Dark energy and nonlocal gravity Michele Maggiore Vacuum 2015, Barcelona
based on Jaccard, MM, Mitsou, PRD 2013, 1305.3034 MM, PRD 2014, 1307.3898 Foffa, MM, Mitsou, PLB 2014, 1311.3421 Foffa, MM, Mitsou, IJMPA 2014, 1311.3435 Kehagias and MM, JHEP 2014, 1401.8289 MM and Mancarella, PRD 2014, 1402.0448 Dirian, Foffa, Khosravi, Kunz, MM, JCAP 2014, 1403.6068 Dirian and Mitsou, JCAP 2014, 1408.5058 Dirian, Foffa, Kunz, MM, Pettorino, JCAP 2015, 1411.7692 MM 1506.06217 Dirian, Foffa, Kunz, MM, Pettorino, Cusin, Foffa, Maggiore, in preparation in preparation cosmological applications (Yves's talk) / theoretical aspects (this talk)
the general idea: modify GR in the infrared using non-local terms non-locality emerges from fundamental local theories in many situations classically, when separating long and short wavelength and integrating out the short wave-length (e.g cosmological perturbation theory) in QFT, when computing the effective action that includes the effect of radiative corrections of massless or light particles.
a natural way of modifying GR in the IR is by introducing a mass scale (e.g. massive gravity, bigravity,...) we will introduce a mass scale as the coefficient of a nonlocal term phenomenological approach. Identify a non-local modification of GR that works well attempt at a more fundamental understanding IR running and strong coupling in R 2 theories strong IR effects from the anomaly-induced effective action
a locality / gauge-invariance duality for massive gauge fields Proca theory for massive photons S = R d 4 x 1 4 F µ F µ 1 2 m2 A µ A µ j µ A µ @ µ F µ m 2 A = j! n m 2 @ A =0 ( m 2 )A µ =0 non-local formulation (Dvali 2006) Stueckelberg trick: A µ! A µ + 1 m @ µ' we add one field and we gain a gauge symmetry A µ! A µ @ µ, '! ' + m
S = R d 4 x 1 4 F µ F µ 1 2 m2 A µ A µ 1 2 @ µ'@ µ ' m A µ @ µ ' j µ A µ @ µ F µ = m 2 A + m @ ' + j, ' + m @ µ A µ =0. If we choose the unitary gauge φ=0 we get back to the original formulation of Proca theory (and loose the gauge sym because of gauge fixing). Instead, keep the gauge sym explicit and integrate out φ using its own equation of motion: '(x) = m 1 (@ µ A µ )
Substituting in the eq of motion for A ν : or 1 m 2 ( m 2 )A = we have explicit gauge invariance for the massive theory, at the price non-locality @ µ F µ = j 1 m 2 @ @ µ A µ + j a sort of duality between explicit gauge-invariance and explicit locality @ µ A µ =0 we can fix the gauge and the non-local term disappears (and we are back to Proca eqs.) with hindsight, the Stueckelberg trick was not needed
massive photon: can be described replacing (Dvali 2006) for gravity, a first guess for a massive deformation of GR could be however this is not correct since (Arkani-Hamed, Dimopoulos, Dvali and Gabadadze 2002) we lose energy-momentum conservation
to preserve energy-momentum conservation: G µ m 2 ( 1 G µ ) T =8 GT µ (Jaccard,MM, Mitsou, 2013) however, instabilities in the cosmological evolution (Foffa,MM, Mitsou, 2013) G µ m 2 (g µ 1 R) T =8 GT µ (MM 2013) stable cosmological evolution! a related model: S NL = 1 Z 16 G d 4 x p g apple R m 2 R 1 2 R (MM and M.Mancarella, 2014)
At the phenomenological level, these two non-local models work very well: no vdvz discontiuity, solar system tests OK generates dynamically a dark energy cosmological perturbations work well passes tests of structure formation comparison with CMB,SNe,BAO with modified Boltzmann code ok higher value of H0! see Yves' talk They are the only existing models, with the same number of parameters as ΛCDM, which are competitive with ΛCDM from the point of view of fitting the data
Where such non-locality comes from? we will explore two related ideas that could lead to strong IR effects in gravity MM 1506.06217 IR running of coupling constants in R 2 gravity IR effects from the anomaly-induced effective action in Einstein gravity
Running coupling constants in R 2 theory in QFT, loops involving massless or light particles give nonlocal terms e.g. in QED S e = 1 4 R d 4 xf µ 1 e 2 ( ) F µ 1 e 2 ( ) = 1 e 2 (µ) 0 log µ 2
R 2 gravity S = Z d 4 x p g apple m 2 Pl 2 R a 1C 2 + a 2 R 2 + a 3 E schematically: L m 2 Pl(h@ 2 h + h@h@h +...) a (2h) 2 + h(2h) 2 +... in an effective action approach, the dof are read from the Einstein-Hilbert term after canonical normalization: L h@ 2 h + 1 h@h@h +... m Pl a m 2 Pl apple (2h) 2 + 1 m Pl h(2h) 2 +...
