A&A 507, 617620 (2009)
Constraints on the size of extra dimensions from the orbital evolution of the blackhole Xray binary XTE J1118+480
T. Johannsen
Physics Department, The University of Arizona, 1118 E. 4th Street, Tucson, AZ 85721, USA
Received 1 July 2009 / Accepted 9 September 2009
Abstract
Context. To constrain RandallSundrum type
braneworld gravity models and the expected rapid evaporation of
astrophysical black holes due to the emission of gravitational modes in
the extra dimension.
Aims. It is argued that the blackhole Xray binary
XTE J1118+480 is suitable for a constraint on the asymptotic
curvature radius of the extra dimension in such braneworld models.
An upper limit on the rate of change of the orbital period of
XTE J1118+480 is obtained.
Methods. The expected blackhole evaporation in the
extra dimension leads to a potentially observable rate of change of the
orbital period in XTE J1118+480. The timechange of the
orbital period is calculated from previous orbital period measurements
from the literature. The lack of observed orbital period evolution is
used to constrain the asymptotic curvature radius of the extra
dimension.
Results. The asymptotic AdS radius of curvature is
constrained to a value comparable to other limits from astrophysical
sources. A unique property of XTE J1118+480 is that the
expected rate of change of the orbital period due to magnetic braking
alone is so large that only one additional measurement of the orbital
period would lead to the first detection of orbital evolution in a
blackhole binary and impose the tightest constraint to date on the
size of one extra dimension of the order of
.
Key words: gravitation  black hole physics  Xrays: binaries  stars: individual: J1118+480  Xrays: stars
1 Introduction
The socalled hierarchy problem is one of the key open questions in the search for a grand theory beyond general relativity and the standard model. The fundamental scale of Einsteinian gravity, the Planck scale, differs from the electroweak scale by 16 orders of magnitude, thus making general relativity and the standard model hard to reconcile in a grand unification scheme. Braneworld gravity, as a candidate for a unified theory, requires the existence of more than three spatial dimensions (see Maartens 2004, for a review), but the size of a potential extra dimension has already been constrained to the submm range by precision tests of Newton's inverse square law (see, e.g., Adelberger et al. 2003, for a review).
One possible solution (ArkaniHamed et al. 1998) is to demand that extra dimensions be compactified at a scale smaller than those probed by experiment. All standardmodel particles are bound to the known fourdimensional world (``the brane''). Only gravity can access the extra dimensions, which accounts for its apparent weakness, because it is spread out through higherdimensional space (``the bulk''). This approach reconciles the gravitational and the electroweak scales if the extra dimensions are large enough, but it is beyond the scope of astrophysical tests.
A second scenario (Randall & Sundrum 1999) embeds the brane in a fivedimensional antide Sitter space, which allows for the extra dimension to be infinite. The bulk is filled with a negative cosmological constant such that the extra dimension only effects the brane on a length scale which is small enough and which is set by the asymptotic curvature radius L of the bulk. This model has striking consequences for astrophysical black holes.
Perturbative solution of the classical bulk equations (Tanaka 2003) indicated that no stable black holes can exist on the brane. A treatment of higherdimensional black holes via the AdS/CFT correspondence showed that black holes are indeed unstable and lose energy in the extra dimension through the emission of CFT modes (Emparan et al. 2003; see, however, Fitzpatrick et al. 2006). The lifetimes of astrophysical black holes are dramatically reduced and can be as short as a Megayear if the asymptotic curvature L of the extra dimension turns out to be in the submm range.
For a binary system consisting of a black hole and a companion star, this effect should be observable and lead to a measurable change of the orbital period (Johannsen et al. 2009). Several blackhole binaries were identified as candidates, and the system SXT A062000 yielded a constraint on the asymptotic curvature radius of at 3 (Johannsen et al. 2009). Similar limits have been obtained from the age of the black hole XTE J1118+480 ( ; Psaltis 2007) and from tabletop experiments of Newton's inverse square law (Adelberger et al. 2007; Geraci et al. 2008). The current 3upper limit on the AdS radius L is of the order of (Kapner et al. 20072007).
In this paper, I extend the analysis of Johannsen et al. (2009) to the blackhole binary XTE J1118+480 and compute an additional constraint on the asymptotic curvature radius L. In Sect. 2, I briefly review the results of Johannsen et al. (2009) for the evolution of the orbital period of a blackhole binary system with nonconservative mass transfer and blackhole evaporation. In Sect. 3, I apply this result to the blackhole binary J1118+480 and I obtain a constraint on the asymptotic curvature radius of in Sect. 4.
2 Blackhole binaries in braneworld gravity
For a binary system with a black hole of mass m_{1}
and a companion star of mass m_{2}
on a circular orbit, the orbital angular momentum,
,
is subject to change because of magnetic braking (e.g., Webbink
et al. 1983) and the evolution of the secondary star (e.g.,
Verbunt 1993). Here, ,
,
and a is the semimajor axis.
A third effect is the emission of CFT modes in the
extra dimension (Johannsen et al. 2009), where the blackhole
mass loss is given by (Emparan et al. 2003)
Here, L is the asymptotic AdS radius of curvature.
This leads to a change of the orbital period of (Johannsen
et al. 2009)
where , and c_{1}, c_{2}, c_{3}, as well as a_{0}, a_{1}, a_{2}, and a_{3} are constants depending on the core composition of the companion star. The parameter governs the strength of magnetic braking (cf. Eq. (36) in Rappaport et al. 1983). The additional quantities are defined by (Johannsen et al. 2009)
Here, P is the orbital period, is the fraction of matter that is accreted by the black hole, is the adiabatic index of the companion star, and is the specific angular momentum in units of that is lost through the stellar wind, which carries away angular momentum J at a rate (Will & Zaglauer 19891989)
(4) 
In this expression, is the angular momentum loss due to stellar wind, m_{1} and m_{2} are the masses of the black hole and the companion star, respectively, and their time derivatives, and q=m_{1}/m_{2} is the mass ratio.
Figure 1: The rate of change of the orbital period P of the binary systems J1118+480 and A062000 versus the asymptotic curvature radius L in the extra dimension. The parameters are , , , and . The transition from predominant magnetic braking (constant rate) to predominant blackhole evaporation (rapidly increasing rate) occurs at (A062000) and (J1118+480), respectively. 

