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# Binary black hole

Computer simulation of the black hole binary system GW150914 as seen by a nearby observer, during its final inspiral, merge, and ringdown. The star field behind the black holes is being heavily distorted and appears to rotate and move, due to extreme gravitational lensing, as space-time itself is distorted and dragged around by the rotating black holes.[1]

A binary black hole (BBH) is a system consisting of two black holes in close orbit around each other. Like black holes themselves, binary black holes are often divided into stellar binary black holes, formed either as remnants of high-mass binary star systems or by dynamic processes and mutual capture, and binary supermassive black holes believed to be a result of galactic mergers.

For many years, proving the existence of BBHs was made difficult because of the nature of black holes themselves, and the limited means of detection available. However, in the event that a pair of black holes were to merge, an immense amount of energy should be given off as gravitational waves, with distinctive waveforms that can be calculated using general relativity.[2][3][4] Therefore, during the late 20th and early 21st century, BBHs became of great interest scientifically as a potential source of such waves, and a means by which gravitational waves could be proven to exist. BBH mergers would be one of the strongest known sources of gravitational waves in the Universe, and thus offer a good chance of directly detecting such waves. As the orbiting black holes give off these waves, the orbit decays, and the orbital period decreases. This stage is called binary black hole inspiral. The black holes will merge once they are close enough. Once merged, the single hole settles down to a stable form, via a stage called ringdown, where any distortion in the shape is dissipated as more gravitational waves.[5] In the final fraction of a second the black holes can reach extremely high velocity, and the gravitational wave amplitude reaches its peak.

The existence of stellar-mass binary black holes (and gravitational waves themselves) were finally confirmed when LIGO detected GW150914 (detected September 2015, announced February 2016), a distinctive gravitational wave signature of two merging stellar-mass black holes of around 30 solar masses each, occurring about 1.3 billion light years away. In its final moments of spiraling inward and merging, GW150914 released around 3 solar masses as gravitational energy, peaking at a rate of 3.6×1049 watts — more than the combined power of all light radiated by all the stars in the observable universe put together - during its last brief moments.[6][7][8] Supermassive binary black hole candidates have been found but as yet, not categorically proven.[9]

## Occurrence

Supermassive black-hole binaries are believed to form during galaxy mergers. Some likely candidates for binary black holes are galaxies with double cores still far apart. An example double nucleus is NGC 6240.[10] Much closer black-hole binaries are likely in single core galaxies with double emission lines. Examples include SDSS J104807.74+005543.5[11] and EGSD2 J142033.66 525917.5.[10] Other galactic nuclei have periodic emissions suggesting large objects orbiting a central black hole, for example in OJ287.[12]

The quasar PG 1302-102 appears to have a binary black hole with an orbital period of 1900 days.[13]

Stellar mass binary black holes have been demonstrated to exist, by the first detection of a black hole merger event GW150914 by LIGO.[14]

## Final parsec problem

When two galaxies collide, the supermassive black holes at their centers do not hit head-on, but would shoot past each other if some mechanism did not bring them together. The most important mechanism is dynamical friction, which brings the black holes to within a few parsecs of each other. At this distance, they form a bound, binary system. The binary system must lose orbital energy somehow, for the black holes to orbit more closely or merge.[15]

Initially, the explanation is easy. The black holes transfer energy to gas and stars between them, ejecting matter at high speed via a gravitational slingshot and thereby losing energy. However, the volume of space subject to this effect shrinks as the orbits do, and when the black holes reach a separation of about one parsec, there is so little matter left between them that it would take billions of years to orbit closely enough to merge - more than the age of the universe. Gravitational waves can be a significant contributor, but not until the separation shrinks to a much smaller value, roughly 0.01–0.001 parsec.

Nonetheless, supermassive black holes appear to have merged, and what appears to be a pair in this intermediate range has been observed, in PKS 1302-102.[16] The question of how this happens is the "final parsec problem".[17]

A number of solutions to the final parsec problem have been proposed. Most involve the interaction of the massive binary with surrounding matter, either stars or gas, which can extract energy from the binary and cause it to shrink. For instance, if enough stars pass close by to the orbiting pair, their gravitational ejection can bring the two black holes together much quicker than would otherwise be the case.[18]

## Lifecycle

### Inspiral

The first stage of the life of a binary black hole is the inspiral which resembles a gradually shrinking orbit. The first stages of the inspiral take a very long time, as the gravitation waves emitted are very weak when the black holes are distant from each other. In addition to the orbit shrinking due to the emission of gravitational waves, extra angular momentum may be lost due to interactions with other matter present, such as other stars.

As the black holes’ orbit shrinks, the speed increases, and gravitational wave emission increases. When the black holes are close the gravitational waves cause the orbit to shrink rapidly.

The last stable orbit or innermost stable circular orbit (ISCO) is the innermost complete orbit before the transition from inspiral to merger.

### Merger

This is followed by a plunging orbit in which the two black holes meet, followed by the merger. Gravitational wave emission peaks at this time.

### Ringdown

Immediately following the merger, the now single black hole will “ring” – oscillating in shape between a distorted, elongated spheroid and a flattened spheroid. This ringing is damped in the next stage, called the ringdown, by the emission of gravitational waves. The distortions from the spherical shape rapidly reduce until the final stable sphere is present, with a possible slight distortion due to remaining spin.

## Observation

The first observation of stellar mass binary black holes merging was performed by the LIGO detector.[14][19][20] As measured from earth, a pair of black holes with estimated masses around 36 and 29 times that of the Sun spun into each other and merged to form a 62 solar mass black hole (approximate) on 14 September 2015, at 09:50 UTC.[21] Three solar masses were converted to gravitational radiation in the final fraction of a second, with a peak power 3.6×1056 ergs/second (200 solar masses per second),[14] which is 50 times the total output power of all the stars in the observable universe.[22] The merger took place at 1.3 billion light years from Earth.[19] The observed signal is consistent with the predictions of numerical relativity.[2][3][4]

## Dynamics modelling

Some simplified algebraic models can be used for the case where the black holes are far apart, during the inspiral stage, and also to solve for the final ringdown.

Post-Newtonian approximations can be used for the inspiral. These approximate the general relativity field equations adding extra terms to equations in Newtonian gravity. Orders used in these calculations may be termed 2PN (second order post Newtonian) 2.5PN or 3PN (third order post Newtonian). Effective-one-body (EOB) solves the dynamics of the binary black hole system by transforming the equations to those of a single object. This is especially useful where mass ratios are large, such as a stellar mass black hole merging with a galactic core black hole, but can also be used for equal mass systems.

For the ringdown, black hole perturbation theory can be used. The final Kerr black hole is distorted, and the spectrum of frequencies it produces can be calculated.

To solve for the entire evolution, including merger, requires solving the full equations of general relativity. This can be done in numerical relativity simulations. Numerical relativity models space-time and simulates its change over time. In these calculations it is important to have enough fine detail close into the black holes, and yet have enough volume to determine the gravitation radiation that propagates to infinity. In order to make this have few enough points to be tractable to calculation in a reasonable time, special coordinate systems can be used such as Boyer-Lindquist coordinates or fish-eye coordinates.

Numerical relativity techniques steadily improved from the initial attempts in the 1960s and 1970s.[23][24] Long-term simulations of orbiting black holes, however, were not possible until three groups independently developed groundbreaking new methods to model the inspiral, merger, and ringdown of binary black holes [2][3][4] in 2005.

In the full calculations of an entire merger, several of the above methods can be used together. It is then important to fit the different pieces of the model that were worked out using different algorithms. The Lazarus Project linked the parts on a spacelike hypersurface at the time of the merger.[25]

Results from the calculations can include the binding energy. In a stable orbit the binding energy is a local minimum relative to parameter perturbation. At the innermost stable circular orbit the local minimum becomes an inflection point.

The gravitational waveform produced is important for observation prediction and confirmation. When inspiralling reaches the strong zone of the gravitational field, the waves scatter within the zone producing what is called the post Newtonian tail (PN tail).[25]

In the ringdown phase of a Kerr black hole, frame-dragging produces a gravitation wave with the horizon frequency. In contrast the Schwarzschild black-hole ringdown looks like the scattered wave from the late inspiral, but with no direct wave.[25]

The radiation reaction force can be calculated by Padé resummation of gravitational wave flux. A technique to establish the radiation is the Cauchy characteristic extraction technique CCE which gives a close estimate of the flux at infinity, without having to calculate at larger and larger finite distances.

