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Response function of the large-scale structure of the

universe to the small scale inhomogeneities

Takahiro Nishimichi, Francis Bernardeau, Atsushi Taruya

To cite this version:

Takahiro Nishimichi, Francis Bernardeau, Atsushi Taruya. Response function of the large-scale

struc-ture of the universe to the small scale inhomogeneities. Physics Letters B, Elsevier, 2016, 762, pp.247

- 252. �10.1016/j.physletb.2016.09.035�. �hal-01496477�

(2)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Response

function

of

the

large-scale

structure

of

the

universe

to

the

small

scale

inhomogeneities

Takahiro Nishimichi

a

,

c

,

d

,

,

Francis Bernardeau

a

,

b

,

Atsushi Taruya

e

,

c

aSorbonneUniversités, UPMCUnivParis6etCNRS,UMR7095,Institutd’AstrophysiquedeParis,98bisbdArago, 75014Paris,France bCEA, CNRS,UMR3681,InstitutdePhysiqueThéorique,F-91191Gif-sur-Yvette,France

cKavliInstituteforthePhysicsandMathematicsoftheUniverse(WPI),TheUniversityofTokyoInstitutesforAdvancedStudy,TheUniversityofTokyo,

5-1-5 Kashiwanoha,Kashiwa277-8583,Japan

dCREST,JST,4-1-8Honcho,Kawaguchi,Saitama 332-0012,Japan

eYukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto606-8502,Japan

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received27December2015 Receivedinrevisedform13July2016 Accepted18September2016 Availableonline22September2016 Editor:S.Dodelson

Keywords:

Gravitationalgrowthofcosmicstructures Perturbationtheory

N-bodysimulation

Inordertoinfertheimpactofthesmall-scalephysicstothelarge-scalepropertiesoftheuniverse,weuse aseriesofcosmological N-bodysimulationsofself-gravitatingmatterinhomogeneitiestomeasure,for thefirsttime,theresponsefunctionofsuchasystemdefinedasafunctionalderivativeofthenonlinear power spectrum with respect toits linearcounterpart. Its measured shape and amplitudeare found to bein good agreementwith perturbation theorypredictionsexcept for the couplingfrom smallto large-scale perturbations. Thelatter isfound tobe significantlydamped,followingaLorentzianform. Theseresultsshedlightonvalidityregimeofperturbationtheorycalculationsgivingausefulguideline for regularization of smallscale effects in analyticalmodeling. Mostimportantly our result indicates thatthe statisticalpropertiesofthe large-scalestructureoftheuniverse areremarkably insensitiveto the detailsofthesmall-scalephysics,astrophysicalorgravitational,paving thewayfor thederivation ofrobustestimatesoftheoreticaluncertainties onthe determinationofcosmologicalparameters from large-scalesurveyobservations.

©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

The cosmic energy fluctuations on large scales provide rich probes of the early universe physics, the mass of neutrinos or the nature of dark energy. Wide-field galaxy surveys are there-forewidelyconsideredforunveilingthedetailsoftheuniverse[1]. Among them are the DES,1 LSST2 andEuclid3 now under devel-opment.Such measurements rely howeverlargely on our under-standingofthestatisticalpropertiesofthecosmicfluctuations.The greatsuccessofthelatestcosmicmicrowavebackground observa-tions inestablishing the standard picture of ourUniverse largely owedto the fact that themeasured temperature fluctuationsare in the linear regime and thus can accurately be predictedusing

*

Correspondingauthorat:KavliInstituteforthePhysicsandMathematicsofthe Universe(WPI),TheUniversityofTokyoInstitutesforAdvancedStudy,The Univer-sityofTokyo,5-1-5Kashiwanoha,Kashiwa277-8583,Japan.

E-mailaddresses:takahiro.nishimichi@ipmu.jp(T. Nishimichi),

francis.bernardeau@iap.fr(F. Bernardeau),ataruya@yukawa.kyoto-u.ac.jp(A. Taruya).

1 https://www.darkenergysurvey.org/. 2 http://www.lsst.org/lsst/. 3 http://www.euclid-ec.org.

lineartheory[2,3].Likewise,weexpectthatthelate-time fluctua-tionsonlargescalesareinamildlynonlinearstage,andthereare robustwaystopredictthempreciselybeyondlinear-theory calcu-lations.

