Table Of ContentMon.Not.R.Astron.Soc.000,1–16(2014) Printed15May2014 (MNLATEXstylefilev2.2)
The Ellipsoidal Universe in the Planck Satellite Era
Paolo Cea1,2⋆
4
1 1Dipartimento di Fisica, Universita`di Bari, I-70126 Bari, Italy
0 2INFN - Sezione di Bari, I-70126 Bari, Italy
2
y
a
M
ABSTRACT
4
1 RecentPlanckdataconfirmthatthecosmicmicrowavebackgrounddisplaysthequadru-
polepowersuppressiontogetherwithlargescaleanomalies.Progressingfromprevious
] results,that focusedon the quadrupole anomaly,we strengthen the proposalthat the
O
slightlyanisotropicellipsoidaluniversemayaccountfortheseanomalies.Wesolvedat
C
large scales the Boltzmann equation for the photon distribution functions by taking
h. intoaccountboththeeffects oftheinflationproducedprimordialscalarperturbations
p andtheanisotropyofthegeometryintheellipsoidaluniverse.Weshowedthatthelow
- quadrupole temperature correlations allowed us to fix the eccentricity at decoupling,
o
r edec = (0.86 ± 0.14)10−2, and to constraint the direction of the symmetry axis. We
t foundthattheanisotropyofthegeometryoftheuniversecontributesonlytothelarge
s
a scale temperature anisotropies without affecting the higher multipoles of the angular
[
power spectrum. Moreover, we showed that the ellipsoidal geometry of the universe
3 induces sizable polarization signal at large scales without invoking the reionization
v scenario. We explicitly evaluated the quadrupole TE and EE correlations. We found
7 an average large scale polarization ∆T = (1.20 ± 0.38) µK. We point out that
pol
2
great care is needed in the experimental determination of the large-scale polarization
6
5 correlations since the average temperature polarization could be misinterpreted as
. foreground emission leading, thereby, to a considerable underestimate of the cosmic
1
microwavebackground polarization signal.
0
4
Key words: cosmic microwaveradiation - cosmology: theory.
1
:
v
i
X
1 INTRODUCTION
r
a The Cosmic Microwave Background (CMB) anisotropy data produced by the final analysis of the Wilkinson Microwave
AnisotropyProbe(WMAP)(Bennett et al.2013;Hinshaw et al.2013)and,morerecently,bythePlancksatellite (Adeet al.
2013a,c,d) confirme the standard cosmological Lambda Cold Matter (ΛCDM) model at an unprecedented level of accuracy.
At large scales, however, several anomalous features have been reported: an unusual alignment of the preferred axes of the
quadrupoleandoctopole(Land & Magueijo 2005;deOliveira-Costa et al.2004;Ralston & Jain2004;Copi et al.2006),non-
Gaussian signatures due to a cold spot (Cruzet al. 2005), an hemispherical power asymmetry at large scales (Eriksen et al.
2004; Hansen, Banday & Gorski 2004). Nevertheless, we feel that one of the most important discrepancy resides in the low
quadrupole moment, which signals an important suppression of power at large scales. In fact, the Planck collaboration re-
ported a statistical significant tension between the best fit ΛCDM model and the large-scale spectrum due to a systematic
lack of power for ℓ . 40 (Adeet al. 2013c) and to anomalies in thestatistical isotropy of the sky maps (Adeet al. 2013e).
Iftheseanomalies should turnouttohaveacosmological origin, thenitcouldhavefarreachingconsequencesforourpresent
understandingof the universe.
Quite recently it has been suggested (Campanelli, Cea & Tedesco 2006, 2007; Cea 2010) that, if one admits that the
large-scale spatial geometry of our universe is only plane-symmetric with eccentricity at decoupling of order 10−2, then the
quadrupole amplitude can be drastically reduced without affecting higher multipoles of the angular power spectrum of the
temperatureanisotropy.Asdiscussed in Campanelli, Cea & Tedesco(2007),theanisotropicexpansion describedbyaplane-
symmetric metric can be generated by cosmological magnetic fields or topological defects, such as cosmic domain walls or
⋆ E-mail:[email protected])
2 Paolo Cea
cosmic strings. Indeed, topological cosmic defects are relic structures that are predicted to be produced in the course of
symmetry breaking in thehot, early universe (e.g., see Vilenkin & Shellard (1994)).
1.1 Outline of the Results
In an isotropic and homogeneous universe the most general metric is the Friedmann-Robertson-Walker (FRW) metric (see,
for instance, Peebles 1993). In particular, the metric of standard cosmological model is given by 1:
ds2= dt2+a2(t)δijdxidxj . (1)
−
Ifweassumethatthelarge-scalespatialgeometryofouruniverseisonlyplane-symmetric,thenthemetricEq.(1)isreplaced
with theellipsoidal universemetric:
ds2= dt2+a2(t)(δij+hij)dxidxj , (2)
−
where hij is a metric perturbation which we assume to be of the form:
hij = e2(t)ninj . (3)
−
In Eq.(3) e(t)= 1 (b(t)/a(t))2 is theellipticity and theunit vector~n determines thedirection of thesymmetry axis.
