Quantum cosmology Quantum cosmology andand

scale invariancescale invariance

quantum gravity with

scalar field –the role of scale

symmetry

fluctuations inducefluctuations inducerunning couplingsrunning couplings

violation of scale symmetryviolation of scale symmetry well known in QCD or standard well known in QCD or standard

modelmodel

functional functional renormalization :renormalization :

flowing actionflowing action

Wikipedia

Quantum scale symmetryQuantum scale symmetry

quantum fluctuations violate scale quantum fluctuations violate scale symmetrysymmetry

running dimensionless couplingsrunning dimensionless couplings at fixed points , scale symmetry is at fixed points , scale symmetry is

exact !exact !

Crossover in quantum Crossover in quantum gravitygravity

Origin of massOrigin of mass UV fixed point : UV fixed point : scale symmetry unbrokenscale symmetry unbroken

all particles are masslessall particles are massless

IR fixed point : IR fixed point :

scale symmetry spontaneously brokenscale symmetry spontaneously broken, ,

massive particles , massless dilatonmassive particles , massless dilaton

crossover : crossover : explicit mass scale explicit mass scale μμ important important

approximate SM fixed point : approximate SM fixed point : approximate scale approximate scale symmetrysymmetry spontaneously brokenspontaneously broken, , massive particles , massive particles , almost massless cosmon, tiny cosmon potentialalmost massless cosmon, tiny cosmon potential

Approximate scale symmetry near fixed points

UV : approximate scale invariance of primordial fluctuation spectrum from inflation

IR : cosmon is pseudo Goldstone boson of cosmon is pseudo Goldstone boson of

spontaneously broken scale symmetry,spontaneously broken scale symmetry,

tiny mass,tiny mass, responsible for dynamical Dark Energy

Asymptotic safetyAsymptotic safety

if UV fixed point exists :if UV fixed point exists :

quantum gravity is quantum gravity is

non-perturbatively non-perturbatively renormalizable !renormalizable !

S. Weinberg , M. S. Weinberg , M. ReuterReuter

Variable GravityVariable Gravity

quantum effective action,quantum effective action,variation yields field equationsvariation yields field equations

MM22 R REinstein gravity :Einstein gravity :

Variable GravityVariable Gravity

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 involves Simple quadratic scalar potential involves

intrinsic mass intrinsic mass μμ Nucleon and electron mass proportional to Nucleon and electron mass proportional to

dynamical Planck massdynamical Planck mass

running coupling Brunning coupling B

B varies if intrinsic scale µ changesB varies if intrinsic scale µ changes similar to QCD or standard modelsimilar to QCD or standard model

Kinetial B : Kinetial B : Crossover between two Crossover between two

fixed pointsfixed points

Kinetial B : Kinetial B : Crossover between two Crossover between two

fixed pointsfixed points

m : scale of crossoverm : scale of crossovercan be exponentially larger than intrinsic scale can be exponentially larger than intrinsic scale μμ

runningrunningcoupling coupling obeysobeysflow flow equation equation

Infrared fixed pointInfrared fixed point

μ→μ→ 0 0BB→→ 0 0

no intrinsic mass scaleno intrinsic mass scalescale symmetryscale symmetry

Ultraviolet fixed pointUltraviolet fixed point

μ→μ→ ∞ ∞

kinetial divergeskinetial diverges

scale symmetry with anomalous scale symmetry with anomalous dimension dimension σσ

Renormalized field at UV Renormalized field at UV fixed pointfixed point

no no massmassscalescaledeviation deviation fromfromfixed pointfixed pointvanishes forvanishes forμ→μ→ ∞ ∞

1 < 1 < σσ

Cosmological solution :Cosmological solution :crossover from UV to IR crossover from UV to IR

fixed pointfixed point Dimensionless functions as B Dimensionless functions as B

depend only on ratio depend only on ratio μμ//χχ . . IR: μ→0 , IR: μ→0 , χ→∞χ→∞ UV: UV: μ→∞μ→∞ , , χ→χ→00

Cosmology makes Cosmology makes

crossover betweencrossover between

fixed points byfixed points by

variation of variation of χχ . .

