Table Of ContentCorrelated Isocurvature Fluctuation in Quintessence and Suppressed CMB
Anisotropies at Low Multipoles
Takeo Moroi(a) and Tomo Takahashi(b)
(a)Department of Physics, Tohoku University, Sendai 980-8578, Japan
(b)Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599
We consider cosmic microwave background (CMB) anisotropy in models with quintessence taking
intoaccountofisocurvaturefluctuationinthequintessence. Itisshownthat,iftheprimordialfluc-
tuation of thequintessence has a correlation with theadiabatic density fluctuations, CMB angular
power spectrum Cl at low multipoles can be suppressed without affecting Cl at high multipoles.
Possible scenario of generating correlated mixture of the quintessence and adiabatic fluctuations is
also discussed.
4
0 Preprint number: TU-695, astro-ph/0308208 has correlation with the adiabatic density fluctuations.
0 (For explicit model of generating correlated mixture of
2 The recent measurement of the cosmic microwave the quintessence and adiabatic fluctuations, see the dis-
n background (CMB) angular power spectrum C by the cussion below.)
l
a WilkinsonMicrowaveAnisotropyProbe(WMAP)[1]has Thus, in this letter, we study the CMB angular power
J greatly improved our understanding of the universe. In spectrum in models with quintessence paying particu-
9 particular,sincetheCMBangularpowerspectrumissen- lar attention to the effects of the correlation of the
sitive to the properties of the origin and evolutionof the quintessence fluctuation with adiabatic density fluctu-
2
cosmicdensityfluctuations,nowweareatthepositionto ations. If the correlation exists, behavior of the CMB
v
8 test various scenarios of generating cosmic density fluc- angular power spectrum becomes different from the un-
0 tuations, which requires some interplay of astrophysics, correlated case. In particular, as we will discuss, Cl at
2 cosmology, and particle physics. In addition, the shape low multipoles may be suppressed without affecting the
8 of the CMB angular power spectrum also depends on shape of the CMB angular power spectrum at high mul-
0 various cosmologicalparameters,and hence many of the tipoles, which may be related to the suppression of the
3 cosmologicalparameters are now precisely determined. measuredvaluesofCl atlowmultipoles. (Forothermod-
0
/ OneoftheimportantmessagesfromtheWMAPisthat els of suppressing Cl at low multipoles, see [6].)
h our universe is well described by a low-density cold dark Let us start with presenting the framework. Here we
p matter (CDM) model with (almost) flat geometry. In consider the scenario where the dark energy of the uni-
o- other words, the WMAP confirmed the existence of the verse is given by a potential energy of a slowly evolving
r dark energy which is also suggested by the observations scalar field, quintessence Q; if the slow-roll condition is
st of high red-shift Type-Ia supernovae [2]. satisfied for the quintessence field Q, equation-of-state
a Althoughthecosmologicalconstantisthemostfamous parameterofQisalmost−1andtheenergydensityofQ
v: candidate of the dark energy, a slowly evolving scalar behaves as the cosmological constant. Although various
i field, dubbed as “quintessence” [3], is another important models of quintessence have been proposed, we adopt a
X
possibility. Inmodelswithquintessence,however,behav- simple approximation of the quintessence potential; we
ar ioroftheCMBangularpowerspectrummaybedifferent use the parabolic potential with a constant term:
from that in the ΛCDM model (where the cosmological
1
constant is assumed as the dark energy). One reason is V(Q)=V + m2Q2. (1)
0 2 Q
that,sincethequintessenceisadynamicalscalarfield,its
amplitude may acquire primordial fluctuation and that We assume that Q has non-vanishing initial amplitude.
