Emergent Gravity: from Condensed Matter Analogues to Phenomenology

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Emergent gravity: From Condensed matter analogues to Phenomenology.

Emergent properties of space-time. CERN.

Stefano Liberati SISSA - INFN Trieste, Italy GR, a beautiful but weird theory…

Some tantalising features of General Relativity

Singularities

Critical phenomena in gravitational collapse

Horizon thermodynamics

Spacetime thermodynamics: Einstein equations as equations of state.

The “dark ingredients” of our universe?

Faster than light and Time travel solutions

AdS/CFT duality, holographic behaviour

Gravity/fluid duality Gravity as an emergent phenomenon?

Emergent gravity idea: quantizing the metric or the connections does not help because perhaps these are not fundamental objects but collective variables of more fundamental structures.

GR ⇒ Hydrodynamics Metric as a collective variable All the sub-Planckian physics is low energy physics Spacetime as a condensate of some more fundamental objects Spacetime symmetries as emergent symmetries Singularities as phase transitions (big bang as geometrogenesis) Cosmological constant as deviation from the real ground state

Many models are nowadays resorting to emergent gravity scenarios

Causal sets Quantum graphity models Group field theories condensates scenarios AdS/CFT scenarios where the CFT is considered primary Gravity as an entropic force ideas Condensed matter analogues of gravity Gravity as an emergent phenomenon?

Emergent gravity idea: quantizing the metric or the connections does not help because perhaps these are not fundamental objects but collective variables of more fundamental structures.

Many models are nowadays resorting to emergent gravity scenarios

The Analogue gravity pathway Emergent spacetimes

And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of Nature, and they ought least to be neglected in Geometry. Johannes Kepler Unruh ’81, Visser ’93 But see also White ‘73 A paradigmatic example: Acoustic Gravity

! ! ∂ρ ! ! dv '∂ v ! ! * ! ! ! Continuity + ∇ ⋅ (ρv) = 0 Euler ρ ≡ ρ +(v ⋅ ∇)v = −∇ p − ρ∇Φ + f viscosity ∂t dt () ∂t +, ! ! 1 1 4 ! ! ! External Forces f = +η∇ 2v +3ζ + η6∇ ∇ ⋅ v viscosity 2 3 5 ( ) p = pressure, η = dynamic viscosity, ζ = bulk viscosity, Φ = potential of external driving force (gravity included)

Irrotational Flow ! ! ! ! ! 2 dp Barotropic € Basic Assumptions ∇ × v = 0 v = ∇ψ c = s d Viscosity free ﬂow

Linearize the above Eq.s around some background € € ρ(t,x) = ρ0(t,x) + ερ1(t,x) And combine then so to get a second order ﬁeld equation

p(t,x) = p0(t,x) + εp1(t,x) ∂ ( −2 ( ∂ψ1 ! ++ ( −2 ! ( ∂ψ1 ! ++ ψ(t,x) =ψ0 (t,x) + εψ1(t,x) * cs ρ0* + v 0 ⋅ ∇ψ1-- = ∇ ⋅* ρ0∇ψ1 − cs ρ0v 0* + v 0 ⋅ ∇ψ1-- ∂ t ) ) ∂t ,, ) ) ∂t ,,

This looks messy but if we introduce the “acoustic metric” 2 2 j € ρ '− c − v -v * g ≡ 0 ) ( s 0 ) 0, We€ get µν i cs () -v0 δij +,

1 µν Δψ1 ≡ ∂ −gg ∂ ψ1 = 0 −g µ ( ν ) €

€ Unruh ’81, Visser ’93 But see also White ‘73 A paradigmatic example: Acoustic Gravity

Irrotational Flow ! ! ! ! ! 2 dp Barotropic € Basic Assumptions ∇ × v = 0 v = ∇ψ c = s d Viscosity free ﬂow

Linearize the above Eq.s around some background € € ρ(t,x) = ρ0(t,x) + ερ1(t,x) And combine then so to get a second order ﬁeld equation

This looks messy but if we introduce the “acoustic metric” 2 2 j € ρ '− c − v -v * g ≡ 0 ) ( s 0 ) 0, We€ get µν i cs () -v0 δij +, 1 ggµν 0 This is the same equation as for a scalar ﬁeld moving in curved Δψ1 ≡ ∂µ ( − ∂νψ1) = −g spacetime, possibility to simulate FRW and Black Holes! €

€ Analogue Black holes v A moving fluid will p the “sound cones” as it moves. Supersonic flow will p the cone past the vercal. A moving fluid can form “trapped regions”” when supersonic flow will p the cone past the vercal.

A moving ﬂuid will drag sound pulses along with it SISSA-Nottingham experiment: an analogue of superradiant scattering of gravity waves (PI: S. Weinfurtner) A prototype quantum analogue model: BEC

A BEC is quantum system of N interacting bosons in which most of them lie in the same single-particle quantum state 5 6 (T

It is described by a many-body Hamiltonian which in the limit of dilute condensates gives a non-linear Schrödinger equation (a=s-wave scattering length) 2 ⇥ 2 2 i ˆ = ˆ µˆ + ˆ ˆ . ⇥t 2m⇥ | | This is still a very complicate system, so let’s adopt a mean ﬁeld approximation ˆ 2 Mean field approximation : Ψ( t,x) =ψ(t,x) + χˆ (t,x) where ψ(t,x) = nc (t,x) = N /V ψ(t,x) = Ψˆ ( t,x) = classical wave function of the BEC , χˆ (t,x) = excited atoms

€

The ground state is the vacuum for the collective excitations of the condensate (quasi-particles) but this an inequivalent state w.r.t. the atomic vacuum. They are linked by Bogolyubov transformations. Bose-Einstein condensate: an example of analogue emergent spacetime

By direct substitution of the mean field ansatz in the non-linear Schrödinger equation gives 2 ⌅ 2 2 Background dynamics i ⇤(t, x)= µ + ⇤ ⇤ +2 (ñ⇤ +˜m⇤) ⌅t 2m⇥ | | ⇥ 2 ⌅ 2 i ⇥ = µ +2nT ⇥ + mT ⇥† ⌅t 2m⇥ ⇥ n ⇤⇥(t, x) 2 ; m ⇤⇥2(t, x); ⇤ Excitations dynamics c | | c n˜ † ;˜m ; ⇥ ⇤ ⇥ ⇤ nT = nc +ñ; mT = mc +˜m. These are the so called Bogoliubov-de Gennes equations The first one encodes the BEC background dynamics The second one encodes the dynamics for the quantum excitations

The equations are coupled via the so called anomalous mass m~ and density ñ. Which we shall neglect for the moment…

Let’s consider quantum perturbations over the 1 n i/ ⇥ c BEC background and adopt the “quantum acoustic ⇥(t, x)=e n1 i 1 2⇥n representation’' (Bogoliubov transformation) c ⇥ ⇤ ⇤ ⇤ for the perturbations one gets the system of equations

Where D2 is a represents a second-order differential operator: the linearized quantum potential Acoustic metric

€ Acoustic metric

For very long wavelengths the terms coming from the linearized quantum potential D2 can be neglected. (− c 2 − v2  − v + (− c 2 − v2  − v + * ( s 0 ) ( 0) j - * ( s 0 ) ( 0) j - The so obtained metric is again the acoustic metric cs n0 gµν (t,x) ≡ *  ⋅  - = *  ⋅  - c m λ * - s * - −(v0 )  δij −(v0 )  δij ) i , ) i , Lessons The collective excitations (phonons) of the BEC propagate on a Lorentzian spacetime determined € by the background velocity and density and they can be meaningfully quantised

The atomic physics enters only in determining the fundamental constants of the low energy phenomenology (e.g. speed of sound cs=analogue speed of light)

The spacetime are stably causal as they inherit the causal structure from lab system: no-time machines possible.

Lorentz symmetry of the phononic theory is an low energy symmetry

Non Local quasi-particles interactions Acoustic Lorentz invariance breaking in BEC analogue gravity

If instead of neglecting the quantum potential we adopt the eikonal approximation (high-momentum approximation) we find, as expected, deviations from the Lorentz

E.g. the dispersion relation for the BEC quasi-particles is

This (Bogoliubov) dispersion relation (experimentally observed) actually interpolates between two different regimes depending on the value of the fluctuations wavelength λ=2π/|k| with respect to the “acoustic Planck wavelength” λC=h/(2mcs)=πξ with ξ=healing length of BEC=1/(8πρa)1/2

For λ»λC one gets the standard phonon dispersion relation ω≈c|k| For λ«λC one gets instead the dispersion relation for an individual gas particle (breakdown of the continuous medium approximation) ω≈(ħ2k2)/(2m) Robustness and Detection of Hawking radiation in Black hole analogues It turned out that Hawking Radiation is robust against LIV (see e.g. Parentani and collaborators recent papers), however you might get (controllable) instabilities such as “black hole laser effect” (superluminal relation in compact supersonic region).

Some facts: In static spacetimes Hawking radiation robustness is generally assured if there is a separation of scales: κBH<<Λ where Λ is parametrically related to the UV LIV scale.

Tentative detection via density-density correlations Recent observation of a characteristic instability for compact ergo regions J. Steinhauer. Nature Physics (2014). Even more recently ﬁrst claim of Hawking detection (J. Steinhauer. 2015)! Open debate on quantum vs classical excitations Hawking Radiation detection via density- density correlation. BEC simulation. (in, out) Carusotto et al.

So does BH thermodynamics survive without Lorentz invariance?

(out, in) This is interesting even if you do not believe Lorentz invariance can be broken in the UV. Where thermodynamics comes from Fig. 3. Observation of Hawking/partner pairs. The horizon is at the origin. The dark bands emanating from the horizon are the correlations between the Hawking and partner particles. The in Gravitational Theories? solid line shows the angle of equal times from the horizon, found in Fig. 4. The Fourier transform along the dashed line measures the entanglement of the Hawking pairs. More soon…

Fig. 4 shows the profile of the correlation band along the dashed line of Fig. 2. The fact that the profile has finite area gives information about the spectrum of the Hawking radiation. We find 〈� � 〉 � that the Fourier transform of the profile gives �� , where �� is the annihilation operator � for a Hawking particle with wavenumber �� localized outside the black hole, and �� is the annihilation operator for a partner particle localized inside the black hole [11]. The relation is [11]

〈� � 〉 �out−�out �in−�in (2)( ) �� + �� + �� = � + FT�� , �′ � (1) �in−�in �out−�out

where �� and �� are the Bogoliubov coefficients for the phonon quasiparticles, which are completely determined by ��. the Fourier transform is of Fig. 4 where � is in units of �, giving a function of � in units of �−1. Here, we have neglected the phonons which occur due to the finite temperature of the condensate. These phonons are negligible in our system [10].

Robustness and Detection of Hawking radiation in Black hole analogues It turned out that Hawking Radiation is robust against LIV (see e.g. Parentani and collaborators recent papers), however you might get (controllable) instabilities such as “black hole laser effect” (superluminal relation in compact supersonic region).

Some facts: In static spacetimes Hawking radiation robustness is generally assured if there is a separation of scales: κBH<<Λ where Λ is parametrically related to the UV LIV scale.

So does BH thermodynamics survive without Lorentz invariance?

〈� � 〉 �out−�out �in−�in (2)( ) �� + �� + �� = � + FT�� , �′ � (1) �in−�in �out−�out

The Analogue gravity pathway From emergent spacetimes to emergent gravity Emergent gravity scenarios Can we emerge a relativistic theory for Spin 2? Obstruction: the Weinberg-Witten theorem

“No spin 2 particle can be emergent if you have Lorentz invariance and local Gauge invariant currents or conserved SET” Hence potential ways out are:

One step emergence

“Spacetime Atoms”+QM No mesoscopic physics Manifold, Metric, LLI, Diffeo on a Spacetime Gravity+SM, Locality

From Manifold to Gravity with Lorentz breaking or non-locality

Differentiable Manifold, “Spacetime Atoms” in LLI, Diffeo, Phase transition or Sum over Fields, Locality, Lorentz breaking Low energies flat spacetime +QM Gravity+Matter microscopic configurations or non-locality

Induced gravity a la Sakarov

Differentiable Manifold, LLI, Diffeo, “Spacetime Atoms” +QM ? Fields + Lorentz+ Metric but no Gravity+Matter, Locality geometrodynamics a la Induced Gravity

While the first “extreme emergence” route might seem the more fundamental it is also the hardest.

