New constraints on primordial gravitational waves from Planck 2015

Physics Letters B 760 (2016) 823–825

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

New

constraints

on

primordial

gravitational

waves

from

Planck

2015

Luca Pagano

a

,

b

,

∗

,

Laura Salvati

b

,

Alessandro Melchiorri

b

a_{Institutd’AstrophysiqueSpatiale,CNRS,Univ.Paris-Sud,UniversitéParis-Saclay,Bât.121,91405Orsaycedex,France} b_{PhysicsDepartmentandINFN,UniversitàdiRoma“LaSapienza”,Ple.AldoMoro2,00185,Rome,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received7September2015 Receivedinrevisedform31May2016 Accepted29July2016

Availableonline3August2016 Editor:S.Dodelson

WeshowthatthenewprecisemeasurementsofCosmicMicrowaveBackground(CMB)temperatureand polarizationanisotropiesmadebythePlancksatellitesignificantlyimprovespreviousconstraintsonthe cosmicgravitationalwavesbackground(CGWB)atfrequencies f>10−15_{Hz.Onscalessmallerthanthe}

horizonatthetimeofdecoupling,primordialgravitationalwavescontributetothetotalradiationcontent ofthe Universe.Considering adiabatic perturbations,CGWB affects temperatureand polarization CMB powerspectraandmatterpowerspectruminamanneridenticaltorelativisticparticles.Consideringthe latestPlanckresultsweconstraintheCGWBenergydensityto



gwh2<1.7×10−6at95%CL.Combining

CMB power spectra with lensing, BAO and primordial Deuterium abundance observations,we obtain gwh2<1.2×10−6 at95% CL,improvingpreviousPlanckbounds byafactor3 andthe recentdirect

upper limitfromthe LIGOand VIRGO experimentsafactor2.Acombined analysis offuture satellite missionsasCOrEandEUCLIDcouldimprovecurrentboundbymorethananorderofmagnitude.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Different processes inthe early Universe mayhave generated aprimordial gravitationalwave background,such as, among oth-ers, quantum perturbations during inflation, cosmic strings, pre-big-bang theories, etc. (for a complete review see [1] and ref-erences therein). Detecting this cosmological gravitational wave background(hereafterCGWB)providesauniquewaytoprobethe primordialUniverseanditsevolution.

The CGWB can be measured at low frequencies constraining a possible tensor-mode contribution to the large-scale tempera-tureandpolarization fluctuationsinthe cosmic microwave back-ground radiation (hereafter CMB). The most recent constrain on thetensor-to-scalarratioistheonepublishedbytheBICEP2/Keck jointanalysis[2],i.e.r

<

0

.

07 at95% CL[3,4],whichcorresponds to



gwh2

<

10−14inthefrequencyrange10−17

−

10−16Hz.

At higher frequencies, in the range 10−9

₋

₁₀−8 _{Hz, pulsars}

canbe usedasnaturalgravitationalwave detectors,e.g.fromthe last European PulsarTiming Array (EPTA) datarelease, Lentatiet al.constrain the amplitude of GW up to



gwh2

<

1

.

1

×

10−9 at

2.8 nHz [5]. At smaller scales, interferometers such as LIGO [6]

andVIRGO[7]arealsolookingforgravitationalwavesignals.A

re-*

Correspondingauthor.

E-mailaddress:luca.pagano@roma1.infn.it(L. Pagano).

cent bound, at

∼

102Hz, is



gwh2

<

2

.

6

×

10−6 from the cross

correlation betweenLIGOandVIRGOdetectors[8].Mostrecently, duringthereviewprocessofthispaper,theLIGO-VIRGO collabora-tiondetectedaGravitationalWavessignalfromaBinaryBlackHole marger[9].The currentbestlimiton stochasticbackground com-ingfromBinaryBlackHolesis



gw

(

f

=

25Hz

)

=

1

.

1+₋20..79

×

10−9at

90% CL[10].

