The Inflationary Gravity Waves in light of recent Cosmic Microwave Background Anisotropies data

5 Pages·0.19 MB·English

by  Alessandro Melchiorri

The Inflationary Gravity Waves in light of recent Cosmic Microwave Background
Anisotropies data.
Alessandro Melchiorri♭ and Carolina J. O¨dman♯
♭ Astrophysics, Denys Wilkinson Building, University of Oxford, Keble road, OX1 3RH, Oxford, UK
♯ Astrophysics Group, Cavendish Laboratory, Cambridge University, Cambridge, U.K.
Oneofthemajor predictionsof inflation is theexistenceof astochastic backgroundof cosmo-
logicalgravitationalwaves(GW).Thesegravitationalwavescaninducesignificanttemperature
anisotropiesintheCosmicMicrowaveBackground(CMB)ontheangularscalesrecentlyprobed
bytheArcheopsexperiment. Here,weperformacombinedanalysisofArcheopstogetherwith
information from other CMB experimentsand/or cosmological datasets, in order to constrain
3 the amplitude of the GW background. We find that, for a scale-invariant GW background,
0 the ratio of tensor/scalar perturbations at the CMB quadrupole is now constrained to be
0 r ≤ 0.43 at 95% c.l., while the bound on the spectral index of primordial density fluctua-
2 tions is n = 0.97+0.10. We discuss the implications for future GW detections through CMB
S −0.12
polarization measurements.
n
a
J I. INTRODUCTION bination, however, unlike the anisotropies generated

8 by scalar fluctuations, those generated by GW damp
Thelastyearshaveseenspectacularadvancesinour like fluctuations in a fluid of massless bosons (see e.g.
3 ability to confront the inflationary scenario of struc- [11]). Since the theoretical spectrum, normalized to
v
ture formation to observational data. The “multi- COBE,isalinearsumofthescalarandtensorcompo-
6
0 ple peaks” observed in the Cosmic Microwave Back- nents,ifthereisarelevantcontributionfromGWthis
6 ground (CMB) angular power spectrum ( [28], [17], would lower the predicted amplitude of the acoustic
0 [23], [32], [35]) are indeed providing strong sup- peaks on sub-degree angular scales.
1 porting evidence for the inflationary predictions of Withthe adventofthenew CMBpeaksdetections,
2 a flat universe and of a primordial background of many authors have therefore addressed the question
0 scale-invariant adiabatic perturbations (see e.g. [39], ofthe GW’s contribution(see e.g.[26],[21],[39],[12],
/
h [29]). More recently, the new CMB results from the [24], [40], [18]). However, despite the different scale
p Archeops experiment ( [1]) have confirmed and re- dependence, robust constraints on tensor modes re-
o- finedthepresentobservationalstatus,samplingangu- main difficult to obtain. The decrease in the ampli-
r larscalesbetweenthoseprobedbytheCOBEsatellite tude of the acoustic oscillations induced by GW can
st and the latest high precision datasets. Again, flat- indeed be compensated by an increase in one of the
a ness, adiabaticity and scale invariance are in agree- unconstrained parameters of the model, like, for ex-
v: ment with the data ( [2]). ample, the spectral index of scalar fluctuations nS.
i It has been argued that the next and probably Therefore, some form of ’cosmic degeneracy’arises in
X
mostconclusiveevidenceforinflationwouldbethede- thetradeoffbetweenthesetwo(andmore)parameters
r tection of a stochastic background of Gravity Waves (see [26], [12]) and only weak constraints on the GW
a
(GW)(seee.g.[7],[44]). Twotypesofspacetimemet- background were obtained.
ric fluctuations are indeed naturally produced during In this context, and before more accurate polariza-
inflation: densityperturbations(scalarmodes),which tiondatabecomeavailable(seediscussionbelow),the
form the “seeds” of structure formation, and gravity newresultsonintermediateangularscales,asrecently
waves (tensor modes) ( [16]). provided by Archeops,can offer an interesting oppor-
The GW background, if detected, would also pro- tunity.
videvaluableinformationontheinflationaryscenario. As we illustrate in Fig.1, this spectral region has a
In particular, in most inflationary models (and cer- particular sensitivity to a GW contribution. In the
tainlyinthesimplestones),theamplitude oftheGW figure, we plot two theoretical power spectra. The
backgroundisproportionaltothesquareoftheenergy models have identical power on sub-degree scales and
scaleofinflation(seee.g.[8]). Furthermore,acomple- on COBE scales (considering cosmic variance), but
mentary measurement of the ’tilt’ of the GW pertur- different tensor contributions, parametrizedby a ten-
bations (and of the scalar as well) can give direct in- sor over scalar ratio of the angular power spectrum
formationup to the second derivatives of the inflaton quadrupole r =C2T/C2S (see e.g. [21]).
