Relic gravitational waves in light of the 7-year Wilkinson Microwave Anisotropy Probe data andimproved prospects for the Planck mission

W. Zhao,1,2,3,* D. Baskaran,1,2,† and L. P. Grishchuk1,4,‡

1School of Physics and Astronomy, Cardiff University, Cardiff, CF24 3AA, United Kingdom2Wales Institute of Mathematical and Computational Sciences, Swansea, SA2 8PP, United Kingdom

3Department of Physics, Zhejiang University of Technology, Hangzhou, 310014, People’s Republic of China4Sternberg Astronomical Institute, Moscow State University, Moscow, 119899, Russia

(Received 25 May 2010; published 6 August 2010)

The new release of data from Wilkinson Microwave Anisotropy Probe improves the observational

status of relic gravitational waves. The 7-year results enhance the indications of relic gravitational waves

in the existing data and change to the better the prospects of confident detection of relic gravitational

waves by the currently operating Planck satellite. We apply to WMAP7 data the same methods of analysis

that we used earlier [W. Zhao, D. Baskaran, and L. P. Grishchuk, Phys. Rev. D 80, 083005 (2009)] with

WMAP5 data. We also revised by the same methods our previous analysis of WMAP3 data. It follows

from the examination of consecutive WMAP data releases that the maximum likelihood value of the

quadrupole ratio R, which characterizes the amount of relic gravitational waves, increases up to R ¼0:264, and the interval separating this value from the point R ¼ 0 (the hypothesis of no gravitational

waves) increases up to a 2� level. The primordial spectra of density perturbations and gravitational waves

remain blue in the relevant interval of wavelengths, but the spectral indices increase up to ns ¼ 1:111 and

nt ¼ 0:111. Assuming that the maximum likelihood estimates of the perturbation parameters that we

found from WMAP7 data are the true values of the parameters, we find that the signal-to-noise ratio S=N

for the detection of relic gravitational waves by the Planck experiment increases up to S=N ¼ 4:04, even

under pessimistic assumptions with regard to residual foreground contamination and instrumental noises.

We comment on theoretical frameworks that, in the case of success, will be accepted or decisively rejected

by the Planck observations.

DOI: 10.1103/PhysRevD.82.043003 PACS numbers: 98.70.Vc, 04.30.�w, 98.80.Cq

I. INTRODUCTION

The Wilkinson Microwave Anisotropy Probe (WMAP)Collaboration has released the results of 7-year (WMAP7)observations [1,2]. In this paper, we apply to WMAP7 datathe same methods of analysis that we have used before [3]in the analysis of WMAP5 data. This is important forupdating the present observational status of relic gravita-tional waves and for making more accurate forecasts forthe currently operating Planck mission [4].

In Sec. II we briefly summarize our basic theoreticalfoundations and working tools. Part of this material waspresent in the previous publication [3], but in order to makethe paper self-contained we briefly repeat it here.Section III exposes full details of our maximum likelihoodanalysis of WMAP7 data. In the focus of attention is theinterval of multipoles 2 � ‘ � 100, where gravitationalwaves compete with density perturbations. We compareall the results that we derived by exactly the same methodfrom WMAP7, WMAP5, and WMAP3 data sets. Thiscomparison demonstrates the stability of data and dataanalysis. On the grounds of this comparison, one can say

that the perturbation parameters found from the consecu-tive WMAP data releases have the tendency of saturatingnear some particular values. The WMAP7 maximum like-lihood (ML) value of the quadrupole ratio R is close toprevious evaluations of R, but increases up to R ¼ 0:264.The interval separating this ML value from the point R ¼ 0(the hypothesis of no gravitational waves) increases up to a2� level. The primordial spectra remain blue, but thespectral indices in the relevant interval of wavelengthsincrease up to ns ¼ 1:111 and nt ¼ 0:111.In Sec. IV we analyze why, to what extent, and in what

sense our conclusions with respect to relic gravitationalwaves differ from those reached by the WMAPCollaboration. The WMAP team has found ‘‘no evidencefor tensor modes.’’ A particularly important issue, whichwe discuss in some detail, is the presumed constancy (orsimple running) of spectral indices. We derive an exactformula for the spectral index nt as a function of wavenumbers and discuss in this context the formula for runningthat was used in WMAP analysis. Another contributingfactor to the difference of conclusions is the difference inour treatments of the inflationary ‘‘consistency relations’’based on the inflationary ‘‘classic result.’’ We do not usethe inflationary theory.A comprehensive forecast for Planck findings in the area

of relic gravitational waves is presented in Sec. V. We

*Wen.Zhao@astro.cf.ac.uk†Deepak.Baskaran@astro.cf.ac.uk‡Leonid.Grishchuk@astro.cf.ac.uk

PHYSICAL REVIEW D 82, 043003 (2010)

1550-7998=2010=82(4)=043003(15) 043003-1 � 2010 The American Physical Society

discuss the efficiency of various information channels, i.e.various correlation functions and their combinations. Weperform multipole decomposition of the calculated S=Nand discuss physical implications of the detection in vari-ous intervals of multipole moments. We stress again thatthe B-mode detection provides the most of S=N only in theconditions of very deep cleaning of foregrounds and rela-tively small values of R. The improvements arising from a28-month, instead of a nominal 14-month, Planck surveyare also discussed. In the center of our attention is themodel with the WMAP7 maximum likelihood set of pa-rameters. For this model, the signal-to-noise ratio S=N inthe detection of relic gravitational waves by Planck experi-ment increases up to S=N ¼ 4:04, even under pessimisticassumptions with regard to residual foreground contami-nation and instrumental noises. Section VI gives Bayesiancomparison of different theoretical frameworks and iden-tifies predictions of R that may be decisively rejected bythe Planck observations.

II. PERTURBATION PARAMETERS AND CMBPOWER SPECTRA

The temperature and polarization anisotropies of CMBare produced by density perturbations, rotational perturba-tions, and gravitational waves. Rotational perturbations areexpected to be very small and are usually ignored, and arein this paper, too. The cosmological perturbations arecharacterized by their gravitational field (metric) powerspectra which are in general functions of time. Here, weintroduce the notations and equations that will be used insubsequent calculations.

As before (see [3,5], and references therein), we areworking with perturbed Friedmann-Lemaitre-Robertson-Walker universes

ds2 ¼ �c2dt2 þ a2ðtÞð�ij þ hijÞdxidxj¼ a2ð�Þ½�d�2 þ ð�ij þ hijÞdxidxj�;

where the functions hijð�;xÞ are metric perturbation fields.

