Entanglement of Matter and Geometry !

Connection of cosmic acceleration with matter fields

Craig Hogan University of Chicago and Fermilab

1

What determines the physical value of cosmic acceleration?

Rough coincidence in Planck units known since 1930’s: H0 ~m3 where H0 is Hubble rate and m is an elementary particle mass

Could now be a clue to the absolute value of cosmic acceleration

The connection may be understood in the framework of emergent, holographic quantum geometries entangled with matter fields

May enable a precise connection of cosmic measurements, e.g.

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with numbers from laboratory measurements of particles

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For information counting purposes, the thing that mat-ters is the radius H�1

⇤ of the event horizon in the asymp-totic future. The value of Einstein’s cosmological con-stant in these units is given by ⇤ = 3H2

⇤. Again assuminga standard (⇤CDM) cosmology, the cited measurementsgive a current estimated value,

H�1⇤ = ⌦�1/2

⇤ H�10 = 1.01± 0.018⇥ 1061. (62)

The total cosmic information I is one quarter of the areaof the event horizon in Planck units,

I⇤ = ⇡H�2⇤ = 3.2± 0.11⇥ 10122. (63)

Also assuming a flat 3-geometry, the mean density ofcosmic information n⇤ per 3-volume is given by

n⇤ = 3H⇤/4 = 7.4± 0.13⇥ 10�62. (64)

At present, the connection of these fundamental numberswith any other physical quantities is not known.

B. Comparison of Cosmic and Field Information

We next estimate the field scale mL(H⇤) that gives thesame information density as the geometry with horizonradius 1/H⇤. For a spin-zero field, the amount of infor-mation is given by the number of modes in a volume,which we count in the standard way as follows.

For a field with a UV cuto↵ at |kmax| = m, the volumeof phase space accessible to modes is Vk = 4⇡m3/3. In aspatial volume of size L in any direction, modes occur ink space with a mean spacing (2⇡/L) in that direction, soin the total field information density for a cubical volumeVL = L3 is

nf (m) = If/VL = Vk(2⇡/L)�3V �1

L = m34⇡/3(2⇡)3,(65)

independent of L or the shape of the volume. (Of course,as argued above, this estimate should only actually holdfor field-like states for volumes up to about LR(m) wherethe geometrical entanglement becomes important.)

Combining these relations, the overall cosmic informa-tion density is equal to the mean field information den-sity, n⇤ = nf (mL), for a field cut o↵ at energy

m3L = H⇤(2⇡)

3(9/16⇡). (66)

This equation provides a more precise estimate than Eq.(59), for the specific case of a scalar field.

For measured cosmological parameters, the particlemass where field information matches cosmic informationis

mL(H⇤) = 1.65± 0.01 ⇥ 10�20. (67)

This value approximately corresponds to the Yukawascale of the strong interactions. For example, consider

a field theory cut o↵ at the mass of the neutral pion, thelightest hadronic state,

m⇡ = 134.98 GeV = 1.11⇥ 10�20. (68)

For this value of the field cuto↵ and for measured cos-mological parameters (Eq. 64), the dimensioness ratio ofcosmic to field information density is

n⇤/nf (m⇡) = I⇤/If (m⇡) = 0.8± 0.014. (69)

The approximate agreement suggests that measured by

total information in position states within the cosmic

horizon, the scale of the cosmological constant is the same

as that of the strong interactions.

The agreement in Eq. (69) is surprisingly close, andbecomes even better if the higher locally-calibrated valueof the Hubble constant is chosen. Indeed the close agree-ment must be regarded as accidental, since we do notknow exactly how the appropriate cuto↵ is related tom⇡. Even so, it motivates us to ask how such a relation-ship could arise in an emergent system. A more exactversion of this theory could provide an exact calculationto compare with the empirical ratio given by Eq. (69).

C. A Proposal for Emergent Cosmic Acceleration

The value of the cosmological constant is arbitrary andessentially unexplainable within the standard frameworkof field theory and classical geometry[14, 15]. A numberof alternative models of cosmic acceleration[41] have beenproposed, such as various forms of modified gravity, andin some cases they lead to significant predicted deviationsfrom standard behavior, for example in the growth ofcosmic structure. These models generally involve addingan extra scale ad hoc for dark energy comparable to thecurrent expansion rate, and in some cases a macroscopicinteraction or screening scale for new forces.

