Abstract
Inspired by real-time computations in AdS black holes, we propose a method to obtain the influence phase of a cosmological observer by calculating the on-shell action on a doubled spacetime geometry. The influence phase is the effective action for an open system: for a dS static patch observer coupled to a scalar field it incorporates the radiation reaction due to the bulk fields and their dS Hawking radiation. For a general extended source in dS, we describe how to account for finite size effects. In the long-time limit, we get a Markovian open quantum system susceptible to cosmological fluctuations, whereas the short-time limit reproduces the worldline theory of flat-space radiation reaction. We also present a fully covariantised form for the cubic corrections to the radiation reaction in even spacetime dimensions, including Hubble contributions, and find an intriguing recursive structure across dimensions.
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References
S.M. Carroll, The Cosmological constant, Living Rev. Rel. 4 (2001) 1 [astro-ph/0004075] [INSPIRE].
P.J.E. Peebles and B. Ratra, The Cosmological Constant and Dark Energy, Rev. Mod. Phys. 75 (2003) 559 [astro-ph/0207347] [INSPIRE].
J. Frieman, M. Turner and D. Huterer, Dark Energy and the Accelerating Universe, Ann. Rev. Astron. Astrophys. 46 (2008) 385 [arXiv:0803.0982] [INSPIRE].
D.H. Weinberg et al., Observational Probes of Cosmic Acceleration, Phys. Rept. 530 (2013) 87 [arXiv:1201.2434] [INSPIRE].
E. Witten, Quantum gravity in de Sitter space, in the proceedings of the Strings 2001: International Conference, Mumbai, India, January 05–10 (2001) [hep-th/0106109] [INSPIRE].
R. Bousso, Flat space physics from holography, JHEP 05 (2004) 050 [hep-th/0402058] [INSPIRE].
D. Anninos, S.A. Hartnoll and D.M. Hofman, Static Patch Solipsism: Conformal Symmetry of the de Sitter Worldline, Class. Quant. Grav. 29 (2012) 075002 [arXiv:1109.4942] [INSPIRE].
R. Nakayama, The World-Line Quantum Mechanics Model at Finite Temperature which is Dual to the Static Patch Observer in de Sitter Space, Prog. Theor. Phys. 127 (2012) 393 [arXiv:1112.1267] [INSPIRE].
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: A Conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
W. Taylor, M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory, Rev. Mod. Phys. 73 (2001) 419 [hep-th/0101126] [INSPIRE].
B. Ydri, Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory, arXiv:1708.00734 [INSPIRE].
A. Strominger, The dS / CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
D. Marolf and I.A. Morrison, Group Averaging for de Sitter free fields, Class. Quant. Grav. 26 (2009) 235003 [arXiv:0810.5163] [INSPIRE].
D. Anninos, T. Hartman and A. Strominger, Higher Spin Realization of the dS/CFT Correspondence, Class. Quant. Grav. 34 (2017) 015009 [arXiv:1108.5735] [INSPIRE].
D. Anninos, F. Denef, R. Monten and Z. Sun, Higher Spin de Sitter Hilbert Space, JHEP 10 (2019) 071 [arXiv:1711.10037] [INSPIRE].
M. Hogervorst, J. Penedones and K.S. Vaziri, Towards the non-perturbative cosmological bootstrap, JHEP 02 (2023) 162 [arXiv:2107.13871] [INSPIRE].
T. Chakraborty et al., The Hilbert space of de Sitter quantum gravity, arXiv:2303.16315 [INSPIRE].
M. Loparco, J. Penedones, K. Salehi Vaziri and Z. Sun, The Källén-Lehmann representation in de Sitter spacetime, JHEP 12 (2023) 159 [arXiv:2306.00090] [INSPIRE].
N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities, JHEP 04 (2020) 105 [arXiv:1811.00024] [INSPIRE].
C. Sleight and M. Taronna, Bootstrapping Inflationary Correlators in Mellin Space, JHEP 02 (2020) 098 [arXiv:1907.01143] [INSPIRE].
H. Goodhew, S. Jazayeri and E. Pajer, The Cosmological Optical Theorem, JCAP 04 (2021) 021 [arXiv:2009.02898] [INSPIRE].
D. Baumann et al., Snowmass White Paper: The Cosmological Bootstrap, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2203.08121] [INSPIRE].
