quantum vacuum in cosmology. what is the vacuum in cosmology ?
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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