CAMB Notes - Antony Lewis

25 Pages·2014·0.63 MB·English

CAMB Notes
Antony Lewis1,∗
1DAMTP/CITA/IoA/Sussex
(Dated: April 2, 2014)
Brief summary of some CAMB definitions, parameterizations, and details of internal implemen-
tation. Plus some random notes.
I. THE CAMB CODE
The physics is contained in the equations.f90 file. Everything is in conformal time and units of megaparsecs. The
dtaudaroutinereturns1/a(cid:48) (scalefactora=S),wheretheconformalHubblerateisa(cid:48)/a. Thisisusedtocomputethe
conformal age of the universe, and background evolution for the pre-computed ionization history and should always
be finite. Background density variables include a factor of a2, like grho≡κa2ρ where κ≡8πG.
The initial conditions for the scalars and tensors are in the initial and initialt routines (see above for details).
The derivs and derivst functions return the differential equations - a vector y(cid:48) of derivatives of the variables being
evolved.
The output and outputt routines calculate the sources for integration against the Bessel functions.
A. Gauge and perturbation definitions
CAMB propagates the covariant equations in the zero-acceleration frame, in which the CDM velocity is zero. This
is equivalent to the synchronous gauge.
Note that CAMB uses the variable etak = kη , where η = −η/2 (in the CDM frame) is the usual synchronous
s s
gauge variable. Here η is the curvature perturbation variable as in the previous section. The fractional density
perturbation variables are called clxc (∆ ), clxg (∆ ), clxr (∆ ), clxb (∆ ) etc. The synchronous gauge variable
c γ ν b
h appearsintheCDMframeash(cid:48) =6h(cid:48) =2kZ. Thel=1momentsarehandledintermsoftheheatfluxesq where
s s i
ρ q =(ρ +p )v . The total heat flux appears as dgq=κa2(cid:80) ρ q in the code, and the total matter perturbation is
digriho=iκa2(cid:80)i iρ ∆ . i i i
i i i
Other variables are the shear σ and perturbation to the expansion rate Z. They are related by
2 (cid:88)
k2(β σ−Z)=κa2 ρ q =dgq
3 2 i i
i
(cid:88)
k2η =κa2 ρ ∆ −2kHZ =dgrho−2kHZ.
i i
i
In terms of synchronous gauge variables σ =(h˙ +6η˙ )/2k. Note that in the non-flat case η above is β η of Kodama
s s s 2
& Sasaki.
II. NEUTRINOS AND DARK RADIATION
We assume a Fermi-Dirac distribution for neutrinos when they are relativistic. For the massless case the result
is independent of the Fermi-Dirac assumption, so massless dark radiation can be modelled as effective neutrinos.
When the particles have non-negligible mass, the Fermi-Dirac assumption is not general (it is a good approximation
for massive active neutrinos, and is also equivalent via parameter redefinition with the Dodelson-Widrow model for
sterile neutrinos (see [1–3]), but may not be correct for more general forms of massive dark radiation).
∗URL:http://cosmologist.info
2
Massive neutrinos are assumed to decouple when relativistic and remain exactly thermal with zero chemical poten-
tial, so eigenstate i has occupation number (in natural units)
1
N (p)= (1)
i p
eTi +1
wherepisthemomentum. Allequationsformassiveneutrinoshavemassdependencedeterminedbythedimensionless
ratiousedinthecodenu masses(i)≡m c2/(k T0), whichdeterminestheparticlevelocitiesasafunctionofredshift.