V.E. Hubeny, S. Minwalla and M. Rangamani, The fluid/gravity correspondence, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: String theory and its Applications: From meV to the Planck Scale, Boulder, U.S.A., June 1–25 (2010), p. 348–383 [arXiv:1107.5780] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Anomaly inflow and thermal equilibrium, JHEP 05 (2014) 134 [arXiv:1310.7024] [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Chern-Simons terms from thermal circles and anomalies, JHEP 05 (2014) 110 [arXiv:1311.2935] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: A holographic description of the black hole interior, Phys. Rev. D 75 (2007) 106001 [Erratum ibid. 75 (2007) 129902] [hep-th/0612053] [INSPIRE].
K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
H. Maxfield, A view of the bulk from the worldline, arXiv:1712.00885 [INSPIRE].
D.L. Jafferis and L. Lamprou, Inside the hologram: reconstructing the bulk observer’s experience, JHEP 03 (2022) 084 [arXiv:2009.04476] [INSPIRE].
V. Chandrasekaran, R. Longo, G. Penington and E. Witten, An algebra of observables for de Sitter space, JHEP 02 (2023) 082 [arXiv:2206.10780] [INSPIRE].
H. Borchers, Über die Vollständigkeit lorentzinvarianter Felder in einer zeitartigen Röhre, Nuovo Cim. (1955-1965) 19 (1961) 787.
H. Araki, A Generalization of Borchers Theorem, Helv. Phys. Acta (Switzerland) 36 (1963) 132.
A. Strohmaier and E. Witten, The Timelike Tube Theorem in Curved Spacetime, arXiv:2303.16380 [INSPIRE].
E. Witten, Algebras, Regions, and Observers, arXiv:2303.02837 [INSPIRE].
W.D. Goldberger and I.Z. Rothstein, An effective field theory of gravity for extended objects, Phys. Rev. D 73 (2006) 104029 [hep-th/0409156] [INSPIRE].
W.D. Goldberger, Les Houches lectures on effective field theories and gravitational radiation, in the proceedings of the Les Houches Summer School — Session 86: Particle Physics and Cosmology: The Fabric of Spacetime, Les Houches, France, July 31 – August 25 (2006) [hep-ph/0701129] [INSPIRE].
R.A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633 (2016) 1 [arXiv:1601.04914] [INSPIRE].
M. Levi, Effective Field Theories of Post-Newtonian Gravity: A comprehensive review, Rept. Prog. Phys. 83 (2020) 075901 [arXiv:1807.01699] [INSPIRE].
L. Barack and A. Pound, Self-force and radiation reaction in general relativity, Rept. Prog. Phys. 82 (2019) 016904 [arXiv:1805.10385] [INSPIRE].
R.A. Porto, L. Senatore and M. Zaldarriaga, The Lagrangian-space Effective Field Theory of Large Scale Structures, JCAP 05 (2014) 022 [arXiv:1311.2168] [INSPIRE].
K. Skenderis and B.C. van Rees, Real-time gauge/gravity duality: Prescription, Renormalization and Examples, JHEP 05 (2009) 085 [arXiv:0812.2909] [INSPIRE].
K. Skenderis and B.C. van Rees, Real-time gauge/gravity duality, Phys. Rev. Lett. 101 (2008) 081601 [arXiv:0805.0150] [INSPIRE].
P. Glorioso, M. Crossley and H. Liu, A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems, arXiv:1812.08785 [INSPIRE].
J. de Boer, M.P. Heller and N. Pinzani-Fokeeva, Holographic Schwinger-Keldysh effective field theories, JHEP 05 (2019) 188 [arXiv:1812.06093] [INSPIRE].
B. Chakrabarty et al., Nonlinear Langevin dynamics via holography, JHEP 01 (2020) 165 [arXiv:1906.07762] [INSPIRE].
C. Jana, R. Loganayagam and M. Rangamani, Open quantum systems and Schwinger-Keldysh holograms, JHEP 07 (2020) 242 [arXiv:2004.02888] [INSPIRE].
B. Chakrabarty and A. P. M., Open effective theory of scalar field in rotating plasma, JHEP 08 (2021) 169 [arXiv:2011.13223] [INSPIRE].
J.K. Ghosh et al., Effective field theory of stochastic diffusion from gravity, JHEP 05 (2021) 130 [arXiv:2012.03999] [INSPIRE].
