Dimensional Reduction in the Early Universe

Transcription

1 Dimensional Reduction in the Early Universe Giulia Gubitosi University of Rome Sapienza References: PLB 736 (2014) 317 PRD 88 (2013) PRD 88 (2013) PRD 87 (2013) (with G. Amelino-Camelia, Michele Arzano, João Magueijo) ESQG - SISSA - September 2014

2 Spectral dimension - intro Heat diffusion on a Riemannian manifold: s K( 0,,s)+ K( 0,,s)=0 (heat equation for diffusion process from to during diffusion time s ) 0 Return probability density P (s) = 1 V K =< e s 0 > d p g K(,,s) P (s) = 1 V X e j (sum over eigenvalues of the Laplacian) js

3 Spectral dimension - intro Heat diffusion on a Riemannian manifold: s K( 0,,s)+ K( 0,,s)=0 (heat equation for diffusion process from to during diffusion time s ) 0 Return probability density P (s) = 1 V d p g K(,,s) K =< e s 0 > P (s) = 1 V X e j (sum over eigenvalues of the Laplacian) js only eigenvalues up to 1/s contribute, so s is related to the scale that is probed

4 Spectral dimension - intro Heat diffusion on a Riemannian manifold: s K( 0,,s)+ K( 0,,s)=0 (heat equation for diffusion process from to during diffusion time s ) 0 Return probability density P (s) = 1 V d p g K(,,s) Spectral dimension d S (s) = 2 d ln P (s) d ln s

5 Spectral dimension and geometry The return probability density is related to the manifold geometrical properties 1 X P (s) = a V (4 s) d/2 n s n n (heat trace expansion) a 0 = p g, a1 p g R, a2 p g 5R 2 2R µ R µ + 2(R µ ) 2,... flat space (d dimensions): P (s) =(4 s) d/2 d S (s) d generic smooth Riemannian manifold (d dimensions): d S (s) =d 2 Pn=1 P na ns n d S (s! 0) = d n=0 a ns n d S (s!1)=0

6 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= fit D s 1.5 D s σ / N 2/3 IR behavior [D. Benedetti and J. Henson PRD 2009] σ UV behavior most QG theories find running spectral dimension in the UV ( ) s! 0 d (QG) (s! 0) 6= d S

7 Spectral dimension from dispersion relation dispersion relation is the momentum space representation of the Laplacian 2 = f(k 2 ) D L = spectral dimension is probed by a fictitious diffusion process on spacetime, governed by the Wick rotated Laplacian operator (in flat ST) 2 t f( r 2 ) apple s + 2 t + f( r 2 ) K( 0,,s)=0 the return probability can be written as P (s) = and the spectral dimension R d D kd 2 + f(k 2 ) e s ( 2 +f(k 2 )) d S (s) =2s R dd kd e s( 2 +f(k 2 )) d D kd (2 ) D+1 e s ( 2 +f(k 2 )) [T. Sotiriou, M. Visser, S. Weinfurtner PRD 2011]

8 From MDR to RSD - an example the ansatz (Euclidean MDR) 2 + p 2 1+( p) 2 =0 gives the general result d s HsL d S (0) = 1 + D 1+ (for D spatial dimensions) spectral dimension for D=3 blue: purple: =1 = s d S (0) = 2 ( for =2and D+1=3+1 )

9 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

10 Momentum space Hausdorff dimension consider again a MDR of the form! 2 + p 2 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: d H =2+ D 1 1+ =1+ D 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!)

11 Momentum space running dimension the momentum space/spectral dimension UV correspondence holds also for general MDRs of the form: (!,p)! 2 + p 2 + `2 t t the return probability reads: P (s) =! 2(1+ t) + `2 x x p 2(1+ x) =0 d D kd! s (!,p) e (2 ) D+1 after changing variables (in the UV)! / E 1+ t P (s) / p / p 1+ x 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 d H = 1 1+ t + D 1+ x

12 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?

13 Dimensional reduction without a preferred frame curved momentum space with de Sitter metric (Euclidean) momentum space metric: ds 2 = de 2 + e 2`E 3X measure: dµ(e,p)= p gded 3 p = e 3 E ded 3 p j=1 dp 2 j the isometries of de Sitter momentum space define the deformed relativistic symmetries Laplacian: where C`(1 + `2 C` ) C = 4 2 sinh2 E 2 + e E p 2 is invariant under the deformed relativistic symmetries P (s) dedp p 2 e 3 E e sc`(1+ 2 C` )

14 Dimensional reduction without a preferred frame change of variables to make the dispersion relation trivial in the UV: Ẽ = e E/2 / = r cos( ), p = e E/2 p = r sin( ) P (s) drr 5 e sr2( +1) ˆr = r +1 P (s) dˆr ˆr e sˆr2 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) d S (0) = d H = 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!)

15 Cosmological perturbations - standard second-order action of scalar cosmological perturbations S (2) = 1 2 equation of motion in Fourier space v 00 + d 4 x modes matching in de Sitter ST solution of EOM at small scales: pressure term dominates applev 02 ( i v) 2 + a00 a v2 applec 2 k 2 a 00 a v v =0 e ik c p ck ( >> 1) solution of EOM at large scales: v F (k)a ( << 1) Jeans instability term dominates de Sitter: a 1/ the power spectrum P (k) k 3 F (k) 1 v a k 3/2 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 is scale invariant outside horizon

16 Cosmological perturbations with MDR start from same EOM for perturbations v 00 + applec 2 k 2 a 00 a v =0 warning: there are a few assumption behind this. (see João s talk) if dispersion relation is modified then c is k-dependent 2 + p 2 1+( p) 2 =0 ( = 2) c = E p ( p)2 k a 2 (in the UV) solution of EOM at small scales: v e ik c p ck a e ik c k 3/2 a the power spectrum P (k) k 3 v a 2 is already scale invariant before the mode exits the horizon! (no need for inflationary expansion)

17 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?

18 Gravity in dimensionally reduced momentum space momentum space dimensional reduction does affect equation for perturbations (in the linearizing variables) quadratic action for perturbations, again: S 2 = d d 3 ka c 2 k 2 2 = v a change of variables (for comoving momenta k): k = k p 1+( k) 2 k 2 dk k 2 1+ d k in these new variables the dispersion relation looks trivial S 2 = d d k k 2 1+ a 2 " 02 + k 2 a 2 2 #

19 Gravity in dimensionally reduced momentum space momentum space dimensional reduction does affect equation for perturbations (in the linearizing variables) quadratic action for perturbations, again: S 2 = d d 3 ka c 2 k 2 2 = v a change of variables (for comoving momenta k): k = k p 1+( k) 2 k 2 dk k 2 1+ d k in these new variables the dispersion relation looks trivial S 2 = d d k k 2 1+ a 2 " 02 + k 2 a 2 2 # the effective speed of light is c a, we redefine time units to make it trivial

20 Gravity in dimensionally reduced momentum space action in the linearizing units S 2 = EOM for perturbations: d d kk 2 1+ z 2 h 2 + k 2 2i (z = a 1 2 ) apple v + k 2 z z v =0 ( = v/z)

21 Gravity in dimensionally reduced momentum space action in the linearizing units S 2 = EOM for perturbations: d d kk 2 1+ z 2 h 2 + v + apple k 2 for =2 z is time independent ( z 1) z z v =0 k 2 2i (z = a 1 2 ) ( = v/z) the effect of expansion disappears and the theory is effectively conformal invariant (more on this in João s talk!)

22 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?

23 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?