i B i
Neutrinos are assumed to be highly non-relativistic today, so
ρ0 =m n0 =f Ω ρ0 (2)
i i i i ν c
wheref isthefractionofthetotalneutrinoenergydensityineigenstatei,andρ isthecriticaldensity. Numberden-
i c
sities are set from the thermal distribution at decoupling while relativistic, where for a Fermion with spin degeneracy
two:
3ζ(3)(k T )3
n = B i . (3)
i 2π2 (cid:126)3c3
Thedecouplingofthestandardactiveneutrinosisnearly,butnotentirely,completebythetimeofelectron-positron
annihilation. This leads to a slight heating of the neutrinos in addition to that expected for the photons and hence
to a small departure from the thermal equilibrium prediction T = (11/4)1/3T between the photon temperature T
γ ν γ
and the reference neutrino temperature T defined by this relation. The actual temperature T for any eigenstate is
ν i
given by reference to the thermal result T , so that
ν
T ≡g1/4T , (4)
i i ν
where this defines g , the neutrino degeneracy factor. Hence while relativistic
i
7 7(cid:18) 4 (cid:19)4/3
n =g3/4n ρ = a T4 =g ρ =g ρ , (5)
i i ν i 8 R i i ν i8 11 γ
wheren andρ aretheresultsforthethermalprediction. ρ (foronemasslessneutrino)isinthecodeasthevariable
ν ν ν
grhor≡8πGρ0/c2, which is used to set the densities of the actual species. The dimensionless mass variable is given
ν
by
m f Ω ρ0 f Ω ρ0 f Ω ρ0
i = i ν c = i ν c ∝ i ν c. (6)
T0 n0T0 g n0T0 g ρ0
i i i i ν ν i ν
For each eigenstate the code only needs m /T0 and ρ =g ρ , the latter being stored in grhormass(i)=g ×grhor
i i i i ν i
and grhornomass for massless neutrinos.
Note that if we fix the total Ω , having N neutrino eigenstates with equal g and f (and hence equal mass), is
ν s i i
(cid:80) (cid:80)
code-equivalent to having one neutrino with g = g , f = f , since m /T (which determines particle velocities)
(cid:80) i i i i i (cid:80)
and ρ (which determines the density) remain the same, even though physically having one state with g = g
i i i i
corresponds to having a smaller number density of hotter more-massive neutrinos. c neutrinos of degenerate mass
i
and the same temperature can therefore modelled simply as a single state with appropriately multiplied geff =c g .
i i i
The physical mass m inferred from Ω h2 does of course depend on the number of physical neutrino eigenstates.
i ν
Assuming the neutrinos are highly non-relativistic today, we have
Ω h2 ≈ 16ζ(3)(cid:0)k T0(cid:1)3 Gh2 (cid:88)c g3/4m ≈ (cid:80)icigi3/4mi . (7)
ν 11π B γ c5(cid:126)3H2 i i i 94.07eV
0 i
(taking T0 =2.7255K).
γ
Energy density in neutrinos or dark radiation is conventionally described using an effective energy density number
(cid:80)
N when relativistic, so N = c g . The default assumption is three active neutrinos with N = 3.046 [4, 5],
eff eff i i i eff
corresponding to g =3.046/3 for each of three neutrinos.
i
3
CAMB input parameters
Thedegeneracyparameterscanbespecifiedexplicitlyforeacheigenstate(andthemasslessneutrino),orthedegen-
eraciescanbesetinternallytogiveatotalN assumingalleigenstateshavethesametemperature,whichisthemost
eff
naturalassumptionforthecaseofthreeactiveneutrinos. Thisisdeterminedbytheshare delta neffparameter, as
described further below. In all cases when there is more than one massive eigenstate nu mass fractions=f f ...
1 2
determines the fraction of the energy density today Ω h2 that is associated with each eigenstate, and should sum
ν
to 1, e.g. nu mass fractions = 0.75 0.25 to have the first state of two eigenstates accounting for three quarters of
the total neutrino density today. When run CAMB prints out the physical eigenstate masses, calculated assuming a
thermal distribution (different in the Dodelson-Widrow re-interpretation).
1. share delta neff=T
In this case all neutrinos are set to have the same g per physical mass eigenstate. massive neutrinos=c c ...
i 1 2
isanarrayofnu mass eigenstatesintegers,specifyingthephysicalnumberofneutrinosper(degenerate)eigenstate.
The non-integer part of the real number massless neutrinos is used to determine the neutrino temperature, so the
(cid:80)
total effective density parameter is N =massless neutrinos+ c and for all species
eff i i
N
g = eff (8)
i [N ]
eff
where [] denotes the integer part.
