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Vacuum Energy versus Dark Energy
Manuel AsoreyUniversidad de Zaragoza
Dual year Russia-Spain
Barcelona, October 2011
VACUUM ENERGY VERSUS DARK ENERGY
• Dark Energy
Tμν = ρ
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
• Conformal matter
Tμν = ρ
1 0 0 0
0 −13 0 0
0 0 −13 0
0 0 0 −13
VACUUM ENERGY VERSUS DARK ENERGY
• Vacuum Energy
E = T00 =1
2
∫
k2√
k2dk =π2
4Ω4
• Vacuum Pressure
Pi = Tii = −1
6
∫
k4
√k2
dk = −1
3E
VACUUM ENERGY VERSUS DARK ENERGY
• Energy Momentum Tensor
Tμν =π2
4Ω4
1 0 0 0
0 −13 0 0
0 0 −13 0
0 0 0 −13
VACUUM ENERGY VERSUS DARK ENERGY
• Point Splitting regularization [Christensen]
Tμν =1
2π2ε(ημν − 4nμnν)
VACUUM ENERGY VERSUS DARK ENERGY
• Point Splitting regularization [Christensen]
Tμν =1
2π2ε(ημν − 4nμnν)
Tμν = − 2
π2ε
−14 + n2
0 n1n2 n1n3 n1n4
n1n214 + n2
1 n2n3 n2n4
n1n3 n2n314 + n2
2 n3n4
n1n4 n2n4 n3n414 + n2
3
VACUUM ENERGY VERSUS DARK ENERGY
• Pauli Villars regularization
Tμν =π2
4Ω4ημν
Tμν =π2
4Ω4
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
FINITE CORRECTIONS TO VACUUM ENERGY
Possible sources IR scales
• Massive Fields M
• Space Curvature R
FINITE CORRECTIONS TO VACUUM ENERGY
Possible sources IR scales
• Massive Fields M
• Space Curvature R
• Finite Temperature T
FINITE CORRECTIONS TO VACUUM ENERGY
Possible sources IR scales
• Massive Fields M
• Space Curvature R
• Finite Temperature T
• Finite Volume Space V
SPACE CURVATURE AND COSMOLOGY
If I presume to present a few remarks that haveneither any real practical applicability nor anypertinent mathematical meaning, my excuse is thatthe topic we are considering has a particularattraction for many of you because it presents anextension of our view of things way beyond that dueto our accessible experience, and opens the moststrange prospects for later possible experiences.
SPACE CURVATURE AND COSMOLOGY
If I presume to present a few remarks that haveneither any real practical applicability nor anypertinent mathematical meaning, my excuse is thatthe topic we are considering has a particularattraction for many of you because it presents anextension of our view of things way beyond that dueto our accessible experience, and opens the moststrange prospects for later possible experiences.
That it requires a total break with the astronomers?deeply entrenched views cannot but seem a furtheradvantage to anyone convinced that all knowledge isrelative.
SPACE TOPOLOGY AND COSMOLOGY
We are considering the possibility of curvature ofspace. The questions as to how far we have pushedback the boundaries of this fairyland can now beasked: how small is the curvature of space? and whatis a lower bound for its radius of curvature?.
SPACE TOPOLOGY AND COSMOLOGY
We are considering the possibility of curvature ofspace. The questions as to how far we have pushedback the boundaries of this fairyland can now beasked: how small is the curvature of space? and whatis a lower bound for its radius of curvature?.
’On the permissible curvature of space?K Schwarzschild
Vierteljahrschrift d. Astronom. Gesellschaft. 35 337-47 (1900)
SPACE TOPOLOGY AND COSMOLOGY
We are considering the possibility of curvature ofspace. The questions as to how far we have pushedback the boundaries of this fairyland can now beasked: how small is the curvature of space? and whatis a lower bound for its radius of curvature?.
’On the permissible curvature of space?K Schwarzschild
Vierteljahrschrift d. Astronom. Gesellschaft. 35 337-47 (1900)
= -0.0125+0.0064
−0.0067
Ωk = -0.0111+0.0060
−0.0063WMAP2011
= -0.0057+0.0067
−0.0068
SPHERICAL MANIFOLDS
• Sphere S3
• Lens spaces S3/ZZ∗q (order ZZ∗q= 2q)
• Prisma spaces S3/D∗
q (order D∗
q= 4q)
• Tetrahedral space S3/T ∗, (order T ∗= 24)
• Octahedral space S3/O∗,(order O∗= 48)
• Poincaré Dodecahedral space S3/Y ∗, (order Y ∗= 120)
SPHERICAL MANIFOLDS
• Sphere S3
• Lens spaces S3/ZZ∗q (order ZZ∗q= 2q)
• Prisma spaces S3/D∗
q (order D∗
q= 4q)
• Tetrahedral space S3/T ∗, (order T ∗= 24)
• Octahedral space S3/O∗,(order O∗= 48)
• Poincaré Dodecahedral space S3/Y ∗, (order Y ∗= 120)
SPACE CURVATURE AND COSMOLOGY
CMB probe Space Topology
• Closed spaces leave their fingerprints in thecontributions to low multipoles
SPACE CURVATURE AND COSMOLOGY
CMB probe Space Topology
• Closed spaces leave their fingerprints in thecontributions to low multipoles
• Supression of low multipoles: quadrupole,octupole, . . .