isocurvature fluctuation may exist in the quintessence In our study, for simplicity, we consider m comparable
Q
sector,whichaffectsbehaviorsoftheCMBangularpower to (or smaller than) the present expansion rate of the
spectrum. universe. Inaddition,V isassumedtobeoftheorderof
0
So far, effects of the primordial fluctuation of the thepresentcriticaldensityorsmaller. Withsuchasmall
quintessence have been studied only for the cases where valueofm (andV ),slow-rollconditionforQissatisfied
Q 0
the primordial fluctuation of the quintessence is uncor- untilveryrecentlyandenergyfractionofthequintessence
related with the adiabatic density fluctuations. In this becomes sizable only at the very recent epoch. In this
case,ifthequintessence-dominateduniverseisrealizedin case,shape of the CMB angularpower spectrum at high
very recent epoch as in the ΛCDM model, effects of the multipoles is almost unaffected. Notice that the above
quintessence fluctuation on Cl are only on very low mul- potential is a good approximation for some quintessence
tipoles. In addition, in the uncorrelated case, Cl at low models, like the cosine-type one [7].
multipoles arealwaysenhanced[4,5]. Itis possible,how- Since the quintessence is a dynamical scalar field, its
ever, that the primordial fluctuation of the quintessence amplitude may fluctuate. In particular, if its (effective)
1
mass during inflation is smaller than the expansionrate,
6000
the quintessence field also acquiresquantum fluctuation.
Such a primordial fluctuation becomes a new source of
5000
the cosmic density fluctuations and affects the CMB an-
gular power spectrum.
4000
Evolution and effects of the primordial fluctuation of π
2
/
Q have been studied for the case where the primordial Cl 3000
fluctuation of the quintessence is not correlated with +1)
the adiabatic fluctuations [8,4,5]. Fluctuation of the l(l 2000
quintessence field can be, however, correlated with the
adiabatic fluctuations. If some correlation exists, effects 1000
of the primordial fluctuation of Q on the CMB angular
power spectrum are expected to be different from those 0
1 10 100 1000
in the uncorrelated case. Hereafter, we study the effects
multipole l
of quintessence fluctuation for the case where the corre-
FIG.1. TheCMBangularpowerspectrumgeneratedfrom
lation between the primordial fluctuation of Q and the
thecorrelatedmixtureofthequintessenceandadiabaticfluc-
adiabatic fluctuations exists. tuations. Wetake(a)mQ =10−42GeV,V0=0andrQ=400
In order to parameterize the relative size of the pri-
with Ωb = 0.046, Ωm = 0.27, h = 0.72, and τ = 0.166 [10]
mordial quintessence fluctuation and the adiabatic fluc- (solid), and (b) mQ = 10−40 GeV, V0 = 3.0×10−47 GeV
tuations, we define and rQ = 2.5 with Ωb = 0.048, Ωm = 0.27, h = 0.7, and
τ =0.2(dashed). (Here,ΩbandΩmaredensityparametersof
δQ
r ≡ init . (2) baryonandnon-relativisticmatter,respectively,htheHubble
Q
M∗ΨRD constant in units of 100 km/sec/Mpc, and τ thereionization
optical depth.) The full correlation between δQ and Ψ
init RD
(Strictly speaking, the above expression is valid only
is assumed. Result for the purely adiabatic ΛCDM model is
for the case where δQ and Ψ are fully corre-
init RD also shown in the dotted line. (For cosmological parameters,
lated. For the case where δQinit and ΨRD are uncor- we use the best-fit values suggested by the WMAP for the
related, for example, it should be understood as rQ = power-law ΛCDM model.) For comparison, we also plot the
hδQ2initi/M∗ hΨ2RDi.) Here, δQinit is the primor- data points measured by the WMAP [12]. (The errors are
dpial fluctuatiopn of Q, Ψ denotes the fluctuation of the measurement errors only.)
(0,0) component of the metric in the Newtonian gauge:
g = a2(1+2Ψ) with a being the scale factor [9], and
00
M∗ is the reduced Planck scale. In addition, ΨRD is the
metricperturbationrelatedtotheadiabaticdensityfluc- δQ and Ψ are fully correlated: hδQ Ψ i2 =
init RD init RD
tuation in the radiation-dominated epoch. We assume hδQ2 ihΨ2 i. Inournumericalanalysis,wealwaystake
thatΨ is(almost)scale-invariant,assuggestedbythe init RD
RD Q > 0. In this case, primordial fluctuation of the en-
init
WMAP [10].
ergydensityofQhaspositivecorrelationwithΨ when
RD
Ifwecalculatethe CMB angularpowerspectrumwith
r >0. (For details of the calculation, see [4,5].)
