Quantum noises and the large scale structure

0.19 MB·English

by  Wo-Lung Lee

February 5, 2008 15:47 WSPC/INSTRUCTION FILE cnoise˙cospa
ModernPhysicsLetters A
(cid:13)c WorldScientificPublishingCompany
7
0
0 QUANTUM NOISES AND THE LARGE SCALE STRUCTURE
2

n
a Wo-LungLee
J
Department of Physics, National Taiwan Normal University
1 Taipei, Taiwan 116, Republic of China
3 [email protected]

1
v Received(DayMonthYear)
6 Revised(DayMonthYear)
8
8
Weproposethat cosmological densityperturbation mayoriginatefrompassivefluctua-
1
tions of the inflaton, which are induced by colored quantum noise due to the coupling
0
oftheinflatontothequantumenvironment.Atsmallscales,thefluctuations growwith
7
time to become nearly scale-invariant. However, the larger-scale modes cross out the
0
horizonearlieranddonothaveenough timetogrow,thus resultinginasuppressionof
/
h the density perturbation. This may explain the observed low quadrupole in the CMB
p anisotropydata andand potentially unveil the initialtime of inflation. We alsodiscuss
- theimplicationstotherunningspectralindexandthenon-Gaussianityoftheprimordial
o
densityperturbation.
r
t
s Keywords:Cosmicmicrowavebackground;Inflationarycosmology;Large-scalestructure
a oftheUniverse.
:
v
PACSNos.:98.80.Cq,98.70.Vc,98.80.Es
i
X
r
a 1. Introduction
TherecentlyreleaseddataoftheWilkinsonMicrowaveAnisotropyProbe(WMAP)
confirmedthe earlierCOBE-DMR’sobservationaboutthe deficiency influctuation
power at the largestangularscales 1,2. The amount of the quadrupole mode of the
CMB temperature fluctuations is anomalously low if comparedto the predictionof
theΛCDMmodel.Theusualexplanationtothislowquadruplemomentistoinvoke
the so-call “cosmic variance”, i.e. at large scales the CMB experiments are limited
by the fact that we only have one sky to measure and so cannot pin down the
cosmic average to infinite precision no matter how good the experiment is. Given
thelargeuncertaintiesduetothecosmicvariance,wemightneverknowwhetherthe
phenomenon of the low CMB quadrupole constitutes a truly significant deviation
from the standard cosmologicalexpectations.
Thoughthe cosmicvarianceimplies thatwesimply liveinauniversewithalow
quadrupole moment for no special reason, the lack of power on the largest scale
can be treated as a physical effect that demands a judicious explanation. There
exists various viable mechanisms for generating this large-scaleanomaly.So far,all
1
February 5, 2008 15:47 WSPC/INSTRUCTION FILE cnoise˙cospa
2 Wo-Lung Lee
the efforts can be classifiedinto three essentialcategories:(1) a possible non-trivial
topology(for example, see J.-P. Luminet et al. 3 and the references therein); (2)
a cut-off due to causality (for example, see A. Berera et al. 4 and the references
5
therein); (3) initial hybrid fluctuations (for example, see Lee & Fang ). Recently
the colored noise has been considered to explain the anomaly in the context of
6
stochasticinflation .InsteadofaHeavisidewindowfunctionusedinthestochastic
7
approach to inflation , Ref.[6] adopts a Gaussian window function with a width
8
characterizingthe sizeofthecoarse-graineddomain whichisthenarbitrarilycho-
sentobecomparabletotheHubbleradius.Thissmoothwindowfunctioneventually
yieldsabluetiltofthepowerspectrumonlargescaleswhichcanbetunedtofitthe
WMAP large-scale anisotropy data. Nevertheless, in this classical approach they
inevitably resort to an ad hoc smoothing window function and therefore the width
of the window function remains undetermined.
Fromthefieldtheoreticalpointofview,ontheotherhand,interactionsbetween
the inflaton and other quantum fields in the very early universe may be inevitable.
