Cosmology and quantum gravities: Where are we?

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Quantum gravity? Cosmological problems Quantum and emergent gravities Final remarks

Cosmology and quantum gravities:

Where are we?

Gianluca Calcagni

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Outline

1 _{Quantum gravity?}

2 Cosmological problems

3 Quantum and emergent gravities

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Outline

3 Quantum and emergent gravities

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Outline

3 _{Quantum and emergent gravities}

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Outline

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01/27–

Goal

Systematic introduction and comparison of the status of the

most prominent theories ofquantumandemergent gravityin

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Outline

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03/27–

Unification and open problems

Theoreticalnecessity, not experimental.

Non-predictivegravity if quantized perturbatively [Goroff & Sagnotti 1985,1986].

Models oftheories of everythingandquantum gravityare

very formaland withlittle contactwith observations. Cosmological problems must be addressed.

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03/27–

Unification and open problems

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03/27–

Unification and open problems

very formaland withlittle contactwith observations.

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03/27–

Unification and open problems

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Outline

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05/27–

Big bang problem

•Singularities typical of classical gravity (black holes,big bang).

•Borde–Guth–Vilenkin theorem(2003):Let(M,g)be a

spacetime with a congruenceuµcontinuously defined along any past-directed timelike or null geodesicvµ(the observer). Letuµ obey the averaged expansion conditionHav>0for almost any vµ. Then(M,g)is geodesically past-incomplete(finite

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06/27–

Inflation

Graceful exit.

Trans-Planckian problem. Model building.

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07/27–

Cosmological constant problems

•Oldproblem: zero-point energy (dim. reg. [Koksma & Prokopec 2011])ρvac∼10−68m4_Pl∼1056ρΛwrong magnitude. E.g., ρeq ≈2.4×10−113m4Pl is calculable.

•New: WhyρΛ=O(ρm)?Coincidence: Why doesΛdominate

atz1?

•Shift symmetry:L_m → L_m+ρ0 ⇒ Tµν →Tµν+ρ0δµν.

E.o.m.s∇_νT_µν =0invariant, Einstein eqs. are not:Why?

•4π puzzle: For the observed value ofρΛ, the duration of the

matter-radiation era (# modes reentered) is4π±10−3e-folds

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07/27–

Cosmological constant problems

•New: WhyρΛ=O(ρm)? Coincidence: Why doesΛdominate

atz1?

•Shift symmetry:L_m → L_m+ρ0 ⇒ Tµν →Tµν+ρ0δµν.

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07/27–

Cosmological constant problems

atz1?

•Shift symmetry:L_m → L_m+ρ0 ⇒ Tµν →Tµν+ρ0δνµ.

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07/27–

Cosmological constant problems

atz1?

•Shift symmetry:L_m → L_m+ρ0 ⇒ Tµν →Tµν+ρ0δνµ.

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Outline

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08/27–

Asymptotic safety: setting

Weinberg, Reuter, Bonanno, Lauscher, Litim, Saueressig, . . .

•All dimensionless couplings approach a UV NGFP

limk→∞¯λi(k) = ¯λ∗i 6=0(existence checked a posteriori).

•Gravity: effective action

Γk = 1 16πGk Z dDx√−g(R−2Λk), δΓk δgµν [hgµνik] =0

•Λand average metric are scale-dependent:

hgµνik =k−2hgµνik0,Λk =k

2_Λk

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09/27–

Asymptotic safety: cosmology

Cutoff identificationk=k(t)∝H(t) ⇒ Λ,G→Λ(t),G(t). RG-improved dynamics: H2= 8πG(t) 3 ρ+ Λ(t) 3 , ρ˙+3H(ρ+P) =− ˙ Λ +8πρG˙ 8πG

•Near the NGFP:G(p˜ 2)' −1/p4forp2 m2_Pl,

hh(t,x)h(t,0)i ∼ln|x|2_,_hδR₍_t,_x)_δR₍_t,₀₎_{i ∼ |}_x_|−4_for_δR_∼_∂2_h_.

