Chameleon scalar fields in relativistic gravitational backgrounds

Chameleon scalar fields in relativistic gravitational backgrounds

Shinji Tsujikawa Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan shinji@rs.kagu.tus.ac.jp    Takashi Tamaki Department of Physics, Waseda University, Okubo 3-4-1, Tokyo 169-8555, Japan tamaki@gravity.phys.waseda.ac.jp Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan    Reza Tavakol Astronomy Unit, School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK r.tavakol@qmul.ac.uk
July 15, 2019
Abstract

We study the field profile of a scalar field that couples to a matter fluid (dubbed a chameleon field) in the relativistic gravitational background of a spherically symmetric spacetime. Employing a linear expansion in terms of the gravitational potential at the surface of a compact object with a constant density, we derive the thin-shell field profile both inside and outside the object, as well as the resulting effective coupling with matter, analytically. We also carry out numerical simulations for the class of inverse power-law potentials by employing the information provided by our analytical solutions to set the boundary conditions around the centre of the object and show that thin-shell solutions in fact exist if the gravitational potential is smaller than 0.3, which marginally covers the case of neutron stars. Thus the chameleon mechanism is present in the relativistic gravitational backgrounds, capable of reducing the effective coupling. Since thin-shell solutions are sensitive to the choice of boundary conditions, our analytic field profile is very helpful to provide appropriate boundary conditions for .

I Introduction

The origin of the so called dark energy responsible for the present cosmic acceleration remains a great mystery. Since the cosmological constant (originating from the vacuum energy) is plagued by a severe fine-tuning problem, many alternative models have been proposed to account for the origin of dark energy (see Refs. review () for reviews). A number of these models, including the quintessence quin (), k-essence kes () and tachyon tac () models, make use of a scalar field with a very light mass ( eV) in order to account for the present cosmic acceleration. If the scalar field originates from candidate theories for fundamental interactions such as string theory or supergravity, it should interact with the standard model particles with a long ranged force (the so called “fifth force”). In string theory, for example, a dilaton field universally couples to matter as well as gravity Lidsey (). Similarly, in modified gravity theories such as gravity fR () and scalar-tensor theories stensor (), the scalar degree of freedom interacts with the matter fluid (except for radiation). This is clearly seen if one transforms the action to the Einstein frame via a conformal transformation Maeda (). For example, it is known that Brans-Dicke theory Brans () (a historically important class of scalar-tensor theories) gives rise to a constant coupling between the scalar field and the matter TUMTY (). In this sense such modified gravity theories can be regarded as a coupled quintessence scenario Amen () in the Einstein frame.

In the absence of a scalar-field potential, the present solar-system tests constrain the strength of the coupling to be smaller than the order of TUMTY (). However, the couplings that appear in string theory Lidsey () and gravity APT () are typically of the order of unity. In such cases it is not possible to satisfy the local gravity constraints, unless a scalar-field potential with a large mass exists to suppress the coupling in the regions of high density. Moreover, if the same field is responsible for the cosmic acceleration today, the potential needs to be sufficiently flat in the regions of low density (i.e., on cosmological scales).

In spite of the above requirements it is possible for the large coupling models to satisfy the local gravity constraints through the chameleon mechanism KW (); KW2 (), while at the same time for the field to have sufficiently small mass to lead to the present cosmic acceleration. The existence of a matter coupling gives rise to an extremum of the scalar-field potential around which the field can be stabilized. In high density regions, such as the interiors of the astrophysical objects, the field mass about the extremum would be sufficiently large to avoid the propagation of the fifth force. Meanwhile, the field would have a much lighter mass in the low-density environments, far away from compact objects, so that it could be responsible for the present cosmic acceleration. In the case of inverse power-law potentials Ratra () with , local gravity constraints can be satisfied for  eV KW2 (). Interestingly, this roughly corresponds to the energy scale required for the cosmic acceleration today. See Refs. Brax (); Gubser (); fRca (); Clifton (); Das (); radion (); Mota (); exper (); TT () for works concerning a number of interesting aspects of the chameleon mechanism.

