Space-borne Gravitational Wave Detector LISA/eLISA

Yan Wang

[Yan Wang is the recipient of the 2014 Stefano Braccini Thesis Prize administered by the Gravitational Wave International Committee (GWIC) for his PhD thesis “On inter-satellite laser ranging, clock synchronization and gravitational wave data analysis” (PDF). His thesis work was carried out at Leibniz University of Hannover, Germany.

The Stefano Braccini Thesis Prize was established to honor the memory of a talented gravitational wave physicist whose promising career was cut short. Stefano worked with the French-Italian Virgo project, and contributed to the superattentuator design, to the integration and commissioning of Virgo and to its data analysis efforts. -- 2Physics.com]

Author: Yan Wang

Affiliation: School of physics, University of Western Australia, Perth, Australia.

Observations of electromagnetic radiation have revolutionized our understanding of the Universe and fundamental physics during the last century. With the advent of the expected first detection of gravitational waves (GWs) in near future, a completely new window onto the Universe will soon be opened by GW astronomy. GWs are spacetime ripples, predicted by Einstein’s theory of General Relativity. Their existence has been indirectly proven by measurements of the orbital decay due to gravitational radiation of the binary pulsar PSR 1913+16 [1], for which Hulse and Taylor won the 1993 Nobel Prize, but, due to their weak coupling with matter (i.e. GWs pass through stars, galaxies, Earth, Sun, and everything), direct detection of GWs has been beyond our technological capabilities until now. This weak coupling does mean, however, that GWs carry uncorrupted physical, astrophysical and cosmological information, enabling us to probe deep into the very early Universe, to test general relativity (GR) with unprecedented precision, and to measure the masses and spins of black holes (BHs) with exquisite accuracy.

Currently, large laser interferometers are the most sensitive GW detectors. There are several existing ground-based interferometric GW detectors: LIGO (Hanford and Livingston) [2], VIRGO [3], GEO600 [4], either operating or being upgraded; and KAGRA [5] under construction. During the past few decades, scientists took all efforts to isolate or mitigate various kinds of disturbances on the Earth, in order to increase the sensitivity of the detectors. These detectors are expected to detect GWs in near future.

Figure 1: Classic LISA configuration.

There are also space-borne interferometric GW detector (planned) missions. Among them, the most mature one is the Laser Interferometer Space Antenna (LISA) [6-7] ‘family’ (e.g. classic LISA, eLISA, and variations). LISA/eLISA consists of three spacecrafts (Fig. 1), each individually following a slightly elliptical orbit around the Sun, trailing the Earth by about 20 degree. These orbits are chosen such that the three spacecrafts retain an equilateral triangular configuration with an arm length of a few million kilometers as much as possible. This is accomplished by tilting the plane of the triangle by about 60 degree out of the ecliptic. Graphically, the triangular configuration does a cartwheel motion around the Sun.

The benefits of sending an interferometric GW detector to space are mainly: (i) Less noise disturbances, (ii) More GW sources. Roughly speaking, laser interferometric GW detectors are most sensitive to celestial systems of a size comparable to the interferometers’ arm length. LISA/eLISA is sitting in a frequency band, where there are most abundant GW sources (Fig. 2). There are several known white dwarf binaries in our galaxy directly visible to LISA/eLISA. In addition, LISA/eLISA can resolve thousands of other white dwarf binaries in our galaxy. LISA/eLISA can also observe massive black hole mergers throughout the entire universe [8]. Extreme mass ratio inspirals (i.e. stellar mass compact object orbiting a massive black hole) and primordial GWs from the birth of the universe add great scientific values to the mission as well [8].

Figure 2: (Click on the image to view with higher resolution) The GW spectrum from extremely low frequency to high frequency [Image courtesy: Chris Henze]

For many years, scientists have been spending great effort -- both theoretically and experimentally -- in preparation of LISA/eLISA. Some of the key techniques required by LISA/eLISA cannot be tested on the ground. LISA pathfinder satellite is going to be launched in this November (2015) to test the drag-free altitude control system in space, laser interferometry with picometer resolution at mHz band, the reliability of the instruments in the space environment, etc.

Unlike the ground-based interferometric GW detectors, the arm lengths of LISA/eLISA are varying significantly with time due to celestial mechanics in the solar system. As a result, the arm lengths differ by about one percent (i.e. tens of kilometers), and the dominating laser-frequency noise will not cancel out. The remaining laser-frequency noise would be stronger than other noises by about 8 orders of magnitude. Fortunately, the coupling between distance variations and the laser-frequency noise is very well known and understood. Therefore, we can use time-delay interferometry (TDI) techniques [9], which combine the measurement data series with appropriate time delays, in order to cancel the laser-frequency noise to the desired level.

However, the performance of TDI depends largely on the knowledge of arm lengths and relative longitudinal velocities between the spacecrafts, which are required to determine the correct delays to be adopted in the TDI combinations. In addition, the raw data are referred to the individual spacecraft clocks, which are not physically synchronized but independently drifting and jittering. This timing mismatch would degrade the performance of TDI variables. Therefore, they need to be referred to a virtual common constellation clock which needs to be synthesized from the inter-spacecraft measurements. Simultaneously, one also needs to extract the inter-spacecraft separations and synchronize the time-stamps properly to ensure the TDI performance. This has been a long existing gap.

Recently [10-11], we have tried to bridge this gap by designing sophisticated first stage data analysis algorithms for LISA/eLISA. The following are the main steps involved in the algorithms: (i) different types of inter-spacecraft measurements (e.g. the pseudo ranging measurements, the beat-notes of the carrier frequencies of the lasers emitted from one spacecraft and a remote spacecraft, the beat-notes of the laser sidebands) are precisely formulated as functions of the system state variables; (ii) several precise and effective dynamic models are designed for the system state variables (these models basically describe how the state variables evolve with time); (iii) the measurement data are pre-processed so that they can be used by a optimal filtering algorithm; (iv) the information of the measurements and the information from the dynamic models of the system state variables are optimally combined via a Kalman-like optimal filter, in order to reduce the noise in the measurements and the clock recording time stamps.

Simulation shows that our algorithms can successfully calibrate and synchronize the phasemeter raw data, estimate the inter-spacecraft distances and the clock errors, hence making the raw measurements usable for TDI techniques and astrophysical data analysis algorithms. This result can significantly increase the robustness of the LISA/eLISA project. The flexible design structure of our algorithms also provides a general framework of first stage LISA/eLISA data preparation, which can be easily extended to deal with various emergent scenarios in the future.

References:
[1] Joel M. Weisberg, Joseph H. Taylor, "The Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis", ASP Conference Series on 'Binary Radio Pulsars', vol. 328, p25 (2005). Article.
[2] http://www.advancedligo.mit.edu/
[3] http://www.cascina.virgo.infn.it/advirgo/
[4] http://www.geo600.org
[5] http://gwcenter.icrr.u-tokyo.ac.jp/en/
[6] https://www.elisascience.org/
[7] LISA International Science Team 2011 (European Space Agency), "LISA Unveiling a hidden universe", LISA Assessment Study Report (Yellow Book), ESA/SRE(2011) 3. Link.
[8] The eLISA Constortium, "The Gravitational Universe", Whitepaper submitted to ESA for the L2/L3 Cosmic Vision call. arXiv:1305.5720 [astro-ph.CO] (2013).
[9] Massimo Tinto, Sanjeev V. Dhurandhar, "Time-Delay Interferometry", Living Review Relativity 17 (2014), 6. Article.
[10] Y. Wang, Thesis: ‘On inter-satellite laser ranging, clock synchronization and gravitational wave data analysis’ (2014). Link.
[11] Yan Wang, Gerhard Heinzel, Karsten Danzmann, "First stage of LISA data processing: Clock synchronization and arm-length determination via a hybrid-extended Kalman filter", Physical Review D, 90, 064016 (2014). Abstract.

