dimensional reduction in the early universe

Dimensional Reduction in the Early Universe

Giulia GubitosiUniversity of Rome ‘Sapienza’

ESQG - SISSA - September 2014

References: PLB 736 (2014) 317

PRD 88 (2013) 103524PRD 88 (2013) 041303PRD 87 (2013) 123532

(with G. Amelino-Camelia, Michele Arzano, João Magueijo)

Spectral dimension - intro

• Heat diffusion on a Riemannian manifold:

• Return probability density

P (s) =1

V

Zd�

p|g|K(�, �, s)

K =< �|e�s�|�0 >

(sum over eigenvalues of the Laplacian)

P (s) =1

V

X

j

e��js

(heat equation for diffusion process from to during diffusion time s )⇠0 ⇠

⇥

⇥sK(�0, �, s) +�K(�0, �, s) = 0



P (s) =1

V

Zd�

p|g|K(�, �, s)

K =< �|e�s�|�0 > P (s) =1

V

X

j

e��js

(sum over eigenvalues of the Laplacian)

only eigenvalues up to 1/s contribute, so s is related to the scale that is probed



⇥

⇥sK(�0, �, s) +�K(�0, �, s) = 0




⇥

⇥sK(�0, �, s) +�K(�0, �, s) = 0


P (s) =1

V

Zd�

p|g|K(�, �, s)

• Spectral dimension

dS(s) = �2d lnP (s)

d ln s

Spectral dimension and geometry

• flat space (d dimensions):

P (s) = (4�s)�d/2 dS(s) ⌘ d

dS(s) = d� 2

Pn=1 nansnPn=0 ans

n

dS(s ! 0) = d

dS(s ! 1) = 0

• The return probability density is related to the manifold geometrical properties

P (s) =1

V (4�s)d/2

X

n

ansn

a0 =

Z p|g|, a1 ⇥

Z p|g|R, a2 ⇥

Z p|g|

⇥5R2 � 2Rµ�R

µ� + 2(Rµ�⇥⇤)2⇤, . . .

(heat trace expansion)

• generic smooth Riemannian manifold (d dimensions):

Spectral dimension in Quantum Gravity

1 2 3 4 5 6 7 8

0.5

1

1.5

2

2.5

3

σ / N2/3

Ds

Figure 2: The Spectral dimension data from CDT simulations with N = 200k against scaled diffusion time

σ̃ = σN−2/3, plotted in black, fitted to the spectral dimension plot for the scaled sphere with r = 1.20,

s = 1.96, superimposed in green. The two-parameter fit agrees with the data for σ̃ ! 1.5. at very low

times the “quantum correction” to the posited classical geometry is negative, but it becomes positive in

an intermediate range.

value, as found in the 4D case (this is investigated more fully in the next section). However, there

is an intermediate regime in which the quantum correction is positive, and persists for a longer

diffusion time than might be expected from looking at the behaviour at very low diffusion times.

5.3 Results at shorter scales

These results have implications for previous methods used to measure the large scale dimension

of CDTs. In [8], in the 4D case, Ambjørn, Jurkiewicz and Loll (AJL hereafter) take a short range

of early diffusion times, and fit the CDT spectral dimension data to the function

Ds(σ) = a−b

σ + c. (5.3)

This fit provides estimates of the dimension as σ → 0 and as σ → ∞. In particular, the estimation

of the large scale dimension rests on two assumptions. The first is that the finite size effects have

a negligible effect on the fit over the range fitted. The second is that the fit is good into the range

13

0 50 100 150 200 250 300

2

2.2

2.4

2.6

2.8

3

σ

Ds

CDT data N=70 000fit

Figure 4: The Spectral dimension data with number of simplices N = 70k, at small diffusion times, is

plotted with error bars in black. The exponential fit is superimposed in a lighter colour.

The large scale dimension is consistent with 3 here, even with the assumptions given above,

and one expects the finite size error to be small and negative. This is similar to the situation

found by AJL in the 4D case, where the large scale dimension is consistent with 4. However, as

one goes to yet higher N , the technique begins to slightly overestimate the dimension in the 3D

case.

