Dark energy accretion onto black holes

DARK ENERGY ACCRETION ONTO BLACK HOLES E.O. BABICHEV, V.I. DOKUCHAEV AND Yu.N. EROSHENKO Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, 117312 Moscow, Russia Solution for a stationary spherically symmetric accretion of the relativistic perfect fluid with an arbitrary equation-of-state p(ρ) onto the Schwarzschild black hole is presented. It is shown that accretion of phantom energy is accompanied with a gradual decrease of the black hole mass. Masses of all black holes tend to zero in the phantom energy universe approaching to the Big Rip.

1

Introduction

Our Universe apparently undergoes a period of accelerated expansion. In the framework of General Relativity this means that a considerable part of the cosmological density consists of dark energy component with a negative pressure 1 . There are several theoretical candidates for a dark energy in the universe: cosmological constant (vacuum energy), dynamically evolving fields such as quintessence 2 or a more general k-essence 3 and geometrical dark energy (deviation from General Relativity including extra-dimensional generalizations). The dynamical dark energy models seem to be more realistic in resolving the problem of fine-tuning as they admit to construct “tracker” 4 or “attractor” 3 solutions without addressing to the antropic principle. One of the peculiar feature of the cosmological dark energy is a possibility of the Big Rip scenario 5 : the infinite expansion of the universe during a finite time. The Big Rip scenario is realized if a dark energy is in the form of the phantom energy with ρ + p < 0. In this case the Big Rip scenario provides the unrestrained growing of the cosmological phantom energy density grows and as a result the disruption of all bound objects up to subnuclear scale. Note, however that the condition ρ + p < 0 is not enough for the realization of the Big Rip 6 . A model independent analysis of the supernova data 7 showed that a presence of the phantom energy with an equation-of-state −1.2 < w < −1 is preferable in the present moment of time. Some analogy between phantom and QFT in curved space-time may be traced 8 . The entropy of the universe with phantom energy is discussed in 9 . On quantum level the phantom dark energy can be described as an unstable scalar ghost field. Possible ultraviolet stabilization of phantom field may be achieved e. g. by combining the scalar and vector fields 10 . Nevertheless the physical origin of dark energy is still controversial. Usually the evolution of quintessence or k-essence are considered in view of cosmological problems. However in the presence of compact objects such as black holes, the behavior of dark energy should be sufficiently different from that in the cosmological consideration. Indeed, what would be the fate of black holes in the universe filled with the phantom energy and approaching to the Big Rip? In contrast to the ordinary bound objects, the black holes cannot be disrupted in any classical processes. Recently we showed that all black holes gradually decrease their

masses and finally disappear very near the Big Rip 11 . Here we study in details the stationary accretion of dynamical dark energy into the black hole. As a model of dark energy we consider the relativistic perfect fluid with a negative pressure. The regular investigation of an accretion of perfect fluid on the compact objects is started since a formulation of the Bondi problem in the newtonian approximation 12 . The corresponding relativistic formulation of the perfect fluid accretion problem onto the Schwarzschild black hole were made by Michel 13 for the case of a polytropic equation-of-state. Below we describe the solution for a stationary accretion of the relativistic perfect fluid with an arbitrary equation-ofstate p(ρ) onto the Schwarzschild black hole. Using this solution we show that the black hole mass diminishes by accretion of the phantom energy. Masses of all black holes gradually tend to zero in the phantom energy universe approaching to the Big Rip. The diminishing of a black hole mass is caused by the violation of the energy domination condition ρ + p ≥ 0 which is a principal assumption of the classical black hole ‘non-diminishing’ theorems 14 . The another consequence of the existence of a phantom energy is a possibility of traversable wormholes 15 . In 16,17,18,19 authors studied the accretion of scalar quintessence field onto black hole, using the specific quintessence potentials V (φ) for obtaining the analytical solutions for black hole mass evolution. We use here an essentially different approach for the description of dark energy accretion onto a black hole. Namely, we model the dark energy by perfect fluid with a negative pressure. 2