Scale-invariant power spectrum. Buthigher-order curvature termsare required, sinceGH2∝G∗Λ∗∼10−3∼107Aobs_t .

•f(R)actionsfrom scale identificationk2∝R, Starobinsky

dynamics:R/Gk ∼R/R−1∼R2near the NGFP.

•OldΛproblem: Why is the perturbative semiclassical regime

of the RG trajectory fine tuned so much? [Reuter & Weyer 2004].

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09/27–

Asymptotic safety: cosmology

Cutoff identificationk=k(t)∝H(t) ⇒ Λ,G→Λ(t),G(t). RG-improved dynamics: H2= 8πG(t) 3 ρ+ Λ(t) 3 , ρ˙+3H(ρ+P) =− ˙ Λ +8πρG˙ 8πG •Near the NGFP:G(p˜ 2)' −1/p4forp2 m2_Pl,

Scale-invariant power spectrum.

Buthigher-order curvature termsare required, sinceGH2∝G∗Λ∗∼10−3∼107Aobs_t .

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09/27–

Asymptotic safety: cosmology

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09/27–

Asymptotic safety: cosmology

dynamics:R/Gk ∼R/R−1∼R2 near the NGFP.

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09/27–

Asymptotic safety: cosmology

dynamics:R/Gk ∼R/R−1∼R2 near the NGFP.

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09/27–

Asymptotic safety: cosmology

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10/27–

Causal dynamical triangulations: setting

Ambjørn, Loll, Jurkiewicz, . . .

Z = Z [Dg]eiS[g] → X T 1 Aut(T)e −SRegge_E (T)_.

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11/27–

Causal dynamical triangulations: cosmology

A:branched-polymer phase, disconnected “lumps” of space, non-Riemannian geometry.

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12/27–

Non-local gravity: setting

Krasnikov, Tomboulis, Mazumdar, Modesto, G.C., . . .

Minimal requirements:(i)continuous spacetime with Lorentz

invariance;(ii)classical local (super)gravity good approximation

at low energy;(iii)perturbative super-renormalizability or

finiteness;(iv)unitary and ghost free;(v)typical classical solutions singularity-free.

Example [G.C. & Modesto 2014]:

Sg = 1 2κ2 Z dDx√−g " R−2Λ−Gµν e−f(/M2₎ −1 R µν # .

Reproduces the linearized effective action ofstring field theory

whenf =/M2. Exponential operators havegoodproperties

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12/27–

Non-local gravity: setting

Reproduces the linearized effective action ofstring field theory

whenf =/M2. Exponential operators havegoodproperties

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12/27–

Non-local gravity: setting

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13/27–

Non-local gravity: cosmology

G.C., Modesto, Nicolini 2014

Typical classicalbouncingprofiles inD=4:

a(t) = a∗ cosh r

ω

2 t

, a(t) = a∗ exp

H1 2 t 2 .

Difficult dynamics, e.o.m.s still under study [G.C., Modesto & Nardelli in progress].

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13/27–

Non-local gravity: cosmology

G.C., Modesto, Nicolini 2014

Typical classicalbouncingprofiles inD=4:

a(t) = a∗ cosh r

ω

2 t

, a(t) = a∗ exp

H1 2 t 2 .

Difficult dynamics, e.o.m.s still under study [G.C., Modesto & Nardelli

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14/27–

Canonical quantum cosmology

DeWitt, Hawking, Vilenkin, Ashtekar, Bojowald, . . .

Hamiltonian formalism (unconstrained):

S=

Z

dt L[q,q]˙ → H[q,p] =pq−L[q,˙ q]˙ → _H[ˆˆ _q,_ˆ_p₌_i_~_∂_q]|ψi=E|ψi

Gravity+matter dynamics (constrained): Wheeler–DeWitt

equationHˆ(g, φ)Ψ[g, φ] =0. Symmetry reduction. FLRW:g_µν = (−1,a2(t),a2(t),a2(t)), p₍a)=−6aa˙,Πφ=a3φ˙: H= 1 2a3 " −a 2_p2 (a) 6κ2 + Π 2 φ # +· · ·=0 → Hˆ= 1 2a3 κ2 6 ∂2 (∂lna)2 − ∂2 ∂φ2 +. . .