Sound Velocity Bound and Neutron Stars

Paulo Bedaque (left) and Andrew W. Steiner (right)

Authors: Paulo Bedaque¹, Andrew W. Steiner^2,3,4 

Affiliation: 
¹Department of Physics, University of Maryland, College Park, USA 
²Institute for Nuclear Theory, University of Washington, Seattle, USA
³Department of Physics and Astronomy, University of Tennessee, Knoxville, USA 
⁴Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA. 

Neutron stars are the final stage in the evolution of a star, the remnants of supernova explosion marking the end of the star’s life. They are incredibly compact objects: masses comparable to the Sun’s are compressed in a region of about 10 miles radius. At these densities, most matter is composed of neutrons. The repulsion between neutrons balances precariously against the strong gravitational fields generated by this high matter concentration: a little less repulsion or a little more mass leads to the collapse of the star into a black hole [1].

It has been possible to measure the mass of several neutron stars, and until recently, all accurate mass measurements were near 1.4 times the mass of our sun. However, within the past few years, two neutron stars have been discovered to have a mass around twice that of our sun [2]. What this discovery means is that the neutron matter composing the star is stiffer than previously expected.

The speed of sound in air is about 346 meters per second, and it tends to increase with either the density or the temperature of the medium in which it travels. Since neutron stars contain the most dense matter in the universe one might wonder how fast the speed of sound is inside neutron stars.

Everywhere else in the universe [3], the speed of sound seems to be limited to the speed of light divided by the square root of 3, that is, v < 0.577 c (see, for example, the figures here: Link to plots >> ). At high enough densities or temperatures, the speed of sound approaches this limiting value. This result comes from quantum chromodynamics (QCD) [4] - the physical theory which describes how neutrons and protons (made of quarks) interact. At high enough densities and temperatures, QCD exhibits "asymptotic freedom", meaning that the interaction becomes weaker [5]. Unfortunately, neutron star densities are not large enough so that quarks are weakly interacting.

In a paper published in Physical Review Letters (as an 'Editor's suggestion') on January 21st [6], we showed that the speed of sound in neutron stars must exceed this value at some point inside a neutron star. The reason is that models where the speed of sound is smaller than the limiting value at all densities (those like the black lines in the figure) are too soft to produce neutron stars with masses twice the mass of the sun. Thus, the only alternative is that the speed of sound must look something like either the blue dotted or red dashed lines.

This result is important because it tells us more about how neutrons and protons interact, not only in neutron stars, but also here on earth [7]. It gives us more insight into how QCD behaves at high densities. Finally, it also helps us understand some of the more extreme neutron star-related processes like core-collapse supernovae, magnetar flares, and neutron star mergers.

Notes & References:
[1] See a diagram of stellar evolution from the Chandra X-ray observatory, their neutron star page, or the wikpedia entry on neutron stars.
[2] P.B. Demorest, T. Pennucci, S.M. Ransom, M.S.E. Roberts, J.W.T. Hessels, "A two-solar-mass neutron star measured using Shapiro delay". Nature, 467, 1081–1083 (2010). Abstract; John Antoniadis, Paulo C. C. Freire, Norbert Wex, Thomas M. Tauris, Ryan S. Lynch, Marten H. van Kerkwijk, Michael Kramer, Cees Bassa, Vik S. Dhillon, Thomas Driebe, Jason W. T. Hessels, Victoria M. Kaspi, Vladislav I. Kondratiev, Norbert Langer, Thomas R. Marsh, Maura A. McLaughlin, Timothy T. Pennucci, Scott M. Ransom, Ingrid H. Stairs, Joeri van Leeuwen, Joris P. W. Verbiest, David G. Whelan, "A Massive Pulsar in a Compact Relativistic Binary". Science, 340, 6131 (2013). Abstract.
[3] The only possible exception is matter inside the event horizon of a black hole, which is not causally connected with the rest of the universe anyway.
[4] See the Wikipedia article on Quantum Chromodynamics.
[5] This finding led to 2004 Nobel prize in physics for David J. Gross, H. David Politzer and Frank Wilczek.
[6] Paulo Bedaque, Andrew W. Steiner, "Sound Velocity Bound and Neutron Stars". Physical Review Letters, 114, 031103 (2015). Abstract. Also available at: arXiv:1408.5116 [nucl-th].
[7] Neutrons and protons are the basic building blocks of all atomic nuclei.

A Rare Middle-Weight Black Hole in a Nearby Galaxy

Dheeraj R. Pasham

Author: Dheeraj R. Pasham^1,2

Affiliation:
¹Astronomy Department, University of Maryland, College Park, USA 
²Astrophysics Science Division and Joint Space-Science Institute, NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA

Black holes are among the most exotic and mysterious objects in the Universe, serving as one-way portals for matter, light, or anything else that gets too close. Our modern conception of black holes stems directly from Albert Einstein's theory of gravity, the general theory of relativity, proposed in 1915. However, it was not until the 1970s that new evidence emerged that these objects not only exist, but actually power the brightest objects in the Universe [1,2].

It is now established that there are at least two classes of black holes in the Universe: (1) the so-called "stellar-mass" black holes which weigh anywhere between 3-50 times the mass of our Sun and (2) "super-massive" black holes that are a million to a few billion times more massive than the Sun. Although we understand that the former are produced by spectacular deaths of the heaviest stars, the formation and the growth of super-massive black holes that are responsible for shaping the nature of many galaxies is still a mystery.

Understanding the formation of super-massive black holes holds a key to understanding the growth of galaxies that are the building blocks of our Universe. Current evidence indicates that super-massive black holes might have grown by accumulation of matter onto middle-weight black holes that are a few hundred to a thousand times more massive than the Sun and formed by the collapse the massive, first generation of stars formed when the Universe was only ~ 5% of its current age [3]. However, although candidates for such middle-weight black holes exist, no definitive mass measurements have yet been made. This is primarily because these objects are faint and thus the traditional methods used to weigh stellar-mass and supermassive black holes have not yet yielded any meaningful results [4,5].

In our recent result published in Nature [6], we used a new technique involving the 3:2 frequency ratio, X-ray resonance oscillations arising from close to the black hole in the galaxy M82, to measure its black hole mass to be 428±105 heavier than the Sun.

Figure 1 (click on the figure to view higher resolution version )

The basic idea behind the measurement is as follows. A subset of stellar-mass black holes exhibit the so-called high-frequency quasi-periodic oscillations (QPOs). Often in these systems the high-frequency QPOs occur in pairs of two, with their frequencies in a 3:2 ratio [7,8]. The power spectra of three such systems showing the twin pair QPOs are shown in Figure 1 [7]. The respective timescales of these oscillations (~0.01 seconds: 100-450 Hz) are comparable to the Keplerain orbital periods of test particles near the innermost stable circular orbit (ISCO) of these black holes. For example, for a non-rotating black hole weighing 10 solar masses, the Keplerian frequency of a test particle at ISCO is 220 Hz. In addition, for a given source, these frequencies appear to be stable to within a few percent for changes in the source luminosity. The fact that their frequencies are stable and appear to be originating from close to the ISCO (for a non-spinning black hole, the ISCO radius is 3 times the radius of the event horizon) suggests that they originate from very near to the black hole where strong gravity dominates and hence tied to the black hole's mass [9,10]. Under the assumption that these oscillations originate from a fixed characteristic radius within the accretion disk around the black hole, their frequencies should scale inversely with the black hole mass, and there is observational support that they do for stellar-mass black holes [7].

Some recent studies on X-ray variability of stellar-mass and supermassive black holes suggest that supermassive black holes behave as scaled-up stellar-mass black hole systems. More specifically, the qualitative nature of the variability of both the smaller stellar-mass and the heavier supermassive black holes appears to be the same (they appear to vary the same way) with the respective timescales of supermassive black holes being longer than than those of their stellar-mass counterparts. McHardy et al. (2006) [11] have demonstrated that these timescales scale inversely with the black hole mass after taking into account the rate at which matter fall onto the black hole. Under this black hole unification paradigm [11], if middle-weight black holes exist, some of them should exhibit these 3:2 pairs but at frequencies scaled down (longer timescales) according to their black hole masses.

