Mimetic Gravity Ali Chamseddine American University of Beirut, Lebanon & Radboud University Excellence Professor Modern Trends in Particle Physics, Nathfest 80, May 17 2019, Northeastern University, Boston, MA. Modified Gravity • Modifications of gravity in most cases involve adding new fields. • Modifications of GR or adding new fields are needs to solve well known problems in cosmology, such as adding an inflaton, curvaton, quintessence, or F( R) gravity (equivalent to φR+ f(φ) ) or as in Hordinski model using scalar with higher safe derivatives. • In some modified theories utilize breaking diffeomorphism invariance such as in unimodular gravity, or Horava‐Lifschitz gravity Dark Matter • Various new particles and interactions are proposed to account for the missing mostly non‐baryonic matter in the universe. • Examples: Supersymmetric neutralino, axions, WIMP, dilaton, • Lack of direct evidence for dark matter made some physicists to propose modification of GR at large scales. • Most modifications of GR are made to solve one problem at a time, and do not address solving the problems simultaneously. Scale factor for metric • GR is not invariant under scale transformations of the metric • Newton constant will be redefined. If we take Ω(x) then it will get kinetic term. It is related to determinant of metric. • Unimodular gravity assumes |det(g)|=1 breaking diffeomorphism Invariance. Scale factor • In string theory the metric is accompanied by a dilaton and axionin the form of antisymmetric tensor. New degrees of freedom. • How can one isolate physics of scaling factor? • Let Then it is scale invariant under This imply Mimetic gravity (with S. Mukhanov) • The system ( has 11 components subject to the constraint leaving 10 degrees, same as GR. Consider the action then equations of motion for metric are Mimetic matter • The constraint could be solved in synchronous gauge where by thus φ represents time coordinate. Initial condition • The scalar field equation is which in the synchronous gauge simplifies to Integrating with respect to time gives Mimetic gravity • Mimetic gravity is a very slight modification of GR where the only difference is that the longitudinal mode of graviton gets excited. • No new degrees of freedom appear. • One realizes that it is possible to add to the Einstein action any function V(φ) without breaking diffeomorphism invariance. In synchronous gauge this becomes V(t). • Equations of motion in this case become Cosmological models (also with Vickman) • For flat Friedman universe, we get in synchronous gauge • A change of variable y=a3/2 gives a linear differential equation Cosmological solutions • By choosing appropriate potentials one can obtain cosmological solutions. • • For V(φ)=αφn then if n<‐2, then y ∝ t for large t so that a(t) ∝ t2/3 as in dust dominated universe. Inflation • If n>‐2, y∝ tn‐4 exp( n/2+1) depends on sign of α as t →∞ • If n=‐2, a(t)∝ t‐1/3 exp( t2) α>0 similar to chaotic inflation with quadratic potential. Without introducing new scalar fields with independent degrees of freedom, the mimetic field is capable of producing cosmological models. Quintessence • In presence of other matter with equation of state p=wε and potential V(φ)=α/φ2 gives a scalar factor a(t) ∝ t2/3(1+α) and ε=‐α/wt2 and mimetic matter acts as quintessence. • For V(φ)=αφ2/(1+eφ) describes an inflationary model with a graceful exit to a matter dominated universe with a(t) ∝ exp(‐ t2) • Cosmological perturbations reveals that Newtonian potential is related to time variation of φ. Avoiding singularities • One of the main problems in GR is the appearance of singularities, e.g. at initial time t=0, or for black hole solutions. • The constraint on scalar field φ is invariant under constant shifts. To solve singularity problem we will use, instead of potential V, which is not invariant under constant shifts, the function Equations of motion • The term Contributes to the energy momentum tensor To avoid singularities we implement the idea of limiting curvature Special function f • Consider the function • If we set χ=sinψ a Taylor expansion gives f(χ) ∝ O(ψ3) • For Friedman Einstein equation gives Non‐singular Friedman solution 2 1/3 • a(t)=(1+3/4 εmt ) is an exact solution. ‐1/2 2/3 • When t<‐(εm) then a(t)~t as in cold dominated universe ‐1/2 ‐1/2 • When ‐(εm) <t< (εm) then we get a regular bounce during short period of time ‐1/2 • When t> (εm) then we get normal dust dominated universe. • Special curvature terms are regular during bounce and all curvature invariants remain bounded. • The multivalued function f satisfy matching conditions at branch points to insure regularity of cosmological evolution. Non‐singular Kasner universe • Kasner metric is ds2=dt2‐t2p dx2 –t2p dy2 –t2p dz2 Where p1+p2+p3=1, p1+p2+p3=1 This is a homogeneous anisotropic solution of Einstein equations In our case