Established probes such as the baryon acoustic oscillations (BAOs; e.g. [4,5]) that give us a robust standard ruler useful for dark energy studies, or the redshift-space distortions (e.g., [6]) as an additional clue to discriminate gravity theories [7], are amongthose that we expect ina mildly nonlinearregime. Alter-natively, wecan accesscosmic fluctuationsonsimilar and some-whatsmallerscales withweak-lensingmeasurements (see[8]for arecentreview).Suchscientificprograms canonlybeachievedif relatedobservablescanbeaccuratelypredictedeitherfrom numer-ical simulationsor analyticallyforany givencosmologicalmodel. In particular it is important that such observables are shielded fromthedetailsofastrophysicsatgalacticorsub-galacticscales.4

4 Forinstancein[9–11]baryoniceffectsareshowntobeconfinedwithinthe

clusterscale,andtheycontributetothematterpowerspectrumatmost∼10% at

k=1hMpc−1anddropsrapidlytowardlargerscales. http://dx.doi.org/10.1016/j.physletb.2016.09.035

0370-2693/©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.

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248 T. Nishimichi et al. / Physics Letters B 762 (2016) 247–252

One way to reformulate this question is to quantify the im-pactofsmall-scalestructuresonthegrowthoflargescalemodes. Perturbation theory (PT) is a powerful framework to predict the growth of structure. Assuming that the system is described by self-gravitating pressure-less fluids, it provides the first-principle approachto thenonlinear growth(see[12] fora review).Its im-portancehas beenheightened afterthe detectionof BAOs inthe clusteringofgalaxies,makingprecisepredictionsofnonlinearities cruciallyimportant.

PT calculations show precisely that mode couplings between differentscalesareunavoidable.Weproposeheretoquantifythese couplings witha two-variable response function,5 definedasthe

linearresponseofthe

nonlinear power

spectrumatwave mode

k

withrespecttothe

linear counterpart

atwavemode

q

6:

K

(

k

,

q

;

z

)

=

q

δ

P

nl

(

k

;

z

)

δ

Plin

(

q

;

z

)

.

(1)

In the context of PT calculations, Refs. [14,15] showed progres-sivebroadeningoftheresponsefunctionwithincreasingPTorder, pointingtotheneedofregularization ofthesmall-scale contribu-tion.

Ifthebroadnessoftheresponsefunctionatlate timesistrue, physics atvery small scale can influencesignificantly the matter distribution onlarge scales, where theacoustic feature is promi-nent.7 It also questions the reliability of simulations, which can follow the evolution of Fourier modes only in a finite dynam-ical range. We here discuss the response function at the non-perturbativelevelutilizingcosmological

N-body

simulations.

2. Methodology

Weheredescribeourmethodtomeasuretheresponsefunction from simulations. We prepare two initial conditions with small modulationsin thelinearspectrum overa finiteinterval ofwave mode

q,

evolvethemtoalatetime,andtakethedifferenceofthe nonlinearspectrameasuredfromthetwo.Thatis

ˆ

Ki,jPlinj

Pnli

[

P+,linj

] −

Pinl

[

P−,linj

]



ln Plin



ln q

,

(2)

wherethetwoperturbedlinearspectraaregivenby

ln



Plin ±,j

(

q

)

Plin

(

q

)



=



±

1 2



ln P lin if q

∈ [

q j

,

qj+1

),

0 otherwise

.

(3)

In the above, the index i ( j) runs over the wave-modebins for thenonlinear(linear)spectrum,andwechooselog-equalbinning, ln qj+1

ln qj

=

ln ki+1

ln ki

= 

ln q.Itisstraightforwardtoshow

that the estimator K approaches

ˆ

to theresponse function K de-fined in Eq.(1), when



ln q and



ln Plin are small. The

defini-tion(1)isadvantageousinthatitallowsthemeasurementinthis wayatthefullynonlinearlevel.8 Notethatasimilarfunctionwas firstdiscussednumericallyinRef.[18]inthecontextoflocal trans-formationsofthedensityfield.

We adopt a flat-



CDM cosmology consistent with the five-yearWMAPresult[19]withparameters

(

m

,



b

/

m

,

h

,

A

s

,

ns

)

=

(

0

.

279

,

0

.

165

,

0

.

701

,

2

.

49

×

10−9

,

0

.