Their spatial Fourier expansions are given by

hijð�;xÞ ¼ C

ð2�Þ3=2Z þ1

�1d3nffiffiffiffiffiffi2n

p Xs¼1;2

½ps ijðnÞhs

nð�Þein�xcsn

þ ps �ijðnÞh

s �nð�Þe�in�xc

syn�: (1)

The polarization tensors psijðnÞ (s ¼ 1; 2) refer either to the

two transverse-traceless components of gravitationalwaves (gw) or to the scalar and longitudinal-longitudinalcomponents of density perturbations (dp). Density pertur-bations necessarily include perturbations of the accompa-nying matter fields (not shown here).

In the quantum version of the theory, the quantities csn

and csyn are the annihilation and creation operators, respec-

tively, of the considered type of perturbations, and the j0i is

the initial vacuum (ground) state of the correspondingtime-dependent Hamiltonian. The metric power spectrumh2ðn; �Þ is defined by the expectation value of the quadraticcombination of the metric field:

h0jhijð�;xÞhijð�;xÞj0i ¼Z 1

0

dn

nh2ðn; �Þ;

h2ðn;�Þ � C2

2�2n2

Xs¼1;2

jhsnð�Þj2: (2)

The mode functions hs

nð�Þ are taken either from gw or dp

equations, and C ¼ ffiffiffiffiffiffiffiffiffi16�

plPl for gravitational waves and

C ¼ ffiffiffiffiffiffiffiffiffi24�

plPl for density perturbations.

The simplest assumption about the initial stage of cos-mological expansion (i.e. about the initial kick that pre-sumably took place soon after the birth of our Universe[6,7]) is that it can be described by a single power-law scalefactor [7–9]

að�Þ ¼ loj�j1þ�; (3)

where lo and � are constants, �<�1. Then the generatedprimordial power spectra (primordial means the interval ofthe spectrum pertaining to wavelengths longer than theHubble radius at the considered moment of time) havethe universal power-law dependence, both for gw and dp:

h2ðnÞ / n2ð�þ2Þ:

It is common to write these power spectra separately for gwand dp:

h2ðnÞðgwÞ ¼ B2t n

nt ; h2ðnÞðdpÞ ¼ B2sn

ns�1: (4)

In accordance with the theory of quantum-mechanicalgeneration of cosmological perturbations [7–9], the spec-tral indices are approximately equal ns � 1 ¼ nt ¼ 2ð�þ2Þ and the amplitudes Bt and Bs are of the order ofmagnitude of the ratio Hi=HPl, where Hi � c=lo is thecharacteristic value of the Hubble parameter during thekick.If the initial stage of expansion is not assumed to be a

pure power-law evolution (3), the spectral indices nt andns � 1 are not constants, but their wave number depen-dence is calculable from the time dependence of the scalefactor að�Þ and its time derivatives [10]. In fact, as we shallargue below, the CMB data suggest that even at a span of2 orders of magnitude in terms of wavelengths the spectralindex ns is not the same. We discuss this issue in detail inSec. IV.In what follows, we use the numerical code CAMB [11]

and related notations for gw and dp power spectra adoptedthere:

PtðkÞ ¼ At

�k

k0

�nt; PsðkÞ ¼ As

�k

k0

�ns�1

; (5)

where k0 ¼ 0:002 Mpc�1. Technically, the power spec-trum PsðkÞ refers to the curvature perturbation called R

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or � , but the amplitudes Bs and ðAsÞ1=2 are equal to eachother up to a numerical coefficient of order 1. The constantdimensionless wave number n is related to the dimension-ful k by k ¼ n=ð2lHÞ, where lH ¼ c=H0 is the present-dayHubble radius. The wave number n ¼ nH ¼ 4� marks thewavelength equal to lH today. The CMB temperature an-isotropy at the multipole ‘ is mostly generated by metricperturbations with wave numbers n � ‘ (see [12] for de-tails). Setting h ¼ 0:704 we obtain ‘ � ð0:85104 MpcÞk, which is consistent with the numerical result‘ � ð1:0 104 MpcÞk derived in [13].

The CMB temperature and polarization anisotropies areusually characterized by the four angular power spectraCTT‘ , CEE

‘ , CBB‘ , and CTE

‘ as functions of the multipole ‘.The contribution of gravitational waves to these powerspectra has been studied, both analytically and numeri-cally, in a number of papers [14–18]. The derivation oftoday’s CMB power spectra brings us to approximate for-mulas of the following structure [17]:

CTT‘ ¼

Z dn

nh2ðn; �recÞ½FT

‘ ðnÞ�2;

CTE‘ ¼

Z dn

nhðn; �recÞh0ðn; �recÞ½FT

‘ ðnÞFE‘ ðnÞ�;

CYY‘ ¼

Z dn

nðh0Þ2ðn; �recÞ½FY

‘ ðnÞ�2; where Y ¼ E; B:

(6)

In the above expressions, h2ðn;�recÞ and ðh0Þ2ðn; �rec) arethe power spectra of the gravitational wave field and itsfirst time derivative, respectively. The spectra are taken atthe recombination (decoupling) time �rec. The functionsFX‘ ðnÞ (X ¼ T; E; B) take care of the radiative transfer of

CMB photons in the presence of metric perturbations. Asalready mentioned, the power residing in the metric fluc-tuations at wave number n mostly translates into the CMBTT power at the angular scales ‘ � n. Similar results holdfor the CMB power spectra induced by density perturba-tions. The actually performed numerical calculations useequations more accurate than Eq. (6). They also include theeffects of the reionization era.

The CMB power spectra needed for the analysis ofWMAP data are calculated in the framework of the back-ground cosmological �CDM model characterized by theWMAP7 best-fit parameters [1]

�bh2 ¼ 0:022 60; �ch

2 ¼ 0:1123;

�� ¼ 0:728; �reion ¼ 0:087; h ¼ 0:704:(7)

To quantify the contribution of relic gravitational wavesto the CMB we use the quadrupole ratio R defined by

R � CTT‘¼2ðgwÞ

CTT‘¼2ðdpÞ

: (8)

Another measure is the so-called tensor-to-scalar ratio r.

This quantity is constructed from the primordial powerspectra (5)

r � At

As

: (9)

Often this parameter is linked to incorrect (inflationary)statements, such as the inflationary consistency relation(more details in Sec. IV). However, if one uses r withoutimplying inflationary claims, one can find a useful relationbetween R and r.In general, the relation between R and r depends on

background cosmological parameters and spectral indicesns and nt. We found this relation numerically using theCAMB code [11]. (For a semianalytical approach see [19].)

