11.2 Constant density star

Let us consider a constant density star with &tidle;ρM = &tidle;ρA. We also assume that the density outside the star is constant, ρ&tidle; = ρ&tidle; M B. We caution that the conserved density ρ&tidle;(c) M in the Einstein frame is given by (c) −Qϕ &tidle;ρM = e ρ&tidle;M [343Jump To The Next Citation Point]. However, since the condition Qϕ ≪ 1 holds in most cases of our interest, we do not distinguish between &tidle;ρ(cM) and &tidle;ρM in the following discussion.

Inside the spherically symmetric body (0 < &tidle;r < &tidle;rc), Eq. (11.12View Equation) gives

( ) −1 2Φ(&tidle;r) 8πG-- 2 e = 1 − 3 &tidle;ρA&tidle;r . (11.14 )
Neglecting the field contributions in Eqs. (11.8View Equation) – (11.11View Equation), the gravitational background for 0 < &tidle;r < &tidle;rc is characterized by the Schwarzschild interior solution. Then the pressure P&tidle;M (&tidle;r) inside the body relative to the density &tidle;ρA can be analytically expressed as
&tidle;P (&tidle;r) ∘1--−-2(&tidle;r2∕&tidle;r2)Φ--− √1--−-2Φ-- -M-----= -√-----------c∘--c-----------c- (0 < &tidle;r < &tidle;rc), (11.15 ) &tidle;ρA 3 1 − 2Φc − 1 − 2(&tidle;r2∕&tidle;r2c)Φc
where Φc is the gravitational potential at the surface of the body:
GMc 1 Φc ≡ -----= -&tidle;ρA &tidle;r2c. (11.16 ) &tidle;rc 6
Here Mc = 4 π&tidle;r3c&tidle;ρA∕3 is the mass of the spherically symmetric body. The density &tidle;ρB is much smaller than ρ&tidle;A, so that the metric outside the body can be approximated by the Schwarzschild exterior solution
GMc &tidle;rc Φ(&tidle;r) ≃ ----- = Φc --, P&tidle;M (&tidle;r) ≃ 0 (&tidle;r > &tidle;rc). (11.17 ) &tidle;r &tidle;r

In the following we shall derive the analytic field profile by using the linear expansion in terms of the gravitational potential Φ c. This approximation is expected to be reliable for Φ < 𝒪 (0.1 ) c. From Eqs. (11.14View Equation) – (11.16View Equation) it follows that

2 &tidle; ( 2) Φ (&tidle;r) ≃ Φc&tidle;r- , PM--(&tidle;r)-≃ Φc- 1 − &tidle;r- (0 < r&tidle;< &tidle;rc). (11.18 ) &tidle;r2c &tidle;ρA 2 &tidle;r2c
Substituting these relations into Eq. (11.13View Equation), the field equation inside the body is approximately given by
2 ( &tidle;r2 ) ( r&tidle;2 ) 3 ( &tidle;r2) ϕ′′ +-- 1 − ---Φc ϕ′ − (V,ϕ + Qρ&tidle;A ) 1 + 2Φc-- + --Qρ&tidle;A Φc 1 − -- = 0. (11.19 ) &tidle;r 2&tidle;r2c r&tidle;2c 2 &tidle;r2c
If ϕ is close to ϕ A at &tidle;r = 0, the field stays around ϕ A in the region 0 < &tidle;r < &tidle;r 1. The body has a thin-shell if &tidle;r1 is close to the radius &tidle;rc of the body.