96

)

,whicharethecurrent mat-ter density parameter, baryon fraction, the Hubble constant in

5 ThisconceptwasrecentlyutilizedinRef.[13]tocomputethedifferenceofthe

nonlinearpowerspectrumforslightlydifferentcosmologicalmodels.

6 ThenormalizationissuchthatK contributestothechangeinPnlwithuniform

weightsperdecade.

7 Notice,however,thatthefeaturecanalsobeaffectedbygalaxybias[16,17]. 8 ThisiscontrastedtothefunctionF

nappearinginPTforthen-thordercoupling.

Table 1

Simulationparameters.Boxsize(box),softeningscale(soft)andmassofthe parti-cles(mass)arerespectivelygiveninunitofh−1Mpc,h−1kpc and1010h−1M

.The numberofq-binsisshowninthe“bins”column,foreachofwhichweruntwo sim-ulationswithpositiveandnegativeperturbationsinthelinearspectrum.The“runs” columnshowsthenumberofindependentinitialrandomphasesoverwhichwe re-peatthesameanalysis.Thetotalnumberofsimulationsareshowninthe“total” column.

name box particles zstart soft mass bins runs total

L9-N10 512 10243 63 25 0.97 5 1 10 L9-N9 512 5123 31 50 7.74 15 4 120 L9-N8 512 2563 15 100 61.95 13 4 104 L10-N9 1024 5123 15 100 61.95 15 1 30 high_ns 512 5123 31 50 7.74 5 4 40 low_ns 512 5123 31 50 7.74 5 4 40

units of 100km

/

s

/

Mpc, thescalar amplitude normalizedat k0

=

0

.

002Mpc−1 andits power index, respectively. We also consider different cosmologies to check the generality ofthe result. Since we can check the dependence of the response function on the overall amplitude of the power spectrum by looking at the re-sults at different redshifts, we here focus on the variety in only the shape of the spectrum. As a representative of the parame-ters that control the shape, we consider the spectral tilt ns. We

run simulations for two additional models, one with a higher (1

.

21;

high_ns

) and the other with a lower (0

.

71;

low_ns

) value of ns. Although the parameter ns has been constrained

very tightly from observations of the cosmic microwave back-ground (with only

1% uncertainty), we choose to give it a rather large (

±

0

.

25) variation to cover a wider class of models with differentlinear powerspectra. The amplitude parameter As

for thesemodelsis determined such that therms linear fluctua-tionat8h−1Mpc iskeptunchanged.Thematter transferfunction is computed for these models using the

CAMB

code [20] with the high-precision mode of the calculation in the transfer func-tion(

transfer_high_precision

issettobetrueand

accu-racy_boost

=

2)upto

k

max

=

100h Mpc−1.Weconfirmthatthe

resultiswellconvergedbytestingmorestrictvaluesinthe param-eterfile.

We run four sets of simulations for the fiducial model with different volume and number of particles as listed in Table 1. Coveringdifferentwavenumberintervals,thesesimulations allow us to examine the convergence of the measured response func-tion. Theinitial conditionsare createdusingacode developedin

[21,22] based on the second-order Lagrangian PT (e.g., [23,24]). The initialredshifts ofthesimulationsare determinedasfollows. A lower starting redshift can induce transient effects associated withhigher-orderdecayingmodes.Ontheotherhand,as increas-ing the initial redshift, the randomly generated particle position generally gets closer to the pre-initial grid, and thiscan lead to discretenessnoiseintheforcecalculation.Tominimizethesumof thesetwo systematiceffects, weset theinitial redshift suchthat the rms displacement is roughly 20% of the inter-particle spac-ing, and thus it depends on theresolution as shownin Table 1. We evolve the matter distribution using a Tree-PM code

Gad-get2

[25].WefinallymeasurethepowerspectrumbyfastFourier transform of the Cloud-in-Cell (CIC) density estimates on 10243 meshwiththeCICkerneldeconvolvedinFourierspace.