No tiny dimensionless No tiny dimensionless parametersparameters

( except gauge hierarchy )( except gauge hierarchy )

one mass scaleone mass scale

one time scale one time scale μμ-1-1 = 10 = 10 1010 yr yr

Planck mass does not appear as Planck mass does not appear as parameterparameter

Planck mass grows large dynamicallyPlanck mass grows large dynamically

μ = 2 μ = 2 10 10 -33 -33

eVeV

Particle massesParticle masseschange with time change with time

At SM fixed point : At SM fixed point : All particle masses ( except for neutrinos ) are All particle masses ( except for neutrinos ) are

proportional to scalar field proportional to scalar field χχ . . Scalar field varies with time – so do particle Scalar field varies with time – so do particle

masses.masses. Ratios of particle masses are independent of Ratios of particle masses are independent of χχ

and therefore remain constant.and therefore remain constant. Compatibility with observational bounds on Compatibility with observational bounds on

time dependence of particle mass ratios.time dependence of particle mass ratios. Dimensionless couplings are independent of Dimensionless couplings are independent of χχ . .

Four-parameter model

model has four dimensionless parameters three in kinetial B : σ ~ 2.5 κ ~ 0.5 ct ~ 14 ( or m/μ ) one parameter for present growth rate of

neutrino mass over electron mass : γ ~ 8 + standard model particles and dark

matter : sufficient for realistic cosmology from inflation to dark energy

no more free parameters than ΛCDM

Cosmological solutionCosmological solution

scalar field scalar field χχ vanishes in the infinite vanishes in the infinite pastpast

scalar field scalar field χχ diverges in the infinite diverges in the infinite futurefuture

Sonntagszeitung Zürich , LaukenmannSonntagszeitung Zürich , Laukenmann

Strange evolution of Strange evolution of UniverseUniverse

Slow UniverseSlow UniverseAsymptotic solution in Asymptotic solution in freeze frame :freeze frame :

Expansion or shrinking always slow ,Expansion or shrinking always slow , characteristic time scale of the order of the characteristic time scale of the order of the age of theage of the Universe : Universe : ttch ch ~~ μμ-1 -1 ~ ~ 10 billion years10 billion years ! !Hubble parameter of the order of Hubble parameter of the order of presentpresent Hubble Hubble parameter for all times , including inflation parameter for all times , including inflation and big bang !and big bang !Slow increase of particle masses ! Slow increase of particle masses !

μμ= 2 = 2 10 10-33 -33 eVeV

Model is compatible with present observations

Together with variation of neutrino mass over electron mass in present cosmological epoch :

model is compatible with all present observations

Do we know that the Do we know that the Universe expands ?Universe expands ?

instead of redshift due to expansion :instead of redshift due to expansion :

smaller frequencies have been emitted in the smaller frequencies have been emitted in the past, because electron mass was smaller !past, because electron mass was smaller !

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

Hot plasma ?Hot plasma ?

Temperature in radiation dominated Temperature in radiation dominated Universe : T ~ Universe : T ~ χχ ½½ smallersmaller than today than today

Ratio temperature / particle mass : Ratio temperature / particle mass : T /m T /mpp ~ ~ χχ --½½ largerlarger than today than today

T/mT/mpp counts ! This ratio decreases with counts ! This ratio decreases with time.time.

Nucleosynthesis , CMB emission as in Nucleosynthesis , CMB emission as in standard cosmology ! standard cosmology !

Einstein frameEinstein frame

““Weyl scaling” maps variable gravity Weyl scaling” maps variable gravity model to Universe with fixed masses model to Universe with fixed masses and standard expansion history.and standard expansion history.

Exact equivalence of different frames !Exact equivalence of different frames !

Standard gravity coupled to scalar Standard gravity coupled to scalar field.field.

Only neutrino masses are growing.Only neutrino masses are growing.

Einstein frameEinstein frame

Weyl scaling :Weyl scaling :

effective action in Einstein frame :effective action in Einstein frame :

Field relativity : Field relativity : different pictures of different pictures of

cosmologycosmology same physical content can be same physical content can be

described by different picturesdescribed by different pictures related by field – related by field – redefinitionsredefinitions , ,

e.g. Weyl scaling , e.g. Weyl scaling , conformal scaling of metricconformal scaling of metric

which picture is usefull ?which picture is usefull ?