General equations

Let us consider the spherical accretion of dark energy onto a black hole. We assume that the dark energy density is sufficiently low, so that the space-time geometry can be described by the Schwarzschild metric (i. e. the test field approximation). We model the dark energy by a perfect fluid with energy-momentum tensor: Tµν = (ρ + p)uµ uν − pgµν , where ρ is the density and p is the pressure of the dark energy and uµ is the four-velocity uµ = dxµ /ds of the fluid. The integration of the time component of the energy-momentum conservation law T µν;ν = 0 gives the first integral of motion 1/2  2 x2 u = C1 , (1) (ρ + p) 1 − + u2 x where x = r/M , u = dr/ds and C1 is a constant determined below. Given the equation-of-state p = p(ρ), one can introduce the function n by the relation: dρ dn = . ρ+p n

(2)

The function n plays a role of an effective particle number density (‘concentration’) in the fluid. At the same time, one can use n for the formal description of the fluid without correlation with any physical particles. In this case n is an auxiliary function. For a general equation-of-state, p = p(ρ), we obtain from (2) the following solution for n: 

n(ρ) = exp  n∞

Zρ

ρ∞



dρ′ , ρ′ + p(ρ′ )

(3)

From the conservation of energy-momentum along the velocity uµ T µν;ν = 0, using (3) we obtain the another first integral: n(ρ) 2 ux = −A, (4) n∞

where n∞ (an effective ‘concentration’ of the dark energy at the infinity) was introduced formally for convenience. In the case of inflow u = (dr/ds) < 0, and the constant A > 0. From (1) and (4) one can easily obtain 1/2  2 ρ+p = C2 , (5) 1 − + u2 n x where ρ∞ + p(ρ∞ ) . (6) C2 = n(ρ∞ ) Let us now calculate the radial 4-velocity component and the fluid density on the event horizon of the black hole, r = 2M . Setting x = 2, we obtain from Eqs. (4), (5) and (6) A ρH + p(ρH ) n2 (ρH ) = 2 , 4 ρ∞ + p(ρ∞ ) n (ρ∞ )

(7)

where ρH is the density on the x = 2 horizon. Thus, having specified the density at infinity ρ∞ , the equation of state p = p(ρ), and the flux A and using definition (2) of the concentration, we can calculate the fluid density ρH on the event horizon of the black hole from (7). Given the density on the horizon ρH , we can easily determine the radial fluid 4-velocity component on the horizon from (4): A n(ρ∞ ) . (8) uH = − 4 n(ρH ) We will see below that the constant A which determines the flux is fixed for fluids with ∂p/∂ρ > 0. This can be done through finding of the critical point. Following Michel 13 we obtain the parameters of critical point: u2∗ 1 , V∗2 = , (9) u2∗ = 2x∗ 1 − 3u2∗ where V2 =

n d(ρ + p) − 1. ρ + p dn

(10)

From this by using (2) it follows that V 2 = c2s (ρ), where c2s = ∂p/∂ρ is the squared effective speed of sound in the media. Combining (5), (6), (9) and (10) we find the following relation: i1/2 ρ + p(ρ ) ρ∗ + p(ρ∗ ) h ∞ ∞ = 1 + 3c2s (ρ∗ ) , n(ρ∗ ) n(ρ∞ )

(11)

which gives the ρ∗ for an arbitrary equation-of-state p = p(ρ). Given ρ∗ one can find n(ρ∗ ) using (3) and values x∗ , u∗ , using (9) and (10). Then substituting the calculated values in (4) one can find the constant A. Note that there is no critical point outside the black hole horizon (x∗ > 1) for c2s < 0 or c2s > 1. This means that for unstable perfect fluid with c2s < 0 or c2s > 1 a dark energy flux onto the black hole depends on the initial conditions. This result has a simple physical interpretation: the accreting fluid has the critical point if its velocity increases from subsonic to trans-sonic values. In a fluid with a negative c2s or with c2s > 1 the fluid velocity never crosses such a point. It should be stressed, however, that fluids with c2s < 0 are hydrodynamically unstable (see discussion in 21,22 ). The Eq. (5) together with (3) and (4) describe the requested accretion flow onto the black hole. These equations are valid for perfect fluid with an arbitrary equation-of-state p = p(ρ), in particular, for a gas with zero-rest-mass particles (thermal radiation) and for a gas with nonzero-rest-mass particles. For a nonzero-restmass gas the couple of equations (4) and (5) is reduced to similar ones found by Michel 13 . One would note that the set of equations (3), (4) and (5) are also correct in the case of dark energy and phantom energy ρ + p < 0. In this case concentration n(ρ) is positive for any ρ and constant C2 in (5) is negative.

