Cosmological Geometrodynamics: from Planck to Modified Gravity

Research Center for Astronomy and Applied Mathematics of the Academy of Athens

Spyros Basilakos CosmologicalCosmological Geometrodynamics:Geometrodynamics: fromfrom PlanckPlanck toto modifiedmodified gravitygravity

Naples January 2014 THE OUTLINE OF THE PRESENT TALK

IntroductionIntroduction ObservationalObservational evidenceevidence ofof thethe cosmiccosmic accelerationacceleration RecentRecent PlanckPlanck resultsresults DarkDark energyenergy asas aa modificationmodification ofof gravitygravity TheThe modifiedmodified gravitygravity modelsmodels CosmologicalCosmological ImplicationsImplications Summary-ConclusionsSummary-Conclusions THE CURRENT COSMOLOGICAL STATUS

ΩDE + ΩDM + ΩBAR =1

~4% luminous matter (baryons) ~70% Dark ~26% Cold energy Dark matter

WeWe livelive inin aa veryvery excitingexciting periodperiod forfor thethe advancementadvancement ofof ourour knowledgeknowledge forfor thethe Cosmos.Cosmos.

• Current observational data strongly support a flat and accelerating Universe with

Ho~67-73Km/sec/Mpc and To~14 Gyr. • The mystery of dark energy poses a challenge of such magnitude that, as stated by the Dark Energy Task Force (DETF –advising DOE, NASA and NSF), ‘’Nothing short of a revolution in our understanding of fundamental physics will be required to achieve a full understanding of the cosmic acceleration’’ (Albrecht et al. 2006). COSMOLOGICAL OBSERVATIONS & SPACETIME S. Perlmutter, A. Reiss & B. Schmidt: Nobel Prize 2011

HubbleHubble diagramdiagram ((SNIa)SNIa)  ΩΩm ++ ΩΩΛ

TemperatureTemperature fluctuationsfluctuations ofof thethe CMBCMB  spatialspatial geometrygeometry ΩΩκκ

Large-ScaleLarge-Scale StructureStructure  ΩΩm (independent(independent fromfrom darkdark energy)energy) ExtragalacticExtragalactic sourcessources (galaxies,(galaxies, AGNs,AGNs, GRBs,GRBs, HII,HII, LBGsLBGs )) atat largelarge redshiftsredshifts  ΩΩm ++ ΩΩΛ

GrowthGrowth datadata ++ gravitationalgravitational lensinglensing toto checkcheck gravitygravity onon cosmologicalcosmological scalesscales

The Friedmann-Lemaitre-Robertosn-Walker (FRLW) spacetime

Assuming that the Universe is homogeneous and isotropic (Cosmological principle) dr 2 ds2=−c2 dt 2 +α2 t  +r 2 dϑ2sin 2 θdϕ2  [ 1− Kr2 ] Spatial part of the metric

The scale factor Time part a˙ t  The expansion H t ≡ rate of the at  Universe, Hubble function

The analysis of the CMB data (Planck: Ade et al. 2013; Spergel et al. 2013) points a spatially flat (K=0) FRW metric.

CMB temperature anisotropies provide a standard ruler. They were produced about 400,000 θ -1/2 Η 2 [ peak(deg)] =1+K/ 0 R0 years after the Big Bang,and should be most prominent at a physical Observation: θ = 1o size of 400,000 light years across. peak l ≈220 The universe is spatially flat (K=0) peak to an accuracy of better than a percent:

b positively m curved f l a t Evidence of cosmic acceleration

Observationally assuming a matter dominated Universe we have: 2 a˙ 8πG H 2≡  ρ  a  3 m

or

S. Perlmutter, A. Reiss & B. Change the cosmic Schmidt: Nobel Prize 2011 Schmidt: Nobel Prize 2011 Change Gravity. fluid. We add a GR is not valid at ‘’dark energy’’ Cosmological scales. Modify the ‘’law of gravity’’ HUBBLE DATA – EXPANSION RATE ‘’‘’DarkDark Energy’’Energy’’ introducesintroduces aa NewNew PhysicsPhysics

Artistic view of a universe filled by a turbulent sea of dark energy R. Caldwell (2005) THE DYNAMICS OF THE UNIVERSE

Friedmann equations:

In the matter dominated era and for spatially flat models we get:

EquationEquation ofof StateState parameterparameter

DE regime w>-1 Λ-vacuum w=-1 Modified Gravity or DM DE interactions w<-1 The cosmological parameters