T. He et al., The timbre of Hawking gravitons: an effective description of energy transport from holography, JHEP 09 (2022) 092 [arXiv:2202.04079] [INSPIRE].
R. Loganayagam, M. Rangamani and J. Virrueta, Holographic open quantum systems: toy models and analytic properties of thermal correlators, JHEP 03 (2023) 153 [arXiv:2211.07683] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
R.P. Feynman and F.L. Vernon Jr., The theory of a general quantum system interacting with a linear dissipative system, Annals Phys. 24 (1963) 118 [INSPIRE].
J.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2 (1961) 407 [INSPIRE].
L.V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [INSPIRE].
A. Kamenev, Field Theory of Non-Equilibrium Systems, Cambridge University Press (2011) [https://doi.org/10.1017/CBO9781139003667].
L.M. Sieberer, M. Buchhold and S. Diehl, Keldysh Field Theory for Driven Open Quantum Systems, Rept. Prog. Phys. 79 (2016) 096001 [arXiv:1512.00637] [INSPIRE].
L. Senatore and M. Zaldarriaga, Redshift Space Distortions in the Effective Field Theory of Large Scale Structures, arXiv:1409.1225 [INSPIRE].
M. Schmittfull et al., Modeling Galaxies in Redshift Space at the Field Level, JCAP 05 (2021) 059 [arXiv:2012.03334] [INSPIRE].
O.H.E. Philcox et al., Cosmology with the redshift-space galaxy bispectrum monopole at one-loop order, Phys. Rev. D 106 (2022) 043530 [arXiv:2206.02800] [INSPIRE].
C. Sleight and M. Taronna, From dS to AdS and back, JHEP 12 (2021) 074 [arXiv:2109.02725] [INSPIRE].
A.O. Caldeira and A.J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121 (1983) 587 [INSPIRE].
H. Breuer, F. Petruccione and S. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002) [https://doi.org/10.1093/acprof:oso/9780199213900.001.0001].
G.W. Gibbons and S.W. Hawking, Cosmological Event Horizons, Thermodynamics, and Particle Creation, Phys. Rev. D 15 (1977) 2738 [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
R. Bousso, Holographic probabilities in eternal inflation, Phys. Rev. Lett. 97 (2006) 191302 [hep-th/0605263] [INSPIRE].
R. Bousso and B. Freivogel, A paradox in the global description of the multiverse, JHEP 06 (2007) 018 [hep-th/0610132] [INSPIRE].
S. Mukohyama, Gauge invariant gravitational perturbations of maximally symmetric space-times, Phys. Rev. D 62 (2000) 084015 [hep-th/0004067] [INSPIRE].
H. Kodama and A. Ishibashi, Master equations for perturbations of generalized static black holes with charge in higher dimensions, Prog. Theor. Phys. 111 (2004) 29 [hep-th/0308128] [INSPIRE].
A. Ishibashi and R.M. Wald, Dynamics in nonglobally hyperbolic static space-times. 3. Anti-de Sitter space-time, Class. Quant. Grav. 21 (2004) 2981 [hep-th/0402184] [INSPIRE].
T.S. Bunch and P.C.W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A 360 (1978) 117 [INSPIRE].
A. Lopez-Ortega, Quasinormal modes of D-dimensional de Sitter spacetime, Gen. Rel. Grav. 38 (2006) 1565 [gr-qc/0605027] [INSPIRE].
S. Chaudhuri, C. Chowdhury and R. Loganayagam, Spectral Representation of Thermal OTO Correlators, JHEP 02 (2019) 018 [arXiv:1810.03118] [INSPIRE].
L.M. Burko, A.I. Harte and E. Poisson, Mass loss by a scalar charge in an expanding universe, Phys. Rev. D 65 (2002) 124006 [gr-qc/0201020] [INSPIRE].
O. Birnholtz, S. Hadar and B. Kol, Theory of post-Newtonian radiation and reaction, Phys. Rev. D 88 (2013) 104037 [arXiv:1305.6930] [INSPIRE].
R. Loganayagam and M. Godwin, An open eft for hawking radiation, to appear.
R. Loganayagam and O. Shetye, Influence phase of a de sitter observer. Part II. Gauge theory and gravity, to appear.
O. Birnholtz and S. Hadar, Action for reaction in general dimension, Phys. Rev. D 89 (2014) 045003 [arXiv:1311.3196] [INSPIRE].