In the region 0 < &tidle;r < &tidle;r1 the field derivative of the effective potential around ϕ = ϕA can be approximated by dVe ff∕dϕ = V,ϕ + Q &tidle;ρA ≃ m2A (ϕ − ϕA). The solution to Eq. (11.19View Equation) can be obtained by writing the field as ϕ = ϕ + δϕ 0, where ϕ 0 is the solution in the Minkowski background and δϕ is the perturbation induced by Φc. At linear order in δϕ and Φc we obtain

[ ( ) ] ′′ 2- ′ 2 2m2A&tidle;r2- &tidle;r- ′ 3- &tidle;r2 δϕ + &tidle;rδϕ − m Aδϕ = Φc &tidle;r2 (ϕ0 − ϕA ) + &tidle;r2ϕ0 − 2 Qρ&tidle;A 1 − &tidle;r2 , (11.20 ) c c c
where ϕ0 satisfies the equation ϕ′0′+ (2∕&tidle;r)ϕ ′0 − m2A (ϕ0 − ϕA ) = 0. The solution of ϕ0 with the boundary conditions dϕ0∕d &tidle;r = 0 at &tidle;r = 0 is given by ϕ0(&tidle;r) = ϕA + A (e−mA &tidle;r − emA &tidle;r)∕&tidle;r, where A is a constant. Plugging this into Eq. (11.20View Equation), we get the following solution for ϕ(&tidle;r) [594Jump To The Next Citation Point]:
A(e−mA &tidle;r − emA&tidle;r) ϕ(&tidle;r) = ϕA + ----------------- [( &tidle;r ) ( ) ] − -AΦc-- 1-m2 &tidle;r2 − 1mA r&tidle;− 1+ --1--- emA&tidle;r + 1m2 &tidle;r2 + 1-mA &tidle;r − 1-− ---1-- e−mA &tidle;r mA r&tidle;2c 3 A 4 4 8mA &tidle;r 3 A 4 4 8mA &tidle;r 3Q &tidle;ρA Φc [ ] − ----4-2- m2A(&tidle;r2 − r&tidle;2c) + 6 . (11.21 ) 2m A&tidle;rc

In the region &tidle;r1 < &tidle;r < &tidle;rc the field |ϕ(&tidle;r)| evolves towards larger values with increasing &tidle;r. Since the matter coupling term Q &tidle;ρA dominates over V,ϕ in this regime, it follows that dVeff ∕dϕ ≃ Q &tidle;ρA. Hence the field perturbation δϕ satisfies

2 [ &tidle;r 1 ( &tidle;r2) ] δϕ′′ +-δϕ ′ = Φc -2ϕ′0 − -Q &tidle;ρA 3 − 7-2 , (11.22 ) &tidle;r &tidle;rc 2 &tidle;rc
where ϕ0 obeys the equation ϕ′0′+ (2∕&tidle;r)ϕ ′0 − Q &tidle;ρA = 0. Hence we obtain the solution
( ) ( ) B &tidle;r2 1 2 3 23 &tidle;r2 ϕ (&tidle;r) = − -- 1 − Φc--2- + C + -Q ρA&tidle;r 1 − -Φc + ---Φc-2 , (11.23 ) &tidle;r 2&tidle;rc 6 2 20 &tidle;rc
where B and C are constants.

In the region outside the body (&tidle;r > &tidle;rc) the field ϕ climbs up the potential hill after it acquires sufficient kinetic energy in the regime &tidle;r < &tidle;r < &tidle;r 1 c. Provided that the field kinetic energy dominates over its potential energy, the r.h.s. of Eq. (11.13View Equation) can be neglected relative to its l.h.s. of it. Moreover the terms that include ρ&tidle;M and &tidle;PM in the square bracket on the l.h.s. of Eq. (11.13View Equation) is much smaller than the term (1 + e2Φ)∕&tidle;r. Using Eq. (11.17View Equation), it follows that

( ) ′′ 2 GMc ′ ϕ + &tidle;r- 1 + --&tidle;r-- ϕ ≃ 0, (11.24 )
whose solution satisfying the boundary condition ϕ(&tidle;r → ∞ ) = ϕB is
( ) ϕ (&tidle;r) = ϕ + D- 1 + GMc-- , (11.25 ) B &tidle;r r&tidle;
where D is a constant.