Forexample,forthesimplecaseofonemassiveneutrinoandtwomasslessneutrinos,allwithstandardtemperatures
so N =3.046, you could use massless neutrinos=2.046, massive neutrinos=1.
eff
Foraninvertedhierarchyofdegeneratemass,youcouldhavemassless neutrinos=0.046,massive neutrinos=
3.
For mass splitting but with two degenerate, e.g.: nu mass eigenstates = 2, massless neutrinos = 0.046,
massive neutrinos=2 1, nu mass fractions=0.7 0.3.
Note that since the non-integer part of N is used to define the temperature, the physical model for massive
eff
neutrinos is discontinuous as the total crosses an integer boundary: N = 2.99 is two physical neutrinos with high
eff
temperature,N =3.046isthreephysicalneutrinoswithstandardtemperature. Howevertheresultsarecontinuous,
eff
only the interpretation in terms of physical mass changes abruptly. Also note that for massless neutrinos<1, no
massless neutrinos are actually included unless massive neutrinos=0.
2. share delta neff=F
This allows an arbitrary specification of different neutrino temperatures and masses, where now
nu mass degeneracies = geff geff ... must be specified explicitly. massless neutrinos determines the g for mass-
1 2
less neutrinos. The physical numbers of neutrinos is still specified by massive neutrinos, but does not affect the
result except when writing out the physical mass: power spectra etc. from massive neutrinos = 2 1 should be the
same as from massive neutrinos=1 1 for the same nu mass degeneracies, but the physical neutrino mass of the
first eigenstate is larger in the second case.
Note that the degeneracy inputs are the total, i.e. geff = c g , so nu mass degeneracies = 2,
i i i
massive neutrinos = 2 corresponds to two neutrinos with thermal temperature T (contributing 2 to the total
ν
N ), nu mass degeneracies = 1, massive neutrinos = 2 corresponds to two neutrinos with reduced temperature
eff
(contributing 1 to the total N )
eff
3. Further examples
To within rounding these combinations of input parameters are equivalent: share delta neff = T,
massive neutrinos = 1, massless neutrinos = 2.046 and share delta neff = F, massive neutrinos = 1,
massless neutrinos=2.0307, nu mass degeneracies=1.0153.
Consider 2 massless active and 1 massive active neutrino all at the standard temperature (3.046/3)1/4T , where
ν
the massive active neutrino has m =0.06eV and hence contributes Ωah2 =0.00064. If there is also one sterile (at a
i ν
different temperature) contributing ∆N to the total so that N =3.046+∆N , and the total density parameter
eff eff eff
4
1
1.4 10
1.2
0
10
1
−1
10
0.8
e e
x x
0.6 −2
10
0.4
−3
10
0.2
−4
0 10
0 5 10 15 20 25 0 5 10 15 20 25
z z
FIG.1: Threerecombinationhistoriesallwithτ =0.09. ThedashedlineisthemodeltypicallyusedbyCMBFASTandCAMB
prior to March 2008 with f =1. The black line is the new model with ∆ =0.5, the red line with ∆ =1.5.
z z
Ω h2 =0.00064+Ωsh2, then use: share delta neff=F, massless neutrinos=2.0307, nu mass eigenstates=
ν ν
2, massive neutrinos = 1 1, nu mass degeneracies = 1.0153 ∆N , nu mass fractions = X Y where X =
eff
0.00064/Ω h2 and Y =Ωsh2/Ω h2.
ν ν ν
III. REIONIZATION
CAMB’s reionization model is described in the appendix of [6] and reproduced here. The reionization.f90
module can easily be changed for different models. The optical depth to reionization is defined by
(cid:90) η0
τ = dηSnreionσ (9)
e T
0
wherenreion isthenumberdensityoffreeelectronsproducedbyreionizationandη istheconformaltimetoday. This
e 0
is not quite the same as the number density of electrons at late times because there is a small residual ionization
fractionfromrecombination. Atthelevelofprecisionrequiredwecanneglectthisdifference((cid:46)10−3),thoughCAMB
keeps the ionization history smooth my mapping smoothly onto the recombination-residual.