The results are plotted in Fig. 1. For this calculation weused the background cosmological parameters (7) and thecondition nt ¼ ns � 1 required by the theory of quantum-mechanical generation of cosmological perturbations. Weverified that the relation rðRÞ only weakly depends on thebackground parameters and does not change significantlywhen the values (7) are varied within the WMAP7 1� errorrange [1]. It is seen from the graph that r ¼ 1:92R in thecase of ns ¼ 1:0, and r � 2R for all considered ns, if R andr are sufficiently small. In other words, one can use r � 2Rfor a quite wide class of models.We do not know enough about the very early Universe to

predict R with any certainty. However, since the theory ofquantum-mechanical generation of cosmological perturba-tions requires that the amplitudes Bt and Bs [as well as At

and As in Eq. (5)] should be of the same order of magni-tude, our educated guess is that R should lie somewhere inthe range R 2 ½0:01; 1�. If R were observationally foundsignificantly outside this range, we would have to concludethat the underlying perturbations are unlikely to be ofquantum-mechanical origin. On the other hand, the most

FIG. 1 (color online). The relation between R and r for differ-ent values of the spectral index ns. From top to bottom the nschanges from ns ¼ 0:8 to ns ¼ 1:2.

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advanced inflationary theories predict the ridiculouslysmall amounts of gravitational waves, something at thelevel of r � 10�24 or less, r 2 ½0; 10�24�. The rapidlyimproving CMB data will soon allow one to decisivelydiscriminate between these theoretical frameworks (letalone the already performed discrimination on the groundsof purely theoretical consistency). We discuss this issue inSec. VI.

III. EVALUATION OF RELIC GRAVITATIONALWAVES FROM 7-YEAR WMAP DATA

A. Likelihood function

Relic gravitational waves compete with density pertur-bations in generating CMB temperature and polarizationanisotropies at relatively low multipoles. For this reasonwe focus on the WMAP7 data at 2 � ‘ � 100. As before[3,12], the quantities DTT

‘ , DTE‘ , DEE

‘ , and DBB‘ denote the

estimators (and also the actual observed data points in thelikelihood analysis) of the corresponding power spectra.Since the WMAP7 EE and BB observations are not par-ticularly informative, we marginalize (integrate) the totalprobability density function (pdf) over the variables DEE

‘

and DBB‘ [3,12]. The resulting pdf [3] is a function of DTT

‘

and DTE‘ :

fðDTT‘ ;DTE

‘ Þ ¼ n2xn�3=2

�21þn��2

�n

2

�ð1� �2

‘Þ

ð�T‘ Þ2nð�E

‘ Þ2��ð1=2Þ

exp

�1

1� �2‘

��‘z

�T‘�

E‘

� z2

2xð�E‘ Þ2

� x

2ð�T‘ Þ2

��: (10)

This pdf contains the variables DXX0‘ (XX0 ¼ TT; TE)

through the quantities x � nðDTT‘ þ NTT

‘ Þ and z � nDTE‘ ,

where NTT‘ and NEE

‘ are total noise power spectra.

Information about the power spectra CXX0‘ is contained in

the quantities �T‘ , �

E‘ , and �‘ (see [3] for details). The

sought-after parameters As, At, ns, and nt enter the pdf

through the CXX0‘ . The quantity n ¼ ð2‘þ 1Þfsky in (10) is

the effective number of degrees of freedom at multipole ‘,where fsky is the sky-cut factor.

In the WMAP7 data release the sky-cut factor is fsky ¼0:783 [2], which is slightly smaller than fsky ¼ 0:85 used

inWMAP5 data analysis [3]. The smaller fsky increases the

uncertainties, but this disadvantage is more than compen-sated by the reduction of overall noises. Therefore, theerror bars surrounding the WMAP7 data points are some-what smaller than those for the WMAP5 data release. Thisfact, together with slightly shifted data points themselves,allows us to strengthen our conclusions (see below) aboutthe presence of gravitational wave signal in the WMAPdata.

We seek the perturbation parameters R, As, and ns (nt ¼ns � 1) along the lines of our maximum likelihood analy-sis of WMAP5 data [3]. The pdf (10) considered as afunction of unknown R, As, and ns with known data points

DXX0‘ is a likelihood function subject to maximization. For

a set of observed multipoles ‘ ¼ 2; . . . ; ‘max, the likelihoodfunction can be written as [3]

� 2 lnL ¼ X‘

�1

1� �2‘

�z2

xð�E‘ Þ2

þ x

ð�T‘ Þ2

� 2�‘z

�T‘�

E‘

�

þ lnðð1� �2‘Þð�T

‘ Þ2nð�E‘ Þ2Þ

�þ C; (11)

where the constant C is chosen to make the maximumvalue of L equal to 1.

B. Results of the analysis of the WMAP7 data

The WMAP7 data points for DTT‘ and DTE

‘ at multipoles

2 � ‘ � ‘max ¼ 100 were taken, with gratitude, from theWeb site [20]. The 3-dimensional likelihood function (11)was probed by the Markov chain Monte Carlo method[21,22] using 10 000 samples. The ML values of the threeperturbation parameters R, ns, and As were found to be

R ¼ 0:264; ns ¼ 1:111; As ¼ 1:832 10�9;

(12)

and nt ¼ 0:111. In Fig. 2 we show the projection of the10 000 sample points on the 2-dimensional planes R� nsand R� As. The color of an individual point signifies thevalue of the 3-dimensional likelihood of the correspondingsample. The projections of the maximum (12) are shownby a black þ. (The value R ¼ 0:264 is equivalent to r ¼0:550.)Before analyzing the 3-parameter results, it is instructive

to consider the 2-parameter and 1-parameter probabilitydistributions of the sought-after parameters. These margi-nalized distributions are obtained by integrating the like-lihood function L (11) (already represented by 10 000points) over one or two parameters. By integrating overAs or ns, we arrive at 2-dimensional distributions for thepairs R� ns or R� As, respectively. The area around themaximum of the resulting distributions is shown in Fig. 3.In the R� ns space, the maximum is located at

R ¼ 0:228; ns ¼ 1:108: (13)

In the R� As space, the maximum is located at

R ¼ 0:253; As ¼ 1:787 10�9: (14)

In the left panel we also reproduce the 2-dimensionalcontours obtained by the WMAP team [1,2] (their r istranslated into our R). In contrast to the WMAP5 paper[23], the WMAP7 paper [2] shows only the uncertaintycontours derived under the assumption of a strictly con-stant spectral index ns, but not for the case of its running.