Scale factor – Density Parameters :

 mQr  =1

Note that for w=-1 or X(α)=1 we get the usual Λ cosmology. Performing a Cosmo-Statistics we can but constraints on the cosmological parameters Planck results Ade et al. 2013 Planck Results: Ade al al. (2013-arXiv:1303.5080 )

H 0=67.3±1.2 Km/ s/ Mpc m=0.315±0.017

2 σ 8=0.829±0.012 b h =0.02207±0.00033

Y =0.2477±0.00012 p z dec=1090. 43±0.54

z eq=3386±69 N eff =3.30±0.27

N= 50−60 ns=0 .9603±0.0073

P If we include dark energy then w= de =−1.09±0.17 Planck+WMAP+SNIa ρde Planck versus Inflation DarkDark EnergyEnergy asas aa modificationmodification ofof gravity?gravity?

AnAn alternativealternative approachapproach toto cosmiccosmic accelerationacceleration isis toto changechange gravitygravity ‘’geometrical‘’geometrical darkdark energy’’energy’’

HereHere wewe alteralter generalgeneral relativityrelativity byby modifyingmodifying thethe usualusual Einstein’sEinstein’s fieldfield equationsequations

The alternative-gravity theories have to satisfy the following criteria: • For systems such as the solar system or galaxies, the alternative gravity must be very close to general relativity

• At cosmological scales we must have gMR

local acceleration of bodies of different composition Eot-Wash: Baessler et al, PRL 83 (1999) 3585

Lunar laser ranging Williams et al, PRL 93 (2004) 261101

Nordtvedt effect: observations of the acceleration of bodies with different gravitational binding energies tests the Strong Equivalence Principle

Mass definitions: TheThe mainmain stepssteps are:are:

1.1. DefineDefine thethe modifiedmodified Einstein-HilbertEinstein-Hilbert actionaction (S)(S) inin whichwhich wewe includeinclude allall thethe ingredientsingredients (modified(modified gravity,gravity, scalarscalar fields,fields, mattermatter etc).etc).

2.2. VaryingVarying thethe actionaction ((δδS=0)S=0) inin orderorder toto obtainobtain thethe modifiedmodified Einstein’sEinstein’s fieldfield equationsequations asas wellwell asas thethe KleinKlein GordonGordon equationequation (if(if aa scalarscalar fieldfield isis present).present).

3.3. UsingUsing thethe FRWFRW metricmetric wewe derivederive thethe soso calledcalled FriedmannFriedmann equationsequations (equations(equations ofof motion)motion) whichwhich describedescribe thethe cosmiccosmic dynamicsdynamics ofof thethe Universe.Universe. SCALARSCALAR TENSORTENSOR THEORIESTHEORIES  These are probably the simplest example of modified gravity models and as such one of the most intensely studied alternatives to General Relativity. The general Einstein-Hilbert action is

1 1 1 S= d 4 x−g f φ,R− ς φ  ∇ φ 2 +S +S 8πG ∫ [ 2 2 ] m r

Is a general Is a function of Are the matter- function of the the scalar field radiation actions that scalar field and the (inflaton, depend on the metric Ricci scalar dilaton string and matter fields theory etc) For more details see Fujiii & Maeda 2003; Amendola & Tsujikawa 2010; Capozziello & de Laurentis 2011 (and references therein) The above general action includes a wide variety of theories. Indeed some of these are: GENERALGENERAL RELATIVITYRELATIVITY  GR is a particular case of the scalar tensor theories. f φ,R =R−2Λ Varying the action we have the Einstein’s equations ς φ =0

Λ is the Einstein’s cosmological constant 1 G ≡R − Rg +Λg =8πG T μν μν 2 μν μν N μν

μ The energy momentum T ν =diag− ρ,P,P,P  tensor ρ=ρm+ρr P=Pm+Pr Λ ρΛ= =ρ Λ0 8πG N

The Bianchi identities ∇ G μ =∇ T μ=0 insure the covariance of μ ν μ ν the theory For the spatially flat FRW metric we get the Friedmann equations of motion

8πG From (00)- 2 N H =  ρm+ρr +ρΛ  components 3 2 From (ii)- 3H 2 H˙ =−8πG N  Pm +Pr− ρΛ  components ρ˙ 3H ρ +P =0 Bianchi identities m m m assuming no = −3 P m 0 ρ =ρ a interactions For m m0 dust / For radiation P r =ρr 3 −4 ρr =ρr0 a