Since n ∝(1+z)3x (z), and reionization is expected to happen during matter domination,
e e
(cid:90) √ (cid:90)
τ ∝ dzx 1+z ∝ d[(1+z)3/2]x . (10)
e e
It is therefore handy to parameterize x as a function of y ≡(1+z)3/2. As of March 2008 CAMB’s default parame-
e
terization uses a tanh-based fitting function
(cid:20) (cid:18) (cid:19)(cid:21)
f y(z )−y
x (y)= 1+tanh re , (11)
e 2 ∆
y
where y(z ) = (1 + z )3/2 is where x = f/2: i.e. z measures where the reionization fraction is half of its
re re e re
maximum. Since the fitting function is antisymmetric about the mid point (and assuming it does not extend out of
matter domination)
(cid:90) η0 (cid:90) zre dη
τ = dηSnreionσ ≈akthom×f dz . (12)
e T dS
0 0
5
Inotherwords, withthisparameterizationtheopticaldepthagreeswiththatforaninstantaneousreionizationmodel
at the same z for all (matter-dominated) values of ∆ (CAMB’s variable akthom is n σ today). Except in early
re y p T
dark energy models this result is quite accurate for the expected range of z . In practice the input parameter is ∆
√ re z
and ∆ is taken to be 1.5 1+z ∆ .
y re z
To keep the implementation general (e.g for early dark energy models), the default reionization module actually
maps τ into z by doing a binary search. This method is also generally applicable for other monotonic models, e.g.
re
themodulealsowillworkfineusingawindowfunctioninadifferentpowerof(1+z)(i.e. Rionization zexp(cid:54)=3/2),
though in this case there is in general a more complicated relation between the optical depth and that of a sharp
reionization model. Changing the exponent allows flexibility in relatively how quickly reionization starts and ends,
with values 0.5−2.5 changing the EE power by a couple of percent at fixed τ.
Ifhydrogenfullyionizesf =1. Howeverthefirstionizationenergyofheliumissimilar,anditisoftenassumedthat
helium first re-ionizes in roughly the same way. In this case f = 1+f , where f = n /n is easily calculated
He He He H
from the input helium mass fraction Y . This is CAMB’s default value of f; typically f ∼1.08
He
In addition at z ∼ 3.5 helium probably gets doubly ionized. Due to the low redshift this only affects the optical
depth by ∼0.001, but for completeness this is included using a fixed tanh-like fitting function (modifying the above
result for τ appropriately). Some reionization histories are shown in Fig. 1.
IV. INITIAL POWER SPECTRA
A. Scalar parameterization
CAMBusesastandardrunningpowerlawmodelfortheprimordialsuper-horizonpowerspectrumP (k)ofcurvature
s
perturbations (and similarly for isocurvature modes), with
n n
lnP (k)=lnA +(n −1)ln(k/k )+ run[ln(k/k )]2+ run,run[ln(k/k )]3.
s s s s 2 s 6 s
The input parameters are pivot scale pivot scalar (k ), and scalar spectral index (n ), plus optional running
s s
parameters and scalar nrun (n ) and scalar nrunrun (n ). The amplitude at the pivot scale is set by
run run,run
scalar power amp (A ). Wavenumber (k) scales are in Mpc−1. You can generate results for multiple initial power
s
spectraatonego,soalltheinputparameter(exceptpivotscales)arefollowedbytheindex,e.g. scalar power amp(1)
for the first set.
B. Tensor parameterizations
CAMB supports three parameterizations of the primordial tensor power spectrum, set in the .ini file
by tensor parameterization integer variable. These use the input pivot scale pivot tensor (k ), and
t
tensor spectral index (n ). You can also specify a tensor running, tensor nrun (n ) so that
t t,run
n
lnP (k)=lnA +n ln(k/k )+ t,run[ln(k/k )]2.
t t t t 2 t
The tensor power amplitude A is set depending on the parameterization, either using initial ratio (r), or
t
tensor amp (A ) directly:
t
• tensor parameterization=1
A =rA
t s
The tensor amplitude here is independent of n and the scalar pivot scale (convenient if generating tensors
s
separately), but does depend on A (scalar amp). The parameterization only gives P (k ) = rP (k ) if the
s t t s t
tensor and scalar pivots are the same. This option is the default, and was used by CAMB prior to April 2014.
