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Consistency of the local Hubble constant with the cosmic microwave background

LOMBRISER, Lucas

Abstract

A significant tension has become manifest between the current expansion rate of our Universe measured from the cosmic microwave background by the Planck satellite and from local distance probes, which has prompted for interpretations of that as evidence of new physics. Within conventional cosmology a likely source of this discrepancy is identified here as a matter density fluctuation around the cosmic average of the 40 Mpc environment in which the calibration of Supernovae Type Ia separations with Cepheids and nearby absolute distance anchors is performed. Inhomogeneities on this scale easily reach 40% and more. In that context, the discrepant expansion rates serve as evidence of residing in an underdense region of δenv≈−0.5±0.1. The probability for finding this local expansion rate given the Planck data lies at the 95% confidence level. Likewise, a hypothetical equivalent local data set with mean expansion rate equal to that of Planck, while statistically favoured, would not gain strong preference over the actual data in the respective Bayes factor. These results therefore suggest borderline consistency between the [...]

LOMBRISER, Lucas. Consistency of the local Hubble constant with the cosmic microwave background. Physics Letters. B , 2020, vol. 803, no. 135303

DOI : 10.1016/j.physletb.2020.135303

Available at:

http://archive-ouverte.unige.ch/unige:131863

Disclaimer: layout of this document may differ from the published version.

1 / 1

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Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Consistency of the local Hubble constant with the cosmic microwave background

Lucas Lombriser

DépartementdePhysiqueThéorique,UniversitédeGenève,24quaiErnestAnsermet,1211Genève4,Switzerland

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received30October2019

Receivedinrevisedform17January2020 Accepted12February2020

Availableonline17February2020 Editor:H.Peiris

AsignificanttensionhasbecomemanifestbetweenthecurrentexpansionrateofourUniversemeasured fromthecosmicmicrowavebackgroundbythePlancksatelliteandfromlocaldistanceprobes,whichhas promptedforinterpretationsofthatasevidenceofnewphysics.Withinconventionalcosmologyalikely sourceofthisdiscrepancyisidentifiedhereasamatterdensityfluctuationaroundthecosmicaverageof the40 MpcenvironmentinwhichthecalibrationofSupernovaeType IaseparationswithCepheidsand nearbyabsolutedistanceanchorsisperformed.Inhomogeneitiesonthisscaleeasilyreach40%andmore.

Inthatcontext,thediscrepantexpansionratesserveasevidenceofresidinginanunderdenseregionof δenv≈ −0.0.1.TheprobabilityforfindingthislocalexpansionrategiventhePlanckdataliesatthe 95%confidencelevel.Likewise,ahypotheticalequivalentlocaldatasetwithmeanexpansionrateequal tothatofPlanck,whilestatisticallyfavoured,wouldnotgainstrongpreferenceoverthe actualdata in therespectiveBayesfactor.Theseresultsthereforesuggestborderlineconsistencybetweenthelocaland PlanckmeasurementsoftheHubbleconstant.Generallyaccountingfortheenvironmentaluncertainty,the localmeasurementmaybereinterpretedas aconstraintonthecosmologicalHubbleconstantofH0= 74.7+54..82km/s/Mpc.Thecurrentsimplifiedanalysismaybeaugmentedwiththeemploymentofthefull availabledatasets,animpactstudyfortheimmediate10 Mpcenvironmentofthedistanceanchors, more proneto inhomogeneities, as wellas expansionrates measuredby quasarlensing, gravitational waves,currentlylimitedtothesame40 Mpcregion,andlocalgalaxydistributions.

©2020TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

DeterminingtheexpansionhistoryofourUniverseisparamount toresolvingitsenergeticcompositionortestingkeyingredientsof standard cosmology such asthe validity ofGeneral Relativity on large scales and the cosmic principle. Measurementsof the cur- rentexpansionratereveal asignificant tensionbetweenthecon- straintsinferredfromthecosmicmicrowavebackground(CMB) [1]

andthose found fromseparations to Supernovae Type Ia (SN Ia) using absolute distance anchors from nearby Cepheids, masers, parallaxes, and ecliptic binaries [2–6]. The 4.4

σ

[6] discrepancy between these probes of the early and late Universe has hence spurredmuch speculation forits interpretation asa signature of newphysics(seeforexampleRef. [7]).Independentmeasurements of the Hubble constant such as from quasar lensing [8] further increase this tension, and important insights are expected from StandardSirens,whichhowevercurrentlydonotgiveaclearpref- erenceforeithervalueoftheexpansionrate [9].

E-mailaddress:lucas.lombriser@unige.ch.

This Letter argues that the local measurement of the Hub- ble constant is likelyimpacted by the matter densityfluctuation aroundthecosmic averageofthe40 Mpcenvironmentsurround- ing us. Such an inhomogeneity is shown here to affect the cal- ibration of SN Ia distances with the Cepheids and the anchors, performed in the same volume. The scale is set by the furthest host galaxy of the sample [5]. Previous studies (see Ref. [10,11]

andreferencestherein)havefocusedonunderdenseenvironments atscales of severalhundreds ofMpc,where inhomogeneitiesare howeverstronglyconstrainedinamplitude.Densityfluctuationson a40 Mpcscaleincontrastmayeasilyreach40%andmore.