At N = 200k we have a similarly good fit with the same function (see Fig. 5). The estimates

in this case are

Ds(∞, N = 200k) = 3.05 ± 0.04 ; (5.7)

Ds(0, N = 200k) = 2.04 ± 0.10 . (5.8)

Here, the value 3 for the large scale dimension begins to looks doubtful. This is explained

by looking at the spectral dimension plot at larger scales, as in figure 2. There we see that the

quantum correction to the spectral dimension function for the conjectured classical geometry

actually becomes positive, and stays significant for some time after the maximum value of Ds is

reached. But the above fits are taken over a short range before the maximum is reached. This

means that the assumption that the fit will have a large scale limit free of quantum effects is

15

• IR limit probes global geometry, intermediate scales probe local (flat) geometry, UV limit probes quantum spacetime properties

example (3d CDT)

UV behaviorIR behavior [D. Benedetti and J. Henson PRD 2009]

• most QG theories find running spectral dimension in the UV ( )s ! 0

d(QG)S (s ! 0) 6= d

3

2

Spectral dimension from dispersion relation• dispersion relation is the momentum space representation of the

Laplacian

spectral dimension is probed by a fictitious diffusion process on spacetime, governed by the “Wick rotated” Laplacian operator (in flat ST)

�2 = f(k2) DL = ��2t � f(�r2)

⇥

⇥s+��⇥2

t + f(�r2)��

K(�0, �, s) = 0

• the return probability can be written as

and the spectral dimension

P (s) =

ZdDk d⇥

(2�)D+1e�s(�2+f(k2))

dS(s) = 2s

RdDk d�

⇥�2 + f(k2)

⇤e�s(�2+f(k2))

RdDk d� e�s(�2+f(k2))

[T. Sotiriou, M. Visser, S. Weinfurtner PRD 2011]

From MDR to RSD - an example

• the ansatz (Euclidean MDR)

gives the general result

⇥2 + p2�1 + (�p)2�

�= 0

dS(0) = 1 +D

1 + �

( for and D+1=3+1 )� = 2dS(0) = 210-4 0.01 1 100 104

s

2.5

3.0

3.5

4.0

dsHsL

spectral dimension for D=3

blue:

purple:

� = 1

� = 2

(for D spatial dimensions)

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

• change the momentum space measure accordingly

• change momentum variables to make the dispersion relation trivial

p̃ = pp

1 + (�p)2�

pD�1dp ⇠ p̃D�1��

1+� dp̃ (in the UV)

energy-momentum space (Hausdorff) dimension is effectively modified in the UV:

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!)

• consider again a MDR of the form !2 + p2�1 + (�p)2�

�= 0

dH = 2 +D � 1� �

1 + �= 1 +

D

1 + �

Momentum space running dimension

• the return probability reads:

• the momentum space/spectral dimension UV correspondence holds also for general MDRs of the form:

after changing variables (in the UV)

standard Laplacian but deformed measure, from which we can read dimension

P (s) =

ZdDk d!

(2⇡)D+1e�s⌦(!,p)

!̃ / E1+�t

p̃ / p1+�x

P (s) /Z

d!̃dp̃ p̃D��

x

�1�

x

+1 !̃� �

t

�

x

+1 e�s(!̃2+p̃2)

dH =1

1 + �t+

D

1 + �x

⌦(!, p) ⌘ !2 + p2 + `2�t

t

!2(1+�

t

) + `2�x

x

p2(1+�

x

) = 0

Dimensional reduction without a preferred frame?

• 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

• 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?

but will the theory run to same values of dimension?

Dimensional reduction without a preferred frame

dµ(E, p) =p�gdEd3p = e3�EdEd3pmeasure:

Laplacian:

C� =4

�2sinh2

✓�E

2

◆+ e�E |⇥p|2

C`(1 + `2�C�` )

P (s) ⇠Z

dEdp p2e3�Ee�sC`(1+�2�C�` )

• curved momentum space with de Sitter metric (Euclidean)

momentum space metric: ds2 = dE2 + e2`E3X

j=1

dp2j

the isometries of de Sitter momentum space define the deformed relativistic symmetries

where is invariant under the

deformed relativistic symmetries

Dimensional reduction without a preferred frame

• change of variables to make the dispersion relation trivial in the UV:

P (s) ⇠Z

drr5e�sr2(�+1)

r̂ = r�+1P (s) ⇠

Zdr̂ r̂

61+� �1e�sr̂2

˜E = e�E/2/⇥ = r cos(�), p̃ = e�E/2p = r sin(�)

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

equation of motion in Fourier space

S(2) =1

2

Zd4x

v02 � (�iv)

2 +a00

av2�

v00 +

c2k2 � a00

a

�v = 0

• modes matching in de Sitter ST

solution of EOM at small scales: v ⇠ e�ik�⇥c

pck

(⇥� >> 1)

solution of EOM at large scales: v ⇠ F (k)a (⇥� << 1)

F (k) ⇠ 1

k3/2

the power spectrum is scale invariant outside horizonP (k) ⇠ k3��v

a

��2

the comoving gauge curvature perturbation is ⇣ = �v

a

k is the comoving momentum, i.e. the conserved charge under translations. p=k/a is the physical momentum

de Sitter: a ⇠ 1/⌘

pressure term dominates

Jeans instability term dominates

Cosmological perturbations with MDR

• start from same EOM for perturbations

v00 +

c2k2 � a00

a

�v = 0

• if dispersion relation is modified then c is k-dependent

the power spectrum is already scale invariant

before the mode exits the horizon!

⇥2 + p2�1 + (�p)2�

�= 0

(� = 2)

• solution of EOM at small scales:

P (k) ⇠ k3��v

a

��2

warning: there are a few assumption behind this.

(see João’s talk)

c =E

p⇠ (�p)2 ⇠

✓�k

a

◆2

(in the UV)

v ⇠ e�ik�⌘c

pck

a ⇠ e�ik�⌘c

�k3/2a

(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

• is there a deeper connection between running to 2 dimensions and scale invariance?

• 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

Gravity in dimensionally reduced momentum space

• momentum space dimensional reduction does affect equation for perturbations (in the linearizing variables)

quadratic action for perturbations, again:

change of variables (for comoving momenta k):

k̃ = kp1 + (�k)2� k2dk ⇠ k̃

2��1+� dk̃

S2 =

Zd⇥dk̃ k̃

2��1+� a2

"� 02 +

k̃2

a2��2#

S2 =

Zd⌘d3k a2

⇥⇣ 02 + c2k2⇣2

⇤⇣ = �v

a

in these new variables the dispersion relation looks trivial


• momentum space dimensional reduction does affect equation for perturbations (in the linearizing variables)

quadratic action for perturbations, again:

the effective speed of light is , we redefine time units to make it trivial

change of variables (for comoving momenta k):

k̃ = kp1 + (�k)2� k2dk ⇠ k̃

2��1+� dk̃

S2 =

Zd⇥dk̃ k̃

2��1+� a2

"� 02 +

k̃2

a2��2#

c ⇠ a��

S2 =

Zd⌘d3k a2

⇥⇣ 02 + c2k2⇣2

⇤⇣ = �v

a

in these new variables the dispersion relation looks trivial


• action in the linearizing units

S2 =

Zd⇥dk̃ k

2��1+� z2

h�̇2 + k̃2�2

i(z = a1�

�2 )

• EOM for perturbations:

v̈ +

k̃2 � z̈

z

�v = 0

(� = �v/z)


• action in the linearizing units

S2 =

Zd⇥dk̃ k

2��1+� z2

h�̇2 + k̃2�2

i

• EOM for perturbations:

v̈ +

k̃2 � z̈

z

�v = 0

the effect of expansion disappears and the theory is effectively conformal invariant

for is time independent ( )� = 2 z z ⌘ 1

(more on this in João’s talk!)

(z = a1��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

- Horava-Lifshitz gravity in 4d with characteristic exponent z=3

- Loop Quantum Gravity

[P. Horava, PRL 2009]

[D. F. Litim, PRL (2004)]

[L. Modesto, CQG 2009]

[J. Ambjorn, J. Jurkiewicz and R. Loll, PRL 2005][D. Benedetti and J. Henson PRD 2009]

Conjecture: is scale invariance of primordial perturbations a common feature of all theories with 2 UV spectral dimensions?

• 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

Conclusions

• Running to spectral dimension of 2 in the UV is a common feature of many Quantum Gravity theories

• Scale invariant spectrum for primordial perturbations iff the running goes to 2 ?

• 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

dimensional reduction in the early universe

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