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0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n s

R

1 1.25 1.5 1.75 2 2.25 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

As * 109

R

FIG. 3 (color online). The ML point (red ) and the 68.3% and 95.4% confidence intervals (red solid lines) for the 2-dimensionaldistributions R� ns (left panel) and R� As (right panel). The left panel shows also the WMAP7 confidence contours (black dashedline) derived under the assumption that the spectral index ns is one and the same constant throughout all measured multipoles [1,2].The WMAP7 papers do not show the confidence contours in the R� As plane.

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ns

R

Likelihood

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

As * 109

R

Likelihood0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FIG. 2 (color online). The projection of 10 000 samples of the 3-dimensional likelihood function, based on the WMAP7 data, ontothe R� ns (left panel) and R� As (right panel) planes. The black þ indicates the maximum likelihood parameters listed in (12).

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The 1-dimensional distributions for R, ns, or As areobtained by integrating the likelihood function L (11)over the sets of two parameters (As, ns), (As, R), or (R,ns), respectively. These distributions are presented inFigs. 4 and 5 (red solid lines). The ML values of theparameters and their 68.3% confidence intervals are foundto be

R ¼ 0:273þ0:185�0:156; ns ¼ 1:112þ0:089

�0:064;

As ¼ ð1:765þ0:279�0:263Þ 10�9:

(15)

For completeness and comparison (see Sec. III C), we also

show in Figs. 4 and 5 the 1-dimensional (1D, for brevity)distributions derived by exactly the same procedure fromWMAP5 and WMAP3 observations. Obviously, theWMAP5 curves are copies of the previously reported dis-tributions [3]; the WMAP3 curves are explained inSec. III C.The constancy of spectral indices is a matter of special

discussion in Sec. IV. In preparation for this discussion, wereport the results of our likelihood analysis of data in otherintervals of multipole moments ‘, in addition to the interval2 � ‘ � 100 that resulted in the ML values (12). First, weanalyzed the WMAP7 data in the interval 101 � ‘ � 220.The ns coordinate of the maximum in 3-dimensional spaceR, As, and ns was found to be ns ¼ 0:951. The 1D margi-nalized distribution for ns gave the ML result

ns ¼ 0:969þ0:083�0:063 ð68:3%C:L:Þ: (16)

Second, we have done the same in the combined interval ofmultipoles 2 � ‘ � 220. The maximum of 3D likelihoodlies at ns ¼ 1:003, whereas the 1D ML result is 1D:

ns ¼ 1:021þ0:043�0:038 ð68:3%C:L:Þ: (17)

The 1D results allow one to make easier comparison ofconfidence intervals. As is seen from (15) and (16), the 1Ddeterminations of ns in the adjacent ranges 2 � ‘ � 100and 101 � ‘ � 220 overlap, but only marginally, in their1� intervals. As expected, if ns is assumed constant in theentire range 2 � ‘ � 220, its value (17) is intermediatebetween (15) and (16). The spreads of wave numbers towhich the 3D spectral indices ns ¼ 1:111, ns ¼ 0:951, andns ¼ 1:003 refer are shown by marked red lines in Fig. 6.

FIG. 5 (color online). The 1-dimensional likelihoods for ns (left) and As (right). In both panels the red (solid), black (dashed), andblue (dotted) curves show the results for WMAP7, WMAP5, and WMAP3 data, respectively.

FIG. 4 (color online). The 1-dimensional likelihoods for R.The results of the analysis of WMAP7, WMAP5, and WMAP3data are shown, respectively, by the red (solid), black (dashed),and blue (dotted) curves. The shaded regions indicate the 68.3%and 95.4% confidence intervals for WMAP7 likelihood.

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C. Comparison of results derived from WMAP7,WMAP5, and WMAP3 data releases

One and the same 3-parameter analysis of WMAP5 andWMAP7 data has resulted in somewhat different MLparameters. From WMAP7 observations we extracted theML parameters (12), whereas the ML parameters extractedfrom WMAP5 observations [3] are

R ¼ 0:229; ns ¼ 1:086; As ¼ 1:920 10�9:

Certainly, the results are consistent and close to each other.The same holds true for marginalized 1-dimensional pa-rameters and distributions shown in Figs. 4 and 5. Thissimilarity of results testifies to the stability of data and dataanalysis. There exist, however, important trends in thesequence of ML parameters extracted from the progres-sively improving WMAP3, WMAP5, and WMAP7 data.We want to discuss these trends.

Specially for this discussion, we derived the parametersR, ns, and As from WMAP3 data in exactly the samemanner as was done here and in [3] with WMAP7 andWMAP5 data releases. Previously [12], we derived theseparameters by a different method: we restricted the like-lihood analysis to TE data and a single parameter R, whilens and As were determined from phenomenological rela-tions designed to fit the TT data. That analysis has led us toR ¼ 0:149þ0:247

�0:149 and ns ¼ 1:002. Our new derivation,

based on 3-dimensional likelihood, gives the followingWMAP3 maximum likelihood parameters:

R ¼ 0:181; ns ¼ 1:045; As ¼ 2:021 10�9:

The corresponding 1-dimensional distributions give

R ¼ 0:205þ0:181�0:157; ns ¼ 1:059þ0:097

�0:066;

As ¼ ð1:894þ0:290�0:307Þ 10�9:

(18)

