= ν +A ν [pν +ρν]−1 ,
− h∂T ∂µνi
Wenotethattheauthorsofref[17]foundthattheirfree-
streaming criterion ruled out couplings down to a value ∂pν
α = . (33)
G 10−7. ν ∂T
≈
Finally,aswillhavebeenobvious,wemadenoattempt
Thequantitiesabovearecalculatedfromtheexpressions,
to include the complications that would arise from three
∞
flavorsofneutrinoswithamasssplittingschemetoagree 1 1
n = p2dp , (34)
with the data, let alone the possible complications of φ ν π2 Z 1+e(E−µν)/T
0
couplingsthatcarryanythingotherthantheunitmatrix
in flavor space. When couplings that allow ν decays are ∞
1 E
included,thebasicphysicsofourdescriptionremainsun- ρ = p2dp , (35)
changed, but the numerology concerning the interesting ν π2 Z0 1+e(E−µν)/T
domain of G’s becomes very different.
I thank Nicole Bell for discussions. This work was 1 ∞ p2/E
p = p2dp , (36)
supported in part by NSF grant PHY-04559 18. ν π2 Z 1+e(E−µν)/T
0
where E = p2+m2. The parameters γ 1 , α are
APPENDIX ν ν − ν
functions ofpthe single variable, T/m . In figs. 3 and 4.
ν
we show plots of the two parameters.
We need the temperature response γ 1 to an adi- Note that γ approaches 4/3 at high temperatures, as
ν ν
−
abatic change in volume, and the pressure response α expected. Wenoteaborderlineinconsistencyinourpro-
ν
tothistemperatureresponse,allofthis withthe interac- cedures, above, in that we will trust interactions that
tions between ν and φ that transfer energy between the equilibrate the neutrino distribution thermally, such as
bathsturnedoff. Acomplicationisthatinthetrueequi- ν ν scattering (fromφ exchange)to do their job, while
−
librium state, in the presence of the annihilation mech- turning off (schematically) the change in neutrino num-
anism, the number of ν’s and ν¯’s (the latter, if we use berandtransferofenergybetweenthetwoseas,although
Diracν’s)arenotconserved. Theaveragestate,towhich all of these effects are of the same order in G.
weapplyourperiodicdisturbance,thusretainsthevalue Forthefunctiong(m /T)wetaketheexpressiongiven
ν
µ = 0 for neutrino chemical potentials. But to conserve in eq(5) of ref. [14], for the ν annihilation rate, and
6
γ
ν [13] G. M. Fuller, R. Mayle, and J. R. Wilson, Astrophys.J.
1.55 332, 826 (1988)
[14] J. F. Beacom, N. F. Bell, and S. Dodelson,
1.5 Phys.Rev.Lett. 93, 121302 (2004), astro-ph/0404585
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1.45 [16] G. Raffelt and J. Silk, Phys.Lett. 192, 65 (1987)
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m T-1
(2005),hep-ph/0509278
2 4 6 8 10
[18] N. F. Bell, E. Pierpaoli, and K. Sigurdson,
1.35
astro-ph/0511410
[19] C. Cutler, L. Lindblom, R. Splinter, Astrophys. J. 363,
603 (1990)
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FIG. 3: Theindex γν(mν/T). [21] P.Haensel and R. Schaeffer, Phys. Rev. D45, 4708
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[22] J. Madsen, Phys. Rev. D46, 3290 (1992) Authors: X.
α χ-1
ν Chen, E. A. Spiegel (Astronomy Department, Columbia
University)Comments: 17pages.Latexwithrefereestyle
0.25
file of MNRAS (mn.sty). MNRAS, in press Journal-ref:
0.225
Mon.Not.Roy.Astron.Soc. 323 (2001) 865
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0.175 (1996)
0.15
0.125
m T-1
2 4 6 8 10
FIG. 4: The function αν(mν/T) expressed as a multiple of
thespecific heat χ=cV.
multiply by the appropriate density ratio to obtain the
estimate for the rate for φ+φ ν+ν,
→
G4 T m T 3/2n
Γ= ν φe−mν/T . (37)
64πm3 2π n
ν(cid:16) (cid:17) ν
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