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14/27–

Canonical quantum cosmology

S=

Z

equationHˆ(g, φ)Ψ[g, φ] =0. Symmetry reduction. FLRW:g_µν = (−1,a2(t),a2(t),a2(t)), p₍a)=−6aa˙,Πφ=a3φ˙: H= 1 2a3 " −a 2_p2 (a) 6κ2 + Π 2 φ # +· · ·=0 → Hˆ= 1 2a3 κ2 6 ∂2 (∂lna)2 − ∂2 ∂φ2 +. . .

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14/27–

Canonical quantum cosmology

S=

Z

equationHˆ(g, φ)Ψ[g, φ] =0. Symmetry reduction. FLRW:gµν = (−1,a2(t),a2(t),a2(t)), p₍a)=−6aa˙,Πφ=a3φ˙: H= 1 2a3 " −a 2_p2 (a) 6κ2 + Π 2 φ # +· · ·=0 → Hˆ= 1 2a3 κ2 6 ∂2 (∂lna)2 − ∂2 ∂φ2 +. . .

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15/27–

WDW quantum cosmology

PDF (nucleation probability) of the initial state of the Universe: ratio of the squared wave-function at the classical turning point

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16/27–

WDW quantum cosmology and

Λ

Probabilisticinterpretation [Baum 1983; Hawking 1984; Wu 2008]:

PV(Λ) =exp −3m 2 Pl 2πΛ , PHH(Λ) =exp 12 κ2_Λ =exp 3m2_Pl 2πΛ .

Hartle–Hawking wave-function:small-ΛUniverses favored.

Problem: aΛ-dependent normalization ofΨmay erase the

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16/27–

WDW quantum cosmology and

Λ

Probabilisticinterpretation [Baum 1983; Hawking 1984; Wu 2008]:

PV(Λ) =exp −3m 2 Pl 2πΛ , PHH(Λ) =exp 12 κ2_Λ =exp 3m2_Pl 2πΛ .

Hartle–Hawking wave-function:small-ΛUniverses favored.

Problem: aΛ-dependent normalization ofΨmay erase the

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17/27–

Loop quantum cosmology

Other canonical variables,p=a2→ˆp,c∼a˙ →ˆh=_e\iµ(p)c_.

Quantum bounce (a=0never):

1 _Bounded_{spectrum of inverse-volume operator:}

[

|v|l−1_|vi₌ 1

2l |v+1|

l_{− |v}₋₁_|l

|vi.

2 _State_|v₌₀_i_{disappears from dynamics:}

cv+2Ψv+4−(cv+2+cv−2)Ψv+cv−2Ψv−4+hv|Hˆφ|viΨv =0.

3 _{Volume expectation value (massless field):}

h|ˆv|i=V∗cosh(κ0φ).

4 _{Effective dynamics}_:_sin2_(¯_µc_{) =} ρ

ρ∗ ↔H 2₌ κ2 3 ρ α−_ρρ ∗ , α=1+δPl=1+Ca−σ.

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17/27–

Loop quantum cosmology

Other canonical variables,p=a2→ˆp,c∼a˙ →ˆh=_e\iµ(p)c_.

1 _Bounded_{spectrum of inverse-volume operator:}

[

|v|l−1_|vi₌ 1

2l |v+1|

l_{− |v}₋₁_|l

|vi.

2 State|v=0idisappears from dynamics:

cv+2Ψv+4−(cv+2+cv−2)Ψv+cv−2Ψv−4+hv|Hˆφ|viΨv =0.

3 _{Volume expectation value (massless field):}

h|ˆv|i=V∗cosh(κ0φ).