Figure 2

Combing 6 years of archival X-ray data, we recently discovered such stable, twin-peak (3.3 and 5 Hz, 3:2 frequency ratio) X-ray oscillations from an object named M82 X-1 (see Figure 2) at frequencies roughly 50 times lower (or at timescales 50 times longer) than stellar-mass black holes. Scaling these frequencies to the oscillations of the black holes of known stellar mass implies that M82 X-1's black hole is 428±105 heavier than our Sun.

References:
[1] C.T. Bolton, "Identification of Cygnus X-1 with HDE 226868". Nature, 235, 271 (1972). Abstract.
[2] B. Louise Webster, Paul Murdin, "Cygnus X-1—a Spectroscopic Binary with a Heavy Companion?" Nature, 235, 37 (1972). Abstract.
[3] Piero Madau and Martin J. Rees, "Massive Black Holes as Population III Remnants". Astrophysical Journal Letters, 551, L27 (2001). Abstract.
[4] T.P. Roberts, J.C. Gladstone, A.D. Goulding, A.M. Swinbank, M.J. Ward, M.R. Goad, A.J. Levan, "(No) dynamical constraints on the mass of the black hole in two ULXs". Astronomische Nachrichten, 332, 398 (2011). Abstract.
[5] D. Cseh, F. Grisé, P. Kaaret, S. Corbel, S. Scaringi, P. Groot, H. Falcke, E. Körding, "Towards a dynamical mass of the ultraluminous X-ray source NGC 5408 X-1". Monthly Notices of the Royal Astronomical Society, 435, 2896 (2013). Abstract.
[6] Dheeraj R. Pasham, Tod E. Strohmayer, Richard F. Mushotzky, "A 400 solar mass black hole in galaxy M82". Nature, 513, 74 (2014). Abstract.
[7] Ronald A. Remillard and Jeffrey E. McClintock, "X-Ray Properties of Black-Hole Binaries". Annual Review of Astronomy & Astrophysics, 44, 49-92 (2006). Abstract.
[8] T. M. Belloni, A. Sanna, M. Méndez,"High-frequency quasi-periodic oscillations in black hole binaries". Monthly Notices of the Royal Astronomical Society, 426, 1701 (2012). Abstract.
[9] Marek A. Abramowicz, Włodek Kluźniak, Jeffrey E. McClintock, Ronald A. Remillard, "The Importance of Discovering a 3:2 Twin-Peak Quasi-periodic Oscillation in an Ultraluminous X-Ray Source, or How to Solve the Puzzle of Intermediate-Mass Black Holes". Astrophysical Journal Letters, 609, L63 (2004). Abstract.
[10] Robert V. Wagoner, "Diskoseismology and QPOs Confront Black Hole Spin". Astrophysical Journal Letters, 752, LL18 (2012). Abstract.
[11] I. M. McHardy, E. Koerding, C. Knigge, P. Uttley, R. P. Fender,"Active galactic nuclei as scaled-up Galactic black holes". Nature, 444, 730 (2006). Abstract. 

Testing the Strong-Field Dynamics of General Relativity

Tjonnie G. F. Li

[Tjonnie G. F. Li is the recipient of the 2013 Stefano Braccini Thesis Prize administered by the Gravitational Wave International Committee (GWIC) for his PhD thesis “Extracting Physics from Gravitational Waves: Testing the Strong-field Dynamics of General Relativity and Inferring the Large-scale Structure of the Universe” (PDF). His thesis work was carried out at Nikhef - Dutch National Institute for Subatomic Physics, the Netherlands and the Ph.D was awarded by Vrije Universiteit, Amsterdam, the Netherlands.

The Stefano Braccini Thesis Prize was established to honor the memory of a talented gravitational wave physicist whose promising career was cut short. Stefano worked with the French-Italian Virgo project, and contributed to the superattentuator design, to the integration and commissioning of Virgo and to its data analysis efforts. -- 2Physics.com]

Author: Tjonnie G. F. Li

Affiliation: Rubicon Postdoctoral Fellow, LIGO Laboratory, California Institute of Technology, USA.

Motion of celestial objects

Humans have been watching the sky for thousands of years. In early times, humans tracked the motion of the Sun and the Moon to make calendars and to associate it with Earthly events such as tides and seasons. By tracking the motion of celestial objects, the early notion of the orbit of the Sun, the Moon and the planets started to form. Building on earlier work developed by Greek astronomers, Claudius Ptolemy (90–168) introduced an accurate model of the planetary orbits by including the notion of a smaller circular orbit (epicycle) augmenting the primary circular orbit.

During the Renaissance, our knowledge of the sky started to change. Johannes Kepler (1571–1630) introduced three laws that described the planetary orbits as ellipses with the Sun at the focus. Later, Isaac Newton (1642-1727) showed that Kepler’s laws of planetary motion can be derived from a law that not only describes the motion of planets, but also describes how all objects are attracted to each other. Newton’s law of universal gravitation states that all objects “pull” on each other through the gravitational force, and the strength of this force is determined by the masses of the two objects.

Despite the success of Newton’s law of universal gravitation, it could not account for the shift in Mercury’s perihelion, the point in Mercury’s orbit that is closest to the Sun. It was Albert Einstein (1879–1955) who refined Newton’s law of universal gravitation by introducing the general theory of relativity. Einstein’s general theory of relativity states that the curvature of spacetime dictates the way in which matter flows through it, and conversely, matter curves spacetime around it. Einstein’s theory explained the shift in Mercury’s perihelion, and so far seems to be the correct description of the motion of planets, stars and even galaxies.

Gravitational waves: a new window into the Universe

The general theory of relativity does more than just predicting the motion of objects. It also predicts a new type of radiation, known as gravitational radiation or gravitational waves. Gravitational waves are ripples in the curvature of spacetime, which propagate at the speed of light. The effect of gravitational waves is the periodic expansion and contraction of space and time (see Fig. 1).

Figure 1: Example of the distortion of spacetime due to a incident gravitational wave onto a ring of test particles. The top and bottom row represent the effects of the two polarisation states as a function of the phase of the gravitational wave.

The existence of gravitational waves has only been inferred indirectly through the motion of two stars orbiting each other. In particular, in 1974, Russell Hulse and Joseph Taylor found two pulsars (neutron stars that emit highly collimated beams of electromagnetic radiation) in a binary system that appeared to behave exactly as if the system was loosing energy and angular momentum in the form of gravitational waves (see Fig. 2). Today this discovery is regarded as the first indirect evidence of gravitational waves, and earned Hulse and Taylor the 1993 Nobel Prize in Physics [1].

Figure 2: Change in the time of the periastron of the binary pulsar “PSR B1913+16” as a function of time (red dots). These observation are compared to the prediction of general relativity (blue line). This data is considered as the first indirect evidence of gravitational waves.

Quest for strong gravity

So does this mean that general relativity has been fully verified? From a theoretical perspective, we might be inclined to say that general relativity cannot be the final answer, because of the current inability to describe it using a complete quantum theory. Therefore, it is currently not possible to unify gravity with the other forces of nature (electromagnetic, strong and weak force) into a grand unified theory, which some might argue is an indication that general relativity cannot be the final answer. In other words, a theory must exist that, in the low-energy regime, behaves like general relativity.

From an experimental perspective, one can argue that all of the tests of general relativity have so far been done in the regime of weak gravity. A figure of merit which describes the strength of gravity is the quantity ϵ ~ GM∕(Rc²), where G is the gravitational constant, M is the total mass of the system, R is the characteristic length scale of the system and c is the speed of light. Near a black hole the strength is ϵ ~ 10^-1, whereas for solar system tests, and for binary pulsar tests, this strength is about ϵ ~ 10^-6 [2]. Therefore, there is a whole new regime of gravity to explore experimentally.

Scientists all over the world are working hard on the quest for strong gravity. Amongst many interesting questions, they also hope to uncover empirical insight into the quantisation of gravity, which could refine or guide new theories of gravity. One of the ways in which we could hope to probe the regime of strong gravity is through the direct measurement of gravitational waves. Such measurements could probe gravity close to black holes and other exotic astrophysical objects.

