1 f(R) Gravity and its relation to the interaction between DE and DM Bin Wang Shanghai Jiao Tong University

SNe Ia The current universe is accelerating! LSS CMB Dark Energy

Simplest model of dark energy Cosmological constant: This corresponds to the energy scale If this originates from vacuum energy in particle physics, Huge difference compared to the present value! (Equation of state: ) Cosmological constant problem

There are two approaches to dark energy. (i) Changing gravity(ii) Changing matter f(R) gravity models, Scalar-tensor models, Braneworld models, Inhomogeneities, ….. Quintessence, K-essence, Tachyon, Chaplygin gas, ….. Are there some other models of dark energy? (Einstein equations)

‘Changing matter’ models To get the present acceleration most of these models are based upon scalar fields with a very light mass: Quintessence, K-essence, Tachyon, phantom field, … Flat In super-symmetric theories the severe fine-tuning of the field potential is required. (Kolda and Lyth, 1999) The coupling of the field to ordinary matter should lead to observable long-range forces. (Carroll, 1998)

‘Changing gravity’ models f(R) gravity, scalar-tensor gravity, braneworld models,.. Dark energy may originate from some geometric modification from Einstein gravity. The simplest model: f(R) gravity model: f(R) modified gravity models can be used for dark energy ? R: Ricci scalar

Field Equations The field equation can be derived by varying the action with respect to satisfies Trace The field equation can be written in the form

Field Equations We consider the spatially flat FLRW spacetime the Ricci scalar R is given by The energy-momentum tensor of matter is given by The field equations in the flat FLRW background give where the perfect fluid satisfies the continuity equation

f(R) gravity GR Lagrangian:(R is a Ricci scalar) Extensions to arbitrary function f (R) f(R) gravity The first inflation model (Starobinsky 1980) Starobinsky Inflation is realized by the R term. 2 Favored from CMB observations Spectral index: Tensor to scalar ratio: N: e-foldings

f(R) dark energy: Example Capozziello, Carloni and Troisi (2003) Carroll, Duvvuri, Trodden and Turner (2003) It is possible to have a late-time acceleration as the second term becomes important as R decreases. In the small R region we have Late-time acceleration is realized. (n>0) Problems: Matter instability, perturbation instability, absence of matter dominated era, local gravity constraints…

Stability of dynamical systems consider the following coupled differential equations for two variables x(t) and y(t): Fixed or critical points (xc, yc) if A critical point (xc, yc) is called an attractor when it satisfies the condition Stability around the fixed points consider small perturbations δx and δy around the critical point (xc, yc), leads to The general solution for the evolution of linear perturbations E. Copeland, M. Sami, S.Tsujikawa, IJMPD (2006)

stability around the fixed points

Autonomous equations We introduce the following variables: Then we obtain and where The above equations are closed. See review: S.Tsujikawa et al (2006,2010) and,

model: The parameter characterises a deviation from the model. The constant m model corresponds to (a) (b) (c) The model of Capozziello et al and Carroll et al: This negative m case is excluded as we will see below.

The cosmological dynamics is well understood by the geometrical approach in the (r, m) plane. (i) Matter point: P M From the stability analysis around the fixed point, the existence of the saddle matter epoch req uires at (ii) De-sitter point P A For the stability of the de-Sitter point, we require m1m1

Viable trajectories Constant m model: (another accelerated point)

Lists of cosmologically non-viable models (n>0) …. many ! Lists of cosmologically viable models (0<n<1) Li and Barrow (2007) Amendola and Tsujikawa. (2007) Hu and Sawicki (2007) Starobinsky (2007) More than 200 papers were written about f(R) dark energy!

Conformal transformation Under the conformal transformation The Ricci scalars in the two frames have the following relation where The action is transformed as for the choice introduce a new scalar field φ defined by the action in the Einstein frame (The scalar is directly coupled to matter)

19 the Lagrangian density of the field φ is given by the energy-momentum tensor The conformal factoris field-dependent. Using matter The energy-momentum tensor of perfect fluids in the Einstein frame is given by

20 consider the flat FLRW spacetime The field equation can be expressed as the scalar field and matter interacts with each other

The f(R) action is transformed to Matter fluid satisfies: Coupled quintessence where Dark matter is coupled to the field Is the model (n>0) cosmologically viable? No! This model does not have a standard matter era prior to the late-time acceleration. (Einstein frame)

The model The potential in Einstein frame is The standard matter era is replaced by ‘phi matter dominated era’ For large field region, Coupled quintessence with an exponential potential : (n>0) Jordan frame: Incompatible with observations L. Amendola, D. Polarski, S.Tsujikawa, PRL (2007).

