Modified Gravity

Modified GravityModified Gravity

Modification of Einstein Modification of Einstein equationequation

replareplacece

keep diffeomorphism keep diffeomorphism symmetry !symmetry !

at least unimodular diffeomorphismsat least unimodular diffeomorphisms

Modification of Einstein Modification of Einstein equationequation

modifimodifiededgravitygravity

DarkDarkEnerEnergygy

Split is ambiguous !Split is ambiguous !example : cosmological example : cosmological constantconstant

Quantum effective actionQuantum effective action

gravitational part:gravitational part: functional of metricfunctional of metric

New degrees of freedomNew degrees of freedom

Modifications of gravity involve new Modifications of gravity involve new degrees of freedomdegrees of freedom

not necessarily new fields beyond metricnot necessarily new fields beyond metric

what matters : degrees of freedom – what matters : degrees of freedom – not choice of field variables to describe themnot choice of field variables to describe them

Cosmological scalar Cosmological scalar fieldsfields

Brans- Dicke Brans- Dicke theory :theory :

QuintessenQuintessence :ce :

turns out to be special form of turns out to be special form of scalar modelscalar modelcoupled to mattercoupled to matter

Weyl scalingWeyl scaling

w can depend on w can depend on fields !fields !

Weyl scaling in Brans-Dicke Weyl scaling in Brans-Dicke theorytheory

In this version ( Einstein In this version ( Einstein frame )frame )no modification of gravitational no modification of gravitational action !action !

FramesFrames

Jordan frame Jordan frame : field dependent gravitational : field dependent gravitational constant ( coefficient of curvature scalar )constant ( coefficient of curvature scalar )

Einstein frame Einstein frame : fixed Planck mass M: fixed Planck mass M on level of quantum effective action: on level of quantum effective action:

both frames are equivalent !both frames are equivalent ! simply different “field – coordinates “ for simply different “field – coordinates “ for

solutions of differential equationssolutions of differential equations no measurement can distinguish the two no measurement can distinguish the two

framesframes only dimensionless quantities can be only dimensionless quantities can be

measuredmeasured

Weyl scaling in matter Weyl scaling in matter sectorsector

fermionfermions :s :

constant mass in Jordan constant mass in Jordan frame :frame :field dependent mass in field dependent mass in Einstein frame !Einstein frame !similar for bosonssimilar for bosons

time variation of ratio nucleon mass / time variation of ratio nucleon mass / Planck mass:Planck mass:strict limits !!! strict limits !!!

How to obey constraints How to obey constraints from time variation of from time variation of

particle massesparticle masses field dependent mass in Jordan framefield dependent mass in Jordan frame

mass ~ mass ~ χχ in Jordan frame : in Jordan frame :constant mass in Einstein frame !constant mass in Einstein frame !similar for bosonssimilar for bosons

only tiny variation of scalar fieldonly tiny variation of scalar field only tiny local variation of scalar field only tiny local variation of scalar field

( chameleon mechanism etc. ) ( chameleon mechanism etc. )

Quantum effective action with Quantum effective action with scale symmetry scale symmetry

( dilatation symmetry, “conformal ( dilatation symmetry, “conformal symmetry” )symmetry” )

all mass scales replaced by all mass scales replaced by χχonly dimensionless couplings only dimensionless couplings FujiiFujii, CW, CW

potential for Higgs scalar hpotential for Higgs scalar h

fixed value offixed value of

scalar – tensor theoriesscalar – tensor theories

violation of scale symmetry if V, F or K contain violation of scale symmetry if V, F or K contain parameters with dimension of massparameters with dimension of mass

Weyl scaling of scalar Weyl scaling of scalar potentialpotential

V’ = wV’ = w22 VV

cosmological cosmological constantconstant in Jordan frame in Jordan frame λλcc

λλcc

Model Model

μμ= 2 = 2 10 10-33 -33 eVeV

m~m~μμ

onlyonly scalescale : :

Universe without Expansion

NATURENATURE | NEWS | NEWS

Cosmologist claims Universe may not be Cosmologist claims Universe may not be expandingexpandingParticles' changing masses could Particles' changing masses could explain why distant galaxies appear to explain why distant galaxies appear to be rushing away.be rushing away.