• tensor parameterization=2
A =rP (k )
t s t
This allows the use of r defined at any given tensor pivot scale, and P (k ) = rP (k ). However for different
h t s t
tensor/scalar pivot scales the result depends on n and the scalar pivot.
s
6
• tensor parameterization=3
Here tensor amp gives the amplitude A directly.
t
This directly parameterizes the tensors in terms of their amplitude, which removes dependence on the shape
and amplitude of the scalar spectrum (and is what tensor BB observations are actually probing directly).
C. Tensors in a closed universe
The power spectrum P =P defined by the transverse traceless part of the metric tensor h so that
t h ab
(cid:88) ν ν2−4
(cid:104)h hab(cid:105)= P (ν)
ab ν2−1 ν2 h
ν
where the ν2−4 factor is the mode sum l=2...ν−1 of 2l+1. Note that ν2/(ν2−1) is no longer the measure over
lnk for tensor modes. The CMB result is then
C = π (l+1)(l+2)(cid:88) ν ν2−3P (ν)[T(l)(ν)]2.
l 32 2l(l−1) ν2−1 ν2 h I
ν
Internally CAMB uses the H , the metric tensor variable, and the shear σ . The relation between H , E (the
k k k k
electric part of the Weyl tensor) and the shear σ is
k
ν2−3
H =− (2E +σ(cid:48)/k)
k ν2−1 k k
and H(cid:48) =−kσ . Using the series solution the relation between the initial E and H is
k k k k
2R +10ν2−1
E =− ν H
k 4R +15ν2−3 k
ν
where R is the ratio of the neutrino and total radiation densities.
ν
V. MULTIPOLE EQUATIONS, HARMONIC EXPANSION AND C
l
Here we summarize the equations of the covariant approach.
The photon multipole evolution is governed by the geodesic equation and Thomson scattering, giving [7]
4 l 4 8
I˙ + ΘI +DbI − D I + IA δ − Iσ δ
Al 3 Al bAl 2l+1 (cid:104)a Al−1(cid:105) 3 a1 l1 15 a1a2 l2
(cid:18) (cid:19)
4 2
=−n σ I −Iδ − Iv δ − ζ δ (13)
e T Al l0 3 a1 l1 15 a1a2 l2
where I is taken to be zero for l<0 and
Al
3 9
ζ ≡ I + E (14)
ab 4 ab 2 ab
is a source from the anisotropic stress and E-polarization. The equation for the density perturbation D I is obtained
a
by taking the spatial derivative of the above equation for l = 0. The corresponding evolution equations for the
polarization multipole tensors are [7]
4 (l+3)(l−1) l 2 2
E˙ + ΘE + DbE − D E − curlB = −n σ (E − ζ δ )
Al 3 Al (l+1)2 bAl 2l+1 (cid:104)al Al−1(cid:105) l+1 Al e T Al 15 a1a2 l2
4 (l+3)(l−1) l 2
B˙ + ΘB + DbB − D B + curlE = 0. (15)
Al 3 Al (l+1)2 bAl 2l+1 (cid:104)al Al−1(cid:105) l+1 Al
For numerical solution we expand the covariant equations into scalar, vector and tensor harmonics. The resulting
equations for the modes at each wavenumber can be studied easily and also integrated numerically.
7
A. Scalar, vector, tensor decomposition
Itisusefultodoadecompositionintom-typetensors, scalar(m=0), vector(m=1)and2-tensor(m=2)modes.
Theydescriberespectivelydensityperturbations,vorticityandgravitationalwaves. Ingeneralarank−(cid:96)PSTFtensor
X can be written as a sum of m−type tensors
Al
l
X = (cid:88) X(m). (16)
Al Al
m=0
Each X(m) can be written in terms of l−m derivatives of a transverse tensor
Al
X(m) =D Σ (17)
Al (cid:104)Al−m Am(cid:105)
where DAl ≡Da1Da2...Dal and ΣAm is first order, PSTF and transverse DamΣmAm−1am =0. The ‘scalar’ component
is X(0), the ‘vector’ component is X(1), etc. Since GR gives no sources for X with m > 2 usually only scalars,
a Am
vectors and (2-)tensors are considered. At linear order they evolve independently.