Let’s see how far we can go with the Analogue Gravity framework… A toy model for emergent gravity: non-relativistic BEC

So let’s go back to the mean ﬁeld approximation of BEC and focus F. Girelli, S.L.,L.Sindoni on the equation for the background: Phys.Rev.D78:084013,2008

2 ⇤ ~ 2 i~ ⇥(t, x)= + Vext(x)+nc ⇥(t, x)+ (2˜n⇥(t, x)+m ˜⇥(t, x)) ⇤t 2mr ✓ ◆

Can this be encoding some form of gravitational dynamics? If yes it must be some form of Newtonian gravity (non relativistic equation) But, in order to have any chance to see this, we need to have some massive ﬁeld

One way to get this is to introduce a soft U(1) breaking term (i.e. pass from massless Goldstone bosons top massive pseudo-Goldstone bosons)

Note: this kind of symm breaking is actually experimentally realized in magnon (quantized spin wave) BEC in 3He-B (see e.g. related work by G.Volovik) 2 ⇤ 2 2 We assume –μ<λ<<μ (soft breaking). i ˆ = ˆ µˆ + ˆ ˆ ⇥ˆ † ⇤t 2m⇥ | | p4 2(p)= + c2p2 + 2c4 E 4m2 s M s It can be checked that the extra term gives massive phonons µ +2 (µ + ) c2 = , 2 =4 m2 which at low momenta propagate on the standard acoustic s m M (µ +2)2 geometry of BEC m if µ M Non-relativistic BEC gravitational potential

So we would now like to cast the equation for the a stationary, homogeneous, condensate background in a Poisson-like form with the quasi-particles moving accordingly to the analogue gravitational potential. F⇤ = ⇤a = ⇤ ⇥ M⇥ grav 1 2 ⇥ =4G ⇥ + ⇥ L2 grav N ⇥

where L=> range of the gravitational interaction, GN => analogue G Newton, Λ => analogue cosmological constant Results

1. It is possible to show by looking at the Newtonian limit of the acoustic geometry that the gravitational potential is encoded in density perturbations

µ + ⇥ 1/2 2. By adopting the ansatz ⇤ = (1 + u(x)) ⇥ and looking at the Hamiltonian for the quasi-particles in the non relativistic limit, one can actually show that the analogue of the gravitational potential is (µ +4)(µ +2) (x) = u(x) grav 2m Non-relativistic BEC gravitational potential

where L=> range of the gravitational interaction, GN => analogue G Newton, Λ => analogue cosmological constant Results

1. It is possible to show by looking at the Newtonian limit of the acoustic geometry that the gravitational potential is encoded in density perturbations

This is the form the gravitational potential affecting the quasi-particle motion for a slightly inhomogeneous BEC. We now want to see if it satisfies some modified Poisson equation… non-relativistic BEC: emergent Newtonian gravity

Let’s consider the equation for a static background with a source term. The latter is given partly by a localized quasi-particle plus a vacuum contribution due to the unavoidable presence/ backreaction of excited atoms above the condensate 2 1 1 2 2(µ + ⇥) u(x) = 2 n¯(x) + m¯ (x) +2 n˜ + m˜ 2m⌅ 2 0 2 0 ⇥ ⇥ ⇥ wheren ¯(x) =n ˜(x) n˜ , m¯ (x) =m ˜ (x) m˜ 0 0 andn ˜ = 0 ⇤ˆ†(x)ˆ⇤(x) 0 , m˜ = 0 ⇤ˆ(x)ˆ⇤(x) 0 0 ⇥ | | ⇤ 0 ⇥ | | ⇤ are the quasi-particle vacuum backreaction terms

Now, knowing what is the analogue gravitational potential, this can be cast in the form of a generalized Poisson equation with a (negative) cosmological constant. non-relativistic BEC: emergent Newtonian gravity

Now, knowing what is the analogue gravitational potential, this can be cast in the form of a generalized Poisson equation with a (negative) cosmological constant.

1 (µ +4⇥)(µ +2⇥)2 1 ⌅ (x) = n¯(x) + m¯ (x) G 2 matter 2 N 4⇤ 2m⇥3/2(µ + ⇥)1/2 ⇥grav =4GN ⇥matter + , where ⇥ 2 2 ⇥ L 2(µ +4⇥)(µ +2⇥) 1 2 ⇥ (˜n0 + m˜ 0),L 2⇥ 2 4m(µ + ⇥) non-relativistic BEC: emergent Newtonian gravity

Now, knowing what is the analogue gravitational potential, this can be cast in the form of a generalized Poisson equation with a (negative) cosmological constant.

1 (µ +4⇥)(µ +2⇥)2 1 ⌅ (x) = n¯(x) + m¯ (x) G 2 matter 2 N 4⇤ 2m⇥3/2(µ + ⇥)1/2 ⇥grav =4GN ⇥matter + , where ⇥ 2 2 ⇥ L 2(µ +4⇥)(µ +2⇥) 1 2 ⇥ (˜n0 + m˜ 0),L 2⇥ 2 4m(µ + ⇥) Weighting Lambda… 4 7 5/2 cs cs 3 = , P = 2 , = E ⇤0a . E 4⇥GN E ~G P ⇥ g⇤0 N E ✓ ◆ The cosmological constant scale is suppressed by a small number (the dilution factor ρa3<<1) w.r.t. the analogue/emergent Planck scale!

The cosmological constant is not the phonons ground state energy, neither it is the atoms grand canonical energy density h, or energy density ε=h+μρ It is just related to the subdominat second order correction to these latter quantities due to quantum depletion (the part related to the excitations) and its scale is the healing scale. week ending PRL 108, 071101 (2012) PHYSICAL REVIEW LETTERS 17 FEBRUARY 2012

d3k 8 evidences. However, it displays an alternative path to the ^ ^ u v a3F ; (21) Cosmological constant in emergent3 k k gravity:0 lessonscosmological from BEC constant, from the perspective of a micro- h i ¼ 0 2 ¼ p g0 Z ð Þ qffiffiffiffiffiffiffiffiffiffi scopic model. The analogue cosmological constant that we where F 0 1 (see Fig. 1, dot-dashedffiffiffiffi line), we obtain have discussed cannot be computed as the total zero-point ð Þ¼ S. Finazzi, S. Liberati and L. Sindoni, energy of the condensed matter system, even when taking 20mg g 3 Phys. Rev. Lett. 108, 071101 (2012) Ã 0ð 0 þ Þ a3F ; (22) into account the natural cutoff coming from the knowledge ¼À 3p 2 0 Ã g qffiffiffiffiffiffiffiffiffiffi 0 of the microphysics [9]. In fact the value of Ã is related @ only to the (subleading) part of the zero-point energy where F 2F ffiffiffiffi3F =5 (see Fig. 1, solid line). Ã ¼ð þ Þ proportional to the quantum depletion of the condensate. Let us now compare the value of Ã either with the This model is too simple to be realistic but still teaches us Thissome holds lessons also in a spinor BEC model, since the reasoning ground-state grand-canonical energy density h [Eq. (11)], there is absolutely identical. The virtue of the single BEC The analogue cosmologicalwhich constant in [9] that was suggestedwe have discussed as the correct cannot vacuum be energy computed as the total zero- point energy of the condensed matter system, even when taking into account themodel natural is to display cutoff the key physical result without obscur- corresponding to the cosmological constant, or with the coming from the knowledge of the microphysics ing it with unnecessary mathematical complications, with- ground-state energy density of Eq. (14). Evidently, Ã In fact the value of Λ is relateddoes notonly correspond to the (subleading) to either of them: part evenof the when zero-point taking energyout loss proportional of generality. Interestingly, this result finds some intoto the account quantum the correct depletion behavior of the at small condensate. scales, the vac- support from arguments within loop quantum gravity mod- uum energy computed with the phonon EFT does not lead els [18], suggesting a BCS energy gap as a (conceptually to the correct value of the cosmological constant appearing rather different) origin for the cosmological constant. 3 The implications for gravity are twofold. First, there in Eq. (17). Noticeably, since Ã is proportional to 0a , it could be no a priori reason why the cosmological constant can even be arbitrarily smaller both than h and thanpffiffiffiffiffiffiffiffiffiffi, if the should be computed as the zero-point energy of the system. condensate is very dilute. Furthermore, Ã is proportional More properly, its computation must inevitably pass only to the subdominant second order correction of h or , through the derivation of Einstein equations emerging which is strictly related to the depletion [see Eq. (12)]. from the underlying microscopic system. Second, the en- Fundamental scales.—Several scales show up in this ergy scale of Ã can be several orders of magnitude smaller system, in addition to the naive Planck scale computed than all the other energy scales for the presence of a very by combining and the emergent constants GN and cs: @ small number, nonperturbative in origin, which cannot be 5 3=4 computed within the framework of an EFT dealing only cs À 3 1=4 LP @ 0a À a: (23) with the emergent degrees of freedom (i.e., semiclassical ¼ sffiffiffiffiffiffiffiffiGN / g0 ð Þ gravity). The model discussed in this Letter shows all this explic- For instance, the Lorentz-violation scale LLV 3 1=2 ¼ / itly: the energy scale of Ã is here lowered by the diluteness 0a À a differs from LP, suggesting that the breaking ðof theÞ Lorentz symmetry might be expected at scale much parameter of the condensate. Furthermore, our analysis longer than the Planck length (energy much smaller than strongly supports a picture where gravity is a collective 3 1=4 phenomenon in a pregeometric theory. In fact, the cosmo- the Planck energy), since the ratio LLV=LP 0a À increases with the diluteness of the condensate./ ð Þ logical constant puzzle is elegantly solved in those scenar- 3 ios. From an emergent gravity approach, the low energy Note that LLV scales with 0a exactly as the range of the gravitational force [see Eq. (18)], signaling that this effective action (and its renormalization group flow) is model is too simple to correctly grasp all the desired computed within a framework that has nothing to do with features. However, in more complicated systems [13], quantum field theories in curved spacetime. Indeed, if we this pathology can be cured, in the presence of suitable interpreted the cosmological constant as a coupling con- symmetries, leading to long range potentials. stant controlling some self-interaction of the gravitational It is instructive to compare the energy density corre- field, rather than as a vacuum energy, it would immediately sponding to Ã to the Planck energy density: follow that the explanation of its value (and of its properties under renormalization) would naturally sit outside the 4 7 5=2 domain of semiclassical gravity. Ãcs cs EÃ 3 À E Ã ; EP 2 ; 0a : (24) For instance, in a group field theory scenario (a general- ¼ 4GN ¼ GN EP / g0 @ ization to higher dimensions of matrix models for two The energy density associated with the analogue cosmo- dimensional quantum gravity [19]), it is transparent that logical constant is much smaller than the values computed the origin of the gravitational coupling constants has noth- from zero-point-energy calculations with a cutoff at the ing to do with ideas like ‘‘vacuum energy’’ or statements Planck scale. Indeed, the ratio between these two quantities like ‘‘energy gravitates’’, because energy itself is an emer- 3 is controlled by the diluteness parameter 0a . gent concept. Rather, the value of Ã is determined by the Final remarks.—Taken at face value, this relatively microphysics, and, most importantly, by the procedure to simple model displays too many crucial differences with approach the continuum semiclassical limit. In this respect, any realistic theory of gravity to provide conclusive it is conceivable that the very notion of cosmological

