xPand

xPand (2012-2025) is a package for Mathematica in which tools are provided to compute formally the cosmological perturbations around a homogeneous spacetime. It is developed by Cyril Pitrou, Xavier Roy and Obinna Umeh. It is based on the xAct distribution for efficient tensor manipulations.

You first need to download and install xAct.
Then you can download the tarball of xPand's last released version (currently 0.4.4).
See the installation notes in order to install it correctly in your local Mathematica distribution.

Once installed, some documentation is available in the Documentation subfolder, and examples in the Examples subfolder. If you wish to have a glance at the package, you can download the pdf dump of the documentation.

The details of the algorithm are available in an associated publication. Please cite it along with the page http://www.xact.es/xPand/ if you use it.

Show the 109 articles citing xPand

• 109. From formation to evaporation: Induced gravitational wave probes of the primordial black hole reheating scenario
  Domènech, Guillem, Tränkle, Jan
  2409.12125
  
• 108. Induced gravitational waves for arbitrary higher orders: vertex rules and loop diagrams in cosmological perturbation theory
  Zhou, Jing-Zhi, Kuang, Yu-Ting, Wu, Di, Lü, H., Chang, Zhe
  2408.14052
  
• 107. Cosmological perturbations with ultralight vector dark matter fields: numerical implementation in CLASS
  Chase, Tomás Ferreira, Leizerovich, Matías, López Nacir, Diana, Landau, Susana
  2408.12052
  
• 106. Inflationary initial conditions for the cosmological gravitational wave background
  Valbusa Dall'Armi, Lorenzo, Mierna, Alina, Matarrese, Sabino, Ricciardone, Angelo
  JCAP, 07 (2024), 043
  2407.09405
  
• 105. Stability of Axion-Saxion wormholes
  Hertog, T., Maenaut, S., Missoni, B., Tielemans, R., Van Riet, T.
  2405.02072
  
• 104. Cosmic Birefringence as a Probe of Fundamental Parity-Violation
  Greco, Alessandro
  Link: View Paper
  PDF: Download PDF
  
• 103. Gravitational Wave Background Anisotropies: Probing Early and Late Universe Cosmology with Interferometers
  Valbusa Dall’Armi, Lorenzo
  Link: View Paper
  
• 102. Induced gravitational waves: the effect of first order tensor perturbations
  Picard, Raphael, Malik, Karim A.
  2311.14513
  
• 101. Ultralight vector dark matter, anisotropies, and cosmological adiabatic modes
  Chase, Tomás Ferreira, López Nacir, Diana
  Phys.Rev.D, 109 (2024), 083521
  2311.09373
  
• 100. New constraints on primordial non-Gaussianity from missing two-loop contributions of scalar induced gravitational waves
  Chang, Zhe, Kuang, Yu-Ting, Wu, Di, Zhou, Jing-Zhi, Zhu, Qing-Hua
  Phys.Rev.D, 109 (2024), L041303
  2311.05102
  
• 99. Unraveling the early universe's equation of state and primordial black hole production with PTA, BBN, and CMB observations
  Zhu, Qing-Hua, Zhao, Zhi-Chao, Wang, Sai, Zhang, Xin
  2307.13574
  
• 98. Primordial black holes from second order density perturbations as probes of the small-scale primordial power spectrum
  Kuang, Yu-Ting, Zhou, Jing-Zhi, Chang, Zhe, Zhang, Xukun, Zhu, Qing-Hua
  2307.02067
  
• 97. Axion-de Sitter wormholes
  Aguilar-Gutierrez, Sergio E., Hertog, Thomas, Tielemans, Rob, van der Schaar, Jan Pieter, Van Riet, Thomas
  JHEP, 11 (2023), 225
  2306.13951
  
• 96. Primordial non-Gaussianity f $_{NL}$ and anisotropies in scalar-induced gravitational waves
  Li, Jun-Peng, Wang, Sai, Zhao, Zhi-Chao, Kohri, Kazunori
  JCAP, 10 (2023), 056
  2305.19950
  