The 1-dimensional WMAP3 distributions are plotted inFigs. 4 and 5 by dotted curves alongside with WMAP5(dashed) and WMAP7 (solid) curves.Looking at all derivations and graphs collectively, we

can draw the following conclusions. First, about R. Themaximum likelihood value of R increases as one goes overfrom 3-year to better quality 7-year data. The ML valueR ¼ 0:264 of WMAP7 data release is 15% larger than theanalogous result R ¼ 0:229 obtained from WMAP5 data.The 3-, 2-, and 1-parameter determinations of R persis-tently concentrate somewhere around the mark R ¼ 0:25.The uncertainties �R, although still considerable, getsmaller as one progresses from 3-year to better quality 7-year data. The R ¼ 0 (no gravitational waves) hypothesisis under increasing pressure. For example, the WMAP7 1-parameter result, Eq. (15) and Fig. 4, excludes the R ¼ 0hypothesis at almost 2� level. To be more precise, the R ¼0 point is right on the boundary of the 94% confidence area(94% of surface area under the WMAP7 curve in Fig. 4)surrounding the 1D ML value R ¼ 0:273. This is an im-provement in comparison with a slightly larger than 1�interval separating R ¼ 0 from the WMAP5 1D ML valueR ¼ 0:266 [3]. Admittedly, the gradual changes in MLvalues of R are small, while uncertainties are still large.It would not be very surprising if random realizations ofnoise moved the consecutive 3-year, 5-year, and 7-year MLvalues of R in arbitrary order. Nevertheless, we observe thetendency for systematic increase and saturation of R along-side with decrease of �R.Second, our conclusions about ns. Together with the

increase of ML R, we observe the tendency for the increaseof ns accompanied by the theoretically expected andunderstood decrease of As (the contribution of relic gravi-tational waves becomes larger, while the contribution ofdensity perturbations becomes smaller). The ML valuens ¼ 1:111 derived from WMAP7 data, Eq. (12), is largerthan ns ¼ 1:086 derived from WMAP5 data, while theWMAP7 value of As is somewhat smaller than that ofWMAP5 [3]. The tendency for increase of ns and decreaseof As is also illustrated by the 1-parameter distributions inFig. 5. The spectral indices ns and nt persistently point outto blue primordial spectra, i.e. ns > 1 for density perturba-tions and nt > 0 for gravitational waves, in the interval ofwavelengths responsible for CMB anisotropies at 2 � ‘ �100. The larger values of ns and nt derived from theWMAP7 data release enhance the doubt on whether theconventional scalar fields could be the driver of the initialkick, since these cannot support �>�2 in Eq. (3) and,consequently, ns > 1 and nt > 0 in Eq. (4). [Since theinflationary theory is capable of predicting virtually any-thing that one can possibly imagine, there exists of courseliterature claiming that inflation can predict blue spectra.But these claims are based on the incorrect (inflationary)formula for density perturbations; see [9].] The questionspertaining to the spectral indices are analyzed in somedetail in the next section.

FIG. 6 (color online). The spectral index ns as a function of thewave number k. The ML results of this work are shown by redlines. Other lines are our plots of WMAP7 findings [1], Table 7.

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IV. COMPARISON OF OUR RESULTS WITHCONCLUSIONS OF WMAP COLLABORATION

Having analyzed the 7-year data release, the WMAPCollaboration concludes that a minimal cosmologicalmodel without gravitational waves and with a constantspectral index ns across the entire interval of consideredwavelengths remains a ‘‘remarkably good fit’’ to ever-improving CMB data and other data sets. The WMAPteam emphasizes: ‘‘We do not detect gravitational wavesfrom inflation with 7-year WMAP data; however the upperlimits are 16% lower. . .’’ [2], p. 11; ‘‘The 7-year WMAPdata combined with BAO and H0 excludes the scale-invariant spectrum by more than 3�, if we ignore tensormodes (gravitational waves)’’ [1], p. 15; ‘‘We find noevidence for tensor modes. . .’’ [1], p. 30; ‘‘We find noconvincing deviations from the minimal model’’ [1], p. 1,etc.

In contrast, our analysis of WMAP3, WMAP5, andWMAP7 data leads us in the opposite direction: the im-proving data make the gw indications stronger. The majorpoints of tension between the two approaches seem to bethe constancy of spectral indices and the continuing use bythe WMAP team of the inflationary theory in data analysisand interpretation. We shall start from the discussion ofspectral indices.

The constancy of spectral indices is a reasonable as-sumption, but not a rule. If the power-law dependence (3)is not a good approximation to the gravitational pump fieldduring some interval of time, the constancy of nt and ns isnot a good approximation to the generated primordialspectra (4) in the corresponding interval of wavelengths.In fact, the future measurements of frequency dependenceof the spectrum of relic gravitational waves will probablybe the best way to infer the ‘‘early history of the Hubbleparameter’’ [10].

The frequency dependence of a gw spectrum is fullydetermined by the time dependence of the function ðtÞ �� _H=H2. [In more recent papers of other authors thisfunction is often denoted ðtÞ.] The function ðtÞ describesthe rate of change of the time-dependent Hubble radiuslHðtÞ � c=HðtÞ:

ðtÞ ¼ d

dt

�1

HðtÞ�¼ 1

c

dlHðtÞdt

:

The function ðtÞ is a constant for power-law scale factors(3): ¼ ð2þ �Þ=ð1þ �Þ, and ¼ 0 for a period ofde Sitter expansion. The interval of time dt during theearly era when gravitational waves were entering the am-plifying regime and their today’s frequency spread d� arerelated by [see Eq. (21) in [10]]

d

dt¼ ½1� ð�Þ�Hð�Þ d

d ln�:

Today’s spectral energy density of gravitational wavesð�Þ is related to the early Universe parameter ðtÞ by [see

Eq. (22) in [10]]

ð�Þ ¼ � ½d lnð�Þ=d ln��2� ½d lnð�Þ=d ln�� : (19)

The spectral index ng of a pure power-law energy den-

sity ð�Þ / �ng is defined as ng ¼ ½d lnð�Þ=d ln��. It isreasonable to retain this definition for more complicatedspectra. Then, Eq. (19) can be rewritten as

ngð�Þ ¼ � 2ð�Þ1� ð�Þ : (20)

Obviously, in the case of pure power laws (3) we return tothe constant spectral index ng ¼ �2=ð1� Þ ¼2ð2þ �Þ.Equation (20) was derived for the energy density of

relatively high-frequency gravitational waves, � >10�16 Hz, which started the adiabatic regime during theradiation-dominated era. In our CMB study we deal withsignificantly lower frequencies. It is more appropriate tospeak about wave numbers n rather than frequencies �, andabout power spectra of metric perturbations h2ðnÞ ratherthan energy density ð�Þ. The k-dependent spectral indexntðkÞ entering primordial spectrum (5) is defined asntðkÞ ¼ ½d lnPtðkÞ=d lnk�. Then the formula for ntðkÞ re-tains exactly the same appearance as Eq. (20):

ntðkÞ ¼ � 2ðkÞ1� ðkÞ : (21)