So far the analyses of the chameleon mechanism have typically concentrated on the weak gravity backgrounds where the spherically symmetric metric is described by a Minkowski spacetime. This amounts to neglecting the backreaction of gravitational potential on the scalar-field equation. In Ref. TT () the field profile in the Minkowski background was analytically derived both inside and outside the object by taking into account the mass of the chameleon field inside the body. In this settings it has been shown that the field would need to be extremely close to the maximum of the effective potential around the centre of the spherically symmetric body in order to allow thin-shell solutions required for consistency with the local gravity constraints.

If we take into account the backreaction of gravitational potential to the field equation, the relativistic pressure is present even in weak gravity backgrounds such as the Sun or the Earth. It is expected that this effect changes the field profile inside the body in order to allow the existence of thin-shell solutions. We shall analytically derive the thin-shell field profile using a linear expansion in terms of the gravitational potential at the surface of compact objects. In fact we show that there exists a region around the centre of the massive objects in which the field evolves toward the maximum of the effective potential because of the presence of the relativistic pressure. In order to realize thin-shell solutions, the driving force along the potential needs to dominate over the pressure for distances larger than a critical value . This distance () is required to be smaller than the distance at which the field enters a thin-shell regime. In spite of such different properties of the field profile inside the body relative to the case of the Minkowski background, the effective coupling outside the body can be reduced by the presence of thin-shell solutions. We confirm this by using numerical simulations for a class of potentials of the form .

To study the viability of theories with large couplings, it is important to determine whether thin-shell solutions can also exist in strong gravitational backgrounds with . We shall derive analytic solutions using linear expansions in terms of and then carry out numerical simulations to confirm the validity of solutions in the regimes with . Our analytic solutions are useful as a way of finding the boundary conditions around the centre of the object in order to obtain thin-shell solutions. By choosing boundary conditions with field values larger than those estimated by the analytic solutions, we shall demonstrate numerically that the thin-shell solutions are present for backgrounds with gravitational potentials satisfying , in the case of the field potentials of the type . This marginally covers the case of neutron stars. In backgrounds with still larger gravitational potentials the relativistic pressure around the centre of the object is so strong that the field typically overshoots the maximum of the effective potential to reach the singularity at , unless the boundary conditions of the field around the centre of the body are chosen to be far from the maximum of the effective potential. This overshoot behaviour is similar to the one recently found by Kobayashi and Maeda Kobayashi () in the context of dark energy models (see also Ref. Frolov ()). We note, however, that our analytic solutions based on the linear expansion of do not cover the field profiles for the really strong gravitational backgrounds with . In such cases we need a separate analysis which incorporates the formation of black holes.

The outline of the paper is as follows. In Section II we discuss our theoretical set up as well as giving the relevant equations for the case of a spherically symmetric central body. In section III we give the analytical thin-shell solutions to the scalar field equations, both inside and outside of the body, and consider in turn the matching of thin-shell solutions. In Section IV we study the analytical field profile in more details and discuss how the field evolves as a function of in the presence of the relativistic pressure. In Section V we integrate the field equation numerically and show the existence of thin-shell solutions for . Finally Section VI contains our conclusions.

Ii Setup

We consider settings in which a scalar field with potential couples to a matter with a Lagrangian density . In particular we shall study theories based on the action

(1)

where is the determinant of the metric , is the reduced Planck mass ( is the gravitational constant), is a Ricci scalar, and are matter fields that couple to a metric related with the Einstein frame metric via

(2)

Here are the strength of couplings for each matter field. In the following we shall consider cases in which the couplings are the same for each matter component, i.e., , and use units such that . We restore when it is needed.

An example of a scalar-tensor theory which gives rise to constant couplings in the Einstein frame is given by the action TUMTY ()

(3)

where a tilde represents quantities in the Jordan frame. The action (3) is equivalent to that in Brans-Dicke theory with a potential . Under the conformal transformation, , we obtain the action (1) in the Einstein frame, together with the field potential . Clearly the metric in Eq. (2) corresponds to the metric in the Jordan frame.