Open with DEXTER 
Observations of the system A062000 have been used previously in conjunction with the above theoretical prediction to constrain the asymptotic curvature radius to a value of (Johannsen et al. 2009). In the following, I apply the formalism above to the system XTE J1118+480.
Figure 2: The rate of change of the orbital period P (in years) of the binary system J1118+480 versus the specific angular momentum removed by the stellar wind , the accretion parameter , and the magnetic braking parameter , for and . On varying one parameter, the others are held constant at the respective values , , and . 

Open with DEXTER 
3 The orbital evolution of XTE J1118+480
In order to obtain a constraint on the asymptotic curvature radius L of the extra dimension, it is essential to select a blackhole binary with an unevolved companion star. In that case, the evolution term in Eq. (2) can be neglected, and the binary can be used to constrain the AdS radius as long as the magnetic braking term is negligible compared to the evaporation term. In the following I argue that this approach can be applied to the blackhole binary J1118+480, and I use previous measurements of its orbital period to place a bound on the rate of change of its orbital period.
Table 1: The observed orbital periods and times of measurement for J1118+480.
The system J1118+480 has been monitored for more than a decade (Remillard & McClintock 2006). The companion star of the black hole resembles a latetype mainsequence star of spectral type K7 VM0 V (Wagner et al. 2001). In addition, the mean density is only 50% higher than for a usual mainsequence M0 star (McClintock et al. 2001), and the mass is only 50% lower than that of such a star (see Charles & Coe 2006).
It is important to note that this is not simply a normal star. It has emerged out of an exeptional evolutionary history (see de Kool et al. 1986, for an example), and it does not evolve on a nuclear timescale. For my analysis, however, it is sufficient that the secondary only behaves like a mainsequence star. Then the evolution term in Eq. (2) is negligible.
Considering only the evaporation term and the magnetic braking term in Eq. (2), I plot in Fig. 1 the rate of change of the orbital period versus the asymptotic curvature radius L in the extra dimension for the binary systems J1118+480 and A062000. The parameters used in this plot are , , , and . For values of the asymptotic curvature radius greater than the evaporation term dominates the evolution of the orbital period in the case of J1118+480. Below that value, the magnetic braking is predominant. For the binary A062000 the transition occurs at (Johannsen et al. 2009). Consequently, these sources are similar in constraining the asymptotic curvature radius of the extra dimension.
A measurement of a positive rate of change of the orbital period automatically constrains the asymptotic curvature radius to an interval, as can be seen by simply evaluating the magnetic braking term. For a binary system with a high mass ratio, i.e. m_{1}/m_{2}>5.5, the effect of magnetic braking can only decrease the orbital period (Johannsen et al. 2009). Thus such a measurement actually determines the AdS radius L and would prove the existence of an extra dimension.
In Fig. 2, I plot the rate of change of the orbital period of J1118+480 as a function of the parameters , , and . On varying one of them, the others are held constant at the respective values , , and . For all plots I choose and , which is the value of the current experimental limit (Kapner et al. 2007). I am interested in the smallest rate of orbital period evolution, so in the following I set the parameters to the respective values (no angular momentum loss due to stellar wind), (no accretion onto the black hole), and .
Figure 3: The times T_{0} of the inferior conjunction and residuals versus the orbital cycle number n for the blackhole binary J1118+480. 