The final mass of the resultant black hole depends on the definition of mass in general relativity. The Bondi mass MB is calculated from the Bondi-Sach mass loss formula. ${\displaystyle {\frac {dM_{B}}{dU}}=-f(U)}$ . With f(U) the gravitational wave flux at retarded time U. f is a surface integral of the News function at null infinity varied by solid angle. The Arnowitt-Deser-Misner (ADM) energy or ADM mass is the mass as measured at infinite distance and includes all the gravitational radiation emitted. ${\displaystyle M_{ADM}=M_{B}(U)+\int _{-\infty }^{U}F(V)dV}$ .

Angular momentum is also lost in the gravitational radiation. This is primarily in the z axis of the initial orbit. It is calculated by integrating the product of the multipolar metric waveform with the news function complement over retarded time.[26]

## Shape

One of the problems to solve is the shape or topology of the event horizon during a black-hole merger.

In numerical models, test geodesics are inserted to see if they encounter an event horizon. As two black holes approach each other, a ‘duckbill’ shape protrudes from each of the two event horizons towards the other one. This protrusion extends longer and narrower until it meets the protrusion from the other black hole. At this point in time the event horizon has a very narrow X-shape at the meeting point. The protrusions are drawn out into a thin thread.[27] The meeting point expands to a roughly cylindrical connection called a bridge.[27]

Simulations as of 2011 had not produced any event horizons with toroidal topology (ring-shaped). Some researchers suggested that it would be possible if, for example, several black holes in the same nearly-circular orbit coalesce.[27]

## Black-hole merger recoil

An unexpected result can occur with binary black holes that merge in that the gravitational waves carry momentum and the merging black-hole pair accelerates seemingly violating Newton's third law. The center of gravity can add over 1000 km/s of kick velocity.[28] The greatest kick velocities (approaching 5000 km/s) occur for equal-mass and equal-spin-magnitude black-hole binaries, when the spins directions are optimally oriented to be counter-aligned, parallel to the orbital plane or nearly aligned with the orbital angular momentum.[29] This is enough to escape large galaxies. With more likely orientations a smaller effect takes place, perhaps only a few hundred kilometers per second. This sort of speed will eject merging binary black holes from globular clusters, thus preventing the formation of massive black holes in globular cluster cores. In turn this reduces the chances of subsequent mergers, and thus the chance of detecting gravitational waves. For non spinning black holes a maximum recoil velocity of 175 km/s occurs for masses in the ratio of five to one. When spins are aligned in the orbital plane a recoil of 5000 km/s is possible with two identical black holes.[30] Parameters that may be of interest include the point at which the black holes merge, the mass ratio which produces maximum kick, and how much mass/energy is radiated via gravitational waves. In a head-on collision this fraction is calculated at 0.002 or 0.2%.[31] One of the best candidates of the recoiled supermassive black holes is CXO J101527.2+625911.[32]

## References

1. ^
2. ^ a b c Pretorius, Frans (2005). "Evolution of Binary Black-Hole Spacetimes". Physical Review Letters. 95 (12): 121101. arXiv:gr-qc/0507014. Bibcode:2005PhRvL..95l1101P. doi:10.1103/PhysRevLett.95.121101. ISSN 0031-9007. PMID 16197061.
3. ^ a b c Campanelli, M.; Lousto, C. O.; Marronetti, P.; Zlochower, Y. (2006). "Accurate Evolutions of Orbiting Black-Hole Binaries without Excision". Physical Review Letters. 96 (11): 111101. arXiv:gr-qc/0511048. Bibcode:2006PhRvL..96k1101C. doi:10.1103/PhysRevLett.96.111101. ISSN 0031-9007. PMID 16605808.
4. ^ a b c Baker, John G.; Centrella, Joan; Choi, Dae-Il; Koppitz, Michael; van Meter, James (2006). "Gravitational-Wave Extraction from an Inspiraling Configuration of Merging Black Holes". Physical Review Letters. 96 (11): 111102. arXiv:gr-qc/0511103. Bibcode:2006PhRvL..96k1102B. doi:10.1103/PhysRevLett.96.111102. ISSN 0031-9007. PMID 16605809.
5. ^ Abadie, J.; LIGO Scientific Collaboration; The Virgo Collaboration; Abernathy, M.; Accadia, T.; Acernese, F.; Adams, C.; Adhikari, R.; Ajith, P.; Allen, B.; Allen, G. S.; Amador Ceron, E.; Amin, R. S.; Anderson, S. B.; Anderson, W. G.; Antonucci, F.; Arain, M. A.; Araya, M. C.; Aronsson, M.; Aso, Y.; Aston, S. M.; Astone, P.; Atkinson, D.; Aufmuth, P.; Aulbert, C.; Babak, S.; Baker, P.; Ballardin, G.; Ballinger, T.; et al. (2011). "Search for gravitational waves from binary black hole inspiral, merger and ringdown". Physical Review D. 83 (12): 122005. arXiv:1102.3781. Bibcode:2011PhRvD..83l2005A. doi:10.1103/PhysRevD.83.122005.
6. ^ "Observation Of Gravitational Waves From A Binary Black Hole Merger" (PDF). LIGO. 11 February 2016. Archived from the original (PDF) on 16 February 2016. Retrieved 11 February 2016.
7. ^ Harwood, W. (11 February 2016). "Einstein was right: Scientists detect gravitational waves in breakthrough". CBS News. Archived from the original on 12 February 2016. Retrieved 12 February 2016.
8. ^ Drake, Nadia (11 February 2016). "Found! Gravitational Waves, or a Wrinkle in Spacetime". National Geographic News. Archived from the original on 12 February 2016. Retrieved 12 February 2016.
9. ^ Liu, Fukun; Komossa, Stefanie; Schartel, Norbert (22 April 2014). "UNIQUE PAIR OF HIDDEN BLACK HOLES DISCOVERED BY XMM-NEWTON". A milli-parsec supermassive black hole binary candidate in the galaxy SDSS J120136.02+300305.5. Retrieved 23 December 2014.
10. ^ a b Gerke, Brian F.; Newman, Jeffrey A.; Lotz, Jennifer; Yan, Renbin; Barmby, P.; Coil, Alison L.; Conselice, Christopher J.; Ivison, R. J.; Lin, Lihwai; Koo, David C.; Nandra, Kirpal; Salim, Samir; Small, Todd; Weiner, Benjamin J.; Cooper, Michael C.; Davis, Marc; Faber, S. M.; Guhathakurta, Puragra; et al. (6 April 2007). "The DEEP2 Galaxy Redshift Survey: AEGIS Observations of a Dual AGN AT z p 0.7" (PDF). The Astrophysical Journal Letters. 660: L23–L26. arXiv:astro-ph/0608380. Bibcode:2007ApJ...660L..23G. doi:10.1086/517968.
11. ^ Hongyan Zhou; Tinggui Wang; Xueguang Zhang; Xiaobo Dong; Cheng Li (26 February 2004). "Obscured Binary Quasar Cores in SDSS J104807.74+005543.5?". The Astrophysical Journal Letters. The American Astronomical Society. 604: L33–L36. arXiv:astro-ph/0411167. Bibcode:2004ApJ...604L..33Z. doi:10.1086/383310.
12. ^ Valtonen, M. V.; Mikkola, S.; Merritt, D.; Gopakumar, A.; Lehto, H. J.; Hyvönen, T.; Rampadarath, H.; Saunders, R.; Basta, M.; Hudec, R. (February 2010). "Measuring the Spin of the Primary Black Hole in OJ287". The Astrophysical Journal. The American Astronomical Society. 709 (2): 725–732. arXiv:0912.1209. Bibcode:2010ApJ...709..725V. doi:10.1088/0004-637X/709/2/725.
13. ^ Graham, Matthew J.; Djorgovski, S. G.; Stern, Daniel; Glikman, Eilat; Drake, Andrew J.; Mahabal, Ashish A.; Donalek, Ciro; Larson, Steve; Christensen, Eric (7 January 2015). "A possible close supermassive black-hole binary in a quasar with optical periodicity". Nature. 518 (7537): 74–6. arXiv:1501.01375. Bibcode:2015Natur.518...74G. doi:10.1038/nature14143. ISSN 0028-0836. PMID 25561176.
14. ^ a b c B. P. Abbott; et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). "Observation of Gravitational Waves from a Binary Black Hole Merger". Physical Review Letters. 116 (6): 061102. arXiv:1602.03837. Bibcode:2016PhRvL.116f1102A. doi:10.1103/PhysRevLett.116.061102. PMID 26918975.
15. ^ More Evidence for Coming Black Hole Collision - The New York Times
16. ^ D'Orazio, Daniel J.; Haiman, Zoltán; Schiminovich, David (17 September 2015). "Relativistic boost as the cause of periodicity in a massive black-hole binary candidate". Nature. 525 (7569): 351–353. arXiv:1509.04301. Bibcode:2015Natur.525..351D. doi:10.1038/nature15262.
17. ^ Milosavljević, Miloš; Merritt, David (October 2003). "The Final Parsec Problem" (PDF). AIP Conference Proceedings. American Institute of Physics. 686 (1): 201–210. arXiv:astro-ph/0212270. Bibcode:2003AIPC..686..201M. doi:10.1063/1.1629432.
18. ^ Merritt, David (2013). Dynamics and Evolution of Galactic Nuclei. Princeton: Princeton University Press. ISBN 978-0-691-12101-7.
19. ^ a b Castelvecchi, Davide; Witze, Witze (February 11, 2016). "Einstein's gravitational waves found at last". Nature News. doi:10.1038/nature.2016.19361. Retrieved 2016-02-11.
20. ^ "Gravitational waves detected 100 years after Einstein's prediction | NSF - National Science Foundation". www.nsf.gov. Retrieved 2016-02-11.
21. ^ Abbott, Benjamin P.; et al. (LIGO Scientific Collaboration and Virgo Collaboration) (11 February 2016). "Properties of the binary black hole merger GW150914". Physical Review Letters. 116 (24): 241102. arXiv:1602.03840. Bibcode:2016PhRvL.116x1102A. doi:10.1103/PhysRevLett.116.241102. PMID 27367378.
22. ^ Kramer, Sarah (11 February 2016). "This collision was 50 times more powerful than all the stars in the universe combined". Tech Insider. Retrieved 12 February 2016.
23. ^ Hahn, Susan G; Lindquist, Richard W (1964). "The two-body problem in geometrodynamics". Annals of Physics. 29 (2): 304–331. Bibcode:1964AnPhy..29..304H. doi:10.1016/0003-4916(64)90223-4. ISSN 0003-4916.
24. ^ Smarr, Larry; Čadež, Andrej; DeWitt, Bryce; Eppley, Kenneth (1976). "Collision of two black holes: Theoretical framework". Physical Review D. 14 (10): 2443–2452. Bibcode:1976PhRvD..14.2443S. doi:10.1103/PhysRevD.14.2443. ISSN 0556-2821.
25. ^ a b c Nichols, David A.; Yanbei Chen (1 September 2011). "Hybrid method for understanding black-hole mergers: Inspiralling case". Physical Review D. 85 (4): 044035. arXiv:1109.0081. Bibcode:2012PhRvD..85d4035N. doi:10.1103/PhysRevD.85.044035.
26. ^ Thibault
27. ^ a b c Cohen, Michael I.; Jeffrey D. Kaplan; Mark A. Scheel (11 October 2011). "On Toroidal Horizons in Binary Black Hole Inspirals". Physical Review D. 85 (2): 024031. arXiv:1110.1668. Bibcode:2012PhRvD..85b4031C. doi:10.1103/PhysRevD.85.024031.
28. ^ Pietilä, Harri; Heinämäki, Pekka; Mikkola, Seppo; Valtonen, Mauri J. (10 January 1996). Anisotropic Gravitational Radiation In The Merger Of Black Holes. Relativistic Astrophysics Conference. CiteSeerX 10.1.1.51.2616.
29. ^ Campanelli, Manuela; Lousto, Carlos; Zlochower, Yosef; Merritt, David (7 June 2007). "Maximum Gravitational Recoil". Maximum Gravitational Recoil. arXiv:gr-qc/0702133. Bibcode:2007PhRvL..98w1102C. doi:10.1103/PhysRevLett.98.231102.
30. ^ Lousto, Carlos; Zlochower, Yosef (2 December 2007). "Hangup Kicks: Still Larger Recoils by Partial Spin-Orbit Alignment of Black-Hole Binaries". Hangup Kicks: Still Larger Recoils by Partial Spin-Orbit Alignment of Black-Hole Binaries. arXiv:1108.2009. Bibcode:2011PhRvL.107w1102L. doi:10.1103/PhysRevLett.107.231102.
31. ^ Pietilä, Harri; Heinämäki, Pekka; Mikkola, Seppo; Valtonen, Mauri J. (1995). "Anisotropic gravitational radiation in the problems of three and four black holes". Celestial Mechanics and Dynamical Astronomy. 62 (4): 377–394. Bibcode:1995CeMDA..62..377P. CiteSeerX 10.1.1.51.2616. doi:10.1007/BF00692287.
32. ^ Kim, D.-C.; et al. (2017). "A Potential Recoiling Supermassive Black Hole CXO J101527.2+625911". Astrophysical Journal. 840: 71–77. arXiv:1704.05549. Bibcode:2017ApJ...840...71K. doi:10.3847/1538-4357/aa6030.