Such interactions essentially convert the potential energy stored in the inflaton
9 10
field into radiation during or after inflation . Consequently, the evolution of
inflationary density perturbations will be altered depending upon the details of
the mechanisms 5,11. Therefore, the interactions between the inflaton and other
fields have not only provided us an opportunity to grasp the underlying physics
of the inflation, but also explored the possibility of generating primordial density
fluctuations with statistical properties deviated from the scale-invariance and the
Gaussianity.
2. Power Spectrum Derived from the Noise-Driven Fluctuations
To mimic the quantum environment, we consider a slow-rolling inflaton φ coupled
12
to a quantum massive scalar field σ, with a Lagrangiangiven by
1 1 m2 g2
L= gµν∂ φ∂ φ+ gµν∂ σ∂ σ−V(φ)− σσ2− φ2σ2, (1)
µ ν µ ν
2 2 2 2
whereV(φ)isthe inflatonpotentialthatcomplieswiththeslow-rollconditionsand
g is a coupling constant. Thus,we can approximatethe space-time during inflation
by a de Sitter metric given by
ds2 =a(η)2(dη2−d~x2), (2)
where η is the conformal time and a(η) = −1/(Hη) with H being the Hubble
parameter.Herewerescalea=1attheinitialtimeoftheinflationera,η =−1/H.
i
Followingtheinfluencefunctionalapproach11,13,wetraceoutσuptotheone-loop
level and thus obtain the semiclassical Langevin equation for φ, given by
φ¨+2aHφ˙−∇2φ+a2 V′(φ)+g2hσ2iφ −g4a2φ ×
φ
d4x′a4(η′)θ(η−η′)i(cid:0)G−(x,x′)φ2(x′)=(cid:1) ξ, (3)
a2
Z
February 5, 2008 15:47 WSPC/INSTRUCTION FILE cnoise˙cospa
Quantum Noises and the Large Scale Structure 3
where the dot and prime denote respectively differentiation with respect to η and
φ. The effects from the quantum field on the inflaton are given by the dissipation
viathekernelG− aswellasastochasticforceinducedbythemultiplicative colored
noise ξ with
hξ(x)ξ(x′)i=g4a4(η)a4(η′)G (x,x′). (4)
+
The kernels G± in Eqs. (3) and (4) can be constructed from the Green’s function
of the σ field with respect to a particular choice of the initial vacuum state which
we will specify later, given by
G±(x,x′)=hσ(x)σ(x′)i2±hσ(x′)σ(x)i2. (5)
To solve Eq. (3), let us first drop the dissipative term which we will discuss
later. Then, after decomposing φ into a mean field and a classical perturbation:
φ(η,~x)=φ¯(η)+ϕ(η,~x), we obtain the linearized Langevin equation,
φ¯
ϕ¨+2aHϕ˙ −∇2ϕ+a2m2 ϕ= ξ, (6)
ϕeff a2
where the effective mass is m2 = V′′(φ¯)+g2hσ2i and the time evolution of φ¯
ϕeff
is governed by V(φ¯). The equation of motion for σ from which we construct its
Green’s function can be read off from its quadratic terms in the Lagrangian(1) as
σ¨+2aHσ˙ −∇2σ+a2m2 σ =0, (7)
σeff
where m2 =m2 +g2φ¯2.
σeff σ
12
The solution to Eq. (6) is obtained as
η
ϕ =−i dη′φ¯(η′)ξ (η′) ϕ1(η′)ϕ2(η)−ϕ2(η′)ϕ1(η) , (8)
~k ~k k k k k
Zηi
(cid:2) (cid:3)
where the homogeneous solutions ϕ1,2 are given by
k
1
ϕ1,2 = (π|η|)1/2H(1),(2)(kη). (9)
k 2a ν
Here H(1) and H(2) are Hankel functions of the first and second kinds respectively
ν ν
and ν2 =9/4−m2 /H2. In addition, we have from Eq. (7) that
ϕeff
1
σ = (π|η|)1/2 c H(1)(kη)+c H(2)(kη) , (10)
k a 1 µ 2 µ
h i
where the constants c and c are subject to the normalization condition, |c |2−
1 2 2
|c |2 =1, and µ2 =9/4−m2 /H2.