D.V. Gal’tsov, Radiation reaction in the Kerr gravitational field, J. Phys. A 15 (1982) 3737 [INSPIRE].
A. Ori, Radiative evolution of orbits around a Kerr black hole, Phys. Lett. A 202 (1995) 347 [gr-qc/9507048] [INSPIRE].
A. Ori, Radiative evolution of the Carter constant for generic orbits around a Kerr black hole, Phys. Rev. D 55 (1997) 3444 [INSPIRE].
T.C. Quinn, Axiomatic approach to radiation reaction of scalar point particles in curved space-time, Phys. Rev. D 62 (2000) 064029 [gr-qc/0005030] [INSPIRE].
E. Poisson, Radiation reaction of point particles in curved spacetime, Class. Quant. Grav. 21 (2004) R153.
L.M. Burko, Instability of scalar charges in (1 + 1)-dimensions and (2 + 1)-dimensions, Class. Quant. Grav. 19 (2002) 3745 [gr-qc/0201021] [INSPIRE].
D.V. Gal’tsov and P.A. Spirin, Radiation reaction in curved even-dimensional spacetime, Grav. Cosmol. 13 (2007) 241 [arXiv:1012.3085] [INSPIRE].
X. Chen et al., Automation of antenna subtraction in colour space: gluonic processes, JHEP 10 (2022) 099 [arXiv:2203.13531].
R. Loganayagam and O. Shetye, Influence phase of a de sitter observer. Part III. Scalar interactions, to appear.
C. Jana, Aspects of open quantum field theory, Ph.D. thesis, Tata Institute of Fundamental Research (TIFR), Bengaluru 560 012, India (2021) [INSPIRE].
Q. Yan, X. Ren, Y. Zhao and E.N. Saridakis, Stochastic gravitational wave background from the collisions of dark matter halos, arXiv:2301.02414 [INSPIRE].
K.Z. Yang et al., Measurement of the cross-correlation angular power spectrum between the stochastic gravitational wave background and galaxy overdensity, Phys. Rev. D 108 (2023) 043025 [arXiv:2304.07621] [INSPIRE].
V. Saeedzadeh et al., Shining Light on the Hosts of the Nano-Hertz Gravitational Wave Sources: A Theoretical Perspective, arXiv:2309.08683 [INSPIRE].
L. Susskind, Black Holes Hint towards De Sitter Matrix Theory, Universe 9 (2023) 368 [arXiv:2109.01322] [INSPIRE].
S. Brahma, R. Brandenberger and S. Laliberte, BFSS Matrix Model Cosmology: Progress and Challenges, arXiv:2210.07288 [INSPIRE].
D. Anninos and D.M. Hofman, Infrared Realization of dS2 in AdS2, Class. Quant. Grav. 35 (2018) 085003 [arXiv:1703.04622] [INSPIRE].
D. Anninos, D.A. Galante and D.M. Hofman, De Sitter horizons & holographic liquids, JHEP 07 (2019) 038 [arXiv:1811.08153] [INSPIRE].
D. Anninos, D.A. Galante and B. Mühlmann, Finite features of quantum de Sitter space, Class. Quant. Grav. 40 (2023) 025009 [arXiv:2206.14146] [INSPIRE].
D. Anninos and E. Harris, Interpolating geometries and the stretched dS2 horizon, JHEP 11 (2022) 166 [arXiv:2209.06144] [INSPIRE].
A. Sahu, P. Simidzija and M. Van Raamsdonk, Bubbles of cosmology in AdS/CFT, JHEP 11 (2023) 010 [arXiv:2306.13143] [INSPIRE].
V. Gorbenko, E. Silverstein and G. Torroba, dS/dS and \( T\overline{T} \), JHEP 03 (2019) 085 [arXiv:1811.07965] [INSPIRE].
A. Lewkowycz, J. Liu, E. Silverstein and G. Torroba, \( T\overline{T} \) and EE, with implications for (A)dS subregion encodings, JHEP 04 (2020) 152 [arXiv:1909.13808] [INSPIRE].
V. Shyam, \( \textrm{T}\overline{\textrm{T}} \) + Λ2 deformed CFT on the stretched dS3 horizon, JHEP 04 (2022) 052 [arXiv:2106.10227] [INSPIRE].