−
In this paper we shall further elaborate on the ellipsoidal universe proposal and extend previous investigations in several
p
directions. Forreader’s convenience, it is useful to summarize the main results of the present paper.
First, we consider the Boltzmann equation for the photon distribution in theellipsoidal universe,discussed for the first time
in Cea (2010), by taking into account also the effects of the cosmological inflation produced primordial scalar perturbations.
In thelarge scale approximation we explicitly show that theCMB temperaturefluctuations can bewritten as:
∆T ∆TI + ∆TA, (4)
≃
where ∆TI and ∆TA are the temperature fluctuations induced by the cosmological scalar perturbations and by the spatial
anisotropy of the metric of the universe, respectively. Since the temperature anisotropies caused by the inflation produced
primordial scalar perturbations are discussed in several textbooks (Dodelson 2003; Mukhanov 2005), we focus on the tem-
peraturefluctuationsinducedbytheanisotropy ofthemetric bysolving therelevantBoltzmann equation.Atlarge scales we
solvethatequationanddeterminethesolutionsrelevantfortheCMBtemperatureandpolarization fluctuations.Indeed,itis
well known (Rees 1968; Negroponte & Silk 1980; Basko & Polnarev 1980) that anisotropic cosmological models give sizable
contributionstothelargescalepolarizationofthecosmicmicrowavebackgroundradiation.Infact,polarizationmeasurements
could providea uniquesignature of cosmological anisotropies.
Wego beyondthe approximations adopted in Cea (2010) and confirm that the main contributions to theCMB temperature
fluctuationsaffectthequadrupolecorrelations.Inaddition,wealsoshowthattheeffectsofthespatialanisotropyofthemetric
of theuniverse extendto low-lying multipoles ℓ 10.
∼
Asis well known, theCMB temperature fluctuationsare fully characterized bythe power spectrum:
+ℓ
ℓ(ℓ+1) 1
(∆T )2 = C , C = a 2 . (5)
ℓ ℓ ℓ ℓ ℓm
≡ D 2π 2ℓ+1 | |
mX=−ℓ
In particular, the quadrupole anisotropy refers to the multipole ℓ = 2. Remarkably, the Planck data (Adeet al. 2013c)
confirmed that the observed quadrupoleanisotropy:
(∆T )2 = 299.5 µK2 , (6)
2 2
D ≃
is much smaller than thequadrupole anisotropy expected according to the best fit ΛCDM model to thePlanck data:
(∆TI)2 = 1150 727µK2 . (7)
2 ±
NotethatinEq.(6)weareneglectingtherathersmallmeasurementerrors,whiletheuncertaintiesduetotheso-calledcosmic
varianceareincludedinthetheoreticalexpectations,Eq.(7).Infact,usingEq.(4)weshowthatthequadrupoletemperature
anisotropy can bereconciled with observations in theellipsoidal universeif the eccentricity at decoupling is:
e = (0.86 0.14)10−2 , (8)
dec
±
irrespectiveofthephysicalmechanismresponsibleforthegenerationofthespatialanisotropyintheearlyuniverse.Moreover,
if we denote with bn and ln the galactic latitude and longitude of the symmetry axis respectively, we also were able to show
that theaxis of symmetry were constrained to:
1 Notethatthroughthepaperweshalluseunitsinwhichc = 1,~ = 1andkB = 1.
Ellipsoidal Universe & Planck 3
bn 17◦ , (9)
≃ ±
while the longitude bn turnsout to be poorly constrained, in qualitative agreement with Campanelli, Cea & Tedesco (2007).
As concern the CMB polarization, we confirm our previous result (Cea 2010) that the ellipsoidal geometry of the universe
inducessizable polarization signal atlargescaleswithoutinvokingthereionization scenario.Inparticular,wefindanaverage
large scale polarization:
1
∆T dΩ ∆TE(θ,φ) = (1.20 0.38)µK , (10)
pol
≡ 4π ±
Z
where∆TE(θ,φ)isthepolarizationoftheCMBtemperaturefluctuations.Moreover,weevaluatethequadrupoletemperature-
polarization cross-correlation (TE) and polarization-polarization (EE) correlation. Wefind:
∆TTE = 3.14 0.76µK , (11)
2 ±
and
∆TEE = 0.83 0.27µK . (12)
2 ±
These values should be compared with the available observational data. Since the Planck collaboration does not yet make
public the large scale polarization data, we must rely on the final analysis of the Wilkinson Microwave Anisotropy Probe
collaboration. The WMAP nine-year full-sky maps of the polarization detected at large scales in the foreground corrected
maps an average E-mode polarization power (Bennett et al. 2013; Hinshawet al. 2013). In particular for the quadrupole
correlations we have (includingonly thestatistical uncertainties):
l(l+1)
CTE = 2.4439 2.2831 µK2 , WMAP nine years (13)
2π l=2 ± −
and
l(l+1)
CEE = 0.0860 0.0247 µK2 , WMAP nine years (14)
2π l=2 − ± −
Using thedefinition in Eq. (5) we estimate:
∆TTE = 1.56 0.73µK , WMAP nine years (15)
2 ± −
which within two standard deviations agrees with our result Eq. (11). On the other hand, as concern the quadrupole EE
correlation, Eq. (14) at best gives an upper bound which, however, is not consistent with our result Eq. (12). We believe
that thisdiscrepancy could be duetothefact that in theellipsoidal universemodel, at variance of thestandard reionization
scenario, there is a non-zero average temperature polarization. In fact, the eventual presence of an average temperature
polarization couldbemisinterpretedasforegroundemissionleadingtoanunderestimateofthecosmicmicrowavebackground
polarization signal.