With the entangled matter/geometry system proposedhere, the value of the cosmological constant could havea quantum-geometrical connection with Standard Modelfields. Cosmic expansion in the classical limit could be-have like a standard cosmological constant ⇤, withouta nonstandard signature in structure formation or otherastronomical signatures. However, a microscopic scalein the field vacuum, connected with localization of mi-croscopic particle states, could directly a↵ect the macro-scopic space-time curvature, with a relationship deter-mined just by the Planck scale. Its value could be relatedto other physical scales, in particular, to the particle scaleof long-lived position states, ⇤QCD ⇡ m⇡.

In the absence of any fields, or any externally im-posed cosmological constant, the most probable zero-temperature space-time configuration is flat space, whichalso has zero energy density. Presumably, the presence offield degrees of freedom in the combined system changesthe most probable emergent metric of the combined sys-tem, which may no longer have zero curvature.

log(mass/Planck

mass)

-60

-40

-20

0

20

40

60

log(radius or wavelength/Planck length)

0 10 20 30 40 50 60

most compact quantum system (particle, m=1/L)

most compact

classica

l geometry

(black hole, M=R/2)

(General Relativ

ity)

(Quantum Field Theory)

Planck scale: minimum size

What are the quantum properties of geometry in

large systems?

all figures in Planck units

Quantum matter and quantum geometry

Matter is a quantum system (field theory); geometry is dynamical but classical (general relativity)

Standard approximations: ignore geometrical dynamics in small systems, or ignore quantum behavior in large systems

Whole quantum system includes matter and geometry

Matter and geometry are subsystems

Their degrees of freedom are entangled, beyond standard theory

4

Quantum geometry in large systems

Quantum geometrical degrees of freedom are unknown, but differ from classical ones

Information is holographic

Classical behavior is “emergent”

Dynamics is thermodynamics (e.g., Jacobson)

Entanglement with matter can have observable consequences on large scales, beyond those predicted from canonical quantization of metric

5

Standard quantum kinematic uncertainty: minimum width of wave function for position compared at two times increases with time interval,

!

Gravity also relates mass, position and duration

Non-relativistic systems have indeterminate metric even with only standard physics (no “quantum gravity”)

Wave function scale is macroscopic at low mass, long times

Standard quantum position uncertainty in long-lived systems

6

6

Although we have used Newtonian gravity to describethe system, there is also a classical solution using relativ-ity. The gravitational potential is replaced by a curvedspace-time metric. The configuration of the potential (orthe metric) depends on the configuration of the bodies.The metric can also be used as a basis for modeling aquantum system.

In the quantum system, the structure and behavior ofthe potential and metric have the same indeterminatecharacter, and the same symmetry, as the trajectories ofthe bodies. The potential has the discrete spectrum ofstates corresponding to energy, En = �2M4/n2, whilethe curvature radius has a discrete spectrum correspond-ing to orbital timescale,

⌧n = n2/2M4. (20)

This non-relativistic approximation is appropriate forM << 1. (Above the Planck mass, the radius is smallerthan a black hole of the same mass.)

The atom is not analytically solvable with more thantwo bodies, in either the classical or quantum systems.However, it is clear from scaling the above argumentsthat for a given total mass, the spatial size of the groundstate scales like the number of particles, so quantum ef-fects in principle operate on even larger spatial scales.However, the e↵ect of each particle on the potential isless. A nearly-uniform matter distribution is consideredbelow, in the context of a perturbed cosmological solu-tion without gravity; that estimate gives the same scaleof indeterminate curvature as the atom.

B. Quantum Kinematic Uncertainty of PositionCompared at Two Times

Consider now a system where gravity and other forcescan be neglected. In this case evolution is governed sim-ply by quantum kinematics, so it can be formulated in ageneral way applicable to a wide variety of systems.