• 95. Second-order Teukolsky formalism in Kerr spacetime: Formulation and nonlinear source
  Spiers, Andrew, Pound, Adam, Moxon, Jordan
  Phys.Rev.D, 108 (2023), 064002
  2305.19332
  
• 94. Synchronizing the consistency relation
  Inomata, Keisuke, Lee, Hayden, Hu, Wayne
  JCAP, 08 (2023), 021
  2304.10559
  
• 93. Vorticity generation in cosmology and the role of shell crossing
  Umeh, Obinna
  JCAP, 12 (2023), 043
  2303.08782
  
• 92. Primordial gravitational waves assisted by cosmological scalar perturbations
  Yu, Yan-Heng, Wang, Sai
  Eur.Phys.J.C, 84 (2024), 555
  2303.03897
  
• 91. FeynGrav 2.0
  Latosh, Boris
  Comput.Phys.Commun., 292 (2023), 108871
  2302.14310
  
• 90. Scalar induced gravitational waves from Chern-Simons gravity during inflation era
  Feng, Jia-Xi, Zhang, Fengge, Gao, Xian
  JCAP, 07 (2023), 047
  2302.00950
  
• 89. Viable massive gravity without nonlinear screening
  Manita, Yusuke, Panpanich, Sirachak, Kimura, Rampei
  2301.09797
  
• 88. Primordial Gravitational Wave- and Curvature Perturbation-Induced Energy Density Perturbations
  Chang, Zhe, Kuang, Yu-Ting, Zhang, Xukun, Zhou, Jing-Zhi
  Universe, 10 (2024), 39
  2211.11948
  
• 87. Mapping the weak field limit of scalar-Gauss-Bonnet gravity
  Elder, Benjamin, Sakstein, Jeremy
  Phys.Rev.D, 107 (2023), 044006
  2210.10955
  
• 86. Wave-optics limit of the stochastic gravitational wave background
  Garoffolo, Alice
  Phys.Dark Univ., 44 (2024), 101475
  2210.05718
  
• 85. Primordial black holes and third order scalar induced gravitational waves*
  Chang, Zhe, Kuang, Yu-Ting, Zhang, Xukun, Zhou, Jing-Zhi
  Chin.Phys.C, 47 (2023), 055104
  2209.12404
  
• 84. Propagation of scalar and tensor gravitational waves in Horndeski theory
  Kubota, Kei-ichiro, Arai, Shun, Mukohyama, Shinji
  Phys.Rev.D, 107 (2023), 064002
  2209.00795
  
• 83. Impact of the free-streaming neutrinos to the second order induced gravitational waves
  Zhang, Xukun, Zhou, Jing-Zhi, Chang, Zhe
  Eur.Phys.J.C, 82 (2022), 781
  2208.12948
  
• 82. The cosmological vector modes from a monochromatic primordial power spectrum
  Chang, Zhe, Zhang, Xukun, Zhou, Jing-Zhi
  JCAP, 10 (2022), 084
  2207.01231
  
• 81. Supercomputers against strong coupling in gravity with curvature and torsion
  Barker, W.E. V.
  Eur.Phys.J.C, 83 (2023), 228
  2206.00658
  
• 80. Circularly polarized scalar induced gravitational waves from the Chern-Simons modified gravity
  Zhang, Fengge, Feng, Jia-Xi, Gao, Xian
  JCAP, 10 (2022), 054
  2205.12045
  
• 79. Consequences of using a smooth cosmic distance in a lumpy universe. I.
  Umeh, Obinna
  Phys.Rev.D, 106 (2022), 023514
  2202.08230
  
• 78. Emergence of smooth distance and apparent magnitude in a lumpy Universe
  Umeh, Obinna
  Class.Quant.Grav., 39 (2022), 235006
  2202.08237
  
• 77. The art of building a smooth cosmic distance ladder in a perturbed universe
  Umeh, Obinna
  JCAP, 08 (2022), 023
  2201.11089
  