The plan of the paper is as follows. In sect. 2 we discuss the Boltzmann equation of thecosmic background radiation in
the ellipsoidal universe. In sect. 3 we determine the solutions of the Boltzmann equation at large scales. Sect. 4 is devoted
totheproblem of thequadrupoleanomaly in thetemperature-temperaturefluctuation correlations. In sect. 5 wediscuss the
large scale polarization. In particular, we determine the quadrupole TE and EE correlations. Finally, our conclusions are
drawninsect.6.Sometechnicaldetailsarerelegated inappendixA,whileinappendixBwediscussthemultipoleexpansion
of thelarge scale temperature anisotropies.
2 THE BOLTZMANN EQUATION IN THE ELLIPSOIDAL UNIVERSE
Weare interested in thetemperaturefluctuationsofthecosmic background radiation inducedbyeccentricity of theuniverse
andbytheinflationproducedprimordialcosmologicalperturbations.Weassumethatthephotondistributionfunctionf(~x,t)
is an isotropically radiating blackbody at asufficiently early epoch. The subsequentevolution of f(~x,t) is determined bythe
Boltzmann equation (Dodelson 2003;Mukhanov 2005):
df ∂f
= , (16)
dt ∂t
(cid:18) (cid:19)coll
where (∂f) is the collision integral which takes care of Thomson scatterings between matterand radiation.
∂t coll
The metric of thestandard FRW universeis:
ds2= dt2 + a2(t)δijdxidxj . (17)
−
Here,weareinterestedinprimordialscalarperturbationsinducedbytheinflation.IntheconformalNewtonian,orlongitudinal
gauge (Mukhanov 2005),the metric Eq.(17) can be written as:
4 Paolo Cea
ds2= [1 + 2Ψ(~x,t)]dt2 + a2(t)δij[1 + 2Φ(~x,t)]dxidxj . (18)
−
In this gauge the perturbations to the metric are determined by the functions Ψ(~x,t) and Φ(~x,t) which correspond to the
Newtonian potential and the perturbation to the spatial curvature, respectively. In the ellipsoidal universe the metric would
be:
ds2= [1 + 2Ψ(~x,t)]dt2 + a2(t)[δij+hij][1 + 2Φ(~x,t)]dxidxj , (19)
−
wherehij isgivenbyEq.(3).However,boththeprimordialperturbationsΨ(~x,t),Φ(~x,t)andtheellipticityaretobeconsidered
smallatthetimesandscalesofinterest.Thereforeinthefollowing weshallneglect alltermsquadraticinthem.Accordingly,
instead of Eq. (19) we have:
ds2= [1 + 2Ψ(~x,t)]dt2 + a2(t) δij[1 + 2Φ(~x,t)] + hij dxidxj . (20)
− { }
We are interested in the anisotropies in the cosmic distribution of photons. To this end, we need to evaluate the photon
distribution function f(~x,t) which satisfies the Boltzmann equation Eq. (16). Actually, the distribution function depends on
thespace-time point xµ and themomentum vector pµ defined by:
dxµ
pµ = , (21)
dλ
where λ parametrizes the particle’s path.For massless particles we, obviously,have:
P2 gµν pµpν = 0. (22)
≡
Using themetric in Eq.(20) and defining:
p2 gij pipj , (23)
≡
from Eq. (22) we easily obtain:
p0 p[1 Ψ]. (24)
≃ −
Itisconvenienttoconsiderthedistributionfunctionasafunctionofthemagnitudeofmomentumpandmomentumdirection
pˆi, δijpˆipˆj =1. Therefore we have:
df ∂f ∂f dxi ∂f dp ∂f dpˆi
= + + + . (25)
dt ∂t ∂xi dt ∂p dt ∂pˆi dt
Now we notethat:
dxi dxi dλ pi
= = . (26)
dt dλ dt p0
Let us write
pi = Cpˆi, (27)
then it is easy to find:
p 1
C [1 Φ hijpipj]. (28)
≃ a(t) − − 2
Sothat we have:
dxi pˆi 1
[1 Φ+Ψ hijpipj]. (29)
dt ≃ a(t) − − 2
Thus, we get:
df ∂f ∂f pˆi 1 ∂f dp ∂f dpˆi ∂f ∂f pˆi ∂f dp
dt ≃ ∂t + ∂xi a(t)[1−Φ+Ψ− 2hijpipj]+ ∂p dt + ∂pˆi dt ≃ ∂t + ∂xi a(t) + ∂p dt (30)
since ∂f is already a first-orderterm. To evaluate dp, we note that thetime component of the geodesic equations gives:
∂xi dt
dp0
= Γ0 pαpβ . (31)
dλ − αβ
Since
dp0 dp0 dt dp0
= = p0 , (32)
dλ dt dλ dt
after using Eq. (24) we obtain:
dp dΨ pαpβ ∂Ψ pˆi ∂Ψ pαpβ
p Γ0 [1+2Ψ] = p[ + ] Γ0 [1+2Ψ], (33)
dt ≃ dt − αβ p ∂t a(t) ∂xi − αβ p
Ellipsoidal Universe & Planck 5
Moreover, a standard calculation (Dodelson 2003) shows that:
Γ0 pαpβ p(1 2Ψ)[∂Ψ +2 pˆi ∂Ψ + ∂Φ + 1pˆipˆj∂hij +H], (34)
αβ p ≃ − ∂t a(t) ∂xi ∂t 2 ∂t
where H =a˙/a is the Hubblerate. Finally, inserting Eqs. (33) and (34) intoEq. (30) and collecting terms we obtain:
df ∂f + pˆi ∂f p∂f [H(t) + ∂Φ + pˆi ∂Ψ + 1pˆipˆj∂hij]. (35)
dt ≃ ∂t a(t) ∂xi − ∂p ∂t a(t) ∂xi 2 ∂t
To go further we expandthe photon distribution about its zero-order Bose-Einstein value:
1
f (p,t) = . (36)
0 p
eT(t) 1
−
Wewrite:
1
f(~x,t,p,pˆ)= , (37)
p
eT(t)[1+Θ(~x,t,p,pˆ)] 1
−
and expand to thefirst order in theperturbation Θ(~x,t,p,pˆ):
∂f
f(~x,t,p,pˆ) f (p,t) p 0 Θ(~x,t,p,pˆ). (38)
≃ 0 − ∂p
Using therelation ∂lnf0 1 which is valid in theRayleigh-Jeans region, we can rewrite Eq. (38) as:
∂lnp ≃−
f(~x,t,p,pˆ) f (p,t)[1 + Θ(~x,t,p,pˆ)]. (39)
0
≃
Ifweneglecttheperturbations,itiseasytoseethatthezero-orderBoltzmannequationissatisfiedbythePlanckdistribution
Eq. (36) with T(t) 1 . To determine the perturbed distribution Θ(~x,t,p,pˆ) we need to evaluate the Boltzmann equation
∼ a(t)
to thefirst order. From Eqs. (35) and (37) it follows that:
df p∂f0 ∂Θ + pˆi ∂Θ + ∂Φ + pˆi ∂Ψ + 1pˆipˆj∂hij . (40)
dt ≃ − ∂p ∂t a(t) ∂xi ∂t a(t) ∂xi 2 ∂t
(cid:18) (cid:19)firstorder (cid:26) (cid:27)
Thusthe first-orderBoltzmann equation becomes:
∂Θ + pˆi ∂Θ + ∂Φ + pˆi ∂Ψ + 1pˆipˆj∂hij 1 ∂f . (41)
∂t a(t) ∂xi ∂t a(t) ∂xi 2 ∂t ≃ f0 (cid:18)∂t(cid:19)coll
Thecollisionintegralisingeneralanonlinearfunctionalofthedistributionfunction.However,inthefirstorderapproximation
it is a linear functional of Θ(~x,t,p,pˆ). Moreover, since we are interested in the solutions of the Boltzmann equation at large
scales,wemayneglecttheeffectsduetothebulkvelocityoftheelectronswhichparticipatetothephotonComptonscatterings.
In this case the collision integral can be considered a linear homogeneous functional of the distribution function Θ(~x,t,p,pˆ).
Asa consequence, if we write
Θ(~x,t,p,pˆ) ΘA(~x,t,p,pˆ) + ΘI(~x,t,p,pˆ), (42)
≃
then we also have:
∂f ∂f ∂f
[Θ] [ΘA] + [ΘI]. (43)
∂t ≃ ∂t ∂t
(cid:18) (cid:19)coll (cid:18) (cid:19)coll (cid:18) (cid:19)coll
Infact,Eq.(41)suggeststhatwemayassociateΘAandΘI withthetemperaturefluctuationsinducedbythespatialanisotropy
of the geometry of the universe and by the scalar perturbations generated during the inflation, respectively. Accordingly we
set:
∂ΘI pˆi ∂ΘI ∂Φ pˆi ∂Ψ 1 ∂f
+ + + [ΘI], (44)
∂t a(t) ∂xi ∂t a(t) ∂xi ≃ f0 (cid:18)∂t(cid:19)coll
and
∂ΘA + pˆi ∂ΘA + 1pˆipˆj∂hij 1 ∂f [ΘA]. (45)
∂t a(t) ∂xi 2 ∂t ≃ f0 (cid:18)∂t(cid:19)coll
We note that from Eq. (37) it follows that the distribution function Θ(~x,t,p,pˆ) is identified with the temperature contrast
function:
∆T(~x,t,p,pˆ)
Θ(~x,t,p,pˆ) = . (46)
T(t)
Therefore, at large scales Eq. (42) implies that:
∆T(~x,t,p,pˆ) ∆TA(~x,t,p,pˆ) + ∆TI(~x,t,p,pˆ). (47)
≃
6 Paolo Cea
Note that Eq. (47) was assumed in Campanelli, Cea & Tedesco (2006, 2007); Cea (2010). Even though this hypothesis was
considered reasonable, an explicit proof was lacking. Our discussion shows that Eq. (47) arises as a natural consequence of
theBoltzmann equation which, however, is valid only at large distances.