Advanced LIGO and Virgo

Large-scale physics experiments such as the USA-based LIGO (see Fig. 3) [3] and the Italy-based Virgo [4] aim to, for the first time in the history of mankind, detect the influences of gravitational waves directly. These experiments are set up to measure tiny changes in distances of about one thousandth of the diameter of a proton. These tiny perturbations of spacetime could lead us down a new path in our quest for strong gravity.

Figure 3: Aerial view of the LIGO-Hanford detector

The motion of the source closely dictates the characteristics of the gravitational waves emitted. So by mapping out the distortions caused by the incident gravitational wave, one could infer a wealth of information about its origins. In other words, where astronomers needed telescopes to determine the motion of planets, stars and galaxies, measurements of gravitational waves can provide an additional way to map the dynamics of celestial objects.

In particular, a promising class of candidates for the first detection of gravitational wave is the compact binary coalescence [5]. Compact binary coalescence typically refers to (especially in the context of LIGO/Virgo) the mergers of binary black holes or neutron stars (see Fig. 4). The components of such systems spiral toward each other as energy and momentum are radiated away through the emission of gravitational waves. Finally, when the objects are sufficiently close to each other, they merge to form a single black hole which then continues to ring down as it reaches a quiescence state. The dynamics of coalescence of a compact binary can be seen through simulations as in Ref. [6].

Figure 4: Image from a binary black hole simulation.

Testing strong-field gravity with compact binary coalescences

Compact binary coalescences are attractive systems to probe strong gravity, because close to black holes and neutron stars the effects of gravity can be considered strong. Moreover, these systems are relatively easy to understand theoretically, because they mainly involve the application of general relativity. In contrast, mechanisms behind, for example, supernovae, which are also candidates to be measured by Advanced LIGO/Virgo, involve a complicated interplay amongst many branches of physics. The direct measurement of gravitational waves emitted from a compact binary coalescence will therefore give us access to the motion of black holes in orbit around each other’s strong gravitational pull.

However, in order to extract this information, we need specialised algorithms to dig deep into the data. One of such algorithms is called Test Infrastructure for GEneral Relativity (TIGER). This algorithm tries to answer the question “is the signal consistent with general relativity?” through the application of Bayesian hypothesis testing [7]. This framework ensures the optimal use of available information, and allows one to combine information across multiple detections of compact binary coalescences. Using this algorithm in a simulation environment, we have shown that the Advanced LIGO-Virgo network is indeed capable of probing gravity in uncharted territories, to an accuracy never seen before.

Of course, many challenges have to be faced. Detection of gravitational waves is a major challenge by itself on which hundreds of scientist are currently working. Moreover, once the Advanced LIGO-Virgo network is making confident detections, we need to analyse the motion of black holes or neutron stars in the presence of noise that can be orders of magnitude louder than the signal. Nevertheless, we are on the brink of the first direct detection with Advanced LIGO coming online as early as 2015. A hundred years after the introduction of general relativity, Advanced LIGO/Virgo could either put the crown on Einstein’s work, or showcase its limitations.

To be continued…

References
[1] “The Nobel Prize in Physics 1993”. Link in: Nobelprize.org.
[2] C. M. Will. “The Confrontation between General Relativity and Experiment”. In: Living Reviews in Relativity 17.4 (2014). Link.
[3] http://www.ligo.org .
[4] http://wwwcascina.virgo.infn.it .
[5] B.S. Sathyaprakash and Bernard F. Schutz. “Physics, Astrophysics and Cosmology with Gravitational Waves”. In: Living Reviews in Relativity 12.2 (2009). Link.
[6] Download from: http://numarch.aei.mpg.de/numrel-webpages/movies/bbh08_small.mov .
[7] T. G. F. Li, W. Del Pozzo, S. Vitale, C. Van Den Broeck, M. Agathos, J. Veitch, K. Grover, T. Sidery, R. Sturani, A. Vecchio, “Towards a generic test of the strong field dynamics of general relativity using compact binary coalescence”. Physical Review D, 85, 082003 (2012). Abstract.

Comparing Matter Waves in Free Fall

[From Left to Right] J. Hartwig, D. Schlippert, E. M. Rasel

Authors: J. Hartwig, D. Schlippert, Ernst M. Rasel

Affiliation: Institut für Quantenoptik and Centre for Quantum Engineering and Space-Time Research (QUEST), Leibniz Universität Hannover, Germany

Introduction to Einstein’s Equivalence Principle

Einstein’s general relativity is based on three fundamental building blocks: local Lorentz invariance, the universality of the gravitational redshift and the universality of free fall. The enormous importance of general relativity in modern science and technology merits a continuous effort in improving experimental verification of these underlying principles.

The universality of free fall is one of the oldest mechanical theories originally proposed by Galileo. Testing can be done by so called free fall experiments, where two bodies with different composition are freely falling towards a third gravitating body.

Figure 1: Goddard Spaceflight Center Laser Ranging Facility. Source: NASA

Amongst the most sensitive measurements of this principle is the Lunar Laser Ranging experiment, which compares the free fall of earth and moon in the solar gravitational potential [1]. This measurement is only surpassed by torsion balance experiments based on the design of Eötvös [2]. In addition, exciting new insights are expected from the MICROSCOPE experiment [3] that is planned to launch in 2016.

Figure 2: Torsion balance experiment as used in the group of E. Adelberger, University of Washington. Source: Eöt-Wash-Group

The emergence of quantum physics and our improved understanding of the basic building blocks of matter increases the interest scientists have in the understanding of gravity. How do gravity and quantum mechanics interact? What`s the connection between different fundamental particles and their mass? Is there a deeper underlying principle combining our fundamental theories? To comprehensively approach these questions a wide array of parameters must be analyzed. The way how certain test materials may act under the influence of gravity can either be parametrized using a specific violation scenario, like the Dilaton scenario by T. Damour [4], or by using a test theory such as the extended Standard Model of particles (SME) [5]. Since the SME approach is not based on a specific mechanism of violating UFF it also does not predict a level to which a violation may occur. Instead, it delivers a model-independent approach to compare methodically different measurements and confine possible violation theories.

Table 1 states possible sensitivities for violations based on the SME framework for a variety of test masses and underlines the importance of complementary test mass choices are. Hence in comparison to classical tests, the use of atom interferometry opens up a new field of previously inaccessible test masses with perfect isotopic purity in a well-defined spin state. Quantum tests appear to differ from previous test also in a qualitative way. They allow to perform test with new states of matter, such as wave packets by Bose Einstein condensates being in a superposition state. The work presented here is just another early step in a quest to understand the deeper connections between the quantum and classical relativistic world.

Table 1: Sample violation strengths for different test masses linked to “Neutron excess” and the “total Baryon number” based on the Standard Model Extension formalism. The test mass pairs are chosen according to the best torsion balance experiment [6] and existing matter wave tests [7]. An anomalous acceleration would be proportional to the stated numerical coefficients. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014).

Measuring accelerations with atom interferometry

Measuring accelerations with a free fall experiment is always achieved by tracking the movement of an inertial mass in free fall in comparison to the lab frame of reference. This is even true when the inertial mass in question needs to be described by a matter wave operating on quantum mechanics. Falling corner cube interferometers operating on this principle are among the most accurate measurements of gravity with classical bodies. They use a continuous laser beam to track the change of velocity of a corner cube reflector due to gravity in a Michelson interferometer with the corner cube changing one of the arm lengths. Acceleration sensors based on free falling matter waves use a similar principle.

Figure 3: A side view of the experimental setup with the two-dimensional (left side) and three-dimensional (right side) magneto-optical traps employed in [Phys. Rev. Lett. 112, 203002 (2014)].

The first demonstration of a true quantum test of gravity with matter waves was performed 1975 with neutrons in the Famous COW experiment [8]. We will focus on atom interferometers using alkaline atoms, since they are most commonly used for inertial sensing and are also employed in the discussed experiment. Experiments of this kind were first used for acceleration measurements in 1992 [9] and have improved in their performance ever since. The first test of the equivalence principle comparing two different isotopes was then performed in 2004 [10]. Research on quantum tests is for example proposed at LENS in Italy [11], in Stanford in an already existing large fountain [12] and in the scope of the French ICE mission in a zero-g plane [13]. All these initiatives show the high interest of testing gravity phenomena with quantum matters as opposed to classical tests.