E. Coleman et al., De Sitter microstates from \( T\overline{T} \) + Λ2 and the Hawking-Page transition, JHEP 07 (2022) 140 [arXiv:2110.14670] [INSPIRE].
G. Torroba, \( T\overline{T} \) + Λ2 from a 2d gravity path integral, JHEP 01 (2023) 163 [arXiv:2212.04512] [INSPIRE].
F. Ferrari, Gauge Theories, D-Branes and Holography, Nucl. Phys. B 880 (2014) 247 [arXiv:1310.6788] [INSPIRE].
F. Ferrari, D-Brane Probes in the Matrix Model, Nucl. Phys. B 880 (2014) 290 [arXiv:1311.4520] [INSPIRE].
F. Ferrari and A. Rovai, Gravity and On-Shell Probe Actions, JHEP 08 (2016) 047 [arXiv:1602.07177] [INSPIRE].
E. Poisson and C. Will, Gravity: Newtonian, Post-Newtonian, Relativistic, Cambridge University Press (2014) [https://doi.org/10.1017/cbo9781139507486].
A. Higuchi, Symmetric Tensor Spherical Harmonics on the N Sphere and Their Application to the De Sitter Group SO(N, 1), J. Math. Phys. 28 (1987) 1553 [Erratum ibid. 43 (2002) 6385] [INSPIRE].
M.A. Rubin and C.R. Ordónez, Symmetric Tensor Eigen Spectrum of the Laplacian on n Spheres, J. Math. Phys. 26 (1985) 65 [INSPIRE].
M.A. Rubin and C.R. Ordónez, Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics, J. Math. Phys. 25 (1984) 2888 [INSPIRE].
C.R. Frye and C.J. Efthimiou, Spherical Harmonics in p Dimensions, arXiv:1205.3548 [INSPIRE].
S. Bhattacharyya et al., Currents and Radiation from the large D Black Hole Membrane, JHEP 05 (2017) 098 [arXiv:1611.09310] [INSPIRE].
S.G. Turyshev and V.T. Toth, Spherical harmonics representation of the gravitational phase shift, Phys. Rev. D 107 (2023) 104031 [arXiv:2303.07270] [INSPIRE].
A. Ross, Multipole expansion at the level of the action, Phys. Rev. D 85 (2012) 125033 [arXiv:1202.4750] [INSPIRE].
S. Foffa and R. Sturani, Tail terms in gravitational radiation reaction via effective field theory, Phys. Rev. D 87 (2013) 044056 [arXiv:1111.5488] [INSPIRE].
M. Spradlin, A. Strominger and A. Volovich, Les Houches lectures on de Sitter space, in the proceedings of the Les Houches Summer School: Session 76: Euro Summer School on Unity of Fundamental Physics: Gravity, Gauge Theory and Strings, Les Houches, France, July 30 – August 31 (2001), p. 423–453 [hep-th/0110007] [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS / CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
P.A.M. Dirac, Classical theory of radiating electrons, Proc. Roy. Soc. Lond. A 167 (1938) 148 [INSPIRE].
S.L. Detweiler and B.F. Whiting, Selfforce via a Green’s function decomposition, Phys. Rev. D 67 (2003) 024025 [gr-qc/0202086] [INSPIRE].
E. Poisson, A. Pound and I. Vega, The motion of point particles in curved spacetime, Living Rev. Rel. 14 (2011) 7 [arXiv:1102.0529] [INSPIRE].
V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
Acknowledgments
We would like to thank Ofek Birnholtz, Tuneer Chakraborty, Chandramouli Chowdhury, Victor Godet, Chandan Jana, Godwin Martin, Shiraz Minwalla, Priyadarshi Paul, Suvrat Raju, Mukund Rangamani, Joseph Samuel, Ashoke Sen, Shivam Sharma, Akhil Sivakumar, Sandip Trivedi and Spenta Wadia for valuable discussions. RL would like to thank the organisers of All Lambdas Holography @ Prague 2021 online workshop for discussions related to this work. We acknowledge support of the Department of Atomic Energy, Government of India, under project no. RTI4001, and would also like to acknowledge our debt to the people of India for their steady and generous support to research in the basic sciences.
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Loganayagam, R., Shetye, O. Influence phase of a dS observer. Part I. Scalar exchange. J. High Energ. Phys. 2024, 138 (2024). https://doi.org/10.1007/JHEP01(2024)138
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DOI: https://doi.org/10.1007/JHEP01(2024)138