The black hole mass changes at a rate M˙ = −4πr 2 T0r due to the fluid accretion. With the help of (4) and (5) this can be expressed as M˙ = 4πAM 2 [ρ∞ + p(ρ∞ )].

(12)

It follows from (12) that a black hole mass increases as it accretes e. g. a gas of particles when p > 0, but decreases as it accretes the phantom energy when p + ρ < 0. In particular, this implies that the black hole masses in the universe filled with phantom energy must decrease. This result is general. It does not depend on the specific form of the equation-of-state p = p(ρ); only the satisfaction of the condition p + ρ < 0 is important. The physical cause of the decrease in the black hole mass is as follows: the phantom energy falls to the black hole, but the energy flux associated with this fall is directed away from the black hole. If we ignore the cosmological evolution of the density ρ∞ , then we find the law of change in the black hole mass from (12) to be   t −1 , (13) M = Mi 1 − τ where Mi is the initial mass of the black hole, and τ is the evolution time scale τ = 1/ {4πAMi [ρ∞ + p(ρ∞ )]} . 3

(14)

Model with a Linear Equation-of-State

Let us consider the model of dark energy with a general linear equation-of-state p = α(ρ − ρ0 ),

(15)

where α and ρ0 are constants. Among the other cases, this model describes an ultrarelativistic gas (p = ρ/3), a gas with an ultra-hard equation-of-state (p = ρ), and the simplest model of dark energy (ρ0 = 0 and α < 0). The quantity α is related to the parameter w = p/ρ of the equation-of-state by w = α(ρ − ρ0 )/ρ. An equation-of-state with w = const < 0 throughout the cosmological evolution is commonly used to analyze cosmological models. The matter with such an equation-of-state is hydrodynamically unstable and can exist only for a short period. Our equation-of-state (15) for α > 0 does not have this shortcoming. For α > 0, it also allows the case of hydrodynamically stable phantom energy to be described, which is not possible when using an equation-of-state with w = const < −1. In the real Universe, the equation-of-state changes with time (i. e., w depends on t). Therefore, Eq. ((15) has a physical meaning of an approximation to the true equation-of-state only in a limited ρ range. From the physical point of view, the condition ρ > 0 must be satisfied for any equation-of-state in a comoving frame of reference. In particular, the state of matter with ρ = 0, but p 6= 0, is physically unacceptable. The corresponding constraints for the equation-of-state (15) are specified by conditions (23) and (24) given below. For α < 0, there is no critical point for the accreted fluid flow. For α > 0, using (9) we obtain the parameters at the critical point x∗ =

1 + 3α , 2α

u2∗ =

α . 1 + 3α

(16)

Note that the parameters of the critical point (16) in the linear model (15) are determined only by ∂p/∂ρ = α and do not depend on ρ0 , which fixes the physical nature of the fluid under consideration: a relativistic gas, dark energy, or phantom energy. Note also that no critical point exists beyond the event horizon of the black hole for α > 1 (this corresponds to a nonphysical

situation with a superluminal speed of sound). Let us calculate the constant A, which defines the energy flux onto the black hole. We find from (3) that 1/(1+α)



ρ n eff = n∞ ρeff,∞

,

(17)

where we introduce an effective density ρeff ≡ ρ + p = −ρ0 α + (1 + α)ρ. Using (11) we obtain ρeff∗ ρeff,∞

!α/(1+α)

= (1 + 3α)1/2 ,

(18)

where ρeff∗ and ρeff,∞ , are the effective densities at the critical point and at infinity, respectively. Substituting (18) into (17) and using (4), we obtain for the linear model (1 + 3α)(1+3α)/2α . (19) 4α3/2 It is easy to see that A ≥ 4 for 0 < α < 1. A = 4 for α = 1 (this corresponds to cs = 1); i.e., the constant A is on the order of 1 for relativistic speeds of sound. Using (19), we obtain from (14) A=

"

(1 + 3α)(1+3α)/2α τ = πMi (ρ∞ + p∞ ) α3/2

#−1

.