Jon CartwrightJon Cartwright 16 July 201316 July 2013

German physicist German physicist stops Universestops Universe25.07.201325.07.2013

SonntagszeitungSonntagszeitungZuerichZuerichLaukenmannLaukenmann

The Universe is The Universe is shrinkingshrinking

The Universe is The Universe is shrinking …shrinking …

while Planck mass and while Planck mass and particle masses are particle masses are

increasingincreasing

What is increasing ?What is increasing ?

Ratio of distance between galaxies Ratio of distance between galaxies over size of atoms !over size of atoms !

atom size constant : expanding geometryatom size constant : expanding geometry

alternative : shrinking size of atomsalternative : shrinking size of atoms

general idea not new : Hoyle, general idea not new : Hoyle, Narlikar,…Narlikar,…

Simple model ofSimple model of “ Variable Gravity “ Variable Gravity

Universe “Universe “ Scalar field coupled to gravityScalar field coupled to gravity Effective Planck mass depends on scalar Effective Planck mass depends on scalar

fieldfield Simple quadratic scalar potential :Simple quadratic scalar potential : Nucleon and electron mass proportional Nucleon and electron mass proportional

to Planck massto Planck mass Neutrino mass has different dependence Neutrino mass has different dependence

on scalar fieldon scalar field

SimplicitySimplicity

simple description of simple description of allall cosmological cosmological epochsepochs

natural incorporation of Dark Energy :natural incorporation of Dark Energy :inflationinflationEarly Dark EnergyEarly Dark Energypresent Dark Energy dominated epochpresent Dark Energy dominated epoch

Time history of the Time history of the UniverseUniverse

Inflation : Universe expandsInflation : Universe expands Radiation : Universe shrinksRadiation : Universe shrinks Matter : Universe shrinksMatter : Universe shrinks Dark Energy : Universe expandsDark Energy : Universe expands

Compatibility with Compatibility with observationsobservations

Almost same prediction for radiation, Almost same prediction for radiation, matter, and Dark Energy domination as matter, and Dark Energy domination as ΛΛCDMCDM

Inflation with:Inflation with:

n=0.97, r=0.13n=0.97, r=0.13

Presence of small fraction of Early Dark Presence of small fraction of Early Dark EnergyEnergy

Large neutrino lumpsLarge neutrino lumps

Cosmon inflationCosmon inflation

Unified picture of inflation and Unified picture of inflation and dynamical dark energydynamical dark energy

Cosmon and inflaton are the Cosmon and inflaton are the same fieldsame field

QuintessenceQuintessence

Dynamical dark energy ,Dynamical dark energy ,

generated by scalargenerated by scalar field field (cosmon )(cosmon )

C.Wetterich,Nucl.Phys.B302(1988)668, C.Wetterich,Nucl.Phys.B302(1988)668, 24.9.8724.9.87P.J.E.Peebles,B.Ratra,ApJ.Lett.325(1988)L17, P.J.E.Peebles,B.Ratra,ApJ.Lett.325(1988)L17, 20.10.8720.10.87

Prediction :Prediction :

homogeneous dark energy homogeneous dark energyinfluences recent cosmologyinfluences recent cosmology

- of same order as dark - of same order as dark matter -matter -

Original models do not fit the present observationsOriginal models do not fit the present observations……. modifications. modifications

Merits of variable gravity Merits of variable gravity modelmodel

Economical settingEconomical setting No big bang singularityNo big bang singularity Arrow of timeArrow of time Simple initial conditions for inflationSimple initial conditions for inflation

Model Model

μμ= 2 = 2 10 10-33 -33 eVeV

Scalar field Scalar field equationequation::

additional force from R additional force from R counteracts potential counteracts potential

gradient : increasing χ !gradient : increasing χ !

Incoherent contribution Incoherent contribution to scalar field equationto scalar field equation

if particle massif particle massproportional to proportional to χ :χ :

Modified Einstein Modified Einstein equationequation

New term with derivatives of scalar fieldNew term with derivatives of scalar field

Curvature scalar and Curvature scalar and Hubble parameterHubble parameter

Scaling solutionsScaling solutions( for constant K )( for constant K )

Four different scaling Four different scaling solutions forsolutions forinflation, radiation inflation, radiation domination, domination, matter domination andmatter domination andDark Energy dominationDark Energy domination

Scalar dominated epoch, Scalar dominated epoch, inflationinflation

Universe expands for K > -4, shrinks for K < -4.Universe expands for K > -4, shrinks for K < -4.