• 76. FeynGrav: FeynCalc extension for gravity amplitudes
  Latosh, Boris
  Class.Quant.Grav., 39 (2022), 165006
  2201.06812
  
• 75. Linear growth of structure in projected massive gravity
  Manita, Yusuke, Kimura, Rampei
  Phys.Rev.D, 105 (2022), 084038
  2112.13855
  
• 74. The momentum constraint equation in parameterised post-Newtonian cosmology
  Anton, Theodore, Clifton, Timothy
  Class.Quant.Grav., 39 (2022), 095005
  2111.10860
  
• 73. Non-Gaussianity in DHOST inflation
  Brax, Philippe, Lazanu, Andrei
  JCAP, 01 (2022), 026
  2110.05913
  
• 72. Theoretical and Numerical Methods for Modified Gravity
  Casalino, Alessandro
  
• 71. Applications of Cosmological Perturbation Theory in the Late Universe
  Fuentes, Jorge L.
  2106.10181
  
• 70. Scale-dependence in DHOST inflation
  Brax, Philippe, Lazanu, Andrei
  JCAP, 08 (2021), 061
  2106.09319
  
• 69. The third order scalar induced gravitational waves
  Zhou, Jing-Zhi, Zhang, Xukun, Zhu, Qing-Hua, Chang, Zhe
  JCAP, 05 (2022), 013
  2106.01641
  
• 68. A topic review on probing primordial black hole dark matter with scalar induced gravitational waves
  Yuan, Chen, Huang, Qing-Guo
  iScience, 24 (2021), 102860
  2103.04739
  
• 67. The effect of finite halo size on the clustering of neutral hydrogen
  Umeh, Obinna, Maartens, Roy, Padmanabhan, Hamsa, Camera, Stefano
  JCAP, 06 (2021), 027
  2102.06116
  
• 66. Cosmological implications of extended massive gravity theories
  Kenna-Allison, Michael Patrick Roland
  Link: View Paper
  
• 65. Galaxy number counts at second order in perturbation theory: a leading-order term comparison
  Fuentes, Jorge L., Hidalgo, Juan Carlos, Malik, Karim A.
  Class.Quant.Grav., 38 (2021), 215008
  2012.15326
  
• 64. xPPN: an implementation of the parametrized post-Newtonian formalism using xAct for Mathematica
  Hohmann, Manuel
  Eur.Phys.J.C, 81 (2021), 504
  2012.14984
  
• 63. Approximate gauge independence of the induced gravitational wave spectrum
  Domènech, Guillem, Sasaki, Misao
  Phys.Rev.D, 103 (2021), 063531
  2012.14016
  
• 62. Generalised Proca theories in teleparallel gravity
  Nicosia, Gianbattista-Piero, Levi Said, Jackson, Gakis, Viktor
  Eur.Phys.J.Plus, 136 (2021), 191
  2012.11959
  
• 61. Testing the equivalence principle on cosmological scales using the odd multipoles of galaxy cross-power spectrum and bispectrum
  Umeh, Obinna, Koyama, Kazuya, Crittenden, Robert
  JCAP, 08 (2021), 049
  2011.05876
  
• 60. Quasi-normal mode of a regular Schwarzschild black hole
  Villani, Mattia
  Class.Quant.Grav., 37 (2020), 215019
  
• 59. On the Gauge Invariance of Scalar Induced Gravitational Waves: Gauge Fixings Considered
  Chang, Zhe, Wang, Sai, Zhu, Qing-Hua
  2010.01487
  
• 58. Gauge Invariant Second Order Gravitational Waves
  Chang, Zhe, Wang, Sai, Zhu, Qing-Hua
  2009.11994
  
• 57. Note on gauge invariance of second order cosmological perturbations
  Chang, Zhe, Wang, Sai, Zhu, Qing-Hua
  Chin.Phys.C, 45 (2021), 095101
  2009.11025
  
• 56. Cosmic acceleration and growth of structure in massive gravity
  Kenna-Allison, Michael, Gumrukcuoglu, A. Emir, Koyama, Kazuya
  Phys.Rev.D, 102 (2020), 103524
  2009.05405
  