Q
non-vanishing values of δQ and Ψ , we obtain the
init RD In Fig. 1, we plot the resultant CMB angular power
form
spectra for several cases with positive values of r : (a)
Q
Cl =Cl(adi)+Cl(corr)+Cl(uncorr). (3) r2Q.5=wi3t5hwmithm=Q10=−4100−G4e2VGeaVndanVdV=0 =3.00×,a1n0d−(4b7)GrQeV=.
Q 0
(We checked that these data points are consistent with
Here, C(adi) is the result with purely adiabatic density
l the recent Type 1a supernovae data [11].) Initial am-
fluctuations. With the cosmological and model parame-
plitude of the quintessence Q is determined so that
ters used in the following analysis, Cl(adi) almost agrees the present energy fraction ofitnhite quintessence becomes
with the CMB angular spectrum for the ΛCDM model Ω =1−Ω =0.73,whichgivesQ =8.2×1018 GeV
Q m init
C(ΛCDM). On the contrary, C(uncorr) is the CMB angu- and7.5×1017GeVforthecases(a)and(b),respectively.
l l
lar power spectrum purely generated from δQ , while As one can see, if r >0, sizable suppression of the low
init Q
C(corr) parameterizes the effects of correlation. multipoles is possible compared to the ΛCDM model.
l
When δQ and Ψ are uncorrelated, C(corr) = 0. Notice that, in the uncorrelatedcase,such a suppression
init RD l of C at the low multipoles cannot be realized. It is also
Furthermore,C(uncorr) isincreasedatlowmultipoles. As l
l notable that, even in the correlated case, the CMB an-
a result, the totalCMB angularpower spectrum may be
gular power spectrum at higher multipoles is almost the
significantly enhanced at low multipoles [4,5].
same as that in the ΛCDM case.
IfδQinitandΨRDhavecorrelation,Cl(corr)playsimpor- In order to see how much C2 can be suppressed rel-
tantroles. Tosee the effects ofthe correlation,wecalcu- ative to the ΛCDM model, in Fig. 2, we plot the ratio
latetheCMBangularpowerspectrumforthecasewhere
2
of C at the low multipoles measured by the WMAP is
l
relatedto the primordialfluctuationof the quintessence,
1.4
positive correlation with the metric perturbation Ψ
RD
(i.e., r >0) is required in the present model.
Q
1.2
M) So far, we have discussed effects of primordial fluctu-
CD ations of quintessence on the CMB angular power spec-
Λ( 1 trum, so we would like to comment on other cosmologi-
2
C
/ calperturbations in our scenario. Since the quintessence
2
C 0.8 dominates the universe at very recent epoch, its effects
areonlyonfluctuations withverylargewavelengthcom-
0.6 parable to the present Hubble distance. In other words,
density fluctuations with smaller wavelength are deter-
0.1 1 10 100 1000
minedbythe adiabaticpartofthe fluctuationandhence
r
Q the results for those perturbations are the same as the
FIG.2. C2/C2(ΛCDM) as a function of rQ for cases (a) and predictions of the simple inflationary paradigm.
(b) given in Fig. 1 (solid and dashed, respectively), and for
Finally,wepresentapossiblescenariowhichgenerates
mQ = 10−43 GeV and V0 = 0 (dotted) with Ωb = 0.046, the correlated fluctuations. We use the fact that fluctu-
Ωm = 0.27, h = 0.72, and τ = 0.166. For Cl(ΛCDM), we use ations of two scalar fields can be correlated if they have
the best-fit values of the cosmological parameters suggested
a mixing during the inflation [13]. We consider the case
by the WMAP. The full correlation between δQ and Ψ
init RD withtwoscalarfields,Qandφ. (Inthefollowing,primor-
is assumed.