potential, sheding light on the physics at ∼ 1016GeV As wecansee,while the twomodelsaredegenerate
(see e.g. [19]). on scales ℓ ≥ 200, the degeneracy is broken on larger
TheGWbackgroundleavesanimprintontheCMB angular scales (see the bottom panel), mostly in the
anisotropies at large scales through the Sachs-Wolfe region sampled by Archeops. Both increasing nS and
effect. On scales smaller than the horizon at recom- adding tensorschangetherateofgrowthofthe scalar
modes fromthe Sachs-Wolfe plateautowardsthe first
1
0.003, ΩΛ = 0.5,...,0.95, in steps of 0.05. Our choice
8000 r = 0 n=0.94 oftheaboveparametersismotivatedbytheBigBang
s
7000 r = 0.4 ns=0.97 Nucleosynthesis bounds on ωb (both from D [6] and
4He +7 Li [9]), from supernovae ( [14]) and galaxy
6000
clustering observations (see e.g. [38]).
2K]µ 5000 Variations inthe tensor and scalarspectralindices,
C/2 [πl4000 nS and nT are not computationally relevant. How-
1) 3000 ever, we restrict our analysis to relevant inflationary
+
l(l 2000 values nS =0.7,...,1.3 and we fix nT =0 (see discus-
sion below for different values of n ).
T
1000
Furthermore, the value of the Hubble constant
0 is not an independent parameter, since h =
10 100 1000
Multipole l p(ωcdm+ωb)/(1−ΩΛ). We also include the further
0,35
0,30 top-hat prior h = 0.7± 0.2 ( [13]) and we consider
Difference 0000,,,,12215050 omneWldyieummaoldlboeywls wvfaoirtryhianaggreettiho0en>iczoa1mt1iopGntyornosf.otphteicailntdeerpgtahlacptaic-
% 0,05
0,00 rameter τc in the range τc = 0.0,...,0.45 in steps of
10 100 1000
0.05. Wenoteherethathighvaluesofτ areinsevere
c
Multipole l
disagreement with recent estimates of the redshift of
FIG. 1. Best-fit models to recent CMB data with and reionization zre ∼ 6±1 (see e.g. [15]) which points
withoutGWcontribution(TopPanel). TheArcheopsdata towards τc ∼ 0.05−0.10. On the other hand, if the
pointsare shown as open circles. In theBottom panelwe reported CBI excess at ℓ ∼ 3000 is due to Sunyaev-
plot the % difference between the two degenerate mod- Zeldovicheffect,thenthiswouldfavourvaluesτ ∼0.3
c
els together with the cosmic variance limit (dashed line) ( [3]).
averaged in bins of ∆ℓ=10. For the CMB data, we use the recent results from
the BOOMERanG-98,DASI,MAXIMA-1, CBI,VSA
and Archeops experiments. The power spectra from
peakandthiscaninprinciplebeusedtoconstrainthe
these experiments were estimated in 19, 9, 13, 14, 10
GW background.
and16binsrespectively(fortheCBI,weusethedata
It is therefore extremely timely to analyze the
from the MOSAIC configuration, [10]), spanning the
Archeops data allowing the possibility of a GW con-
range2≤ℓ≤1500. WealsousetheCOBEdatafrom
tributioninorderto seeifthe amplitude ofthis back-
the RADPACK compilation ( [34]).
ground can now be better constrained than in the
For the CBI, DASI, MAXIMA-I and VSA exper-
past.
iments we use the publicly available correlation ma-
Furthermore, the GW background produces a
trices and window functions. For the Archeops and
unique statistical signature in the polarization of the
BOOMERanGexperimentsweassignaflatinterpola-
CMB by inducing a curl component ( [33], [20]), of-
tionforthespectrumineachbinℓ(ℓ+1)C /2π=C ,
ℓ B
ten defined as B mode, while scalar (but also ten-
and we approximate the signal C inside the bin to
B
sor)perturbationsproducesagradientcomponent(E
be a Gaussian variable. The likelihood for a given
mode). Given the large number of future and ongo- theoretical model is defined by −2lnL = (Cth −
ifnorgeCcaMstBfrpoomlatrhizeatpiroenseenxtpCerMimBenttesm,piteirsatinutreeredsattinagthtoe CBex)MBB′(CBth′ −CBex′) where MBB′ is the GausBsian
curvature of the likelihood matrix at the peak.
expected amplitude of the B modes and/or if the E
We consider 5%, 10%, 4%, 5%, 3.5% and 5%
modesproducedby tensorscanbe distinguishedfrom
Gaussian distributed calibration errors (in ∆T) for
those produced by scalar perturbations only.
the Archeops, BOOMERanG-98, DASI, MAXIMA-
We pursue this investigation in the present Rapid
1, VSA, and CBI experiments respectively and we
Communication as follows: in Section II we illustrate
include the beam uncertainties by the analytical
our analysis method. In section III we present our
marginalization method presented in ( [4]).
results. Finally,insectionIV,wediscussourfindings.