This work presents an estimation of the likelihood of resid- ing in a sufficiently underdense 40 Mpc environment to give risetotheobserveddiscrepancybetweenthemeasuredlocaland cosmologicalHubble constants. The numerical computationsper- formed forthispurpose will assume Planck [1] mean cosmologi- calparameters andstandarddeviations,specificallythetotalmat- ter density parameter m=0.315±0.007, the Hubble constant H0=(67.0.5) km/s/Mpc, and the matter fluctuation ampli- tude

σ

8 =0.811±0.006. The local expansion rate is given by

https://doi.org/10.1016/j.physletb.2020.135303

0370-2693/©2020TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.

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2 L. Lombriser / Physics Letters B 803 (2020) 135303

Hˆ0=(74.03±1.42) km/s/Mpc [6], where hats will denote local quantitiesforclarity.Thespeedoflightinvacuumissettoc=1.

2. CalibrationofSN Iadistances

AmeasurementofthelocalHubbleconstantHˆ0wasconducted in Ref. [2], employing a distanceladder that isbased on precise maserseparationstoNGC 4258tocalibrateCepheiddistancesand from that separations to SN Ia using suitable host galaxies. Im- provedmeasurements adoptingthistechniquewere performedin Refs. [3–6],includingmorehostsforthe Cepheid-calibratedSN Ia separations as well as additionalindependent calibrations ofthe absoluteCepheid distances fromMilky Way parallaxes and from Cepheids and detached eclipsing binary separations in the Large MagellanicCloud. The combinationof thesedistanceprobes cur- rentlyyieldsameasurementofHˆ0=(74.03±1.42)km/s/Mpc [6], whichisin4.4

σ

tensionwiththeCMBmeasurementofPlanck.

Morespecifically,themeasurementofthelocalHubbleconstant isobtainedfromtherelation [2] (App.A.3)

logH

ˆ

0

=

1

5

(

m0x,N4258

μ

0,N4258

) +

ax

+

5

logd

ˆ

1

,

(1) where m0x,N4258 is the expected peak magnitude of a SN Ia in NGC 4258and

μ

0,N4258 denotestheindependentprecise distance modulusfromitsmasers,whichrelatestotheluminositydistance dˆL,N4258 as

μ

=5log(dˆL/dˆ1)+25. The peak magnitudem0x,N4258 isfound froma fitto thejointCepheid/SN Iadata,where the j- thmeasured Cepheid magnitudein the i-thhost mWH,i,j provides thedifferenceinthedistancemoduli

μ

0,i

μ

0,N4258,whichcorre- spondstothedifferencem0x,im0x,N4528 fortheSN Iamagnitudes m0x,i [5]. The term ax is the intercept of the magnitude-redshift relation for the SN Ia, given by ax=log(H0dL)0.2m0x, which importantlyisindependentofanabsoluteseparationscale.Thelu- minositydistancedL inax isfitto alarge SN Iasample covering theredshifts0.023<z<0.15,∼(100–650) Mpc,andtheindivid- ualm0x aredeterminedbyalight-curvefitter.

Notethatthedistancenormalisationdˆ1 inEq. (1) hasbeenkept explicithereandrefers toa1 Mpc absolutedistanceinthe local referenceframe. Fordˆ1 to simplycancel it mustbe given inthe samereferenceframeasdˆL,N4258.Forahomogeneousuniverse,the cosmologicalrulerequals thelocalone,d1= ˆd1,such thatEq. (1) alsoimpliesameasurementofthecosmologicalH0.Inthecaseof a local inhomogeneity,however, thelocal andcosmological met- ricsarerelatedbyaconformaltransformationgˆμν=(ˆa/a)2gμν for the respectivescale factors a,ˆ a (App. A.2). Unlikefor coordinate transformations,conformaltransformations donotpreserve phys- icaldistances,anditiseasy toshow that Hˆ0/H0=dL/dˆL.Hence, Eq. (1) implies a measurementof theexpansion rateofthelocal environment, defining the frame in which dˆL,N4258 is measured.

GivendˆL,N4258butnotdL,N4258,onealsoarrivesatthisconclusion if expressing Eq. (1) in terms of H0 and d1 instead. The use of anabsolutedistanceanchorinthecosmologicalframe incontrast wouldprovideameasurementof H0,consistentwiththefindings ofRef. [12].

In principle the absolute distance scale only enters through NGC 4258 at7.5 Mpc andother more nearby anchors. One may thereforeconsider the relevantenvironment as setby that scale.

However, the calibration of the Cepheid magnitudes mWH,i,j from whichthescalein

μ

0,i isinferredisperformedwithaninvolved fittingprocedureofCepheid parametersto thefullsample ofthe jointCepheid/SN Ia hosts at40 Mpc.The environment forthe absolute distance anchors is therefore conservatively set here to 40 Mpc. The effectof anenvironmental densityfluctuation for theabsolutedistanceanchorsat10 Mpcliesbeyondthescope

of thisworkandis left forfuturestudy.Whereasmatter density fluctuations on the scales of (100–650) Mpc ofthe high-z SN Ia samplesarelimitedtostandarddeviationsof(5-20)%aroundthe mean,thesereachabout40%onscalesof40 Mpcandmayhence impactthelocallymeasuredexpansionrate.