dialfluctuation ofφ becomes the dominantsourceof the
adiabatic density fluctuations, as we will see below.) Q
andφ aredefined as mass eigenstates in the presentuni-
verse. In the early universe (i.e., for example, during in-
C /C(ΛCDM) as a function of r . Since C(corr) ∝r and flation), however,Hubble-induced interaction may cause
2 2 Q l Q
C(uncorr) ∝ r2, C(uncorr) is more important than C(corr) a mixing between Q and φ and hence the mass eigen-
l Q l l states may be linear combinations of them. We denote
when r is large. In this case, the ratio C /C(ΛCDM)
Q 2 2 the mass eigenstates as ξ and η, and define the mixing
becomes larger than 1. If rQ is smaller, however, Cl(corr) angle θ as
becomes sizable and C can be suppressed. As one can
2
see, with the correlated primordial fluctuation in the ξ cosθ(t) −sinθ(t) Q
= . (4)
quintessence amplitude, C /C(ΛCDM) can be as small as (cid:18)η (cid:19) (cid:18) sinθ(t) cosθ(t) (cid:19)(cid:18) φ (cid:19)
2 2
∼ 0.6 if r is properly chosen. Importantly, effects of
Q
Inourmodel,θ(t)variesfromnon-vanishingvalueduring
δQ is limited to C with small l and hence the shape
init l
inflation θ to the present value θ =0. If the (effec-
of the CMB angular power spectrum at higher multi- inf now
tive) mass of η is large during inflation while that of ξ
poles is unchanged. Thus, the correlated fluctuation in
is negligible, then only the ξ field acquires the quantum
thequintessencesectorprovidesanewmechanismtosup-
fluctuation:
pressC atverylowmultipoleswithoutaffectingthehigh
l
multipoles. This may have some relevance with the sup-
H
inf
pressed values of the CMB angular power spectrum at δξinf = , δηinf =0, (5)
2π
low multipoles measured by the WMAP.
SuppressionofC atlowmultipolesimprovestheagree- where H is the expansion rate during the inflation.
l inf
ment of the theoretical prediction with the observation. Assuming that θ(t) rapidly changes from θ to 0 at the
inf
To discuss this issue quantitatively, we calculate the end of inflation, primordial fluctuation of Q and φ are
goodness-of-fit parameter χ2 using the numerical pro- given by
gram provided by the WMAP collaboration with the
WMAP data [12]. Compared to the ΛCDM model, we δQinit =δξinfcosθinf, δφinit =δξinfsinθinf, (6)
found that the χ2 variable can decrease in models with
quintessence. Forexample,forthecases(a)and(b)given and correlated fluctuations are generated in the
in Fig. 1, changes of the goodness-of-fit parameter are quintessence and φ fields.
∆χ2 = −2.1 and −1.2, respectively. As can be under- Theabovesituationmaybe realizedifthe potentialof
stood from the figure, the changes in χ2 are only from the scalar fields is of the form
the low multipoles.
1 1
In our study, we have also calculatedCl with negative V =V0+ 2m2QQ2+ 2m2φφ2+VHubble, (7)
values of r . Since C(corr) is proportional to r , the
Q l Q
resultant CMB angular power spectrum is enhanced at where
thelowmultipoleswhenr <0. Thus,ifthesuppression
Q V =H2 (Qsinθ +φcosθ )2. (8)
Hubble vac inf inf
3
Here, V is the Hubble-induced interaction which is Acknowledgments: We acknowledge the use of CMB-
Hubble
effective only during the inflation with H being the FAST [17] package for our numerical calculations. T.T.
vac
expansion rate of the universe induced by the “vacuum thanks High Energy Theory Group in Tohoku Univer-
energy;” H = H and H ≃ 0 for during and after sity, where this work has been done, for their hospitality
vac inf vac
theinflation,respectively. Withthispotential,oneofthe during the visit. The work of T.M. is supported by the
masseigenstatesη ≃φ+θ Qacquiresaneffectivemass Grant-in-AidforScientificResearchfromtheMinistryof
inf
comparable to the expansion rate and its quantum fluc- Education, Science, Sports, and Culture of Japan, No.
tuation during inflation becomes negligibly small. Other 15540247. The work of T.T. is supported by the US De-
mass eigenstate ξ ≃ Q − θ φ, on the contrary, stays partment of Energy under Grant No. DE-FG02-97ER-
inf
almostmasslessanditacquiresthequantumfluctuation. 41036.
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m , slow roll condition is satisfied for φ at the time of
φ
the inflaton decay. In this case, δφ may become the
init
dominantsourceoftheadiabaticdensityfluctuations. In
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Q
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4