(20)

To determine the fluid density on the event horizon of the black hole, we substitute (17) into (7) to yield    (1+α)/(1−α) αρ0 A αρ0 + ρ∞ − , (21) ρH = 1+α 1+α 4 where A is given by (19). For 0 < α < 1, the effective density on the horizon ρeff,H cannot be lower than ρeff,∞ . A radial 4-velocity component on the horizon can be found from (8) and (21): A uH = − 4 

−α/(1−α)

,

(22)

The value of uH changes from 1 to 1/2 for 0 < α < 1. A linear model (15) describes the phantom energy when ρ∞ /ρ0 < α/(1 + α). In this case, ρ + p < 0. However, the requirement that the density ρ be nonnegative should be taken into account. This parameter can formally be negative in the range 0 < α ≤ 1. Such a nonphysical situation imposes a constraint on the linear model ((15) under consideration. For a physically proper description of the accretion process, we must require that the density ρ be nonnegative. We obtain the following constraint on the validity range of the linear model from (21) for hydrodynamically stable phantom energy: "

α 1− 1+α



A 4

−(1+α)/(1−α) #

<

α ρ∞ . < ρ0 1+α

(23)

As follows from (23), at a given α, we can always choose the parameters ρ0 and ρ∞ in such a way that ρ > 0 for any r > 2M . On the other hand, the model (15) describes the quintessence (not the phantom energy) for the entire r range only if p < 0. Consequently, a physically proper description of the quintessence can be obtained from (21) if "

ρ∞ α 1 α < + < 1+α ρ0 1+α α



A 4

−(1+α)/(1−α) #

.

(24)

For some specific choices of α in linear model (more specifically, for α = 1/3, 1/2, 2/3, and 1) and also in the case of Chaplygin Gas with p = −α/ρ, the corresponding radial functions ρ(x) and u(x) can be calculated analytically 11 .

4

Dark Energy Accretion onto a Moving and Rotating Black Hole

Let us consider the accretion onto a moving and rotating black hole in the special case of a linear equation-of-state with α = 1. The condition α = 1 allows an exact analytical expression to be derived for the accretion rate of dark energy onto a black hole. For α = 1, we easily find from (17) that 1/2



ρ n eff = n∞ ρeff,∞

.

(25)

We have the continuity equation for the particle number density (nuµ );µ = 0. Let us introduce the scalar field φ in terms of which the fluid velocity can be expressed as follows (there is no torsion in the fluid): ρ+p uµ = φ,µ . (26) n Using (25) and (26) we derive an equation for the auxiliary function φ: φµ;µ = 0.

(27)

Exactly the same equation arises in the problem of the accretion of a fluid with the equationof-state p = ρ. Thus, we reduce the problem of a black hole moving in dark energy with an equation-of-state p = ρ − ρ0 to the problem of a fluid with an extremely hard equation-of-state, p = ρ. Using the method suggested in 23 , we obtain the mass evolution law for a moving and rotating black hole immersed in dark energy with the equation-of-state p = ρ − ρ0 : 2 M˙ = 4π(r+ + a2 )[ρ∞ + p(ρ∞ )]u0BH ,

(28)

where r+ = M + (M 2 − a2 )1/2 is an event horizon radius of a rotating black hole, a = J/M is a specific angular momentum of the black hole (rotation parameter), and u0∞ is the zeroth 4-velocity component of the black hole relative to the fluid. Expression (28) for u0BH = 0 reduces to (12) for a Schwarzschild (a = 0) black hole at rest. 5

Black Holes in the Big Rip Universe

Now we turn to the problem of the black hole evolution in the universe with the Big Rip when a scale factor a(t) diverges at finite time 5 . For simplicity we will take into account only dark energy and will disregard all others forms of energy. The Big Rip solution is realized for in the linear model (15) for ρ + p < 0 and α < −1. From the Friedman equations for the linear equation-of-state model one can obtain: |ρ + p| ∝ a−3(1+α) . Taking for simplicity ρ0 = 0 we find the evolution of the density of a phantom energy in the universe: ρ∞ = ρ∞,i

t 1− τ



−2

,

(29)

where

1/2 3(1 + α) 8π (30) ρ∞,i τ =− 2 3 and ρ∞,i is an initial density of the cosmological phantom energy and the initial moment of time is chosen so that the ‘doomsday’ comes at time τ . From (29) and (30) it is easy to see that the Big Rip solutionis realized for α ≡ ∂p/∂ρ < −1. In general, the satisfying the condition ρ + p < 0 is not enough for the possibility for Universe to come to Big Rip. From (12) using (29) we find the black hole mass evolution in the universe coming to the Big Rip:



