Talk given at Cambridge DAMTP on Friday, 20 June 2008. Describes recent work on understanding what is necessary to embed accelerating cosmology in higher-dimensional theory.
7.
Theorem (Gibbons ’84)
To obtain a four-dimensional de Sitter universe from a static warped
reduction on closed compact manifold M, if the Einstein equations
hold, one must violate the Strong Energy Condition (SEC).
SEC requires
8.
Maldacena and Nunez ‘01
One cannot obtain four-dimensional de Sitter space from static
compactifications with all the following properties:
1. Einstein-Hilbert gravity in higher-D
2. Potential energies all non-positive: V ≤ 0
3. Fields in the theory:
a. ... have no mass terms
b. ... have positive definite kinetic terms
4. Extra dimensions M:
a. .... give finite four-dimensional GN
(M has finite warped volume)
b. ... compact, no boundary
9.
Maldacena and Nunez ‘01
One cannot obtain four-dimensional de Sitter space from static
compactifications with all the following properties:
1. Einstein-Hilbert gravity in higher-D
2. Potential energies all non-positive: V ≤ 0
imply 3. Fields in the theory:
SEC
a. ... have no mass terms
b. ... have positive definite kinetic terms
4. Extra dimensions M:
a. .... give finite four-dimensional GN
(M has finite warped volume)
b. ... compact, no boundary
10.
A warping no-go theorem
positive definite
1
Warping requires NEC
p
ar
W
SE
NEC violated if... 0 C
SEC violated if...
-1
-> Obtaining a warp factor requires
neither SEC nor NEC violation.
-2 -1 0 1
12.
Improvements on previous results
• Time-dependent compactifications
• Four-dimensional cosmology is not exactly de Sitter
• Weakening of energy conditions <--> strengthening of theorems.
• Claim: transient acceleration is allowed, and explicit bounds on the
number of allowed e-foldings can be constructed.
Statement
• For each number k of extra dimensions
• There exists a threshold wthresh
• For w < wthresh there is a bound N(w) on the number of e-foldings
• Varying w (1): for any w(t) the bound N[w(t)] obtained by quadrature.
• Varying w (2): if w < w✻ then N[w(t)] < N(w✻).
• Exceeding the bound violates an energy condition
• Can be the strong energy condition (SEC)
• Can also be the null energy condition (NEC)
13.
Curved Curvature-free CRF
R≠0 on M R=0 on M
Non-Abelian Special holonomy Klebanov-Strassler
continuous warped throat
Sp(n)
isometries** Spin(7)
SU(n) (Calabi-Yau) Giddings-Kachru-
... includes models which G2 (M theory)
Polchinski flux
obtain 4D gauge
symmetries by KK solutions
One-dimensional
reduction
Original Kaluza-Klein
Randall-Sundrum
Rugby-ball SLED
Tori
Rn / Zn
** We only know these cannot be
with R ≥ 0
Ricci-flat, which is a stronger
condition than “curvature-free.”
16.
4D universe with constant w embedded with k extra dimensions,
which are Ricci-flat and evolve by breathing mode only.
a(η) ∼ η 2/(1+3w)
2
1+w dψ
ρ+P = H=
2
3 dη
1+w
√
ψ(η) = ± ln η 6 + ψ0
V (ψ) = V0 e 3(1+w)ψ
1 + 3w
2c
ds2 = A(η) −dη + dx2 + exp ψ(η) ds2
2 2
4+k 3 k
k
2k
a(η) = ecψ/2 A(η)
c=
k+2
17.
Reconstruct (4+k)D metric and use Einstein
equations to compute stress-energy tensor
w
NEC OK NEC OK
TMN nMnN
P = -2 ρ
NEC violated
NEC violated
w k
√
V (ψ) = V0 e 3(1+w)ψ
This unusual (even pathological) behavior
is completely invisible from 4D
18.
For a more general case...
• Extra dimensions could distort
•Anisotropically
•Inhomogeneously
• Warped extra dimensions
• Non-Ricci-flat extra dimensions
• Scalar field might not be breathing mode
• Metric moduli may not act like quintessence
• Completely different scalar could drive accel.
• ... etc
19.
Assumptions
• Higher-dimensional action has Einstein-Hilbert form
...includes g(φ)R and F(R) models
• All four-dimensional statements refer to the Einstein
frame metric and its associated cosmology.
• M closed and compact, or a quotient of such, as M =M’/G
• Arbitrary other matter fields present in the action
20.
Three tools:
1. Gauge choice
2. Averaging
3. NEC lemma
21.
A purely scalar metric transformation which leaves the total
volume of M invariant is gauge-equivalent to zero.
Local: Global:
The last term integrates to zero,
and can only cancel out the piece
of the first term which integrates
to zero -- eg, the non-zero mode
23.
If either then NEC violated
1. Diagonalise τab obtain real eigenvalues λ1, λ2,... λk.
2. To each λj, associate null vector na = (1,0,0,0,εnλj )
3. But PDk is the average of the λj. Choose j* with smallest λj*.
27.
1. Metric deformations
enter with negative sign
Must be 2. Negative
3. Coefficients depend on
non-negative definite when
averaging parameter A.
to satisfy NEC accelerating
4. “Arbitrary” warp
function enters
the NEC condition
5. kN depends on dξ/dt
28.
Non-negative for all A
Positive or negative depending on A
“Optimising” A: choice of A for which all coefficients are non-negative
An optimising A always exists.
29.
“Optimal solution”: saturates the inequalities
ρD+PD3 ≥ 0 and ρD+PDk ≥ 0
& any other solution has fewer e-foldings consistent with the NEC.
Further assume that w = constant
These equalities define the differential equation
obeyed by the optimal solution for constant w.
30.
“Optimal solution”: saturates the inequalities
ρD+PD3 ≥ 0 and ρD+PDk ≥ 0
& any other solution has fewer e-foldings consistent with the NEC.
Further assume that w = constant
These equalities define the differential equation
obeyed by the optimal solution for constant w.
34.
k=1-4 k=5-10 and k = 11 ...
Number of e-foldings
w w
35.
(Ricci-flat only)
w
w
(Curved and not CRF only)
number of extra dimensions
36.
Varying w(t): if w(t) < w✻ then N[w(t)] < N(w✻).
Therefore ζ goes faster, and has less to go, for the w(t).
37.
“Loopholes”
1. Curvature of M
could go either way.
what metric should we use?
2. Quantum effects
If we are unable to impose the Einstein equations,
how literally should we take the extra dimensions?
3. Higher-derivative corrections to GR
if effectively NEC-violating, do we avoid usual problems?
4. Negative-tension objects (not really a loophole)
braneworld boundaries, O-planes
38.
Bubbles of nothing Accelerating to decelerating
transition after inflation
Kate Marvel’s talk & work-in-progress
Are additional sources
of NEC violation required?
M
negative
inﬂating
tension
boundary
nothing
M
radiation
dominated
39.
Observational prospects
(work-in-progress with Paul Steinhardt)
More precise measurements of w(a) and bounds on
variation of fundamental constants can prove surprisingly
powerful probes of extra-dimensional physics.
Simple families of models give
promising results. Study of more
complicated models requires a
little more computing power.
Stay tuned....
[Preliminary]
40.
Conclusions
• We can make very general statements about extra-
dimensional physics from the observation that the
Universe is/was accelerating
• For a broad class of theories, this implies violation of
energy conditions, more severe than already known
• Raises interesting questions about the role of
corrections to GR, explicit higher-D solutions, and
exotic negative-tension objects
• Can make refined measurements of w and variation
of fundamental constants into powerful probes of
extra-dimensional physics.
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