Inertia of Energy Meshchersky’s equation M v dm vtvt v Momentum Inertial drag Momentum transfer energy Rocket He, Wang, Abdalla PRD(2010)

24 Physical meaning of The conformal transformation The equation of motion under such a transformation where we have used D.20 in Wald’s book for perfect fluid and drop pressure where it reduces to comparing with the equation of motion of particles with varying mass We have introduce a scalar field Γ which satisfies This Γ can be rewritten as mass dilation rate due to the conformal transformation. J.H.He, B.Wang, E.Abdalla, PRD(11)

For the FRW background with a scale factor a, we have Pressure-less Matter: Radiation: To avoid: matter instability, instability in perturbation, absence of MD era, inability to satisfy local gravity constraints What are general conditions for the cosmological viability of f(R) dark energy models? S.Tsujikawa et al (07); W.Hu et al (07)

26 (0<n<1) Li and Barrow (2007) Amendola and Tsujikawa. (2007) Hu and Sawicki (2007) Starobinsky (2007) Lists of cosmologically viable models constructed To avoid: matter instability, instability in perturbation, absence of MD era, inability to satisfy local gravity constraints

27 Construct the f(R) model in the Jordan frame FRW metric take an expansion history in the Jordan frame that matches a DE model with equation of state w For w=-1: C and D are coefficients which will be determined by boundary conditions J.H.He, B.Wang (2012)

28 f(R) model should be “chameleon” type go back to the standard Einstein gravity in the high curvature region need to set C = 0. The solution turns D above is a free parameter which characterizes the different f(R) models which have the same expansion history as that of the LCDM model. is the complete Euler Gamma function analytic f(R) form and D are two free dimensionless parameters, avoid the short-timescale instability at high curvature, D<0 is required is satisfied J.H.He, B.Wang (2012)

29 Construct the f(R) model from the Einstein frame Conformal transformation Motion of particle with varying mass freedom in choosing Solving dynamics in the Einstein frame J.H.He, B.Wang,E.Abdalla, PRD(11)

30 conformal dynamics in the Jordan frame J.H.He, B.Wang,E.Abdalla, PRD(11)

31 Perturbation theory The Jordan frame The perturbed line element in Fourier space The perturbed form of the modified Einstein equations Inserting the line element, we can get the perturbation form of the modified Einstein equation J.H.He, B.Wang (2012)

32 perturbed form of the modified Einstein equations Under the infinitesimal transformation, we can show that the perturbed quantities in the line element Inserting into the perturbation equation, we find that under the infinitesimal transformation, the perturbation equations are covariant. They go back to the standard form when F → 1,δF → 0.

33 The Newtonian gauge is defined by setting B=E=0 and these conditions completely fix the gauge The perturbations in this gauge can be shown as gauge invariant. the Synchronous gauge is not completely fixed because the gauge condition ψ = B = 0 only confines the gauge up to two arbitrary constants C1,C2 Usually, C2 is fixed by specifying the initial condition for the curvature perturbation in the early time of the universe and C1 can be fixed by setting the peculiar velocity of DM to be zero, v_m = 0. After fixing C1,C2, the Synchronous gauge can be completely fixed. perturbations in different gauges are related

34 in the Newtonian gauge use the Bardeen potentials Φ= φ, Ψ= ψ to represent the space time perturbations. Consider the DM dominated period in the f(R) cosmology, we set P_m = 0 and δP_m = 0 The perturbed Einstein equations Where From the equations of motion matter perturbation evolves

35 Perturbation theory The Einstein frame In the background in perturbed spacetime the symbols with “tilde” indicate the quantities in Einstein frame Under the infinitesimal coordinate transformation, the perturbed quantities ˜ψ, ˜φ behave as in a similar way as in the Jordan frame In the Newtonian gauge, the gauge conditions B=E=0 in the Synchronous gauge, the gauge conditions in the Einstein frame reads

36 The perturbed equation In the Einstein frame J.H.He, B.Wang (2012)

37 SUBHORIZION APPROXIMATION When the modified gravity doesn’t show up When the modified gravity becomes important J.H.He, B.Wang (2012)

Modified Gravity G Jordan frameEinstein frame if we compare it with the Einstein gravity. This is effectively equivalent to rescale the gravitational mass the inertial mass in the Jordan frame is conserved so that the equation of motion for a free particle in the Jordan frame is described by change in the gravitational mass changes the gravitational field change the inertial frame. inertial “frame- dragging” inertial frame unchanged, inertial mass rescaled Gravity Probe B Mach principle inertial “mass- dragging” Understand the mass dilation

Conclusions 1.We reviewed the relation between the f(R) gravity and the interaction between DE and DM 2.We discussed viability condition for the f(R) model 3.We discussed the perturbation theory for the f(R) model 4.We further discussed the physical connection between the Jordan frame and the Einstein frame and the physical meaning of the mass dilation