Foreach setofsimulations, we preparemultipleinitial condi-tionswithlinearspectraperturbedby

±

1% over

q

j

q

<

qj+1.The

amplitude of perturbation should be sufficiently small such that the correction fromthe higher-orderderivative (

δ

2Pnl

Plin

δ

Plin) does not contaminate the result. We tested different amplitudes (

±

3% and

±

5%), and confirmed that the result is almost un-changed. Weset thebin widthas



ln q

=

ln

(

2

)

andeach sim-ulation set covers different wavenumber range corresponding to

(4)

Fig. 1. Responsefunctionmeasuredfrom simulations.We plot|K(k,q)|Plin

(q)as afunctionofthelinearmodeq forafixednonlinearmodeatk=0.161hMpc−1

indicatedbytheverticalarrow.Thefilled(open)symbolsshowL9-N9(L10-N9), thelinesdepictL9-N8,whilethebighatchedsymbolsonsmallscalesareL9-N10. Positive(negative)valuesareindicatedastheupward(downward)trianglesorthe solid(dashed)lines.

theboxsizeandresolutionlimit.Thebinning effectwillbetaken intoaccount intheanalyticalcalculationsforfaircomparison.For thebestresolutionrun,

L9-N10

,westudyonlyfivebinsonsmall scales.Further,weperformfourrealizationsfor

L9-N9

and

L9-N8

ateachwave-modebintoestimatethestatisticalscatter.Thesame random phases are used for initial conditions with positive and negativeperturbationsateach

q bins

foreachrealization.Sincethe estimator(2)takesthedifferenceofthetwospectra,thishelpsus to reduce the statistical scatter on the response function signifi-cantly.

3. ShapeoftheresponsefunctionandcomparisonwithPT

Wearenowinapositiontopresenttheresponsefunction mea-suredfromsimulations.The combination K

(

k

,

q

)

Plin

(

q

)

is plotted atafixed

k shown

bytheverticalarrowasafunctionof

q in

Fig. 1. Thestrongoverlapamongdifferentsymbolsandlinesensuresthe convergenceoftheresultsagainstresolutionandvolume.

At high redshifts, we can see a prominent peak at k

=

q as expectedfromlineartheory(i.e.,nomodetransfer).Nonlinear cou-pling then gradually grows with time and the peak feature gets lesssignificant.One ofthekey features hereisthe larger contri-butionfromsmallerwavemodes(q

<

k); thegrowthofstructure isdominated by mode flows fromlarge to smallscales. Not sur-prisingly,theformationofastructureismoreefficientlyamplified whenitispartofalargerstructurethan whenitcontains small-scalefeatures.

SuchfindingsarefullyinlinewithexpectationsfromPT calcu-lations.WeshowthepredictionsinFig. 2uptothetwo-looplevel (i.e.,next-to-next-to-leadingorder)ignoringbinningeffectsatthis stage.We presentthe contributionfrom Pi j

(

k

)

∝ δ

(i)

δ

(j)

,where

δ

(i)isthe

ith-order

overdensityinthePTexpansion.Thetermsat thesameloopordercancelatsmall

q due

totheGalilean invari-anceofthesystemasdiscussedine.g.,[26–30].Ontheotherhand, smallscalesaredominatedbyonetermateachorder, P13

(

k

)

and

P15

(

k

)

.Similarly, it has been shownthat, atthe p-loop orderin

Fig. 2. ResponsefunctionpredictedbyPT(un-binned)uptoone- (thinsolid)and two-loop(thicksolid)orderat k=0.161hMpc−1 at z=1.Dashed(dotted)lines showeachoftheone- (two-)loopcontributionswiththelegend(i j)showingthe perturbativeorderofthecalculation.Weshowanegativesigninthelegendwhen

K isnegative.ThesimulationdataL9-N9arealsoshownbytriangles.

Fig. 3. Rescaledresponsefunction,T(k,q)≡ [K(k,q)Klin(k,q)]/[q Plin(k)].PT

cal-culations areshown bylines, whereasthe symbols areL9-N9 (seelegend for detail).Thenonlinearwave-modebinisfixedatk=0.161hMpc−1(verticalarrow).

Binningistakenintoaccounttotheanalyticalcalculationsconsistentlytothe sim-ulations.

PT, thetermoriginatingfromthe

(

2p

+

1

)

th-order densitykernel function, F2p+1, dominates the mode-coupling effect from small

scales[14].