Open with DEXTER 
Figure 4: The minimum rate of change of the orbital period P of the binary J1118+480 versus the asymptotic curvature radius L in the extra dimension. The intersection point of the observed 3upper limit on (horizontal line) with the minimum rate of change of the orbital period marks the constraint on the AdS curvature radius of (vertical line). 

Open with DEXTER 
The orbital period P of J1118+480 has been measured several times over the past decade. Based on those measurements, I calculate the rate of change of the orbital period. Table 1 shows the orbital period measurements for J1118+480. The values are for the most part consistent with each other, indicating an at most small change of the period over the last years.
Assuming a constant rate of change of the orbital period, the
time of the nth orbital cycle is given by
(e.g., Kelley et al. 1983)
where P is the orbital period at time , is its derivative, and n is the orbital cycle number. In order to calculate the rate of change of the orbital period, I fitted the times T_{0} in Table 1 using the IDL routine curvefit. The value from reference (2) was omitted, and instead the more precise reference (3) was used. The fit yields which is still consistent with zero quoting 3errors. The fit and residuals are shown in Fig. 3.
4 Results
Using the values of the orbital period derivative calculated in Sect. 3, I compute a limit on the asymptotic curvature radius of the extra dimension. For the mass of the black hole I use the best fit value of (cf. Charles & Coe 2006).
The 3upper limit on the observed rate of change of the orbital period and the smallest theoretical expectation thereof determine an upper bound on the AdS curvature radius L. In Fig. 4, I plot the rate of change of the orbital period of J1118+480 for the set of parameters that minimizes it (cf. Sect. 3). The intersection point of this curve with the 3upper limit on the orbital period evolution yields a constraint on the AdS curvature radius of .
This value is comparable to the constraint obtained from A062000 (Johannsen et al. 2009). A refinement of that upper bound would be easy to obtain and simply requires an additional measurement of the orbital period. The fact that the period change due to magnetic braking is so large for J1118+480 in the case where the blackhole evaporation is negligible ( ; cf. Fig. 1) has two important consequences. First, even one additional measurement of the orbital period in 2009 (n=17 000) with an error in the ephemeris T_{0} of the order of would prove the period derivative to be negative at , hence measuring the first orbital period evolution of a blackhole Xray binary. Second, such a measurement would show that blackhole evaporation is negligible against magnetic braking in this binary, which would in turn impose a constraint of the order of on the size of one extra dimension, the tightest constraint to date.
The constraint on the asymptotic curvature radius also depends on the mass of the primary, but since the blackhole mass has been measured quite precisely (m_{1}=6.8 ; from Charles & Coe 2006), its effect is small.
5 Conclusions
 1.
 The blackhole binary XTE J1118+480 is wellsuited for a constraint on the asymptotic curvature radius of the extra dimension in RandallSundrum type braneworld gravity models.
 2.
 An upper limit on the rate of change of the orbital period of XTE J1118+480 is calculated based on previous measurements of the orbital period from the literature. The lack of observed orbital evolution of this binary imposes a constraint on the asymptotic curvature radius of .
 3.
 This constraint can be significantly improved by only one additional measurement of the orbital period of XTE J1118+480. This would be the first detection of orbital evolution in a blackhole binary. The expected predominance of magnetic braking would provide the best constraint to date on the asymptotic curvature radius of the extra dimension of the order of .