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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 Radial oscillation of linear differential equation Bull. Korean Math. Soc. 2012 Vol. 49, No. 5, 911-921 https://doi.org/10.4134/BKMS.2012.49.5.911Printed September 30, 2012 Zhaojun Wu Hubei University of Science and Technology Abstract : In this paper, the radial oscillation of the solutions of higher order homogeneous linear differential equation $$f^{(k)}+A_{n-2}(z)f^{(k-2)}+\cdots+A_{1}(z)f'+A_{0}(z)f=0$$ with transcendental entire function coefficients is studied. Results are obtained to extend some results in [Z. Wu and D. Sun, Angular distribution of solutions of higher order linear differential equations, J. Korean Math. Soc. $\bf 44$ (2007), no. 6, 1329--1338]. Keywords : radial exponent of convergence, iterated order, sectorial oscillation, Borel direction MSC numbers : Primary 30D30; Secondary 30D35, 34M20 Downloads: Full-text PDF

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## The Geometry of Special Relativity

### Errata

(Last update: 2/28/19)

• Page 10, last paragraph, line 2: “observer a rest” should be “observer at rest”.
• Page 16, Equation (3.12): The matrix should be transposed (the minus sign should be in the upper right).
• Page 26, first two full paragraphs: The unit conversions with $c$ are backward. The best fix is to swap “…measure distance in seconds” with “… measure time in meters”, and probably also swap the two subsequent sentences (each starting with “Thus,”) in their entirety.
• Pages 30–35: The examples in §5.5 assume that both vectors lie on the same branch of the hyperbola. For instance, all timelike vectors are assumed to be future-pointing. In most cases, this restriction can be removed by replacing $\cosh\alpha$ by $\pm\cosh\alpha$.
• Page 34, Figure 5.6: The labels are reversed; (a) shows timelike vectors, and (b) shows spacelike vectors.
• Page 35, Figure 5.7: The caption is misleading; both figures show projections between a timelike and a spacelike vector.
• Page 38, first full paragraph, line 9: $\beta$ is an angle, not a distance, so “$\beta$ is precisely” should be “$\rho\beta$ is precisely”.
• Page 43, Equation (6.9): Both $t$ and $t'$ should be preceded by $\Delta$; the equation should read $\Delta t'=\frac{\Delta t}{\cosh\beta}$.
• Page 53, Equation (7.18): The penultimate expression should be $\frac{\tan\theta}{1-\frac{v}{u\cos\theta}}$; the full equation should read:

$\tan\theta' = \frac{u_y'}{u_x'} = \frac{u_y}{u_x-v} = \frac{\tan\theta}{1-\frac{v}{u\cos\theta}} = \frac{u\sin\theta}{u\cos\theta-v}$