Einstein equations give Exact Kasner solution • Exact solution Non‐Singular Schwarzschild • Same method gives also a non‐singular Schwarzschild solution. Complicated behavior at Even horizon. • One can show that the solution is regular near the bounce and that all curvatures invariants are finite. • For Spherical symmetry, Schwarzschild solution Metric is regular for both r>rg and inside black hole 0<r<rg Non‐singular black hole • Inside black hole r, t exchange roles. By change of variables we write Non‐singular • Solution is regular near the bounce, there is no singularity and all curvature invariants are finite. • Difficult to obtain exact solution near the horizon because one must find analogue os Lemaitre coordinate system for synchronous gauge. • Excellent accuracy can be found by using successive approximations. Inside black hole 2 2 2 • For χ >>χ m theory coincides with GR. Near horizon τ →1 we get a coordinate singularity at horizon. • Deep inside black hole τ2<<1 and close to horizon, spatial curvature is negligible. • Near bounce Δt ~ we obtain solution Mimetic massive gravity • Can the graviton have mass? Main problem is that a massive graviton will have 5 degrees of freedom in contrast to a massless graviton with 2 degrees. • The dynamical fields for a graviton are the 6 components, the space‐ space part of the metric. Diffeomorphism invariance eliminate four degrees of freedom keeping only two dynamical. When diffeomorphism invariance is broken, the 4 broken generators will force 6 degrees of freedom to propagate, one of which is ghost like. • This manifest itself that the zero mass limit of massive gravity is singular. Ghost‐Free mimetic massive gravity • Known models of massive gravity with a Fierz‐Pauli mass term, although termed as ghost free, suffer from the problem that the ghost sixth mode, gets excited in a non‐trivial time dependent background. • Four scalar fields are employed where three modes are absorbed by the massless graviton, while the fourth mode is tuned to decouple. • We have seen that in mimetic gravity the scale factor, a scalar mode, is constrained. Declaring this scalar field as part of the four scalars employed in making the graviton massless, we found that when using a non‐Fierz Pauli mass term, the ghost mode is removed to all orders. Ghost‐free action Let Then the action is The massive graviton is the field The other fields are dependent and expressible in terms of Ghost free • Also The expression for is complicated • The factor ½ is important because the equations of motion imply an equation that resembles the harmonic gauge of the massless graviton Asymptotically free mimetic gravity • The idea here is to make both the gravitational constant and cosmological constant functions of ﬦφ=κ the intrinsic curvature. • The action is Signifying a running Newton constant. Non‐singular solutions • Make the simplifying assumptions • Equations of motion imply that De Sitter limit • For contracting universe dominated by matter with equation of state p=wε we have the solution a∝2/3(1+w) for large negative t. • When curvature approaches limiting value κ0 the gravitational constant 2 begins to decrease and for 1‐(κ/κ0) <<1 equation is approximated by 2 2 2 κ =κ 0(1‐κ 0/ε) • In a contracting universe the scale factor a(t) decreases while the energy density grows as ε ∝ a‐3(1+w) hence the solution of equation of motion approaches contracting flat de Sitter universe with constant curvature where the scale factor decreases as a∝ exp(‐κ0t/3) for κ0t>>1. The gravitational constant G ∝1/f vanishes as 1/ε as ε→∞. Singularity is thus avoided irrespective of matter content of universe. What does it mean? • What is mimicking dark matter here are contributions f(χ) where χ=ﬦφ. In the synchronous gauge this is scalar extrinsic curvature. One can have more complicated variables which are functions of extrinsic curvature, or spatial curvature. The vector could be used as the normal vector in the 3+1 splitting such as We can add functions of intrinsic and spatial curvature, and these will act as sources of dark matter. Possible origin of mimetic gravity • In the volume quantization we encountered in noncommutative geometry, we have the mapping fields Y, Y’ from four manifold to four spheres. If we need to have three volume quantization instead, we to map R times three hypersurface to the three spheres. This will look like Conclusions • Mimetic gravity is a modest modification of GR with enormous consequences. • It eliminates the need to introduce arbitrary new scalar fields and could solve multitude of cosmological problems. • It offers a very nice solution to the problem of dark matter as a simple contribution of the curvature of geometry. • Provides a natural way to 3+1 splitting of space‐time as the vector • provides the normal vector to project into the three dimensional hypersurface.