Emergent Gravity according to Verlinde

EMERGENT GRAVITYErik Verlinde’s Proposal

Stephen Perrenod, Ph.D.

darkmatterdarkenergy.com

Initial Concept

• Erik Verlinde, in 2011, published a paper on Emergent Gravity

• “On the Origin of Gravity and the Laws of Newton”

• Motivations - connections to string theory, need for quantum description of gravity and the uncertainty around dark matter

• He derived Newton’s laws in an emergent gravity scenario, but assumed an unrealistic Anti-deSitter space


Quantum Entanglement

• Quantum entanglement is when pairs (or groups) of particles have their quantum states coupled

• The quantum states of the two particles remain coupled as they are separated and observing a property of one is tantamount to observing both

• This has been observed in the laboratory over long distances of ~ 100 kilometers


Emergence• Space is defined by the connectivity between

quantum entangled particles

• Gravity is an emergent, statistical feature of the system

• Just as thermodynamics is an emergent feature of the particles of air in the atmosphere

• The entropy (and information) content of space is due to the excitations of the vacuum state that manifest dark energy


de Sitter and Anti-de Sitter

• de Sitter space has a positive Cosmological Constant (Λ > 0), as does our universe

• Anti-de Sitter space (AdS) with negative Λ has been popular with string theorists

• because easier to solve and they can show a correspondence between AdS and conformal field theory (CFT)


Anti-de Sitter

• In the AdS/CFT correspondence quantum entanglement of the microscopic state leads to:

• Spacetime geometry is due to the entanglement structure of the microscopic quantum state

• Entanglement is the glue for the connectivity of spacetime!

• Gravity is then an emergent phenomenon rather than a fundamental force

• Determined by the entropy on the surface surrounding the matter


de Sitter

• Our universe is not AdS, but roughly a dark energy dominated de Sitter solution (dS)

• Verlinde has now developed a partial solution for the dS case also

• And he finds that there is a volume (bulk) contribution as well as a surface contribution


Entropy

• Entropy is the information content reflecting the range of states, the number of microscopic degrees of freedom

• The volume contribution grows relative to the surface (area) contribution as one looks at a larger and larger volume (R3 vs. R2)

• The Hilbert space of a full system has entanglement scaling with volume, while ground states scale with area

• Newtonian dynamics / general relativity reflect the surface contribution

• There is an additional term, resulting in a stronger gravitational field than predicted by GR, when the volume contribution comes into play

• This ‘spoofs’ the existence of dark matter, but there is no DARK MATTER !

GibbsBoltzmann


Black Holes

• The connection between gravity and thermodynamics has been around for 3 decades

• This is originally from the black hole research of Jacob Bekenstein and Stephen Hawking

• The Entropy is proportional to the black hole area divided by the Gravitational constant (G)

• The AdS/CFT correspondence in string theory derives the same formula for quantum entanglement in a vacuum


Cosmic Horizon• The cosmic horizon is at radius ~ c/H0 where H0 is the Hubble

parameter (~ 70 kilometers/sec/Megaparsec and a parsec is 3.26 light-years), and it is around 4.1 Gigaparsecs

• So it is only at very large scales, in the outer regions of galaxies and at larger scales that departures from GR can readily be seen

• Generally these departures from standard gravitational theory are interpreted as an unseen matter contribution: Dark Matter

• But MOND (Modified Newtonian Dynamics) has been around as a phenomenological interpretation due to Milgrom since the 1980s, who argued that at very low accelerations of order a centimeter/sec/year one should modify the gravitational law

• Verlinde’s proposal is not MOND, it provides an actual theoretical basis, unlike MOND, but it does predict similar behavior


Dark Energy

• Dark Energy (Λ) has been around since early in the history of general relativity

• It was observed through plotting the redshift-luminosity relation for distant supernovae in 1998, and subsequently in fitting to the cosmic microwave background measurements

• The CMB measurements are well fit by the simple cosmological constant model in GR with an unchanging value of Λ (equation of state pressure = - energy density)

• In the canonical ΛCDM model with cold dark matter (CDM) it represents about 69% of the universe’s mass-energy and CDM represents about 26%

• However, if Verlinde is right, then there is only ordinary matter at 5% and then Λ would be around 95% of the mass-energy!