We may conclude that to determine the CMB temperature fluctuations at large scales we need to solve the Boltzmann
equations Eqs. (44) and (45). Eq. (44) is the Boltzmann equation of the standard ΛCDM cosmological model, and it has
been extensively discussed in several textbooks (Dodelson 2003; Mukhanov 2005). Therefore, in the following we focus on
the Boltzmann equation Eq. (45), derived for the first time in Cea (2010), which allows us to find the CMB temperature
fluctuationscaused by theanisotropy of thegeometry of theuniverse.
3 LARGE SCALE SOLUTIONS OF THE BOLTZMANN EQUATION
In thissection we discuss the Boltzmann equation in theellipsoidal universe Eq.(45):
∂Θ(~x,t,p,pˆ) + pˆi ∂Θ(~x,t,p,pˆ) + 1pˆipˆj∂hij 1 ∂f . (48)
∂t a(t) ∂xi 2 ∂t ≃ f0 (cid:18)∂t(cid:19)coll
where, for simplicity, the superscript A has been dropped. In Cea (2010) we have discussed the solutions of Eq. (48) by
neglecting thespatial dependenceof thetemperature contrast function Θ(~x,t,p,pˆ). Here we try to solve Eq. (48) in general.
To dothis, we introducetheFourier transform of thetemperature contrast function:
Θ(~x,t,p,pˆ) = d3k ei~k·~x Θ(~k,t,p,pˆ). (49)
(2π)3
Z
Taking intoaccount that thecollision integral dependslinearly on Θ, we easily obtain:
∂Θ(~k,t,p,pˆ) + i~k·pˆΘ(~k,t,p,pˆ) + 1pˆipˆj∂hij 1 ∂f [Θ(~k,t,p,pˆ)]. (50)
∂t a(t) 2 ∂t ≃ f0 (cid:18)∂t(cid:19)coll
To determine the polarization of the cosmic microwave background we need the polarized distribution function which, in
general, is represented by a column vector whose components are thefour Stokesparameters (Chandrasekhar1960). In fact,
due to the axial symmetry of the metric only two Stokes parameters need to be considered, namely the two intensities of
radiationwithelectricvectorsintheplanecontaining~pand~nandperpendiculartothisplanerespectively.Asaconsequence,
instead of Eq. (39) we have:
1
f(~x,t,p,pˆ) f (p,t) + Θ(~x,t,p,pˆ) , (51)
≃ 0 1
(cid:20)(cid:18) (cid:19) (cid:21)
where Θ(~x,t,p,pˆ) is a two component column vector. Using Eq. (3) and defining
~k pˆ
µ = cosθp~~n , cosθ~kp~ = k· , (52)
we get from Eq. (50):
∂Θ(~k,t,µ) ik 1 d 1
+ cosθ Θ(~k,t,µ) e2(t) µ2
∂t a(t) ~kp~ ≃ 2 dt 1
(cid:20) (cid:21) (cid:18) (cid:19)
3 1 2(1 µ2)(1 µ′2)+µ2µ′2 µ2
−σT ne Θ(~k,t,µ) − 8 − µ−′2 1 Θ(~k,t,µ′)dµ′ (53)
(cid:20) Z−1(cid:18) (cid:19) (cid:21)
where σT is theThomson cross section and ne(t) the electron numberdensity (Chandrasekhar 1960).