• Page 54, Figure 7.8(b): The side labeled “$\Delta x$” should be labeled “$\Delta x'$”.
• Page 68, Figure 9.1: The hyperbolic angle $\beta$ should be $\alpha$.
• Page 72, Figure 9.2: The hyperbolic angle $\beta$ should be $\alpha$.
• Page 72, penultimate paragraph, line 1: “makes sense” should be “make sense”.
• Page 75, lines 3–4: “elastic” should be “inelastic” (or the first part of the sentence should be omitted entirely).
• Page 83, just before (11.2): “current density” should be “current”.
• Page 85, Equation (11.12): $\lambda$ should be $\lambda_0$ (2x).
• Page 86, Equation (11.20): There should be no minus sign on the RHS.
• Page 91, Equation (11.54): $u$ should be $\mathbf{u}$.
• Page 91, line 9: The extra “is” should be deleted.
• Page 93, Equation (11.71): $F^{uv}$ should be $F^{\mu\nu}$.
• Page 94, Equation (11.80): $\partial t$ should be $c\,\partial t$. The full equation should read:

$\Partial{F^{\mu\nu}}{x^\nu} = \frac{1}{c}\Partial{F^{\mu0}}{t} + \Partial{F^{\mu1}}{x} + \Partial{F^{\mu2}}{y} + \Partial{F^{\mu3}}{z}$

• Page 95, Equation (11.92): The last instance of $G_{\mu\nu}$ should be $G^{\mu\nu}$. The full equation should read:

$\frac12 F_{\mu\nu} F^{\mu\nu} = -\frac{1}{c^2}\,{|\EE|}^2 + |\BB|^2 = - \frac12 G_{\mu\nu} G^{\mu\nu}$

• Page 96, Equation (11.96): Delete $m$ on the LHS (or replace $p^\mu$ by $u^\mu$).
• Page 109, Equation (14.3): The minus sign is in the wrong place; the correct equation is:

$d_H(U,V) = \rho\,\cosh^{-1}\left(-\frac{\uu\cdot\vv}{\rho^2}\right)$

• Page 109, next sentence: The sentence should end with a period.
• Page 109, Equation (14.6): There should be absolute values in the last expression; the full equation should read:

\begin{align*} d_H(U,V) &= \rho\,\cosh^{-1}(-\sinh\alpha\sinh\beta+\cosh\alpha\cosh\beta) \\ &= \rho\,\cosh^{-1}(\cosh(\beta-\alpha)) \\ &= \rho\,|\beta-\alpha| \end{align*}

• Page 112, Equation (14.15): There should be an overall minus sign; the correct equation is:

$X + iY = -2\frac{i(x+iy)+\rho}{(x+iy)+\rho\,i}$

• Page 112, next sentence: “unit disk” should be “disk of radius $\rho$”, and $y\ge0$ should be $Y\ge0$.
• Page 112, Equation (14.16): The first denominator is incorrect; the correct equation is:

$ds^2 = \frac{4\,\rho^4\left(dx^2+dy^2\right)}{(\rho^2-x^2-y^2)^2} = \rho^2\>\frac{dX^2+dY^2}{Y^2}$

• Page 115, last paragraph: The closed interval $[0,\pi/2]$ should be the half-open interval $(0,\pi/2]$.
• Pages 120–125: The points $A$ and $B$ have been interchanged here compared to §4.3. One fix is to interchange $A$ and $B$ throughout §4.3, including in Figure 4.2, also in Figure 5.2, and finally in §6.2.
• Page 123, line -4: “$\rho\,\sinh\beta$” should be “$\rho\,\sinh\alpha$”.
• Page 123, Equation (15.24): $A$ should be $B$ in the derivative; the full equation should read $B^\perp = (\rho\,\cosh\beta,\rho\,\sinh\beta)=\frac{dB}{d\beta}$.

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# Chapter 5 - Section 5.3 - The Definite Integral - Exercises: 43

$-2$

#### Work Step by Step

According to Rule (4) in Table 5.4 : $\int_{0}^{2} (2t-3)dt=\int_{0}^{2} 2tdt+\int_{0}^{2} (-3)dt$ According to Rule (3) in Table 5.4 and because $0\lt2$, by Equation (2): $\int_{0}^{2} 2tdt+\int_{0}^{2} (-3)dt=2\int_{0}^{2} tdt-3\int_{0}^{2} 1dt=2\cdot(\frac{2^2}{2}-\frac{0^2}{2})-3\cdot(2-0)=2\cdot(2-0)-3\cdot2=4-6=-2$

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i-Hack 2018 Qualification: Reverse and Pwn Write-up

This write-up are focus on reverse engineering and pwn category for i-Hack 2018 Qualification.

First, run file command on the binary to check what type of data is it. So based on the output above, it is an ELF 32bit file. So, let’s run it on our terminal.

The program ask us for the password and we test the password with “TEST” string and the program said “Wrong!!” then terminate. So, from above behavior we can guess this is a CrackMe Challenge.

Let’s go deeper into the program using this tool, GDB debugger. Type gdb ./<filename>. In case of this, I’ll be using gdb with peda extension.

Type pdisas main then enter to disassemble the main function. You will see lot of Assembly Language code.

From the code, we can see there is C library function which is strcmp() function at address 0x080485f7. I think it will compare our input strings with the real password.
So here, we use a program call ltrace tool to see what is the strings that the program compare with.

Run it and try with any strings on the fgets() function line.

Then, it will compare our strings with the real password.

The password is ihack2018^:. So, we got the password and we can try it on the program.

Yeah. We get the flag by use the password.

By the way, you can also run the strings command on the binary to get the flag. I forgot to use it on the first try of the challenge, then I realize it more easier and save time than debug the program but atleast I sharps my skill using gdb and ltrace.

Reverse Engineering – Another Password Challenge

Soon we got the binary. I run it first and yes same concept like previous challenge. Crackme challenge.

File command the program first and it said a 32bit ELF file. Okay let’s move on. Strings command the program and we found a strings “Hackf00t” and “StaR”.

Try both of it, and it’s wrong. So I try to join the words became Hackf00tStaR and yeah, It is the password! And we got the flag.

But during the game. Again, I forgot to strings command the program first. There will be lot of time I saved just by strings it. Hahaha.

So, during the game, I do static analysis code using IDA Pro and I debug it using gdb and ltrace, and the program was so tricky to be debugging.

So based on static analysis on it using IDA pro, I see a function call ptrace which is make this program tricky to debug. By seeing this graph on IDA, I think we can skip the ptrace function by set the jump into the loc_8048626 which is a function that we want.

So using GDB, I make a brakepoint before the ptrace function and set $eip to function loc_8048626’s address which is 08048626. Then by continue the program we can see that the program compare our strings with Hackf00tStaR strings. I repeat again the steps, • Breakpoint at any place before ptrace function. • Breakpoint on strcmp. • run program using r • jump to 0x08048626 address using jmp *0x08048626 • Type “ni” until you see the strings comparison Reverse Engineering – What is the protection used by the chal2? 3rd challenge, they ask us what the protection used by the chal2 that make the program tricky to debug. By observing and decompile the program using IDA Pro, the API call or protection that been used by the chal2 is ptrace. Reverse Engineering – Keygen Me For Your Flag Soon got the binary. I run it first and it ask us for a key. So, try random key and it not correct. So, we fire up our gdb and try to debug it. From the code that we assemble the main function, the program making comparison (the check_with_mod function) with my computer system. Decompile it using IDA maybe can get the better understanding for this program. Let’s do. So here it is, in yellow circle, it is the some of check_with_mod functions that we must bypass, and the blue one is succeed function, our target. When we decompile it, we see a function call succeed which is maybe a flag are store in this function. So all we need is skip all the check_with_mod function and go to succeed function. Fire up gdb again. Disassemble it. So, • I breakpoint on the first instruction on the main • run the program • I set the $eip to the “succeed” function address which is 0x080487bd.

And… we got the flag.

Reverse Engineering – Dynamic is my style

Assalamualaikum.

First I run the program, and it compare my computer name with their variable.

File command the program, and it ELF 32bit file. So I immidiately fire up my IDA to see what strings that the program compare with.

Yeah from above, we can see that it check the hostname of our computer with ihackreverser!~ strings and few other strcmp function.

Hmm, come to my mind “do we need to change our hostname?”

So, from above decompile code, at the bottom of the code there is genrandom() function. Double click on it. And we can see that is a flag but it is encrypted with some algorithm of code.

So, by using gdb tool, I think we can go direct to the genrandom() by ignore all the strcmp function.

From above code, breakpoint before of any function calls, then jump to the genrandom() function address which is 0x08048c99 and we got the flag.