071101-4 week ending PRL 108, 071101 (2012) PHYSICAL REVIEW LETTERS 17 FEBRUARY 2012

d3k 8 evidences. However, it displays an alternative path to the ^ ^ u v a3F ; (21) Cosmological constant in emergent3 k k gravity:0 lessonscosmological from BEC constant, from the perspective of a micro- h i ¼ 0 2 ¼ p g0 Z ð Þ qffiffiffiffiffiffiffiffiffiffi scopic model. The analogue cosmological constant that we where F 0 1 (see Fig. 1, dot-dashedffiffiffiffi line), we obtain have discussed cannot be computed as the total zero-point ð Þ¼ S. Finazzi, S. Liberati and L. Sindoni, energy of the condensed matter system, even when taking 20mg g 3 Phys. Rev. Lett. 108, 071101 (2012) Ã 0ð 0 þ Þ a3F ; (22) into account the natural cutoff coming from the knowledge ¼À 3p 2 0 Ã g qffiffiffiffiffiffiffiffiffiffi 0 of the microphysics [9]. In fact the value of Ã is related @ only to the (subleading) part of the zero-point energy where F 2F ffiffiffiffi3F =5 (see Fig. 1, solid line). Ã ¼ð þ Þ proportional to the quantum depletion of the condensate. Let us now compare the value of Ã either with the This model is too simple to be realistic but still teaches us Thissome holds lessons also in a spinor BEC model, since the reasoning ground-state grand-canonical energy density h [Eq. (11)], there is absolutely identical. The virtue of the single BEC The analogue cosmologicalwhich constant in [9] that was suggestedwe have discussed as the correct cannot vacuum be energy computed as the total zero- point energy of the condensed matter system, even when taking into account themodel natural is to display cutoff the key physical result without obscur- corresponding to the cosmological constant, or with the coming from the knowledge of the microphysics ing it with unnecessary mathematical complications, with- ground-state energy density of Eq. (14). Evidently, Ã In fact the value of Λ is relateddoes notonly correspond to the (subleading) to either of them: part evenof the when zero-point taking energyout loss proportional of generality. Interestingly, this result finds some intoto the account quantum the correct depletion behavior of the at small condensate. scales, the vac- support from arguments within loop quantum gravity mod- uum energy computed with the phonon EFT does not lead els [18], suggesting a BCS energy gap as a (conceptually to the correct value of the cosmological constant appearing rather different) origin for the cosmological constant. 3 The implications for gravity are twofold. First, there This impliesin Eq. ( 17for). Noticeably, emergent since Ãgravityis proportional scenarios to 0a , it thatcould be no a priori reason why the cosmological constant can even be arbitrarily smaller both than h and thanpffiffiffiffiffiffiffiffiffiffi, if the should be computed as the zero-point energy of the system. condensate is very dilute. Furthermore, Ã is proportional More properly, its computation must inevitably pass only to the subdominant second order correction of h or , through the derivation of Einstein equations emerging which is strictly related to the depletion [see Eq. (12)]. from the underlying microscopic system. Second, the en- Fundamental scales.—Several scales show up in this ergy scale of Ã can be several orders of magnitude smaller system, in addition to the naive Planck scale computed than all the other energy scales for the presence of a very by combining and the emergent constants GN and cs: @ small number, nonperturbative in origin, which cannot be 5 3=4 computed within the framework of an EFT dealing only cs À 3 1=4 LP @ 0a À a: (23) with the emergent degrees of freedom (i.e., semiclassical ¼ sffiffiffiffiffiffiffiffiGN / g0 ð Þ gravity). The model discussed in this Letter shows all this explic- For instance, the Lorentz-violation scale LLV 3 1=2 ¼ / itly: the energy scale of Ã is here lowered by the diluteness 0a À a differs from LP, suggesting that the breaking ðof theÞ Lorentz symmetry might be expected at scale much parameter of the condensate. Furthermore, our analysis longer than the Planck length (energy much smaller than strongly supports a picture where gravity is a collective 3 1=4 phenomenon in a pregeometric theory. In fact, the cosmo- the Planck energy), since the ratio LLV=LP 0a À increases with the diluteness of the condensate./ ð Þ logical constant puzzle is elegantly solved in those scenar- 3 ios. From an emergent gravity approach, the low energy Note that LLV scales with 0a exactly as the range of the gravitational force [see Eq. (18)], signaling that this effective action (and its renormalization group flow) is model is too simple to correctly grasp all the desired computed within a framework that has nothing to do with features. However, in more complicated systems [13], quantum field theories in curved spacetime. Indeed, if we this pathology can be cured, in the presence of suitable interpreted the cosmological constant as a coupling con- symmetries, leading to long range potentials. stant controlling some self-interaction of the gravitational It is instructive to compare the energy density corre- field, rather than as a vacuum energy, it would immediately sponding to Ã to the Planck energy density: follow that the explanation of its value (and of its properties under renormalization) would naturally sit outside the 4 7 5=2 domain of semiclassical gravity. Ãcs cs EÃ 3 À E Ã ; EP 2 ; 0a : (24) For instance, in a group field theory scenario (a general- ¼ 4GN ¼ GN EP / g0 @ ization to higher dimensions of matrix models for two The energy density associated with the analogue cosmo- dimensional quantum gravity [19]), it is transparent that logical constant is much smaller than the values computed the origin of the gravitational coupling constants has noth- from zero-point-energy calculations with a cutoff at the ing to do with ideas like ‘‘vacuum energy’’ or statements Planck scale. Indeed, the ratio between these two quantities like ‘‘energy gravitates’’, because energy itself is an emer- 3 is controlled by the diluteness parameter 0a . gent concept. Rather, the value of Ã is determined by the Final remarks.—Taken at face value, this relatively microphysics, and, most importantly, by the procedure to simple model displays too many crucial differences with approach the continuum semiclassical limit. In this respect, any realistic theory of gravity to provide conclusive it is conceivable that the very notion of cosmological

071101-4 week ending PRL 108, 071101 (2012) PHYSICAL REVIEW LETTERS 17 FEBRUARY 2012

d3k 8 evidences. However, it displays an alternative path to the ^ ^ u v a3F ; (21) Cosmological constant in emergent3 k k gravity:0 lessonscosmological from BEC constant, from the perspective of a micro- h i ¼ 0 2 ¼ p g0 Z ð Þ qffiffiffiffiffiffiffiffiffiffi scopic model. The analogue cosmological constant that we where F 0 1 (see Fig. 1, dot-dashedffiffiffiffi line), we obtain have discussed cannot be computed as the total zero-point ð Þ¼ S. Finazzi, S. Liberati and L. Sindoni, energy of the condensed matter system, even when taking 20mg g 3 Phys. Rev. Lett. 108, 071101 (2012) Ã 0ð 0 þ Þ a3F ; (22) into account the natural cutoff coming from the knowledge ¼À 3p 2 0 Ã g qffiffiffiffiffiffiffiffiffiffi 0 of the microphysics [9]. In fact the value of Ã is related @ only to the (subleading) part of the zero-point energy where F 2F ffiffiffiffi3F =5 (see Fig. 1, solid line). Ã ¼ð þ Þ proportional to the quantum depletion of the condensate. Let us now compare the value of Ã either with the This model is too simple to be realistic but still teaches us Thissome holds lessons also in a spinor BEC model, since the reasoning ground-state grand-canonical energy density h [Eq. (11)], there is absolutely identical. The virtue of the single BEC The analogue cosmologicalwhich constant in [9] that was suggestedwe have discussed as the correct cannot vacuum be energy computed as the total zero- point energy of the condensed matter system, even when taking into account themodel natural is to display cutoff the key physical result without obscur- corresponding to the cosmological constant, or with the coming from the knowledge of the microphysics ing it with unnecessary mathematical complications, with- ground-state energy density of Eq. (14). Evidently, Ã In fact the value of Λ is relateddoes notonly correspond to the (subleading) to either of them: part evenof the when zero-point taking energyout loss proportional of generality. Interestingly, this result finds some intoto the account quantum the correct depletion behavior of the at small condensate. scales, the vac- support from arguments within loop quantum gravity mod- uum energy computed with the phonon EFT does not lead els [18], suggesting a BCS energy gap as a (conceptually to the correct value of the cosmological constant appearing rather different) origin for the cosmological constant. 3 The implications for gravity are twofold. First, there This impliesin Eq. ( 17for). Noticeably, emergent since Ãgravityis proportional scenarios to 0a , it thatcould be no a priori reason why the cosmological constant can even be arbitrarily smaller both than h and thanpffiffiffiffiffiffiffiffiffiffi, if the should be computed as the zero-point energy of the system. condensate is very dilute. Furthermore, Ã is proportional More properly, its computation must inevitably pass There could be no a priorionly reason to the why subdominant the cosmological second order constant correction should of h or be, computed as the zero- point energy of the system. More properly, its computation must inevitably throughpass through the derivation the of Einstein equations emerging which is strictly related to the depletion [see Eq. (12)]. derivation of Einstein equations emerging from the underlying microscopicfrom thesystem. underlying microscopic system. Second, the en- Fundamental scales.—Several scales show up in this ergy scale of Ã can be several orders of magnitude smaller the energy scale of Λ can system,be several in addition orders to of the magnitude naive Planck smaller scale than computed all the other energy scales than all the other energy scales for the presence of a very for the presence of a veryby combining small number,and thenonperturbative emergent constants in Gorigin,and cwhich: cannot be computed within the framework of an EFT dealing only with the emergentN degreess of freedom (i.e., @ small number, nonperturbative in origin, which cannot be semiclassical gravity). computed within the framework of an EFT dealing only c5 3=4 L s À a3 1=4a: (23) with the emergent degrees of freedom (i.e., semiclassical From an emergent gravity approach, Pthe low@G energyg effective0 Àaction (and its renormalization group flow) is computed within a framework¼ thatsffiffiffiffiffiffiffiffi Nhas/ nothing0 toð doÞ with quantum fieldgravity). theories in curved spacetime. Indeed, the explanation of its value (and of its properties under renormalization)The model discussed would in this Letter shows all this explic- naturallyFor instance, sit outside the the Lorentz-violation domain of semiclassical scale LLV gravity. 3 1=2 ¼ / itly: the energy scale of Ã is here lowered by the diluteness 0a À a differs from LP, suggesting that the breaking ðof theÞ Lorentz symmetry might be expected at scale much parameter of the condensate. Furthermore, our analysis longer than the Planck length (energy much smaller than strongly supports a picture where gravity is a collective 3 1=4 phenomenon in a pregeometric theory. In fact, the cosmo- the Planck energy), since the ratio LLV=LP 0a À increases with the diluteness of the condensate./ ð Þ logical constant puzzle is elegantly solved in those scenar- 3 ios. From an emergent gravity approach, the low energy Note that LLV scales with 0a exactly as the range of the gravitational force [see Eq. (18)], signaling that this effective action (and its renormalization group flow) is model is too simple to correctly grasp all the desired computed within a framework that has nothing to do with features. However, in more complicated systems [13], quantum field theories in curved spacetime. Indeed, if we this pathology can be cured, in the presence of suitable interpreted the cosmological constant as a coupling con- symmetries, leading to long range potentials. stant controlling some self-interaction of the gravitational It is instructive to compare the energy density corre- field, rather than as a vacuum energy, it would immediately sponding to Ã to the Planck energy density: follow that the explanation of its value (and of its properties under renormalization) would naturally sit outside the 4 7 5=2 domain of semiclassical gravity. Ãcs cs EÃ 3 À E Ã ; EP 2 ; 0a : (24) For instance, in a group field theory scenario (a general- ¼ 4GN ¼ GN EP / g0 @ ization to higher dimensions of matrix models for two The energy density associated with the analogue cosmo- dimensional quantum gravity [19]), it is transparent that logical constant is much smaller than the values computed the origin of the gravitational coupling constants has noth- from zero-point-energy calculations with a cutoff at the ing to do with ideas like ‘‘vacuum energy’’ or statements Planck scale. Indeed, the ratio between these two quantities like ‘‘energy gravitates’’, because energy itself is an emer- 3 is controlled by the diluteness parameter 0a . gent concept. Rather, the value of Ã is determined by the Final remarks.—Taken at face value, this relatively microphysics, and, most importantly, by the procedure to simple model displays too many crucial differences with approach the continuum semiclassical limit. In this respect, any realistic theory of gravity to provide conclusive it is conceivable that the very notion of cosmological

071101-4 Relativistic BEC and emergent LLI

Of course the BEC model is not Lorentz invariant at all scales as it is fundamentally non relativistic. But even breaking Lorentz invariance can be done in different ways S.Fagnocchi, S. Finazzi, SL, M. Kormos, A. Trombettoni: New. J. Phys.