Introducingtheconformal time:
t dt′
η(t) = , (54)
a(t′)
Z0
we rewrite Eq. (53) as:
∂Θ(~k,η,µ) 1 d 1 1
+ ik cosθ Θ(~k,η,µ) e2(η) (µ2 )
∂η ~kp~ ≃ 2 dη − 3 1
(cid:20) (cid:21) (cid:18) (cid:19)
3 1 2(1 µ2)(1 µ′2)+µ2µ′2 µ2
−a(η)σTne Θ(~k,η,µ) − 8 − µ−′2 1 Θ(~k,η,µ′)dµ′ (55)
(cid:20) Z−1(cid:18) (cid:19) (cid:21)
withasuitableoverallnormalizationoftheblackbodyintensity.TodeterminethegeneralsolutionsofEq.(55)wewrite(Basko & Polnarev
1980; Cea 2010):
1 1 1
Θ(~k,η,µ) = θa(~k,η)(µ2 ) + θp(~k,η)(1 µ2) . (56)
− 3 1 − 1
(cid:18) (cid:19) (cid:18)− (cid:19)
Ellipsoidal Universe & Planck 7
From Eq. (56) it is evident that θa measures the degree of anisotropy, while θp gives the polarization of the primordial
radiation. With theaid of Eq. (56) we rewrite Eq. (55) as:
∂θa∂(~kη,η) + ik cosθ~kp~θa(~k,η) ≃ ∆H(η) − a(η)σTne 190θa(~k,η) + 35θp(~k,η)
(cid:20) (cid:21)
(57)
∂θp∂(~kη,η) + ik cosθ~kp~ θp(~k,η) ≃ −a(η)σTne 110θa(~k,η) + 52θp(~k,η)
(cid:20) (cid:21)
where we introduced thecosmic shear (Negroponte & Silk 1980;Cea 2010):
1 d
∆H(η) e2(η) . (58)
≡ 2 dη
The solution of the linear differential system Eq. (57) is the sum of the general solution of the homogeneous system and a
particular solution. The solution of thehomogeneous system (i.e. ∆H(η)=0) is:
θa(~k,η) = θp(~k,η) = 0, (59)
forthearenoanisotropies withoutcosmological perturbations.TodeterminetheparticularsolutionofEq.(57),wenotethat
thelinear combination:
θ¯(~k,η) θa(~k,η) + θp(~k,η) (60)
≡
satisfies thefollowing equation:
∂θ¯(~k,η)
∂η + ik cosθ~kp~ θ¯(~k,η) ≃ ∆H(η) − a(η)σTneθ¯(~k,η). (61)
Introducingtheoptical depth:
η
τ(η,η′)= σT nea(η′′)dη′′ , (62)
Zη′
it is easy to verify that thesolution of Eq. (61) is given by:
θ¯(~k,η)= η∆H(η′)e−τ(η,η′) eikcosθ~kp~(η′−η)dη′, (63)
Zηi
where ηi is an early conformal time such that θ¯(~k,ηi)=0. It is now easy to determineθa and θp. Weget:
θa(~k,η)= 1 η∆H(η′) 6e−τ(η,η′) + e−130τ(η,η′) eikcosθ~kp~(η′−η) dη′, (64)
7
Zηi h i
θp(~k,η)= 1 η∆H(η′) e−τ(η,η′) e−130τ(η,η′) eikcosθ~kp~(η′−η) dη′. (65)
7 −
Zηi h i
Insummary,wehavefoundthatthetemperaturefluctuationsinducedbythespatialanisotropyofthegeometryoftheuniverse
atlargescalesisgivenbyEqs.(56),(64)and(65).Obviously,weareinterestedinthetemperatureanisotropiesforη=η (η
0 0
is the conformal time now). As will be evident later on, the main contributions to the integrals in Eqs. (64) and (65) come
from conformal times near thedecoupling conformal time η . Moreover, observing that η η we may write:
d d 0
≪
Θ(~k,η0,µ,pˆ) ≃ θa(µ2− 31)e−ikcosθ~kp~η0 11 + θp(1−µ2)e−ikcosθ~kp~η0 11 , (66)
(cid:18) (cid:19) (cid:18)− (cid:19)
where:
θa 1 η0∆H(η′) 6e−τ(η0,η′) + e−130τ(η0,η′) dη′, (67)
≃ 7
Zηi h i
θp 1 η0∆H(η′) e−τ(η0,η′) e−130τ(η0,η′) dη′. (68)
≃ 7 −
Zηi h i
In appendix A we evaluate thetwo parameters θa and θp. We find(see Eqs. (A15) and (A8) ):
1
θa ≃ − 2 ×0.944e2dec, (69)
θp ≃ 8.9210−3 e2dec . (70)
8 Paolo Cea
4 THE QUADRUPOLE ANOMALY
We are, now, in position to discuss the low quadrupole anomaly in the CMB temperature anisotropies detected by WMAP
andrecentlyconfirmedbyPlanck.Thetemperatureanisotropiesofthecosmicbackgrounddependonthepolarangleθ,φ,so
that one usually expandsin terms of spherical harmonics:
∞ +ℓ
∆T(θ,φ)
= a Y (θ,φ), (71)
ℓm ℓm
T
0
Xℓ=1mX=−ℓ
whereT 2.7255K (Fixsen2009)istheactual(average)temperatureoftheCMBradiation.Notethatthea ’sinEq.(71)
0 ℓm
≃
aredimensionlessandareobtainedfromthecorrespondingcoefficientsinEq.(5)bydividingbyT .Afterthat,oneintroduces
0
thepower spectrum:
∆T 1 ℓ(ℓ+1)
( ℓ)2 = a 2 , (72)
ℓm
T0 2π 2ℓ+1 | |
m
X
thatfullycharacterizesthepropertiesoftheCMBtemperatureanisotropy.Inparticular,wefocusonthequadrupoleanisotropy
ℓ=2:
∆T
2 ( 2)2 . (73)
Q ≡ T0
In thestandard model theCMB temperature fluctuations are induced by the cosmological perturbations of theFRW homo-
geneous and isotropic background metric generated by the inflation-produced potentials. In the ellipsoidal universe we must
also consider the effects on the CMB anisotropies induced by the anisotropic expansion of the universe. In fact, as discussed
in sec. 2, at large scales the observed anisotropies in the CMB temperature are due to the linear superposition of the two
contributions according to Eq. (47).Therefore, we may write:
a = aA + aI . (74)
ℓm ℓm ℓm
In the previous section we have determined the contributions to temperature contrast function induced by the anisotropic
expansion of theuniverse:
ΘA(~k,η0,µ,pˆ) ≃ θa(µ2− 31)e−ikcosθ~kp~η0 , µ = cosθp~n~ . (75)
In appendix B, starting from Eq. (75) we perform the multipole expansion of the temperature fluctuation correlations and
obtain the multipole coefficients aA . However, it is evident from Eq. (75) that the main contribution to the temperature
ℓm
fluctuations is for k 0. It is easy to see that this corresponds to solve the Boltzmann equation Eq. (48) by neglecting the
≃
spatial dependenceon thetemperature contrast function. In this case we obtain at once:
∆TA(θ,φ) 1
T0 ≃ θa(cos2θp~~n− 3), (76)
where θa is given by Eq.(69) and θ,φ are the polar angles of thephoton momentum p~.