Binary Exploitation – Feed Me

During competition we got five challenge for pwn challenge, and we managed to solved only three from the five challenges. Chal1 which is my friend solved it. While me solve the chal4 and chal5 challenges.

Here is the chal4 challenge.

So,

They gave us chal4 binary to us to decompile it and understand the code. We try to run it and input anything. And it terminate itself said segmentation fault.

By analyse it using gdb-peda. Everything we input for the fgets function, the input will overwrite the EIP register. By looking at the title’s challenge which is “Feed me”, we think let’s feed the program with a shellcode.

Use your Google-fu skill, google any shellcode and make the exploit for the program. Here the one that we used during the competition.

\x31\xc9\xf7\xe1\xb0\x0b\x51\x68\x2f\x2f\x73\x68\x68\x2f\x62\x69\x6e\x89\xe3\xcd\x80

Source: http://shell-storm.org/shellcode/files/shellcode-752.php

Ok now let’s do some python script using pwntool. Copy the shellcode into our code.

Run it and we got our sweet flag.

Binary Exploitation – Vanilla AlephOne

Last pwn challenge. They gave us chal5 binary to us to decompile it and understand the code.

We try to run it and it ask us to input something. And it said ECHO: [our input].

Okay let’s make the program crash.

We use cyclic tools to generate random strings pattern to easier to us to detect which line that overwrite the EIP.

Copy it to the program and the program crash.

Let’s make it in gdb to know which is the line that overwrite the EIP.

And the “laaa” characters overwrite the EIP value.

Use cyclic again to know the lookup value of laaa character.


text:
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Emergence of pseudogap from short-range spin-correlations in electron-doped cuprates

Abstract

Complex electron interactions underlie the electronic structure of several families of quantum materials. In particular, the strong electron Coulomb repulsion is considered the key ingredient to describing the emergence of exotic and/or ordered phases of quantum matter, from high-temperature superconductivity to charge- and magnetic-order. However, a comprehensive understanding of fundamental electronic properties of quantum materials is often complicated by the appearance of an enigmatic partial suppression of low-energy electronic states, known as the pseudogap. Here we take advantage of ultrafast angle-resolved photoemission spectroscopy to unveil the temperature evolution of the low-energy density of states in the electron-doped cuprate Nd2-xCexCuO4, an emblematic system where the pseudogap intertwines with magnetic degrees of freedom. Using an optical excitation we drive the electronic system across the pseudogap onset temperature T*, and we report the direct relation between the momentum-resolved pseudogap spectral features and the spin-correlation length with a remarkable sensitivity. This transient approach, corroborated by mean-field model calculations, allows us to establish the pseudogap in electron-doped cuprates as a precursor to the incipient antiferromagnetic order even when long-range antiferromagnetic correlations are not established, as in the case of optimal doping.

Introduction

In the presence of strong correlations, the interactions within and among various degrees of freedom often obfuscate the microscopic origin of exotic electronic phenomena.1,2,3 As an example, the interplay between intertwined orders4 continues to preclude a thorough understanding of the pseudogap (PG) phenomenon, a mysterious state of correlated matter ubiquitous to systems as diverse as unconventional superconductors,5,6,7 dichalcogenides,8,9 and ultracold atoms.10,11,12 Broadly speaking, the PG in condensed matter is associated with a partial suppression of the electronic spectral weight in the vicinity of the Fermi level (ω = 0), and evidence for the PG has been widely reported.13 This behavior may be anticipated in the presence of long-range (or mesoscopic) order, e.g. spin- or charge-order, which breaks the translational symmetry of the crystal: the loss of spectral weight in particular momentum-energy regions would be simply a consequence of the avoided crossings in the symmetry-reduced bandstructure.8,9,14,15 However, this argument may be unsatisfactory in the presence of strong electronic correlations and short-range orders with correlation length of few unit cells. Copper-oxide high-temperature superconductors are a paradigmatic example where the origin of the PG—which presents different phenomenology for hole and electron doping—is still debated, and a universal understanding has yet to emerge.1,5,6,7,14,15,16,17,18

In the specific case of electron-doped cuprates, the PG is stable above the entire antiferromagnetic (AF) and superconducting (SC) domes, with its onset temperature indicated by T* as measured by spectroscopic and transport probes,18,19,20,21,22 as illustrated in Fig. 1a (orange shadow). Scattering experiments on electron-doped cuprates have shown that the long-range AF order disappears when entering the narrow SC dome,23,24 and that the commonly reported charge-order in cuprates does not exhibit a clear connection to the AF order,25,26 although a coupling to dynamic magnetic correlations has been recently shown.27 In addition, 3D collective charge modes, which may play a substantial role in mediating high-temperature superconductivity, have been reported.28

The PG in electron-doped cuprates is believed to be related to the AF order:18,19,20,21,22,24,29,30,31,32 in the presence of long-range AF order, i.e. when the instantaneous spin-correlation length (ξspin) diverges at low temperature and a Néel temperature is defined, T* has been proposed to be a temperature crossover for which the quasiparticle de Broglie wavelength (λB ~ vFπT, where vF is the Fermi velocity) becomes comparable to ξspin.24,33 However, these considerations seem to fail at optimal doping, where only short-range spin-fluctuations (ξspin 20–25 a, where a is the unit cell size) are detected by inelastic neutron scattering.24 Indeed, for dopings where the long-range AF order disappears, i.e. when the short-range ξspin does not diverge at low temperature, an unambiguous identification of a temperature crossover with λB is prevented. In addition, the underlying SC phase has been proposed to limit the development of ξspin.24 Finally, and most importantly, a momentum-resolved study connecting explicitly the PG spectral features and short-range AF correlations in electron-doped cuprates is still missing.

In order to tie together the observations of the PG and short-range AF correlations via a unique experimental approach, we performed a time- and angle-resolved photoemission (TR-ARPES) study of optimally doped Nd2-xCexCuO4 (NCCO, Tc ~ 24 K, yellow arrow in Fig. 1a), which is characterized by ξspin ~ 20 a at low temperatures (T ~ Tc).24 TR-ARPES provides an alternative, more effective and controlled experimental approach to measure a detailed temperature-dependence than the standard equilibrium ARPES, which is often complicated by surface degradation as well as coarse and uncorrelated sampling. As in standard pump-probe spectroscopy, a near-infrared pump pulse is used to perturb the system, whose subsequent relaxation is studied by varying the temporal delay of a UV probe pulse. Photoexcited quasiparticles in cuprates—and in general in strongly correlated electron systems—release the energy deposited by the pump pulse through electron–electron and electron–boson scattering, thermalizing and reaching a state of quasi-equilibrium on an ultrafast time scale of about 100 fs.34,35,36,37,38,39 After this initial relaxation, an effective electronic temperature Te is defined at each point in time, allowing a temperature-dependent scan to be performed continuously and with remarkable accuracy and efficiency.40 Since the acquisition of TR-ARPES data is performed by cycling continuously the pump-probe delays, each time delay—and consequently each electronic temperature—is acquired in the same experimental conditions (details in the Methods section). By applying this transient approach (which minimizes aging and sample drift) and computing differential curves (which remove any extrinsic background contribution, see Eq. (2)), we demonstrate the direct relation between the subtle momentum-resolved spectroscopic features of the PG and short-range ξspin(Te), as extracted from inelastic neutron scattering.24 In particular, we identify T* as the crossover temperature above which the spectral broadening due to the reduction of ξspin exceeds the PG amplitude, establishing the PG as a precursor of the underlying AF order.