Moreover,atfrequenciesgreaterthan

∼

10−10_Hz,the

stochas-tic background can be constrained through big-bang nucleosyn-thesis(BBN).Infact,atthesefrequencies,primordialgravitational wavescontribute to the totalradiation energydensity,increasing theexpansionrateoftheUniverse.Inthisscenario,theCGWB be-havesasa free-streaminggasofmassless particles.Bymeasuring primordial abundances of light elements is possible to constrain the total number of relativistic degrees of freedom and, conse-quently,the gravitationalwavesenergydensity,forscales smaller thanthecomovinghorizonattheendoftheBBN[11].

Straightforwardly, it is possible to constrain the total radia-tion densitythrough theCMB, reaching evensmaller frequencies,

∼

10−15_{Hz, corresponding to the comoving horizon at the}

de-coupling.In particular,if theCGWB energydensityperturbations are adiabatic, the extra energy contribution due to gravitational waves is indistinguishable from the one due to relativistic neu-trinos. Therefore, if we fix the relativisticdegrees of freedom to its standardvalue, Neff

=

3

.

046,andparametrizeall theextra

ra-diation astheeffective numberofgravitationalwaves degreesof http://dx.doi.org/10.1016/j.physletb.2016.07.078

0370-2693/©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

824 L. Pagano et al. / Physics Letters B 760 (2016) 823–825

freedom,

N

gw,itistheneasytotranslate

N

gwintoaCGWBenergy

density,aspointedoutin[12].

InthispaperweupdatepreviousconstraintsontheCGWB en-ergy density, asthose presented in [13,14], inlight ofthe latest Planckdatareleaseandwepresentthebound reachable combin-ingthefuturesatellitemissionsCOrEandEuclid.

2. CosmologicalconstraintsonGWbackground

In this section we discuss the datasets used in the analysis and the obtained results. We make use of the latest CMB Tem-peratureandPolarizationpowerspectraofthePlancksurvey[15, 16]together withthe Plancklensinglikelihood[17],theBaryonic AcousticOscillationsobservationsof[18] andthemostrecent pri-mordial Deuterium abundance observation by Cooke et al. [19]. Regarding CMBpolarization dataatlarge angularscales, we con-sideredboththelowPlikelihoodbasedon70 GHzdata,described in[16],andthenewSimLowlikelihoodbasedon100and143 GHz HFIchannelsasdescribedin[20]byincludinganexternalprioron theopticaldepth

τ

=

0

.

055

±

0

.

009 at68% CL.

We explore the cosmological parameters space with the July 2015version ofthepubliclyavailable

cosmomc

package[21].We adopt the following parametrization for the



CDM model: the baryonandcolddarkmatterdensities

ω

b

≡ 

bh2 and

ω

c

≡ 

ch2,

theratioofthesoundhorizontotheangulardiameterdistanceat decoupling

θ

MC,there-ionizationopticaldepth

τ

,thescalar

spec-tral indexnS, and the overall normalizationof the spectrum AS

at

k

=

0

.

05Mpc−1.Furthermoreweassumeadiabaticinitial condi-tionsandweimposespatialflatness.Wefixtherelativisticdegrees of freedom, parametrized as Neff, to its standard value of 3

.

046

and, foreach MCMC sample, we compute theprimordial Helium abundanceassumingstandardBBN[22],usingarecentfitting for-mulabasedonresultsfromthe

PArthENoPE

BBNcode[23].

Asmentionedabove,weparametrizetheextrarelativistic con-tent due to the CGWB as Ngw, addingit to the total amount of

masslessneutrinos.Totranslate

N

gw totheCGWBenergydensity,

weassumethatthecontributionfromasinglemasslessparticleto amonochromaticCGWBis5

.

6

×

10−6_{,therefore(following[1,12]}

andwhatisdonein[24]):



gwh2

≡

h2 ∞



0

d

(ln f

)

gw

(

f

)

=

5.6

×

10−6Ngw (1)

Theeffectivenumberofgravitationalwavesdegreesoffreedom, actingasextraradiation,iscompletely degeneratewiththeextra relativistic degreesof freedom. Therefore,our boundsare model dependent, in fact, any process, that modifies the standard ra-diation content of the Universe, will affectthe Ngw bounds. For

example,thepresenceofextra relativisticparticlesat recombina-tion(e.g.sterileneutrinos[25])willtightentheboundsup.Onthe otherhand,areheatingphaseat MeV temperaturecanproducea valueof

N

efflowerthanthestandardone[26],relaxingthebounds

on

N

gw.