−1



M = Mi 1 +

t Mi ˙ τ − t M0 τ



−1

,

(31)

where

M˙ 0 = (3/2) A−1 |1 + α|,

(32)

and Mi is the initial mass of the black hole. For α = −2 and typical value of A = 4 (corresponding to uH = −1) we have M˙ 0 = 3/8. In the limit t → τ (i.e. near the Big Rip) the dependence of black hole mass on t becomes linear, M ≃ M˙ 0 (τ − t). While t approaches to τ the rate of black hole mass decrease does not depend on both an initial black hole mass and the density of the phantom energy: M˙ ≃ −M˙ 0 . In other words masses of all black holes in the universe tend to be equal near the Big Rip. This means that the phantom energy accretion prevails over the Hawking radiation until the mass of black hole is the Planck mass. However, formally all black holes in the universe evaporate completely at Planck time before the Big Rip due to Hawking radiation. 6

Scalar field accretion

In remaining let us confront our results with the calculations of (not phantom) scalar field accretion onto the black hole 16,17,18,19 . The dark energy is usually modelled by a scalar field φ with potential V (φ). The perfect fluid approach is more rough because for given ’perfect fluid variables’ ρ and p one can not restore the ’scalar field variables’ φ and ∇φ. In spite of the pointed difference between a scalar field and a perfect fluid we show below that our results are in a very good agreement with the corresponding calculations of a scalar field accretion onto the black hole. The standard-form Lagrangian of a scalar field is L = K − V , where K is a kinetic term of a scalar field φ and V is a potential. For the standard choice of a kinetic term K = φ;µ φ;µ /2 the energy flux is T0r = φ,t φ,r . Jacobson 16 found the scalar field solution in Schwarzschild metric for the case of zero potential V = 0: φ = φ˙ ∞ [t + 2M ln(1 − 2M/r)], where φ∞ is the value of the scalar field at the infinity. In 18 it was shown that this solution remains valid also for a rather general form of runaway potential V (φ). For this solution we have T0r = −(2M )2 φ˙ 2∞ /r 2 and correspondingly M˙ = 4π(2M )2 φ˙ 2∞ . The energy-momentum tensor constructed from Jacobson solution completely coincides with one for perfect fluid in the case of ultra-hard equation-of-state p = ρ under the replacement p∞ → φ˙ 2∞ /2, ρ∞ → φ˙ 2∞ /2. It is not surprising because the theory of a scalar field with a zero potential V (φ) is identical to perfect fluid consideration 24 . In a view of this coincidence it is easily to see the agreement of our result (12) for M˙ in the case of p = ρ and the corresponding result of 16,18 . To describe the phantom energy the Lagrangian of a scalar field must have a negative kinetic term 5 , e. g., K = −φ;µ φ;µ /2 (for a more general case of negative kinetic term see 20 ). In this case the phantom energy flux onto black hole has the opposite sign, T0r = −φ,t φ,r , where φ is the solution of the same Klein-Gordon equation as in the case of standard scalar field, however with a replacement V → −V . For zero potential this solution coincides with that obtained by Jacobson 16 for a scalar field with the positive kinetic term. Lagrangian with a negative kinetic term and V (φ) = 0 does not describe, however, the phantom energy. At the same time, the solution for scalar field with potential V (φ) = 0 is the same as with a positive constant potential V0 = const, which can be chosen so that ρ = −φ˙ 2 /2 + V0 > 0. A corresponding scalar field represents the required accreting phantom energy ρ > 0 and p < −ρ and provides a black hole decreasing with the rate M˙ = −4π(2M )2 φ˙ 2∞ . A simple example of phantom cosmology without a Big Rip 25 is realized for a scalar field with the potential V = m2 φ2 /2, where m ∼ 10−33 eV. After short transition phase this cosmological model tends to the asymptotic state with H ≃ mφ/31/2 and φ˙ ≃ 2m/31/2 . In the Klein-Gordon equation the m2 term (with a mentioned replacement V → −V ) is comparable to other terms only at the cosmological horizon distance. This means that the Jacobson solution is valid for this case also. Calculating the corresponding energy flux one can easily obtain M˙ = −4π(2M )2 φ˙ 2∞ = −64M 2 m2 /3. For M0 = M⊙ and m = 10−33 eV the effective time of black hole mass decrease is τ = (3/64)M −1 m−2 ∼ 1032 yr.