Let θn,φn bethe polar angles of the direction of the axis of symmetry~n, then:
∆TA(θ,φ) 2 2 4π +2
T0 ≃ 3 θa P2(cosθp~~n) = 3 θa 5 Y2m(θ,φ)Y2∗m(θn,φn). (77)
m=−2
X
Sincefrom Eq. (71) it follows that:
∆TA(θ,φ)
aA = dΩ Y∗ (θ,φ), (78)
ℓm T ℓm
Z 0
we obtain immediately:
4π
aA2m ≃ − 15 ǫ2 Yℓ∗m(θn,φn) , ǫ2 ≡ 0.944e2dec, (79)
while aA = 0 for ℓ = 2. In other words, at large scales the anisotropy of the metric contributes mainly to the quadrupole
ℓm 6
CMB temperatureanisotropies. From Eq. (79) we find:
4π 5
aA20 ≃ − 15 ǫ2 16π [1−3cos2θn],
r
4π 15
aA21 = (aA2,−1)∗ ≃ +i 15 ǫ2 8πe−iφnsinθn cosθn ,
r
4π 15
aA22 = (aA2,−2)∗ ≃ 15 ǫ2 32π e−2iφnsin2θn . (80)
r
Ellipsoidal Universe & Planck 9
Aftera little algebra we rewrite Eq.(80) as:
1 π
aA20 ≃ +6ǫ2 5 [1+3cos2(2θn)],
r
π
aA21 = (aA2,−1)∗ ≃ +i 30ǫ2e−iφnsin(2θn),
r
π
aA22 = (aA2,−2)∗ ≃ + 30ǫ2 e−2iφnsin2θn . (81)
r
Definingthequadrupoleanisotropy:
∆TA
2 ( 2 )2, (82)
QA ≡ T0
we find:
2
A ǫ2 . (83)
Q ≃ 5√3
To determine thecoefficients a , Eq. (74), we need to known theaI ’s. First we observethat thetemperature anisotropies
ℓm ℓm
are real functions, so that we must have a = ( 1)m(a )∗. Observing that aA = ( 1)m(aA )∗ (see Eq. (81)), we
ℓ,−m − ℓ,m ℓ,−m − ℓ,m
havetheconstraints aI =( 1)m(aI )∗. Moreover, becausethestandard inflation-producedtemperature fluctuationsare
ℓ,−m − ℓ,m
statisticallyisotropic,wereasonablyassumethattheaI coefficientsareequalsuptoaphasefactor.Therefore,wecanwrite:
2m
π
aI20 ≃ 3 QI,
r
π
aI21 = −(aI2,−1)∗ ≃ +i 3 eiφ1 QI , (84)
r
π
aI22 = (aI2,−2)∗ ≃ 3 eiφ2 QI ,
r
where 06φ ,φ 62π are unknown phases. It is easy tocheck that:
1 2
∆TI
2 = ( 2)2 . (85)
QI T0
Using Eq.(7) we obtain theestimate:
I (12.44 3.93)10−6 . (86)
Q ≃ ±
Taking intoaccount Eqs. (73), (74), (81) and (84) we get for the total quadrupole:
Q2=Q2A + Q2I + 2f(θn,φn,φ1,φ2)QAQI , (87)
where
1 3 3
f(θn,φn,φ1,φ2) = 4√5[1+3cos(2θn)] + r10 sin(2θn) cos(φ1+φn) + r10 sin2θn cos(φ2+2φn). (88)
Eqs. (87) and (88) show that, indeed, if the space-time background metric is not isotropic, the quadruple anisotropy may
become smaller than the one expected in the standard isotropic ΛCDM cosmological model of temperature fluctuations. In
fact, from Eq. (74) and using Eqs. (81) and (84), we get:
π 1 π
a20 ≃ + 3 QI + 6ǫ2 5 [1+3cos2(2θn)], (89)
r r
π π
a21 = +i 3 eiφ1 QI + i 30ǫ2e−iφnsin(2θn), (90)
r r
π π
a22 ≃ + 3 eiφ2 QI + 30ǫ2 e−2iφnsin2θn . (91)
r r
Eqs.(89),(90)and(91)giveasystemoffiveequationswhichcanbesolvedtogetthefiveunknownparameterse2dec,θn,φn,φ1,φ2.