Totranslatethevalueof



gwh2 intoacorrespondingvalue of

thetensor-to-scalar ratior is not straightforwardsince theshape ofthetensorspectrum needstobespecified.Forrecentandvery comprensivediscussionsonthisspecifictopicsee[27,28].

We test different combinations of data, starting from Planck 2015CMB data alone (bothusing lowP and SimLowlikelihoods) andthen addingBAO observations,Lensing dataandthe primor-dial Deuterium abundance measurement. We show in Fig. 1 the 95% CLupperlimitsfortheCGWBenergydensity



gwh2.

InTable 1 wereport theobtainedupperlimitson



gwh2 and

Ngw for all the data combination that we consider. As expected

Fig. 1. 95%CLupperlimitsfortheCGWBenergydensity,gwh2andfortheeffective numberofgravitationalwavesdegreesoffreedom,Ngwforthedifferentanalyzed datasets.WithPlanck-SLwerefertothePlancktemperatureandpolarization Like-lihoodincombinationwiththenewvalueof

τ

publishedin[20].Wereporthere alsothevaluequotedbytheLIGO-VIRGOcollaboration[8],theprevious cosmolog-icalconstraintobtainedcombiningWMAP7[29]andSPT[30],andthePlanck2013 constraint[14].

Table 1

95%CLupperlimitsfortheCGWBenergydensitygwh2andtheeffectivenumber ofgravitationalwavesdegreesoffreedom(Ngw)for theconsideredcosmological datasets.WereportalsoboundsfromLIGO-VIRGOcollaboration,inthefrequency range41.5–169.25 Hz[8].WithlowPwerefertothelikelihoodpublishedbyPlanck in2015[16]whilewithSimLowwerefertothenewvalueofopticaldepth pub-lishedbyPlanckcollaboration[20].

Datasets gwh2 Ngw

PlanckTTTEEE−lowP <2.1×10−6

<0.37 PlanckTTTEEE−lowP+BAO <1.9×10−6 _<0.34 PlanckTTTEEE−lowP+BAO+Lensing <1.6×10−6 _<0.29 PlanckTTTEEE−lowP+BAO+Lensing+Deut. <1.2×10−6 _<0.22 PlanckTTTEEE−SimLow <1.7×10−6 _<0.31 PlanckTTTEEE−SimLow+BAO <1.7×10−6 _<0.30 PlanckTTTEEE−SimLow+BAO+Lensing <1.5×10−6

<0.27 PlanckTTTEEE−SimLow+BAO+Lensing+Deut. <1.2×10−6

<0.22

LIGO-VIRGO[8] <2.6×10−6

COrE <0.50×10−6 _<0.089

COrE+Euclid <0.076×10−6 _<0.013

using the new value of

τ

from Planck we reach more stringent constrains. Ontheotherhanditisworthtonoticethat,including theDeuteriumabundancemeasurements,thisdifferenceis practi-callynegligible.

ThePlanck-SimLowaloneconstraintis50%betterthanthe up-per bounds estimated by the LIGO-VIRGO collaboration in 2014

[8]. Combining CMB power spectra with the Lensing likelihood andthe BAO datawe obtain aslightly morestringentconstraint,



gwh2

<

1

.

5

×

10−6; finally,assuming standard BBN [22,23], we

addalsotheprimordialDeuteriumabundancemeasurement reach-ing



gwh2

<

1

.

2

×

10−6at95%CL,improvingtheprecedentPlanck

constraint by a factorofabout 3 [14], thepre-Planck cosmologi-calconstraint[13](basedonWMAP[29]andSouthPoleTelescope

[30] results)bya factorof6 andthecurrentinterferometer mea-surementsbyabout2.