4 _{Effective dynamics}_:_sin2_(¯_µc_{) =} ρ

ρ∗ ↔H 2₌ κ2 3 ρ α−_ρρ ∗ , α= +δ = + −σ_.

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18/27–

Quantum gravity and superconductivity

S. Alexander & G.C. PLB 672 (2009) 386; Found. Phys. 38 (2008) 1148

LQGwithΛin vacuum,Chern–Simonsstate annihilates

the constraints.

Different gravity vacua connected vialarge gauge

transformations.

Gravityin a degenerate sectordescribed withfermionic

variables, behaves as aFermi liquid(BCS):

Λ = Λ0exp(−jz5) = Λ0exp(−ψγ¯ 5γzψ), exponentially

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19/27–

Group field theory: setting

Freidel, Oriti, Rovelli, . . .

SGFT= Z G d4g Z G d4g0ϕ∗(g)K(g,g0)ϕ(g0) +V .

•Fock quantization: [ ˆϕ(g),ϕˆ†(g0)] =1G(g,g0), vacuum|∅i

“no-spacetime” configuration, one-particle state|gi:= ˆϕ†(g)|∅i

4-valent spin-network vertex or dual tetrahedron, . . .

• ∞many particles,continuity!All GFT quanta in the same

state,homogeneity! Condensate (coherent state):

|ξi:=Aeξˆ|∅i, ξˆ:=

Z

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19/27–

Group field theory: setting

SGFT= Z G d4g Z G d4g0ϕ∗(g)K(g,g0)ϕ(g0) +V . •Fock quantization: [ ˆϕ(g),ϕˆ†(g0)] =1G(g,g0), vacuum|∅i

Z

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19/27–

Group field theory: setting

state,homogeneity!

Condensate (coherent state):

Z

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19/27–

Group field theory: setting

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20/27–

Group field theory: cosmology

Gielen, Oriti & Sindoni 2014; G.C. Phys. Rev. D 90 (2014) 064047

Gross–Pitaevskii equation: 0=hξ|1C|ξiˆ =R d4g0K(g,g0)ξ(g0) + _δϕδV∗₍_g₎ ϕ=ξ. G=SU(2),isotropy: 2χ(1−χ)ξ00(χ) + (3−4χ)ξ0(χ) +mξ(χ) =0, χ=sin2 µc¯ 2 WKB approx.ξWKB(χ, φ) =A(χ, φ)eiS(χ,φ)/` 2 Pl: 2χ(1−χ)(S,χ)2 =−E2₍_S ,φ)2+m4. Classically,

pχ ∼a/(¯µ2H), pφ∼a3φ ,˙ (a¯µ)−2∝ −E2φ˙2.

E2_<₀_{, l.h.s. is}_H2 _if_a_∝_eHt _{(de Sitter) in the}_{LQC improved}

quantization schemen=1/2.

LQC dynamics? 4χ(1−χ) =sin2(¯µc), l.h.s. of LQC Friedmann

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20/27–

Group field theory: cosmology

Gross–Pitaevskii equation: 0=hξ|1C|ξiˆ =R d4g0K(g,g0)ξ(g0) + _δϕδV∗₍_g₎ ϕ=ξ. G=SU(2),isotropy: 2χ(1−χ)ξ00(χ) + (3−4χ)ξ0(χ) +mξ(χ) =0, χ=sin2 µc¯ 2 WKB approx.ξWKB(χ, φ) =A(χ, φ)eiS(χ,φ)/` 2 Pl: 2χ(1−χ)(S,χ)2 =−E2₍_S ,φ)2+m4. Classically,

pχ ∼a/(¯µ2H), pφ∼a3φ ,˙ (a¯µ)−2∝ −E2φ˙2.

E2_<₀_{, l.h.s. is}_H2 _if_a_∝_eHt _{(de Sitter) in the}_{LQC improved}

quantization schemen=1/2.

