Dimensional Reduction in the Early Universe

Dimensional Reduction in the Early
Universe
Giulia Gubitosi
University of Rome ‘Sapienza’
References:
PLB 736 (2014) 317
PRD 88 (2013) 103524
PRD 88 (2013) 041303
PRD 87 (2013) 123532
(with G. Amelino-Camelia, Michele Arzano, João Magueijo)
ESQG - SISSA - September 2014
Spectral dimension - intro
• Heat diffusion on a Riemannian manifold:
⇥
K( 0 , , s) +
⇥s
K( 0 , , s) = 0
(heat equation for diffusion process from ⇠0 to ⇠ during diffusion time s )
• Return probability density
1
P (s) =
V
K =< |e
s
|
0
>
Z
p
d |g|K( , , s)
1 X
e
P (s) =
V j
js
(sum over eigenvalues of the Laplacian)
Spectral dimension - intro
• Heat diffusion on a Riemannian manifold:
⇥
K( 0 , , s) +
⇥s
K( 0 , , s) = 0
(heat equation for diffusion process from ⇠0 to ⇠ during diffusion time s )
• Return probability density
1
P (s) =
V
K =< |e
s
|
0
>
Z
p
d |g|K( , , s)
1 X
e
P (s) =
V j
js
(sum over eigenvalues of the Laplacian)
only eigenvalues up to 1/s contribute, so
s is related to the scale that is probed
Spectral dimension - intro
• Heat diffusion on a Riemannian manifold:
⇥
K( 0 , , s) +
⇥s
K( 0 , , s) = 0
(heat equation for diffusion process from ⇠0 to ⇠ during diffusion time s )
• Return probability density
1
P (s) =
V
Z
p
d |g|K( , , s)
• Spectral dimension
dS (s) =
d ln P (s)
2
d ln s
Spectral dimension and geometry
• The return probability density is related to the manifold geometrical
properties
X
1
n
P (s) =
an s
(heat trace expansion)
d/2
V (4 s)
n
a0 =
Z p
|g|,
Z p
a1 ⇥
|g|R,
a2 ⇥
Z p
⇥
|g| 5R
2
2Rµ R
µ
+ 2(Rµ
⇥⇤ )
2
⇤
, ...
• flat space (d dimensions):
P (s) = (4 s)
d/2
dS (s) ⌘ d
• generic smooth Riemannian manifold (d dimensions):
P
n
dS (s ! 0) = d
n
a
s
n
n=1
dS (s) = d 2 P
n
a
s
n
dS (s ! 1) = 0
n=0
Spectral dimension in Quantum Gravity
• IR limit probes global geometry, intermediate scales probe local (flat)
geometry, UV limit probes quantum spacetime properties
example (3d CDT)
CDT data N=70 000
fit
3
3
3
2.5
2.8
2
D
s
Ds
2.6
1.5
2.4
2
1
2.2
0.5
2
1
2
3
4
σ / N2/3
5
IR behavior
6
7
8
0
50
100
150
200
250
300
σ
[D. Benedetti and J. Henson PRD 2009]
UV behavior
2: The Spectral dimension data from CDT simulations with N = 200k against scaled diﬀusion time Figure 4: The Spectral dimension data with number of simplices N = 70k, at small diﬀusion times,
N −2/3 , plotted in black, fitted to the spectral dimension plot for the scaled sphere with r = 1.20, plotted with error bars in black. The exponential fit is superimposed in a lighter colour.
6, superimposed in green. The two-parameter fit agrees with the data for σ
˜ ! 1.5. at very low
he “quantum correction” to the posited classical geometry is negative, but it becomes positive in
The large scale dimension is consistent with 3 here, even with the assumptions given abov
rmediate range.
• most QG theories find running spectral dimension in the UV ( s ! 0 )
and one expects the finite size error to be small and negative. This is similar to the situatio
found by AJL in the 4D case, where the large scale dimension is consistent with 4. However, a
as found in the 4D case (this is investigated more fully in the next section). However, there
one goes to yet higher N , the technique begins to slightly overestimate the dimension in the 3
ntermediate regime in which the quantum correction is positive, and persists for a longer
(QG)
dS (s
! 0) 6= d
Spectral dimension from dispersion relation
• dispersion relation is the momentum space representation of the
Laplacian
2
2
= f (k )
2
t
DL =
2
f( r )
spectral dimension is probed by a fictitious diffusion process
on spacetime, governed by the “Wick rotated” Laplacian
operator (in flat ST)