Cosmological Constant Λ

• The cosmological constant (dark energy for an equation of state w = -1 as observed) has units of inverse length squared

• The Hubble parameter H0 has units of inverse time

• Thus a cosmic acceleration scale in a dS universe dominated by Λ is:

• H2 / sqrt (Λ)

• The observed values are:

• Λ =1.2 x 10-56 cm-2 and H0 = 2.2 x 10-18 sec-1

• 4 x 10-8 cm / sec / sec


Cosmic Horizon & Acceleration

• The cosmological horizon is at distance of order c/H0 (currently the ‘particle horizon’ is at about 3.2 times that distance in the standard cosmological model)

• With a cosmological acceleration scale defined by a0 = c*H0

• For H0 = 70, this works out to be a0 ~ 20 kilometers2/sec

2/parsec (in

astronomer units) or ~ 2 centimeters/sec/year (everyday units) or ~ 6 x 10

-8 cm / sec / sec

• Over 15 billion times weaker than Earth’s surface gravity !

• This is same order of magnitude as MOND which finds a characteristic acceleration of about 10

-8 cm / sec / sec as the key

parameter to use for rotation curves of galaxies and other dynamics, it also works for gravitational lensing observations


Cosmic Acceleration Scale

• Coincidence?

• Not really, we live in a Dark Energy dominated (dS) universe

• ΩΛ ~ 0.7 if general relativity w/ dark matter, where the Ω notation is energy/mass density in units of the critical density (9 x 10-30 gm/cc or 5 protons per cubic meter). Standard ΛCDM model.

• ΩΛ ~ 0.95 if no dark matter and emergent gravity

• In a fully dS (ΩΛ = 1) universe Λc2 = 3 H02 so the two estimates are

equivalent to within sqrt(3) which is roughly their numerical difference


String Theoretic View

• The dark energy excitations are fundamental

• Matter particles are bound states of those excitations that have sprung forth from the DE medium

• They can move freely and have much smaller entropy

• Matter creation removes energy and entropy from the underlying DE medium

• GR describes response of area law entanglement of the vacuum to matter


Dark Energy

• Verlinde proposes that Λ and the accelerated expansion of the universe are due to the slow rate at which the emergent spacetime thermalizes

• High degeneracy and very slow dynamics means the universe is not in a ground state, therefore there should be a volume contribution to entropy

• At large distances, the volume contribution overwhelms, and since the time scale for dynamics is 1/H0 then a distance scale of c/H0 is natural

• It is also interesting that since a dS universe has a constant Λ , as well as H, then if Λ is 95% of mass-energy in our universe, we are very close to this pure dS solution


Surface Mass Density

• When the surface mass density falls below c*H0/(8πG) things change

• The spacetime medium becomes elastic

• The effective ‘dark’ gravity is calculated resulting from an effective surface density additional contribution proportional to the square root of the baryon (ordinary matter) surface density


Modified Acceleration

• With this new effective surface density the acceleration for dark gravity scales with the Hubble acceleration as:

• gD = sqrt (gB*a0/6) and the total acceleration is thus g = gB + gD (B baryons, D dark)

• And a0/6 is 10-8 cm/sec/sec, this approximate cutover is quite close to the MOND favored value 1.2 x 10-8

• Example, suppose gB is 10-8 cm/sec/sec, then gD is also 10-8 and the total gravity is g = gB + gD is doubled to g = 2 * 10-8 cm/sec/sec, spoofing the appearance of additional dark matter equal to ordinary matter

• But it is not DM , it is additional gravity due to the volume contribution to entropy which ordinary baryonic matter has replaced and displaced


Glassy Dynamics, Memory Effects

• Extremely long-range correlations of the underlying microscopic degrees of freedom / entanglement

• Slow dynamics of many delocalized microscopic states whose presence and dynamics invisible at small scales