I would like to thank Dimitrios Psaltis for carefully reading the manuscript. This work was supported by the NSF CAREER award NSF 0746549.
References
 Adelberger, E. G., Heckel, B. R., Hoedl, S., et al. 2007, PRL, 98, 131104 [CrossRef] [NASA ADS]
 ArkaniHamed, N., Dimopoulos, S., & Dvali, G. 1998, Phys. Lett. B, 429, 263 [CrossRef] [NASA ADS]
 Charles, P. A., & Coe, M. J. 2003, in Compact Stellar Xray Sources ed. W. H. G. Lewin, & M. van der Klis (Cambridge University Press) [arXiv:0308020]
 de Kool, M., van den Heuvel, E. P. J., & Pylyser, E. 1987, A&A, 183
 Emparan, R., GarcíaBellido, J., & Kaloper, N. 2003, JHEP, 0301, 079 [CrossRef] [NASA ADS]
 Fitzpatrick, A. L., Randall, L., & Wiseman, T. 2006, JHEP, 0611, 033 [CrossRef] [NASA ADS]
 Gelino, D. M., & Harrison, T. E. 2003, ApJ, 599, 1254 [CrossRef] [NASA ADS]
 Geraci, A. A., Smullin, S. J., Weld, D. M., Chiaverini, J., & Kapitulnik, A. 2008, Phys. Rev. D, in press [arXiv:0802.2350.v1]
 Gonzalez Hernandez, J. I., Rebolo, R., Israelian, G., et al. 2008, ApJ, 679, 732 [CrossRef] [NASA ADS]
 Johannsen, T., Psaltis, D., & McClintock, J. E. 2009, ApJ, 691, 997 [CrossRef] [NASA ADS]
 Kalogera, V., & Webbink, R. F. 1996, ApJ, 458, 301 [CrossRef] [NASA ADS]
 Kapner, D. J., Cook, T. S., Adelberger, E. G., et al. 2007, Phys. Rev. Lett., 98, 021101 [CrossRef] [NASA ADS]
 Kelley, R. L., Rappaport, S., Clark, G. W., & Petro, L. D. 1982, ApJ, 268, 790 [CrossRef] [NASA ADS]
 Maartens, R. 2004, LRR, 7, 7 [NASA ADS]
 McClintock, J. E., & Remillard, R. A. 1986, ApJ, 308, 110 [CrossRef] [NASA ADS]
 McClintock, J. E., Garcia, M. R., Caldwell, N., et al. 2001, ApJ, 551, 147 [CrossRef] [NASA ADS]
 Psaltis, D. 2007, Phys. Rev. Lett., 98, 181101 [CrossRef] [NASA ADS]
 Randall, L., & Sundrum, R. 1999, Phys. Rev. Lett., 83, 4690 [CrossRef] [NASA ADS]
 Rappaport, S., Verbunt, F., & Joss, P. C. 1983, ApJ, 275, 713 [CrossRef] [NASA ADS]
 Tanaka, T. 2003, Prog. Theor. Phys. Suppl., 148, 307 [CrossRef] [NASA ADS]
 Torres, M. A. P., Callanan, P. J., Garcia, M. R., et al. 2004, ApJ, 612, 1026 [CrossRef] [NASA ADS]
 Verbunt, F. 1993, ARA&A, 31, 93 [CrossRef] [NASA ADS]
 Wagner, R. M., Foltz, C. B., Shahbaz, T., et al. 2001, ApJ, 556, 42 [CrossRef] [NASA ADS]
 Webbink, R. F., Rappaport, S., & Savonije, G. J. 1983, ApJ, 270, 678 [CrossRef] [NASA ADS]
 Will, C. M., & Zaglauer, H. W. 1989, ApJ, 346, 366 [CrossRef] [NASA ADS]
 Yungelson, L., & Lasota, J.P. 2008, New Astron. Rev., 51, 860 [CrossRef] [NASA ADS]
 Zurita, C., Casares, J., Shahbaz, T., et al. 2002, MNRAS, 333, 791 [CrossRef] [NASA ADS]
All Tables
Table 1: The observed orbital periods and times of measurement for J1118+480.
All Figures
Figure 1: The rate of change of the orbital period P of the binary systems J1118+480 and A062000 versus the asymptotic curvature radius L in the extra dimension. The parameters are , , , and . The transition from predominant magnetic braking (constant rate) to predominant blackhole evaporation (rapidly increasing rate) occurs at (A062000) and (J1118+480), respectively. 

Open with DEXTER  
In the text 
Figure 2: The rate of change of the orbital period P (in years) of the binary system J1118+480 versus the specific angular momentum removed by the stellar wind , the accretion parameter , and the magnetic braking parameter , for and . On varying one parameter, the others are held constant at the respective values , , and . 

Open with DEXTER  
In the text 
Figure 3: The times T_{0} of the inferior conjunction and residuals versus the orbital cycle number n for the blackhole binary J1118+480. 

Open with DEXTER  
In the text 
Figure 4: The minimum rate of change of the orbital period P of the binary J1118+480 versus the asymptotic curvature radius L in the extra dimension. The intersection point of the observed 3upper limit on (horizontal line) with the minimum rate of change of the orbital period marks the constraint on the AdS curvature radius of (vertical line). 

Open with DEXTER  
In the text 
Copyright ESO 2009