As in the case of the atom, observables are representedby operators. Components of spatial position x̂i and mo-mentum p̂i of a system are described by conjugate oper-ators with commutator

[x̂i, p̂i] = ih̄�ij . (21)

These operators can refer to the position and momen-tum of a body, or to some other degrees of freedom of asystem, characterized by equations of motion.

The state of the system can be described, for exam-ple, by a wavefunction that represents a complex ampli-tude for any configuration, e.g., (xi). The wave func-tions obey the standard Heisenberg uncertainty relationof standard deviations, �xi�pi > h̄�ij , that follows di-rectly from the commutator. The equations of motioncan also be used to derive other uncertainty relationsthat relate wave functions of other observable quanti-tates, such as observables at di↵erent times. These re-

lations describe the relations between preparation andmeasurement.In the force-free case (potential U = some constant),

the motion of a system of mass M is governed by sim-ple kinematics, @x̂i/@t = p̂i/M . The standard quantumuncertainty of position di↵erence measured at two timesseparated by an interval ⌧ is then[7–9]

�xq(⌧)2 ⌘ h(x̂(t)� x̂(t+ ⌧))2i|t > 2h̄⌧/M. (22)

It may seem surprising at first that this uncertaintygrows with time, since intuitively it seems that uncer-tainty should get small with a longer average. The ex-planation is that position after a long time is susceptibleto momentum uncertainty. The minimal uncertainty cor-responds to states prepared in such a way that �xq(⌧)gets equal uncertainty from position and momentum un-certainty after time ⌧ . As a result, in any system thatevolves slowly and lasts a long time, the scale of quantumuncertainty gets surprisingly large. This result approxi-mately applies to any system over timescales short com-pared to its natural dynamical timescale, since it assumesonly force-free kinematics (see Fig. 1) .

C. Macroscopic Quantum Trajectories

1. Bound Systems

A typical real gravitating system is composed of mas-sive bodies whose individual wave function widths aremuch smaller than the system size. The quantum ef-fects on their orbits can then be neglected. However, anisolated system with mass and size comparable with theground state of the gravitational atom, and dominatedby gravitational forces, displays quantum characteristics.This situation could actually apply, for example, in thereal universe in deep intergalactic space, far from con-centrations of matter. In the real universe, such systemsare also a↵ected by new physics of cosmic accelerationor dark energy not included in this model, as discussedbelow.

2. Quantum Uncertainty of Black Hole Position over anEvaporation Time

An exotic application of these ideas is the motion of ablack hole. The center of mass should behave in the sameway as any other massive body. This illustration is in-teresting because seemingly reasonable assumptions, forexample about locality of information, have been shownto lead to apparent paradoxes.Consider the motion of a black hole of mass M over

a timescale of the order of the time it takes for itsmass to evaporate by Hawking radiation, ⌧evap ⇡ M3

in Planck units. The position of the hole is indetermi-nate by �xq(⌧) ⇡ (⌧evap/M)1/2 ⇡ M , that is, by about

Example: gravitational atom

Quantum system of two masses bound by gravity instead of electricity

Standard atomic solution

Ground state much larger than Bohr radius

In energy eigenstate, orbits and potential (curvature) are indeterminate

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log(reduced atom

ic mass/Planck

mass)

-60

-40

-20

0

20

40

60

log(radius/Planck length)

0 10 20 30 40 50 60

gravitational atom, M=R-1/3

black hole

indeterminate gravitational orbits

log(reduced

sytem

mass/Planck

mass)

-60

-40

-20

0

20

40

60

log(radius or wavelength/Planck length)

0 10 20 30 40 50 60

gravitational atom ground state wave function

black hole

quantum

solid

den

sity,

orbi

tal t

ime

~ ho

urs

quantum position uncertainty

in orbital time (~ hours)

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nanoscale “artificial gravitational atom” ~10-7 meters

Another example: quantum-classical boundary for cosmic expansion

define cosmic expansion by measuring position of expanding material over some time interval

quantum indeterminacy defines a scale of size, mass, and precision

same size as gravitational atom of cosmic density

For current universe, the scale is macroscopic (about 60 meters)

10

log(mass/Planck

mass)

-60

-40

-20

0

20

40

60

log(radius/Planck length)