No big bang singularityNo big bang singularity

Scaling solution is Scaling solution is attractiveattractive

Scaling solution ends Scaling solution ends when K gets closer to -6when K gets closer to -6

Radiation dominationRadiation domination

UniverseUniverseshrinks !shrinks !

Early Dark EnergyEarly Dark Energy

Energy density in radiation Energy density in radiation increases , increases , proportional to cosmon proportional to cosmon potentialpotential

fraction in early dark fraction in early dark energyenergy

requires large requires large α >10α >10

scaling of particle scaling of particle massesmasses

mass of electron or nucleon is proportionalmass of electron or nucleon is proportionalto variable Planck mass to variable Planck mass χχ ! !

effective potential for Higgs doublet heffective potential for Higgs doublet h

cosmon coupling to cosmon coupling to mattermatter

qqχχ=-(ρ-3p)/χ=-(ρ-3p)/χ

Matter dominationMatter domination

Universe Universe shrinks !shrinks !

Neutrino massNeutrino mass

seesaw seesaw andandcascadecascademechanismechanismm

omit generation omit generation structurestructure

triplet expectation value ~ doublet squaredtriplet expectation value ~ doublet squared

Neutrino massNeutrino mass

assume that singlet scale has not yet reached assume that singlet scale has not yet reached scaling limit ~ scaling limit ~ χχ

Dark Energy dominationDark Energy domination

neutrino masses scalesneutrino masses scalesdifferently from electron massdifferently from electron mass

new scaling solution. not yet reached.new scaling solution. not yet reached.at present : transition periodat present : transition period

Why now problemWhy now problem

Why does fraction in Dark Why does fraction in Dark Energy increase Energy increase in present cosmological epoch ,in present cosmological epoch ,and not much earlier or much and not much earlier or much later ?later ? neutrinos becomeneutrinos become

non-relativisticnon-relativisticat z = 5at z = 5

ObservationsObservations

simplest description in Einstein simplest description in Einstein frameframe


Kinetial Kinetial

scalar scalar σσ with withstandard normalizationstandard normalization

conclusionsconclusions

Variable gravity cosmologies can give a Variable gravity cosmologies can give a simple and realistic description of Universesimple and realistic description of Universe

Compatible with tests of equivalence Compatible with tests of equivalence principle and bounds on variation of principle and bounds on variation of fundamental couplings if nucleon and fundamental couplings if nucleon and electron masses are proportional to electron masses are proportional to variable Planck massvariable Planck mass

Different cosmon dependence of neutrino Different cosmon dependence of neutrino mass can explain why Universe makes a mass can explain why Universe makes a transition to Dark Energy domination transition to Dark Energy domination nownow

characteristic signal : neutrino lumpscharacteristic signal : neutrino lumps

f (R) - theoriesf (R) - theories

f (R) – theories , examplef (R) – theories , example

Equivalent scalar modelEquivalent scalar model

solve solve scalarscalarfield field equationequation

insert solution intoinsert solution intoeffective actioneffective action

Equivalent scalar modelEquivalent scalar model


Canonical scalar kinetic Canonical scalar kinetic term term

Expansion for small Expansion for small φφ

c = c = 1 :1 :

scalar scalar mass mass

order Planck mass , unless order Planck mass , unless αα is is huge !!!!huge !!!!

Higher order terms in Higher order terms in effective gravitational effective gravitational

actionaction similar situation if f( R ) admits Taylor similar situation if f( R ) admits Taylor

expansion around R=0expansion around R=0 additional fields with mass close to additional fields with mass close to

Planck mass are not relevant for late Planck mass are not relevant for late cosmology ( but inflation…)cosmology ( but inflation…)

holds also for more complicated holds also for more complicated effective actionseffective actions

Universal coupling to Universal coupling to massive particlesmassive particles

Scalar field is allowed to change Scalar field is allowed to change only by tinyonly by tinyamount on cosmological and local amount on cosmological and local scales !scales !