B. Harmonic expansion
For numerical work we perform a harmonic expansion in terms of zero order eigenfunctions of the Laplacian Qm ,
Am
k2
D2Qm = Qm , (18)
Am S2 Am
where Qm is transverse on all its indices, DamQm =0. So a scalar would be expanded in terms of Q0, vectors
Am Am−1am
in terms of Q1, etc. We usually suppress the labelling of the different harmonics with the same eigenvalue, but when
a
a function depends only on the eigenvalue we write the argument explicitly, e.g. f(k).
For m > 0 there are eigenfunctions with positive and negative parity, which we can write explicitly as Qm± when
Am
required. Since
k2
D2(curlQ )= curl(D2Q )= curlQ (19)
Am Am S2 Am
they are related by the curl operation. Using the result
k2 (cid:20) K(cid:21)
curl curlQm = 1+(m+1) Qm (20)
Am S2 k2 Am
we can choose to normalize the ± harmonics the same way so that
(cid:114)
k K
curlQm± = 1+(m+1) Qm∓. (21)
Am S k2 Am
A rank-(cid:96) PSTF tensor of either parity may be constructed from Qm± as
Am
(cid:18)S(cid:19)l−m
Qm ≡ D Qm (22)
Al k (cid:104)Al−m Am(cid:105)
and a X(m) component of X may be expanded in terms of these tensors. They satisfy
Al Al
k2 (cid:18) K(cid:19)
D2Qm = 1−[l(l+1)−m(m+1)] Qm
Al S2 k2 Al
k (l2−m2)
DalQm = βm Qm
Al−1al l S l(2l−1) Al−1
curlQm± = (cid:112)βmmkQm∓ (23)
Al 0 l S Al
8
where βm ≡1−(cid:8)l2−(m+1)(cid:9)K/k2 and l≥m.
l
Dimensionless harmonic coefficients are defined by
σ(m) =(cid:88) kσ(m)±Qm± H(m) =(cid:88) k2H(m)±Qm±
ab S ab ab S2 ab
k,± k,±
q(m) =(cid:88)q(m)±Qm± E(m) =(cid:88) k2E(m)±Qm±
a a ab S2 ab
k,± k,±
π(m) =(cid:88)Π(m)±Qm± I(m) =ρ (cid:88)I(m)±Qm±
ab ab Al γ l Al
k,± k,±
(cid:88) k (cid:88) k
Ω = Ω±Q1± A(m) = A(m)±Qm±
a S a a S a
k,± k,±
(cid:88) k
(D X)(m) = (δX)(m)±Qm± (24)
a S a
k,±
where the k dependence of the harmonic coefficients is suppressed. We also often suppress m and ± indices for
clarity. The other multipoles are expanded in analogy with I . The heat flux for each fluid component is given by
Al (cid:80)
q =(ρ +p )v , where v is the velocity, and the total heat flux is given by q = q . In the frame in which Ω =0
i i i i i i i a
¯ (1)
gradients are purely scalar (δX) =0.
C. Harmonic multipole equations
Expanded into harmonics, the photon multipole equations (13) become
k (cid:20) (l+1)2−m2 (cid:21)
I(cid:48)+ βm I −lI =
l 2l+1 l+1 l+1 l+1 l−1
(cid:18) (cid:19)
4 2 8 4
−Sn σ I −δ I − δ v− ζδ + kσδ −4h(cid:48)δ − kAδ (25)
e T l l0 0 3 l1 15 l2 15 l2 l0 3 l1
where l ≥ m, I = δρ /ρ , I = 0 for l < m, and m superscripts are implicit. The scalar source is h(cid:48) = (δS/S)(cid:48).