To study chameleon fields in the relativistic gravitational background of a spherically symmetric body, we consider the following spherically symmetric static metric in the Einstein frame:

(4)

where and are functions of the distance from the centre of symmetry. For the action (1) the energy momentum tensors for the scalar field and the matter are given, respectively, by

(5)
(6)

Under the gravitational background (4), the (00) and components for the energy momentum tensors are

(7)

where a prime represents a derivative with respect to and

(8)

Here and are the energy momentum tensors of matter in the Jordan frame. Denoting the energy density and the pressure of the matter in the Jordan frame as and , the matter energy-momentum tensor in this frame takes the form . The corresponding expressions for the energy density and pressure in the Einstein frame are then given by and .

The evolution equation for the scalar field is given by

(9)

where the derivative of in terms of is

(10)

We then obtain

(11)

The Einstein equations give:

(12)
(13)
(14)

From the conservation equation, , we also obtain

(15)

which is the generalization of the Tolman-Oppenheimer-Volkoff equation. Note that this equation can also be derived by combining Eqs. (11)-(14).

Our main interest is the case in which the field potential is responsible for dark energy. In that case both and are negligible relative to in the local regions whose density is much larger than the cosmological one ( g/cm). Then Eq. (12) can be integrated to give

(16)

Substituting Eqs. (12) and (13) into Eq. (11) gives

(17)

We assume that the energy density is constant inside () and outside () of the spherically symmetric body with a radius . Strictly speaking the conserved density in the Einstein frame is given by KW (); KW2 (); TT (). However, since the condition holds in most cases of interest, we do not need to distinguish between and .

Inside the spherically symmetric body () we have and Eq. (16) gives

(18)

With the neglect of the scalar-field contributions in Eqs. (12)-(15) it is known that the background gravitational field for corresponds to the Schwarzschild interior solution. In this case the pressure inside the body relative to the density can be analytically expressed as

(19)

where is the gravitational potential at the surface of body:

(20)

Here is the mass of the spherically symmetric body, and in the last equality in Eq. (20) we have used units such that . Equation (19) shows that the pressure vanishes at the surface of the body ().

In the following we shall derive analytic solutions for Eq. (11), under the conditions and . We neglect the terms higher than the linear order in and . From Eqs. (18)–(20) it then follows that

(21)

At the centre of the body we have , which shows that the effect of the pressure becomes important in strong gravitational backgrounds.

Outside the body we assume that the density is very much smaller than with a vanishing pressure. Then the metric outside the body can be approximated by the Schwarzschild exterior solution:

(22)

Iii Matching solutions of the chameleon scalar field

In this section we solve the scalar-field equation (17) in the relativistic gravitational backgrounds discussed in Sec. II.

In the nonrelativistic gravitational background where the pressure as well as the gravitational potential are negligible, the effective potential for the scalar field is defined as KW (); KW2 ()

(23)

This potential has a minimum either when (i) and or (ii) and . An example of class of potentials satisfying (i) is provided by the inverse power-law potentials (). Since gravity corresponds to the coupling , the effective potential has a minimum for the case (as in the case of the models proposed in Refs. AGPT (); Li (); Hu (); Star (); Appleby (); Tsuji08 ()).

For constant matter densities, and , inside and outside of the body, the effective potential (23) has two minima at the field values and characterized by the conditions

(24)
(25)

The former corresponds to the region with a high density (interior of the body) that gives rise to a heavy mass squared , whereas the latter corresponds to the lower density region (exterior of the body) with a lighter mass squared .

The following boundary conditions are imposed at and :

(26)

We need to consider the potential in order to find the “dynamics” of with respect to . This means that the effective potential has a maximum at . The field is at rest at and begins to roll down the potential when the matter-coupling term becomes important at a radius . If the field value at is close to , the field stays around in the region . The body has a thin-shell if is close to the radius of the body.