⇥
+
⇥s
⇥t2 + f ( r2 )
K( 0 , , s) = 0
• the return probability can be written as P (s) =
and the spectral dimension
R
D
d kd
R
dS (s) = 2s
⇥
2
2
⇤
Z
s(
dD k d⇥
e
D+1
(2 )
2
+ f (k ) e
dD k d e s( 2 +f (k2 ))
s(
2
+f (k2 ))
+f (k2 ))
[T. Sotiriou, M. Visser, S. Weinfurtner PRD 2011]
From MDR to RSD - an example
• the ansatz (Euclidean MDR)
⇥ 2 + p2 1 + ( p)2
=0
gives the general result
D
dS (0) = 1 +
1+
ds HsL
(for D spatial dimensions)
4.0
3.5
spectral dimension for D=3
blue:
3.0
purple:
=1
=2
2.5
10
-4
s
0.01
1
100
10
4
( dS (0) = 2
for
= 2 and D+1=3+1 )
Spectral dimension or something else?
• Physical characterization of spectral dimension suffers from a few
issues
Euclideanization
Return probability not always well defined
[Calcagni, Eichhorn, Saueressig, PRD 2013]
we would like to have a more physically compelling
characterization of ‘dimensionality’
(which could be robustly linked to observable features of a given QG theory)
let’s use again the toy model of MDR to look for an
alternative
Momentum space Hausdorff dimension
• consider again a MDR of the form ! 2 + p2 1 + ( p)2
=0
• change momentum variables to make the dispersion relation trivial
p˜ = p
p
1 + ( p)2
• change the momentum space measure accordingly
p
D 1
dp ⇠ p˜
D
1
1+
d˜
p
(in the UV)
energy-momentum space (Hausdorff) dimension is effectively
modified in the UV:
dH = 2 +
D
1
1+
D
=1+
1+
the UV momentum space Hausdorff dimension matches the UV spectral
dimension of spacetime with modified Laplacian of the same kind
(true only in the UV!)
Momentum space running dimension
• the momentum space/spectral dimension UV correspondence holds also for
general MDRs of the form:
2
2
⌦(!, p) ⌘ ! + p +
• the return probability reads:
P (s) =
Z
2
`t
t
! 2(1+
t)
+ `2x x p2(1+
D
d k d!
e
(2⇡)D+1
x)
=0
s⌦(!,p)
after changing variables (in the UV)
!
˜/E
p˜ / p
1+
1+
t
P (s) /
x
Z
d˜
! d˜
p p˜
D
x 1
x +1
!
˜
t
x +1
e
s(˜
! 2 +p˜2 )
standard Laplacian but deformed measure, from which we can read dimension
dH
1
=
1+
t
D
+
1+
x
Dimensional reduction without a preferred frame?
• until now we have considered a dispersion relation that can hold in
only one preferred frame
does this mean that running spectral dimensions implies breakdown of
relativistic symmetries, even for more fundamental QG theories?
• interplay between dispersion relation and measure can be used to
build a relativistic theory
in fact one can make the return probability density invariant by introducing a
non-trivial measure on momentum space
but will the theory run to same values of dimension?
Dimensional reduction without a preferred frame
• curved momentum space with de Sitter metric (Euclidean)
momentum space metric:
measure:
dµ(E, p) =
p
ds2 = dE 2 + e2`E
3
X
dp2j
j=1
gdEd3 p = e3
E
dEd3 p
the isometries of de Sitter momentum space define the
deformed relativistic symmetries
Laplacian:
2
C` (1 + ` C` )
✓ ◆
4
E
2
+e
where C = 2 sinh
2
E
|⇥
p| 2
is invariant under the
deformed relativistic symmetries
P (s) ⇠
Z
2 3 E
dEdp p e
e
sC` (1+
2
C` )
Dimensional reduction without a preferred frame
• change of variables to make the dispersion relation trivial in the UV:
˜=e
E
rˆ = r
E/2
/⇥ = r cos( ), p˜ = e
E/2
p = r sin( )
+1
P (s) ⇠
P (s) ⇠
Z
Z
5
drr e
dˆ
r rˆ
6
1+
sr 2(
1
+1)
e
sˆ
r2
from the integration measure in the variables with standard dispersion relation
one can read the UV spectral dimension
(and the UV Hausdorff momentum space dimension)
dS (0) = dH
2D
=
1+
(for D spatial dimensions)
it is possible to get running to 2 spectral/Hausdorff
dimensions in the UV when D=3 and
= 2 (remember for later!)
Cosmological perturbations - standard
• second-order action of scalar cosmological perturbations
S (2)
1
=
2
Z