2 2 1 ⌅⇤ˆ† ⌅⇤ˆ m c Bose–Einstein condensation, may ˆ = ⇤ˆ† ⇤ˆ + V (t, x) ⇤ˆ†⇤ˆ U(⇤ˆ†⇤ˆ; ) , L c2 ⌅t ⌅t ⇥ · ⇥ 2 i occur also for relativistic bosons. ✓ ~ ◆ So far only theoretical model. 2 2 3 3 with U(⇤ˆ†⇤ˆ; )= ⇥ˆ + ⇥ˆ + where⇥ ˆ = ⇤ˆ†⇤ˆ i 2 6 ··· The associated dispersion relation has a gapped and gapless mode 2 µ 2 mcc 2 µ µ 2 mcc 2 ⇤2 = c2 k2 +2 1+ 0 2 k2 + 1+ 0 . 8 v 9 ± c~ " µ # ± c~ u c~ " µ # <> ⇣ ⌘ ✓ ◆ ⇣ ⌘ u ⇣ ⌘ ✓ ◆ => t ~2 :> where c = ⇥U”(⇥, ) ,µ= relativistic chemical potential;> . 0 2m The gapless/massless mode is the interesting one as it admits an effective metric in the phononic regime mcc In the limit of very relativistic atoms b 0 1 ⌘ µ 2 2mc0 2 2 2 c cs vµv Long wavelengths limit k = ⇤ = csk with gµ = ⇥ µ + 1 ⌧ ) c c2 c2 ~ s  ✓ ◆ 2 2 LLI+Gravity cs = c b/(1 + b)

2mc0 2 2 2 SR Short wavelengths limit k = ⇤ = c k with gµ = µ ~ ) 4 2 2 2 k No LLI+Raibow metric/Finsler ? Intermediate ⇤ = cs k + 2 " 4b µ (1 + b)2 # ~c Lesson from rBEC: one can have IR and UV relativistic physics but nonetheless Lorentz violation at intermediate scales. You have to understand the classical/continuous limit to be sure about LI. decomposition in eq. (14) and taking the expectation value we get the equation of motion for the condensate to relativity groups with two limit speeds (cs in the IR limit, c in the UV one) and the

( m2)' 2'3 2'3 3 2 + 2 =0, relativity group(15) remain always the same at any energy. This is hence an example of a ⇤ 0 0 0 h 1i h 2i ⇥ ⇤ discrete model of emergent space-time where the low and high energy regime share the same where we have assumed that the cross-correlation of the fluctuations vanish, i.e., 1 2 =0. Lorentzh invariance.i This point is not trivial since, as far as we know, there is no toy model of This is justified a posteriori by equations (19), which show that 1 and 2 do not interact From the form of the e↵ective potential it is clear that at a givenemergent if µ>m spacetimethen the system in which Lorentz violation is screened in this way at the lowest order of with each other at the order of approximation we are working. Eq. (15) determines the is in the broken U(1) phase and the condensate has formed. It can be shown [23] that this dynamics of the condensate taking into account the backreaction of the fluctuations.perturbation It theory. is Our case show that this can be possible at the price of some non-trivial phase transition is second order and the critical temperature is given by conditions on the background system. the relativistic generalisation of the Gross–Pitævskii equationto relativity [27]. groups with two limit speeds (c in the IR limit, c in the UV one) and the 3 s T = µ2 m2 . (10) c relativity group remain always the same at any energy. This is hence an example of a where the subscript numbers in the Lagrangians represent the number of the perturbation B. DynamicsLater of perturbations: we shall be interested acoustic in the metric masslessfieldsdiscrete limitin each for of model which their ofthe terms. emergentV. critical Having EMERGENT temperature space-time done tis splitting is where given GENERAL one the can low show and COVARIANCE high how to energy rewrite regime the share the same to relativity groups with two limit speeds (cs in the IR limit, c in the UV one) and the 2 2 actionLorentz in a geometric invariance. form, i.e. This asa point function is not of the trivial conformal since, metric as farg asµ⌫ = we' know,⌘µ⌫. there is no toy model of by Tc =3µ /. Thus, in the massless case,relativity a non-zero group chemical remain potential always the is samenecessary at any in energy. This is hence0 an example of a Having determined the dynamics of the condensateThe we now final result want is to derived calculate in Eq. the (?? equa-)and,usingtheredefinedfieldswithappropriate order for the U(1) symmetry to be brokenemergentdiscrete and the model spacetime condensate of emergent inIn to which thebe space-time formed last Lorentz section where at violationa finite the we low saw and is screened high that energy the in regime thisfluctuations way share at the the of same lowest the condensate, order of also called the quasi- tions of motion for the perturbations themselves. dimensions, is given by non-zero critical temperature. perturbationLorentz invariance. theory.particle This Our point case excitations, is not show trivial that since, arethis as oblivious can far beas we possible know, of the there at flat the is no price background toy model of some of non-trivial metric. They instead experience a c3 to relativity1 groups1 with two limit speeds (c in the IR limit, c in the UV one) and the To this end, we insert ' = '0(1 + 1 + i 2)ineq.(14)andexpandittolinearorderinSemergent= S + S spacetime+ S + in= which Lorentzd4xp violationg R is screened⇤ in this way at the lowests(26) order of conditions0 1 on the2 curved background··· G geometry system. dictated6 12 by the condensate and the background. On the other hand, they N relativity⇢ group remain always the same at any energy. This is hence an example of a ’s. Using the Gross–Pitævskii equation to that order and separatingperturbation the theory. real Our and case imaginaryZ show that this can be possible at the price of some non-trivial III. RELATIVISTIC BEC AS AN ANALOGUEc3 GRAVITY MODEL1 1 4 back-react ondiscrete the model condensate of emergent through space-time the where relativistic the low and generalization high energy regime of share the the Gross-Pitaevskii same conditions+ ond thexp backgroundg R( system.+ ⇤) ⇤( ⇤ + ) parts we get the equation of motion for 1 and 2, sGN 6 6 V. EMERGENTZ ⇢ GENERALLorentz COVARIANCE invariance. This point is not trivial since, as far as we know, there is no toy model of A. Dynamics of the condensate: Gross–Pitævskii4 equationequation1 (15).1 It is natural toµ ask⌫ if it is possible to have a geometric description of the + d xp g R( ⇤ ) ⇤emergent[ + spacetime⇤ ⇤ +4 ⇤ in] whichg @ Lorentzµ ⇤@⌫ violation+ ... is screened in this way at the lowest order of +2⌘µ⌫@ (ln ' )@ 4'2 =0, 6 12(16a) ⇤ 1 µ 0 ⌫ 1 0 V.1 Z EMERGENT⇢ dynamics GENERAL of COVARIANCE the condensate too? In the last section we saw thatperturbation the fluctuations theory. Our of case the show condensate, that this can also be called possible the at the quasi- price of some non-trivial The e↵ective Lagrangian ofµ eq.⌫ (8) canLet be usrewritten now define in terms the analogue of the complex gravitational valued stress-energy fields tensorwhere for the the perturbation subscript numbers in the Lagrangians represent the number of the perturbation ⇤ 2 +2⌘ @µ(ln '0)@⌫ 2 =0. (16b) fieldsparticle as In the excitations, lastSo sectionthe cosmological weareThe saw oblivious Ricci that constantconditions the tensor of fluctuations the appear on offlat the the to background of background be the acoustic proportionalBelenchia, SL, Mohd: arXiv:1407.7896 condensate,fields system. metric. metric into also each the They called(17) Plank of their insteadis the mass calculated terms.quasi- squared. experience Having It to is done be a tis splitting one can show how to rewrite the as Emergent gravity in relativistic BEC Phys.Rev. D90 (2014) 10, 104015 1 (p g 2) 2 particle excitations,anyway true are thatobliviousTµ⌫ in the of definition the flat background ofL the, cosmological metric.action They constant in instead a geometric appears experience(27) form, also the a i.e. coupling asa function of the conformal metric gµ⌫ = ' ⌘µ⌫. We therefore see that is the massless mode, which is thecurved Goldstone geometry boson dictated of the by broken thep condensateg gµ⌫ and the background. On the other hand, they 0 2 µ⌫ 2 2 2 ⌘ ⇤ '0 e↵ = ⌘ @µ⇤@⌫ m ⇤ (⇤) + µ ⇤ + iµ(⇤@t @V.t⇤)(11) EMERGENT GENERALThe COVARIANCER final= result6 is derived in Eq. (??)and,usingtheredefinedfieldswithappropriate(21) L where wecurved will take geometryconstant into account dictated that only by can the the be condensate quadratic chosen to part beand small, the of the background. meaning action in that the On we perturbations, theg are other looking hand, at3 they a weakly inter- U(1) symmetry,where while the subscript1 is the numbers massive in mode the Lagrangians with mass represent 2back-react'0p. the We number on now the define of condensate the perturbation a “acoustic” through the relativistic generalization of the'0 Gross-Pitaevskii since theback-react linear oneacting on it the can system. condensate be show This to through give can no maybe contribution the relativistic give us usingthe generalization separation the backgrounddimensions, of of scales the and is Gross-Pitaevskiithat given pertur- one by could wish to find fieldsThe in each equation of their of terms. motion Having for is done obtained tis splitting by variation one can with show respect how to to rewrite and the weIn get,the last section we saw that the fluctuations of the condensate, also called the quasi- metric, which is conformal to the background Minkowski,equation (15). It is natural⇤ to ask if it is possible to have a geometric description3 c3 of the 1 1 Let us again decomposebation’s φequation equations.as φ = (15). andφ The(1 that It calculation is+Dividing was naturalψ2), found where to is the in explicitly ask [20] relativisticifφ due, itis is shown in possiblethe that in condensed case, Gross-Pitaevskii Appendix to have to matter a C, geometric here vacuum partS we= equationdescriptionS report polarizationof+ S the the+ S of byfield+ e the↵'ects.0=, eq. (15)d4xp cang beR written⇤ as (26) action in a geometric form, i.e. asa function of the conformal metric0gµ⌫ = '0⌘µ⌫. particle0 excitations, are oblivious of the flat0 background1 2 metric. They instead experience a dynamics of the condensate too? ··· GN 6 12 (⟨φ⟩ = φ ) and ψ is 2the2 fractionaldynamics@ of fluctuation theIn ourcondensate case the too? matter which vacuum can fluctuations be written contribution in terms to the cosmological of its real constant willZ ⇢ The final result is derived0 in Eq.⇤ + (??m2)and,usingtheredefinedfieldswithappropriateµfinal2 resultiµ for +2 the Gibbs(⇤) average =0. of the trace of the above(12) defined stress tensor 2 gµ⌫ = '0 ⌘µ⌫. @t curved(17) geometry dictated by the condensatem c3 and the background.1 On the other1 hand, they TheTheand Ricci Ricci imaginarybe tensor tensor given by of of the the parts acoustic acoustic metricψ = metric ψ (17)+ is (17)iψ calculated is calculated to be to be 4 dimensions, is given by ✓ ◆ ⇤ 2 2 1 2 2 2 Rg + 2 + ⇤ d=xpTqpg , R( + ⇤) ⇤( ⇤ + ) (22) T = 3 1 + 2 back-react= 2 3 on 1 the+ condensate 2 =6T throughqp . thesG relativisticN (28) h generalization i 6 of the Gross-Pitaevskii6 h i 6 h i h i h Gi h 2i h2 i Z ⇢ We can factor out explicitly3 the dependence on the chemical potential and write the field as⇤ 3 1 + 2 , (30) c 4 1 1 3⇤ '0 The relation between the d’Alembertian operators for gµ⌫ and ⌘µ⌫ is given by,⇥ equation⇤ ⇥ (15).c It ish⇤ naturali⇤'0 h toi ask6G if it is possible4 to1 have a geometric1 description of the µ⌫ S = S0 + S1 + S2 + = d xp gGivenR this last⇤ expression one sees that(26) the RHSRg = of6 eq. (25)3 is given by T+ andd x hencep g (21)R( ⇤ ) ⇤ [ + ⇤ ⇤ +4 ⇤ ] g @µ ⇤@⌫ + ... Crucial··· point:G in some 6 suitable 12 regime (neutral backgroundRg =' 6 3 field,c3 cs=c) you can (21) 2 2 N ⇢ where ⇤ := 12 is the “cosmological0⇥ ' ⇤ constant”h i and we6 have defined 12 Tqp := 12 [3 1 + 2 ], 1 Z2 = 'eiµt. so that the full cosmologicaldynamics constant of(13) the will condensate be 0 too? Z ⇢ h i h i h i 3 completelyµ⌫ our mask emergent the Nordström lorentz gravity breaking. equation will beIn exactlythis ofregime3 the form (23)one if finds one defines c 4g = 1+ ⌘ @µ(ln1 '0)Dividing@⌫. the relativistic Gross-Pitaevskii(18) equation by ' , eq. (15)Let can us be now written define as the analogue gravitational stress-energy tensor for the perturbation + ⇤d xp g2 ⇤ R( 2+ ⇤) ⇤( ⇤ + ) the “qp” subscript here stands0 for3 “quasiparticle” as this quantity is clearly dependent on ' ' eft Dividing the5 relativistic2 Gross-PitaevskiiThe Ricci equation tensor4 of⇡~ the by acoustic'2 , eq. metric2 (15) can (17) be is calculated written as to be sGN 0 6 0 G6 N = G/4⇡ = ~c /(4⇡µ ). This corresponds⇤ to an⇤ emergent1+ analogue3 0 Planck+ scale. MPl = (31) Z ⇢ R 2 c2M 2 fields1 as2 This gets rid of the µ dependent terms and we2 get m⌘ Pl h i h i Excitations1 Eq. 1 µp4⇡/c . the presenceR of the+ excitation✓⇤2= T , of our condensate.◆ '0 (22) 1 (p g 2) Equations (16) can be written4 p in terms of the d’Alembertian of gµ⌫ asµ⌫ g 2 m qp ⇥ ⇤ ⇤ T L , (27) + d x g R( ⇤ ) ⇤ [ + ⇤ ⇤ +4 ⇤ ]Backgroundg @µ ⇤@⌫ +Eq.... h i Rg = 6 3 µ⌫ µ⌫ (21) 6 12 Note that the contribution ofR theg vacuum+ fluctuations⇤ = Tqp , appears to be positive ' and not negative⌘(22)p g g Z ⇢ 2 2 2 0 ⇤ m ' 2 ' ' =0. This equation is clearly(14) reminiscenth wherei of we the will takeEinstein–Fokker into2 account2 only equation the quadratic describing part of the Nordström action in the perturbations, Let us now define the analogue gravitational stress-energywhere| ⇤| := tensoras 12 one is for thecould the “cosmological wishperturbation in order constant” to diminish and wethe have “bare” defined cosmologicalTqp := constant.12 [3 1 +3 2 ], 4 =0, (19a)Dividing the relativistic Gross-Pitaevskiih i equationh byi '0,h eq.i (15) can be written as ⇤g 1 1 B. Cosmological Constant 2 2 fields as wherethe “qp”⇤For:= subscript 12 m->is0 (allowed by non zero chemical the here “cosmological stands for “quasiparticle” constant” as and this we quantity havesince defined isthe clearly linearT dependentqp one:= it can12 be on show[3 to give+ no contribution], using the background and pertur- Nonetheless,gravity as [29, in [20], 30]. is worth to analyse the ratio between the2 energy density associated1 2 Let us now decompose2 ' as ' = 1'0(1 (p + g), where) '0 is the condensed part of the field mh i h i h i gμν =φ 0ημν g 2 =0. 2 potential μ) Equivalent(19b) to Einstein- bation’sR equations.+ ⇤ = TheT calculation, is explicitly shown in(22) Appendix C, here we report the Tµ⌫ ⇤ Athe dedicatedLthe “qp”, presence discussion subscriptto of the the bare isexcitation here deserved cosmological stands of by(27) our the for constant condensate. cosmological “quasiparticle” and the constant Planck as energy this term quantity that densityg arise2 of is in this clearly this model.qp dependent The former on ( ' = ' ), which we take to⌘ bep real,g andgµ⌫ isFokker the fractional equation fluctuation. of Nordström Reality of gravity!' R +⇤=24⇡Gh N T,i (23) 0 0 final result for the Gibbs average of the trace of the above defined stress tensor h i This equationis(but given4 is clearly bynot truly reminiscent background of the Einstein–Fokker equation describing Nordström where we will take into account only the quadraticmodel partthe due ofpresence to the the action oftype in the the of excitation perturbations, interaction. ofwhere First our⇤ of condensate.:= all 12 is worthis the “cosmological notice that other constant” interaction and we have defined T := 12 [3 2 + 2 ], amounts to considering the condensate at rest. Note that is complexindependent) and =0.It ⇤c4 ⇤ qp 1 2 7 gravity [29, 30]. h i 2 h 2 i h2 i h2 i since the linear one it can be show to give no contributionterms, polynomial using the in background the field,where will and notR pertur- giveand inT generalare contributions respectively✏⇤ such the that Ricci one can scalar identifyT = and the3 1 trace+ 2 of= the2 stress3 1 + energy 2 =6 tensorTqp . (28) Nordström gravity (1913) is theThis only equation other is clearly theory reminiscent inthe 3+1 “qp” dimensions of subscript the Einstein–Fokker/ here G standswhichN for satisfies “quasiparticle” equationh describingthei as Strong this6 h quantityNordströmi h isi clearly dependenth i h oni h i can be written in terms of its real and imaginary parts = 1 + i 2. SubstitutingR + ⇤ =24 this⇡G T, ✓ ◆ (23) bation’s equations. The calculationEquivalence is explicitlyan shownprinciple. acoustic in Appendix conformally However, C, flat hereof metric matter. weit reportis asnot in Unfortunately,the ourtruly case. background ReinstatingN the dimensional gravitational independent quantities analogy the of⇥ our equation⇤ is⇥ spoiled by⇤ the mass6G and the latter by the presence of the excitationGiven of our this condensate. last expression one sees that the RHS of eq. (25) is given by c3 T and hence gravity [29, 30]. 7 h i final result for the Gibbs average of the trace ofvalue the above of the(fixed defined would-be Minkowski stress cosmological tensor causal constant instructure) the model is c where R and T are respectively theThis Ricci equation scalar and is✏ clearly the trace reminiscent.our of emergent the stress of the Nordström energyEinstein–Fokker tensor gravity equation equation describing will be exactlyNordström of the form (23) if one defines term. Also we shall need toP show2 that the above defined Tqp is indeed the trace of the SET of 2 R + ⇤2 =242 /⇡G~NGNT, (23) ⇤ 2 2 6of2 matter.2 Unfortunately, the gravitationalµ analogyMPc of our equationeft is spoiled by5 the mass2 T = 3 + = 2 3 + =6Tqp . ⇤ 12(28)gravity=12 [29, 30]. . GN = G/4⇡ = ~c(29)/(4⇡µ ). This corresponds to an emergent analogue Planck scale MPl = h i 6 h 1i h 2i h 1i h 2i Thenh thei ratio is⌘ givenc by2 4⇡ ~ ~ 2 2 term. Also we shall need to show that the above defined✏ Tqp isc indeedp the2 trace of the SET of ⇥ ⇤ ⇥ where R and⇤ T are6G respectively2 the Ricci scalar⇤ and~ theµ 4 trace⇡small/cR.+ of⇤ the=24⇡ stressGN T, energy tensor (23) Given this last expression one sees that the RHS of eq. (25) is givenMPl = byµpc3 4T⇡/cand hence 2 (32) h i 11 ✏P ' µ our emergent Nordström gravity equation will be exactlyof matter. of the Unfortunately, form (23) if one the defineswhere gravitationalR and T analogyare respectively of our the equation Ricci9 scalar is spoiled and the by trace the of mass the stress energy tensor From this last expression is clear that this ratio is pretty small since it is suppressed by the eft 5The “bare”2 Λ is in this case small and positive but it will9 generically receivesB. Cosmological a (negative) Constant GN = G/4⇡ = ~c /(4⇡µ ). This corresponds to an emergentterm. Also analogue we shall Planck need scale2 to2 showMPlof= matter. that the Unfortunately, above defined theT gravitationalqp is indeed analogy the trace of our of the equation SET is of spoiled by the mass correction from the fraction of atomssmall ratio in the~ /µ non-condensate. phase, the depletion factor. 2 µp4⇡/c . term. Also we shall need to show that the above defined Tqp is indeed the trace of the SET of A dedicated discussion is deserved by the cosmological constant term that arise in this 4 VI. SUMMARY AND OUTLOOK9 model due to the type of interaction. First of all is worth notice that other interaction B. Cosmological Constant terms, polynomial9 in the field, will not give in general contributions such that one can identify In this paper we have studied the relativistic Bose-Einsteinan acoustic conformally condensation flat in metric a massless as in our case. Reinstating dimensional quantities the A dedicated discussion is deserved by the cosmological constant term that arise in this complex scalar field theory with a quartic coupling.value Below of the the critical would-be temperature cosmological the U constant(1) in the model is model due to the 4 type of interaction. First of all is worth notice that other interaction symmetry is broken and the charge resides in the non-zero value of the expectation value µ2 M 2c2 terms, polynomial in the field, will not give in general contributions such that one can identify ⇤ 12 =12 P . (29) of the field– the condensate. We showed that the dynamics of the condensate is described⌘ c2~ 4⇡~ an acoustic conformally flat metric as in our case. Reinstating dimensional quantities the by the relativistic generalization of the Gross-Pitaevskii equation given in eq. (15). The 11 value of the would-be cosmological constant in the model is µ2 M 2c2 12 ⇤ 12 =12 P . (29) ⌘ c2~ 4⇡~ 11 What next?