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20/27–

Group field theory: cosmology

Gross–Pitaevskii equation: 0=hξ|1C|ξiˆ =R d4g0K(g,g0)ξ(g0) + _δϕδV∗₍_g₎ ϕ=ξ. G=SU(2),isotropy: 2χ(1−χ)ξ00(χ) + (3−4χ)ξ0(χ) +mξ(χ) =0, χ=sin2 µc¯ 2 WKB approx.ξWKB(χ, φ) =A(χ, φ)eiS(χ,φ)/` 2 Pl:2χ(1−χ)(S,χ)2 =−E2₍_S ,φ)2+m4. Classically,

pχ ∼a/(¯µ2H), pφ∼a3φ ,˙ (a¯µ)−2∝ −E2φ˙2.

E2_<₀_{, l.h.s. is}_H2_if_a_∝_eHt _{(de Sitter) in the}_{LQC improved}

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20/27–

Group field theory: cosmology

Gross–Pitaevskii equation: 0=hξ|1C|ξiˆ =R d4g0K(g,g0)ξ(g0) + _δϕδV∗₍_g₎ ϕ=ξ. G=SU(2),isotropy: 2χ(1−χ)ξ00(χ) + (3−4χ)ξ0(χ) +mξ(χ) =0, χ=sin2 µc¯ 2 WKB approx.ξWKB(χ, φ) =A(χ, φ)eiS(χ,φ)/` 2 Pl:2χ(1−χ)(S,χ)2 =−E2₍_S ,φ)2+m4. Classically,

pχ ∼a/(¯µ2H), pφ∼a3φ ,˙ (a¯µ)−2∝ −E2φ˙2.

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21/27–

Causal sets: setting

Bombelli, Dowker, Sorkin, . . .

•“Order and number”: discrete structure of partially ordered

points (xy) reproducing the causal structure of continuous

Lorentzian geometries.

•Volume measurement=point counting: N≈α2mD_PlV.

•Sprinkling: random generation of points on a Lorentzian

manifoldMrespecting light-cone structure. Discreteness and

Lorentz invariance require a Poisson distribution(i):

P(N=n) = _n1_!(ρV)n_e−ρV_{. Standard deviation, kinematical}

fluctuations: ∆N :=σN = √ N, ∆V = (αmD/_Pl2)−1 √ V. Regular lattices/graphsdo not satisfy this property.

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21/27–

Causal sets: setting

•Volume measurement=point counting: N≈α2_mD

PlV.

•Sprinkling: random generation of points on a Lorentzian

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21/27–

Causal sets: setting

PlV. •Sprinkling: random generation of points on a Lorentzian

manifoldMrespecting light-cone structure.

Discreteness and

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21/27–

Causal sets: setting

fluctuations:

∆N :=σ =√N, ∆V = (αmD/2)−1 √

V.

Regular lattices/graphsdo not satisfy this property.

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21/27–

Causal sets: setting

fluctuations: ∆N :=σN = √ N, ∆V = (αmD/_Pl2)−1 √ V.

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21/27–

Causal sets: setting

fluctuations:

∆N :=σ =√N, ∆V = (αmD/2)−1 √

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22/27–

Causal sets: cosmology

(ii) Statistical uncertainty and volume fixing imply

∆V∆Λ&κ2.

(iii) Λ =action ofα2fundamental elements; if each contribution is independent and fluctuates in sign by±α2_{, then}_h_Λ_i₌₀_.

(iv) (i)–(iii) imply aneverpresentΛ(only inD=4):

ρΛ= ∆Λ κ2 ∼ 1 ∆V = αm2_Pl √ V ∼αm 2 PlH2.

(v) To explain dark energy,α=O(10−2)÷O(1). Forα=O(1),

N0≈m4_PlV0∼10244andρΛ ∼10−122m4_Pl.

(vi) Fine tuning problem!

∆T/T ∼∆Φ/Φ∼δρΛ/ρtot∼α∼10−5.

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22/27–

Causal sets: cosmology

∆V∆Λ&κ2.

N0≈m4_PlV0∼10244andρΛ ∼10−122m4_Pl.