3. Expansionrateofthelocalenvironment

The luminositydistancesdL entering Eq. (1) arerelatedtothe background expansionhistory H(z) asdL(z)=(1+z)z

0z/H(˜z), whereaspatially flatstatisticallyhomogeneousandisotropicuni- verse with standard cosmological components of baryons, cold darkmatter, anda cosmologicalconstant is assumedthrough- out.Radiationwillbeneglectedforthelate-timeuniverseofinter- est.From theresultingtime-timecomponentoftheEinsteinfield equationswiththeFriedmann-Lemaître-Robertson-Walker(FLRW) metric,itfollowsthat

H2

d lna

dt

2

=

8

π

GN 3

ρ ¯

m

+

3

,

(2)

where a(t)=(1+z)1 is the scalefactor ofthe FLRWmetric in termsofcosmologicaltimet normalisedtoday,GN denotesNew- ton’sgravitationalconstant,

ρ

¯m representsthetotalmatterdensity inthecosmologicalbackground,andH0H(z=0).

ThelocalenvironmentmaybetreatedasaseparateFLRWuni- verseembeddedinthelargerCosmos [13].Thisansatziswellmo- tivated [14] since withrvir0.2 MpcthevirialradiioftheMilky Way andNGC 4258aremuchsmallerthanthewavelengthofthe 40 Mpc density fluctuation. Energy conservationin this environ- mentwithrespecttocosmologicaltimetimpliesalocalexpansion rateof

H

ˆ = −

1 3

d ln

ρ ˆ

m

dt

,

(3)

where

ρ

ˆm istheaverageenvironmentalmatterdensity,whichre- latestothatofthecosmologicalbackgroundas

ρ

ˆm≡ ¯

ρ

m(1+δenv). This defines the local environmental matter density fluctuation δenv.

TheevolutionofδenvisdescribedbythesolutionofEq. (4) and can be castasafunction of Hˆ0/H0.Thus, themeasured discrep- ancyof Hˆ0/H01.1 betweentheHubbleconstant obtainedfrom Planck and from the local probes may be interpreted as a mea- surementofthelocalenvironmentalinhomogeneity,averagedover a 40 Mpc scale. For simplicity keeping cosmological parameters fixed, except forthe expansionrates, andcasting dataandprob- abilitiesinto Gaussians,one infers fromthemeasurements of Hˆ0 and H0 alocal densitywithmeanandstandard deviationof ap- proximately δenv,0≈ −0.0.1,whereerrorshavebeenaddedin quadratureand δenv,0=∂(∂δHˆenv,0

0/H0) (Hˆ0/H0).

In the following an estimate for the likelihood of residing in suchalocalenvironmentwillbeprovided,notingthattherequired underdensityliesinthenonlinearregimeofstructureformation.

4. Nonlinearevolutionofmatterdensities

For simplicity, the environmental density fluctuation will be treatedasasphericallysymmetrictophat.Fromenergy-momentum conservation,∇μTμν|env=0,one derivesthenonlinear evolution equation [13,15]

y

+

2

+

H H

y

+

1 2

m

(

a

)

y3

1

y

=

0 (4)

for the dimensionless physical top-hat radius y=(

ρ

ˆm/

ρ

¯m)1/3, whereprimesdenotederivativeswithrespecttolnaandm(a)

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Fig. 1.Peak-normalisedprobabilitiesfortheexpectedexpansionrateHˆ0inourlocal40 MpcenvironmentgivenacosmologicalvalueH0(leftpanel)andforacosmological H0giventhelocaldataDˆ [6] castintoaGaussian(rightpanel).Shadedregionsillustrate68%,95%,and99%confidencelevelsaroundthemedian.TheGaussianisedlocal (leftpanel)andPlanck[1] (rightpanel)measurementsareshownforcomparison(dashedcurves),adoptingthemeanH0ofPlanckforHˆ0/H0.Theexpansionratesmarginally agreeatthe95%confidencelevel,andtherewouldbenostrongevidenceintherespectiveBayesfactorforahypotheticalequivalentlocalmeasurementwithequalmeanto thatofPlanckcomparedtotheactualdata.

8

π

GN

ρ

¯m/(3H2). The evolution of y can be determined setting initial conditions inthe matter-dominated regime ai1, where yiy(ai)=1δenv,i/3 and yi= −δenv,i/3 for an initial top-hat fluctuationδenv,i.Foranenvironmentthatundergoessphericalcol- lapsetoday, one can define the current linear spherical collapse density,whichforthePlanck cosmologicalparametersisgivenby δc =1.678. Note that for small fluctuations y1, Eq. (4) can belinearised, yieldingthe usual evolutionequation forthelinear growthfunctionofcosmologicalstructureLL,i)(a).