ρ˙ r 3H ρr +Pr =0  =8πG ρ /3H 2 2 m N m 0 2 H a −3 −4 E  a= 2 =m a r a Λ 2 H 0 Λ=8πG N ρΛ/3H0 2 This is the Λ Cosmology r =8πGN ρr /3H 0 TheThe scalarscalar fieldfield darkdark energyenergy modelsmodels These dark energy models adhere to General Relativity. The corresponding Einstein-Hilbert action includes: 1 1 1 S= d 4 x−g f φ,R− ς φ  ∇ φ 2 +S +S 8πG ∫ [ 2 2 ] m r

f φ,R =R−2V φ

∇ φ 2=φ˙ 2t  For a homogeneous scalar field 1 ζ φ˙ 2−V φ  2 The dark energy w= equation of state 1 ζ φ˙ 2 +V φ  parameter 2

The scalar field dark includes a V φ  wide variety of theories ζ φ  depending on I)I) f(R)f(R) gravitygravity modelsmodels

f φ,R =f  R  ς φ =0

VaryingVarying thethe actionaction wewe derivederive thethe modifiedmodified EinsteinEinstein fieldfield equations.equations. InIn thethe contextcontext ofof FRWFRW geometrygeometry (spatially(spatially flat)flat) thethe FriedmannFriedmann equationsequations become:become:

f R− f 3f H 2− R 3Hf R˙ =16 πG ρ +ρ R 2 RR  m r 

˙ ¨ ˙ −2f R H =16πG  ρm4ρr /3  f R−H f R

μν 2 R=g Rμν=62H H˙ 

Notice that the concordance Λ-cosmology can be found for f  R =R−2Λ TheThe effectiveeffective Newton’sNewton’s constantconstant dependsdepends alsoalso fromfrom scale.scale.

2 2 Geff a,k  1 14 k f RR /a f R = G f 2 2 N R 13 k f RR/ a f R

k=1/ λ≈0.1hMpc−1

Obviously the concordance Λ cosmology admits General Relativity f  R =R−2Λ

Geff a,k =G N

For more details see: Sotiriou & Faraoni 2008; Amendola & Tsujikawa 2010; Capozziello & de Laurentis 2011 (and references therein) InIn thethe literatureliterature wewe usuallyusually useuse eithereither thethe HuHu && SawickiSawicki model:model:

n c  R /m2  f  R =R−m2 1 2 n 1+c2  R/m 

oror thethe StarobinskyStarobinsky model:model:

1 f  R =R−c m2 1− 1 [ 1+R2 /m4 n ]

In Basilakos, Nesseris & Perivolaropoulos (Phys. Rev. D. 2013) we prove that the f(R) models can be written as perturbations around the Λ cosmology II)II) GeneralizedGeneralized Brans-DickeBrans-Dicke TheoriesTheories

f (φ, R) = F(φ)R −2U (φ) ς(φ) =(1−6Q2 )F(φ)

F Q ≡− ,φ F

VaryingVarying thethe actionaction (based(based onon aa spatiallyspatially flatflat FRW)FRW) wewe havehave

3FH 2 1 = (1−6Q2 )Fφ˙ 2 +U −3HF˙ +ρ +ρ 8πG 2 m r

− 2 [φ˙˙ + φ˙ + ˙ φ˙] + + = (1 6Q )F 3H (F / 2F) U,φ QFR 0 TheThe effectiveeffective Newton’sNewton’s constantconstant becomesbecomes

Geff a 1 42ωBD = 1 32ω BD= 2 G N F 0 32ωBD 2Q

Using the Solar System constrain namely at very small scales the modified gravity tends to General Relativity we have

4 −3 ω BD4.3×10 ∣Q∣2. 4×10

For more details see: Nojiri et al. 2005; Damour & Nordvedt 1993; Fujii & Maeda 2003; Nesseris & Perivolaropoulos 2006; Amendola & Tsujikawa 2010 (and references therein) III)III) Gauss-BonnetGauss-Bonnet gravitygravity

2 f φ,R =R−2V φ−2gφ  RGB ς φ =1

2 2 μν μναβ Gauss-BonnetGauss-Bonnet termterm RGB=R −4R μν R +Rμναβ R

VaryingVarying thethe actionaction (based(based onon aa spatiallyspatially flatflat FRW)FRW) wewe havehave