0 10 20 30 40 50 60

gravitational atom

dens

ity =

H0

2

quantum position

uncertainty M=2/L 2H0

uncertainty in perturbation mass

M = L = H0-1

Quantum-classical boundary for cosmic expansion: ~ 60 meters

measure acceleration in 1 year

A new principle is needed on large scales: quantum field theory is inconsistent with gravity

Energy density of quantum field states with unit occupation ~m4, independent of volume

In a large volume, “typical” field states exceed the mean density of a black hole of the same size

With Planck cutoff and Hubble volume, this is the classic ~ x10122 dark energy problem; but is not solved by resetting zero point

This geometry is impossible and must be forbidden by some new principle

Problem is not confined to the Planck scale, indeed is worse on large scales

12

Entanglement of Quantum Fields with Quantum Geometry

Black hole catastrophe requires radical solution

One proposed solution (Cohen, Kaplan, Nelson 1999): there is a maximum extent of field states (IR cutoff), which depends on UV cutoff of effective theory

(consistent with all current experiments)

Can be implemented as a nonstandard, nonlocal entanglement of field states with geometrical states

13

log(particle

mass/Planck

mass)

-60

-40

-20

0

20

40

60

log(system size/Planck length)

0 10 20 30 40 50 60

single quantum

Field states of mass >m exceed black hole mass in

same volume

Field theory OK

Field modes of mass m exceed Planck angular

resolutiongeometrical entanglement, L= m-2

black hole

Directional Entanglement of fields with emergent quantum geometry

A geometrical model: locality and direction emerge by hierarchical entanglement

Angular resolution is limited by Planck diffraction, L-1/2 instead of (mL)-1 as in standard theory

Transverse phases of fields change by ~ mL1/2 , non locally correlated

Radial information is standard, but directional information is less than in standard fields (at most Idirectional~ L, not (mL)2)

Total information is holographic: I~ Lradial x Idirectional ~ L2

Solves black hole catastrophe: limits number of independent states

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Interferometers can directly measure or constrain correlated position noise from Planckian directional uncertainty at separation L,

!

Precise prediction for signal noise can be tested

An operating experiment, the Fermilab Holometer, is designed to measure or rule out this effect

Experimental probe of Planckian directional entanglement

16

10

Since density couples to gravity (via Eq. 5 etc.), the sys-tem is only consistent if the fields live in a metric thatcorresponds to the same mean density of matter as itssource. This consistency becomes impossible to achievein systems with large V and field modes with large m.Even if the zero of the potential (states with n = 0) isset to have approximately zero gravitational density (asit must to agree with cosmology), in a su�ciently largevolume, the energy density of general field states (thatis, all those with n > 1 in the bulk of the modes) exceedsthat of a black hole, ⇢BH ⇡ L�2, which is independentof m.

Motivated by this conundrum, Cohen, Kaplan and Nel-son (CKN)[6] posited an IR cuto↵ to field states— a limitto the box size allowed in quantum field theory. This hy-pothesis goes outside of the e↵ective field theory frame-work. It does not address the nature of the physicalrelationship of field states to quantum geometry directly,but it does allow an estimate of the e↵ects on renormal-ization group flow and other measurable e↵ects on thethe fields. CKN showed that this modification of fieldstates does not produce an observable e↵ect in currentparticle experiments, which generally measure e↵ects inmuch smaller volumes.

In this model, the standard description of field statesis only valid up to a finite range, such that the sum of theenergies of field states in a volume does not have moreenergy than a black hole of the same size. The energydensity of field modes with a frequency cuto↵ m is aboutm4, and a black hole of size L has a mean density L�2

in Planck units, so for a system of size L and field modesof mass < m with occupation n > 1, the bound on thespatial extent of field modes of mass m is about

L < LR(m) = m�2, (39)

where we use LR to denote an entanglement with gravity.This relation is shown in Figure (6). Note that in thiscase, the massm refers to particle mass, not system mass.

This bound suggests that there is a maximum systemsize, above which standard local quantum field theory atscale m breaks down. Somehow, the degrees of freedomof the fields and geometry do not act like independentsubsystems, and this introduces a nonlocal behavior notdescribable by a Lagrangian density in large volumes.