0 γ γ l
Theequationfortheneutrinomultipoles(afterneutrinodecoupling)isthesamebutwithouttheThomsonscattering
terms (for massive neutrinos see Ref. [8]). The polarization multipole equations (15) become
Em±(cid:48)+k(cid:20)βm (l+3)(l−1)(l+1)2−m2Em±− l Em±− 2m (cid:112)βmBm∓(cid:21)=−Sn σ (Em±− 2 ζm±δ )
l l+1 (l+1)3 (2l+1) l+1 2l+1 l−1 l(l+1) 0 l e T l 15 l2
Bm±(cid:48)+k(cid:20)βm (l+3)(l−1)(l+1)2−m2Bm±− l Bm±+ 2m (cid:112)βmEm∓(cid:21)=0. (26)
l l+1 (l+1)3 (2l+1) l+1 2l+1 l−1 l(l+1) 0 l
D. Integral solutions
Solutions to the Boltzmann hierarchies can be found in terms of line of sight integrals. The flat vector and tensor
resultsaregiveninRef.[9]. Generalscalarandtensorresultsarein[7](thoughI inthatpaperdiffersbyacurvature
l
factor).
E. Power spectra
Using the harmonic expansion of I the contribution to the C from type-m tensors becomes
Al l
CTT(m) = π (2l)! (cid:88) (cid:104)I±I± (cid:105)Q± QAl±. (27)
l 4(−2)l(l!)2 l,k l,k(cid:48) Alk k(cid:48)
k,k(cid:48),±
9
ThemultipolesI canberelatedtosomeprimordialvariableX =(cid:80) (cid:0)X+Qm++X−Qm−(cid:1)viaatransferfunction
l Am k Am Am
TX defined by I =TXX. Statistical isotropy and orthogonality of the harmonics implies that
l l l
(cid:104)X±X±(cid:105)=f (k)δ (28)
k k(cid:48) X kk(cid:48)
where (cid:80) δ Y =Y and f (k) is some function of the eigenvalue k. The normalization of the Qm is given by
k kk(cid:48) k k(cid:48) X Al
(cid:90) (cid:90) (cid:18)−S(cid:19)l−m (−2)l−m(l+m)!(l−m)!
dVQmQmAl = dV Qm DAl−mQm =αm N (29)
Al k Am Al l (2l)!
where we have integrated by parts repeatedly, then repeatedly applied the identity for the divergence (23). The
normalization is N ≡(cid:82) dVQAmQAmand αml ≡(cid:81)ln=m+1βnm. By statistical isotropy Cl =(1/V)(cid:82) dVCl and hence
CTT(m) = π(l+m)!(l−m)!(cid:88)Nαm|TX(k)|2f (k) (30)
l 4 (−2)m(l!)2 V l l X
k,±
We choose to define a power spectrum P (k) so that the real space isotropic variance is given by
X
(cid:88)|N| (cid:90)
(cid:104)|X XAm|(cid:105)= f (k)≡ dlnk P (k) (31)
Am V X X
k,±
so the CMB power spectrum becomes
π(l+m)!(l−m)!(cid:90)
CTT(m) = dlnk P (k)αm|TX(k)|2. (32)
l 4 2m(l!)2 X l l
(cid:82)
For a non-flat universe dlnk is replaced by some some other measure which should be specified when defining the
powerspectrum. Inacloseduniversetheintegralbecomesasumoverthediscretemodes. Notethatwehavenothad
(cid:80)
to choose a specific representation of Q or .
Am k
The polarization C are obtained similarly [7] and in general we have
l
CJK(m) = π (cid:20)(l+1)(l+2)(cid:21)p/2 (2l)! (cid:104)JAlKAl(cid:105) (33)
l 4 l(l−1) (−2)l(l!)2 ρ2
γ
π (cid:20)(l+1)(l+2)(cid:21)p/2 (l+m)!(l−m)!(cid:90)
= dlnk P (k)αmJXKX (34)
4 l(l−1) 2m(l!)2 X l l l
where JK is TT (p = 0), EE or BB (p = 2) or TE (p = 1). We have assumed a parity symmetric ensemble, so
CTB =CEB =0.
lFortenlsorsweuseH whereh =(cid:80) 2H Q2 andh isthetransversetracelesspartofthemetrictensor. This
T ij k,± T ij ij
introduces an additional factor of 1/4 into the result for the C in terms of P and THT.
l h l
The numerical factors in the hierarchy and C equations depend on the choice of normalization for the (cid:96) and k
l
expansions. Neither e nor Q are normalized, so there are compensating numerical factors in the expression for
(cid:104)Al(cid:105) Al
the C . If desired one can do normalized expansions, corresponding to an (cid:96)- and k-dependent re-scaling of the I and
l l
other harmonic coefficients, giving expressions in more manifest agreement with Ref. [10].