The spectral index ntðkÞ reduces to a constant nt ¼ 2ð2þ�Þ in the case of power-law functions (3).The spectral index nsðkÞ � 1 for density perturbations is

defined by nsðkÞ � 1 ¼ ½d lnPsðkÞ=d lnk�. The formula fornsðkÞ � 1 is more complicated than Eq. (21) as it containsalso dðtÞ=dt as a function of k. However, it is important tostress that adjacent intervals of power-law evolution (3)with slightly different constants � will result in slightlydifferent pairs of constant indices nt and ns � 1 in thecorresponding adjacent intervals of wavelengths. Ofcourse, the spectrum itself is continuous at the wavelengthmarking the transition between the two regions.Extending the minimal model, the WMAP

Collaboration works with the power spectrum [1]

PsðkÞ ¼ Psðk0Þ�k

k0

�nsðk0Þ�1þð1=2Þ�s lnðk=k0Þ

; (22)

which means that the k-dependent (running) spectral indexnsðkÞ is assumed to be a constant plus a logarithmic cor-rection:

nsðkÞ ¼ nsðk0Þ þ �s lnðk=k0Þ: (23)

The aim of WMAP data analysis is to find �s, unless it ispostulated from the very beginning, as is done in thecentral (minimal) model, that �s � 0. We note thatalthough logarithmic corrections do arise in simple situ-ations and can even be termed ‘‘natural’’ [10], they are not

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unique or compulsory, as we illustrated by exact formula(21). Nevertheless, we do not debate this point. We acceptWMAP’s definitions, and we want to illustrate their resultsgraphically, together with our evaluations of ns in thispaper.

The main result of WMAP7 determination is ns ¼0:963 0:012 (68% C.L.) derived under the assumptionof no gravitational waves and constant ns throughout allwavelengths included in the considered data sets [1,2].When the presence of gravitational waves is allowed, butns is still assumed constant, the ns rises to ns ¼ 0:982þ0:020

�0:019

from WMAP7 data alone. Finally, from WMAP7 dataalone, the WMAP team finds nsðk0Þ ¼ 1:027, �s ¼�0:034 in the case of no gw but with running of ns, andnsðk0Þ ¼ 1:076, �s ¼ �0:048 in the case of running andallowed gravitational waves (we quote only central valueswithout error bars; see Table 7 in [1]). All the resultingvalues of nsðkÞ derived by the WMAP team are shown inFig. 6. For comparison, we also plot by red lines ourevaluations of ns; see Sec. III B.

The lines in Fig. 6 show clearly that our finding of a blueshape of the spectrum, i.e. ns ¼ 1:111, at longest acces-sible wavelengths is pretty much in the territory of WMAPfindings, if one allows running, even as simple as Eq. (23),and especially when running is combined with gravita-tional waves. On the other hand, as was already explainedin [3], the attempt of constraining relic gravitational wavesby using the data from a huge interval of wavelengths andassuming a constant ns (or its simple running) across allwavelengths is unwarranted. The high-‘ CMB data, as wellas other data sets at relatively short wavelengths, havenothing to do with relic gravitational waves, and their useis dangerous. As we argued in Sec. III B, the spectral indexns appears to be sufficiently different even at the span oftwo adjacent intervals of wave numbers. The restriction toa relatively small number of multipoles 2 � ‘ � 100 isaccompanied by relatively large uncertainties in R, butthere is nothing we can do about it to improve the situation;this is in the nature of efforts aimed at measuring R. Thedifference in the treatment of ns is probably the mainreason why we do see indications of relic gravitationalwaves in the data, whereas the WMAP team does not.

Another contributing factor to the difference of conclu-sions is the continuing use by the WMAP Collaboration ofthe inflationary theory and its (incorrect) relation nt ¼�r=8, which automatically sends r to zero when nt ap-proaches zero. This formula is a part of the inflationaryconsistency relations

r ¼ 16 ¼ �8nt:

Only one equality in this formula, 16 ¼ �8nt, is correctbeing an approximate version (for small , � ) of ourexact formula, Eq. (21). The consistency relation r ¼ 16is incorrect. It is an immediate consequence of the so-called classic result of inflationary theory, namely, the

prediction of arbitrarily large amplitudes of density pertur-bations generated in the limit of de Sitter inflation ( ¼ 0,ns ¼ 1), regardless of the strength of the generating gravi-tational field (curvature of space-time) regulated by theHubble parameter H [24]. Certainly, it would be incon-sistent, even by the standards of inflationary theory, not touse the relation r ¼ �8nt in data analysis, if the infla-tionary ‘‘classic’’ is used in derivation of power spectra andinterpretation of results (see, for example, Fig. 19 in [1]).Obviously, in our analysis we do not use the inflationarytheory and its relations.

V. FORECASTS FOR THE PLANCK MISSIONBASED ON THE RESULTS OF ANALYSIS OF

WMAP7 DATA

The Planck satellite [4] is currently making CMB mea-surements and is expected to provide data of better qualitythan WMAP. We hope that the indications of relic gravi-tational waves that we found in WMAP3, WMAP5, andWMAP7 data will become a certainty after Planck obser-vations. We quantify the detection ability of the Planckexperiment by exploring the vicinity of the WMAP7 maxi-mum likelihood parameters (12).It is seen from Fig. 2 that the samples with relatively

large values of the likelihood (red, yellow, and green) areconcentrated along the curve which projects into relativelystraight lines (at least, up to R � 0:5) in the planes R� nsand R� As:

ns ¼ 0:98þ 0:49R; As ¼ ð2:30� 1:77RÞ 10�9:

(24)

The ML model (12) is a specific point on these lines,corresponding to R ¼ 0:264. The parameterization (24)is close to the result derived from WMAP5 data: ns ¼0:98þ 0:46R, As ¼ ð2:27� 1:53RÞ 10�9, and ML R ¼0:229 [see Eq. (15) in [3]]. We use Eq. (24) in formulatingour forecast, thus reducing the task of forecasting to a 1-parameter problem in terms of R.Following [3], we define the signal-to-noise ratio as

S=N � R=�R; (25)

where the numerator is the true value of the parameter R (orits ML value, or the input value in a numerical simulation)while �R in the denominator is the uncertainty in deter-mination of R from the data.We estimate the uncertainty �R using the Fisher matrix

formalism. We take into account all available informationchannels, i.e. TT, TE, EE, and BB correlation functions,and their various combinations. The uncertainty �R de-pends on instrumental and environmental noises, on thestatistical uncertainty of the CMB signal itself, and onwhether other parameters, in addition to R, are derivedfrom the same data set. All our input assumptions aboutPlanck’s instrumental noises, number and specification offrequency channels, foreground models and residual con-

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tamination, sky coverage, and lifetime of the mission, etc.,are exactly the same as in our previous paper [3]. We do notrepeat the details here and refer the reader to the text andappendices in [3] which contain necessary references.Technically, our present forecast is somewhat different(better) than that in the previous analysis [3] because of aslightly different family of preferred perturbation parame-ters (24), with slightly higher than before the ML value ofR, R ¼ 0:264. We have added only one new calculation,reported in Figs. 11 and 12, which is the S=N for the surveyof 28 months duration, instead of the nominal assumptionof 14 months.