2. Directional Entanglement

The IR limit just discussed can be accounted for ina simple model of how field states relate to geometricalones, based on a Planckian bound on directional informa-tion. In this “directional entanglement” model[13], thecuto↵ has a purely geometrical origin. It gives the cor-rect IR cuto↵ scale independently of m or other assumedproperties of the fields, and makes some other uniquepredictions.

In this model, the total information in a volume is bro-ken into radial information, which describes the causal

structure of a spacetime in terms of light cones aroundeach event on a world-line, and directional informa-tion, which describes two-dimensional angular orienta-tion. The directional information is bounded by a Planckdi↵raction limit imposed by the quantum geometry. Inthis way, directions in space emerge from new physics atthe Planck scale in such a way that angular resolution offield states never exceeds the angular resolution of Planckfrequency radiation.This model does not impose an abrupt limit on the

spatial extent of field states. Instead, states of fields areentangled with states of quantum geometry in a partic-ular way[13]. The transverse phases of field wave func-tions are convolved at separation L from an observer’sworld line with a quantum geometrical directional phase.The spread in direction �✓P or transverse position �x?is given by the Planck di↵raction resolution limit, fromstandard wave optics, for states of extent L:

�✓P ⇡ �x?/L ⇡ L�1/2. (40)

The amount of directional information— the number ofdistinguishable angles of propagation in the system offields— is bounded by the resolution of Planck wave-length states. At this angular resolution, directional in-formation is all geometrical, and is shared among fieldmodes.The geometrical contribution to the transverse phase

invalidates the standard count of independent field states.The number of field states of mass m is significantly re-duced beyond the separation L where the geometricale↵ect on the transverse phase causes phase changes oforder unity in typical orientations. That happens when�x? ⇡ m�1, or

L(�✓P = (mL)�1) ⇡ m�2. (41)

Thus, field theory is cut o↵ at the same scale LR thatreconciles virtual field states with gravity (Eq. 39).However, in other respects the e↵ects of directional en-

tanglement are not the same as a simple volume cuto↵.Since the e↵ect on fields is purely transverse, it has neg-ligible e↵ect on longitudinal phase, so the modificationsof standard field theory in particle experiments are likelyto be di↵erent in detail from those estimated by CKN.Because the geometrical e↵ect is purely transverse, thereare no dispersive e↵ects, even at frequencies up to thePlanck scale, of the kind that can be measured in astro-nomical observations (e.g., [32–34]).On large scales where geometry dominates, the num-

ber of directional degrees of freedom in this model is L,instead of (Lm)2 as in standard field theory. At the sametime, the number of radial degrees of freedom is still⇡ L, so the total information agrees with holographicinformation from gravitational theory. An exact calcula-tion of directional uncertainty[35] normalized to gravity,based on a quantum commutator of position analogousto angular-momentum algebra, yields the formal Planckuncertainty of direction at separation L:

h�✓2P i ⌘ hx̂2?i/L2 = lP /

p4⇡L. (42)

Fermilab Holometer

Two 40-meter power-recycled Michelson interferometers, located next to each other

Signals are cross-correlated at ~ few Megahertz, the inverse light crossing time

Geometrical states of two machines are directionally entangled by proximity

Will detect or rule out in-common transverse fluctuations with Planck spectral density

Predicted motion is xrms ~ L1/2~ (40m x lPlanck)1/2 ~ attometers

17

Holometer((E+990)(Opera2ons(Status:(!

•  Beginning!1(year!opera.ons!phase!July,!2014!•  Interferometers(running(stably(and(high(quality(data(being(taken(at(near(full(power(•  Uncorrelated(shot(noise(is(integra2ng(away(nicely(

•  Opera.ons!phase!tasks!•  Develop(in(situ(signal(calibra2on(schemes(•  Inves2gate(and(mi2gate(any(sources(of(MHz(frequency(noise(which(may(be(uncovered(by(

increased(sensi2vity(levels!•  Seismic/acous.c!stability!is!s.ll!an!issue!