Einstein gravity from restricted Einstein gravity from restricted coordinate invariancecoordinate invarianceW. BuchmüllerW. Buchmüllera, ba, b, N. Dragon, N. Dragona, c a, c ,1988,1988

We show that Einstein's equations for the We show that Einstein's equations for the gravitational field can be derived from an action gravitational field can be derived from an action which is invariant only under restricted coordinate which is invariant only under restricted coordinate transformations which preserve the volume. The transformations which preserve the volume. The only difference compared to a general covariant only difference compared to a general covariant theory concerns the cosmological constant, which theory concerns the cosmological constant, which becomes an arbitrary initial condition.becomes an arbitrary initial condition.

Big bang or freeze ?Big bang or freeze ?

just two ways of looking at same physicsjust two ways of looking at same physics

asymptoticallyasymptotically vanishing vanishing cosmological „constant“cosmological „constant“

What matters : Ratio of potential What matters : Ratio of potential divided by fourth power of Planck divided by fourth power of Planck massmass

vanishes for vanishes for χχ →→ ∞∞ ! !

Einstein frameEinstein frame

Weyl scaling :Weyl scaling :

effective action in Einstein frame :effective action in Einstein frame :

conclusionsconclusions

Quantum gravity may be observable in Quantum gravity may be observable in

dynamics of present Universe dynamics of present Universe

Fixed points and scale symmetry crucialFixed points and scale symmetry crucial

Big bang singularity is artefact Big bang singularity is artefact

of inappropriate choice of field of inappropriate choice of field variables – variables –

no physical singularityno physical singularity

conclusions conclusions

crossover in quantum gravity is reflected in crossover in quantum gravity is reflected in crossover in cosmologycrossover in cosmology

quantum gravity becomes testable by quantum gravity becomes testable by cosmologycosmology

quantum gravity plays a role not only for quantum gravity plays a role not only for primordial cosmologyprimordial cosmology

crossover scenario explains different crossover scenario explains different cosmological epochscosmological epochs

simple model is compatible with present simple model is compatible with present observationsobservations

no more parameters than no more parameters than ΛΛCDM CDM : : tests possibletests possible

endend

small dimensionless small dimensionless number ?number ?

needs two intrinsic mass scalesneeds two intrinsic mass scales V and M ( cosmological constant and V and M ( cosmological constant and

Planck mass )Planck mass ) variable Planck mass moving to variable Planck mass moving to

infinity , with fixed V: infinity , with fixed V: ratio vanishes ratio vanishes asymptotically asymptotically !!

Big bang or freeze ?

Infinite past : slow Infinite past : slow inflationinflation

σσ = 2 : field equations = 2 : field equations

solutionsolution

Eternal Universe

solution valid back to the infinite past in physical time

no singularity

physical time to infinite past is infinite

Asymptotic solution in Asymptotic solution in freeze frame :freeze frame :

Physical timePhysical time

field equation for scalar field modefield equation for scalar field mode

determine determine physicalphysicaltimetime by counting by countingnumber of oscillationsnumber of oscillations

( m=0 )( m=0 )

Big bang singularity Big bang singularity in Einstein frame isin Einstein frame isfield singularity !field singularity !

choice of frame with constant particlechoice of frame with constant particlemasses is not well suited ifmasses is not well suited ifphysical masses go to zero !physical masses go to zero !

Inflation

solution for small solution for small χχ : inflationary epoch : inflationary epoch

kinetial characterized by kinetial characterized by anomalous dimension anomalous dimension σσ

Primordial fluctuationsPrimordial fluctuations

inflaton field : inflaton field : χχ primordial fluctuations of inflaton become primordial fluctuations of inflaton become

observable in cosmic microwave observable in cosmic microwave backgroundbackground

Anomalous dimension Anomalous dimension determines spectrum of determines spectrum of primordial fluctuationsprimordial fluctuations

spectral index nspectral index n

tensor amplitude rtensor amplitude r

relation between n and rrelation between n and r

rr = 8.19 ( 1 – = 8.19 ( 1 – nn ) - 0.1365 ) - 0.1365

Amplitude of density Amplitude of density fluctuationsfluctuations

small because of logarithmic small because of logarithmic running running near UV fixed point !near UV fixed point !