In the case of atom interferometers, coherent beam splitting is performed by absorption and stimulated emission of photons. Which atomic transitions are used is dependent on the specific application but, in the case of alkaline atoms two photon transitions coupling either two hyperfine and respective momentum states (Raman transitions) or just momentum states (Bragg transitions) are employed. The point of reference for the measurement is then given by a mirror reflecting the laser beams used to coherently manipulate the atoms, since the electromagnetic field is vanishing at the mirror surface. This results in a reliable phase reference of the light fields and constitutes the laboratory frame. The role of the retroreflecting mirror is similar to the one of the mirror at rest in in the case of the falling corner cube experiment.

The atomic cloud acts as the test mass, which in an ideal case, is falling freely without any influence by the laboratory, except during interaction with the light fields employed as beam splitters or mirrors. During the interaction, the light fields drive Rabi-oscillations in the atoms between the two interferometer states |g> and |e> with a time 2τ needed for a full oscillation. This allows for the realization of beam splitters with a τ/2 pulse length resulting in an equal superposition of |g> and |e>. Mirrors can be realized the same way by applying the beam splitter light fields for a time of τ which leads to an inversion of the atomic state. These pulses are generally called π/2 (for the superposition) and π (for the inversion pulses) in accordance with the Rabi-oscillation phase. The simplest geometry used to measure acceleration is a Mach- Zehnder-like geometry. This is produced by applying a π/2-π-π/2 sequence with free evolution times T placed between pulses. The resulting geometry can be seen in Figure 4.

Figure 4: Space-time diagram of a Mach-Zehnder-like atom interferometer. An atomic ensemble is brought into a coherent superposition of two momentum states by a stimulated Raman transition (π/2 pulse). The two paths I+II propagate separated, are reflected by a pi-pulse after a time T and superimposed and brought to interference with a final π/2 pulse after time 2T. The phase difference is encoded in the population difference of the two output states.

During the interaction with the light fields, the lattice formed by the two light fields imprints its local phase onto the atoms. This results in an overall phase scaling with the relative movement between the atomic cloud and the lattice. Calculating the overall phase imprinted on the atoms results in first order term, Φ=a*T²*k_eff, where k_eff is the effective wave vector of the lattice and a is the relative acceleration between lattice and atoms. This immediately shows the main feature of free fall atom interferometry: the T² scaling of the resulting phase. This is of particular interest for future experiments aiming for much higher free evolution times than currently possible. The phase Φ also shows another key feature. As the acceleration between atoms and lattice approaches zero, the phase also goes to zero, independently of the interferometry time T. This yields a simple way to determine the absolute acceleration of the atomic sample by accelerating the lattice until the lattice motion is in the same inertial reference frame as the freely falling atoms.

Lattice acceleration is achieved by chirping the frequency difference between the two laser beams used for the two photon transition. This transforms the measurement of a relatively large phase, spanning many thousand radians, to a null measurement. The signal produced is the population difference between the interferometer states |g> and |e> as a function of lattice acceleration and thus frequency sweep rate, α. The sweep rate corresponding to a vanishing phase directly leads to the acceleration experienced by the atoms according to lattice acceleration formula a=α/k_eff. Taking into account Earth’s gravitational field and a lattice wavelength of 780/2 nm (the factor of ½ is introduced due to the use of a two-photon transition) this leads to a sweep rate of around 25 MHz/s. The advantage of this method is that the acceleration measurement is now directly coupled to measurement of the wavelength of the light fields and frequencies in the microwave regime, which are easily accessible.

Our data

Figure 5: Determination of the differential acceleration of rubidium and potassium. The main systematic bias contributions do not change their sign when changing the direction of momentum transfer. Hence, the mean acceleration of upward and downward momentum transfer direction greatly suppresses the aforementioned biases. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014)

Figure 6:  the wave nature of ⁸⁷Rb and ³⁹K 
atoms and their interference are exploited 
to measure the gravitational acceleration.

In order to test the universality of free fall, we simultaneously chirp the Raman frequencies to compensate for the accelerations a(Rb,±)(g) and a(K,±)(g) experienced by rubidium and potassium that were previously identified (Figure 6). Here, the observed phase shift exhibits contributions due to additional perturbations, such as magnetic field gradients. We make use of a measurement protocol based on reversing the transferred momentum (upward and downward directions ±). This technique makes use of the fact that many crucial perturbations do not depend on the direction of momentum transfer. Thus, by computing the half-difference of the phase differences determined in a single momentum direction, phase shifts induced by, e.g., the AC-Stark effect or Zeeman effect, can be strongly suppressed [14].

Figure 7: Allan deviations of the single species interferometer signals and the derived Eötvös ratio. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014)

The data presented in this work [15] was acquired in a data run that was ~4 hours long. By acquiring 10 data points per direction of momentum transfer, and species and then switching to the opposite direction, we were able to determine the Eötvös ratio of rubidium and potassium to a statistical uncertainty of 5.4 x 10^-7 after 4096s; the technical noise affecting the potassium interferometer is the dominant noise source.

Taking into account all systematic effects, our measurement yields η(Rb,K)=(0.3 ± 5.4) x 10^-7.

Outlook

In our measurement, the performance was limited both by technical noise and the limited free evolution time T. In order to improve these parameters, we are currently extending the free fall time in our experiment. Furthermore, in an attempt to increase the contrast of our interferometers and thus the signal-to-noise ratio, we are working on implementing state preparation schemes for both species.

We expect to constrain our uncertainty budget (which currently is on the 10 ppb level for the Eötvös ratio) on the ppb level and below through the use of a common optical dipole trap applied to both species. By using Bose-Einstein-condensed atoms, we gain the ability to precisely calculate the ensembles, as well as carefully control the input state. This technique will also be able to reduce uncertainty factors linked to the transverse motion of the cloud, in addition to spatial magnetic field and gravitational field gradients.

Improving the precision of a true quantum test into the sub-ppb regime is the focus of current research. For example we are currently planning a 10m very long baseline atom interferometer (VLBAI) in Hannover [16]. In the framework of projects funded by the German Space Agency (DLR), we moreover develop experiments that are suitable for microgravity operation in the ZARM drop tower in Bremen and on sounding rocket missions [17].

Parallel to the development done in the LUH and at a national level, we are also involved in projects on an international level looking into extending the frontier of atom interferometry and especially the test of the equivalence principle. A major project investigating the feasibility of a space borne mission is the STE-Quest Satellite Mission proposed by a European consortium including nearly all major research institutions working in the field of inertial sensing with atom interferometry, as well as a variety of specialist of other fields [18]. This mission is aimed towards doing a simultaneous test of the equivalence principle with two rubidium isotopes and a clock comparison with several ground based optical clocks, pushing the sensitivity to the Eötvös ratio into the 10^-15 regime.