1 σeff
Now, the power spectrum of the perturbation ϕ can be obtained by employing
Eqs. (4) and (8) as
2π2
hϕ (η)ϕ∗ (η)i= ∆ξ(η)δ(~k−~k′), (11)
~k ~k′ k3 k
February 5, 2008 15:47 WSPC/INSTRUCTION FILE cnoise˙cospa
4 Wo-Lung Lee
where the noise-driven power spectrum is given by
∆ξ(η)= g4z2 zdz zdz φ¯(η )φ¯(η ) sinz− z−1sin2Λz− −1 F(z )F(z ),
k 8π4 Zzi 1Zzi 2 1 2 z1z2z− (cid:20) − k (cid:21) 1 2
(12)
wherez− =z2−z1,z =kη,zi =kηi =−k/H,Λisthemomentumcutoffintroduced
in the evaluation of the ultraviolet divergent Green’s function in Eq. (5), and F(z)
12
is given by
1 1 1
F(y)= 1+ sin(y−z)+ − cos(y−z). (13)
yz y z
(cid:18) (cid:19) (cid:18) (cid:19)
We plot ∆ξ(η) at the horizon-crossing time given by z = −2π versus k/H in
k
Fig. 1. The figure shows that the noise-driven fluctuations depend on the onset
time of inflation and approach asymptotically to a scale-invariant power spectrum
∆ξ ≃ 0.2g4φ¯2/(4π2) at large k. Note that if g4φ¯2 ≃ H2, ∆ξ will be comparable
k 0 0 k
to that of the known scale-invariant quantum fluctuation of the inflation which is
given by ∆q =H2/(4π2) 14.
k
0.2
δξ
k
0.1
0.0
0 100 200 300 400
k/H
Fig.1. Powerspectrumofthenoise-driveninflatonfluctuations δkξ ≡4π2∆ξk/g4φ¯20.Thestarting
point,k/H=2π,correspondstothek-modethatleavesthehorizonatthestartofinflation.
3. Effects on the Large-Scale Structure
Incontrasttothe scale-invariantdensitypowerspectruminducedby∆q,thenoise-
k
driven density power spectrum Pξ induced by ∆ξ is blue-tilted on large scales.
k k
Assuming a dominant Pξ and using the set of cosmological parameters measured
k
2 15
byWMAP ,wehaveruntheCMBFASTnumericalcodes tocomputetheCMB
anisotropy power spectrum as shown in Fig. 2. We can see that the known result
February 5, 2008 15:47 WSPC/INSTRUCTION FILE cnoise˙cospa
Quantum Noises and the Large Scale Structure 5
of the scale-invariant power spectrum at small scales can be achieved by properly
choosing the initial time of inflation. Meanwhile,the suppressed large-scaledensity
perturbation can account for the low CMB low-l multipoles.
5000
4000
2Κ)
µ 3000
π (
2
C/l 2000
1)
+
l(l 1000
0
−1000
10 100 1000
l
Fig.2. CMBanisotropyintheΛCDMmodelwiththenoise-drivendensitypowerspectrumPξ.
k
The curves from below represent respectively different Pξ’s withk/H =200π, 500π, 1000π, and
k
2000π correspondingtothephysicalscaleof0.05Mpc−1.ThetopcurveistheΛCDMmodelwith
a scale-invariant power spectrum. We normalize all the anisotropy spectra at the first Doppler
peak. Alsoshownarethethree-year WMAPdata includingerrorbarsand thefirstyear WMAP
datadenotedbystars.