This is all interesting but is clearly open to several criticisms:

1. In the non-relativistic BEC you get only Newtonian gravity +Lorentz breaking 2. In the relativistic case, fine tuning of the system gives you Nordstrom gravity which is remarkable but it is still “just” scalar gravity+it is not really background independent What next?

This is all interesting but is clearly open to several criticisms:

Marolf’s theorem (Phys.Rev.Lett. 114 (2015) no.3, 031104): What next?

This is all interesting but is clearly open to several criticisms:

Marolf’s theorem (Phys.Rev.Lett. 114 (2015) no.3, 031104): gravitational theories with universal coupling to energy are characterized by Hamiltonians that are pure boundary terms on shell. In order for this to be the low energy effective description of a field theory with local kinematics, all bulk dynamics must be frozen and thus irrelevant to the construction. The result applies to theories defined either on a lattice or in the continuum, and requires neither Lorentz-invariance nor translation-invariance. Emergence of Gravity a la Analogue Model+ background independence requires kinematical non-locality (different microcausality) to start with… Ok, but can we test some of these ideas? From Analogue Models to Phenomenology (bite the bullet) QG phenomenology a la carte ex pluribus quattuor

Broken or deformed Symmetries • Lorentz Locality • • Translations QG induced non-locality • • SUSY (LHC searches but well below QG scale) Uncertainty Principle->GUP (no strong constraints) • Diffeomorphism (strong bounds from pulsar timing • Donoghue et al. PhysRevD.81.084059) Non-commutative geometries

Dimensions QG Modiﬁed • Extra dimensions (still missing obs. evidence so far) gravitational dynamics • Dimensional reduction in QG (early • E.g. Bouncing Universes universe?) • Regular Black holes, Fuzballs QG phenomenology a la carte ex pluribus quattuor

Let’s start with the PULP stuff..

Suggestions for Lorentz violation searches (at low or high energies) were not inspired only by Analogue models of emergent gravity. They came also from several QG models

String theory tensor VEVs (Kostelecky-Samuel 1989, ...) Cosmological varying moduli (Damour-Polyakov 1994) Spacetime foam scenarios (Ellis, Mavromatos, Nanopoulos 1992, Amelino-Camelia et al. 1997-1998) Some semiclassical spin-network calculations in Loop QG (Gambini-Pullin 1999) Einstein-Aether Gravity (Gasperini 1987, Jacobson-Mattingly 2000, …) Some non-commutative geometry calculations (Carroll et al. 2001) Some brane-world backgrounds (Burgess et al. 2002) Ghost condensate in EFT (Cheng, Luty, Mukohyama, Thaler 2006) Horava-Lifshiftz Gravity (Horava 2009, …)

Quote: “How you dare to Violate Lorenz Invariance?”

