Quantum vacuum Quantum vacuum in cosmologyin cosmology

What is the vacuum What is the vacuum in cosmology ?in cosmology ?

Vacuum in cosmologyVacuum in cosmology

state of the Universe in absence of state of the Universe in absence of matter and radiation ?matter and radiation ?

gravity + scalar fieldgravity + scalar field characterized by mean fields and characterized by mean fields and

fluctuationsfluctuations no time translation invarianceno time translation invariance no conserved energy , state of minimal no conserved energy , state of minimal

energy not meaningfulenergy not meaningful attractor of time evolutionattractor of time evolution

vacuum in cosmologyvacuum in cosmology==

(space averaged ) (space averaged ) state of Universestate of Universe

vacuum is quantum object and vacuum is quantum object and not “empty”not “empty”

Three questions

Can we observe quantum vacuum properties in cosmology ?

Can we compute quantum vacuum properties in gravity ?

What is the role of scale symmetry What is the role of scale symmetry and its spontaneous breaking ?and its spontaneous breaking ?

Does inflation allow us to Does inflation allow us to observe quantum observe quantum

vacuum properties ?vacuum properties ?

correlation function for correlation function for different initial conditionsdifferent initial conditions

u = k u = k ηη

no loss of memoryno loss of memory at horizon crossingat horizon crossing( u = -1 )( u = -1 )

inflation processes inflation processes fluctuationsfluctuations

fluctuations are not “ generated “ fluctuations are not “ generated “ during inflationduring inflation

every statistical system has fluctuationsevery statistical system has fluctuations fluctuations already present at fluctuations already present at

“beginning” of inflation ( or at “beginning” of inflation ( or at extremely early stages if inflation lasts extremely early stages if inflation lasts from the infinite past )from the infinite past )

fluctuations get fluctuations get processedprocessed by scale by scale violating effects from inflaton potential violating effects from inflaton potential near horizon crossingnear horizon crossing

memory of initial memory of initial spectrumspectrum

equilibration ?equilibration ?

for interacting theories : there could for interacting theories : there could be processes that bring arbitrary be processes that bring arbitrary initial fluctuations close to universal initial fluctuations close to universal formform

no such loss of memory found within no such loss of memory found within present approximationspresent approximations

equilibration time will be very long equilibration time will be very long since interactions are tinysince interactions are tiny

scalar correlation scalar correlation functionfunction

correlation function fromcorrelation function fromquantum effective actionquantum effective action

no operators neededno operators needed no explicit construction of vacuum no explicit construction of vacuum

statestate no distinction between quantum and no distinction between quantum and

classical fluctuations classical fluctuations one relevant quantity : correlation one relevant quantity : correlation

functionfunction

quantum theory as functional quantum theory as functional integralintegral

background geometry given by vierbein or metricbackground geometry given by vierbein or metric

homogeneous and isotropic cosmologyhomogeneous and isotropic cosmology

M:M: E:E:

quantum effective actionquantum effective action

exact field equationexact field equation

exact propagator equationexact propagator equation

correlation function and correlation function and quantum effective actionquantum effective action

propagator equation propagator equation for interacting scalar field infor interacting scalar field inhomogeneous and isotropic homogeneous and isotropic

cosmologycosmology

GG

general solution and general solution and mode functionsmode functions

Bunch – Davies vacuumBunch – Davies vacuum

αα = 1 , = 1 , ζζ = 0 for all k = 0 for all k

initial conditionsinitial conditions

ββ2 2 + + γγ2 2 = 4 = 4 ζζ ζζ**

mixed state : positive probabilities pmixed state : positive probabilities pii

absence of loss of memoryabsence of loss of memoryof initial correlationsof initial correlations

each k-mode has threeeach k-mode has threefree integration constants !free integration constants !

memory of initial memory of initial spectrumspectrum

αα = 1 + A = 1 + App

example :example :

spectralspectralindex : index :

predictivity of inflation ?predictivity of inflation ?

initial spectrum at beginning of initial spectrum at beginning of inflation : gets only inflation : gets only processedprocessed by by inflaton potentialinflaton potential

small tilt in initial spectrum is small tilt in initial spectrum is not not distinguishable distinguishable from small scale from small scale violation due to inflaton potentialviolation due to inflaton potential

long duration of inflation long duration of inflation before before horizon crossing of observable modes : horizon crossing of observable modes : one sees UV-tail of initial spectrum .one sees UV-tail of initial spectrum .