Acknowledgments This work was supported in part by the Russian Foundation for Basic Research grants 04-0216757 and 06-02-16342, and the Russian Ministry of Science grant 1782.2003.2. References 1. N. Bahcall, J.P. Ostriker, S. Perlmutter and P.J. Steinhardt, Science 284, 1481 (1999); C.L. Bennett et al., Astrophys. J. Suppl. Ser. 148, 1 (2003). 2. C. Wetterich, Nucl. Phys. B 302, 668 (1988); P.J.E. Peebles and B. Ratra, Astrophys. J. 325, L17 (1988); A. Albrecht and C. Skordis, Phys. Rev. Lett. 84, 2076 (2000). 3. C. Armendariz-Picon, T. Damour and V. Mukhanov, Phys. Lett. B 458, 209 (1999); Takeshi Chiba, Takahiro Okabe and Masahide Yamaguchi, Phys. Rev. D 62, 023511 (2000). 4. I. Zlatev, L. Wang, P. Steinhardt, Phys. Rev. Lett. 82, 895 (1998); P. Steinhardt, L. Wang, I. Zlatev, Phys. Rev. D 59, 123504 (1999). 5. R.R. Caldwell, Phys. Lett. B 545, 23 (2002); R.R. Caldwell, M. Kamionkowski and N.N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003). 6. B. McInnes, JHEP 0208, 029 (2002); M. Bouhmadi-Lopez, J.A.J. Madrid, JCAP 0505, 005 (2005). 7. U. Alam, V. Sahni, T.D. Saini, A.A. Starobinsky, Mon. Not. Roy. Astron. Soc. 354, 275 (2004). 8. S. Nojiri, S.D. Odintsov, Phys. Lett. B 562, 147 (2003). 9. I. Brevik, S. Nojiri, S.D. Odintsov, L. Vanzo, Phys. Rev. D 70, 043520 (2004). 10. V.A. Rubakov, arXiv:hep-th/0604153. 11. E.O. Babichev, V.I. Dokuchaev and Yu.N. Eroshenko, Phys. Rev. Lett. 93, 021102 (2004); E.O. Babichev, V.I. Dokuchaev, and Yu.N. Eroshenko, JETP 100, 528 (2005). 12. H. Bondi, Mon. Not. Roy. Astron. Soc. 112, 195 (1952)). 13. F.C. Michel, Ap. Sp. Sc. 15, 153, (1972). 14. S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, (Cambridge University Press, England, 1973), Chapter 4.3. 15. M. Visser, S. Kar and N. Dadhich, Phys. Lett. B 90, 201102 (2003); P.F. Gonzalez-Diaz, Phys. Rev. D 68, 084016 (2003). 16. T. Jacobson, Phys. Lett. B 83, 2699 (1999). 17. R. Bean and J. Magueijo, Phys. Rev. D 66, 063505 (2002). 18. A. Frolov and L. Kofman, J. Cosmology Astrop. Phys. 5, 9 (2003). 19. W.G. Unruh, Phys. Rev. D 14, 3251 (1976); L.A. Urena-Lopez and A.R. Liddle, Phys. Rev. D 66, 083005 (2002); M.Yu. Kuchiev and V.V. Flambaum, Phys. Rev. D 70, 044022 (2004). 20. P.F. Gonzalez-Diaz, Phys. Lett. B 586, 1 (2004). 21. J.C. Fabris and J. Martin, Phys. Rev. D 55, 5205 (1997). 22. S.M. Carroll, M. Hoffman and M. Trodden, Phys. Rev. D 68, 023509 (2003). 23. L.I. Petrich, S.L. Shapiro and S.A. Teukolsky, Phys. Rev. Lett. 60, 1781 (1988). 24. V.N. Lukash, JETP 52, 5 (1980). 25. M. Sami and A. Toporensky, Mod. Phys. Lett. A 19, 1509 (2004).