• 55. Cosmological perturbations in modified teleparallel gravity models: Boundary term extension
  Bahamonde, Sebastian, Gakis, Viktor, Kiorpelidi, Stella, Koivisto, Tomi, Levi Said, Jackson, Saridakis, Emmanuel N.
  Eur.Phys.J.C, 81 (2021), 53
  2009.02168
  
• 54. Analytic solutions of scalar perturbations induced by scalar perturbations
  Inomata, Keisuke
  JCAP, 03 (2021), 013
  2008.12300
  
• 53. FieldsX -- An extension package for the xAct tensor computer algebra suite to include fermions, gauge fields and BRST cohomology
  Fröb, Markus B.
  2008.12422
  
• 52. Gravitational waves induced by the local-type non-Gaussian curvature perturbations
  Yuan, Chen, Huang, Qing-Guo
  Phys.Lett.B, 821 (2021), 136606
  2007.10686
  
• 51. Gauge transformation of scalar induced gravitational waves
  Lu, Yizhou, Ali, Arshad, Gong, Yungui, Lin, Jiong, Zhang, Fengge
  Phys.Rev.D, 102 (2020), 083503
  2006.03450
  
• 50. Stable, nonsingular bouncing universe with only a scalar mode
  Kumar, K. Sravan, Maheshwari, Shubham, Mazumdar, Anupam, Peng, Jun
  Phys.Rev.D, 102 (2020), 024080
  2005.01762
  
• 49. Perturbations in Regularized Lovelock Gravity
  Casalino, Alessandro, Sebastiani, Lorenzo
  Phys.Dark Univ., 31 (2021), 100771
  2004.10229
  
• 48. Massive gravity with nonminimal coupling
  Gumrukcuoglu, A. Emir, Kimura, Rampei, Koyama, Kazuya
  Phys.Rev.D, 101 (2020), 124021
  2003.11831
  
• 47. Note on nonsingular Einstein-Aether cosmologies
  Casalino, Alessandro, Sebastiani, Lorenzo, Zerbini, Sergio
  Phys.Rev.D, 101 (2020), 104059
  2003.08204
  
• 46. Gravitational Waves from Inflation
  Maria, Mylova
  Link: View Paper
  
• 45. Stable cosmology in generalized massive gravity
  Kenna-Allison, Michael, Gümrükçüoglu, A. Emir, Koyama, Kazuya
  Phys.Rev.D, 101 (2020), 084014
  1912.08560
  
• 44. Scalar induced gravitational waves in different gauges
  Yuan, Chen, Chen, Zu-Cheng, Huang, Qing-Guo
  Phys.Rev.D, 101 (2020), 063018
  1912.00885
  
• 43. Gauge Independence of Induced Gravitational Waves
  Inomata, Keisuke, Terada, Takahiro
  Phys.Rev.D, 101 (2020), 023523
  1912.00785
  
• 42. Chiral primordial gravitational waves in extended theories of Scalar-Tensor gravity
  Mylova, Maria
  1912.00800
  
• 41. Non-linear effects in early Universe cosmology
  Carrilho, Pedro
  1911.08313
  Link: View Paper
  
• 40. Multi-scale perturbation theory. Part I. Methodology and leading-order bispectrum corrections in the matter-dominated era
  Gallagher, Christopher, Clifton, Timothy, Clarkson, Chris
  JCAP, 03 (2020), 011
  1910.04894
  
• 39. Bianchi spacetimes as supercurvature modes around isotropic cosmologies
  Pereira, Thiago S., Pitrou, Cyril
  Phys.Rev.D, 100 (2019), 123534
  1909.13688
  
• 38. Galaxy number counts at second order: an independent approach
  Fuentes, Jorge L., Hidalgo, Juan Carlos, Malik, Karim A.
  Class.Quant.Grav., 38 (2021), 065014
  1908.08400
  