References:
[1] James G. Williams, Slava G. Turyshev, Dale H. Boggs, "Progress in Lunar Laser Ranging Tests of Relativistic Gravity". Physical Review Letters, 93, 261101 (2004). Abstract.
[2] S. Schlamminger, K.-Y. Choi, T. A. Wagner, J. H. Gundlach, E. G. Adelberger, "Test of the Equivalence Principle Using a Rotating Torsion Balance". Physical Review Letters, 100, 041101 (2008). Abstract.
[3] P. Touboul, G. Métris, V. Lebat, A Robert, "The MICROSCOPE experiment, ready for the in-orbit test of the equivalence principle". Classical and Quantum Gravity, 29, 184010 (2012). Abstract.
[4] Thibault Damour, "Theoretical aspects of the equivalence principle". Classical Quantum Gravity, 29, 184001 (2012). Abstract.
[5] M.A. Hohensee, H. Müller, R.B. Wiringa, "Equivalence Principle and Bound Kinetic Energy". Physical Review Letters, 111, 151102 (2013). Abstract.
[6] S. Schlamminger, K.-Y. Choi, T.A. Wagner, J.H. Gundlach, E.G. Adelberger, "Test of the Equivalence Principle Using a Rotating Torsion Balance". Physical Review Letters, 100, 041101 (2008). Abstract.
[7] A. Bonnin, N. Zahzam, Y. Bidel, A. Bresson, "Simultaneous dual-species matter-wave accelerometer". Physical Review A, 88, 043615 (2013). Abstract ; S. Fray, C. Alvarez Diez, T. W. Hänsch, M. Weitz, "Atomic Interferometer with Amplitude Gratings of Light and Its Applications to Atom Based Tests of the Equivalence Principle". Physical Review Letters, 93, 240404 (2004). Abstract ; M. G. Tarallo, T. Mazzoni, N. Poli, D. V. Sutyrin, X. Zhang, G. M. Tino, "Test of Einstein Equivalence Principle for 0-Spin and Half-Integer-Spin Atoms: Search for Spin-Gravity Coupling Effects". Physical Review Letters, 113, 023005 (2014). Abstract.
[8] R. Colella, A. W. Overhauser, S. A. Werner, "Observation of Gravitationally Induced Quantum Interference". Physical Review Letters, 34, 1472 (1975). Abstract.
[9] M. Kasevich, S. Chu, "Measurement of the gravitational acceleration of an atom with a light-pulse atom interferometer". Applied Physics B, 54, 321–332 (1992). Abstract.
[10] Sebastian Fray, Cristina Alvarez Diez, Theodor W. Hänsch, Martin Weitz, "Atomic Interferometer with Amplitude Gratings of Light and Its Applications to Atom Based Tests of the Equivalence Principle". Physical Review Letters, 93, 240404 (2004). Abstract.
[11] M. G. Tarallo, T. Mazzoni, N. Poli, D. V. Sutyrin, X. Zhang, G. M. Tino, "Test of Einstein Equivalence Principle for 0-Spin and Half-Integer-Spin Atoms: Search for Spin-Gravity Coupling Effects". Physical Review Letters, 113, 023005 (2014). Abstract.
[12] Susannah M. Dickerson, Jason M. Hogan, Alex Sugarbaker, David M. S. Johnson, Mark A. Kasevich, "Multiaxis Inertial Sensing with Long-Time Point Source Atom Interferometry". Physical Review Letters, 111, 083001 (2013). Abstract. 2Physics Article.
[13] G Varoquaux, R A Nyman, R Geiger, P Cheinet, A Landragin, P Bouyer, "How to estimate the differential acceleration in a two-species atom interferometer to test the equivalence principle". New Journal of Physics, 11, 113010 (2009). Full Article.
[14] J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and,M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry". Physical Review A, 65, 033608 (2002). Abstract; Anne Louchet-Chauvet, Tristan Farah, Quentin Bodart, André Clairon, Arnaud Landragin, Sébastien Merlet, Franck Pereira Dos Santos, "The influence of transverse motion within an atomic gravimeter". New Journal of Physics, 13, 065025 (2011). Full Article.
[15] D. Schlippert, J. Hartwig, H. Albers, L. L. Richardson, C. Schubert, A. Roura, W. P. Schleich, W. Ertmer, E. M. Rasel, "Quantum Test of the Universality of Free Fall". Physical Review Letters, 112, 203002 (2014). Abstract.
[16] http://www.geoq.uni-hannover.de/350.html
[17] http://www.iqo.uni-hannover.de/quantus.html
[18] D N Aguilera, H Ahlers, B Battelier, A Bawamia, A Bertoldi, R Bondarescu, K Bongs, P Bouyer, C Braxmaier, L Cacciapuoti, C Chaloner, M Chwalla, W Ertmer, M Franz, N Gaalou, M Gehler, D Gerardi, L Gesa, N Gürlebeck, J Hartwig, M Hauth, O Hellmig, W Herr, S Herrmann, A Heske, A Hinton, P Ireland, P Jetzer, U Johann, M Krutzik, A Kubelka, C Lämmerzah, A Landragin, I Lloro, D Massonnet, I Mateos, A Milke, M Nofrarias, M Oswald, A Peters, K Posso-Trujillo, E Rase, E Rocco, A Roura, J Rudolph, W Schleich, C Schubert, T Schuldt, S Seide, K Sengstock, C F Sopuerta, F Sorrentino, D Summers, G M Tino, C Trenkel, N Uzunoglu, W von Klitzing, R Walser, T Wendrich, A Wenzlawski, P Weßels, A Wicht, E Wille, M Williams, P Windpassinger, N Zahzam,"STE-QUEST—test of the universality of free fall using cold atom interferometry". Classical Quantum Gravity, 31, 115010 (2014), Abstract.

The Formation of Two Supermassive Black Holes from A Single Collapsing Supermassive Star

From left to right: (top row) Christian Reisswig, Christian D. Ott, Ernazar Abdikamalov; (bottom row) Roland Haas, Philipp Mösta, Erik Schnetter

Authors: Christian Reisswig^1,*, Christian D. Ott^1,2,+, Ernazar Abdikamalov¹, Roland Haas¹, Philipp Moesta¹, Erik Schnetter^3,4,5

Affiliation:
¹TAPIR, California Institute of Technology, Pasadena, CA, USA
²Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), The University of Tokyo, Kashiwa, Japan
³Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
⁴Department of Physics, University of Guelph, Guelph, ON, Canada
⁵Center for Computation & Technology, Louisiana State University, Baton Rouge, LA, USA

^*NASA Einstein Fellow
⁺Alfred P. Sloan Research Fellow

The existence of supermassive black holes with masses a billion times the mass of our sun at high redshifts z>7 [1] is one of the mysteries in our understanding of the early history of the universe. At redshift z=7, the universe was less then one billion years old. This leads to a serious problem: how is it possible for black holes to acquire this tremendous amount of mass over a short timescale of just one billion years? A common theory of black hole growth assumes as a starting point the collapse of the very first stars, so called Population III stars. Population III stars may have had masses around 100 times the mass of our sun. The collapse of such a star can leave behind a black hole of similar mass that then grows via subsequent accretion of material from its surroundings. This process can yield quite massive black holes, but in order to reach supermassive scales within only one billion years, the accretion process must be rather extremely rapid to enable fast black hole growth. These high required accretion rates, however, seem to be difficult to be maintained due to, e.g. strong outgoing radiation that can blow away the surrounding gas that otherwise would be accreted onto the black hole [2]. The model, therefore, has difficulties of explaining the existence of very massive black holes in the early universe.

Another model which has recently regained attention is supermassive star collapse. Supermassive stars have originally been proposed by Hoyle & Fowler in the 1960s as a model for strong distant radio sources [3]. Such stars have masses up to a million times the mass of our sun and potentially formed in the monolithic collapse of primordial gas clouds that existed in the early universe [4, 5]. Unlike ordinary stars, which are mainly powered by nuclear burning, supermassive stars are mainly stabilized against gravity by their own photon radiation field that originates from the very high interior temperatures generated by gravitational contraction. During their short lives, they slowly cool due to the emitted photon radiation that keeps the stars in hydrostatic equilibrium. The colder stellar gas can be more easily compressed by the inward gravitational pull, and as a consequence, the stars slowly contract and become more compact. This process continues for a few million years until the stars reach sufficient compactness for gravitationally instability to set in. This general relativistic instability inevitably leads to gravitational collapse. One possible outcome of the collapse is a massive black hole containing most of the original mass of the star. Since the 'seed' mass of the nascent black hole is already pretty large, subsequent growth via accretion from the surroundings can easily push the black hole to supermassive scales within the available time without the need of extreme accretion rates and thus without any strong photon radiation that may blow away the surrounding accreting gas.

In our recent article published in Physical Review Letters [6], we study non-axisymmetric effects in the collapse of supermassive stars. The starting point of our models are supermassive stars which are at the onset of gravitational collapse. We use general relativistic hydrodynamic supercomputer simulations with fully dynamical non-linear space-time evolution to investigate the behavior and dynamics of collapsing supermassive stars. Such computer models have been considered in previous studies [7,8,9,10], however, mostly in axisymmetry.

