The influence of quintessence on the motion of a binary system in cosmology
aa r X i v : . [ a s t r o - ph . C O ] M a r The influence of quintessence on the motion of a binary system in cosmology
Fei Yu, ∗ Molin Liu, † and Yuanxing Gui ‡ School of Physics and Optoelectronic Technology,Dalian University of Technology, Dalian, 116023, P. R. China
We employ the metric of Schwarzschild space surrounded by quintessential matter to study thetrajectories of test masses on the motion of a binary system. The results, which are obtained throughthe gradually approximate approach, can be used to search for dark energy via the difference of theazimuth angle of the pericenter. The classification of the motion is discussed.
PACS numbers: 04.62.+v, 04.20.Cv, 97.60.LfKeywords: quintessence; trajectories of test masses.
I. INTRODUCTION
Several astrophysical observational data have shown that our universe is undergoing an era of accelerated expansion[1–3]. Therefore, in order to explain this bizarre phenomenon, various models of cosmology have been put forward,ranging from the simplest one of a cosmological constant to the scalar field theory of dark energy and modifiedgravitational theory as well [4–6]. On the other hand, taking dark energy for the presumed cosmological componentis also used, combined with the Einstein field equations, to deal with some local gravitational issues [7, 8], such asthe relevant features of black holes. Recently, Kiselev has brought forward new static spherically symmetric exactsolutions of the Einstein equations, either for a charged or uncharged black hole surrounded by quintessential matteror free from it, which satisfy the condition of the additivity and linearity [9]. Otherwise, using a linear post-Newtonianapproach, Kerr et al. considered the orbital differential equations for test bodies of a binary system in the Kerr-deSitter spacetime and gave the elliptically orbital solution [10]. The solutions for parabolic and hyperbolic orbits canbe obtained via the formulae of [11]. In this paper, we study the trajectories of test masses in a binary system underthe general metric mentioned in [9]. Four discrete theoretical values of the state parameter ω from 0 to − , − , − , − ) and make G , ~ and c equal to unity. II. THE METRIC FOR SCHWARZSCHILD SPACE SURROUNDED BY THE QUINTESSENTIALMATTER
To begin with, Kiselev’s work [9] is reviewed briefly. The interval of a spherically symmetric static gravitationalfield is ds = e ν dt − e µ dr − r (cid:0) dθ + sin θdϕ (cid:1) , (1)where ν and µ are functions of r . Due to the condition of the additivity and linearity, which implies µ + ν = 0, theenergy-momentum tensor of quintessence can be written as T tt = T rr = ρ q , (2) T θθ = T ϕϕ = − ρ q (3 ω + 1) , (3) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] where ρ q is the density of the quintessential matter while ω is the state parameter. By setting µ = − ln(1 + f ), we get r f ′′ + 3(1 + ω ) rf ′ + (3 ω + 1) f = 0 , (4)whose solutions are of the forms f q = λr ω +1 , (5) f BH = − r g r , (6)where λ and r g are the normalization factors. Then, according to the relation ρ q = λ ωr ω ) , (7) λ should be negative for quintessence because the density of energy is positive. As a resulting, the metric for thespace surrounded by the quintessential matter can be expressed by ds = (cid:18) − Mr − λr ω +1 (cid:19) dt − (cid:18) − Mr − λr ω +1 (cid:19) − dr − r (cid:0) dθ + sin θdϕ (cid:1) , (8)where M is the black hole mass and r g = 2 M . Obviously, it can be reduced to the Schwarzschild and Schwarzschild-deSitter metrics by λ = 0 and ω = −
1, respectively.
