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The Inflationary Gravity Waves in light of recent Cosmic Microwave Backgroundprovided by CERN Document Server Anisotropies data.
Alessandro Melchiorri[ and Carolina J. Odman¨ ] [ Astrophysics, Denys Wilkinson Building, University of Oxford, Keble road, OX1 3RH, Oxford, UK ] Astrophysics Group, Cavendish Laboratory, Cambridge University, Cambridge, U.K.
One of the major predictions of inflation is the existence of a stochastic background of cosmo- logical gravitational waves (GW). These gravitational waves can induce significant temperature anisotropies in the Cosmic Microwave Background (CMB) on the angular scales recently probed by the ARCHEOPS experiment. Here, we perform a combined analysis of ARCHEOPS to- gether with information from other CMB experiments and/or cosmological datasets, in order to constrain the amplitude of the GW background. We find that, for a scale-invariant GW back- ground, the ratio of tensor/scalar perturbations at the CMB quadrupole is now constrained to be r 0.43 at 95% c.l., while the bound on the spectral index of primordial density fluctua- ≤ +0.10 tions is nS =0.97 0.12 . We discuss the implications for future GW detections through CMB polarization measurements.−
I. INTRODUCTION bination, however, unlike the anisotropies generated by scalar fluctuations, those generated by GW damp The last years have seen spectacular advances in our like fluctuations in a fluid of massless bosons (see e.g. ability to confront the inflationary scenario of struc- [11]). Since the theoretical spectrum, normalized to ture formation to observational data. The “multi- COBE, is a linear sum of the scalar and tensor compo- ple peaks” observed in the Cosmic Microwave Back- nents, if there is a relevant contribution from GW this ground (CMB) angular power spectrum ( [27], [17], would lower the predicted amplitude of the acoustic [22], [31], [34]) are indeed providing strong sup- peaks on sub-degree angular scales. porting evidence for the inflationary predictions of With the advent of the new CMB peaks detections, a flat universe and of a primordial background of many authors have therefore addressed the question scale-invariant adiabatic perturbations (see e.g. [38], of the GW’s contribution (see e.g. [25], [20], [38], [12], [28]). More recently, the new CMB results from the [23], [39]). However, despite the different scale de- ARCHEOPS experiment ( [1]) have confirmed and re- pendence, robust constraints on tensor modes remain fined the present observational status, sampling angu- difficult to obtain. The decrease in the amplitude of lar scales between those probed by the COBE satellite the acoustic oscillations induced by GW can indeed and the latest high precision datasets. Again, flat- be compensated by an increase in one of the uncon- ness, adiabaticity and scale invariance are in agree- strained parameters of the model, like, for example, ment with the data ( [2]). the spectral index of scalar fluctuations nS.There- It has been argued that the next and probably fore, some form of ’cosmic degeneracy’ arises in the most conclusive evidence for inflation would be the de- tradeoff between these two (and more) parameters tection of a stochastic background of Gravity Waves (see [25], [12]) and only weak constraints on the GW (GW) (see e.g. [7], [43]). Two types of spacetime met- background were obtained. ric fluctuations are indeed naturally produced during In this context, and before more accurate polariza- inflation: density perturbations (scalar modes), which tion data become available (see discussion below), the form the “seeds” of structure formation, and gravity new results on intermediate angular scales, as recently waves (tensor modes)([16]). provided by ARCHEOPS, can offer an interesting op- The GW background, if detected, would also pro- portunity. vide valuable information on the inflationary scenario. 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 tainly in the simplest ones), the amplitude of the GW figure, we plot two theoretical power spectra. The background is proportional to the square of the energy models have identical power on sub-degree scales and scale of inflation (see e.g. [8]). Furthermore, a comple- on COBE scales (considering cosmic variance), but mentary measurement of the ’tilt’ of the GW pertur- different tensor contributions, parametrized by a ten- bations (and of the scalar as well) can give direct in- sor over scalar ratio of the angular power spectrum T S formation up to the second derivatives of the inflaton quadrupole r = C2 /C2 (see e.g. [20]). potential, sheding light on the physics at 1016GeV As we can see, while the two models are degenerate ∼ on scales ` 200, the degeneracy is broken on larger (see e.g. [18]). ≥ The GW background leaves an imprint on the CMB angular scales (see the bottom panel), mostly in the anisotropies at large scales through the Sachs-Wolfe region sampled by ARCHEOPS. Both increasing nS effect. On scales smaller than the horizon at recom- and adding tensors change the rate of growth of the scalar modes from the Sachs-Wolfe plateau towards
1 0.003, ΩΛ =0.5, ..., 0.95, in steps of 0.05. Our choice 8000 r = 0 ns=0.94 of the above parameters is motivated by the Big Bang r = 0.4 n =0.97 7000 s Nucleosynthesis bounds on ωb (both from D [6] and 4 7 6000 He + Li [9]), from supernovae ( [14]) and galaxy ]
2 clustering observations (see e.g. [37]).