•  One(of(the(interferometers(s2ll(leaks(30%(more(power(than(the(detectors(can(handle.(•  Inves2ga2ng(electronic(and(mechanical(fixes((alignment(control,(tethering(hut(down)(

7/28/2014(

Photocurrent(cross+correla2on(averaged(down(over(~1M(samples(reveals(clean,(uncorrelated(spectrum(Real,((

out+of+signal+(band(data(at(near(holographic(noise(sensi2vity(

Report from A. Chou and the Holometer team at Fermilab All-Experimenter’s Meeting:

shot-noise-limited data

Curvature entanglement

Directional entanglement modifies field states in large volume to match geometrical information

Field (matter) vacuum also affects emergent curvature in combined system

Radius of entangled curvature ~ size of gravitational atom of particle mass

much bigger than directional entanglement scale

can be derived from information equipartition

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Mean curvature in emergent space-time

In thermodynamic model, macro state (classical metric and matter vacuum) corresponds to maximum number of micro states

Whole system includes matter and geometrical degrees of freedom

Curvature radius is determined by equipartition between matter and geometrical position degrees of freedom

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Equipartition of cosmic position information

de Sitter symmetry violated by localized long lived matter states (comoving world lines)

Scale of matter localization defines position state information density, ~m3

Matter position information density in each dimension given by strong interactions ~ pion mass

Information density in matter matches overall geometrical information density, ~ asymptotic H

21

connects micro scale of fields to macro geometry

Equipartition of information could make rigorous and calculable the well known coincidence between QCD scale and Hubble scale,

!

The actual measured information density ratio is

Information equipartition connects cosmological constant with the QCD scale

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15

The most probable macrostate is the one with thelargest number of microstates, which in turn is associatedwith equipartition of energy among the largest numberof degrees of freedom. The total includes both geomet-rical and field degrees of freedom. A larger number offield degrees of freedom corresponds to a smaller curva-ture radius (Eqs. 59,60), hence fewer geometrical degreesof freedom, so there is some optimum value of curvatureassociated with a given field frequency cuto↵ m. Theequipartition happens when the field positional informa-tion density matches that of geometry. As the above es-timate shows, the scale of the QCD vacuum is matchedpretty well with the curvature scale of cosmic accelera-tion. The proposal is that this coincidence is not acci-dental, but arises because QCD sets the scale of maximalpositional information in all field states that define long-lived timelike trajectories.

The e↵ective temperature of the emergent geometrymatches the heat flux and horizon area via Eq. (13). Thelost information may be thought of as the information offield states being absorbed into the space-time degreesof freedom through the horizon. From the field pointof view, it appears as an entanglement with directionalgeometrical degrees of freedom.

1. What is Special about the Strong Interaction Scale

In QCD, strongly interacting particles are confined inspatially localized hadrons. The Lorentz invariance of theLagrangian is spontaneously broken by the rest frame de-fined by these massive hadronic states. The spatial sizeof the states, and the range of the strong interactions, isgiven approximately by the pion mass m⇡. Thus in therest frame, the fundamental particles are always in col-lective quantum states that are only localized to withinabout a length scale m�1

⇡ . It is not possible to preparea state with a smaller rest-frame uncertainty for particleposition, for an extended duration.

Lorentz invariance is also broken by cosmology. In cos-mological space-times with matter, such as Friedman-Robertson-Walker models, world lines of matter, orisotropy of radiation, define a cosmic rest frame at anyposition.

The proposal is that the value of positional informa-tion limited by the maximum mass of stable particles de-termines the amount of cosmic positional information infields, and that equipartition between the field and geom-etry states sets the value of the e↵ective, emergent cos-mological constant that gives the same overall positionalinformation in the cosmic system as a whole. In the mostprobable configurations of the entangled system, the totalinformation encoded by the global geometry is fixed bythe confinement scale: cosmology information matchesparticle information.

In a thermodynamic model of gravity (i.e. Eq. 13), thecausal structure of the space-time adjusts by curvature,in order to make this so. As described by Jacobson[19],

“the gravitational lensing by matter energy distorts thecausal structure of spacetime so that the Einstein equa-tion holds.” Here we go one step further, and posit thatthe mysterious constant of integration in that theory isfixed by the field vacuum.The measured value of the cosmological constant sug-

gests that baryons are indeed the heaviest long livedstable particles, which in turn suggests that the as-yet-unidentified stable dark matter particles much be lighterthan about 1 GeV.