III. TRAJECTORY OF THE TEST MASS AND THE MOTION OF A BINARY SYSTEM
In relativistic dynamics, the contravariant 4D momentum p µ is defined as p µ = m dx µ dτ , where m is the rest mass.Then the covariant momentum, which is more important in dynamics, can be introduced by p µ = g µν p ν . Whenparticles move in the gravitational field, dynamical conserved quantities are determined by the space-time symmetryof the metric field. From the metric (8), we may see clearly that t and ϕ are both cyclic coordinates; therefore,conserved quantities of the test mass are the t and ϕ components of p µ , i.e. p = m (cid:18) − Mr − λr ω +1 (cid:19) dtdτ = const. , (9) p = − mr sin θ dϕdτ = const. . (10)Due to the symmetry, the test mass moves on the symmetric plane formed by the initial velocity vector (3D) andcenter of force. We regard the direction vertical to the orbital plane as the polar axis. Then the motion of the testmass satisfies θ = π , dθdτ = 0 . (11) p and p are conserved quantities written in the form (cid:18) − Mr − λr ω +1 (cid:19) dtdτ = E, (12) r dϕdτ = L, (13)where E and L are constants of integration that represent energy and angular momentum per unit mass, respectively.Furthermore, the normalization condition for the 4D velocity, g µν u µ u ν = 1, also provides the equation (cid:18) − Mr − λr ω +1 (cid:19) (cid:18) dtdτ (cid:19) − (cid:18) − Mr − λr ω +1 (cid:19) − (cid:18) drdτ (cid:19) − r (cid:18) dϕdτ (cid:19) = 1 . (14)Equations (11) to (14) are four first integrals of the geodesic equation, which compose a set of self-contained differentialequations of particle dynamics.After being cleared up, these equations can lead to the relativistic extension of the Binet formula of Newtonianmechanics, d udϕ + u = ML + 3 M u + λ (3 ω + 1)2 L u ω + λ (3 ω + 3)2 u ω +2 , (15)where drdϕ = 0, because the orbit is not a circle and the second term on the right-hand side is a relativistic correction,while the last two are quintessential contributions. We have introduced the dimensionless variable u = GM/r , andfor convenience u = 1 /r has been used in the process of the calculation.Note that u = GM/r is a small quantity. When ω < u also appears in the denominator. So we now evaluate themagnitude of the last two terms of Eq. (15). First, we make the transformation α = − ω and β = λ /α ; then Eq.(15) turns to d udϕ + u = ML + 3 M u + 1 − α L (cid:18) βu (cid:19) α + 3 − α (cid:18) βu (cid:19) α u . (16)The transformation parameter α satisfies α ∈ [0 , ω ∈ [ − , β , we know that the state parameter ω = − β ∼ Λ / . From recent cosmological observations, themagnitude of the cosmological constant is Λ ≈ H /c ≈ − m − with H ≈
70 km s − Mpc − . As a result, themagnitude of β is β ∼ (10 − ) / < − . Furthermore, due to the definition u = GM/r , we have the expression βu = βrGM . (17)By way of a practical evaluation, we take the orbital radius of Mercury, r = 5 × m, which is the smallest in thesolar system and for the other parameters we get GM = 1 . × m. As a result, Eq. (17) yields βu ≈ − × × − . × ≈ . × − ≪ u ∼ − . (18)Furthermore, from the Binet equation in Newtonian mechanics, d udϕ + u = ML , (19)whose solution is u ( ϕ ) = ML (1 + e cos ϕ ) , (20)we find that M/L ∼ u . Therefore, when α > ω <
0) the last two terms of Eq. (15) are far less than u and M/L and can be regarded as small corrections.According to the linear perturbation scheme put forward in [10], the last three terms could be treated as pertur-bations being a function of ϕ by substituting for u the unperturbed solution. Due to the supernova’s dimming, thedark energy should satisfy ω ≤ − / ω ∈ [ − , A. case of ω = 0 When ω = 0, Eq. (15) reduces to d udϕ + u = M + λ L + (cid:18) M + 3 λ (cid:19) u , (21)where u is a small quantity, resulting in a slight correction of u . So we ignore it in the first place, and we find thezeroth-order approximate solution u = M + λ L (1 + e cos ϕ ) . (22)We presume that the bound orbit is an ellipse with eccentricity e <
1. Via substituting u into the right-hand side ofEq. (21), we can get the first-order approximate solution u = M + λ L ! " e cos ϕ + 3 (cid:18) M + λ (cid:19) M + λ L ! eϕ sin ϕ , (23)where small higher-order quantities, which are impossible to measure very accurately, are neglected for their insignif-icant effects. Furthermore, Eq. (22) means that ( M + λ ) /L ∼ u , and in the weak-field regime, M ≪ r . Introducing ε ≡ M + λ )[( M + λ ) /L ], namely ε ≪
1, makes cos εϕ ∼ εϕ ∼ εϕ . Then Eq. (23) reduces to u ( ϕ ) ≈ M + λ L ! [1 + e cos (1 − ε ) ϕ ] . (24)When considering precession, we may get the difference of the azimuth angle of the pericenter △ ϕ = 6 π (cid:18) ML (cid:19) + 6 π (cid:18) λM + λ / L (cid:19) , (25)where we ignore a quantity of ε , and the second term on the right-hand side belongs to quintessence. B. case of ω = − / When ω = − /