K 5000 µ [
Variations in the tensor and scalar spectral indices, π
2 4000 / l nS and nT are not computationally relevant. How- C ) 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 nT ). 1000 Furthermore, the value of the Hubble constant 0 10 100 1000 is not an independent parameter, since h = Multipole l p(ω + ω )/(1 Ω ). We also include the further 0,35 cdm b − Λ 0,30 top-hat prior h =0.7 0.2 ( [13]) and we consider e
c 0,25 ±
n only models with age t0 > 11 Gyrs. e 0,20 r e 0,15
ff We allow for a reionization of the intergalactic i 0,10 D medium by varying the compton optical depth pa-
% 0,05 0,00 rameter τ in the range τ =0.0, ..., 0.45 in steps of 10 100 1000 c c 0.05. We note here that high values of τ are in severe Multipole l c disagreement with recent estimates of the redshift of FIG. 1. Best-fit models to recent CMB data with and reionization z 6 1 (see e.g. [15]) which points re ∼ ± without GW contribution (Top Panel). The Archeops data towards τc 0.05 0.10. On the other hand, if the points are shown as open circles. In the Bottom panel we reported CBI∼ excess− at ` 3000 is due to Sunyaev- ∼ plot the % difference between the two degenerate mod- Zeldovich effect, then this would favour values τc 0.3 els together with the cosmic variance limit (dashed line) ([3]). ∼ averaged in bins of ∆` = 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 the first peak and this can in principle be used to from these experiments were estimated in 19, 9, 13, constrain the GW background. 14, 10 and 16 bins respectively (for the CBI, we use It is therefore extremely timely to analyze the the data from the MOSAIC configuration, [10]), span- ARCHEOPS data allowing the possibility of a GW ning the range 2 ` 1500. We also use the COBE contribution in order to see if the amplitude of this data from the RADPACK≤ ≤ compilation ( [33]). background can now be better constrained than in For the CBI, DASI, MAXIMA-I and VSA experi- the past. ments we use the publicly available correlation matri- Furthermore, the GW background produces a ces and window functions. For the ARCHEOPS and unique statistical signature in the polarization of the BOOMERanG experiments we assign a flat interpola- CMB by inducing a curl component ( [32], [19]), of- tion for the spectrum in each bin `(`+1)C /2π = C , tendefinedasB mode, while scalar (but also ten- ` B and we approximate the signal CB inside the bin to sor) perturbations produces a gradient component (E be a Gaussian variable. The likelihood for a given mode). Given the large number of future and ongo- th theoretical model is defined by 2ln =(CB ing CMB polarization experiments, it is interesting to Cex)M (Cth Cex)whereM − isL the Gaussian− B BB0 B0 B0 BB0 forecast from the present CMB temperature data the 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% Gaus- modes produced by tensors can be distinguished from sian distributed calibration errors (in ∆T )forthe those produced by scalar perturbations only. 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, in section IV, we discuss our findings. Finally, we parametrize the GW contribution by the T S tensor over scalar quadrupole ratio r = C2 /C2 and we rescale the sum spectrum by a prefactor C10,as- II. ANALYSIS: METHOD COBE sumed to be a free parameter, in units of C10 . As a first step, we consider a template of flat, adi- abatic, Λ-CDM scalar and tensor spectra computed III. ANALYSIS: RESULTS with CMBFAST ( [35]), sampling the various param- 2 eters as follows: Ωcdmh ωcdm =0.05, ...0.25, in The main results of our analysis are plotted in Fig.2. steps of 0.02; Ω h2 ω =0≡.009, ..., 0.024, in steps of b ≡ b In the left top panel we plot the likelihood contours
2 10000 TE-Scalar 95% Max 99 % 2 1000
K] TE-Scalar 95% Min µ [ 100
95 % π
/2 10
| c T l 99 % E 1 C ARCHEOPS | ) 95 % 1 0,1 w/o + ARCHEOPS l ( TE-Tensor 95% C.L. l 0,01 w/o 68 % 10 100 1000 68 % Multipole l 0,1 2
S S K] µ g. lensing [
π 0,01 /2 l B C ) 1 + l
( 1E-3 l B-Modes Tensor 95% C.L.