The results of our forecast for the Planck mission arepresented in figures. In Fig. 7 we show the total S=N for thecase TT þ TEþ EEþ BB, i.e. when all correlation func-tions are taken into account, and at all relevant multipoles2 � ‘ � 100. The possible levels of foreground cleaningare marked by �fg. The pessimistic case is the case of noforeground removal, �fg ¼ 1, and the nominal instrumen-tal noise of the BB information channel at each frequencyincreased by a factor of 4. Three frequency channels at 100,143, and 217 GHz are considered as providing data onperturbation parameters R, ns, and As. The more severe,Dust A, model is adopted for evaluation of residual fore-ground contamination.

In the left panel of Fig. 7 we consider the idealizedsituation, where only one parameter R is unknown andtherefore the uncertainty �R is calculated from the FRR

element of the Fisher matrix, �R ¼ 1=ffiffiffiffiffiffiffiffiffiFRR

p. In the right

panel of Fig. 7 we consider a more realistic situation,where all perturbation parameters R, ns, and As are un-known and therefore the uncertainty �R increases and is

calculated from the element of the inverse matrix, �R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðF�1ÞRRp

.From the right panel of Fig. 7 follows our main con-

clusion: the relic gravitational waves of the maximumlikelihood model (12) will be detected by Planck at thehealthy level S=N ¼ 4:04, even in the pessimistic case.This is an anticipated improvement in comparison with ourevaluation S=N ¼ 3:65 based on WMAP5 data analysis[3]. The detection will be more confident, at the levelS=N ¼ 7:62; 6:91, if �fg ¼ 0:01; 0:1 can be achieved.Even in the pessimistic case, the signal-to-noise ratioremains at the level S=N > 2 for R> 0:11.Further insight in the detection ability of Planck and

interpretation of future results is gained by breaking up thetotal S=N into contributions from different informationchannels and individual multipoles. It is easier to do thisfor the idealized situation, �R ¼ 1=

ffiffiffiffiffiffiffiffiffiFRR

p, exhibited in the

left panel of Fig. 7. In Fig. 8 we show how the TT þ TEþEE and BB contribute to the total S=N based on allcorrelation functions TT þ TEþ EEþ BB. [The ðS=NÞ2for the full combination TT þ TEþ EEþ BB is the sumof ðS=NÞ2 for TT þ TEþ EE and BB.] It is seen from theupper left panel of Fig. 8 that the Bmode of polarization isa dominant contributor to the total S=N only in the con-ditions of very deep cleaning, �fg ¼ 0:01, and relativelysmall values of the parameter R. On the other hand, in thepessimistic case, the BB channel provides only S=N ¼2:02 for the benchmark case R ¼ 0:264. Most of the totalS=N ¼ 7:32 in this case comes from the TT þ TEþ EEcombination.In Fig. 9 we illustrate the decomposition of the total S=N

into contributions from individual multipoles ‘:

FIG. 7 (color online). The total signal-to-noise ratio S=N as a function of R. The left panel shows S=N with �R ¼ 1=ffiffiffiffiffiffiffiffiffiFRR

p, while

the right panel shows S=N with �R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðF�1ÞRRp

.

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FIG. 8 (color online). The decomposition of the total S=N into contributions from different information channels. Four panelsdescribe different assumptions about foreground cleaning and noises. The S=N line from the upper left panel for the optimistic case�fg ¼ 0:01 is copied as a broken line in other panels.

FIG. 9 (color online). The individual terms ðS=NÞ2‘ as functions of ‘ for various combinations of information channels and twoopposite assumptions about residual foreground contamination and noises. The calculations are done for the ML model (12) withR ¼ 0:264.

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ðS=NÞ2 ¼ X‘

ðS=NÞ2‘: (26)

We show the contributing terms ðS=NÞ2‘ for three combi-

nations of information channels, TT þ TEþ EEþ BB,TT þ TEþ EE, and BB alone, and for two opposite ex-treme assumptions about foreground cleaning and noises,namely, �fg ¼ 0:01 and the pessimistic case. The calcula-tions are done for the benchmark model (12) with R ¼0:264. Surely, the total S=N exhibited in the upper left andlower right panels of Fig. 8 is recovered with the help ofEq. (26) from the sum of the terms ðS=NÞ2‘ shown in the leftand right panels of Fig. 9, respectively.

It is seen from the left panel of Fig. 9 that in the case ofdeep cleaning the BB channel is particularly sensitive tothe very low multipoles ‘ ’ 10 associated with the reioni-zation era. At the same time, the right panel of Fig. 9demonstrates that in the pessimistic case most of S=Ncomes from TT þ TEþ EE combination and, specifically,from the interval of multipoles ‘� ð20–60Þ associatedwith the recombination era.Finally, we want to discuss possible improvements in

our forecasts. They will be achievable, if 7 frequencychannels, instead of 3, can be used for data analysis, and/or if the less restrictive Dust B model, instead of the Dust Amodel, turns out to be correct, and/or if 28 months of

FIG. 11 (color online). The improved S=N for 28 months of observations. Other assumptions are the same as in Fig. 7.

FIG. 10 (color online). The improved signal-to-noise ratio S=N for the scenario where the Dust B model is correct and theinstrumental noises are smaller because of the 7 frequency channels used, instead of 3, in data analysis. The left panel shows S=N with

�R ¼ 1=ffiffiffiffiffiffiffiffiffiFRR

p, while the right panel shows S=N with �R ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðF�1ÞRR

p.

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observations, instead of 14 months, can be reached. InFig. 10 we show the results for the scenario where 7frequency channels are used and the Dust B model iscorrect. Similarly to Fig. 7, the left panel shows S=Ncalculated assuming that only R is being determinedfrom the data, whereas the right panel shows a morerealistic case in which all perturbation parameters areunknown. In the later case, the S=N for the ML model(12) with R ¼ 0:264 increases up to the level S=N ¼ 5:56as compared with S=N ¼ 4:04 that we found in the rightpanel of Fig. 7 for 3 frequency channels and the Dust Amodel.