L h@ 2 h + 1 h@h@h +... m Pl a m 2 Pl apple (2h) 2 + 1 m Pl h(2h) 2 +... standard lore: the higher-derivative terms give corrections ae 2 /m 2 Pl, irrelevant in the IR however, in QFT, a 2 =a 2 (E) runs with energy. can it become large in the IR? can it develop a IR singularity a(e) / E 4? Then in the IR a 2 R 2! Ra 2 (2)R / R 1 2 2 R
running due to loops of massles (or light) scalar, spinor and vector field in a fixed curved background: well understood (e.g. Birrel-Davies 1982) S 1 loop NL = graviton loops 1 2(4 ) 2 Z d 4 xg 1/2 apple R log µ 2 + R µ log R + µ 2 R µ log R µ µ 2 R µ in GR, the one-loop divergences vanish on-shell, and can be set to zero with a redefinition of the metric ('t Hooft-Veltman 1974) off-shell, the beta functions depends on the choise of gauge fixing (Kallosh-Tarasov-Tyutin 1978) In GR, there is no sense in which we can compute the physical running of the coupling associated to the R 2 term!
Stelle theory: consider again S = Z d 4 x p g apple m 2 Pl 2 R a 1C 2 + a 2 R 2 + a 3 E but now the R 2 terms are not treated in an effective action approach, but as a full UV completion L m 2 Pl(h@ 2 h + h@h@h +...) a (2h) 2 + h(2h) 2 +... G(k) 1 m 2 Pl k2 + ak 4 the theory is renormalizable! (Stelle 1977)
however: the spectrum of the theory read from the full propagator is massless spin-2 graviton massive spin-2 ghost, m 2 = m Pl2 /(2a 1 ) massive spin 0, m 2 = -m Pl2 /(12a 2 ) because of the ghost, it is not a consistent theory the beta functions of a 1,a 2 are well-defined and gauge-inv Fradkin-Tseytlin 1982, Avramidi-Barvinski 1985 the limit a 1,a 2! 0 of the beta functions is singular, since the structure of the divergences changes (even for graphs with sub-planckian external momenta! nice example of IR/UV mixing)
take-home lessons: in GR the beta functions associated to R 2 and C 2 are not well defined. in a theory with improved UV behavior, as Stelle theory, they are well defined. However, we cannot take literally this result, because the theory has a ghost (and furthermore the beta functions have been computed only at E >> m Pl ) The result depends on the UV completion. In a fundamental UV complete theory of quantum gravity, we expect that the beta functions will be well defined depending on the sign of the beta function, there will be basically two options, a 2 (E) growing or decreasing in the IR we explore the scenario where a 2 (E) grows (when it is is positive)
consider the scenario where the beta function of a 2 (E) corresponds to asymptotic freedom in the UV: a 2 (E) g 2 (E), g 2 (E) = g 2 (µ) 1+g 2 (µ) 0 log(e/µ) 0 > 0 as in QCD, this generates dynamically a IR scale, RR = µe 1/[ 0g 2 (µ)] (dimensional transmutation) in the IR the form factor a 2 ( ) will be non-trivial functions of /Λ RR. Can we get a 2 ( )=(Λ 2 RR/ ) 2? generally speaking, power-like behavior can emerge from resummation of leading logs 1X n=0 ( ) n n! log 2 RR n = 2 RR
a more concrete argument from analogy with QCD? in QCD a term m 2 2 Z d 4 xf a µ apple 1 D 2 ab Fµ b in the effective action is know to reproduce the non-perturbative behavior of the gluon propagator derived from OPE and lattice simulations e.g. review by Boucaud et al. 2011 is a gauge-inv and non-local mass term for the gluon writing g µ = e 2 (x) µ R = 6e 2 ( + @ µ @ µ ) = 6 + O( 2 ) m 2 R 1 2 R = 36m 2 2 + O( 3 ) is a diff-inv and non-local mass term for the conformal mode!