3H 2 1 = φ˙ 2 +V φ 24 g H 3 φ˙ +ρ +ρ 8πG 2 ,φ m r

2 2 φ¨ 3Hφ+V ,φ24 g ,φ H  H  H˙ =0 λφ AA particularparticular formform isis usedused g φ = g0 / λe

For more details see: Nojiri et al. 2005; Carter & Neupane 2006; Amendola et al. 2006; Amendola & Tsujikawa 2010; Capozziello et al. 2013 (and references therein) IV)IV) TheThe braneworldbraneworld -- Dvali,Dvali, GabadadzeGabadadze && PoratiPorati (DGP)(DGP) gravitygravity 1 5  1 4 5 S= ∫ d X −g R ∫ d x −g R−∫ d X −g  Lm Lr  16 πG eff 16 πGN

g AB g μν Is the usual 4D metric which is Is the metric in embedded into the 5D brane. the 5D brane This space is maximally symmetric with a constant curvature K(=0, flat). 2 2 2 2 i j 2 ds =−n τ,ydτ +α τ,yγ ij dx dx +dy

 In the context of a DGP cosmological model the ‘’accelerated’’ expansion of the universe can be explained by a modification of the gravitational interaction in which gravity becomes weak at cosmological scales owing to the fact that our 4D brane survives into an extra dimensional manifold. TheThe 5D5D EinsteinEinstein equationsequations areare givengiven byby

1 G ≡ R − R g =8πG T U  AB AB 2 AB eff AB AB

A brane  T B =δ y diag − ρm ,P m ,Pm ,Pm ,0

AlsoAlso thethe componentscomponents fromfrom thethe scalarscalar curvaturecurvature ofof thethe branebrane areare

3 a˙ 2 n2 U =− +K δ  y 00 8πG  a2 a2  N

1 a2 a2 a n a U =− − ˙ 2 ˙ ˙ −2 ¨ −K γ δ  y ij 2 2 ij 8πG N [ n  a a n a  ] TheThe FriedmannFriedmann equationequation (for(for flatflat K=0)K=0) becomesbecomes

2 2 H a −3 −3 E  a= 2 =m a 2bw2 bw m a bw H 0

2 1− 2 Geff a 24ma  = m = bw 4 G 2 N 33ma

 a−3   a= m m E 2 a V)V) FinslerFinsler –– RandersRanders GeometryGeometry The Finslerian geomtery extends the usual Riemannian geometry. Note that the Riemannian geometry is also a Finslerian. Generally a Finsler space is derived from a generating function F(x,y) on a tangent bundle on a manifold M (Randers 1941, Goenner & Bogoslovsky 1999; Stavrinos & Diakogiannis 2004; Kouretsis et al. 2009; Skakala & Visser 2011; Vacaru 2012 F :TM  ℜ Particular attention has been paid on the so called Finsler- Randers cosmological model. μ σ  x,y = g y μ yν F  x,y =σ x,y+u μ  x  y  μν u =u ,0,0,0 u =2C μ 0 0 000 g μν =Fg μν /σ

Is a weak Cartan primordial vector tensor field 1 R − R g =8πG T μν 2 μν N μν TheThe FriedmannFriedmann equationequation (for(for flatflat K=0)K=0) becomesbecomes 2 2 H a −3 −3 E  a= 2 =m a 2Z2z ma z H 0 2 u˙ 2  = 0 1−  z 2  = m = 4H0 z 4 bw

In Basilakos & Stavrinos (2013) we prove that the Finsler-Randers model is cosmologically equivalent with that of the DGP gravity, despite the fact that the two models have a completely different geometrical origin. ConclusionsConclusions -- FutureFuture workwork TestingTesting gravitygravity onon cosmologicalcosmological scalesscales isis oneone ofof thethe mainmain problemsproblems inin cosmologycosmology

WeWe comparecompare thethe aboveabove modelsmodels withwith expansionexpansion datadata inin orderorder toto putput constrainsconstrains onon thethe freefree parametersparameters ofof thethe models.models.

ThenThen wewe havehave toto comparecompare thethe modelsmodels againstagainst growthgrowth datadata inin orderorder toto checkcheck possiblepossible departuresdepartures fromfrom GRGR

HowHow manymany ofof thethe aboveabove geometricalgeometrical modelsmodels cancan provideprovide exactlyexactly thethe samesame HubbleHubble function?function?

OfOf coursecourse therethere areare alsoalso otherother geometricalgeometrical modelsmodels whichwhich explainexplain cosmiccosmic accelerationacceleration suchsuch asas [f(T),[f(T), massivemassive gravitygravity etc]etc]