General f(R) theories as General f(R) theories as scalar modelsscalar models

Non – local gravityNon – local gravity

Effective Nonlocal Euclidean Effective Nonlocal Euclidean GravityGravity C. Wetterich ,C. Wetterich ,

Gen.Rel.Grav. 30 Gen.Rel.Grav. 30 (1998) 159(1998) 159

more general more general modifications of gravitymodifications of gravity

can often be written in form where can often be written in form where effective degrees of freedom are effective degrees of freedom are more easily visiblemore easily visible

massive gravitymassive gravity two- metric theoriestwo- metric theories ““MOND”MOND”


Modified gravity often easier understood Modified gravity often easier understood in terms of additional fieldsin terms of additional fields

No basic distinction between modified No basic distinction between modified gravity and Dark energy ( except gravity and Dark energy ( except massive gravity )massive gravity )

Is the picture of modified gravity useful ?Is the picture of modified gravity useful ? Sometimes , if important features are Sometimes , if important features are

more easily visible ( scale symmetry, more easily visible ( scale symmetry, absence of singularities )absence of singularities )

EnEndd

Cosmon inflationCosmon inflation

Inflation : Slow roll Inflation : Slow roll parametersparameters

End of End of inflationinflationat at εε = 1 = 1

Number of e-foldings Number of e-foldings before end of inflationbefore end of inflation

εε, η, N can all be , η, N can all be computed computed from kinetial alonefrom kinetial alone

Spectral index and Spectral index and tensor to scalar ratiotensor to scalar ratio

Amplitude of density Amplitude of density fluctuationsfluctuations

Properties of density Properties of density fluctuationsfluctuations

conclusion conclusion

cosmon inflation :cosmon inflation : compatible with observationcompatible with observation simplesimpleno big bang singularityno big bang singularitystability of solution singles out arrow stability of solution singles out arrow of timeof timesimple initial conditionssimple initial conditions

Growing neutrino Growing neutrino quintessencequintessence

connection between dark connection between dark energy energy

and neutrino propertiesand neutrino properties

present present equationequationof state given of state given bybyneutrino mass neutrino mass !!

present dark energy density given by neutrino masspresent dark energy density given by neutrino mass

= 1.27= 1.27

Neutrino cosmon Neutrino cosmon couplingcoupling

realized by dependence of neutrino realized by dependence of neutrino massmass on value of cosmon field on value of cosmon field

β ≈ 1 β ≈ 1 : cosmon mediated attractive force : cosmon mediated attractive force between neutrinos has similar strength between neutrinos has similar strength as gravityas gravity

growing neutrinos growing neutrinos change cosmon evolutionchange cosmon evolution

modification of conservation equation for neutrinosmodification of conservation equation for neutrinos

growing neutrino mass growing neutrino mass triggers transition to triggers transition to

almost static dark energyalmost static dark energy

growinggrowingneutrinoneutrinomassmass

L.Amendola, M.Baldi,…L.Amendola, M.Baldi,…

effective cosmological effective cosmological triggertrigger

for stop of cosmon for stop of cosmon evolution :evolution :

neutrinos get non-neutrinos get non-relativisticrelativistic

this has happened recently !this has happened recently ! sets scales for dark energy !sets scales for dark energy !

cosmon evolutioncosmon evolution

scaliscalingng

““stopped”stopped”

neutrino lumpsneutrino lumps

neutrino fluctuationsneutrino fluctuations

neutrino structures become nonlinear neutrino structures become nonlinear at z~1 for supercluster scalesat z~1 for supercluster scales

stable neutrino-cosmon lumps exist stable neutrino-cosmon lumps exist N.Brouzakis , N.Tetradis ,… ; O.Bertolami ; Y.Ayaita , N.Brouzakis , N.Tetradis ,… ; O.Bertolami ; Y.Ayaita , M.Weber,…M.Weber,…

D.Mota , G.Robbers , V.Pettorino , …D.Mota , G.Robbers , V.Pettorino , …

N-body code with fully N-body code with fully relativistic neutrinos and relativistic neutrinos and

backreactionbackreaction

Y.Ayaita,M.WebY.Ayaita,M.Weber,…er,…

one has to resolve local value of one has to resolve local value of cosmon fieldcosmon fieldand then form cosmological and then form cosmological average;average;similar for neutrino density, dark similar for neutrino density, dark matter and matter and gravitational fieldgravitational field