• 37. The galaxy bias at second order in general relativity with Non-Gaussian initial conditions
  Umeh, Obinna, Koyama, Kazuya
  JCAP, 12 (2019), 048
  1907.08094
  
• 36. Probing primordial–black-hole dark matter with scalar induced gravitational waves
  Yuan, Chen, Chen, Zu-Cheng, Huang, Qing-Guo
  Phys.Rev.D, 100 (2019), 081301
  1906.11549
  
• 35. Gravitational alternatives to dark matter with tensor mode speed equaling the speed of light
  Skordis, Constantinos, Złośnik, Tom
  Phys.Rev.D, 100 (2019), 104013
  1905.09465
  
• 34. Slow-Roll Inflation in Scalar-Tensor Models
  Granda, L.N., Jimenez, D.F.
  JCAP, 09 (2019), 007
  1905.08349
  
• 33. Observational Constraints on Two-field Warm Inflation
  Wang, Yang-yang, Zhu, Jian-Yang, Zhang, Xiao-Min
  Phys.Rev.D, 99 (2019), 103529
  1905.02414
  
• 32. Magnetogenesis from isocurvature initial conditions
  Carrilho, Pedro, Malik, Karim A.
  JCAP, 04 (2019), 028
  1902.00459
  
• 31. General relativistic effects in the galaxy bias at second order
  Umeh, Obinna, Koyama, Kazuya, Maartens, Roy, Schmidt, Fabian, Clarkson, Chris
  JCAP, 05 (2019), 020
  1901.07460
  
• 30. Linear cosmological perturbations in almost scale-invariant fourth-order gravity
  Fuentes, Jorge L., Gillani, Usman A., Malik, Karim A.
  1812.00938
  Link: View Paper
  
• 29. Gravity in mimetic scalar-tensor theories after GW170817
  Ganz, Alexander, Bartolo, Nicola, Karmakar, Purnendu, Matarrese, Sabino
  JCAP, 01 (2019), 056
  1809.03496
  
• 28. Computer algebra in gravity research
  MacCallum, Malcolm A.H.
  Living Rev.Rel., 21 (2018), 6
  
• 27. Equivalence between Scalar-Tensor theories and $f(R)$-gravity: from the action to cosmological perturbations
  Velásquez, Joel, Castañeda, Leonardo
  J.Phys.Comm., 4 (2020), 055007
  1808.05615
  
• 26. Relativistic Euler equations in cosmologies with nonlinear structures
  Gallagher, Christopher S., Clifton, Timothy
  Phys.Rev.D, 98 (2018), 103516
  1807.01655
  
• 25. The observed galaxy bispectrum from single-field inflation in the squeezed limit
  Koyama, Kazuya, Umeh, Obinna, Maartens, Roy, Bertacca, Daniele
  JCAP, 07 (2018), 050
  1805.09189
  
• 24. Isocurvature initial conditions for second order Boltzmann solvers
  Carrilho, Pedro, Malik, Karim A.
  JCAP, 08 (2018), 020
  1803.08939
  
• 23. Two-field warm inflation and its scalar perturbations on large scales
  Wang, Yang-Yang, Zhu, Jian-Yang, Zhang, Xiao-Min
  Phys.Rev.D, 97 (2018), 063510
  1803.10066
  
• 22. Two-parameter Perturbation Theory for Cosmologies with Non-linear Structure
  Goldberg, Sophia Rachel
  Link: View Paper
  
• 21. Cosmic structures and gravitational waves in ghost-free scalar-tensor theories of gravity
  Bartolo, Nicola, Karmakar, Purnendu, Matarrese, Sabino, Scomparin, Mattia
  JCAP, 05 (2018), 048
  1712.04002
  
• 20. General relativistic weak-field limit and Newtonian N-body simulations
  Fidler, Christian, Tram, Thomas, Rampf, Cornelius, Crittenden, Robert, Koyama, Kazuya, Wands, David
  JCAP, 12 (2017), 022
  1708.07769
  