SPACE CURVATURE AND COSMOLOGY
CMB probe Space Topology
• Closed spaces leave their fingerprints in thecontributions to low multipoles
• Supression of low multipoles: quadrupole,octupole, . . .
• Quadrupole and octupole alignment is associatedwith Southern hemisphere cool fingers
SPACE CURVATURE AND COSMOLOGY
CMB probe Space Topology
• Closed spaces leave their fingerprints in thecontributions to low multipoles
• Supression of low multipoles: quadrupole,octupole, . . .
• Quadrupole and octupole alignment is associatedwith Southern hemisphere cool fingers
• Asymmetry between even and odd multipoles
SPACE CURVATURE AND COSMOLOGY
CMB probe Space Topology
• Closed spaces leave their fingerprints in thecontributions to low multipoles
• Supression of low multipoles: quadrupole,octupole, . . .
• Quadrupole and octupole alignment is associatedwith Southern hemisphere cool fingers
• Asymmetry between even and odd multipoles
• Gaussianity of likelihood estimates starts forl > 32
SPACE CURVATURE AND COSMOLOGY
CMB probe Space Topology
• Closed spaces leave their fingerprints in thecontributions to low multipoles
• Supression of low multipoles: quadrupole,octupole, . . .
• Quadrupole and octupole alignment is associatedwith Southern hemisphere cool fingers
• Asymmetry between even and odd multipoles
• Gaussianity of likelihood estimates starts forl > 32
• Circles in the Sky
CLOSED SPACES
Closed spaces provide weaker contributions to lowmultipoles. In general suppress low multipolarcomponents (quadrupole, octupole, . . . )
CLOSED SPACES
Asymmetry between even and odd multipoles
0 100 200 300 400 500 600Multipole l
-0.3
-0.2
-0.1
-0.0
0.1
0.2
Eve
n E
xces
s
Monte Carlo mean & scatter
0 100 200 300 400 500 600Multipole l
-3
-2
-1
0
1
2
3
Sig
nific
ance
(un
its o
f σ)
Deconstructing Vacuum energy
Evac =1
2
∞∑
n=0
dn λn
Evac = Eloc + Eanom + Etop.
• Sphere S3 (radius a)
Spectral modes of conformal scalar field
λk =(k + 1)
a; dk = (k + 1)2 (degeneracy)
ES3 =1
240
1
a=
1
480
1
a+
1
480
1
a
Eloc =1
480
1
a; Eanom =
1
480
1
a; Etop = 0
Spherical spaces
dk(II) =(k+1)2
dk(ZZ2q+1)=(k+1) ([(k+1)/(2q+1)]+ (1+ (-1)k - [(k+1)/(2q+1)](2q+1))/2)
d2l (ZZ2q) = (2l+1) (2 [(2l+1)/(2q)]+1)
d2l (DD∗
q )=(2l+1)([l/q]+1/2 (1+ (-1)l))
d2l (TT∗) =(2l+1)([l/3]+ 2[l/2]+ 1 - l ); l6=1,2,5
d2l (OO∗)= (2l+1)([l/4]+ [l/3]+ [l/2]+ 1 - l); l 6= 1,2,3,5,7,11
d2l (YY∗) =(2l + 1)([l/5]+[l/3]+[l/2]+1-l) ; l 6=1,2,3,4,5,7,8,9,11,13,14,17,19,23,29
Spherical spaces
dk(II) =(k+1)2
dk(ZZ2q+1)=(k+1) ([(k+1)/(2q+1)]+ (1+ (-1)k - [(k+1)/(2q+1)](2q+1))/2)
d2l (ZZ2q) = (2l+1) (2 [(2l+1)/(2q)]+1)
d2l (DD∗
q )=(2l+1)([l/q]+1/2 (1+ (-1)l))
d2l (TT∗) =(2l+1)([l/3]+ 2[l/2]+ 1 - l ); l6=1,2,5
d2l (OO∗)= (2l+1)([l/4]+ [l/3]+ [l/2]+ 1 - l); l 6= 1,2,3,5,7,11
d2l (YY∗) =(2l + 1)([l/5]+[l/3]+[l/2]+1-l) ; l 6=1,2,3,4,5,7,8,9,11,13,14,17,19,23,29
Luminet et al 2003
VACUUM ENERGIES
• Lens spaces S3/ZZq
EZq = −q4+ 10q2 − 14
720q
1
a
• Prisma spaces S3/Dq
EDq = −20q4+ 8q2
+ 180q − 7
1440q
1
a
• Polyhedric spaces S3/T , S3/O, S3/Y
ET = −3761
8640
1
aEO = −11321
17280
1
aEY = −43553
43200
1
a
Vacuum Energy: Spherical Spaces
q0 2 4 6 8 10
Eo
-10
-8
-6
-4
-2