Lorentz invariance is rooted via Einstein equivalence principle in GR and it is a fundamental pillar of the SM. The more fundamental is an ingredient of your theory the more needs to be tested observationally! You do not need Planck scale observations to constraint Planck suppressed Lorentz violations. In any quantum/Discrete gravity model it is a non-trivial task to recover exact Local Lorentz Invariance and/or background independence. Hence it is very important to understand what is needed in order to conciliate LLI and forms of hard or quantum discreteness at the Planck scale. Lorentz violation: a possible ﬁrst glimpse of QG?

Suggestions for Lorentz violation searches (at low or high energies) were not inspired only by Analogue models of emergent gravity. They came also from several QG models

Quote: “How you dare to Violate Lorenz Invariance?”

Lorentz invariance is rooted via Einstein equivalence principle in GR and it is a fundamental pillar of the SM. The more fundamental is an ingredient of your theory the more needs to be tested observationally! You do not need Planck scale observations to constraint Planck suppressed Lorentz violations. In any quantum/Discrete gravity model it is a non-trivial task to recover exact Local Lorentz Invariance and/or background independence. Hence it is very important to understand what is needed in order to conciliate LLI and forms of hard or quantum discreteness at the Planck scale. But what we mean by Lorentz Invariance violation? Breaking of Local Lorentz Invariance von Ignatowsky theorem (1911): Axiomatic Special Relativity

Principle of relativity ➔ group structure Homogeneity ➔ linearity of the Lorentz transformations with transformations unfixed limit speed C Isotropy ➔ rotational invariance and C=∞ ➔ Galileo Riemannian structure C=clight ➔ Lorentz Precausality ➔ observer independence of co- Experiments determine C! local time ordering

W. von Ignatowsky (Tiblisi 1875-Leningrad 1942) Breaking of Local Lorentz Invariance von Ignatowsky theorem (1911): Axiomatic Special Relativity

Lorentz breaking does not necessarily mean to have a preferred frame!

Break Precausality ➔ Hell breaks loose, better not!

Break Principle of relativity ➔ Preferred frame effects

Break Isotropy ➔ Finsler geometries. E.g. Very Special Relativity (Glashow, Gibbons et al.).

Break Homogeneity ➔ no more linear transformations ➔ No Locally Euclidean Space. ➔ tantamount to give up operative meaning of coordinates W. von Ignatowsky (Tiblisi 1875-Leningrad 1942) Breaking of Local Lorentz Invariance von Ignatowsky theorem (1911): Axiomatic Special Relativity

Lorentz breaking does not necessarily mean to have a preferred frame!

Break Precausality ➔ Hell breaks loose, better not!

Break Principle of relativity ➔ Preferred frame effects

Break Isotropy ➔ Finsler geometries. E.g. Very Special Relativity (Glashow, Gibbons et al.).

Break Homogeneity ➔ no more linear transformations ➔ No Locally Euclidean Space. ➔ tantamount to give up operative meaning of coordinates W. von Ignatowsky (Tiblisi 1875-Leningrad 1942)

Let’s relax the Relativity Principle ﬁrst and study the phenomenology. To do this we need a framework… Dynamical frameworks for LIV

Frameworks for preferred frame effects See e.g. SL. CQG Topic Review (2013)

See e.g. Amelino-Camelia. Living Reviews of Relativity EFT+LV Non EFT proposals: Spacetime foam models DSR/Relative Locality

local EFT with LIV Non-renormalizable ops, CPT ever or odd Minimal Standard Model Extension (no anisotropic scaling), Renormalizable ops. (UV LIV – QG inspired LIV) (IR LIV - LI SSB) NOTE: CPT violation implies Lorentz violation but LV does not imply CPT violation in local EFT. “Anti-CPT” theorem (Greenberg 2002 ). So one can catalogue LIV by behaviour under CPT E.g. QED, rot. Inv. dim 3,4 operators E.g. QED, dim 5 operators

(Colladay-Kosteleky 1998, Colemann-Glashow 1998) (Myers-Pospelov 2003) Constraints on QED dim 5 CPT Odd QED extension

Jacobson, SL, Mattingly: Nature (2003) L.Maccione, SL, A.Celotti and J.G.Kirk: JCAP 0710 013 (2007) L.Maccione, SL, A.Celotti and J.G.Kirk, P. Ubertini:Phys.Rev.D78:103003 (2008)

The Crab nebula a supernova remnant (1054 A.D.) distance ~1.9 kpc from Earth. Spectrum (and other SNR) well explained by synchrotron self-Compton (SSC) 9 10 Electrons are accelerated to very high energies at pulsar: in LI QED γe≈10 ÷10 High energy electrons emit synchrotron radiation Synchrotron photons undergo inverse Compton with the high energy electrons Synchrotron Inverse Compton Constraints on QED dim 5 CPT Odd QED extension

Jacobson, SL, Mattingly: Nature (2003) L.Maccione, SL, A.Celotti and J.G.Kirk: JCAP 0710 013 (2007) L.Maccione, SL, A.Celotti and J.G.Kirk, P. Ubertini:Phys.Rev.D78:103003 (2008)

The synchrotron spectrum is strongly aﬀected by LIV: maximum gamma factor for subliminal leptons and vacuum Cherekov limit for superluminal ones (there are both electrons and positrons and they have opposite η). Spectrum very well know via EGRET, now AGILE+FERMI Constraints on QED dim 5 CPT Odd QED extension

Jacobson, SL, Mattingly: Nature (2003) L.Maccione, SL, A.Celotti and J.G.Kirk: JCAP 0710 013 (2007) L.Maccione, SL, A.Celotti and J.G.Kirk, P. Ubertini:Phys.Rev.D78:103003 (2008)

The polarization of the synchrotron spectrum is strongly affected by LIV: there is a rotation of the angle of linear polarization with different rates at different energies. Strong, LIV induced, depolarization effect.

Polarization recently accurately measured by INTEGRAL mission: 40±3% linear polarization in the 100 keV - 1 MeV band + angle θobs= (123±1.5)∘ from the North Constraints on QED dim 5 CPT Odd QED extension

Jacobson, SL, Mattingly: Nature (2003) L.Maccione, SL, A.Celotti and J.G.Kirk: JCAP 0710 013 (2007) L.Maccione, SL, A.Celotti and J.G.Kirk, P. Ubertini:Phys.Rev.D78:103003 (2008)

Polarization recently accurately measured by INTEGRAL mission: 40±3% linear polarization in the 100 keV - 1 MeV band + angle θobs= (123±1.5)∘ from the North EFT with Lorentz breaking Ops. Matter Sector Constraints Terrestrial tests: Astrophysical tests:

Penning traps Cosmological variation of couplings, CMB Clock comparison experiments Cumulative effects in astrophysics Cavity experiments Anomalous threshold reactions Spin polarised torsion balance Shift of standard thresholds reactions with new Neutral mesons threshold phenomenology Slow atoms recoils LV induced decays not characterised by a threshold Reactions affected by “speeds limits”

n 2 2 (n) k E = k + ⇠ n 2 photons ± Mpl n 2 2 2 (n) p Ematter = m + p + ⌘ n 2 leptons/hadrons , ± Mpl

(n) (n) n (n) (n) (n) n (n) where, in EFT, ⇠ ⇠+ =( ) ⇠ and ⌘ ⌘+ =( ) ⌘ . ⌘ ⌘

SL, CQG Topic Review 2013 EFT with Lorentz breaking Ops. Matter Sector Constraints Terrestrial tests: Astrophysical tests:

n 2 2 (n) k E = k + ⇠ n 2 photons ± Mpl n 2 2 2 (n) p Ematter = m + p + ⌘ n 2 leptons/hadrons , ± Mpl

(n) (n) n (n) (n) (n) n (n) where, in EFT, ⇠ ⇠+ =( ) ⇠ and ⌘ ⌘+ =( ) ⌘ . ⌘ ⌘

Warning GZK ISSUE! 0 p+γCMB -> p+π + p+γCMB -> n+π

SL, CQG Topic Review 2013 Caveat: A potential problem with the UHECR data?

With increased statistics the composition of UHECR beyond 1019 eV seems more and more dominated by iron ions rather than protons at AUGER. But Telescope Array (TA) in Utah is instead Ok with purely proton composition. Confusion.

With improved statistic the correlated AUGER UHECR-AGN events have decreased from 70% to 40%: large deﬂections? i.e. heavy (high Z) ions?

Also no evidence at the TA for AGN correlation. But some hint of correlation with LLS for E>57 EeV

Ions do photodisintegration rather than the GZK reaction, this may generate much less protons which are able to create pions via GZK and hence UHE photons.

Have we really seen the GZK cutoﬀ or sources exhaustion? See e.g. arXiv:1408.5213.

If not all the constraints on dim 6 CPT even operators would not be robust…

Furthermore puzzling cut oﬀ above 2 PeV in UHE neutrinos at IceCube maybe 4 15 consistent with p LIV at MLIV~10 GeV. F.W. Stecker, S.T. Scully, SL, D. Mattingly. JCAP 2015 More bad news? The flies in the Ointment…

Lorentz breaking theories suffers two main theoretical problems

• Naturalness problem

• Possible breakdown of black hole thermodynamics The “un-naturalness” of small LV in EFT

[Collins et al. PRL93 (2004), Lifshitz theories (anisotropic scaling): Iengo, Russo, Serone (2009)] Gambini, Rastgoo, Pullin Class.Quant.Grav. 28 (2011) 155005 . Polcinski (2011)

Unsuppressed Dim>4 No new Radiative corrections to LiV physics up to dim 4 operators Corrections Operators ELIV (low energy propagators)

Given the strong constraints on dim 3,4 LIV operators (low energy effects) this is BAD Note: some interesting exceptions apply. See Belenchia, SL, Gambassi: JHEP 2016 The “un-naturalness” of small LV in EFT

[Collins et al. PRL93 (2004), Lifshitz theories (anisotropic scaling): Iengo, Russo, Serone (2009)] Gambini, Rastgoo, Pullin Class.Quant.Grav. 28 (2011) 155005 . Polcinski (2011)

Unsuppressed Dim>4 No new Radiative corrections to LiV physics up to dim 4 operators Corrections Operators ELIV (low energy propagators)

Given the strong constraints on dim 3,4 LIV operators (low energy effects) this is BAD Note: some interesting exceptions apply. Three main Ways out See Belenchia, SL, Gambassi: JHEP 2016 The “un-naturalness” of small LV in EFT

[Collins et al. PRL93 (2004), Lifshitz theories (anisotropic scaling): Iengo, Russo, Serone (2009)] Gambini, Rastgoo, Pullin Class.Quant.Grav. 28 (2011) 155005 . Polcinski (2011)

Unsuppressed Dim>4 No new Radiative corrections to LiV physics up to dim 4 operators Corrections Operators ELIV (low energy propagators)

Given the strong constraints on dim 3,4 LIV operators (low energy effects) this is BAD Note: some interesting exceptions apply. Three main Ways out See Belenchia, SL, Gambassi: JHEP 2016

Custodial symmetry

One needs EFT range to be bonded by EEFT

E.g. gr-qc/0402028 (Myers-Pospelov) or hep-ph/0404271 (Nibblink-Pospelov) or gr- qc/0504019 (Jain-Ralston), SUSY QED:hep-ph/0505029 (Bolokhov, Nibblink-Pospelov). See also Pujolas- Sibiryakov (arXiv:1109.4495) for SUSY Einstein-Aether gravity.