5. Distributionofenvironmentaldensities

The probability for a linear environmental density fluctuation δL, env, definedby the Eulerian(physical) radius ζ=40 Mpc,can beestimatedusing excursionsettheory,where theresultingdis- tributionisapproximatelydescribedbytheexpression [16]

Pζ

L, env

) = β

ω/2

2

π

exp

β

ω 2

δ

2L, env

(

1

δ

L, env

c

)

ω

×

1

+ ( ω

1

) δ

L, env

δ

c

1

δ

L, env

δ

c

ω/21 (5)

withβ=(ζ /8)3c/

σ

82/ω,

ω

=δc

γ

,and

γ

= −d lnd lnMSenvξ =1+ ˜ns/3.

TheLagrangian(orinitial comoving)radiusissetby ξ=8 Mpc/h suchthat =

σ

82.Theslopeofthelinearmatterpowerspectrum PL(k) on large scales atan initial time ai1 in the matter era aftertheturnoverissetton˜s= −1.6.

TheprobabilityforthenonlineardensityfluctuationsPζNL, env) can be determined from employing the linear growth function LL,i)(a) to evolve the range of δL, env back to an initial time ai1 during the matter-dominated epoch, which is well de- scribed by linear theory, and then evolve the initial fluctuations δi forward to the present time with the nonlinear solutions of Eq. (4). The resulting probability distribution P(Hˆ0|H0) for a lo- calHubbleconstant Hˆ0 given H0 inthecosmologicalbackground inferred from PζNL, env) using Eq. (3) is shown in Fig. 1 and is in good agreement with the distribution measured from N- body simulations in Ref. [17] (also see Ref. [18]), where with 0.88Hˆ0/H01.15 at2

σ

thesimulationsallowforevenlarger, lessconservative,fluctuationsattherelevantupperbound.

6. ConsistencyoflocalHˆ0withCMBH0

Thelevelofagreementordisagreementbetweenthelocaland CMBmeasurementsoftheHubbleconstantcannowbequantified.

Forasimplecomparison, thelocal Hˆ0 measurementofRef. [6] is shownalongsidewithP(Hˆ0|H0)inFig.1,wherebothdistributions arenormalisedbytheirpeakamplitudesandHˆ0 hasbeendivided bythemeanPlanckvalueofH0 forillustration.Thelocalmeasure- mentisfoundheretolieatthe95%confidencelevelofP(Hˆ0|H0), suggestingborderlineconsistencywiththePlanck value.However, forabetterassessmentoftheconsistencybetweenthetwomea- surements, one may wish to directly estimate the probability of onemeasurement giventheother.Planck providestheprobability P(H0|D) of a cosmological background value H0 given the CMB data D. From the localmeasurement one obtains the probability P(Hˆ0| ˆD) ofa value Hˆ0 in ourenvironment giventhe localmea- surement Dˆ.For simplicity,both probabilities shall be treatedas Gaussiansadopting themeans andstandarddeviations quoted in Refs. [1,6].Atthispoint itshouldthereforebe stressedthat fora morereliablelikelihoodanalysis,amorecarefulimplementationof thedatabeyondthesimplifiedGaussiantreatmentshouldbeper- formed. This is left for futurework. The analysispresented here shall only serve asan estimate ofthe effect ofour environment on the localmeasurement of the Hubble constant.Non-Gaussian featurescanbeexpectedtobesubdominant.

From the measurements, employing Bayes’Theorem, one may findthe probability ∝P(Dˆ|H0) ofgetting thelocalmeasurement Dˆ whengivenH0 inthebackgroundandfromthattheprobability

P(Dˆ|D)foralocalmeasurementDˆ giventheCMBdataD.More precisely,

P

(

D

ˆ |

D

) =

dH0P

(

D

ˆ |

H0

)

P

(

H0

|

D

) ,

(6) P

(

D

ˆ |

H0

) =

dH

ˆ

0P

(

D

ˆ | ˆ

H0

)

P

(

H

ˆ

0

|

H0

) ,

(7)

andP(Dˆ| ˆH0)=P(Hˆ0| ˆD)P(Dˆ)/P(Hˆ0),whereaflatpriorwillbeas- sumedforP(Hˆ0).Eqs. (6) and(7) usethat P(Dˆ|H0,D)=P(Dˆ|H0) andP(Dˆ| ˆH0,H0)=P(Dˆ| ˆH0)sincethemeasurement Dˆ isindepen- dentofD givenH0 andindependentofH0 givenHˆ0,respectively.

Note that P(Dˆ|H0) and P(Dˆ|D) can only be determined up to thefactor P(Dˆ).From P(Dˆ|H0)andEq. (7), onecancompute the probability P(H0| ˆD)fora cosmologicalHubbleconstant H0 given

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4 L. Lombriser / Physics Letters B 803 (2020) 135303

thelocalmeasurementup toa factor P(H0)1,alsoassumedflat here. P(H0| ˆD)normalisedatitspeakisshowninFig.1,providing alongsideacomparisontothePlanckmeasurement,foundhereto be consistent withexpectationsatthe 95% confidencelevel. Itis worth notingthat the impact ofenvironment suggeststhe inter- pretationofP(H0| ˆD)astheactuallocalmeasurementoftheHub- bleconstantofthecosmologicalbackgroundwithmeanandstan- darddeviation H0=76.5.5,ormoreaccurately H0=74.7+54..82 in termsof the median and68% confidence bounds. Finally, one can assessto what extent a hypotheticallocal measurement Dˆeq with the same uncertainties of Dˆ but equal mean to that mea- suredbyPlanck, Hˆ0=H0,wouldbemorelikelythantherealdata