The coefficients A, B, C, D are known by matching the solutions (11.21View Equation), (11.23View Equation), (11.25View Equation) and their derivatives at &tidle;r = &tidle;r1 and &tidle;r = r&tidle;c. If the body has a thin-shell, then the condition Δ &tidle;rc = &tidle;rc − &tidle;r1 ≪ &tidle;rc is satisfied. Under the linear expansion in terms of the three parameters Δ &tidle;rc∕&tidle;rc, Φc, and 1 ∕(mA &tidle;rc) we obtain the following field profile [594Jump To The Next Citation Point]:

Q &tidle;ρ &tidle;r ( m &tidle;r3Φ Φ &tidle;r2) −1 ϕ (r&tidle;) = ϕA + --2--A----1 1 + --A--1-c − --c-1 (emA &tidle;r − e− mA&tidle;r) m AemA &tidle;r1r&tidle; 3&tidle;r2c 4&tidle;r2c 3Q &tidle;ρ Φ [ &tidle;r2 6 ] Φ &tidle;r Q &tidle;ρ ( m &tidle;r3Φ Φ &tidle;r2 )− 1 + ----A2-c 1 − -2 − -------2 + ---c12--2-mA-&tidle;r- 1 + --A-12-c− --c12- [(2m A &tidle;rc (mA r&tidle;c) mA) &tidle;rcm Ae( A 1 3r&tidle;c 4&tidle;rc ) ] 1 2 2 1 1 1 m &tidle;r 1 2 2 1 1 1 − m &tidle;r × --m A&tidle;r − -mA &tidle;r − -+ ------ e A + -m Ar&tidle; + --mA &tidle;r − --− ------ e A 3 4 4 8mA &tidle;r 3 4 4 8mA &tidle;r (0 < r&tidle;< r&tidle;1),[ (11.]26 ) 2 ( 2) ( ) ( )2 ( 2 ) ϕ (r&tidle;) = ϕA + Q-&tidle;ρA&tidle;rc- 6𝜖th + 6C1r&tidle;1 1 − Φc&tidle;r-- − 3 1 − Φc- + -&tidle;r 1 − 3Φc + 23Φcr&tidle;- 6 &tidle;r 2 &tidle;r2c 4 &tidle;rc 2 20&tidle;r2c (&tidle;r1 < &tidle;r < &tidle;rc)[, ( ) ] (11.27 ) 2 &tidle;rc &tidle;rc ϕ (r&tidle;) = ϕA + Q &tidle;ρA&tidle;rc 𝜖th − C2 &tidle;r 1 + Φc &tidle;r (&tidle;r > &tidle;rc), (11.28 )
where 𝜖th = (ϕB − ϕA)∕(6Q Φc) is the thin-shell parameter, and
[ ( 2) ( 2 ) 2 ( 2) ] C1 ≡ (1 − α ) − 𝜖th 1 + Φc&tidle;r1- + 1- 1 − Φc- + Φcr&tidle;1- − -&tidle;r1- 1 − 3Φc + 7Φc&tidle;r1- 2&tidle;r2c 2 4 2&tidle;r2c 2r&tidle;2c 2 4&tidle;r2c &tidle;r2 ( 3 9Φc&tidle;r2) + --12 1 − -Φc + ---21- , (11.29 ) 3r&tidle;c [ 2 ( 5&tidle;rc ) ( ) ( ) ] &tidle;r1 Φc-&tidle;r21 3Φc- -&tidle;r1 7- Φc-&tidle;r21 &tidle;r31-- 7-Φc&tidle;r21 C2 ≡ (1 − α ) 𝜖th&tidle;r 1 + 2&tidle;r2 − 2 − 2&tidle;r 1 − 4Φc + 2&tidle;r2 + 2&tidle;r3 1 − 3Φc + 4&tidle;r2 ( c) 3 (c c2) c c c 1- 6- -&tidle;r1- 9Φc-&tidle;r1 + 3 1 − 5Φc − 3r&tidle;3c 1 − 3Φc + 5 &tidle;r2c , (11.30 )
where
------------(&tidle;r21∕3&tidle;r2c)Φc-+-1∕(mA-&tidle;r1)------------- α ≡ 1 + (&tidle;r2∕4&tidle;r2)Φ + (m &tidle;r3Φ ∕3 &tidle;r2)[1 − (&tidle;r2∕2&tidle;r2)Φ ]. (11.31 ) 1 c c A 1 c c 1 c c
As long as mA r&tidle;1Φc ≫ 1, the parameter α is much smaller than 1.