N : number of e – foldings at N : number of e – foldings at horizon crossing horizon crossing

σσ=1=1

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 ( different growth of neutrino mass )( different growth of neutrino mass )

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 scalar fieldsame scalar field

Second stage of Second stage of crossovercrossover

from SM to IRfrom SM to IR in sector Beyond Standard Modelin sector Beyond Standard Model affects neutrino masses first affects neutrino masses first

( seesaw or cascade mechanism )( seesaw or cascade mechanism )

Varying particle massesVarying particle massesat onset of second at onset of second

crossovercrossover All particle masses All particle masses except for neutrinos except for neutrinos

are proportional to are proportional to χχ . . Ratios of particle masses remain constant.Ratios of particle masses remain constant. Compatibility with observational bounds Compatibility with observational bounds

on time dependence of particle mass on time dependence of particle mass ratios.ratios.

Neutrino masses show stronger increase Neutrino masses show stronger increase with with χχ , such that , such that ratio neutrino mass ratio neutrino mass over electron mass grows .over electron mass grows .

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= 1.27L.Amendola,L.Amendola,M.Baldi, …M.Baldi, …

Oscillating neutrino Oscillating neutrino lumpslumps

Y.Ayaita, Y.Ayaita, M.Weber,…M.Weber,…

Ayaita, Baldi, Ayaita, Baldi, Fuehrer,Fuehrer,Puchwein,…Puchwein,…

Evolution of dark energy Evolution of dark energy similar to similar to ΛΛCDMCDM

Compatibility with Compatibility with observationsobservations

and possible testsand possible tests Realistic inflation modelRealistic inflation model Almost same prediction for radiation, Almost same prediction for radiation,

matter, and Dark Energy domination as matter, and Dark Energy domination as ΛΛCDMCDM

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

Large neutrino lumpsLarge neutrino lumps

SimplicitySimplicity

simple description of simple description of allall cosmological cosmological epochsepochs

natural incorporation of Dark Energy :natural incorporation of Dark Energy : inflationinflation Early Dark EnergyEarly Dark Energy present Dark Energy dominated epochpresent Dark Energy dominated epoch

conclusions (2)conclusions (2)

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

Cosmon dependence of ratio neutrino mass/ Cosmon dependence of ratio neutrino mass/ electron mass can explain why Universe electron mass can explain why Universe makes a transition to Dark Energy domination makes a transition to Dark Energy domination nownow

characteristic signal : neutrino lumpscharacteristic signal : neutrino lumps

First step of crossover First step of crossover ends inflationends inflation

induced by crossover in Binduced by crossover in B

after crossover B changes only very after crossover B changes only very slowlyslowly

Scaling solutions Scaling solutions near SM fixed pointnear SM fixed point

( approximation for constant B ( approximation for constant B ))

Different scaling solutions Different scaling solutions forforradiation domination and radiation domination and matter dominationmatter domination

Radiation dominationRadiation domination

UniverseUniverseshrinks !shrinks !

K = B - K = B - 66

solution exists for B < 1 solution exists for B < 1 or K< -5or K< -5

Varying particle massesVarying particle massesnear SM fixed pointnear SM fixed point

All particle masses are proportional All particle masses are proportional to to χχ . .

( scale symmetry )( scale symmetry ) Ratios of particle masses remain Ratios of particle masses remain

constant.constant. Compatibility with observational Compatibility with observational

bounds on time dependence of bounds on time dependence of particle mass ratios.particle mass ratios.

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

F = F = χχ22

Matter dominationMatter domination

Universe shrinks Universe shrinks !!

solution exists forsolution exists forB < 4/3 ,B < 4/3 ,

K = B - K = B - 66

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

observation requires observation requires B < 0.02 B < 0.02 ( at CMB ( at CMB emission )emission )

or or mm

Dark Energy dominationDark Energy domination

neutrino masses scaleneutrino masses scaledifferently from electron massdifferently from electron mass

new scaling solution. not yet new scaling solution. not yet reached.reached.at present : transition at present : transition periodperiod