We then rescale the response function at various redshifts as T

(

k

,

q

)

= [

K

(

k

,

q

)

Klin

(

k

,

q

)

]/[

q Plin

(

k

)

]

, where Klin is the linear

contribution,andplottheminFig. 3.Theyarecomparedwiththe one-loop PT calculation (solid), which is time-independent with this normalization. The simulation data indeed show little time dependenceat

q



k in remarkableagreement withtheone-loop

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250 T. Nishimichi et al. / Physics Letters B 762 (2016) 247–252

Fig. 4. Responsefunctiondividedbythetwo-loopPTatthethreewavemodesk

showninthelegend.Weplotdatapointsonlyatq2k fordefiniteness.The over-plottedsolidlinescorrespondtotheempirical form(4),smallsolidsymbolsare

L9-N9whilethebighatchedareL9-N10.

calculation, reproducing the expected q dependence,9 as well as

thechangeofsignbetweenlargeandsmallscales.The smallbut non-negligible z-dependence at k

q is further reproduced by thetwo-loop calculation (see thefigure legend).Notethat atthe wave-modek plotted here (i.e., 0

.

161h Mpc−1), the two-loop PT predictionforthenonlinear powerspectrum agrees with simula-tions within 1% at z



1 and theagreement gets worse atlower redshiftreachingto

5% at

z

=

0 (seee.g.,[15]).

At q



0

.

3h Mpc−1, however, the measured response function isdamped compared tothe PT. The one-loop PT predicts the re-sponse function to reach a constant10; at the two-loop order, it

growsinamplitudewithtime.Thenumericalmeasurementsshow on the other hand that the scaled response function is strongly dampedwithdecreasingredshift.Itissuchthatthecouplingstake placeeffectivelybetweenmodesofsimilarwavelengths.Thiseffect isparticularlyimportantatlatetime.Atredshiftzero,the discrep-ancybetweenthemodelandsimulationsisstriking.Furthermore, analysisofthe responsestructure atthree andhigherloop order (see e.g., [14]) suggests that PT calculations, at any finite order, predict an evenlarger amplitude of theresponse function inthe high

q region.

Thisstronglysuggeststhatthisanomalyisgenuinely non-perturbative.

Wepropose aneffectivedescription ofthisobservedbehavior. AsillustratedinFig. 4itcanbemodeledwithaLorentzian:

Teff.

(

k

,

q

)

−−−−→

high−q T1+2−loop

(

k

,

q

)

1

1

+ (

q

/

q0

)

2

(4)

characterizedbyacriticalwavemode,

q

0,whichdoesnotdepend

onthe nonlinearwave mode

k.

Wenaively expect thatthisscale correspondstothescaleatwhichthefluctuationisorderunityand thus perturbative expansion is not valid. Indeed,we numerically foundthatasimplefittingformula

σlin

(

R

;

z

)

|

R=1/q0

=

1

.

35

,

(5)

9 Ithasthesmall-q asymptote[2519/4410− (23/84)n+ (1/20)n2]q2/π2foran

Einstein–deSitterbackground,withn beingthelocalslopeofthelinearspectrum.

10 Thisconstantis61k2

/(630π2

)foranEinstein–deSitterbackground.

Fig. 5. SameasFig. 4,butforcosmologicalmodelswithdifferentspectralindices,

high_nsandlow_ns.

where

σ

2

lin

(

R

)

is the variance of the linear density fluctuation

smoothedwithaGaussian filteroftheformexp

[−(

0

.

46kR

)

2

/

2

]

,11

canexplain reasonably wellthedatapointsnot onlyforthe fidu-cial modelbutforthe modelswithdifferentspectralindices(see

Fig. 5).Onecanfindthatthefitisnotasaccurateat

z

=

1 forthe

low_ns

model, suggestingthe limitation of the fitting formula. Nevertheless, a simple form (4) witha single parameter 1

.

35 in Eq.(5)seemstocapturethedampingtailoftheresponsefunction in a rather wide range of cosmological models at different red-shifts.Notethat,the

k-dependence

oftheresponse functionatthe high-q limitispreservedinperturbative calculations(itisalways proportional to k2 independent of the perturbative order due to

theasymptoteofthe Fn kernelfunction).Theindependenceof

q

0

on

k is

thusinfullagreementwithPTpredictions.

4. Discussion

The simulation results give a clear evidence that the mode transferfromsmalltolarge scalesissuppressedcomparedto the PTpredictionwhenthemode

q enters

thenonperturbativeregime. However, theoriginofthesuppressionisyettobeunderstood.In particularitisnotclearwhetherthishasitsrootsinshellcrossing ornot. Forinstance,wecan findin[32] that vorticityand veloc-ity dispersion generatedby shell crossingcan alter the evolution ofdensityfluctuationsthrough nonlinearcoupling.Especially,the latter is shown to give non-negligible corrections to the density power spectrum even on very large (

0

.