Analogue model in 2-BEC: SL, Visser, Weinfurtner, Phys.Rev.Lett. 96 (2006) 151301 The “un-naturalness” of small LV in EFT

[Collins et al. PRL93 (2004), Lifshitz theories (anisotropic scaling): Iengo, Russo, Serone (2009)] Gambini, Rastgoo, Pullin Class.Quant.Grav. 28 (2011) 155005 . Polcinski (2011)

Unsuppressed Dim>4 No new Radiative corrections to LiV physics up to dim 4 operators Corrections Operators ELIV (low energy propagators)

Given the strong constraints on dim 3,4 LIV operators (low energy effects) this is BAD Note: some interesting exceptions apply. Three main Ways out See Belenchia, SL, Gambassi: JHEP 2016

Gravitational conﬁnement

Assume only gravity LIV with M*<

Analogue model in 2-BEC: SL, Visser, Weinfurtner, Phys.Rev.Lett. 96 (2006) 151301 The “un-naturalness” of small LV in EFT

[Collins et al. PRL93 (2004), Lifshitz theories (anisotropic scaling): Iengo, Russo, Serone (2009)] Gambini, Rastgoo, Pullin Class.Quant.Grav. 28 (2011) 155005 . Polcinski (2011)

Unsuppressed Dim>4 No new Radiative corrections to LiV physics up to dim 4 operators Corrections Operators ELIV (low energy propagators)

Given the strong constraints on dim 3,4 LIV operators (low energy effects) this is BAD Note: some interesting exceptions apply. Three main Ways out See Belenchia, SL, Gambassi: JHEP 2016

Gravitational conﬁnement

Assume only gravity LIV with M*<

Violations of the Generalised Second Law in Lorentz breaking scenarios

A and B fields interacts only gravitationally

cB > cA RB < RA TB,Haw > TA,Haw

RA Surround the BH with two shells of A and B fields It is possible to choose the temperatures of the shells such that

TB,Haw >TB,shell>TA,shell>TA,Haw RB

Then TA,shell>TA,Haw implies flux from Shell A to BH B,shell But TB,Haw >TB,shell implies flux from BH to shell B One can choose the temperatures of the shells in such a way that the two energy fluxes compensate each other. So BH mass, radius, entropy stay constant. A,shell But TB,shell>TA,shell hence the net effect is to bring heat from cooler shell to hotter one!

Note: split in horizons can be used also to generate classical violation of GSL (region between radii is like ergoregion for B field: possible energy extraction) Conclusion: Violation of LLI seems to lead to violation of the Generalized Second Law (GSL).

S.L.Dubovsky, S.M.Sibiryakov, Phys. Lett. B 638 (2006) 509. C. Eling, B. Z. Foster, T. Jacobson and A. C. Wall, “Lorentz violation and perpetual motion”, Phys. Rev. D 75 (2007) 101502. T. Jacobson and A. C. Wall, “Black hole thermodynamics and Lorentz symmetry”, Found. Phys. 40 (2010) 1076. Violations of the Generalised Second Law in Lorentz breaking scenarios

A and B fields interacts only gravitationally

cB > cA RB < RA TB,Haw > TA,Haw

RA Surround the BH with two shells of A and B fields It is possible to choose the temperatures of the shells such that

TB,Haw >TB,shell>TA,shell>TA,Haw RB

However all of this was not taking into account LIV in gravity and UV completion… Gravity VS Local Lorentz invariance (what does not kill you makes you stronger) Lorentz breaking gravity

Einstein-Aether Rotationally invariant Lorentz violation in the gravity sector via a vector field. (Jacobson-Mattingly 2000) Take the most general theory for a unit timelike vector field coupled to gravity which is second order in derivatives.

Horava-Lifshitz (Horava 2009)

λ=1, ξ=1, η=0 in General Relativity (GR). IR limit L2 is Einstein-Aether with hypersurface orthogonal aether field. Observationally constrained but not ruled out arXiv:1311.7144 [gr-] Yagi et al

Blas,Pujolas,Sibiryakov, Phys. Lett. B 688, 350 (2010). 16 The condition M*<10 GeV is a consequence of the need to protect perturbative renormalizability w.r.t. the mass scale of the Horava scalar mode A new hope? Universal horizons

Bh, spherically symmetric solution are the same in AE and Horava gravity. The aether is hypersurface orthogonal and there is a compact constant Kronon hypersurface from which even infinite velocity signals cannot escape. Alternatively the UH occurs when the Killing field χ associated to energy at infinity becomes orthogonal to the aether field: (χu)=0. (note: Open issue with stability)

Eternal: D. Blas and S. Sibiryakov (2011), E. Barausse, T. Jacobson, T. P. Sotiriou (2011) and Collapse solutions: M.Saravani, N. Afshordi, Robert B. Mann., (2014).

Analogue model of AE-BH with UH: B.Cropp, S.L and R. Turcati, arXiv:1606.01044 [gr-qc].

Low energy

v UH KH 40

FIG. 1: Conformal diagram of black hole with Universal horizon, showing lines of constant khronon ﬁeld, with the Universal horizon shown in red. 10 extensively throughoutRaytracing this paper, we will brieﬂy of summarize of superluminal some of the relevant details matter of the solutions. For more information and background we refer the reader to Ref. [12]. Both 0 r solutions, in Eddington–FinkelsteinCropp, SL, Mohd, coordinates, Visser. can be written as 0.5 1.0 1.5 2.0 Phys.Rev. D89 (2014) no.6, 064061 High energy ds2 = e(r)dv2 +2dv dr + r2 d⌦2. (6) Here the form of the æther is

ua = ↵(r), (r), 0, 0 ; u = (r) e(r)↵(r), ↵(r), 0, 0 . (7) { } a { } Note from the normalization condition, u2 = 1, there is a relation between ↵(r)and(r): e(r)↵(r)2 1 (r)= . (8) 2↵(r)

We can also deﬁne a spacelike vector sa, such that

a 2 s ua =0; s =1. (9)

Explicitly

e(r)↵(r)2 +1 sa = ↵(r),e(r)↵(r) (r), 0, 0 = ↵(r), , 0, 0 , (10) { } 2↵(r) ⇢ which clearly ensures s2 =1.Thetwoknownexactblack-holesolutionstoEinstein–Æther theory correspond to the special combinations of coecients c123 =0andc14 =0.

Solution 1: c =0. • 123 Universal Horizon Thermodynamics?

Berglund, Bhattacharyya, Mattingly UH thermodynamics Laws Phys.Rev. D85 (2012) 124019 Arif Mohd. e-Print: arXiv:1309.0907 c q =(1 c ) + 123 K Gist Status Math UH 13 UH 2 UH| |UH

✔ c13 = c1 + c3 c123 = c1 + c2 + c3 surface gravity is µ oth (but trivial in µ UH =0 constant on UH r | µ ⌫ spherically symm. sols) UH = µ⌫ qUHAUH rµ r KUH = µu Energy conservation ✔ Mæ = p 1st 8⇡Gæ r k peeling instead is Non decreasing 1 d dr ✔? (GSL?) ¯UH 2nd AUH 0 ⌘ 2 dr dv entropy UH Unattainability of 1 3rd ? ? ¯ ua (u ) T=0 state UH ⌘ 2 ra · UH Does the UH radiate? Berglund, Bhattacharyya, Mattingly, Phys.Rev.Lett. 110 (2013) 7, 071301 Tunneling method a la Parikh—Wilczekc leads to predict a thermal spectrum with temperature  UH (1 c13)cæAUH TUH = from this and 1st law SUH = 4⇡c 2Gæ Notes æ

1. The calculation assumes vacuum at UH for infalling observers (like Unruh for UH) 2. The above temperature obtained is not κUH/2π but it is instead k(peeilng)UH/2π. 3. F. Michel and R.Parentani, Phys. Rev. D91, no. 12, 124049 (2015) get from shell collapse different vacuum state (Unruh for KH) and find radiation at KH temperature. Lorentz regained: Searching Non-locality Non-locality as an alternative to symmetry breaking?

What about other mesoscopic physics without Lorentz violation?

We do have concrete QG models of emergent gravity like Causal Sets or String Field Theory or Loop Quantum Gravity which generically seem to predict exact Lorentz invariance below the Planck scale in spite of (fundamental or quantum) discreteness at the price to introduce non-local EFT. Conjecture: Discreetness + Lorentz Invariance = Non-Locality

Note also Marolf’s theorem: Emergence of Gravity a la Analogue Model+ background independence requires diﬀerent notion of locality between fundamental and emergent dof.

Several forms of non-locality

Νon-local kinetic terms Non-local interactions DSR-like non-locality …

These theories involve a very subtle phenomenology very hard to constraint, still they do show novelties. Differently from UV Lorentz breaking physics it will be here a matter of PRECISION instead of HIGH ENERGIES… Non-local D’Alambertians f( ) ⇤ ! ⇤ Generic expectation if you want to introduce length or energy scale in ﬂat spacetime KG equation without giving up Lorentz invariance. Concrete examples of kinematical non-locality respectively with non-analytic or analytic function ↵ 2 2 2 Causal Set Theory ⇤⇢ ⇤ + ⇤ + ⇤ ln ⇤ + ... ⇡ p⇢ p⇢ ⇢ 2 ⇤+m ✓ ◆ String Field Theory 2 2 ( + m )exp ⇤ ⇤ =1/`nl ⇤ ! ⇤ Non-local D’Alambertians f( ) ⇤ ! ⇤ Generic expectation if you want to introduce length or energy scale in ﬂat spacetime KG equation without giving up Lorentz invariance. Concrete examples of kinematical non-locality respectively with non-analytic or analytic function ↵ 2 2 2 Causal Set Theory ⇤⇢ ⇤ + ⇤ + ⇤ ln ⇤ + ... ⇡ p⇢ p⇢ ⇢ 2 ⇤+m ✓ ◆ String Field Theory 2 2 ( + m )exp ⇤ ⇤ =1/`nl ⇤ ! ⇤ A typical signature of non-analytical non-local propagators are violations of the Huygen principle: The propagator of massless particles can have support inside the light cone in 3+1 3

6 44

⌘ TIMELIKE " ! 5 54 3 21 3 " 4 ) 3 ! ⌘fB 3 2 2 B Bob 1 1.5 ! " ⌘ 6 2 A ! iB 2 ! C

5 2 a ! " 1.0 ! " C ! ! " (10 ! " ! 1 ! 1 ! ⌘fA 0.5 C ! " LIGHTLIKE " " " " " " " " " ! SPACELIKE ! " Alice " " " " 0.0 !!!"""0!" ! ! ! ! ! ! ! ! ! !!!!!!!!!!! !" !" !" !" ⌘iA 0 - xxx 1000 2000 1030001234400020 5000 600030 7000 40 R R (/30)

" " FIG. 1. Di↵erent causal relationships between Alice and 22.5.5 " ! Bob’s detectorsPossibly switching very periods. relevant These cases for are explicitly " " 12" 3 4 5 relativistic quantumspeciﬁed in information Table I. Recall that tests⌘i⌫ ⌘as(Ti⌫ ),detectors⌘f⌫ ⌘(Tf⌫ ). can inﬂuence22.0.0 ! " ⌘ ⌘ ) ! " " 2 B

each other at timelike separations 1.5 1.5 ! ! 2 A 1.5 ! ! ! C min (⌘ + R, ⌘ ) max (⌘ + R, ⌘ ) 5 b fA fB iA iB ! ! z = ,z= . " 1 2 1.0 1.0 ! ! " C R R 1.0 ! !