ˆ

DobtainedinRef. [6].ThiscanbeestimatedfromtheBayesfactor B

ˆ

eq

=

P

(

D

ˆ |

Deq

)

P

(

D

ˆ |

D

)

(8)

withtheunknownevidence P(Dˆ)cancellingout.TheBayesfactor ofthishypotheticalscenariototherealmeasurement isgivenby Bˆeq=4,whichwhilestatisticallypreferringsuchascenariowould not amount to strong evidence on the Jeffreys scale for equality overtheobservedcase.

TheseconsiderationssuggestthatthemeasurementoftheHub- bleconstant inthe local40 Mpcenvironment (Hˆ0) ismarginally consistentwiththatinferredfromtheCMBbyPlanck (H0).

7. Conclusions

MeasurementsofthecurrentexpansionrateofourCosmosre- vealasignificanttensionbetweentherateinferred fromtheCMB by Planck and that found from SN Ia separations employing ab- solutedistanceanchors fromnearbyCepheids,masers,parallaxes, andecliptic binaries. It was argued here that the localmeasure- ment of the Hubble constant is likely impacted by a 40 Mpc underdense local region of δenv≈ −0.0.1 that contains the Cepheids/SN Iacalibration sample,andhence inthiscontext,the Hubbletensionmaybeinterpreted asa5

σ

measurementofthat.

This isdue to the absolutedistance ruler beingprovided by the localuniverse,whichisnotapplicabletocosmologicalscaleswith- outappropriatetransformation.Consistenttoexpectations,theuse ofa cosmologicaldistanceruler insteadyields a measurementof Hˆ0 that is in agreement with H0 [12]. The environmental un- certaintiesalso suggest thereinterpretation asa measurement of thecosmologicalHubbleconstantofH0=74.7+54..82km/s/Mpc.The probabilityforfindingthegivenlocalexpansionrateforthegiven cosmologicalrateorPlanck dataliesat the95% confidence level.

Moreover,a hypotheticaldatasetwithlocalexpansionrateequal tothatofPlanck,whilestatisticallyfavoured,wouldnotfindstrong preferenceintheBayesfactorovertheactualmeasurement.These resultsthereforesuggest borderlineconsistencybetweenthelocal andCMBmeasurementsoftheHubbleconstant.

The simplified one-dimensional Gaussian treatment of the lo- cal and CMB data is a caveat to these results. But importantly, the analysis has also been conservative with the adoption of a 40 Mpc environment, as the absolute distance anchors reside in a10 Mpc region,wherematter densityfluctuationsmaybeeven morepronounced.Duetotheinvolvedfittingprocedureinthecal- ibrationoflocalmagnitudes,therelevantscaleoftheenvironment was set to that containing all of the Cepheid/SN Ia samples. A morelocal environmentset by theimmediatevolume ofthe an- chors would further increase the likelihood of encountering the measureddiscrepancybetween H0 andHˆ0 (also seeRefs. [19,20]

for perturbative contributions). Accounting for this effect would requireajointanalysisoftheCepheidmagnitudesinvarying en- vironments,whichisbeyondthescopeofthecurrentanalysisand

leftforfuturework.SuchastudymayalsoincludetheLIGO/Virgo gravitationalwavemeasurement [9] ofHˆ0 fromNGC 4993,alsore- siding inthe ∼40 Mpcenvironment.Repeatedgravitationalwave measurementsinthatvolumewouldbeexpectedtoagreewiththe localexpansionrateratherthanwiththatinferredfromtheCMB.

The jointanalysiscould furthermoreincludethe measurementof the Hubble constant by H0LiCOW quasar lensing [8]. An impor- tant pointworth notinginthiscontextis thatthetime delaysof theimagesaremeasuredwithalocalclockinthereferenceframe setbythelocalunderdensity, andthechangetothecosmicrefer- enceframeyieldsacosmicHubbleparameterconsistentwiththat inferred fromthe CMB (App. A.4). Finally, note that a consistent local underdensity has independently been confirmed by CLAS- SIX X-raygalaxy cluster distributions, findingδenv,0(40 Mpc)=

0.0.2 [21].ThelocalUniverseisalsofoundtobeconsistently underdense bothat40 Mpc and10 Mpc fromrecentall-sky cataloguesofgalaxygroupswitharangefor−δenv,0(40 Mpc)of 0.56–0.71 [22].

TheauthorthanksRuthDurrerandSteenHansenforusefuldis- cussions andPierreFleury,AlexHall,DraganHuterer,D’ArcyKen- worthy, andAdamRiess forusefulcommentson themanuscript.

Thiswork was supportedby aSwissNationalScienceFoundation Professorshipgrant(No. 170547).Pleasecontacttheauthorforac- cesstoresearchmaterials.