Formation of neutrino Formation of neutrino lumpslumps

Y.Ayaita,M.Weber,…Y.Ayaita,M.Weber,…

φφ - dependent neutrino – - dependent neutrino – cosmon couplingcosmon coupling

neutrino lumps form and are disrupted by neutrino lumps form and are disrupted by oscillations in neutrino massoscillations in neutrino masssmaller backreaction smaller backreaction

oscillating neutrino massoscillating neutrino mass

oscillating neutrino oscillating neutrino lumpslumps

small oscillations in dark small oscillations in dark energyenergy

quantum fluctuations quantum fluctuations and and

dilatation anomalydilatation anomaly

Dilatation anomalyDilatation anomaly

Quantum fluctuations responsible both Quantum fluctuations responsible both forfor

fixed point and dilatation anomaly close fixed point and dilatation anomaly close to fxed pointto fxed point

Running couplings:Running couplings: hypothesishypothesis

Renormalization scale Renormalization scale μμ : ( momentum : ( momentum scale )scale )

λλ~(~(χχ//μμ) ) –A–A

Asymptotic behavior of Asymptotic behavior of effective potentialeffective potential

λλ ~ ( ~ (χχ//μμ) ) –A–A

V ~ (V ~ (χχ//μμ) ) –A –A χχ44

VV ~ ~ χχ 4–A 4–A

crucial : behavior for large χ !crucial : behavior for large χ !

Without dilatation – anomaly :V= const. Massless Goldstone boson = dilaton

Dilatation – anomaly :V (φ )Scalar with tiny time dependent mass : cosmon

Dilatation anomaly and Dilatation anomaly and quantum fluctuationsquantum fluctuations

Computation of running couplings Computation of running couplings ( beta functions ) needs unified theory !( beta functions ) needs unified theory !

Dominant contribution from modes Dominant contribution from modes with momenta ~with momenta ~χχ ! !

No prejudice on “natural value “ of No prejudice on “natural value “ of location of fixed point or anomalous location of fixed point or anomalous dimension should be inferred from tiny dimension should be inferred from tiny contributions at QCD- momentum scale contributions at QCD- momentum scale !!

quantum fluctuations and quantum fluctuations and naturalnessnaturalness

Jordan- and Einstein frame completely Jordan- and Einstein frame completely equivalent on level of effective action equivalent on level of effective action and field equations ( and field equations ( afterafter computation computation of quantum fluctuations ! )of quantum fluctuations ! )

Treatment of quantum fluctuations Treatment of quantum fluctuations depends on frame : Jacobian for depends on frame : Jacobian for variable transformation in functional variable transformation in functional integralintegral

What is natural in one frame may look What is natural in one frame may look unnatural in another frameunnatural in another frame

quantum fluctuations quantum fluctuations and framesand frames

Einstein frame : quantum fluctuations Einstein frame : quantum fluctuations make zero cosmological constant look make zero cosmological constant look unnaturalunnatural

Jordan frame : quantum fluctuations Jordan frame : quantum fluctuations can be the origin of dilatation can be the origin of dilatation anomaly;anomaly;

may be key ingredient for may be key ingredient for solutionsolution of of cosmological constant problem !cosmological constant problem !

fixed points and fluctuation fixed points and fluctuation contributions of individual contributions of individual

componentscomponents

If running couplings influenced by fixed points:If running couplings influenced by fixed points:individual fluctuation contribution can be huge individual fluctuation contribution can be huge

overestimate !overestimate !

here : fixed point at vanishing quartic coupling and here : fixed point at vanishing quartic coupling and anomalous dimension anomalous dimension VV ~ ~ χχ 4–A 4–A

it makes no sense to use naïve scaling argument to it makes no sense to use naïve scaling argument to infer individual contribution infer individual contribution VV ~ h ~ h χχ 4 4


naturalness of cosmological constant naturalness of cosmological constant and cosmon potential should be and cosmon potential should be discussed in the light of dilatation discussed in the light of dilatation symmetry and its anomaliessymmetry and its anomalies

Jordan frameJordan frame higher dimensional settinghigher dimensional setting four dimensional Einstein frame and four dimensional Einstein frame and

naïve estimate of individual naïve estimate of individual contributions can be very misleading !contributions can be very misleading !


cosmic runaway towards fixed point cosmic runaway towards fixed point maymay

solve the cosmological constant solve the cosmological constant problemproblem

andand

account for dynamical Dark Energyaccount for dynamical Dark Energy

Documents

Modified Gravity