(10 ! Opportunity for Phenomenology? (11) ! ! R. H. Jonsson, E. Martin-Martinez, and A. Kempf, Phys.Rev.Lett. 114, 110505 (2015). Ana Blasco, Luis J. Garay, Mercedes Martin-Benito, Eduardo Martin-Martinez. Phys.Rev.Lett. 114 (2015) 14, 141103 0.5 C 00.5.5 ! " ! " " " "" "" "" " " " " 0.0 !!!"""!" !!!!!!!!!! For simplicity, in Eqs. (7)-(8) we have already par- 00.0.0 !" !" !" !" ! ! ! ! ! ! ! ! ! ! ticularized the study to the case of zero-gap detectors, 1000 20001000123456730002000 40003000 50004000 60005000 70006000 7000 ⌦⌫ = 0. This choice is arbitrary and has no e↵ect on TiB (10/3) our main results. Moreover, it is not uncommon to find relevant atomic transitions between degenerate (or quasi- FIG. 2. Channel capacity (in bits) and its -term as func- degenerate) atomic energy levels, for example, atomic tions of (a) the spatial separation between Alice and Bob, for electron spin-flip transitions. TfA TiA = TfB TiB = , TiA = /30, and TiB =10, Channel capacity.— Let us now compute the capac- (b) the temporal separation between Alice and Bob. In (b), we vary TiB while keeping TfA TiA = TfB TiB = ity of a communication channel between an early Uni- constant and we fix TiA = /30 and R = /10. Di↵erent verse observer, Alice, and a late-time observer, Bob. To regions are labelled according to the case numbers of Fig. 1 obtain a lower bound to the capacity, we use a simple and Table I. Since both detectors remain switched on during communication protocol: Alice encodes “1” by coupling the same amount of proper time, only cases 1 to 5 occur. The her detector A to the field, and “0” by not coupling it. violation of strong Huygens can be seen in region 5 (timelike Later, Bob switches on his detector B and measures its separation). state. If B is excited, Bob interprets a “1”, and a “0” otherwise. The capacity of this binary asymmetric channel (i.e., the number of bits per use of the channel that given, at leading order, by Alice transmits to Bob with this protocol) was proven to 2 S 2 be non-zero [5], no matter the level of noise, and it is C 2 2 2 + (6 ). (12) ' A B ln 2 4 ↵ O ⌫ ✓ | B|| B|◆ Figures 2a and 2b show the behavior of the channel ca- TABLE I. Cases of causal relationships. See Fig. 1. pacity C. For comparison, we also display the channel capacity in the conformally coupled case, C.Wehave Case Conditions selected initial detector states that, in our case, maxi- 1 ⌘fB ⌘iA + R mize the channel capacity (i.e. ↵ = =1/p2,  | A| | A| 2 ⌘iB < ⌘iA + R<⌘fB ⌘fA + R arg(↵ ) arg( )=⇡, arg(↵ ) arg( )=⇡/2).  A A B B 3 ⌘iB ⌘iA + R, ⌘fB ⌘fA + R Let us first analyze how the ability of Alice to signal  4 ⌘fB > ⌘fA + R>⌘iB ⌘iA + R Bob depends on their time separation. From the -term 5 ⌘iB ⌘fA + R of Eq. (6) we see that the information transmitted by ‘rays of light’ decays with the distance between A and B, 6 ⌘iB < ⌘iA + R, ⌘fB > ⌘fA + R becoming negligible for long times. This yields the unsur- in the local limit (assuming a1 =1). In the SFT inspired case we explicitly have the non-local Schroedinger operator n 1 1 2m 1 1 n+1 . (5) n! 2 ⇤2(n 1) ⇤2 S ⌘ SNL n=0 ~ Testing non-local EFT with X A. Belenchia,✓ D. ◆Benincasa, SL, F. Marin, F. Marino, A. Ortolan. Phys.Rev.Lett.an 116 (2016) no.16, 161303 optomechanical oscillators | {z } in the local limit (assuming a1 =1). 1.1 Quantum harmonic oscillator In the SFT inspired case we explicitly have the non-local Schroedinger E.g. let’s consider its non-relativistic limit of a non-local KG with analytic f(). operator We are interested1 in studying the following equation n 1 1 2m 1 1 n+1 NL. (5) n! 2 ⇤2(n 1) ⇤2 S ⌘ S So we get ( NL V ) (t, x)=0. (6) n=0 ✓ ~ ◆ S X Double Wheel Oscillator - DWO an Given the diculty of resolving such an equation exactly we could try a | {z } Whereperturbative can approach.be Newtest version this? ofIn DWO particular with torsional joints I in will the central consider part that it is the following ansaz for 1.1 Quantum harmonic oscillator actuallyused in the find experiment. a length scale to which compare the non-locality one. Indeed HUMOR the ground state1 wave - Front view of DWO function (SEM image) with the central coating 2 - Back view DWO (SEM image) with the insulation we would like to have wheel We are interested1 in studying the following equation l2 Heisenberg Uncertainty Measured with Opto- ✏ = nl , = 0 + ✏1, (7) mechanical Resonators( (LENSV ) - (Florence,t, x)=0 Italy). (6) SNL where is some length scale. One could think about the linear size of the where is the ground state for the local case and is a perturbatively Given the diDesignedculty of to resolvingtest generalised such uncertainty an equation principle exactly 0 wesystem could or try even a the DeBroglie wavelength. However1 there is actually another Macroscopic harmonic oscillator.small correction correction to it. Here the parameter that I am considering perturbative approach. In-11 particular-5 I+5 will consider+3 the followingscale ansaz in the for system that is the frequency of the oscillator. One could then m~10 / 10 Kg ω~10 / 10 Hz 2 1 the ground state wave function as small is ✏construct= ⇤2 . Given now the form of the non-local operator our equation at the order ✏ gives Withm! ϵ the small1 In order to solve the non-local Schroedinger, one needs to adopt a 1 n dimensionless, = + ✏ , (INFN Gruppo Collegato(7) Trento) HUMOR 05-07-2012 17 / 29 0 1 = 0 + ✏ n ⌘ perturbative expansion around a “local” Sch. solution parameter~ for this that moreover is( then=1 varianceV )1 = of theproblem.0, ground state of the oscillator.(8) Then one where 0 is the ground state for the local case and 1 is a perturbatively S X D And at the lowest order we can solve could identify (t,x) small correction correction to it. Here the parameter that I am considering J m! as small2 is ✏ = 1 . Given now the form of the non-local operator our ✏ = ⇤2 n+1 ⇤2 2 where = 1 an . Now we have to| solve{z } the~ local Schodinger equation equation at the order ✏ gives ~ 2 D 2 nas=12 theS small1 dimensionless2 2 parameter in which doing the expansion. i~@t + @xx for+ ✏ ana2 harmonic oscillator= m! withx . a source term dependent on the non-local 2m ~!P S NOTE:2 One could be tempted to do an expansion and a similar analysis ✓ ( V )1 = ◆ operator,0, ✓ the◆ local ground(8) state, space and time. For solving this we should S D also in the free case, i.e. without a potential. In that case I don’t know what (t,x) J be able to useparemeter the Green could function be identify method, for i.e. an expansion. This maybe is good, since in n+1 where = 1 an . Now we have to| solve{z } the local Schodingerthat case equation we know thatt an expansion, with the corresponding truncation of D n=1 S 1 1 for an harmonic oscillator with a source term dependent onthe the1(t, operator non-localx)= willdx only0 introducedt0 K(x, spurious t; x0,t0) corrections(x0,t0), given the(9) fact that a P ~ J operator, the local ground state, space and time. For solvingsolution this we should ofZ the1 localZ equation1 is solution also of the non-local one. be able to use the Green function method, i.e.where I am using the notation that can be found on Wikipedia for the Green t 1 1 function of the2.0.1 quantum Some harmonic numbers oscillator. Note the causal aspect of the 1(t, x)= dx0 dt0 K(x, t; x0,t0) (x0,t0), (9) 2 ~ 1 J 34 Kgm Z1 Z1 In the following I will perform the calculations~ in10 2D. where I am using the notation that can be found2 onActually Wikipedia this for is a the dimensional Green parameter. Is it⇡ ok to use its as small parameter in the perturbative expansion?Suppose Note that this is the non-locality scale that we were assuming function of the quantum harmonic oscillator. Note the causal aspect of the 9 playing the game of the small parameter last time.m =1µg =10 Kg 1In the following I will perform the calculations in 2D. 2Actually this is a dimensional parameter. Is it ok to use it as smalland parameter in the 4 perturbative expansion? Note that this is the non-locality scale that we were assuming 2 ! 5 10 Hz. ⇡ · playing the game of the small parameter last time. Then our parameter will be ✏ 5 1029l2 2 ⇡ · nl that means 14 ✏ 1 l p2 10 m ⌧ , nl ⌧ · This is clearly reasonable since it means that the expansion is justified for a non-locality scale below the fermi4. 4That however is the raw extimate of the causal set non-locality by Rafael. Note that here we are not taking into account that model.

5 rule we shall deﬁne the probability density as

(t, x) (t, x) ⇢(t, x)= ⇤ , (4.12) 1 2dx 1 | | R such that 1 dx ⇢(x) = 1. It should be noted that, for the ground state this normalization factor is one1 at order ✏,i.e. = 0, while in the case of a generic coherent state an R h 0| 1i order ✏ time dependent correction will be present. The above normalisation factor ensures that even in this case we a have a meaningful probability distribution.

4.1 Spontaneous Squeezing of States Given our probability distribution (4.12) we can now compute the mean and variance of Spontaneousthe position andsqueezing momentum of the particle. We ﬁnd from non-locality

1 x = p2↵ cos(t) 1+ ✏↵2a [cos(2t) 1] + (✏2), (4.13) h i 4 2 O ✓ ◆ 1 p = p2↵ sin(t) 1+ ✏ a ↵2(7 + 3 cos(2t)) 2 + (✏2), (4.14) h i 4 2 O ✓ ◆ 1 ⇥ ⇤ Results Var(x)= 1 ✏a 6↵2 1 sin2(t) + (✏2), (4.15) 2 2 O 1 Var(p)= 1+✏a ⇥6↵2 1 sin2(t))⇤ + (✏2). (4.16) 2 2 O ⇥ ⇤ It is interesting to note that the expectation values of x and p in the ground state, i.e. Let’s consider Wigner quasi whenprobability↵ = 0, are leftdistribution unchanged from for the a standard coherent local case state to first of order our inquantum✏9. However, harmonic oscillator, the variance of x and p is modified to order ✏ always, except for the peculiar case where 1 1 2ipy ↵ = 1/p6. Given that alsoP in( thisx, casep; t the) mean valuesdy of x and(x p+stilly, show t)⇤ corrections(x y, t) e ± of order ✏ we are led to consider this just an⇡ accident. Indeed, this is confirmed by going 2 Z1 to order ✏and, where confront the variances its acquire evolution again corrections for a coherent with respect state to the local(easier result to experimental realise 2 — this time of order ✏ , consistently. 2 Significantly, we observethan that Var( thex)Var( groundp)=1/ state)4+O(✏2 ),in thus the the case perturbed of stateS and S+εS is still a state of minimum uncertainty. It undergoes a spontaneous, cyclic, time dependent squeezing in position and momenta, where the name squeezingThe is justified Coherent in view state Wigner function of the previous observation on its minimal uncertainty. This is shown in Figure 2 AB: dovremmo citare da qualche parte anche Fig.1....forse in sezineshows 5? a periodic almost perfect squeezing. 5 Present Constraints and Forecasts Very difficult to produce spontaneously… DB: I have temporarily added the section from the PRL. I will leave the struc- turing of this whole experimental section to the Franceschi and Antonello. We now consider the constraints imposed by both existing opto-mechanical experiments and experiments that will be performed in the near future. Let us begin by noting

9The same happens also at the next order.

– 10 –

−19 Current best bounds on the non-locality scale by comparing nonlocal relativistic EFTs to the 8 TeV LHC data lnl≤ 10 m

−29 Forecast with experiment in preparation (in absence of periodic squeezing) imply lnl≤ 10 m ! rule we shall deﬁne the probability density as

9The same happens also at the next order.

– 10 –

−19 Current best bounds on the non-locality scale by comparing nonlocal relativistic EFTs to the 8 TeV LHC data lnl≤ 10 m

−29 Forecast with experiment in preparation (in absence of periodic squeezing) imply lnl≤ 10 m ! Conclusions Conclusions

Emergent gravity scenarios seem to be hinted by many features of gravity and seem to have necessary implications towards the nature of Lorentz invariance and/or Locality Conclusions

Emergent gravity scenarios seem to be hinted by many features of gravity and seem to have necessary implications towards the nature of Lorentz invariance and/or Locality Analogue models of Gravity have proved to be not only a work bench for Lab testing QFT in curved spacetime (e.g. Hawking radiation) but they have also taught us lessons about the emergence of spacetime an its dynamics. However it is also clear that Analogue-Inspired Emergent Gravity scenarios currently face strong limits from LIV constraints (although see UHCR crisis) and issues with background independence vs microcausality (but unclear if and how this apply to realistic implementations like those in GFT — see works by Oriti and collaborators.). They are just toy models Also it is unclear the fate of gravitational thermodynamics in LIV theories. This again might be a beacon or… be its doom but it might also teach us what is really at the core of thermodynamics in gravity. Conclusions

The emergence of Emergent Gravity is still in the making…