2. Astrophysical Coincidences

In various forms, conjectures about cosmic coinci-dences with elementary particle masses have a long his-tory, going back to Weyl, Eddington[42] and Dirac[43].For example, why should a “gravitational atom” of ap-proximately atomic mass have approximately a Hubblesize? Dirac’s answer[43] was that the constants of nature,including G, vary with time such that this is always thecase. Later, others, including Dicke[44] and Carter[45]had a di↵erent answer: they argued that the current ageof the universe— the time when we show up— is de-termined by the lifetimes of stars, which in turn is longbecause gravity is (and always has been) so weak.

The relation m⇡ ⇡ H1/30 between the pion mass

(or other strong-interaction scale) and the present Hub-ble scale was also noted long ago by Zeldovich[46, 47].Others[48–51] have argued further that the strong inter-actions may have a direct physical relationship with thenow-observed rate of cosmic acceleration. That is alsothe point of view here.The arguments here suggest that this relationship ac-

tually has a geometrical origin outside of field theory, thatcan be tested by laboratory experiments. The physicalmotivation comes from emergent gravity, and a specificconcrete physical interpretation: in this scenario, there isa cosmological constant, whose value is fixed to the mi-croscopic scale where the matter vacuum spontaneouslycondenses into massive hadronic particles.If this is indeed the case, the astrophysics of stars ex-

plains the Dicke/Carter-type coincidence of the cosmicacceleration time with stellar lifetimes. That is, if thescale of cosmic acceleration is physically connected withthat of strong interactions by H⇤ ⇡ m3

⇡, its timescale au-tomatically coincides with the typical lifetime of a Sun-like star. That lifetime can be roughly estimated[52] bythe time it takes to radiate at nuclear e�ciency (aboutone percent of rest mass) at solar luminosity (about onepercent of Eddington luminosity),

⌧star ⇡ ↵2m�1protonm

�2electron, (70)

where ↵ is the fine structure constant. Since this prod-uct of constants approximately coincides with m�3

⇡ (atleast in the Standard Model), the coincidence H⇤ ⇡ m3

⇡

naturally implies that ⌧star ⇡ 1/H⇤.

14

For information counting purposes, the thing that mat-ters is the radius H�1

⇤ of the event horizon in the asymp-totic future. The value of Einstein’s cosmological con-stant in these units is given by ⇤ = 3H2

⇤. Again assuminga standard (⇤CDM) cosmology, the cited measurementsgive a current estimated value,

H�1⇤ = ⌦�1/2

⇤ H�10 = 1.01± 0.018⇥ 1061. (62)

The total cosmic information I is one quarter of the areaof the event horizon in Planck units,

I⇤ = ⇡H�2⇤ = 3.2± 0.11⇥ 10122. (63)

Also assuming a flat 3-geometry, the mean density ofcosmic information n⇤ per 3-volume is given by

n⇤ = 3H⇤/4 = 7.4± 0.13⇥ 10�62. (64)

At present, the connection of these fundamental numberswith any other physical quantities is not known.

B. Comparison of Cosmic and Field Information

We next estimate the field scale mL(H⇤) that gives thesame information density as the geometry with horizonradius 1/H⇤. For a spin-zero field, the amount of infor-mation is given by the number of modes in a volume,which we count in the standard way as follows.

For a field with a UV cuto↵ at |kmax| = m, the volumeof phase space accessible to modes is Vk = 4⇡m3/3. In aspatial volume of size L in any direction, modes occur ink space with a mean spacing (2⇡/L) in that direction, soin the total field information density for a cubical volumeVL = L3 is

nf (m) = If/VL = Vk(2⇡/L)�3V �1

L = m34⇡/3(2⇡)3,(65)

independent of L or the shape of the volume. (Of course,as argued above, this estimate should only actually holdfor field-like states for volumes up to about LR(m) wherethe geometrical entanglement becomes important.)