In order to derive the above field profile we have used the fact that the radius &tidle;r1 is determined by the condition m2 [ϕ (&tidle;r1) − ϕA ] = Q &tidle;ρA A, and hence

[ ( ) ( ) ] 2 Δ-&tidle;rc 1Δ-&tidle;rc --1--- Δ-&tidle;rc ϕA − ϕB = − Q &tidle;ρAr&tidle;c &tidle;r 1 + Φc − 2 &tidle;r + m &tidle;r 1 − &tidle;r (1 − β) , (11.32 ) c c A c c
where β is defined by
3 2 2 2 β ≡ ----(mA-r&tidle;1Φc-∕3&tidle;rc)(&tidle;r1∕&tidle;rc)Φc----, (11.33 ) 1 + (mA r&tidle;31Φc ∕3&tidle;r2c) − (&tidle;r21∕4&tidle;r2c)Φc
which is much smaller than 1. Using Eq. (11.32View Equation) we obtain the thin-shell parameter
Δr&tidle;c ( 1Δ &tidle;rc) 1 ( Δr&tidle;c) 𝜖th = ---- 1 + Φc − ----- + ------ 1 − ---- (1 − β ). (11.34 ) &tidle;rc 2 r&tidle;c mA &tidle;rc r&tidle;c

In terms of a linear expansion of α, β,Δ &tidle;r ∕&tidle;r ,Φ c c c, the field profile (11.28View Equation) outside the body is

GMc ( GMc ) ϕ(&tidle;r) ≃ ϕB − 2Qe ff----- 1 + ----- , (11.35 ) &tidle;r &tidle;r
where the effective coupling is
[ ( ) ( ) ] Δ-&tidle;rc Δ-&tidle;rc --1--- Δ-&tidle;rc Qeff = 3Q &tidle;r 1 − &tidle;r + m &tidle;r 1 − 2 &tidle;r − Φc − α − β . (11.36 ) c c A c c
To leading-order this gives Qeff = 3Q [Δ &tidle;rc∕&tidle;rc + 1∕(mA &tidle;rc)] = 3Q 𝜖th, which agrees with the result (5.45View Equation) in the Minkowski background. As long as Δ &tidle;rc∕&tidle;rc ≪ 1 and 1∕(mA &tidle;rc) ≪ 1, the effective coupling |Qeff| can be much smaller than the bare coupling |Q|, even in a strong gravitational background.

From Eq. (11.26View Equation) the field value and its derivative around the center of the body with radius &tidle;r ≪ 1∕m A are given by