The above results do not include the intrinsic fluctuation of inflaton. To take
that into account, an additional white noise term ξ in the free field case must be
w
involved on the right hand side of the Langevin equation Eq. (3) as
φ¨+2aHφ˙−∇2φ+a2 V′(φ)+g2hσ2iφ −g4a2φ×
φ
d4x′a4(η′)θ(η−η′)i(cid:0)G−(x,x′)φ2(x′)=(cid:1) a2ξ+ξw. (14)
Z
Accordingly, the effective mass of the σ-field in Eq. (7) is modified as m2 =
σeff
m2 + g2(φ¯2 + hϕ2i), where hϕ2i represents the active quantum fluctuation with
σ q q
a scale-invariant power spectrum given by ∆q = H2/(4π2) 14. The total density
k
power spectrum then is given by
P =Pq+Pξ =X(φ¯) ∆q +∆ξ , (15)
k k k k k
(cid:16) (cid:17)
whereX(φ¯)isdeterminedbytheslow-rollkinematics.Underthecircumstances,the
power spectrum of the noise-driven inflaton fluctuations is not affected. However,
the CMB anisotropy in the ΛCDM model are modified accordingly, as shown in
Fig. 3 where the kinematic condition X(φ¯=φ¯ ) has been set to constant.
0
February 5, 2008 15:47 WSPC/INSTRUCTION FILE cnoise˙cospa
6 Wo-Lung Lee
6000
ΛCDM
5000
ΛCDM+noise (k/H=500π)
ΛCDM+noise (k/H=50π)
4000
2)
Κ
µ
π ( 3000
2
C/l
1) 2000
+
l(l
1000
0
-1000
1 10 100 1000
l
Fig. 3. CMB anisotropy in the ΛCDM model with the density power spectrum Pk =Pkq+Pkξ.
The solidcurve isthe ΛCDM model witha scale-invariant Pq induced by quantum fluctuations.
k
The dashed and dotted curves represent respectively noise-driven Pξ’s with k/H = 500π and
k
k/H=50π correspondingto0.05Mpc−1.
The passive fluctuation induced by the quantum noise not only suppresses the
CMB power on large scales,it also impresses on the trajectory of spectral index. If
we rewrite the total density power spectrum as
∆ξ
P =Pq 1+ k , (16)
k k ∆q
k!
the energy scale of inflation inferred from measurements of the density power
spectrum will be lower than the standard slow-roll prediction by a factor (1 +
∆ξ/∆q)−1/4 ≃0.92. The spectal index is then given by
k k
dlnP dlnPq d ∆ξ
n(k)−1≡ k = k + ln 1+ k . (17)
dlnk dlnk dlnk ∆q
k!
Apparently, the additional term containing ∆ξ in Eq. (17) offers a new dynamical
k
source for breaking the scale invariance. It is worthwhile to study the overalleffect
of the corrections induced by the dissipation, the static mass approximation, and
the static mean field approximation to the spectral index. Especially, we expect a
running spectral index at large k due to dissipational effects on the fluctuations at
latetimes.Furthermore,wehavemadeaGaussianapproximationtoderivethetwo-
point function of the colored noise in Eq. (4). Actually, the noise-induced density
fluctuations are intrinsically non-Gaussian. So, it is interesting to go beyond the
Gaussian approximation and consider the three-point function hξ(x )ξ(x )ξ(x )i,
1 2 3
February 5, 2008 15:47 WSPC/INSTRUCTION FILE cnoise˙cospa
Quantum Noises and the Large Scale Structure 7
which would directly lead to non-zero hϕ ϕ ϕ i without invoking higher-order
~k1 ~k2 ~k3
terms.
4. Conclusion
wehaveproposedanewscenarioinwhichtheinflatonisviewedasaphenomenolog-
ical classical field. In this model, the mean field drives the kinematics of inflation,
while its fluctuations are dynamically generated by the colored quantum noise as
a result of the coupling of the inflaton to a quantum environment. This model is
a better fit to the WMAP anisotropy data than the ΛCDM model. Moreover, the
observedlow CMB quadrupole may unveil the initial condition for inflation. These
results are derived by solving approximately the Langevin equation (3) of the in-
flaton which is coupled to a massive quantum field. The noise term then generates
a stochastic force which drives the inflaton fluctuations. It is worth studying the
overalleffect of the correctionsinduced by the dissipation,the static mass approxi-
mation,andthestaticmeanfieldapproximationtothespectralindexofthedensity
power spectrum. Furthermore, the density power spectrum driven by the colored
noise may be non-Gaussian.