Finally,weparametrizetheGWcontributionbythe
tensor over scalar quadrupole ratio r = CT/CS and
2 2
II. ANALYSIS: METHOD we rescale the sum spectrum by a prefactor C10, as-
sumed to be a free parameter, in units of CCOBE.
10
As a first step, we consider a template of flat, adi-
abatic, Λ-CDM scalar and tensor spectra computed III. ANALYSIS: RESULTS
with CMBFAST ( [36]), sampling the various param-
eters as follows: Ω h2 ≡ ω = 0.05,...0.25, in
cdm cdm ThemainresultsofouranalysisareplottedinFig.2.
steps of 0.02; Ω h2 ≡ω =0.009,...,0.024,in steps of
b b Inthelefttoppanelweplotthelikelihoodcontoursin
2
10000
TE-Scalar 95% Max
99 % 2K]1000 TE-Scalar 95% Min
OPS 95 % 2 [πµ 11000
w/o ARCHEOPS 9959 %% c w/o ARCHE 68 % ETl(l+1)|C|/l00,0,111 TE1-0Tensor 95% C.L1.00 1000
68 % Multi pole l
0,1
S S 2K]µ g. lensing
[π0,01
2
BC/l
1)
l(l+1E-3 B-Modes Tensor 95% C.L.
c 68 % 95 % 99w/ o% AR CHEOPS 22 [K]πµ 110100 EE--SS1cc0aallaarr 9955M%%u MMltiai pnxol1e0 l0 1000
EC/l
1) 0,1
+
FIG. 2. 68%, 95% and 99% confidence regions in the l(l0,01 E-Tensor 95% C.L.
n −r (TopPanel,Left),n −τ (TopPanel,Right),r−τ 10 100 1000
S S Multipole l
(Bottom Panel) planes for the models considered in our
analysis(seetext). Thelinecontoursareconfidencelevels FIG. 3. Maximum and minimum levels of tempera-
without theArcheopsdata. ture-polarization cross correlation (Top Panel), B-modes
(CentralPanel),E-modes(BottomPanel)allowed at95%
C.L. from present CMB temperature data under the as-
the n −r plane, maximizing overthe remainingnui-
S sumption of themodels described in the text.
sanceparameters. As we cansee,in the frameworkof
models we considered, the gravitational wave contri-
bution is constrained to be r ≤0.2 (r ≤0.43) at 68% As we can see from the center panel of Figure 3,
C.L. (95% C.L.), with n = 0.97+0.06 (68% C.L.). the level of the B-modes, is expected to be of ∼ 0.2
S −0.07
While the inclusionofthe Archeopsdatahas little ef- µK, at maximum. The signal is out of the reach of
fect on n , it drastically improves the constraint on mostofthecurrentpolarizationexperimentlikeDASI
S
r. Removingthe Archeopsdatayields r ≤0.6at95% or POLATRON which are sensitive to few µK. Near
C.L.. futureexperimentslikeB2KorQUEST,willprobably
In the right top panel of Fig.2, we plot the likeli- have enough sensitivity to have a statistical B-mode
hood contours in the n −τ plane. As we can see, detection. However, the B-signal in the angular re-
S c
thepresentCMBconstraintonτ isratherweak,with gion sampled by these experiments (ℓ > 50), can be
c
τ <0.25(τ <0.36)at68%C.L.(95%C.L.). Itisin- contaminated by a foreground component due to the
c c
terestingtonotethattheinclusionoftheARCHEOPS conversionofE modestoB modesfromgravitational
datapoints has little effect. lensing (see Fig. 3) ( [41]). Higher-order correlations
Finally, in the bottom panel of Fig.2, we plot will be necessary to map the cosmic shear and sub-
the likelihood contours in the r −τ plane. An in- tract this contribution to the B mode ( [30]).
c
crease in τ or r produces a similar damping on the Tensor perturbations produce E modes as well.