The position of the minimum given in Eq. (24) is shifted in the relativistic gravitational background. In the following we shall derive the field profile by taking into account the corrections coming from the gravitational potential. Inside the body, Eq. (17) to the the linear order in reduces to:

(27)

In the region the field derivative of the effective potential around may be approximated by . The solution to Eq. (27) can be obtained by writing the field as , where is the solution in the Minkowski background and is the perturbation induced by . At the linear order in and we obtain

(28)
(29)

The solution to Eq. (28) that is regular at is given by , where is a constant. Substituting this solution into Eq. (29) we obtain the following solution for :

(30)

One can easily show that this solution satisfies the first of the boundary conditions (26).

In the region the field evolves towards larger values with increasing . Since in this regime one has . In this case and satisfy

(31)
(32)

We then find the following solution

(33)

where and are constants.

The field acquires sufficient kinetic energy in the thin-shell regime, in order to allow it to climb up the potential hill towards larger absolute values in the region outside the body. As long as the kinetic energy of the field dominates over its potential energy, the right hand side of Eq. (17) can be neglected relative to its left hand side. Also the term that includes and in the square bracket on the left hand side of Eq. (17) can be neglected relative to the term . Using Eq. (22), the field equation reduces to

(34)

The solution to this equation is

(35)

where is a constant. Note that here we have used the second boundary condition in Eq. (26).

Having obtained the solutions (30), (33) and (35) in the three regions inside and outside the central body, we proceed to match these solutions at and . The thin-shell corresponds to the region defined by

(36)

namely .

It is possible to satisfy the local gravity constraints as long as the field inside the body is sufficiently massive, i.e., (or ). For example, in the case of the Earth with the class of potentials , we have the constraints , for , and , for , from the experimental tests of the equivalence principle TT (). Since the mass becomes larger in higher density regions, the quantity inside a strong gravitational body becomes even larger than in the case of the Sun or the Earth. We use the approximation that is negligible relative to in Eq. (30).

We recall that - for neutron stars, - for white dwarfs, for the Sun and for the Earth. In the following we shall use linear expansions in terms of the three parameters , and (or ). We drop terms of higher order in these parameters relative to 1. We caution that our approximation loses its accuracy under the really strong gravitational backgrounds with .

Using the continuity of and at and we obtain

(37)
(38)
(39)
(40)

The value of can be derived from Eqs. (39) and (40) keeping terms to linear order in only. Substituting into Eq. (37) and using Eq. (III) we can obtain expressions for and . The coefficient is then obtained from Eq. (40). Using this procedure we find

(41)
(42)
(43)
(44)

where

(45)

Since the denominator in Eq. (45) is larger than 1, the parameter is much smaller than 1.

The distance is determined by the condition

(46)

where

(47)

Substituting Eq. (47) into Eq. (46) gives

(48)

From Eqs. (III) and (48) we then obtain

(49)

where

(50)

Note that .

The thin-shell parameter introduced in Refs. KW (); KW2 () is in this case given by

(51)
(52)

To the first-order in expansion parameters one has , which is identical to the corresponding value derived in the Minkowski background TT (). The effect of the gravitational potential appears as a second-order term to the thin-shell parameter. Substituting Eq. (49) into Eq. (III), we obtain the following approximate solution

(53)

where we have carried out a linear expansion in terms of , , and . The solution outside the body is then given by

(54)

where the effective coupling is

(55)

To leading-order this gives , which agrees with the corresponding result in the Minkowski background TT (). Thus provided that , the effective coupling becomes much smaller than the bare coupling . The gravitational potential appears as a next-order term. As can be seen from Eq. (55) the presence of the gravitational potential leads to a small decrease in compared to the nonrelativistic gravitational background.

Iv The field profile

In this section we shall discuss the analytical field profile derived in the previous section in more details. The coefficient and the field difference are determined by fixing the value of , see Eqs. (48) and (49). From Eqs. (30), (33), (35), (III)-(III), (48) and (49) the thin-shell field profile is given by

(56)
(57)
(58)

where