Appendix A. Derivations

Thefollowingdiscussionprovidesderivationsforthearguments presented around Eq. (1). Forsimplicity, z1 will be assumed throughout. Deviations of this approximation only enter through thedependenceofthemagnitude-redshiftrelationonthecosmo- logical matter densityparameterfor largeredshifts ofthe high-z SN Iasample.Computationscanthereforeeasilybegeneralisedto accountforthesecorrections.

A.1. Luminositydistanceinalocalinhomogeneity

The distance travelled by a light ray emitted by a source at redshift z before observation can be illustrated by Fig. 2. For zz40z(40 Mpc),itisgivenbydˆL= ˆH01z,whereasforz>z40, dL=H01z+ dwith

d

=

H01

H0

H

ˆ

0

1

z40

.

(A.1)

At large distances, dL d, the distance correction can be ne- glected.Forthedistancemodulus,however,itimpliesthatwhereas atlowredshifts

m

M

= ˜ μ

low

=

5 logd

ˆ

L d

ˆ

1

+

25

=

5 log z

z1

+

25 (A.2)

withdˆ11 Mpc,athighredshifts,oneinsteadobtains

˜

μ

high

=

5 logdd

ˆ

1L

+

25

=

5 log z

z1

+

25

+

M

,

(A.3)

whereacorrectionoftheabsolutemagnitudewasdefinedas M

=

5 log

H

ˆ

0

H0

+

H

ˆ

0 d z

=

5 log

H

ˆ

0

H0

+

1

H

ˆ

0 H0

z40

z

.

(A.4)

Thus, whereas at z40, M=0, for zz40, one obtains a shift of M=5log(Hˆ0/H0).This shiftcan beavoided ifchanging the measurementofdistancesfromthelocalreferenceframeofdˆ1 to thecosmologicaloneofd1.

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Fig. 2.Schematicdistance-redshiftrelationforauniformlocalinhomogeneityina homogeneouscosmologicalbackground.ThedistancescalesasdˆL= ˆH01zinthe interioranddL=H01z+ dintheexteriorwithadistancecorrection dsetat theedgeoftheinhomogeneity.

A.2.Referenceframes

ThedifferencebetweentheabsolutedistancescalesdˆL anddL canbe phrased interms oftwo conformally relatedmetrics, one withascalefactora thathasundergoneanevolutionwithanav- eragecosmologicalmatterdensityandonethathasbeengoverned bythelocalmatterdensity,a.ˆ Let

gμν

=

a2

η

μν (A.5)

denotetheflatFLRWmetricoftheaverageuniverse,where

η

μν is theMinkowskimetric.TheFLRWmetricofthelocaluniverseshall insteadbegivenby

ˆ

gμν

= ˆ

a2

η

μν

,

(A.6)

whereaˆ=a foralocalmatterdensitythat deviatesfromitscos- mologicalaverage.Ametricapplyingtobothregimesmayfurther- morebedefinedas

˜

gμν

= ˜

a2

(

x

, η ) η

μν

,

(A.7)

wherea˜=a for|xx0|>R and|xx0|≤R withR definingthe sizeoftheinhomogeneity.

Thetwo homogeneous metrics canbe related by a conformal factorC as

ˆ

gμν

=

a

ˆ

2

a2gμν

C2gμν

.

(A.8)

Foranemissionatthecosmologicaltimetem ing,onecanwrite

aem

a0

=

tem

t0

dta

˙ =

tem

t0

dt aH

a0H0

(

tem

t0

) .

(A.9)

Hence,aH0t.Similarly,aˆ≈ ˆH0t forHˆ suchthatC≈ ˆH0/H0. Importantly,observablequantitiessuchasapparentmagnitudes m and redshifts z remain invariant under the conformal trans- formation.Absolute distancerulers,incontrast,arenotpreserved underthetransformation,andasarule ofthumb,oneshould be carefulwithanyquantitythatcarriesunits.

Inspecific,luminositydistancesdL rescaleas

d

ˆ

L

=

C1dL

=

HH

ˆ

00dL

.

(A.10) Thisrescalingcanalsobeinferreddirectlyfrom

dL

= (

1

+

z

)

z

0

dz H

(

z

)

z

H0

(A.11)

anddˆL≈ ˆH01z suchthat

d

ˆ

L

=

HH

ˆ

00dL

.

(A.12)

Itisimportanttonotethatwhiledistancemoduliareinvariant, ˆ

μ

=

μ

, the adopted normalisation of distance is not (App. A.1).