Results

Fermi surface mapping and modeling

Figure 1b displays the equilibrium Fermi surface mapping of NCCO acquired with 6.2 eV probe pulsed-light. A tight-binding constant energy contour at ω = 041 (blue solid line), and the AF zone boundary (AFZB, red dashed line), are superimposed over the experimental Fermi surface. The intersection point between the tight-binding at ω = 0 and AFZB is commonly referred to as the hot-spot (HS), and coincides with the location where an AF-driven PG is expected to be particle-hole symmetric.18 In a mean field description, the commensurate q = (π,π) folding of the Fermi surface is driven by a strong quasi-2D AF order in the copper-oxygen plane.18 The Green’s function can then be written as:15,22,33

$${G}^{-1}({\bf{k}},\omega )\ =\ \omega -{\epsilon }_{{\bf{k}}}+i\eta -\frac{{\Delta }_{{\rm{PG}}}^{2}}{\omega -{\epsilon }_{{\bf{k+q}}}+i\Gamma },$$
(1)

where ϵk is the bare energy dispersion, ΔPG the AF-driven pseudogap spectroscopic amplitude determined by the local Coulomb interaction and spin susceptibility,33 η the single-particle scattering rate, and Γ a broadening term that leads to a filling of the pseudogap via the reduction of ξspin.32,33 Using Eq. (1) we can calculate the spectral function $$A({\bf{k}},\omega )\ =\ -\frac{1}{\pi }{\rm{Im}}[G({\bf{k}},\omega )]$$42 and compute the Fermi surface (Fig. 1c), which agrees well with our and previous ARPES data.29,30 We used ΔPG = η = Γ = 85 meV for simulation purposes, as suggested by our experimental results (see Fig. 1d–f and Supplementary Discussion C, D, E), and in agreement with previous optical and ARPES studies.19,20,21,29,30,43

Tracking the pseudogap spectral weight in an ultrafast fashion

Before moving to a detailed discussion of the temperature evolution of the PG in NCCO, we illustrate our experimental strategy for tracking the PG via TR-ARPES. The simulated and experimental energy distribution curves (EDCs) integrated along the momentum direction through HS for two (transient) electronic temperatures are presented in Fig. 1d1–d2, respectively; however, the PG is best visualized by the symmetrized EDCs (SEDCs, Fig. 1e1–e2). The symmetrization procedure removes the dependence of the photoemission signal on the Fermi-Dirac distribution function, providing direct access to underlying modifications of the density of states (DOS) along that particular momentum cut.44 Note that this procedure is strictly valid only at the HS where particle-hole symmetry is satisfied, as indicated by model calculations.15,22,32,33 Experimental SEDCs highlight the filling of the PG at high temperature and allow to extract a PG amplitude ΔPG ~ 85 meV. The filling of the PG can be well modeled by increasing the spin-fluctuation spectral broadening term Γ from 85 meV (50 K, black curves) to 160 meV (130 K, red curves), while ΔPG = η = 85 meV are kept fixed (details in Supplementary Discussion D, E, F). However, we remark that any experimental estimate of the temperature dependence of the PG spectral weight by fitting of SEDCs may be affected by intrinsic and uncorrelated noise, as well as extrinsic electron background (arising from irregular cleaves, secondary electrons, and electrons scattered in the detection process). We overcome this limitation by computing the difference between the photoemission intensity for high temperature (130 K) and its counterpart for low temperature (50 K), as shown in Fig. 1f1–f2. As discussed later in more detail in Eq. (2), this quantity is proportional to the differential momentum-integrated EDCs (dEDCs) and, by removing spurious contributions, highlights the temperature evolution of the PG spectral features and Fermi-Dirac distribution. While the latter would lead to a symmetrical suppression (increase) of the photoemission intensity for all ω < 0 (ω > 0), independently of the explored momentum region, it is evident that a filling of the PG may lead to an increase of the photoemission intensity at ω ~ −50 meV (blue arrows in Fig. 1f1–f2). Since thermal contributions are negligible at ω ~ −50 meV within the range of electronic temperatures explored in this work (4kBT ~ 50 meV for T = 150 K), this approach allows to track the evolution of the PG with high sensitivity. We note that a similar approach was recently used to track the electron–boson interaction in hole-doped cuprates via a transient analysis/modeling of the band renormalization (kink);45 in the specific case of NCCO, only a very weak kink was reported (see ref. 43 and Supplementary Discussion A), which does not affect the following analysis and conclusions.

Direct relation of the pseudogap to the spin-correlation length

Having defined the framework for our investigation, we now track the temperature-dependent modification of the low-energy DOS at the HS in optimally-doped NCCO by introducing thermal excitations via optical pumping. To estimate an effective electronic temperature Te,40 we fit the Fermi edge of the momentum-integrated EDCs along the near-nodal direction (φ ~ 39o, green solid line in Fig. 1b), as shown in Fig. 2a for various pump-probe delays; the resulting Te is then plotted versus time delay in Fig. 2b for the two pump fluences employed in this work, here labeled as low fluence (LF) and high fluence (HF). The transient Te can be phenomenologically fit by a double exponential function (solid lines in Fig. 2b), reminiscent of the two-temperature-model framework.34,35,36,37,40 Note that near-nodal momentum-integrated EDCs are well described by a thermal Fermi-Dirac distribution for time delays τ ≥ 0.3 ps, a timescale comparable to the temporal resolution of our TR-ARPES system (~250 fs). Non-thermal features are instead detected for τ ~ 0 ps, i.e. when pump and probe pulses are overlapped in time, and data corresponding to those time delays are highlighted in gray in Fig. 2b, c (and omitted in Fig. 2d, see also Supplementary Discussion G).

Figure 2c displays the transient enhancement of the photoemission intensity at the HS in a 20 meV energy window about ω = −50 meV ($${I}_{\omega \ =\ -50}^{PG}$$, given by the momentum-integrated EDC along φ ~ 26.5°, direction indicated by the black solid line in Fig. 1b). This particular choice of energy window was motivated by the dEDCs at the HS shown in Fig. 1f1–f2 and Fig. 3b. We note that the temporal response of $${I}_{\omega \ =\ -50}^{PG}$$ is dependent on the pump fluence: while the enhancement of $${I}_{\omega \ =\ -50}^{PG}$$ recovers exponentially within 2 ps for the LF regime, in the HF regime $${I}_{\omega \ =\ -50}^{PG}$$ saturates for ~2 ps and does not fully recover within the domain of pump-probe delays studied here. This saturation of $${I}_{\omega \ =\ -50}^{PG}$$ in the HF regime suggests a vanishing of the PG. To further investigate the origin of the suppression of the PG spectral weight, we plot $${I}_{\omega \ =\ -50}^{PG}$$ directly as a function of Te in Fig. 2d (the non-thermal points—grey circles in Fig. 2b, c—are omitted). One can see a close resemblance between the temperature dependence of $${I}_{\omega \ =\ -50}^{PG}$$ and $${\xi }_{spin}^{-1}$$ reported in ref. 24 for optimal doping, shown as a green line and shadow in Fig. 2c (with appropriate offset and scaling). However, for temperatures Te > 110 K, $${I}_{\omega \ =\ -50}^{PG}$$(Te) is found to saturate, departing from $${\xi }_{spin}^{-1}$$; this onset of saturation is in good agreement with T* reported by other experimental probes.19,20,21,22 Note that the deviation of $${I}_{\omega \ =\ -50}^{PG}$$(Te > T*) from $${\xi }_{spin}^{-1}$$ is not associated with a phase transition, but is rather a crossover stemming from the filling—not the closure—of the PG.

Discussion

To further clarify the origin of T*, we now present a comprehensive analysis and modeling of the photoinduced thermal modification of the PG for both LF and HF pump regimes. The spectral features of NCCO are inherently broad and their intensity is comparable to the underlying background level, precluding the sort of detailed quantitative modeling of the transient spectral function which has been achieved for hole-doped cuprates.46 Alternately, here we focus our analysis on the temporal evolution of dEDCs, defined as:

$${\rm{dEDC}}(\omega )\propto {\rm{DOS}}(\omega ,\tau )\cdot f(\omega ,\tau )-{{\rm{DOS}}}_{0}(\omega )\cdot {f}_{0}(\omega ),$$
(2)

where DOS0(ω) and the Fermi-Dirac electronic distribution f0(ω) are the unperturbed quantities (see refs 45,47 and Supplementary Discussion B). Note that the photoemission matrix-elements, not included in Eq. (2), represent a constant multiplicative factor and can be neglected to the first approximation. Figure 3a1 displays experimental dEDCs as a function of pump-probe delay (τ) and binding energy (ω), at HS (HF regime). The TR-ARPES data reported here can be simulated remarkably well using the simple model of Eq. (1) (see Fig. 3a2). The experimental dEDCs are reproduced through a substantial increase of the broadening term Γ alone, which phenomenologically describes the filling of the PG due to the reduction of the spin-correlation length.32,48 Note that $$\Gamma ({{\rm{T}}}_{{\rm{e}}})\propto {\xi }_{spin}^{-1}({{\rm{T}}}_{{\rm{e}}})$$ in Eq. (1) is assumed for simulation purposes, in agreement with theoretical predictions of Vilk and Tremblay.33

In further support of our interpretation of the HS data, in Fig. 3b1, b2 we compare single experimental and simulated dEDCs along the near-nodal direction and HS, for τ = +0.6 ps. Along the near-nodal direction we find an approximately symmetric transient population/depletion (increase/decrease of the photoemission intensity for ω >/< 0), characteristic of a mere thermal broadening effect (along the near-nodal region the PG is located well above the Fermi level, thus only mildly affecting the symmetry of the near-nodal dEDCs; see also Supplementary Discussion B, G). In contrast, at the HS we observe the increment of the intensity for ω ~ −50 meV and a modest (null) depletion signal for ω ~ −20 meV for LF (HF), in good agreement with the model and data of Fig. 1d–f and Fig. 2. We also emphasize that a full closure of the gap fails to reproduce our TR-ARPES data. Figure 3c compares simulated dEDCs at the HS for a complete gap closure (ΔPG → 0, black line) and gap filling (Γ = 2.5 Γ0, red line, where Γ0 = 85 meV), and only the latter matches the experimental curve shown in Fig. 3b1 (red line, HF).