2Q &tidle;ρ &tidle;r ( m &tidle;r3Φ Φ &tidle;r2)− 1[ 1 Φ ] ϕ(&tidle;r) ≃ ϕA + -----Am-1&tidle;r- 1 + --A--12-c − --c21 1 + --(mA &tidle;r)2 + ----c---2 mAe [A 1 3&tidle;rc ] 4&tidle;rc 6 2(mA &tidle;rc) 3Q &tidle;ρAΦc r&tidle;2 6 + -2m2---- 1 − r&tidle;2 − (m--&tidle;r-)2- , (11.37 ) A[ (c A c ) ] ′ 2 2mA &tidle;r1 mA &tidle;r31Φc Φc&tidle;r21 −1 3Φc &tidle;r ϕ (&tidle;r) ≃ Qρ&tidle;A &tidle;rc --mA-&tidle;r1- 1 + ----2---− --2-- − -------2 -2. (11.38 ) 3e 3&tidle;rc 4&tidle;rc (mA &tidle;rc) &tidle;rc
In the Minkowski background (Φ = 0 c), Eq. (11.38View Equation) gives ϕ ′(&tidle;r) > 0 for Q > 0 (or ϕ′(&tidle;r) < 0 for Q < 0). In the strong gravitational background (Φc ⁄= 0) the second term in the square bracket of Eq. (11.38View Equation) can lead to negative ′ ϕ(&tidle;r) for Q > 0 (or positive ′ ϕ (&tidle;r) for Q < 0), which leads to the evolution of |ϕ(&tidle;r)| toward 0. This effects comes from the presence of the strong relativistic pressure around the center of the body. Unless the boundary conditions at &tidle;r = 0 are appropriately chosen the field tends to evolve toward |ϕ(&tidle;r)| = 0, as seen in numerical simulations in [349Jump To The Next Citation Point350Jump To The Next Citation Point] for the f (R) model (4.84View Equation). However there exists a thin-shell field profile even for ′ ϕ (r&tidle;) > 0 (and √ -- Q = − 1∕ 6) around the center of the body. In fact, the derivative ′ ϕ (&tidle;r) can change its sign in the regime 1∕mA < &tidle;r < &tidle;r1 for thin-shell solutions, so that the field does not reach the curvature singularity at ϕ = 0 [594Jump To The Next Citation Point].
View Image

Figure 10: The thin-shell field profile for the model V = M 6ϕ −2 with Φc = 0.2, Δ &tidle;rc∕&tidle;rc = 0.1, mA r&tidle;c = 20, and Q = 1. This case corresponds to 4 &tidle;ρA∕ρ&tidle;B = 1.04 × 10, −3 ϕA = 8.99 × 10, − 1 ϕB = 1.97 × 10 and −1 𝜖th = 1.56 × 10. The boundary condition of φ = ϕ∕ϕA at xi = &tidle;ri∕r&tidle;c = 10− 5 is φ (xi) = 1.2539010, which is larger than the analytic value φ (xi) = 1.09850009. The derivative φ′(xi) is the same as the analytic value. The left and right panels show φ (&tidle;r) for 0 < &tidle;r∕&tidle;r < 10 c and 0 < &tidle;r∕&tidle;r < 2 c, respectively. The black and dotted curves correspond to the numerically integrated solution and the analytic field profile (11.26View Equation) – (11.28View Equation), respectively. From [594Jump To The Next Citation Point].

For the inverse power-law potential 4+n −n V (ϕ) = M ϕ, the existence of thin-shell solutions was numerically confirmed in  [594Jump To The Next Citation Point] for Φc < 0.3. Note that the analytic field profile (11.26View Equation) was used to set boundary conditions around the center of the body. In Figure 10View Image we show the normalized field φ = ϕ∕ϕA versus &tidle;r∕&tidle;rc for the model V (ϕ ) = M 6ϕ−2 with Φc = 0.2, Δ &tidle;rc∕&tidle;rc = 0.1, mA &tidle;rc = 20, and Q = 1. While we have neglected the term V,ϕ relative to Q &tidle;ρA to estimate the solution in the region &tidle;r1 < &tidle;r < &tidle;rc analytically, we find that this leads to some overestimation for the field value outside the body (&tidle;r > &tidle;rc). In order to obtain a numerical field profile similar to the analytic one in the region &tidle;r > &tidle;rc, we need to choose a field value slightly larger than the analytic value around the center of the body. The numerical simulation in Figure 10View Image corresponds to the choice of such a boundary condition, which explicitly shows the presence of thin-shell solutions even for a strong gravitational background.


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