Considerthedistancemodulusin g,whichisgivenby m

M

= μ =

5 logdL

d1

+

25

,

(A.13)

whered1=dL(z1)H01z1=1 Mpc definestheinvariant redshift thatcorresponds to1 Mpcinthecosmologicalframe.Incontrast, ing,ˆ forthesameredshiftonefinds

ˆ

μ =

5 logd

ˆ

L d

ˆ

1

+

25

,

(A.14)

wheredˆ1= ˆdL(z1)≈ ˆH01z1=H0d1/Hˆ0.Hence, thedistance nor- malisationisnolonger1 Mpc.Alternatively,onemaydefineanew scale such that 1 Mpc corresponds to z1, or one may choose to absorb the change ofnormalisation into a shift of the reference magnitude M by M=5log(Hˆ0/H0) as in Eq. (A.4). These is- sues are avoided in the distance modulus as long as luminosity distancesaregiveninthesamereferenceframeasthenormalisa- tionemployed.Itisworthnotingatthispointthatdistancesinthe localuniversearemeasuredinamegaparsecscaledefinedbydˆ1. A.3. MeasurementoftheHubbleconstant

The high-z SN Iaprovide afit tothe magnitude-redshift rela- tion [2]

m0x

=

5 logz

5ax

,

(A.15)

whereas the low-z SN Iaareused to calibrate aconversion ofm intoabsolutedistancedˆ usingthemeasurementofdˆN4258withits calibratedmN4258.Inthecontextoftheconformalrelationbetween the local and cosmologicalmetrics, m and z of the high-z sam- ples canthereforebe usedto attributea redshifttoa distanced,ˆ from whichthe local Hubble constant Hˆ0=z/ˆd can be inferred.

Inversely, dˆ canbe extrapolatedto thehigh-z regime, butd and thereforeH0 remainundeterminedaslongasthereisnoabsolute distancemeasurementprovidedinthisreferenceframe(seeFig.2).

To perform the calculation more carefully, the magnitude- redshiftrelationshallfirstbefollowedfromlargeredshiftsto the edgeofthelocalinhomogeneity atz40,whereitismatchedwith themagnitude-distance calibrationfromthelow-z samples.More specifically,

ax

=

logz40

1

5m0x,40

=

logH

ˆ

0d

ˆ

40

1 5m0x,40

=

logH

ˆ

0d

ˆ

40

1

5

( μ ˆ

0,40

− ˆ μ

0,N4258

+

m0x,N4258

)

=

logH

ˆ

0

5

1

5

(

m0x,N4258

− ˆ μ

0,N4258

) +

logd

ˆ

1

.

(A.16)

(7)

6 L. Lombriser / Physics Letters B 803 (2020) 135303

Hence,oneobtainsEq. (1), logH

ˆ

0

=

1

5

(

m0x,N4258

− ˆ μ

0,N4258

) +

ax

+

5

logd

ˆ

1

.

(A.17) Notethatthedistancenormalisationdˆ1 iskepthereexplicit.This allowsforaframe-invariantexpressionofthemeasurementofthe Hubbleconstant.Atransformationintothecosmologicalreference framegives

logH0

=

1

5

(

m0x,N4258

μ

0,N4258

) +

ax

+

5

logd1

.

(A.18) While the expression is frame invariant, the Hubble constant is a frame-dependent quantity. The distance anchors are obtained fromthelocal referenceframe,andthe Hubbleconstant inferred istherefore thelocalone. A measurementof H0 isonlypossible byagivenabsolutemeasurementofthecosmologicalruler,orin- versely,atransformationtothecosmologicaldistancescaleisonly possiblebyadditionalknowledgeofthecosmologicalHubblecon- stant H0.However, neitherof thosequantities are available from the dataemployed. More explicitly, to cancellogd1 inEq. (A.18) one needs to adopt the distance normalisation d1 in

μ

0,N4258. However, since we measure dˆN4258, this leads to the conversion

logdN4258= −log(Hˆ0/H0)logdˆN4258,whichchangesthemea- surementinEq. (A.18) backintooneoflogHˆ0.

Importantly, the local environment was chosen as a uniform matter density perturbation with respect to the cosmological average density of the 40 Mpc region that contains all joint SN Ia/Cepheidsamplesanddistanceanchors.Thisisaconservative choice that was adoptedto avoidany contamination ofchanging reference frames in the calibration of the SN Ia magnitudes ex- pectedatthelocationofthedistanceanchors,whichusesthejoint SN Ia/Cepheiddata.Inthissetting,distancesintheMilkyWay or totheLargeMagellanicCloudaremeasuredinthesamelocalref- erence frame defined by Eq. (A.6) with constant aˆ= ˆa0. Due to thedifferentmatter densities aˆ hasevolveddifferentlyto a from earlytimestiwhentheuniversewasmorehomogeneous(aˆiai).

Hence,such distancesalso needtobe transformedforuseinthe cosmological reference frame. Neglecting possible contamination in the calibration of SN Ia magnitudes, absolute distance scales maydirectlybesetbythedistanceanchors.Realistically,theirre- spectivenearbyexpandingenvironmentswilldeviatefromthat of the local40 Mpcregion. The environment averaged over a more nearbydistancemaynotablybeexpectedtodepartmorestrongly from the cosmologicalaverage, which further improves the con- sistencybetweenthelocalandcosmologicalmeasurements ofthe Hubbleconstant.Importantly, theexpandingenvironments ofthe differentdistanceanchors are stronglycorrelated duetooverlap- ping volumes, andone may henceexpect only small differences betweentherespectivereferenceframes.