 In an axisymmetric configuration, a supermassive star maintains a spherical shape during its collapse, which is possibly flattened due to rotation. In these previous studies, it has been shown that the possible outcome is either a single massive rotating black hole, or, alternatively, a powerful supernova explosion which completely disrupts the star. In our case, we select an initial stellar model which is rapidly rotating and leads to black hole formation. In fact, it is so rapidly rotating that the shape of our star is no longer spheroidal, but rather resembles the shape of a 'quasi'-torus where the maximum density is off-center and thus forming a central high-density ring (see upper left panel of Figure 1).

Figure 1: (To view higher resolution click on the image) The various stages encountered during the collapse of a supermassive star with an initial m=2 standing density wave perturbation. Each panel shows the density distribution in the equatorial plane.

Such a configuration is unstable to tiny density perturbations that may be present at the onset of collapse [10]. This instability is particularly strong for perturbations in the form of standing poloidal density waves with one (m=1) or two (m=2) maxima. Due to this instability, these perturbations grow exponentially during the collapse, and can lead to significant deformations away from axisymmetry. The nature of the instability typically leads to the formation of orbiting high-density clumps of matter inside the collapsing star (see upper right panel of Figure 1). Since the m=1 and m=2 perturbations grow fastest, either one or two high-density clumps will form, depending on the initial perturbation of the stellar density. These high density fragments continue to grow rapidly during the collapse, thus becoming denser and hotter. 

Once temperatures of more than one billion Kelvins are reached, a process sets, which is called electron-positron pair creation. The creation of particle pairs is possible because there is enough energy available in the surrounding gas to spontaneously create a particle and its anti-particle, in this case electrons and positrons. The pair creation process has the effect of taking out energy from the gas fragments, thus dramatically reducing their local pressure. The reduction in pressure support leads to a rapid increase in the central density within each fragment up to the point at which the fragments become so dense that event horizons appear around each of them (center left panel of Figure 1). In the case of an initial m=2 density perturbation, two black holes form that orbit each other. Since two black holes in close orbit emit very powerful gravitational radiation - ripples of space-time that travel at the speed of light - , the associated loss of energy causes the black hole orbits to shrink, leading to an inspiralling motion (red lines in the center left panel of Figure 1). The leading order mode of the corresponding emitted gravitational wave signal is shown in the lower panel Figure 2.

Figure 2: (To view higher resolution click on the image) The upper panel shows the time evolution of the density maximum until black hole formation. The center panel shows the mass and spin evolution of the black holes. The lower panel shows the emitted leading order gravitational wave signal.

It resembles the typical quasi-sinusoidal oscillatory signal expected from binary black hole mergers: as the orbit shrinks, the emitted radiation becomes higher in frequency. The inspiral continues until a common event horizon appears, marking the merger of the two black holes (lower left panel of Figure 1). The black hole merger remnant is initially deformed into a peanut shape, which quickly relaxes into a spherical shape by emitting exponentially decaying gravitational ring-down radiation. This is shown in the lower panel of Figure 2. The peak amplitude of the waveform corresponds to the black hole merger. From there, the signal quickly decays due to black hole ring-down. By the end of our simulation, the remnant black hole is rapidly rotating and is surrounded by a massive accretion disk (lower right panel of Figure 1).

The formation of two merging black holes requires a particular choice of initial stellar model parameters at the onset of collapse: (i) we require rapid rotation and (ii) a poloidal m=2 standing density wave perturbation must be present. This naturally leads to the question of the likelihood of our model. Recent cosmological simulations of collapsing primordial gas clouds - the potential birth sites for supermassive stars - indicate that rapid rotation is very likely [4]. Curiously, the same simulations also show that an m=2 deformation arises at the center of the clouds where the supermassive star will eventually form. Unfortunately, these simulations currently do not offer sufficient spatial resolution to investigate the formation of supermassive stars in the collapse of primordial gas clouds in detail. Further research will be necessary to self-consistently model the formation of supermassive stars that may inform us about the stellar conditions at the onset of collapse.


The new and exciting prediction that two black holes can form in the collapse of a single star gives rise to very efficient gravitational wave emission compared to models where only one black hole forms. The emitted gravitational radiation in our model configuration is so powerful that future space-borne gravitational wave observatories might see the signal from the edge of our universe. This has important implications for cosmology. If detected, the signal will inform us about the formation processes of supermassive stars and supermassive black holes in the early universe and will allow us to test the validity of the supermassive star collapse pathway to supermassive black hole formation.

Acknowledgements: This research is partially supported by NSF grant nos. PHY-1151197, AST-1212170, PHY-1212460, and OCI-0905046, by the Alfred P. Sloan Foundation, and by the Sherman Fairchild Foundation. CR acknowledges support by NASA through Einstein Postdoctoral Fellowship grant number PF2-130099 awarded by the Chandra X-ray center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. RH acknowledges support by the Natural Sciences and Engineering Council of Canada. The simulations were performed on the Caltech compute cluster Zwicky (NSF MRI award No. PHY-0960291), on supercomputers of the NSF XSEDE network under computer time allocation TG-PHY100033, on machines of the Louisiana Optical Network Initiative under grant loni_numrel08, and at the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the US Department of Energy under contract DE-AC02-05CH11231.

References:
[1] Daniel J. Mortlock, Stephen J. Warren, Bram P. Venemans, Mitesh Patel, Paul C. Hewett, Richard G. McMahon, Chris Simpson, Tom Theuns, Eduardo A. Gonzáles-Solares, Andy Adamson, Simon Dye, Nigel C. Hambly, Paul Hirst, Mike J. Irwin, Ernst Kuiper, Andy Lawrence, Huub J. A. Röttgering, "A luminous quasar at a redshift of z = 7.085", Nature, 474, 616 (2011). Abstract.
[2] Marcelo A. Alavarez, John H. Wise, Tom Abel, "Accretion onto the first stellar-mass black holes", Astrophysical Journal Letters, 701:L133 (2009). Abstract.
[3] F. Hoyle, William A. Fowler, "Nature of strong radio sources", Nature, 197, 533 (1963). Abstract.
[4] Jun-Hwan Choi, Isaac Shlosman, Mitchell C. Begelman, "Supermassive black hole formation at high redshifts via direct collapse: physical processes in the early stage", Astrophysical Journal, 774:149, 18 (2013). Abstract.
[5] M. A. Latif, D. R. G. Schleicher, W. Schmidt, J. Niemeyer, "Black hole formation in the early Universe", Monthly Notices of the Royal Astronomical Society, 433, 1607-1618 (2013). Abstract.
[6] Christian Reisswig, Christian D. Ott, Ernazar Abdikamalov, Roland Haas, Philipp Moesta, "Formation and Coalescence of Cosmological Supermassive-Black-Hole Binaries in Supermassive-Star Collapse", Physical Review Letters, 111, 15, 151101 (2013). Abstract.
[7] Pedro J. Montero, Hans-Thomas Janka, Ewald Mueller, "Relativistic collapse and explosion of rotating supermassive stars with thermonuclear effects", Astrophysical Journal, 749:37, 14 (2012). Article.
[8] Motoyuki Sajio, Ian Hawke, "Collapse of differentially rotating supermassive stars: post black hole formation", Physical Review D, 80, 064001 (2009). Abstract.
[9] Masaru Sibata, Stuart L. Shapiro, "Collapse of a rotating supermassive star to a supermassive black hole: fully relativistic simulations", Astrophysical Journal, 572:L39 (2002). Article.
[10] Burkhard Zink, Nikolas Stergioulas, Ian Hawke, Christian D. Ott, Erik Schnetter, Ewald Mueller, "Nonaxisymmetric instability and fragmentation of general relativistic quasitoroidal stars", Physical Review D, 76, 024019 (2007). Abstract.