0
T* O* Y*I
D
Z
D*q
q
M. A. , I. Cavero, J. M. Munoz-Castaneda [2009]
Vacuum Energy: Spherical Spaces
• Energy Momentum Tensor
Tμν = Evac
1 0 0 0
0 −13 0 0
0 0 −13 0
0 0 0 −13
is traceless
VACUUM ENERGIES
Flat spaces
•Torus T 3
ET 3 = − 1
2π2a
∫
∞
0
dt t(θ33(e−t) − 1) = −0.8375
a
• Twisted Six-Turn Torus E5
E5 = −0.991
a
• Hantzsche-Wendt Space E6
E6 = −0.321
a
Vacuum Energy: Flat Spaces
0 1 2 3 4
Eo
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
T
E
E
3
5
6
E
M. A. , I. Cavero, J. M. Munoz-Castaneda [2009-2011]
Vacuum Energy Density
Twisted Torus
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
-25.1
-24.7
-24.5
Quarter Space
Scale
Vacuum Energy Density
Twisted Torus
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
-25.1
-24.7
-24.5
Quarter Space
Scale
Gravitational effective action
• Effective action S(g)
Sren(g) = Sloc(g) + Sanom(g) + Stop(g)
Sloc(g) =
∫
d4x√
−g{
α1C2+ α2E + α3 R
}
Gravitational effective action
• Effective action S(g)
Sren(g) = Sloc(g) + Sanom(g) + Stop(g)
Sloc(g) =
∫
d4x√
−g{
α1C2+ α2E + α3 R
}
Sanom(g)=b
8(4π)2
∫
d4x
∫
d4x′√
−g
(
E +2
3R
)
(x) −14 (x, x′)
√
−g
[(
E +2
3R
)]
(x′) +
(
c − 2
3b
)
1
12(4π)2
∫
d4x√
−g R2
4 ≡ 2 − 2Rμν∇μ∇ν +2
3R − 1
3(∇μR)∇μ ,
Gravitational effective action
•• Stop(g) depends on the topology of space
•• R2 term can appear in the three sectors
•• Sloc(g) is dependent on renormalization scheme
•• Sanom(g) is ambiguous
•• Stop(g) is universal and hard to calculate.
VACUUM ENERGY VERSUS DARK ENERGY
• Except for the S3 sphere vacuum reactive force isattractive
• Gauge Wilson lines will introduce repulsiveforces, but break homogeneity and isotropy
VACUUM ENERGY VERSUS DARK ENERGY
• Except for the S3 sphere vacuum reactive force isattractive
• Gauge Wilson lines will introduce repulsiveforces, but break homogeneity and isotropy
• Topological vacuum energy is universal
VACUUM ENERGY VERSUS DARK ENERGY
• Except for the S3 sphere vacuum reactive force isattractive
• Gauge Wilson lines will introduce repulsiveforces, but break homogeneity and isotropy
• Topological vacuum energy is universal
• Ambiguities in Vacuum Energy
VACUUM ENERGY VERSUS DARK ENERGY
• Except for the S3 sphere vacuum reactive force isattractive
• Gauge Wilson lines will introduce repulsiveforces, but break homogeneity and isotropy
• Topological vacuum energy is universal
• Ambiguities in Vacuum Energy
• Vacuum Energy and Particle Creation
VACUUM ENERGY VERSUS DARK ENERGY
• Except for the S3 sphere vacuum reactive force isattractive
• Gauge Wilson lines will introduce repulsiveforces, but break homogeneity and isotropy
• Topological vacuum energy is universal
• Ambiguities in Vacuum Energy
• Vacuum Energy and Particle Creation
• Cosmological Backreaction and Space Topology
VACUUM ENERGY VERSUS DARK ENERGY
• Except for the S3 sphere vacuum reactive force isattractive
• Gauge Wilson lines will introduce repulsiveforces, but break homogeneity and isotropy
• Topological vacuum energy is universal
• Ambiguities in Vacuum Energy
• Vacuum Energy and Particle Creation
• Cosmological Backreaction and Space Topology
• No relation with Cosmological Constant Problem