3, Eq. (15) reduces to d udϕ + (1 − λ ) u = ML + 3 M u . (26)Similar to the case of ω = 0, the zeroth-order approximate solution is u = ML h e cos (cid:16) ϕ √ − λ (cid:17)i ; (27)then the first-order approximate solution reads u = ML (cid:20) e cos (cid:16) ϕ √ − λ (cid:17) + 3 M L √ − λ eϕ sin (cid:16) ϕ √ − λ (cid:17)(cid:21) . (28)Introducing ε ≡ M L √ − λ makes Eq. (28) reduce to u ( ϕ ) = ML n e cos h(cid:16) √ − λ − ε (cid:17) ϕ io . (29)Furthermore, the difference of the azimuth angle is △ ϕ = 6 π M L (1 − λ ) / + 2 π (cid:18) − √ − λ √ − λ (cid:19) . (30)Here, ε is ignored, and quintessence also affects the first term, because the zeroth-order approximate solution ischanged for the component ϕ . C. case of ω = − / When ω = − /
3, Eq. (15) reduces to d udϕ + u = (cid:18) ML + λ (cid:19) + 3 M u − λ L u . (31)Obviously, the form turns out to be complicated, because u appears in the denominator. But as mentioned before, λ ∼ Λ, so the last two terms are both perturbation ones. Here we could deal with it by the linear perturbation schemein [10], where the motion of test bodies in the Kerr-de Sitter spacetime was studied. Substituting the zeroth-orderapproximate solution u = p (1 + e cos ϕ ) with p = (cid:0) ML + λ (cid:1) − for u into the r.h.s. of Eq. (31), we can divide theresulting linear equation into two parts: d u dϕ + u = 3 M p − (1 + e cos ϕ ) , (32) d u dϕ + u = p − − λp L (1 + e cos ϕ ) . (33)After being transformed, Eq. (33) can be integrated once to the form ddξ h(cid:0) − ξ (cid:1) − U i = − qe (1 − ξ ) / (1 + eξ ) + C (1 − ξ ) / , (34)where ξ = cos ϕ , U = u − p − , q = − λp L and C is a constant of integration. If once more we integrate it, usingformulae (2.264), (2.266) and (2.269) given in [15], then we get U = q − e (cid:20) (1 + e Ψ sin ϕ ) − e cos ϕ (cid:21) + C cos ϕ + S sin ϕ, (35)where S is a constant of integration and Ψ = 1 √ − e arcsin (cid:18) e + cos ϕ e cos ϕ (cid:19) . (36)Combining with a particular solution of Eq. (32), the final solution for this case is u = U + p − + 3 M p − eϕ sin ϕ ,which can be rewritten in the form u ( ϕ ) ≈ q − e (cid:20) (1 + e Ψ sin ϕ ) − e cos ϕ (cid:21) + (cid:18) ML + λ (cid:19) { e cos [(1 − ε ) ϕ ] } , (37)with ε ≡ M (cid:0) ML + λ (cid:1) introduced. Here u may reduce to the result of Schwarzschild case [16] as λ → D. case of ω = − When ω = −
1, Eq. (15) reduces to d udϕ + u = ML + 3 M u − λL u . (38)The unperturbed solution is u = p (1 + e cos ϕ ) with p = (cid:0) ML (cid:1) − . Similar to the case of ω = − /
3, according to theresult (49) in [10] the approximate solution is u ( ϕ ) ≈ q − e ) (cid:20) e Ψ sin ϕ ) − − e e cos ϕ − (cid:18) e + 1 e (cid:19) cos ϕ (cid:21) + ML { e cos [(1 − ε ) ϕ ] } , (39)where q = − λp /L , ε ≡ M /L and Ψ is represented by Eq. (36). When λ →
0, this result reduces to theSchwarzschild solution. It is different from the two former cases in that the difference of the azimuth angle of thepericenter hardly comes out when ω = − / − V r FIG. 1: The effective potential V for timelike geodesics for variable L with λ = 0 and M = 1. Solid line for L >
4, dash linefor L = 4, dash-dot line for L = 2 √ IV. DISCUSSION AND CONCLUSION
Here, we turn to a discussion of the classification of the motion. First, we introduce the relativistic effective potential V = (cid:18) − Mr − λr ω +1 (cid:19) (cid:18) L r (cid:19) . (40)Thus, the radial equation derived from (14) can be rewritten as (cid:0) drdτ (cid:1) = E − V , which implies that the possiblemotions only occur when E > V . Expand Eq. (40), V = 1 − Mr + L r − M L r − λr ω +1 − λL r ω +3 . (41)Because the effect of λ on V is very small, the curves of V are almost the same as that in the Schwarzschild space[16]. So we illustrate the effective potential V for variable L with λ = 0 in Fig. 1.From Fig. 1, we see that for L >
4, there is a potential barrier whose peak value is larger than 1 when r is verysmall, while there is also a potential well when r is large. Thus, the possible motions can be classified as three kinds E < , the bound state;1 E < V , the scattering state; E > V , the absorbing state . With decreasing L , the central potential barrier decreases in altitude. In the range of 2 √ L V
1, whichindicates that the appearance of the scattering state is impossible, with the bound and absorbing states being leftover. If
L < √
3, both the peak and the hollow of the effective potential will disappear, and only the absorbing stateis left over. But note that the above analysis is based on one assumption that the radius of the gravitation source isso small that the exterior of the source is applicable for very small r .In summary, we have studied the trajectories of test bodies on the motion of a binary system in the presence ofquintessence. There are four cases of the state parameter ω of quintessence, and we obtain four corresponding orbitalequations, where it is assumed that the test mass moves round the center of force with eccentricity e <
1. Thedifference of the azimuth angle of the pericenter can be cast in formulae in the two former cases, while the other twocomplicated ones can hardly be done, considering their different forms. Moreover, the effect caused by dark energyis too small to be detectable in the solar system. But the common feature of our four cases is that the resultantsolutions reduce to the outcomes in Schwarzschild space naturally as the parameter λ vanishes. Acknowledgments