Although we are based on a simple inflaton-scalar model, our results should be
generic in any interacting model. We have considered a quantum scalar field with
massoftheorderoftheHubbleparameterduringinflation.Infact,onecanconsider
a scalar field with much higher mass. So it is mandatory to perform a full analysis
of the cosmologicaleffects due to the quantum environment in a viable interacting
inflation model such as the hybrid or O(N) model.
Acknowledgments
This paper is based on the work, astro-ph/0604292, collaborated with Chun-Hsien
Wu, Kin-Wang Ng, Da-Shin Lee and Yeo-Yie Charng. The author thanks them all
for many inspiring discussions during the pleasant times we workedtogether.
References
1. C. L. Bennett et al., Astrophys.J. 464, L1 (1996).
2. C. L. Bennett et al., Astrophys. J. Suppl. Ser. 148, 1 (2003); G. Hinshaw et al.,
astro-ph/0603451.
3. J.-P. Luminet et al., Nature425, 593 (2003).
4. L.-Z.FangandY.P.Jing,Mod.Phys.Lett.A11,1531(1996); A.Berera,L.-Z.Fang,
G. Hinshaw, Phys. Rev.D 57, 2207 (1998).
5. W.LeeandL.-Z.Fang,Class.QuantumGra.17,4467(2000);W.LeeandL.-Z.Fang,
Phys.Rev.D 69, 023514 (2004).
6. S. Matarrese, M. A. Musso, and A. Riotto, J. Cosmol. Astropart. Phys. 05 (2004) 8;
M. Liguori et al.,J. Cosmol. Astropart. Phys. 08 (2004) 11.
7. A. A. Starobinsky, in Field Theory, Quantum Gravity and Strings, edited by H. J. de
Vega and N. Sanchez (Springer, Berlin, 1986); A. S. Goncharov, A. D. Linde, and V.
F. Mukhanov,Int.J. Mod. Phys.A 2, 561 (1987).
February 5, 2008 15:47 WSPC/INSTRUCTION FILE cnoise˙cospa
8 Wo-Lung Lee
8. S. Winitzki and A. Vilenkin,Phys. Rev.D 61, 084008 (2000).
9. For examples, W. Lee and L.-Z. Fang, Int.J. Mod. Phys. D 6, 305 (1997); Phys.Rev.
D 59, 083503 (1999).
10. Forreviewssee:A.Linde,hep-th/0503195;B.A.Bassett,S.Tsujikawa,andD.Wands,
astro-ph/0507632;D.Boyanovsky,H.J.deVega,andN.G.Sanchez,astro-ph/0601132.
11. M. Morikawa, Phys. Rev. D 42, 1027 (1990); B. L. Hu, J. P. Paz, and Y. Zhang, in
The OriginofStructure inthe Universe,editedbyE.GunzigandP.Nardone(Kluwer,
Dordrecht, 1993); E. Calzetta and B. L. Hu, Phys. Rev. D 49, 6636 (1994); B. L. Hu
and S. Sinha, Phys. Rev. D 51, 1587 (1995); E. Calzetta and B. L. Hu, Phys. Rev.
D 52, 6770 (1995); C. Greiner and B. Muller, Phys. Rev. D 55, 1026 (1997); E. A.
CalzettaandS.Gonorazky,Phys.Rev.D55,1812(1997);H.Kubotani,T.Uesugi,M.
Morikawa,andA.Sugamoto,Prog.Theor.Phys.98,1063(1997);F.C.Lombardoand
D. L. Nacir, Phys. Rev.D 72 063506 (2005).
12. C. H.Wu,N.K.Ng, W.Lee, D.S.Lee, and Y.Y.Charng, astro-ph/0604292,JCAP
in press.
13. R.P. Feynman and F. L. Vernon,Ann.Phys. 24, 118 (1963).
14. S.W.Hawking,Phys.Lett.B115,295(1982);A.A.Starobinsky,Phys.Lett.B117,
175 (1982); A.H. Guth and S. Y.Pi, Phys.Rev.Lett. 49, 1110 (1982).
15. U.Seljak and M. Zaldarriaga, Astrophys. J. 469, 437 (1996).