c
small/intermediate angular scales. It is interesting to However,the amplitude ofthe E tensormodesis pre-
notice that the present data is allowing just a well dicted to be generally much smaller than those from
defined amount of small-scale damping. Values of thescalarmodes(seebottompanel). Awindowofop-
τ ∼ 0.3 are in disagreement with the presence of a portunitymayappearinthetemperature-polarization
c
tensor component. If τ > 0.2 then r < 0.05 at 68% (<TE >) cross-correlationspectra, where,at ℓ∼50,
c
C.L.. the amplitude from tensor can be larger than those
To each theoretical model in the likelihood planes from scalar modes, leaving a possible detectable ex-
producedinFig.2,ispossibletoassociateatheoretical cess for experiments like QUEST or B2K.
polarization power spectrum and translate the confi- In order to cross-check if any information can be
dence contours into an expected maximum and min- obtainedonn weperformedthe analysisonjustone
T
ima polarization signal. cosmological model defined by ΩΛ = 0.7, ωb = 0.022,
We do this in the 3 panels of Fig.3, where we plot Ω = 1, τ = 0.04. We then considered ten-
tot c
the envelope of the minima and maxima polarization sor contributions by varying the scalar and tensor
spectra that, in the panels of Fig.2, are at 95% c.l. spectral indices independently: n = 0.7,...,1.3 and
S
consistent with the CMB temperature data. n = −0.3,...,0.0, step 0.01. We found that the ten-
T
3
sor spectral index is not constrained by the present Ekpyrotic ( [37]) or Pre-Big Bang (see e.g. [27]) sce-
data, but that a value of n =0 is preferred. narios. However, extremely blue spectra (n ∼ 2)
T T
are excluded by constraints on the GW energy den-
sity background from timing milli-second binary pul-
IV. CONCLUSIONS sars [31]. Allowing for extra primordial perturbation
modes like isocurvature, will probably tight our con-
In this Rapid Communication we have presented straintsonGW,sincetheshapeofCDMscalarisocur-
newconstraintsonthestochasticbackgroundofgravi- vaturemodesissimilartothosefromadiabatictensor
tationalwavesfromrecentmicrowaveanisotropydata. modes. However, considering the most general initial
Thanks to Archeops, our results improve the con- conditions scheme and including cross correlations,
straints on tensor modes from previous analyses (see will certainly enlarge our constraints ( [5]). Includ-
e.g. [40], [21]). ing curvature (Ωtot 6=1) would relax our bounds on r
Intheframeworkofmodelsweconsidered,wefound (see e.g. [39]). Non-flatmodels in agreementwith the
(at 95% C.L.) r < 0.43 and n = 0.97+0.10. The CMBdataareingeneralclosedmodels,which,apart
S −0.12
energyscale of inflation E canbe relatedto tensor from a few exceptions ( [25]), are difficult to obtain
inf
by E4 = 0.65CTm4 . The above bound translates frominflation. Finally,includingadifferentmodelfor
inf 2 Pl
therefore in E ≤1.6×1016GeV. dark energy like quintessence would change the large
inf
scale anisotropy through the Integrated Sach Wolfe
Whencomparingwiththeresultspresentedin[2],a
effect (see e.g. [47]), affecting our constraints as well.
partfromthe differenttemplate oftheoreticalmodels
Even if the results presented here do not hint for a
considered,ouranalysisdiffersmainlyinthefollowing
presence of GW background, the data is still consis-
points: we assumed the low-ℓ Archeops bins as gaus-
tent with a sizable tensor contribution. It will there-
sian distributed, we included the COBE data using
forebethe dutyoffuture andongoingexperimentsto
the RADPACK compilation, we have a strong upper
scrutinize this fundamental prediction from inflation.
limitonω <0.025fromBBNand,finally,wenumer-
b
Acknowledgements
ically computed the models with τ > 0 (while in [2]
c
We wish to thank Anthony Challinor, Asantha
an analytical formula was used).
Cooray, Will Kinney, Rocky Kolb, Mike Hobson,
The GW backgroundinduces a unique signature in
RobertIzzard,Anthony Lasenby,Antonio Riotto and
thepolarizationoftheCMBbyproducingacurlcom-
FrancoisXavier-Desert. AMissupportedbyPPARC.
ponent,notpresentinthecaseofscalarperturbations.
CJOissupportedbyaGirtonCollegeScholarshipand
Inthe set of models we considered(andunder the as-
an Isaac Newton Studentship.
sumption of a bayesian method of statistical analy-
sis) we found that the maximum expected level of B
modes allowed by current data is of about ∼ 0.2µK,
whichcanbepartiallyattainablebynearfutureexper-
iments and severly contaminated by lensing E → B
conversion.
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