Vacuum Energy versus Dark Energy Manuel Asoreyicc.ub.edu/congress/ESP-RUS2011/Talks... · Vacuum...

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


• 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

E = T00 =1

2

∫

k2√

k2dk =π2

4Ω4


• Vacuum Energy

E = T00 =1

2

∫

k2√

k2dk =π2

4Ω4

• Vacuum Pressure

Pi = Tii = −1

6

∫

k4

√k2

dk = −1

3E


• Energy Momentum Tensor

Tμν =π2

4Ω4

1 0 0 0

0 −13 0 0

0 0 −13 0

0 0 0 −13


• Point Splitting regularization [Christensen]

Tμν =1

2π2ε(ημν − 4nμnν)


• 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


• Pauli Villars regularization

Tμν =π2

4Ω4ημν


• 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 Temperature T





• 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.


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?.



’On the permissible curvature of space?K Schwarzschild

Vierteljahrschrift d. Astronom. Gesellschaft. 35 337-47 (1900)



’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)


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







• 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

Quadrupole and octupole alignment associated withSouthern hemisphere cool fingers

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.


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

Hantzsche-Wendt Space

Gravitational effective action

• Effective action S(g)



Sren(g) = Sloc(g) + Sanom(g) + Stop(g)

Sloc(g) =

∫

d4x√

−g{

α1C2+ α2E + α3 R

}



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 ,


•• 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.


• 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







• Cosmological Backreaction and Space Topology

• No relation with Cosmological Constant Problem

Vacuum Energy versus Dark Energy Manuel Asoreyicc.ub.edu/congress/ESP-RUS2011/Talks... · Vacuum...

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