The two panels in Fig. 11 illustrate the improvements, ascompared with the two panels in Fig. 7, arising from thelonger, 28-month, survey instead of the nominal 14-monthsurvey. The S=N for the benchmark model R ¼ 0:264increases up to S=N ¼ 5:39 even in the pessimistic case,right panel. The decomposition of S=N appearing in theleft panel of Fig. 11 into contributions from differentinformation channels is presented in Fig. 12. One can seethat the relative role of the BB channel has increased. (Theprospects of B-mode detection in the conditions of lowinstrumental and foreground noises have been analyzed in[25].)

VI. THEORETICAL FRAMEWORKS THAT WILLBE ACCEPTED OR REJECTED BY PLANCK

OBSERVATIONS

Our forecast for the Planck mission, as any forecast innature, can prove its value only after actual observation.We predict sunny days of confident detection of relicgravitational waves, but the reality can turn out to begloomy days of continuing uncertainty. One can illustratethis point with the help of probability distributions in

Fig. 4. We expect that Planck observations will continuethe trend of exhibiting narrower likelihoods with maxi-mum in the area near the WMAP7 ML value R ¼ 0:264.Then the detailed analysis in Sec. V explains the reasonsfor our optimism. But, in principle, the reality can happento be totally different. From the position of pure logic it isstill possible that the Planck data, although making the newlikelihood curve much narrower, will also shift the maxi-mum of this curve to the point R ¼ 0. In this case, insteadof confident detection we will have to speak about ‘‘tightupper limits.’’ We do not think, though, that this is going tohappen. What kind of conclusions about theoretical modelscan we make if the likelihood curve comes out as weanticipated?As was already mentioned in Sec. II, the theory of

quantum-mechanical generation of cosmological perturba-tions implies a reasonable guess for the true value of R:R 2 ½0:01; 1�. We shall call it a model M1. At the sametime, the most advanced string-inspired inflationary theo-ries predict R somewhere in the interval R 2 ½0; 10�24�.We shall call it a model M2. The inflationary calculationscan be perfectly alright in their stringy part, but the obser-vational predictions are entirely hanging on the inflationaryclassic result and therefore should fall together with it, onpurely theoretical grounds. But this is not the point of ourpresent discussion. We wish to conduct a Bayesian com-parison of models M1 and M2, regardless of the motiva-tions that stayed behind these models.The model M1 suggests that the quadrupole ratio R

should lie in the range ½0:01; 1� with a uniform prior inthis range: PpriorðR j M1Þ ¼ c1 for R 2 ½0:01; 1�. The

model M2 suggests that R should lie in the range½0; 10�24� with a uniform prior in this range: PpriorðR jM2Þ ¼ c2 for R 2 ½0; 10�24�. The constants c1 and c2 aredetermined from normalization of the prior distributions,c1 ¼ ð0:99Þ�1 and c2 ¼ 1024. The predicted interval ofpossible values of R is much wider for M1 than for M2.Therefore, M1 is penalized by a much smaller normaliza-tion constant c1 than c2. The observed data allow one tocompare the two models quantitatively with the help of theBayes factor K12 [26]:

K12 �R10 PpriorðR j M1ÞLðRÞdRR10 PpriorðR j M2ÞLðRÞdR ; (27)

where L is the likelihood function of R derived from theobservation.We shall start from the existing WMAP7 data. The

likelihood function as a function of R is shown by a redsolid line in Fig. 4. Calculating the Bayes factor accordingto Eq. (27) we find K12 ¼ 1:61, i.e. lnK12 ¼ 0:48. Asexpected, this value of the Bayes factor, although indica-tive, does not provide sufficient reason for the rejection ofmodel M2 in favor of M1.

FIG. 12 (color online). Decomposition of the total S=N figur-ing in the left panel of Fig. 11 into contributions from differentinformation channels. In the assumed conditions of deep clean-ing �fg ¼ 0:01, the BB alone is more informative than thecombination TT þ TEþ EE.

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With more accurate Planck observations the situationwill change dramatically, if the WMAP7 maximum like-lihood set of parameters (12) is correct. To make calcula-tion in Eq. (27), we adopt a Gaussian shape of thelikelihood function L with maximum at R ¼ 0:264 andR � 0. The standard deviation is taken as �R ¼ 0:065.This value follows from the analysis of S=N for the Planckmission and corresponds to the derived S=N ¼ 4:04 in thepessimistic case. Then the calculation of K12 gives thevalue K12 ¼ 579:23, i.e. lnK12 ¼ 6:36. In accordancewith Jeffreys’ interpretation [26], this result shows thatthe Planck observations will decisively reject the modelM2 in favor of M1.

If Planck observations are as accurate as expected, and ifour assumptions about Planck’s likelihood function arecorrect, the models much less extreme than M2 will alsobe decisively rejected. We introduce the model M3 with aflat prior in the range R 2 ½0; x� and ask the question forwhich x the Bayes factor K13 exceeds the critical valueK13 ¼ 100 ( lnK13 ¼ 4:61) [26] which is required for de-cisive exclusion of the model. The calculation gives x ¼0:05. This means that under the conditions listed above thePlanck experiment will be able to decisively reject anytheoretical framework that predicts R< 0:05. In terms ofthe parameter r this means the exclusion of all models withr < 0:095.

VII. CONCLUSIONS

The analysis of the WMAP7 data release amplifies ob-servational indications in favor of relic gravitational wavesin the Universe. The WMAP3, WMAP5, and WMAP7temperature and polarization data in the interval of multi-poles 2 � ‘ � 100 persistently point out to one and thesame area in the space of perturbation parameters. It in-cludes a considerable amount of gravitational waves ex-pressed in terms of the parameter R ¼ 0:264, andsomewhat blue primordial spectra with indices ns ¼1:111 and nt ¼ 0:111. If the maximum likelihood set ofparameters that we derived from this analysis is a fairrepresentation of the reality, the relic gravitational waveswill be detected more confidently by Planck observations.Even under pessimistic assumption about hindering fac-tors, the expected signal-to-noise ratio should be at thelevel S=N ¼ 4:04.

ACKNOWLEDGMENTS

We acknowledge the use of the LAMBDA and CAMBWeb sites. W. Z. is partially supported by Chinese NSFGrants No. 10703005 and No. 10775119 and theFoundation for University Excellent Young Teacher bythe Ministry of Zhejiang Education.

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