Finally, we plot in Fig. 3d the simulated analog to Fig. 2d, noting a clear correspondence between the two figures. In particular, assuming the direct relationship between the filling of the PG and ξspin (as predicted for 2D spin-fluctuations33), we find that the simulated filling of the PG saturates for temperatures T ~ T*, when Γ(T*) ~ 2ΔPG ~ 170 meV. This empirical observation agrees well with recent theoretical investigations of the vanishing of the PG in the electron-doped cuprate Pr1.3-xLa0.7CexCuO4.49

In conclusion, we have reported the direct relation between the partial suppression of the electronic spectral weight, a.k.a pseudogap, and short-range magnetic correlations in electron-doped cuprates. In particular, by performing a detailed ultrafast ARPES study at the hot-spot of the optimally-doped NCCO electron-doped cuprate, we have demonstrated that the temperature dependence of the low-energy DOS is closely related to the spin-correlation length ξspin. We identified two different temperature regimes for the PG, moving from low to high temperature: (i) T < T*, in which the PG begins to fill alongside with the reduction of ξspin; (ii) T > T*, where ξspin ~ 10–15 a and the PG is completely filled-up. Our results show that the PG phenomenology in optimally-doped NCCO originates from short-range AF correlations, parametrized by ξspin(T), and T* is a crossover temperature above which the spectral broadening driven by the reduction of ξspin overcomes the PG amplitude ΔPG. This suggests that the frequently reported onset temperature T* does not represent a thermodynamic phase transition, i.e. a sharp quenching of a well-defined order parameter; rather, T* is associated with the weakening of the short-range AF correlations and incipient (π,π)-folding.32,48 In addition, the observation of the filling—not closure—of the PG suggests that the energy scale associated with the PG survives to temperatures well above T*, possibly reminiscent of scattering results23,24 which show that the spectral weight associated with magnetic excitations remains finite up to much higher temperatures than the spin-correlation length itself. This phenomenology may bear a relation to the underlying Mott physics,33 or to the recent proposal of a crossover of the SU(2) gauge theory for fluctuating spin-density-waves near optimal-doping.50 Finally, we note that our transient momentum-resolved study demonstrates that even underlying orders with correlation lengths of about ten unit cells may play a significant role in shaping the Fermi surface topology and associated transport properties of complex materials.18,19,20,21,29,30,31,32,43,51,52

Methods

Experimental design

Our TR-ARPES setup exploits the classic pump-probe scheme. A 6.2 eV probe beam is generated by fourth-harmonic generation of the fundamental wavelength (800 nm, 1.55 eV) of a Ti:sapphire laser (Vitesse Duo and RegA 9000 by Coherent, 250 kHz repetition rate). The 1.55 eV pump beam is split from the source before harmonic generation. The pump and probe beam diameters on the sample are 300 μm and 150 μm, respectively. The angle of incidence on the sample is approximately normal for both beams. The two pump incident fluences were 28 ± 5 μJ/cm2 for LF and 50 ± 10 μJ/cm2 for HF. The pump and probe beams were s-polarized. The sample was cleaved and measured at <5 × 10−11 torr base pressure and 10 K temperature. Overall thermal effects have been taken into account and we estimate a temperature of approximately 40 K and 60 K for negative pump-probe delays for LF and HF excitations, respectively. Photoemitted electrons were detected by an hemispherical analyzer (Specs Phoibos 150). Energy and temporal resolutions were 17 meV and 250 fs, respectively. The acquisition time of a full delay/temperature cycles is about 5 min and this is repeated for a total of 100–150 cycles for each momentum-direction/fluence, corresponding to a total acquisition time of 8–13 h.

Sample

High quality electron-doped single crystals Nd2-xCexCuO4+δ were grown by the container-free traveling solvent floating zone (TSFZ) technique. The optimally doped Nd2-xCexCuO4+δ crystals (x ~ 0.15) exhibit an oxygen surplus directly after growth which has to be removed by an additional post-growth annealing treatment. Such annealed crystals exhibit a transition temperature Tc of 23.5 K with transition widths of 1 K.53,54 The very high crystal quality of Nd2-xCexCuO4+δ is manifest particularly well in magnetic quantum oscillations observed on several samples at different doping levels.51

Data availability

Data shown in the main text are available at the Harvard Dataverse.55 Additional data may be requested from the authors.

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Acknowledgements

We gratefully acknowledge M. Greven, A.-M. Tremblay, L. Taillefer, N. P. Armitage and G. A. Sawatzky for useful discussions. This research was undertaken thanks in part to funding from the Max Planck-UBC-UTokyo Centre for Quantum Materials and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. This project is funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF4779 to A.D. and D.J.J.; the Killam, Alfred P. Sloan, and Natural Sciences and Engineering Research Council of Canada’s (NSERC’s) Steacie Memorial Fellowships (A.D.); the Alexander von Humboldt Fellowship (A.D.); the Canada Research Chairs Program (A.D.); NSERC, Canada Foundation for Innovation (CFI); British Columbia Knowledge Development Fund (BCKDF); and CIFAR Quantum Materials Program. E.R. acknowledges support from the Swiss National Science Foundation (SNSF) grant no. P300P2_164649. C.G. acknowledges financial support from MIUR through the PRIN 2015 Programme (Prot. 2015C5SEJJ001) and from Università Cattolica del Sacro Cuore through D.1, D.2.2, and D.3.1 grants.

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F.B. and M.Z. equally contributed to this work. F.B., M.Z., E.R. and A.D. conceived the investigation. F.B. and M.Z. performed TR-ARPES measurements with the assistance of E.R. and M.M., and F.B., M.Z., E.R., R.P.D., M.M., B.Z., P.N., M.S., S.Z., A.K.M., G.L. were responsible for operation and maintenance of the experimental system. F.B., M.Z., E.R., E.H.d.S.N., C.G., D.J.J., and A.D. were responsible for data analysis and interpretation. A.E. provided NCCO samples. All of the authors discussed the underlying physics and contributed to the paper. F.B., M.Z., R.P.D., and A.D. wrote the paper. A.D. was responsible for the overall direction, planning, and management of the project.

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Correspondence to Fabio Boschini or Andrea Damascelli.

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Boschini, F., Zonno, M., Razzoli, E. et al. Emergence of pseudogap from short-range spin-correlations in electron-doped cuprates. npj Quantum Mater. 5, 6 (2020). https://doi.org/10.1038/s41535-020-0208-6

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## Tuesday, May 26, 2020

### Book: At least two philosophers understood background independence

While the fake news media are still obsessed (but less so) with the fake Chinese cough apocalypse, among similar things (and with fake scientists who promote fake theories or irrational hysteria), Tiziana Vistarini, a philosopher at the University of Rome, has published a new book about one of the most important topics in the Universe. Yes, it is
The Emergence of Spacetime in String Theory (Routledge Studies in the Philosophy of Mathematics and Physics Book 5) (click to buy at Amazon)
You may be rather sure that the fake science journalists won't promote this book because it actually has some valuable beef. Hardcover and eVersions of the book are available.