1hMpc−1) scales. This indicates that multistreaming physics, whichtake place on small scales, are somehow related to the growth of large-scale fluctu-ations, andthisis exactly the coupling oflarge and smallscales that we discuss here. Effective Field Theory (EFT) approaches as advocated originally in[33–35] wouldbea naturalframework to invokeforaccountingforsuchmultistreamingeffects.Inthese ap-proaches,however,theresponsefunctionisultimatelyencodedin freecoefficientsforwhichnotheoryexists.

It might be possiblethat such dampingeffect originatesfrom simplermechanismsinsingle-streamphysics.Ithasbeenshownin

11 Thefactor 0

.46 hereischosenfor abettercorrespondencewiththetop-hat radiusintermsofthevariance.See[31]formoredetail.

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particularthat thenonlinear densitypropagator,which expresses the evolution of a givenwave mode withtime, is exponentially damped by the large-scale displacements. This is the standard result on which the Renormalized Perturbation Theory is based

[36,37].Asexplicitlyshownin[38]equal-timespectraarehowever insensitiveto displacements of the globalsystem, that originates fromwave modes smaller than

k.

Displacementsat intermediate scales are nonetheless expected to induce some effective damp-ing for equal-time spectra. The physical idea behind that is that the force driving the collapse of a large-scale perturbation (e.g., a clusterofgalaxies)isaffectedbythesmallscaleinhomogeneities withinthestructure(saygalaxies),butthatthisdependencemight be damped when such small scale inhomogeneities are actually moving within the structure. It is however beyondthe scope of thispresentationtoevaluatetheimportanceofthiseffect.

5. Summary

We have presented the first direct measurement of the re-sponse function that governs the dependence of the nonlinear power spectrum on the initial spectrum during cosmic structure formation.Thismeasurementwasdone usingalargeensembleof N-body simulations that differ slightlyin their initial conditions. Theresultswerefoundtoberobusttothesimulationresolution– asshownin Table 1– supportingthe ideathat measured shapes weregenuinefeaturesinthedevelopmentofgravitational instabil-ities.

The response functions were computed concurrently at next andnext-to-nextleadingorderinPT. Comparisonswith measure-ments show a remarkable agreement over a wide rangeof scale andtime.We found howevermode transfers fromsmallto large scales to be strongly suppressed compared to theoretical expec-tations especially at late time. We propose a description of the dampingtailwithaLorentzianshape.

Theseresultsare of far-reachingconsequences. Theyfirst give insights into the mode coupling structure of cosmological fluids and show that PT approaches capture most of their properties. The small scale damping signalsthe validity limit of the PT be-yondnext-to-leadingorder.Itprovidesinparticularindicationson howtoregularize theircontributions. Theobserveddampingalso marks the irruption of collective non-linear effects although the underlyingmechanismsareyettobeuncovered.Mostimportantly the damped response suggests that small scale physics, whether from the initial metric perturbations or late-time processes, can be effectively controlled. It paves the way for solid estimates of the theoretical uncertainties on the determination of cosmologi-cal parameters(such asinflationaryprimordial non-Gaussianities, neutrinomassesordarkenergyparameters)fromlarge-scale sur-veys.

Acknowledgments

WethankPatrickValageasforfruitfuldiscussionsonanalytical calculationsoftheresponse function.WealsothankYasushiSuto forinsightfulcomments on the originofour findings. This work is supported in part by grant ANR-12-BS05-0002 of the French AgenceNationale de la Recherche. TN is supported by Japan So-cietyforthePromotion ofScience (JSPS)PostdoctoralFellowships forResearchAbroad. ATissupportedby aGrant-in-Aidfor Scien-tificResearchfromtheJapanSocietyforthePromotionofScience (JSPS)(No. 24540257).Numericalcalculationsforthepresentwork havebeencarried out onCray XC30 atCenter forComputational Astrophysics,CfCA,ofNationalAstronomicalObservatoryofJapan.

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Figure

Fig. 3. Rescaled response function, T ( k , q ) ≡ [ K ( k , q ) − K lin ( k , q )]/[ q P lin ( k )]
Fig. 5. Same as Fig. 4, but for cosmological models with different spectral indices, high_ns and low_ns .

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