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22/27–

Causal sets: cosmology

∆V∆Λ&κ2.

N0≈m4_PlV0∼10244andρΛ ∼10−122m4_Pl.

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22/27–

Causal sets: cosmology

∆V∆Λ&κ2.

N0≈m4_PlV0∼10244andρΛ ∼10−122m4_Pl.

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22/27–

Causal sets: cosmology

∆V∆Λ&κ2.

N0≈m4_PlV0∼10244andρΛ ∼10−122m4_Pl.

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22/27–

Causal sets: cosmology

∆V∆Λ&κ2.

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23/27–

Emergent gravity: setting

Padmanabhan

LocalRindler observer(constant proper acceleration)

Total heat withinV:

Q[n] := 1 8π Z σ2 σ1 dσ Z ∂V d2y√γ Q+κ2Tµνnµnν , Q:=∇µnν∇νnµ−(∇µnµ)2

Dynamics (forallRindler observers):

δQ δnµ =0 ⇒ Gµν+ Λgµν−κ 2_T µν nµnν =0

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23/27–

Emergent gravity: setting

Padmanabhan

Total heat withinV:

Q[n] := 1 8π Z σ2 σ1 dσ Z ∂V d2y√γ Q+κ2Tµνnµnν , Q:=∇_µnν∇_νnµ−(∇_µnµ)2

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23/27–

Emergent gravity: setting

Padmanabhan

Total heat withinV:

Q[n] := 1 8π Z σ2 σ1 dσ Z ∂V d2y√γ Q+κ2Tµνnµnν , Q:=∇_µnν∇_νnµ−(∇_µnµ)2

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24/27–

Emergent gravity: cosmology

•Metric not fundamental,other quantum d.o.f.

•Example ofunimodular gravity, e.o.m.s invariant undershift

symmetry:Λis anarbitrary constant.

•If afundamental principlefixed the value ofΛ, the shift symmetry would not change it.

•Holography and statistical mechanics?N_c=# modes

accessible to our causal patchVHduring radiation-dust era.

Emergent gravity:Nc =# d.o.f. populating the Hubble sphere

∂VH. Expansion rate of radiation-dust era is the same as of the

inflationary era and4πis precisely the number of d.o.f. of the boundary of an elementary Planck ball,

N_∂VPl = (4π`

2

Pl)/`2Pl=4π.

Nc =? N∂VPl, Λ∝e

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24/27–

Emergent gravity: cosmology

N_∂VPl = (4π`

2

Pl)/`2Pl=4π.

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24/27–

Emergent gravity: cosmology

N_∂VPl = (4π`

2

Pl)/`2Pl=4π.

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24/27–

Emergent gravity: cosmology

•Holography and statistical mechanics?

N_c=# modes

N_∂VPl = (4π`

2

Pl)/`2Pl=4π.

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24/27–

Emergent gravity: cosmology

N_∂VPl = (4π`

2

Pl)/`2Pl=4π.

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24/27–

Emergent gravity: cosmology

∂VH.

Expansion rate of radiation-dust era is the same as of the inflationary era and4πis precisely the number of d.o.f. of the boundary of an elementary Planck ball,

N_∂VPl = (4π`

2

Pl)/`2Pl=4π.

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24/27–

Emergent gravity: cosmology

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Outline

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25/27–

Comparison

: How far from realistic cosmology?

Asymptotic safety: types off(R)actions naturally

produced. Λproblem reformulated.

Multi-scale spacetimes:Λproblem reformulated.

WDW QC: probabilistic interpretation forΛproblem.

Causal dynamical triangulations: de Sitter universe emerges from full quantum gravity.

Group field theory: cosmology from full theory, LQC dynamics possibly obtained.

Non-local gravity: big bang removed.

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26/27–

More can be found in . . .

Classical and Quantum

Cosmology

(Gradu-ate Texts in Physics,

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Discussion

Thank you! ¡Muchas gracias! Grazie! Danke schön!