To do this, however, we need the observed values of the a ’s. In fact, Campanelli, Cea & Tedesco (2007) used the cleaned
ℓ,m
CMBtemperaturefluctuationmapsoftheWMAPdataobtainedusingtheinternallinearcombinationwithgalacticforeground
subtraction. In particular, these authors used three different maps (Hinshaw et al. 2007; de Oliveira-Costa & Tegmark 2006;
Park, Park & Gott 2007).Actually,thesameprocedurecan beapplied totheforeground-cleaned CMB mapsobtainedfrom
thePlanck dataas detailed in Adeet al.(2013b).However, irrespectivefrom theadopted CMB cleaned map thequadrupole
anomalies detected byWMAP and confirmed byPlanck are accounted for if:
10 Paolo Cea
a 0, (92)
21
≈
and
a 2 2 a 2 . (93)
20 22
| | ≪ | |
Infact,itiseasytocheckthattheseequationsimplyboththealmostplanarityandthesuppressionofpowerofthequadrupole
moment.Remarkably,itturnsoutthatEqs.(92)and(93)allowustodeterminetheeccentricityatdecouplingandconstraint
thepolar angles of thesymmetry axis. InsertingEq. (92) into Eq. (90) we readily obtain:
ǫ2 √10QI , (94)
≃ sin(2θn)
| |
where φn+φ1 0◦, 360◦ if sin(2θn)<0, or φn+φ1 180◦, 540◦ if sin(2θn)>0. Moreover, from Eq. (89) and taking into
≃ ≃
account Eq. (93) we find:
cos(2θn) ≃ −31 − 2 53 Qǫ2I . (95)
r
Combining Eqs. (94) and (95) we obtain:
√6
θn arctan + 2 73◦, 107◦ . (96)
≃ ± 2 ≃
(cid:18) (cid:19)
This last equation together with Eqs. (86) and Eq.(94) gives theeccentricity at decoupling:
e (0.86 0.14)10−2 . (97)
dec
≃ ±
Finally, using Eqs. (73),(92) and (93) we get:
Q2 ≃ 56π |a22|2 ≃ 256π Q2I(cid:20)1 + sisnin2(42θθnn) + |2sisni(n22θθnn)| cos(φ2+2φn)(cid:21) . (98)
Using theobserved value of thequadrupole temperatureanisotropy Eq. (6) we estimate from Eq. (98) :
cos(φ2+2φn) 0.92 0.12. (99)
≃ − ±
Tosummarize, ouralmost modelindependentanalysis allowed tofixtheeccentricity at decoupling, Eq.(97).Asconcern the
symmetry axis, using thegalactic coordinates bn,ln, we found:
bn 17◦ , (100)
≃ ±
whilethelongitudeln turnedout tobepoorly constrained inqualitativeagreement with Campanelli, Cea & Tedesco(2007).
5 THE LARGE SCALE POLARIZATION
In this section we discuss thelarge scale polarization in the primordial cosmic background. In our previous work (Cea 2010)
wearguedthattheellipsoidal geometry oftheuniverseinducessizablepolarization signal atlarge scale withoutinvokingthe
CMB reionization mechanism. If we assume that early CMB reionization is negligible, then it is well known that at large
scale the primordial inflation induced cosmological perturbations do not produce sizable polarization signal (Dodelson 2003;
Mukhanov2005).Inthiscasethepolarization ofthetemperaturefluctuationsarefully givenbytheanisotropic expansionof
theuniverse. According toour discussion in sect. 3 we may write:
ΘE(~k,η0,µ,pˆ) θp(1 cos2θp~~n)e−ikcosθ~kp~η0 , (101)
≃ −
wherethesuperscriptEindicatesthatthetemperaturepolarizationcontributesonlytotheso-calledE-modes.Infact,Eq.(66)
shows that the anisotropy of the metric of the universe gives rise only to a linear polarization of the cosmic background
radiation.InappendixBwediscussthemultipoleexpansionofthetemperaturepolarization correlations.Aswehavealready
observed,themaincontributiontothepolarizationtemperaturecontrastfunctionsisfork 0,whichcorrespondstoneglect
≃
thespatial dependenceof thesolutions of theBoltzmann equation. Thus, we have:
∆TE(θ,φ) 2 2
T0 ≃ θp(1 − cos2θp~n~) = 3θp − 3 θp P2(cosθp~~n). (102)
Wemay,now,expandintermsofsphericalharmonicsasinEq.(71).Itisevidentfrom Eq.(102)thatthenon-zeromultipole
coefficients aE arefor themonopole ℓ=0andthequadrupoleℓ=2.Themonopole term determinestheaverage large scale
ℓm
polarization of thecosmic microwave background:
∆T 1 ∆TE(θ,φ) 2
pol dΩ θp . (103)
T0 ≡ 4π Z T0 ≃ 3
