# Quantum Gravity and the Quantum Gravity and the Cosmological Constant Cosmological Constant

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## Cosmological Constant Cosmological Constant

### Enikő Regős Enikő Regős

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Astrophysical observations and quantum physics Astrophysical observations and quantum physics

Explain Explain ΛΛ from from

quantum fluctuations quantum fluctuations

in gravity in gravity

induce ΛΛ

Quantum gravity and Quantum gravity and accelerator physics

accelerator physics

Extra dimensional Extra dimensional models (strings) models (strings)

Particle astrophysics : Particle astrophysics : dark matter search, dark matter search,

mass of particles mass of particles

Quantum black holesQuantum black holes

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### Effective potential for the curvature Effective potential for the curvature

Effective actionEffective action: :

S [ g ] = - S [ g ] = - κκ² ∫ dx √g ( R – 2 ² ∫ dx √g ( R – 2 λλ ) )

One-loop approximation :One-loop approximation :

ΓΓ [g] = S [g] + Tr ln [g] = S [g] + Tr ln ∂∂² S [g] / ∂g ∂g / 2² S [g] / ∂g ∂g / 2

Gauge fixing and regularizationGauge fixing and regularization

Sharp cutoffSharp cutoff : - D² < : - D² < ΛΛ² ²

Spin projectionSpin projection : :

metric tensormetric tensor fluctuation fluctuation : TT, LT, LL, Tr : TT, LT, LL, Tr

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### Background space Background space

Background Background : maximally symmetric spaces :: maximally symmetric spaces : de Sitterde Sitter

Spherical harmonics to solve Spherical harmonics to solve spectrumspectrum ( ( λλ_l )_l ) for potential :for potential :

γγ1 ( R ) = 1 ( R ) =

D / 2 ln [ D / 2 ln [ κκ² R / ² R / Λ4 ( a Λ4 ( a λλ_l + d - c _l + d - c λλ / R )] / R )]

D_l : degeneracy, sum over multipoles l and D_l : degeneracy, sum over multipoles l and spins

spins

g = < g > + h g = < g > + h

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### Casimir effect Casimir effect

In a boxIn a box : :

ΓΓ  = (L  = (L ΛΛ)^4 ( ln )^4 ( ln μμ² / ² / ΛΛ² – ½ ) / 32 ² – ½ ) / 32 ΠΠ²²

Fit numerical results for gravityFit numerical results for gravity : :

γγ ( R ) = - v ( R ) = - v κκ² / R + c1 ² / R + c1 ΛΛ^4 ( 1 / R² –^4 ( 1 / R² –

1 / R² (1 / R² (ΛΛ) ) ln ( c2 ) ) ln ( c2 κκ² / ² / ΛΛ² )² )

v = 3200 v = 3200 ΠΠ² / 3² / 3

R ( R ( ΛΛ) = c3 ) = c3 ΛΛ²²

Metric tensor controls geometryMetric tensor controls geometry

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### Energetically preferred curvature Energetically preferred curvature

Minimize effective potentialMinimize effective potential

Quantum phase transitionQuantum phase transition at : at :

κκ² = ² = ΛΛ² / c2 : critical coupling² / c2 : critical coupling

Low cutoff phaseLow cutoff phase, below :, below :

R_min = 2 c1 ( R_min = 2 c1 ( ΛΛ^4 / v ^4 / v κκ² ) ln ( c2 ² ) ln ( c2 κκ² / ² / ΛΛ² ) ² )

High cutoff phaseHigh cutoff phase : :

R_min = 0R_min = 0 : : flatflat

2 phases : flat and strongly curved space-time2 phases : flat and strongly curved space-time

Condensation of metric tensorCondensation of metric tensor

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### Running Newton constant Running Newton constant

κκ² ( R ) = ² ( R ) = κκ² - ( R / v ) ² - ( R / v ) γγ1 ( R )1 ( R )

G ( R ) = 1 / ( 16 G ( R ) = 1 / ( 16 ΠΠ κκ² ( R ) )² ( R ) )

Infrared Landau pole in low-cutoff phaseInfrared Landau pole in low-cutoff phase : :

R_L = R_min /2 :R_L = R_min /2 :

Confinement of gravitons Confinement of gravitons ( experiments )( experiments )

G ( R ) increasing in high-cutoff phaseG ( R ) increasing in high-cutoff phase

Savvidy vacuumSavvidy vacuum

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### Induced cosmological constant Induced cosmological constant

ΓΓ [g] = [g] = κκ²_eff ∫ dx √g (x) F ( R (x) )²_eff ∫ dx √g (x) F ( R (x) )

F ( R ) = R – 2 F ( R ) = R – 2 λλ – g R² – g R²

κκ_eff = _eff = κκ

λλ = c1 ( = c1 ( ΛΛ^4 / 2 v ^4 / 2 v κκ² ) ln ( c2 ² ) ln ( c2 κκ² / ² / ΛΛ² )² )

ΛΛ > 0 : curved phase > 0 : curved phase

ΛΛ < 0 : flat phase < 0 : flat phase

Or running GOr running G

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### Stability and matter fields Stability and matter fields

λλ_bare -> 2D _bare -> 2D phase diagramphase diagram

stabilitystability

include matter fieldsinclude matter fields : :

1.1. scalarscalar

2.2. strong interaction : strong interaction :

influence of confinement in gauge andinfluence of confinement in gauge and

gravitational sectors on each othergravitational sectors on each other

gravitational waves gravitational waves

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## References

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