c 10 100 1000 w/o Multi pole l ARCHEOPS
2 100 E-Scalar 95% Max
95 % K]
µ E-Scalar 95% Min 99 % [ 10 68 % π
/2 1 l E C )
1 0,1 + l ( FIG. 2. 68%, 95% and 99% confidence regions in the l 0,01 E-Tensor 95% C.L. n r (Top Panel, Left), n τ (Top Panel, Right), r τ 10 100 1000 S S Multipole l (Bottom− Panel) planes for the− models considered in− our analysis (see text). The line contours are confidence levels FIG. 3. Maximum and minimum levels of tempera- without the ARCHEOPS data. ture-polarization cross correlation (Top Panel), B-modes (Central Panel), E-modes (Bottom Panel) allowed at 95% C.L. from present CMB temperature data under the as- in the nS r plane, maximizing over the remaining sumption of the models described in the text. nuisance parameters.− As we can see, in the frame- work of models we considered, the gravitational wave contribution is constrained to be r 0.2(r 0.43) As we can see from the center panel of Figure 3, ≤ +0.06≤ at 68% C.L. (95% C.L.), with nS =0.97 0.07 (68% the level of the B-modes, is expected to be of 0.2 − C.L.). While the inclusion of the ARCHEOPS data µK, at maximum. The signal is out of the reach∼ of has little effect on nS, it drastically improves the con- most of the current polarization experiment like DASI straint on r. Removing the ARCHEOPS data yields or POLATRON which are sensitive to few µK.Near r 0.6 at 95% C.L.. future experiments like B2K or QUEST, will probably ≤ 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 nS τc plane. As we can see, detection. However, the B-signal in the angular re- − the present CMB constraint on τc is rather weak, with gion sampled by these experiments (`>50), can be τc < 0.25 (τc < 0.36) at 68% C.L. (95% C.L.). It is in- contaminated by a foreground component due to the teresting to note that the inclusion of the ARCHEOPS conversion of E modes to B modes from gravitational datapoints has little effect. lensing (see Fig. 3) ( [40]). 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 τc plane. An in- tract this contribution to the B mode ( [29]). − crease in τc or r produces a similar damping on the Tensor perturbations produce E modes as well. small/intermediate angular scales. It is interesting to However, the amplitude of the E tensor modes is 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 the scalar modes (see bottom panel). A window of op- τc 0.3 are in disagreement with the presence of a portunity may appear in the temperature-polarization ∼ tensor component. If τc > 0.2thenr<0.05 at 68% (
3 sor spectral index is not constrained by the present Ekpyrotic ( [36]) or Pre-Big Bang (see e.g. [26]) sce- data, but that a value of nT = 0 is preferred. narios. However, extremely blue spectra (nT 2) are excluded by constraints on the GW energy∼ den- sity background from timing milli-second binary pul- IV. CONCLUSIONS sars [30]. Allowing for extra primordial perturbation modes like isocurvature, will probably tight our con- In this Rapid Communication we have presented straints on GW, since the shape of CDM scalar isocur- new constraints on the stochastic background of gravi- vature modes is similar to those from adiabatic tensor tational waves from recent microwave anisotropy data. 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. [39], [20]). ing curvature (Ωtot = 1) would relax our bounds on r 6 In the framework of models we considered, we found (see e.g. [38]). Non-flat models in agreement with the +0.10 CMB data are in general closed models, which, a part (at 95% C.L.) r<0.43 and nS =0.97 0.12.The − energy scale of inflation Einf can be related to tensor from a few exceptions ( [24]), are difficult to obtain 4 T 4 from inflation. by Einf =0.65C2 mPl. The above bound translates 16 Even if the results presented here do not hint for a therefore in Einf 1.6 10 GeV . A part from the≤ different× template of theoretical presence of GW background, the data is still consis- models considered, our analysis differs mainly in the tent with a sizable tensor contribution. It will there- following points: we assumed the low-` ARCHEOPS fore be the duty of future and ongoing experiments to bins as gaussian distributed, we included the COBE scrutinize this fundamental prediction from inflation. data using the RADPACK compilation, we have a Acknowledgements We wish to thank Will Kinney, Rocky Kolb, Mike strong upper limit on ωb < 0.025 from BBN and, finally, we numerically computed the models with Hobson, Robert Izzard, Anthony Lasenby, and Anto- nio Riotto. AM is supported by PPARC. CJO is sup- τc > 0 (while in [2] an analytical formula was used). The GW background induces a unique signature in ported by a Girton College Scholarship and an Isaac the polarization of the CMB by producing a curl com- Newton Studentship. ponent, not present in the case of scalar perturbations. In the set of models we considered (and under the as- 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, which can be partially attenaible by near future∼ exper- [1] A. Benoit et al. [Acheops Collaboration], A & A, sub- iments and severly contaminated by lensing E B mitted, 2002 [arXiv:astro-ph/0210305] conversion. → [2] A. Benoit et al. [Acheops Collaboration], A & A, sub- The E modes expected from gravity waves are lower mitted, 2002 [arXiv:astro-ph/0210306] than the E modes expected from scalar perturbations. [3]J.R.Bondet al., arXiv:astro-ph/0205386. However, the tensor
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