Finally,theseresultsareindependentofperforming thecalcu- lationsintheseparatedhomogeneous frames. Thesamerelations areobtainedifadoptingtheinhomogeneousmetric˜g.Morespecif- ically,forthehigh-zsamplesonefinds(Sec.A.1)

˜

μ

0

− ˜ μ

0,40

=

m0x

m0x,40

+

M 5 log dL

d40

=

5 log z

z40

+

M

.

(A.19)

Thisimpliesthat 1

5

(

m0x,N4258

− ˆ μ

0,N4258

) =

1

5

(

m0x,40

− ˜ μ

0,40

)

=

1

5

(

m0x

− ˜ μ

0

+

M

)

=

1

5

(

m0x

+

M

)

5

logdL

+

logd

ˆ

1

=

logz

ax

+

1

5 M

5

log

(

H01z

+

d

) +

logd

ˆ

1

=

logH

ˆ

0

ax

5

+

logd

ˆ

1

,

(A.20) wherethelastequalityfollowsfromEq. (A.4).Hence,onerecovers Eq. (A.17).

A.4. CommentonH0LiCOWmeasurement

Measuring the time delay between two images of gravita- tionally lensed quasars, H0LiCOW [8] reported a 2.4% precision measurement of a highvalue of the Hubble constant consistent with that from the local distance ladder. This delay is speci- fied by ti j=D t[. . .], where the bracket is a function of an- gular positions of the images i and j and the source, D t = (1+zd)DdDs/DdsH01 with angulardiameter distances D and indicesd/sreferringtothelensandsource.

Importantly, this time interval is measured with a local clock inthelocalreferenceframe, ti j= ˆa0

η

i j where

η

istheconfor- maltimeandthenormalisationaˆ0=1 hasimplicitlybeenadopted in the measurement. Thisneeds to be transformed into the cos- micreferenceframeforinferenceofthecosmicHubbleparameter, wheretheconformalfactorchangingthereferenceframecausesa slowdownorspeedupbetweenthelocalandcosmicclocks.More specifically,

η

i j

1

ˆ

a0

a0 a

DdDs Dds

=

aa

ˆ

00

D t

=

C1D t

H0 H

ˆ

0

H01

= ˆ

H01

.

(A.21)

Hence, H0LiCOW measured the local Hˆ0 rather than the cosmic H0.Applying theconformalfactorC forthechangetothecosmic referenceframe,oneobtains

C1 ti j1

|

aˆ01

=

C1

η

i j1

HH

ˆ

00H

ˆ

0

=

H0

.

(A.22)

Withthevalueoftheconformalfactorinferredfromthelocaldis- tanceladdertheH0LiCOWmeasurement Hˆ0=73.3+11..78 km/s/Mpc thereforeyieldsacosmic H0 consistentwiththatofPlanck.

References

[1]Planck,N.Aghanim,etal.,arXiv:1807.06209.

[2]A.G.Riess,etal.,Astrophys.J.699(2009)539.

[3]A.G.Riess,etal.,Astrophys.J.730(2011)119,Astrophys.J.732(2011)129, Erratum.

[4]G.Efstathiou,Mon.Not.R.Astron.Soc.440(2014)1138.

[5]A.G.Riess,etal.,Astrophys.J.826(2016)56.

[6]A.G.Riess, S. Casertano, W.Yuan, L.M.Macri, D. Scolnic, Astrophys. J. 876 (2019)85.

[7]W.L.Freedman,Nat.Astron.1(2017)0121.

[8]K.C.Wong,etal.,arXiv:1907.04869.

[9]LIGOScientific,Virgo,1M2H,DarkEnergyCameraGW-E,DES,DLT40,LasCum- bresObservatory,VINROUGE,MASTER,B.P.Abbott,etal.,Nature551(2017)85.

[10]W.D.Kenworthy,D.Scolnic,A.Riess,Astrophys.J.875(2019)145.

[11]H.-Y.Wu,D.Huterer,Mon.Not.R.Astron.Soc.471(2017)4946.

[12]DES,E.Macaulay,etal.,Mon.Not.R.Astron.Soc.486(2019)2184.

[13]J.E.Gunn,J.R.GottIII,Astrophys.J.176(1972)1.

[14]L.Dai,E.Pajer,F.Schmidt,J.Cosmol.Astropart.Phys.1510(2015)059.

[15]P.J.E.Peebles,TheLarge-ScaleStructureoftheUniverse,PrincetonUniversity Press,1980.

[16]T.Y.Lam,R.K.Sheth,Mon.Not.R.Astron.Soc.386(2008)407.

[17]R.Wojtak,etal.,Mon.Not.R.Astron.Soc.438(2014)1805.

[18]E.L.Turner,R.Cen,J.P.Ostriker,Astron.J.103(1992)1427.

[19]I.Ben-Dayan,R.Durrer,G.Marozzi,D.J.Schwarz,Phys.Rev.Lett.112(2014) 221301.

[20]J.Yoo,E.Mitsou,N.Grimm,R.Durrer,A.Refregier,J.Cosmol.Astropart.Phys.

1912(2019)015.

[21]H.Boehringer,G.Chon,C.A.Collins,Astron.Astrophys.633(2020)A19.

[22]I.D.Karachentsev,K.N.Telikova,Astron.Nachr.339(2018)615.

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