• These carry the dS entropy in a non-local way

• Glassy systems have slow relaxation and memory effects

Galaxy Rotation Curves• Dark Matter was first proposed by

Zwicky in the 1930s to explain clusters of galaxies

• And later in the 1970s Vera Rubin and others did work on galaxy rotation curves

• In both cases there should be of order ~ 5 times more matter or ~ 5 times more gravity than seen in stars, gas

• It has also been known since the 1980s that MOND gives a good phenomenological description of rotation curves


Tully-Fisher Relation

• Tully and Fisher observed that the luminosity of a spiral galaxy goes as the 4th power of angular velocity (or emission line width)

• Since luminosity is basically proportional to ordinary baryonic mass of a galaxy, and since with Newtonian gravity the relationship should be M ~ v2 rather than v4

it suggests

either dark matter or a deviation from standard gravity

• MOND predicts v4 for low acceleration regimes

• Verlinde is able to derive a Tully-Fisher relationship for emergent gravity by modeling the displacement of the entropy content of a dS space


Clusters of Galaxies• It is well known that the MOND formula

falls short by a factor of 2 to 3 in clusters of galaxies

• Emergent gravity can do better, for a general mass distribution, rather than a point mass

• And one picks up a factor of 1.5 to 3.5 times MOND


Universe

• To first order he calculates what the universal value of apparent dark matter should be

• ΩD2 = (4/3) ΩB ,

• Here B is for the baryonic matter, and for ΩB ~ 0.05 one finds ΩD ~ 0.26, these are very close to the actual measured values with the canonical ΛCDM interpretation of the Planck CMB results


Approximations• Pure dS

• Isolated, Static region (e.g. a bound system)

• Spherical geometry

• Not clear how to explain the CMB density fluctuations and galaxy formation


Breaking News

• Margot Brouwer et al. 2016 “First test of Verlinde’s theory of Emergent Gravity using Weak Gravitational Lensing Measurements”

• Prediction from EG, using no free parameters, is in good agreement with galaxy-galaxy lensing across a sample of surface mass density profiles of 33,613 isolated galaxies

• GR can also fit the lensing data, but has more parameters


Summary

“The observed phenomena that are currently attributed to dark matter are the consequence of the emergent

nature of gravity and are caused by an elastic response [of the dark energy medium] due to the volume law

contribution to the entanglement entropy in our universe.” - Erik Verlinde

[my addition to statement, my underlining]


References

• Erik Verlinde 2011 “On the Origin of Gravity and the Laws of Newton” arXiv:1001.0785

• Stephen Perrenod, 2013, 2nd edition, “Dark Matter, Dark Energy, Dark Gravity” Amazon (read Dark Matter chapter with skepticism!)

• Erik Verlinde 2016 “Emergent Gravity and the Dark Universe arXiv:1611.02269v1

• Margot Brouwer et al. 2016 “First test of Verlinde’s theory of Emergent Gravity using Weak Gravitational Lensing Measurements” arXiv:1612.03034v


Friedmann Equation• In general relativity the relation between the critical matter and energy density and

the Hubble constant (expansion parameter) is:

• (8π/3) G ρ = H0

2

• where ρ is the total mass-energy density written in the form of a matter density

• And for a dark energy dominated universe of energy density ρde

• (8π/3) G (ρde/c2

) = H0

2

• or, G ρde = (3/8π) (c H0)2

= (3/8π) (a0)2

written in terms of the Hubble acceleration, defined to be c*H0 (since c has units cm/sec and H0 units sec-1)

• This suggests that a0 , the Hubble acceleration, is the characteristic acceleration scale for the interaction between ordinary gravity G and dark energy ρde


Stress and Strain

• The displacement of entropy is not erased, but leaves a memory imprint, a residual stress and strain

• The medium exerts a reaction force on the matter, which is the excess or dark, gravity (normally blamed on ‘dark matter’)

• Elastic back-reaction, he calculates shear modulus ~ a0

2 / (16 πG)

Science

Emergent Gravity according to Verlinde