We also verified the stability of our results with respect to assumptions on massive neutrinos. By opening



mν as extra parameter, we found, as expected, an overall relaxation on Ngw

L. Pagano et al. / Physics Letters B 760 (2016) 823–825 825

bounds.Nevertheless forall thedata combinationswe have con-sideredabove,theupperlimitsincreaselessthan10%,not chang-ingourconclusions.

Itisalsointerestingtoforecastthefuturesensitivityon



gwh2

achievablewithfuturesatellitemissionssuchasCOrE [31,32]and Euclid [33]. To this end we simulate mock data for the COrE missionusing the5 channelsin therange 100–220 GHz, follow-ing the approach described in [34], assuming perfect foreground removaland ignoring correlationsbetween multipoles.Analyzing thisdataset we find that the COrE mission will be able to con-straintheCGWBenergydensityto



gwh2

<

5

.

0

×

10−7 at95%CL.

FortheEuclid mission we usethe fishermatrix approach as de-scribedin[35,36];wethencombinetheinversecovariancematrix producedby

cosmomc

forCOrEwiththeEuclidfishermatrix ob-taining



gwh2

<

7

.

6

×

10−8 at95%CL.

3. Conclusions

We have used the latest Planck data to constrain a possible cosmologicalgravitationalwave backgroundatfrequenciesgreater than 10−15_{Hz. Our tighter constraint is}

_

gwh2

<

1

.

2

×

10−6 at

95% CL,obtainedcombiningCMBwithBAO, Lensingand primor-dial Deuterium observations. This result improves previous cos-mologicalboundsby afactor3 andtherecentLIGO-VIRGOdirect measurementsby 2.

Wealsoshowthatwiththenextgenerationcosmological satel-lite missions (COrE and Euclid) would be possible to shrink the bounds by more than one order of magnitude with respect to current limits. The constraints presented here are probably not significant for slow roll inflation that produces essentially scale invariant spectra. Those models are already strongly constrained by currentlarge scalebounds onprimordial CMBpolarization B-modes.However,phasetransitions,pseudoscalarinflationorother exoticmechanismsthatproduceaCGWBathigherfrequencies(see forexample[37] and[38]) canbeconstrainedbythebounds pre-sentedhere.

Acknowledgements

We are grateful to G. Cabass and M. Lattanzi for useful dis-cussions andsuggestions.We acknowledge theuse ofcomputing facilities at NERSC (USA). We acknowledge partial financial sup-portbytheresearchgrantTheoreticalAstroparticlePhysicsnumber 2012CPPYP7undertheprogramPRIN2012fundedbyMIURandby TASP,iniziativaspecificaINFN.

References

[1]M. Maggiore, Stochastic backgrounds of gravitational waves, arXiv:gr-qc/ 0008027.

[2]P.A.R.Ade,et al.,BICEP2andKeckArrayCollaborations,Phys.Rev.Lett.116 (2016)031302.

[3]P.A.R.Ade,etal.,PlanckCollaboration,Planck2015results.XIII.Cosmological parameters,arXiv:1502.01589[astro-ph.CO].

[4]P.A.R.Ade,etal.,PlanckCollaboration,Planck2015results.XX.Constraintson inflation,arXiv:1502.02114[astro-ph.CO].

[5]L.Lentati,etal.,Europeanpulsartimingarraylimitsonanisotropicstochastic gravitational-wavebackground,arXiv:1504.03692[astro-ph.CO].

[6]B.P. Abbott,et al., LIGO Scientific Collaboration,Rep. Prog.Phys. 72 (2009) 076901,arXiv:0711.3041[gr-qc].

[7]T.Accadia,etal.,VIRGOCollaboration,Int.J.Mod.Phys.D20(2011)2075. [8]J.Aasi,etal.,LIGOScientificandVIRGOCollaborations,Phys.Rev.Lett.113 (23)

(2014)231101.

[9]B.P. Abbott,et al., LIGO Scientific and Virgo Collaborations, Phys. Rev.Lett. 116 (6)(2016)061102.

[10]B.P. Abbott,et al., LIGO Scientific and Virgo Collaborations, Phys. Rev.Lett. 116 (13)(2016)131102.