VI. QUINTESSENCE
CAMB uses a fluid model by default (see below), but a quintessence module is available separately.
The background field equation for a scalar field ψ can be written
d (cid:0)S2ψ(cid:48)(cid:1)+ S3V,ψ =0.
dS H
This allows ψ(S) and ψ(cid:48)(S) to be evaluated, and hence the background evolution since H relates the time and S
derivatives. The density and pressure are
S2ρ = 1ψ(cid:48)2+S2V S2p = 1ψ(cid:48)2−S2V
ψ 2 ψ 2
10
The linearized exact field equation is
ψ(cid:48)(cid:48)+2Hψ(cid:48)+DaD ψ =−S2V .
a ,ψ
where D is the spatial covariant derivative. Defining the first order perturbation
a
V ≡SD ψ
a a
and fixing to the zero-acceleration frame (the CDM frame) this implies the following equation for the perturbations
V(cid:48)(cid:48)+2HV(cid:48) +SZ ψ(cid:48)+S2D DbV =−S2V V .
a a a a b a ,ψψ
This corrects the equation in [11], where other definitions are given. Performing the harmonic expansion one gets the
scalar equation
V(cid:48)(cid:48)+2HV(cid:48)+kψ(cid:48)Z +k2V =−S2VV .
,ψψ
Thisiswhattellsyouthatyoucan’tconsistentlyhaveV =0inanevolvingbackgroundifthereareotherperturbations
present (Z (cid:54)=0). The heat flux and density perturbation X(ψ) =SD ρ give the scalar relations
a a ψ
S2q =kψ(cid:48)V S2X =ψ(cid:48)V(cid:48)+S2VV .
ψ ,ψ
There is no contribution to the anisotropic stress. CMBFAST/CAMB propagate quantities like κS2ρ, hence it is
√
useful to work with φ≡ κψ and similarly for V.
Constant w
A useful parameterization is w ≡p /ρ =constant, which implies
ψ ψ
1−w
ρ =ρ S−3(1+w) V = 1ψ˙2 .
ψ 0 2 1+w
This potential and ψ(cid:48) are easily obtained in terms of S
1−w
S2V = S2ρ ψ(cid:48)2 =|1+w|S2ρ .
2 ψ ψ
For w <−1 we take the action to have a negative kinetic term, so
ψ(cid:48)(cid:48)+2Hψ(cid:48)+DaD ψ±S2V =0
a ,ψ
etc, where ± is the sign of 1+w. The derivatives needed are then
3(1−w) 3(1−w)
V =∓ Hψ˙ S2V =∓ Hψ(cid:48),
,ψ 2 ,ψ 2
(cid:20) (cid:21)
V =∓3(1−w) H˙ − 3H2(1+w) S2V =∓3(1−w)(cid:2)H(cid:48)− 1H2(5+3w)(cid:3).
,ψψ 2 2 ,ψψ 2 2
In many cases constant w is actually a very good approximation as far as the CMB is concerned, with [12]
(cid:82)
daΩ (a)w(a)
ψ
weff ≡ (cid:82) .
daΩ (a)
ψ
In quintessence domination ρ→ρ and V →±9(1−w2)H2/2.
ψ ,ψψ
Fluid equations
Following [13] the default CAMB module actually uses the fluid equations. For varying w these are
δ(cid:48)+3H(cˆ2 −w )(δ +3H(1+w )v /k)+(1+w )kv +3Hw(cid:48)v /k =−3(1+w )h(cid:48) (35)
i s,i i i i i i i i i i
v(cid:48)+H(1−3cˆ2 )v +kA=kcˆ2 δ /(1+w ), (36)
i s,i i s,i i i
wherehattedquantitiesareevaluatedintheframeco-movingwiththedarkenergy. Theseequationsareimplemented
in the code with w(cid:48) =0, and are equivalent to using the above, with the additional possibility of using the rest-frame
sound speed cˆdifferent from one for generalized dark energy models (cˆ=1 for quintessence)