3-rd lecture: Modified gravity and local gravity constraints

3-rd lecture: Modified gravity and local gravity constraints

Local gravity tests If we change gravity from General Relativity, there are constraints coming from local gravity tests. Solar system tests, violation of equivalence principle, I will discuss the compatibility of modified gravity models (including f(r) gravity, Brans-Dicke theory, dilaton gravity, ) with such experiments.

There are many modified gravity models other than f(r) gravity. A few examples are: (i) Brans-Dicke theory (with a potential V) w BD is called a Brans-Dicke parameter. (ii) Dilaton gravity A scalar field couples to R. (iii) Scalar-tensor theory

These modified gravity models can be written in the form Matter action We perform the conformal transformation where Let us consider scalar-tensor theories Introducing a new scalar field the action in the Einstein frame is where We set " 2 =1

In f(r) gravity we obtain the same action with We introduce the following quantity i.e., In f (R) gravity we have For scalar-tensor models with constant coupling Q, we find Then the Jordan frame action is

The Jordan frame action with constant Q. where (i) Q = 0 Quintessence (GR with a scalar field) (ii) Nonzero constant Q Setting we obtain the Brans-Dicke action: with the correspondence When Q goes to 0 w BD goes to infinity. (GR case)

Orginal Brans-Dicke theory (1961) The potential V is absent. Massless scalar field In this case the scalar field can freely propagate. The current solar-system tests give the bound Using the relation, this bound translates into The constant Q has a meaning of a coupling between the scalar field and matter in the Einstein frame. In the absence of the potential V, such a coupling needs to be suppressed ( Q <<1).

Theories with large couplings Q In f(r) gravity the coupling is large: If the potential is absent, it is not possible to satisfy solar system constraints ( ). However the potential is present in f(r) gravity: (gravitational origin) It is possible to satisfy local gravity constraints if the model is designed so that the mass of the field is heavy in the region where gravity experiments are carried out. This property holds for large coupling models with a scalar-field potential.

Chameleon mechanism (Khoury and Weltman, 2003) The effective coupling between the field and matter can be made much smaller than Q through a chameleon mechanism. Consider the action in Jordan frame: with The action in the Einstein frame is given by where In the Einstein frame dark energy couples to matter with the coupling Q. We are basically interested in the case where the potential V of the field is responsible for dark energy, while at the same time the model is consistent with local gravity tests.

Two demands for large-coupling scalar fields (i) The field mass needs to be small in order to realize the acceleration today on cosmological scales. Massless chameleon (ii) The field mass needs to be large in the region of high density to avoid the propagation of the fifth force. The field changes its mass depending on the environment it is in. Chameleon field Massive chameleon

The scalar-field equation in the Einstein frame Taking the variation of the Einstein frame action ( ) with respect to the field, we obtain The trace of the matter is where (non-relativistic matter) The energy density in the Einstein frame is Instead we use the energy density in the Einstein frame. that is conserved The scalar field directly couples to matter.

An effective potential has a minimum in the presence of a matter coupling. where With a coupling Q Runaway potential (used often in quintessence) such as The coupling induces a potential minimum.

The field mass about the potential minimum gets larger for increasing energy density. Massive Massless Large " m Small " m (The local region with high density) (The cosmological region with low density)!!

Spherically symmetric configuration The field equation in the Einstein frame (for weak gravity) is where Inside and outside the body, the effective potential has maxima at U," (" A ) + Qe Q" A # A = 0,!! " B << " A!! U, ( B ) + Qe Q B # B = 0 The field values at the maxima are different inside/outside the body.

The spherical symmetric configuration Inside the star Outside the star (r < r c ) (r > r c ) The body has a thin-shell if the field is almost frozen around in the most region of the inside of the star and if it evolves around the surface of the star.

The field profile There are three regions of interest. (i) 0 < r < r 1 in this region The field is nearly frozen. The field exists around. (ii) r < r < r 1 c (r is the radius of star) c in this region under the boundary condition The field begins to evolve. The field begins to evolve because of the dominance of the matter coupling. (iii) r > r c in this region The kinetic energy is dominant. under the boundary condition

The coefficients A, C, D, E are known by connecting three solutions at r=r and r=r (T. Tamaki and S.T.) 1 c The field solution outside the body, for m << m, is B A The radius r is determined by the following condition 1 This corresponds to where at the surface of body. is the gravitational potential

Thick-shell and thin-shell solutions The solution outside the body is (i) Thick-shell solutions If the field is away from at r=0, the field rapidly rolls down the potential. This corresponds to r 1 =0 and then The coupling is of the order of Q. It is not possible to satisfy local gravity constraints unless Q <<1.

(ii) Thin-shell solutions Thin-shell If r is close to r and m r >>1, then 1 c A c r 1 r c where Q eff is the effective coupling given by Q eff becomes much smaller than Q when the body has a thin-shell. Using the previous relation we have where

Using the thin-shell parameter, the effective coupling is Q becomes smaller than Q for eff The upper bound on the thin-shell parameter can be obtained by solar-system tests and by the violation of equivalence principle.

Solar-system constraints The spherically symmetric metric in the Jordan frame is The Einstein frame metric is where Under the weak gravity background we have (because ) where (thin-shell solutions)

Under the condition we have The post Newtonian parameter is The tightest solar-system bound coming from the Shapiro time delay effect is This translates into As long as the thin-shell parameter is much smaller than 1, the solar system constraints are satisfied even for Q =O(1).

The fifth-force with The fifth force that exerts on a particle with a unit mass (i.e., acceleration) is (suppressed for ) The presence of the fifth force leads to the difference of accelerations of Earth and Moon toward the Sun. A detailed calculation gives Stronger than solar system constraints

The equivalence constraint gives where we used

(i) The potential Concrete models This runaway potential is often used in the context of dark energy. Solving the equation U," (" B ) + Qe Q" B # B = 0 gives! The constraint gives where we used When n=1, When n=2, Compatible with the energy scale responsible for dark energy

(ii) f(r) gravity In f(r) gravity the potential in the Einstein frame is where Consider the model In this case we have The equivalence constraint gives For the existence of a late-time de Sitter point we require Taking and Indistinguishable from the LCDM model

Models that can deviate from the LCDM model Hu Starobinsky Hu and Sawicki: (R /R f (R) = R " #R 0 ) 2n 0 (May, 2007) (R /R 0 ) 2n +1 2 Starobinsky: $ f (R) = R " #R 0 1" 1+ R 2 2 ( /R 0 ) "n ' R 0 " H 0 %& () (June, 2007)! Cosmological! constant disappears in a flat space. f (R = 0) = 0! R >> R 0 and!! The solar-system constraints are satisfied for n > 0.5 The equivalence principle constraints are satisfied for n >1 (Capozziello and S.T.) In these models the deviation from the LCDM model becomes significant around the present epoch on cosmological scales.!