[11]B. Allen, The stochastic gravity wave background: sources and detection, in: Les Houches 1995, Relativistic Gravitation and Gravitational Radiation, pp. 373–417,arXiv:gr-qc/9604033.

[12]T.L.Smith,E.Pierpaoli,M.Kamionkowski,Phys.Rev.Lett.97(2006)021301. [13]I.Sendra,T.L.Smith,Phys.Rev.D85(2012)123002.

[14]S.Henrot-Versille,etal.,Class.QuantumGravity32 (4)(2015)045003. [15]R.Adam,etal.,PlanckCollaboration,Planck2015results.I.Overviewof

prod-uctsandscientificresults,arXiv:1502.01582[astro-ph.CO].

[16]N.Aghanim,etal.,PlanckCollaboration,Planck2015results.XI.CMBpower spectra, likelihoods, and robustness ofparameters, arXiv:1507.02704 [astro-ph.CO].

[17]P.A.R.Ade,et al.,PlanckCollaboration,Planck2015results.XV.Gravitational lensing,arXiv:1502.01591[astro-ph.CO].

[18]L.Anderson,etal.,BOSSCollaboration,Mon.Not.R.Astron.Soc.441(2014)24. [19]R.Cooke,M.Pettini,R.A.Jorgenson,M.T.Murphy,C.C.Steidel,Precision mea-sures of the primordial abundance of deuterium, arXiv:1308.3240 [astro-ph.CO].

[20]N.Aghanim,etal.,PlanckCollaboration,arXiv:1605.02985[astro-ph.CO]. [21]A.Lewis,S.Bridle,Phys.Rev.D66(2002)103511.

[22]F.Iocco,G.Mangano,G.Miele,O.Pisanti,P.D.Serpico,Phys.Rep.472(2009)1. [23]O.Pisanti,A.Cirillo,S.Esposito,F.Iocco,G.Mangano,G.Miele,P.D.Serpico,

Comput.Phys.Commun.178(2008)956. [24]M.Maggiore,Phys.Rep.331(2000)283.

[25]K.N.Abazajian,etal.,Lightsterileneutrinos:awhitepaper,arXiv:1204.5379 [hep-ph].

[26]M. Kawasaki,K. Kohri,N.Sugiyama,Phys. Rev.Lett.82 (1999)4168,arXiv: astro-ph/9811437.

[27]P.D.Meerburg,R.Hloek,B.Hadzhiyska,J.Meyers,Phys.Rev.D91 (10)(2015) 103505.

[28]G.Cabass,L.Pagano,L.Salvati,M.Gerbino,E.Giusarma,A.Melchiorri,Phys. Rev.D93 (6)(2016)063508.

[29]E.Komatsu,K.M.Smith,J.Dunkley,etal.,Astrophys.J.Suppl.Ser.192(2011) 18.

[30]R.Keisler,C.L.Reichardt,K.A.Aird,etal.,Astrophys.J.743(2011)28. [31]TheCOrECollaboration,C.Armitage-Caplan,M.Avillez,etal.,arXiv:1102.2181,

2011.

[32] J.A.Rubino-Martin,TheCOrE+(CosmicOriginsExplorer)mission,in:Highlights ofSpanishAstrophysicsVIII, ProceedingsoftheXI ScientificMeetingofthe SpanishAstronomicalSocietyHeldonSeptember8–12,2014.

[33]R. Laureijs, J. Amiaux, S. Arduini, et al., ESA/SRE(2011)12, arXiv:1110.3193, 2011.

[34]A.Lewis,Phys.Rev.D71(2005)083008.

[35]M. Martinelli, E. Calabrese, F. De Bernardis, A. Melchiorri, L. Pagano, R. Scaramella,Phys.Rev.D83(2011)023012.

[36]L.Amendola,M.Kunz,D.Sapone,J.Cosmol.Astropart.Phys.0804(2008)013. [37] K.Kadota,M.Kawasaki,K.Saikawa,J.Cosmol.Astropart.Phys.1510 (10)(2015)

041,http://dx.doi.org/10.1088/1475-7516/2015/10/041,arXiv:1503.06998 [hep-ph].