this gives a new perspective on the naturalness problem for the cosmological constant: 1. the scale Λ RR emerges from dimensional transmutation, similarly to Λ QCD. There is no issue of naturalness for such a quantity, which is determined by the logarithmic running of a dimensionless coupling constant
2. numerical value of the DE scale the action in the IR is we can rewrite it as S ' Z d 4 x p S ' m2 Pl 2 m2 Pl 2 Z Z g apple m 2 Pl 2 R 4 RRR 1 2 R d 4 x p d 4 x p g g apple R apple R 2 4 RR m 2 Pl R 1 2 R m 2 R 1 2 R cosmology requires m ' 2 RR m Pl m H 0! RR mev we do not need to introduce a particle with astonishingly small mass m H 0 10 32 ev, as in massive gravity m ~10-32 ev is just a derived quantity. The fundamental scale is Λ RR
Strong IR effects from conformal-mode dynamics? In pure Einstein gravity + massless matter, higher-derivative terms are generated by the conformal anomaly in D=2, this gives the Polyakov action ht µ µ i = N 24 R! S anom = N 96 Z d 2 x p gr 1 2 R writing g µ (x) =e 2 (x) ḡ µ (x) we get S anom = N 24 Z d 2 x p ḡ ( + R ) the anomaly gives dynamics to the conformal mode
in 4D ht µ µ i = bc 2 + b 0 E 2 3 2R + b 00 2R S anom = 1 8 Z d 4 x p g E 23 2R 4 1 appleb 0 E 23 2R 2bC 2 4 = 2 2 +2R µ r µ r 2 3 R2 + 1 3 (rµ R)r µ restricting to the conformal mode S anom = R Q2 16 d 4 x ( ) 2 2 Q 2 = 1 180 (N S + 11 2 N F + 62N V 28) + Q 2 grav
In the classical theory σ is a constrained variable. At the quantum level it acquires dynamics because of the conformal anomaly 4D-quantum gravity in the IR is dominated by the conformal mode, because of the 1/k 4 propagator Antoniadis and Mottola 1992 Antoniadis, Mazur and Mottola 2007 fluctuations in σ become large in the IR ( ) 2! G(x, x 0 )= 1 2Q 2 log µ 2 (x x 0 ) 2
situation similar to the BKT transition in D=2 in D=2, G(k) =1/k 2! G(x) / log x 2 Antoniadis, Mazur and Mottola 2007 vortex solutions S E = 1 2 Q2 q 2 ln L a (x) =q ln x x 0 F = 1 2 Q2 q 2 4 ln L a if if Q 2 > 8/q Q 2 < 8/q the free energy diverges as we remove the cutoffs and these configurations are irrelevant they dominate this is exactly what happens in the XY model in D=2! BTK phase transition and generation of a mass gap
Thus, once again, we expect a dynamical generation of a mass gap for the conformal mode! Recalling that m 2 R 1 R = 36m 2 2 + O( 3 ) 2 diff invariance requires the non-local term with the mass term: G(x) = 8 2 Q 2 Z d 4 k 1 (2 ) 4 k 4 + µ 4 e ikx / log x µx 1 1/(µx) 3 µx 1 physical picture: the log growth of the propagator derived from the anomaly-induced effective action is a sign of a IR instability the generation of a mass gap is the response of the vacuum to this instability
Conclusions a term R2 2 R could in principle be generated by (possibly related) non-perturbative IR effects running of an asymptotically free R 2 coupling large-distance fluctuations of the conformal mode In both cases the mass scale associated to dark energy would be generated dynamically, similarly to QCD, or to the generation of a mass gap in strongly-coupled theories
Thank you!
Absence of vdvz discontinuity and of a strong coupling regime write the eqs of motion of the non-local theory in spherical symmetry: for mr <<1: low-mass expansion A. Kehagias and MM 2014 ds 2 = A(r)dt 2 + B(r)dr 2 + r 2 (d 2 +sin 2 d 2 ) for r>>r S : Newtonian limit (perturbation over Minowski) match the solutions for r S << r << m -1 (this fixes all coefficients)
result: for r>>r s for r s <<r<< m -1 : m! 0 A(r) = 1 the limit is smooth! r S r B(r) = 1+ r S r A(r) ' 1 r S r apple 1+ 1 (1 cos mr) 3 apple 1 1 (1 cos mr mr sin mr) 3 1+ m2 r 2 6 By comparison, in massive gravity the same computation gives A(r) =1 4 3 vdvz discontinuity r S r 1 r S 12m 4 r 5 breakdown of linearity below r V =(r s /m^4) 1/5