00
a
2
( i v) +
v2
a
d4 x v 02
equation of motion in Fourier space

a00
v=0
a
v 00 + c2 k 2
• modes matching in de Sitter ST
e
v⇠
solution of EOM at large scales:
v ⇠ F (k)a
Jeans instability term dominates
de Sitter:
a ⇠ 1/⌘
F (k) ⇠
the power spectrum P (k) ⇠ k
3
k is the comoving momentum, i.e. the
conserved charge under translations.
p=k/a is the physical momentum
ik ⇥c
solution of EOM at small scales:
pressure term dominates
the comoving gauge curvature
v
perturbation is ⇣ =
a
p
ck
(⇥ >> 1)
(⇥ << 1)
1
k 3/2
v 2
is scale invariant outside horizon
a
Cosmological perturbations with MDR
• start from same EOM for perturbations
00

00
a
a
2 2
v + c k
warning: there are a few
assumption behind this.
(see João’s talk)
v=0
• if dispersion relation is modified then c is k-dependent
2
⇥ +p
2
1 + ( p)
2
( = 2)
E
c=
⇠ ( p)2 ⇠
p
=0
• solution of EOM at small scales:
e
v⇠
the power spectrum P (k) ⇠ k 3
v
a
2
before the mode exits the horizon!
ik ⌘c
p
ck
a⇠
e
✓
k
a
◆2
(in the UV)
ik ⌘c
k 3/2
a
is already scale invariant
(no need for inflationary expansion)
Should we blame dimensional reduction?
• we got scale invariance with the modified dispersion relation that
presents running of spectral dimension (and momentum space
Hausdorff dimension) to 2
• scale invariance is achieved thanks to the very peculiar UV time
dependence of the speed of perturbations. Only MDRs with same UV
behavior would work
• is there a deeper connection between running to 2 dimensions and
scale invariance?
Gravity in dimensionally reduced momentum space
• momentum space dimensional reduction does affect equation for
perturbations (in the linearizing variables)
quadratic action for perturbations, again:
S2 =
Z
3
d⌘d k a
⇥
2
02
2 2
⇣ +c k ⇣
change of variables (for comoving momenta k):
p
k˜ = k 1 + ( k)2
⇤
2
k dk ⇠ k˜
2
⇣=
2
1+
in these new variables the dispersion relation looks trivial
S2 =
Z
d⇥dk˜ k˜
2
1+
a
2
"
˜2
k
02
+ 2
a
2
#
dk˜
v
a
Gravity in dimensionally reduced momentum space
• momentum space dimensional reduction does affect equation for
perturbations (in the linearizing variables)
quadratic action for perturbations, again:
S2 =
Z
3
d⌘d k a
⇥
2
02
2 2
⇣ +c k ⇣
change of variables (for comoving momenta k):
p
k˜ = k 1 + ( k)2
⇤
2
k dk ⇠ k˜
2
2
1+
in these new variables the dispersion relation looks trivial
S2 =
Z
the effective speed of light is
d⇥dk˜ k˜
c⇠a
2
1+
a
2
"
˜2
k
02
+ 2
a
⇣=
2
v
a
dk˜
#
, we redefine time units to make it trivial
Gravity in dimensionally reduced momentum space
• action in the linearizing units
S2 =
Z
d⇥ dk˜ k
• EOM for perturbations:

˜2
v¨ + k
2
1+
z
2
h
˙2
˜2 2
+k
z¨
v=0
z
i
(z = a1
( =
2
)
v/z)
Gravity in dimensionally reduced momentum space
• action in the linearizing units
S2 =
Z
d⇥ dk˜ k
2
1+
z
• EOM for perturbations:

˜2
v¨ + k
for
2
h
˙2
˜2 2
+k
i
(z = a1
( =
z¨
v=0
z
= 2 z is time independent ( z ⌘ 1 )
the effect of expansion disappears and the theory is effectively
conformal invariant
(more on this in João’s talk!)
2
)
v/z)
So, is running to 2 special?
• many QG theories favor UV running of spectral dimension to 2
- Causal Dynamical Triangulation in 3d and 4d
- asymptotically safe gravity in 4d
[J. Ambjorn, J. Jurkiewicz and R. Loll, PRL
2005]
[D. Benedetti and J. Henson PRD 2009]
[D. F. Litim, PRL (2004)]
- Horava-Lifshitz gravity in 4d with characteristic exponent z=3
[P. Horava, PRL 2009]
- Loop Quantum Gravity
[L. Modesto, CQG 2009]
• we have seen that running to 2 dimensions is also compatible with
relativistic invariance
• the MDR which produces running to 2 dimensions is also the one that
gives a scale invariant spectrum
Conjecture:
is scale invariance of primordial perturbations a common feature of all
theories with 2 UV spectral dimensions?
Conclusions
• Running to spectral dimension of 2 in the UV is a common feature of many
Quantum Gravity theories
• UV running of spectral dimension is associated to UV running of momentum
space Hausdorff dimension to the same value
• Running dimensions don’t necessarily break Lorentz invariance
• Scale invariant spectrum for primordial perturbations iff the running goes to 2 ?

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