I learned about the book because of the chapter 9 written by Nick Huggett and Christian Wüthrich, professors of philosophy in Chicago and Geneva, respectively, who have clearly understood some essential ideas about the background independence in quantum gravity and string theory.
Chapter 9: The ‘emergence’ of spacetime (Pittsburgh Philosophy Archive)
If you want to quickly get to some of the most crucial ideas in that 40-page-long essay, you may search for the sequence of four letters "Motl". ;-)

OK, among many other things, these wise men have read a 2012 blog post by your humble correspondent, What is background independence and how important is it?, and they noticed that I elaborated upon some rough ideas expressed in the textbooks by Green+Schwarz+Witten and by the late Joe Polchinski – who didn't do a great enough job, however.

I think that they really couldn't because our (and my) understanding of the role of the background in string theory kept on evolving after 1987 when GSW was published and even after 1999 when Joe's Big Book of String was released. Here is one passage that leads me to believe that they understood the background independence much more properly than virtually anyone (or strictly everyone) else who likes to talk about the topic:
First some clarification of its terms: up to now, we have, in line with standard use, referred to $$g_{\mu\nu}$$ as the ‘background’ metric, while in the passage quoted, Motl refers to its metric part as the background! (Of course, because he is addressing the question of whether string theory is independent of this particular ‘background’ structure.) For the purposes of our discussion of this proposal it will help keep things clear to adopt a different terminology: we will call $$g_{\mu\nu}$$ the ‘full’ metric, and its components the ‘partial’ metric and graviton/tensor field. Then we have argued that the full metric is emergent because of its graviton field part; but the question currently addressed is whether the partial metric is a non-emergent spacetime structure. The answer we take from Motl is that the partial metric is no structure at all, just an arbitrary component in a split of the only physical quantity, the emergent full metric. This split is necessitated, not by nature, but by the perturbative formalism currently used to describe string theory.
Exactly right. The spacetime is an emergent structure but the geometric spacetime background (and background values of other fields) is not an emergent structure because it is not an objectively physical structure at all. Instead, it is a choice that the user of the laws of physics has to make. By making this choice, he or she contributes to the "emergence of the spacetime" but "spacetime" has another subjective component because a user-dependent choice was made. He or she has a freedom to do so – and in fact, (as long as he or she wants to do a nonzero amount of science) he or she has a duty to exploit this freedom. Once this separation to the background and excitations is made, physical processes may be quantified. The quantification depends on the exact separation. But there is a one-to-one map between the descriptions that we obtain assuming different (allowed) choices of the background.

The background may be "a lot of things" if not "almost everything" and the theory has the remarkable ability to work for all such choices. These comments incorporate lots of dualities and freedom that we know in string theory. In particular, the coherent condensate of gravitons upon one background must be physically identified with the vacuum state (the state with no gravitons or other particles) of another background. Various dualities (S, T, U, M-IIA, string-string duality, K3/heterotic duality etc.) are also manifestations of a broader principle, and so is the AdS/CFT and ER-EPR correspondence. The theory allows us to make lots of choices.

A primary point that these authors understand – and many others don't – is that the freedom if not the duty to pick a background is a virtue of the theory because the theory is capable of producing answers for many chosen questions by the user of the theory: the multitude of these questions is enhanced by the large number of backgrounds that must be considered a part of the data that define the question that the user of string theory wants string theory to answer (the question has the form "what is the probability amplitude for an outcome of a process").

You shouldn't feel guilty if you choose a background and work with it. On the contrary, you should feel proud because it's really necessary to do anything that may be called physics or even natural science in the conventional sense. Physics and science require the spacetime (or at least time!). The theory remains unique despite the freedom of the user of the theory to make these choices. By the way, in their text, the two philosophers say that the choice of the background is needed for our current perturbative formulations of string theory. Well, I think it holds more generally: even rather nonperturbative definitions of string theory do implicitly contain a particular identification of the classical background and/or the vacuum state.

At some point, I tried to coin another phrase, the "background indifference", to emphasize that the freedom of the user of a theory to choose the background is a good thing. They don't use it, I think, but I am not a kind of person who would care about the choice of the language too much. They got the idea right which is what matters.


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# How to find statistical significance between two years of data

I was asked to determine if the number of HAI infections in 2011 in statistically significant from the number of HAI infection in 2012. I performed a paired T-test using the actually numbers, but am now questioning myself as to whether I should be using the standardized incident rate instead. This is for a 6 hospital network so each hospital is reporting a number that I am adding up for a total yearly number.

• Do you have just one number? Do you have, maybe, an infection count per month? I wonder if this might be modeled better as a time-series, or with a mixed-effects model or the GEE. – gung - Reinstate Monica Jan 21 '13 at 19:49
• I am provided with an actual infection number and a rate per 10,000 patient days for each hospital at the end of 2011 and the end of 2012. I also have these numbers for the entire network eg all 6 hospitals combined. I ultimately want to know if the infection rate for 2012 is significantly different from 2011. – Katie Yohnke Jan 21 '13 at 21:16
• So you have just one number per hospital per year, then? If the number of patient days varies across the hospitals in the network, I would use the rates instead of the raw infection numbers. – gung - Reinstate Monica Jan 21 '13 at 21:43
• Yes, you are correct. So a total of 6 numbers per year. Would a paired test be the correct one to use then since realistically I am using the same hospitals just different years or should I just use a two sample t-test? – Katie Yohnke Jan 21 '13 at 21:57
• I don't know how much power you'll have with so few data. Another issue is that you can't assess the reasonableness of the normality assumption, eg. You do want to respect the correlated nature of your data, though. So you may want to look into a non-parametric test for correlated data, eg the Friedman test. – gung - Reinstate Monica Jan 21 '13 at 22:31

In R code would look like
mod  <-  glm(Ninfections ~ offset(log(exposure))+Ihospital+<other predictors>, family=poisson(link="log"), data=<your data frame>, ...)


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# Homework Help: Harmonic wave equation

1. Jun 28, 2015

### SuperSpasm

1. The problem statement, all variables and given/known data
I'm using a couple different textbooks on waves, and it seems they've possibly contradicted one another.
I think the problem may be that one deduced the equation for displacement in a transverse harmonic wave
y(x,t) through the equation for displacement at t=0 [y(x,0)] and the other for displacement at x=0 [y(0,t)]
they both use the example of a rope shaken in set time intervals to make a harmonic wave, with an x axis set at the position of the rope "at rest", and seem to agree when the wave is going in the negative x direction.
I'm trying to cram this for a test tomorrow and don't really fully understand it, so I'll try to put whatever equations they used to derive this in so maybe you can understand the difference, sorry if this gets long or redundant.

2. Relevant equations
1st textbook (this one I was able to follow):

$t=0:$
$y = Asin(\frac{2\pi x}{\lambda} + \phi)$

$t\neq 0:$
$y = Asin(\frac{2\pi (x \pm vt)}{\lambda} + \phi)$
$y = Asin(\frac{2\pi x}{\lambda} \pm \frac{2\pi vt}{\lambda} +\phi)$

$v = \lambda f,\ \frac{v}{\lambda} = f$
$\omega = 2\pi f,\ k = \frac{2\pi}{\lambda}$

finally:
(1) $y = Asin(kx\pm \omega t + \phi)$

2nd textbook (in the beginning they refer only to a wave moving in the positive x direction):

$x=0$
$y(x,t) = y_{0} (t-\frac{x}{v})$ <- this is where I lost them.

then they reference a previous chapter, siting that harmonic motion is given by this:
$y_0(t) = Acos(\omega t + \phi)$
$y(x,t) = Asin[\omega (t - \frac{x}{v})]$
(2) $y(x,t) = Asin(\omega t - kx)$

and for waves moving in the negative x direction:
$y(x,t) = Asin(\omega t + kx)$

the discrepency here is that (1) = -(2) [since $\omega t - kx = -(kx - \omega t)$ and $sin(-a) = -sin(a)$]

as I write this I'm starting to think it maybe has to do with the second book omitting the phase constant $\phi$? Nonetheless I'm kinda lost with the second book's reasoning. Can anyone explain this to me?

2. Jun 28, 2015

### PeroK

The second equation clearly assumes that $y(0,0) = 0$; whereas, the first equation allows $y(0,0)$ to be non-zero. The two equations are, however, equivalent apart from this. What you're missing is that $A$ is an arbitrary constant. So, for a particular solution, you would have $A$ in one case and $-A$ in the other.

3. Jul 6, 2015

### SuperSpasm

I think I get it now, thanks!