• 19. Perturbation theory for cosmologies with nonlinear structure
  Goldberg, Sophia R., Gallagher, Christopher, Clifton, Timothy
  Phys.Rev.D, 96 (2017), 103508
  1707.01042
  
• 18. Tackling non-linearities with the effective field theory of dark energy and modified gravity
  Frusciante, Noemi, Papadomanolakis, Georgios
  JCAP, 12 (2017), 014
  1706.02719
  
• 17. Failures of homogeneous and isotropic cosmologies in extended quasidilaton massive gravity
  Anselmi, Stefano, Kumar, Saurabh, López Nacir, Diana, Starkman, Glenn D.
  Phys.Rev.D, 96 (2017), 084001
  1706.01872
  
• 16. How Gaussian can our Universe be?
  Cabass, Giovanni, Pajer, Enrico, Schmidt, Fabian
  JCAP, 01 (2017), 003
  1612.00033
  
• 15. Imprint of non-linear effects on HI intensity mapping on large scales
  Umeh, Obinna
  JCAP, 06 (2017), 005
  1611.04963
  
• 14. Primordial gravitational waves in supersolid inflation
  Ricciardone, Angelo, Tasinato, Gianmassimo
  Phys.Rev.D, 96 (2017), 023508
  1611.04516
  
• 13. Cosmology on all scales: a two-parameter perturbation expansion
  Goldberg, Sophia R., Clifton, Timothy, Malik, Karim A.
  Phys.Rev.D, 95 (2017), 043503
  1610.08882
  
• 12. Primordial black holes as a novel probe of primordial gravitational waves. II: Detailed analysis
  Nakama, Tomohiro, Suyama, Teruaki
  Phys.Rev.D, 94 (2016), 043507
  1605.04482
  
• 11. Cosmological perturbations in mimetic Horndeski gravity
  Arroja, Frederico, Bartolo, Nicola, Karmakar, Purnendu, Matarrese, Sabino
  JCAP, 04 (2016), 042
  1512.09374
  
• 10. Vector and tensor contributions to the curvature perturbation at second order
  Carrilho, Pedro, Malik, Karim A.
  JCAP, 02 (2016), 021
  1507.06922
  
• 9. Extreme parameter sensitivity in quasidilaton massive gravity
  Anselmi, Stefano, López Nacir, Diana, Starkman, Glenn D.
  Phys.Rev.D, 92 (2015), 084033
  1506.01000
  
• 8. On Separate Universes
  Dai, Liang, Pajer, Enrico, Schmidt, Fabian
  JCAP, 10 (2015), 059
  1504.00351
  
• 7. Weak-lensing by the large scale structure in a spatially anisotropic universe: theory and predictions
  Pitrou, Cyril, Pereira, Thiago S., Uzan, Jean-Philippe
  Phys.Rev.D, 92 (2015), 023501
  1503.01125
  
• 6. Conformal Fermi Coordinates
  Dai, Liang, Pajer, Enrico, Schmidt, Fabian
  JCAP, 11 (2015), 043
  1502.02011
  
• 5. The effective theory of fluids at NLO and implications for dark energy
  Ballesteros, Guillermo
  JCAP, 03 (2015), 001
  1410.2793
  
• 4. Nonlinear relativistic corrections to cosmological distances, redshift and gravitational lensing magnification. II - Derivation
  Umeh, Obinna, Clarkson, Chris, Maartens, Roy
  Class.Quant.Grav., 31 (2014), 205001
  1402.1933
  
• 3. xTras : A field-theory inspired xAct package for mathematica
  Nutma, Teake
  Comput.Phys.Commun., 185 (2014)
  1308.3493
  
• 2. A Simplified Approach to General Scalar-Tensor Theories
  Bloomfield, Jolyon
  JCAP, 12 (2013), 044
  1304.6712
  
• 1. Nonlinear relativistic corrections to cosmological distances, redshift and gravitational lensing magnification: I. Key results
  Umeh, Obinna, Clarkson, Chris, Maartens, Roy
  Class.Quant.Grav., 31 (2014), 202001
  1207.2109
  

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