Atom Interferometry in a 10 Meter Atomic Fountain

[From left to right] Mark Kasevich; the 10 m fountain team in 2013: Alex Sugarbaker, Tim Kovachy, Jason Hogan, Susannah Dickerson, Sheng-wey Chiow; and Dave Johnson

Authors: Alex Sugarbaker, Susannah M. Dickerson, Jason M. Hogan, David M. S. Johnson, and Mark A. Kasevich

Affiliation: Department of Physics, Stanford University, USA

Link to Kasevich Group Website >>

The equivalence principle states that all objects fall with the same acceleration under the influence of gravity. It is the conceptual foundation of Einstein’s general relativity, but is it true exactly? If not, there are profound implications for our understanding of gravity and the nature of the universe. It is therefore important to continue to test the equivalence principle as precisely as we can.

Galileo reportedly tested it by dropping spheres from the Leaning Tower of Pisa. Apollo astronauts tested it by dropping a hammer and a feather on the moon. More recent measurements have shown that the accelerations of two falling objects differ by no more than one part in 10¹³ [1, 2]. We aim to test the equivalence principle to one part in 10¹⁵ by dropping atoms of two different isotopes of rubidium in a 10 meter tower.

We will precisely measure acceleration differences between the two isotopes using atom interferometry. According to quantum mechanics, atoms are waves. Just as in optical interferometry, it is possible to split and recombine them to form an interference pattern [3, 4, 5]. In our interferometer, we send each atom along two different paths through space – each is in two places at once. When the atom waves are brought back together, the interference pattern depends on the phase difference between the two paths taken.

This phase difference in turn depends sensitively on the forces that act differently on the two parts of the atom while they are separated. This sensitivity to forces is what makes atom interferometry so useful. Compact atom interferometers have been made that can precisely measure rotation and acceleration, which can aid in navigation, mineral exploration, and geophysics. Atom interferometers have also measured the gravitational and fine-structure constants [6, 7]. They could also be used to search for gravitational waves [8].

The sensitivity of an atom interferometer increases with longer interferometer durations. Therefore, as recently described in Physical Review Letters [9, 10, 11], we have built an atom interferometer in which ⁸⁷Rb atoms are separated for 2.3 seconds before being recombined and interfered (Fig. 1). Three times longer than previous records [12], this multiple-second duration is well into the range of macroscopic, human-perceivable timescales. Furthermore, the two halves of each atom are separated by 1.4 centimeters before recombination – that's enough for you to swing your hand between them!

Fig. 1 Photograph of the 10 meter atomic fountain in a pit in the basement of the physics building at Stanford University.

How do we make a long-duration atom interferometer? We prepare a cloud of atoms at the bottom of a 10 meter vacuum tower and then launch them to the top. The interferometry is performed while the atoms rise up and fall back down to the bottom of the tower. The atoms are in free-fall, isolated from the noisy environment.

The cloud of atoms used must be very cold – a few billionths of a degree above absolute zero. At room temperature, the atoms in a gas move at speeds of hundreds of meters per second. Room-temperature rubidium atoms would collide with the walls of our vacuum chamber long before they fell back to the bottom. We therefore cool a few million rubidium atoms to a few nanokelvin before launching them into the tower. (The cooling process builds upon the same techniques used to generate Bose-Einstein condensates [13].)

Even at a few nanokelvin, individual atoms follow slightly different trajectories through the interferometer (like the droplets in a fountain of water), experiencing different position- and velocity-dependent forces. This yields a spatially-dependent phase, which in turn yields a spatial variation in the output atom density distribution that we can observe directly with a CCD camera (Fig. 2). This might at first appear undesirable, but it actually reveals rich details about the forces that generate the spatial interference pattern. Similar spatial fringe patterns have been used to great benefit in optical interferometers for centuries, but it is only recently that the effect has been leveraged in atom interferometry.

Fig. 2 Atomic interference patterns observed at the output of the interferometer. The images are sorted by phase, which can be measured for each experimental shot.

The long drift time of our interferometer enables it to have an acceleration sensitivity of 7 X 10^-12 g for each experimental shot, a hundredfold improvement over previous limits [14]. This is roughly the same as the gravitational attraction you would feel towards a person 10 meters away from you. We have used the sensitive interferometer and the spatial fringe patterns mentioned above to make precise measurements of Earth's rotation [9, 10].

The high sensitivity of the interferometer also holds great promise for our design goal – testing the equivalence principle (as mentioned above). By averaging more measurements or implementing advanced interferometry techniques, we can achieve the desired 10^-15 g sensitivity. Adding a simultaneous ⁸⁵Rb interferometer and comparing the results for the two isotopes will then enable us to make a new precision test of the equivalence principle. This will probe the fundamental assumptions of our current theory of gravity.

References:
[1] S. Schlamminger, K.Y. Choi, T.A. Wagner, J.H. Gundlach, and E.G. Adelberger, “Test of the Equivalence Principle Using a Rotating Torsion Balance”, Physical Review Letters, 100, 041101 (2008). Abstract.
[2] James G. Williams, Slava G. Turyshev, Dale H. Boggs, “Progress in Lunar Laser Ranging Tests of Relativistic Gravity”, Physical Review Letters, 93, 261101 (2004). Abstract.
[3] Mark Kasevich and Steven Chu, “Atomic interferometry using stimulated Raman transitions”, Physical Review Letters, 67, 181 (1991). Abstract.
[4] Alexander D. Cronin, Jörg Schmiedmayer, David E. Pritchard, “Optics and interferometry with atoms and molecules”, Reviews of Modern Physics, 81, 1051 (2009). Abstract.
[5] We focus on light-pulse atom interferometry, where pulses of laser light are used to split, recombine, and interfere the atoms.
[6] G. Lamporesi, A. Bertoldi, L. Cacciapuoti, M. Prevedelli, and G.M. Tino, “Determination of the Newtonian Gravitational Constant Using Atom Interferometry”, Physical Review Letters, 100, 050801 (2008). Abstract.
[7] Rym Bouchendira, Pierre Cladé, Saïda Guellati-Khélifa, François Nez, and François Biraben, “New Determination of the Fine Structure Constant and Test of the Quantum Electrodynamics”, Physical Review Letters, 106, 080801 (2011). Abstract.
[8] Jason Hogan, “A new method for detecting gravitational waves”, SPIE Newsroom, 6 May (2013). Article.
[9] Susannah M. Dickerson, Jason M. Hogan, Alex Sugarbaker, David M. S. Johnson, Mark A. Kasevich, “Multiaxis Inertial Sensing with Long-Time Point Source Atom Interferometry”, Physical Review Letters, 111, 083001 (2013).  Abstract.
[10] Alex Sugarbaker, Susannah M. Dickerson, Jason M. Hogan, David M. S. Johnson, Mark A. Kasevich, “Enhanced Atom Interferometer Readout through the Application of Phase Shear”, Physical Review Letters, 111, 113002 (2013). Abstract.
[11] P. Bouyer, “Viewpoint: A New Starting Point for Atom Interferometry”, Physics, 6, 92 (2013). Article.
[12] H. Müntinga, H. Ahlers, M. Krutzik, A. Wenzlawski, S. Arnold, D. Becker, K. Bongs, H. Dittus, H. Duncker, N. Gaaloul, C. Gherasim, E. Giese, C. Grzeschik, T. W. Hänsch, O. Hellmig, W. Herr, S. Herrmann, E. Kajari, S. Kleinert, C. Lämmerzahl, W. Lewoczko-Adamczyk, J. Malcolm, N. Meyer, R. Nolte, A. Peters, M. Popp, J. Reichel, A. Roura, J. Rudolph, M. Schiemangk, M. Schneider, S. T. Seidel, K. Sengstock, V. Tamma, T. Valenzuela, A. Vogel, R. Walser, T. Wendrich, P. Windpassinger, W. Zeller, T. van Zoest, W. Ertmer, W. P. Schleich, E. M. Rasel, “Interferometry with Bose-Einstein Condensates in Microgravity”, Physical Review Letters, 110, 093602 (2013). Abstract.
[13] “Bose-Einstein condensate”. Past 2Physics Article.
[14] Holger Müller, Sheng-wey Chiow, Sven Herrmann, Steven Chu, Keng-Yeow Chung, “Atom-Interferometry Tests of the Isotropy of Post-Newtonian Gravity”, Physical Review Letters, 100, 031101 (2008). Abstract.