If flat , predictivity retained !If flat , predictivity retained !

vacuum in cosmology

simply the ( averaged ) state of the Universe

result of time evolution minimal “particle number” ? depends on definition of

particle number as observable

Can we compute the Can we compute the vacuum for gravity ?vacuum for gravity ?

Can we compute the Can we compute the vacuum for gravity ?vacuum for gravity ?

Can we compute theCan we compute thequantum effective action ?quantum effective action ?

Functional renormalization :Functional renormalization :Exact renormalization group Exact renormalization group

equationequation

M.Reuter M.Reuter +…+…for gravityfor gravity

flowing actionflowing action

Wikipedia

asymptotic safety for gravity ?

S. Weinberg, M. Reuter, …

Dilaton quantum gravityDilaton quantum gravity

Functional renormalization flow,Functional renormalization flow,

with truncation :with truncation :

Functional renormalization Functional renormalization equationsequations

Percacci, NarainPercacci, Narain

Fixed point for large Fixed point for large scalar fieldscalar field

This fixed point describes already realistic gravity !This fixed point describes already realistic gravity !Limit k → 0 can be taken !Limit k → 0 can be taken !

Fixed point for large Fixed point for large scalar fieldscalar field

Vicinity of fixed pointVicinity of fixed point

solution :solution :

Cosmology with dynamical dark energy !Cosmology with dynamical dark energy !Cosmological constant vanishes asymptotically !Cosmological constant vanishes asymptotically !

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 →→ ∞∞ ! !similar for mass term :similar for mass term :

How well motivated How well motivated are guesses on theare guesses on the

“natural value” of the “natural value” of the cosmological constant ?cosmological constant ?

Quantum fluctuationsQuantum fluctuationsinduce cosmological induce cosmological

constantconstant

Same argument leads to Same argument leads to very different physical very different physical effects when applied in effects when applied in

different framesdifferent frames

V / MV / M44 oror

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 !!

How well motivated How well motivated are guesses on theare guesses on the

“natural values” of “natural values” of masses and couplings ?masses and couplings ?

properties of fixed points properties of fixed points will not be seen bywill not be seen by

naïve order of magnitude naïve order of magnitude estimatesestimates

Scale symmetry and its Scale symmetry and its spontaneous breakingspontaneous breaking

Crossover in quantum Crossover in quantum gravitygravity

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

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 exact !at fixed points , scale symmetry is exact !

scale symmetry includes gravity ( different scale symmetry includes gravity ( different from approximate scale symmetry of from approximate scale symmetry of particle physics with fixed Planck mass )particle physics with fixed Planck mass )

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

fixed pointfixed point Dimensionless functions Dimensionless functions

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 . .

Variable GravityVariable Gravity

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

Variable GravityVariable Gravity

Scalar field coupled to gravityScalar field coupled to gravity Effective Planck mass depends on scalar fieldEffective Planck mass depends on scalar field 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 Neutrino mass has different dependence on Neutrino mass has different dependence on

scalar fieldscalar field

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

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

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

conclusions from conclusions from variable gravityvariable gravity

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

Quantum gravity is obervable in Quantum gravity is obervable in

dynamics of present Universedynamics of present Universe

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

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

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 →→ ∞∞ ! !

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

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

relation between n and rrelation between n and r

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

Compatibility with Compatibility with observationsobservations

and possible testsand possible tests

Realistic inflation model:Realistic inflation model:

n = 0.97 , r = 0.1n = 0.97 , r = 0.1 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

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

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 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

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 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

propagator equation inpropagator equation inhomogeneous and isotropic homogeneous and isotropic

cosmologycosmology

ηη > > ηη’ : G’ : G> > = G= Gs s + G+ Gaa

evolution equation forevolution equation forequal time correlation equal time correlation

functionfunction

massless scalar in massless scalar in de Sitter space :de Sitter space :

u = k u = k ηη

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

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.

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 ?

PrimordialPrimordial flat frame flat frame

Minkowski space in infinite pastMinkowski space in infinite past absence of any singularityabsence of any singularity geodesic completenessgeodesic completeness

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=0m=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 !

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 scale symmetry spontaneously brokenbroken, ,

massive particles , massless dilatonmassive particles , massless dilaton

crossover : crossover : explicit mass scale explicit mass scale μμ or m important or m important

SM fixed point : SM fixed point : approximate scale symmetryapproximate scale symmetry spontaneously brokenspontaneously broken, , massive particles , almost massive particles , almost massless cosmon, tiny cosmon potentialmassless cosmon, tiny cosmon potential

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 !

Infinite past : slow Infinite past : slow inflationinflation

σσ = 2 : field equations = 2 : field equations

solutionsolution

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

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 < < 22