Combining these relations, the overall cosmic informa-tion density is equal to the mean field information den-sity, n⇤ = nf (mL), for a field cut o↵ at energy

m3L = H⇤(2⇡)

3(9/16⇡). (66)

This equation provides a more precise estimate than Eq.(59), for the specific case of a scalar field.

For measured cosmological parameters, the particlemass where field information matches cosmic informationis

mL(H⇤) = 1.65± 0.01 ⇥ 10�20. (67)

This value approximately corresponds to the Yukawascale of the strong interactions. For example, consider

a field theory cut o↵ at the mass of the neutral pion, thelightest hadronic state,

m⇡ = 134.98 GeV = 1.11⇥ 10�20. (68)

For this value of the field cuto↵ and for measured cos-mological parameters (Eq. 64), the dimensioness ratio ofcosmic to field information density is

n⇤/nf (m⇡) = I⇤/If (|kmax| = m⇡) = 0.8± 0.014. (69)

The approximate agreement suggests that measured by

total information in position states within the cosmic

horizon, the scale of the cosmological constant is the same

as that of the strong interactions.

The agreement in Eq. (69) is surprisingly close, andbecomes even better if the higher locally-calibrated valueof the Hubble constant is chosen. Indeed the close agree-ment must be regarded as accidental, since we do notknow exactly how the appropriate cuto↵ is related tom⇡. Even so, it motivates us to ask how such a relation-ship could arise in an emergent system. A more exactversion of this theory could provide an exact calculationto compare with the empirical ratio given by Eq. (69).

C. A Proposal for Emergent Cosmic Acceleration

The value of the cosmological constant is arbitrary andessentially unexplainable within the standard frameworkof field theory and classical geometry[14, 15]. A numberof alternative models of cosmic acceleration[41] have beenproposed, such as various forms of modified gravity, andin some cases they lead to significant predicted deviationsfrom standard behavior, for example in the growth ofcosmic structure. These models generally involve addingan extra scale ad hoc for dark energy comparable to thecurrent expansion rate, and in some cases a macroscopicinteraction or screening scale for new forces.

With the entangled matter/geometry system proposedhere, the value of the cosmological constant could havea quantum-geometrical connection with Standard Modelfields. Cosmic expansion in the classical limit could be-have like a standard cosmological constant ⇤, withouta nonstandard signature in structure formation or otherastronomical signatures. However, a microscopic scalein the field vacuum, connected with localization of mi-croscopic particle states, could directly a↵ect the macro-scopic space-time curvature, with a relationship deter-mined just by the Planck scale. Its value could be relatedto other physical scales, in particular, to the particle scaleof long-lived position states, ⇤QCD ⇡ m⇡.

In the absence of any fields, or any externally im-posed cosmological constant, the most probable zero-temperature space-time configuration is flat space, whichalso has zero energy density. Presumably, the presence offield degrees of freedom in the combined system changesthe most probable emergent metric of the combined sys-tem, which may no longer have zero curvature.

log(particle

mass/Planck

mass)

-60

-40

-20

0

20

40

60

log(system length scale/Planck length)

0 10 20 30 40 50 60

directional entanglement, L=m-2

dens

ity =

H0

2

position uncertainty L 2=2/mH0

QCD localization

M = L = 1/H0

Cosmic = field information

curvature entanglement, L=m-3=H-1

radius of curvature

entanglement

Cosmic coincidences

H~mproton3 is a well known coincidence (e.g., Eddington)

Dirac: gravity is weak because universe is old

Dicke/Carter: weak gravity means stars last a long time

Zeldovich: vacuum density connects with strong interactions

Here: concrete connection of QCD with curvature of emergent quantum geometry, via entanglement

Naturally leads to emergent cosmic acceleration with a stellar evolution timescale

24

Conclusions

Geometrical quantum states are entangled with matter on all scales

Standard physics predicts macroscopic quantum states of geometry in low density systems

Geometrical entanglement may also create new macroscopic effects that are not present in standard theory

In some models these are measurable

Operating experiment will soon detect or constrain them

In some models of geometrical entanglement, cosmic acceleration could be directly linked to the QCD scale

25