The other data products underlying this article will be shared upon a reasonable request to the corresponding author.

[Abbott et al., 2022]
Abbott, T. M. C., Aguena, M., Alarcon, A., et al. 2022, , 105,
023520, 10.1103/PhysRevD.105.023520

[Adami et al., 2018]
Adami, C., Giles, P., Koulouridis, E., et al. 2018, , 620, A5,

[Aihara et al., 2018]
Aihara, H., Arimoto, N., Armstrong, R., et al. 2018a,
, 70, S4, 10.1093/pasj/psx066

[Aihara et al., 2018]
Aihara, H., Armstrong, R., Bickerton, S., et al. 2018b,
, 70, S8, 10.1093/pasj/psx081

[Aihara et al., 2019]
Aihara, H., AlSayyad, Y., Ando, M., et al. 2019, , 71, 114,

[Allen et al., 2011]
Allen, S., Evrard, A., & Mantz, A. 2011, , 49, 409,

[Amendola et al., 2018]
Amendola, L., Appleby, S., Avgoustidis, A., et al. 2018, Living Reviews
in Relativity, 21, 2, 10.1007/s41114-017-0010-3

[Applegate et al., 2014]
Applegate, D., von der Linden, A., Kelly, P., et al. 2014, , 439, 48,

[Astropy Collaboration et al., 2013]
Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013,
, 558, A33, 10.1051/0004-6361/201322068

[Bahé et al., 2012]
Bahé, Y. M., McCarthy, I. G., & King, L. J. 2012, , 421, 1073,

[Bartelmann & Schneider, 2001]
Bartelmann, M., & Schneider, P. 2001, , 340, 291,

[Becker & Kravtsov, 2011]
Becker, M. R., & Kravtsov, A. V. 2011, , 740, 25,

[Bellagamba et al., 2018]
Bellagamba, F., Roncarelli, M., Maturi, M., & Moscardini, L. 2018,
, 473, 5221, 10.1093/mnras/stx2701

[Bellagamba et al., 2019]
Bellagamba, F., Sereno, M., Roncarelli, M., et al. 2019, , 484,
1598, 10.1093/mnras/stz090

[Bergé et al., 2010]
Bergé, J., Amara, A., & Réfrégier, A. 2010, , 712, 992,

[Bernstein & Jarvis, 2002]
Bernstein, G. M., & Jarvis, M. 2002, , 123, 583, 10.1086/338085

[Bleem et al., 2015]
Bleem, L., Stalder, B., de Haan, T., et al. 2015, , 216, 27,

[Bleem et al., 2020]
Bleem, L. E., Bocquet, S., Stalder, B., et al. 2020, , 247, 25,

[Bocquet et al., 2016]
Bocquet, S., Saro, A., Dolag, K., & Mohr, J. 2016, , 456, 2361,

[Bocquet et al., 2019]
Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, , 878,
55, 10.3847/1538-4357/ab1f10

[Bocquet et al., 2024]
Bocquet, S., Grandis, S., Bleem, L. E., et al. 2024, arXiv e-prints,
arXiv:2401.02075, 10.48550/arXiv.2401.02075

[Böhringer et al., 2004]
Böhringer, H., Schuecker, P., Guzzo, L., et al. 2004, , 425, 367,

[Bosch et al., 2018]
Bosch, J., Armstrong, R., Bickerton, S., et al. 2018, , 70, S5,

[Bulbul et al., 2019]
Bulbul, E., Chiu, I. N., Mohr, J. J., et al. 2019, , 871, 50,

[Cash, 1979]
Cash, W. 1979, , 228, 939, 10.1086/156922

[Chambers et al., 2016]
Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv
e-prints, arXiv:1612.05560.

[Chen et al., 2024]
Chen, K.-F., Chiu, I.-N., & Oguri, M. 2024, in preparation.

[Chen et al., 2020]
Chen, K.-F., Oguri, M., Lin, Y.-T., & Miyazaki, S. 2020, , 891,
139, 10.3847/1538-4357/ab74d3

[Chisari et al., 2019]
Chisari, N. E., Alonso, D., Krause, E., et al. 2019, , 242, 2,

[Chiu et al., 2023]
Chiu, I. N., Klein, M., Mohr, J., & Bocquet, S. 2023, , 522,
1601, 10.1093/mnras/stad957

[Chiu et al., 2020]
Chiu, I. N., Okumura, T., Oguri, M., et al. 2020a, ,
498, 2030, 10.1093/mnras/staa2440

[Chiu et al., 2020]
Chiu, I. N., Umetsu, K., Murata, R., Medezinski, E., & Oguri, M.
2020b, , 495, 428, 10.1093/mnras/staa1158

[Chiu et al., 2018]
Chiu, I.-N., Umetsu, K., Sereno, M., et al. 2018, , 860, 126,

[Chiu et al., 2022]
Chiu, I. N., Ghirardini, V., Liu, A., et al. 2022, , 661, A11,

[Clowe et al., 2004]
Clowe, D., De Lucia, G., & King, L. 2004, , 350, 1038,

[Clowe et al., 2000]
Clowe, D., Luppino, G. A., Kaiser, N., & Gioia, I. M. 2000, , 539,
540, 10.1086/309242

[conda-forge community, 2015]
conda-forge community. 2015, The conda-forge Project: Community-based Software
Distribution Built on the conda Package Format and Ecosystem,  Zenodo,

[Corless & King, 2007]
Corless, V. L., & King, L. J. 2007, , 380, 149,

[Costanzi et al., 2021]
Costanzi, M., Saro, A., Bocquet, S., et al. 2021, , 103, 043522,

[Coupon et al., 2018]
Coupon, J., Czakon, N., Bosch, J., et al. 2018, , 70, S7,

[Cromer et al., 2022]
Cromer, D., Battaglia, N., Miyatake, H., & Simet, M. 2022, ,
2022, 034, 10.1088/1475-7516/2022/10/034

[Dalal et al., 2023]
Dalal, R., Li, X., Nicola, A., et al. 2023, , 108, 123519,

[Diemer & Kravtsov, 2015]
Diemer, B., & Kravtsov, A. V. 2015, , 799, 108,

[Dietrich & Hartlap, 2010]
Dietrich, J. P., & Hartlap, J. 2010, , 402, 1049,

[Dolag et al., 2016]
Dolag, K., Komatsu, E., & Sunyaev, R. 2016, , 463, 1797,

[Euclid Collaboration & et al., 2024]
Euclid Collaboration, & et al. 2024, , 681, A67,

[Fan et al., 2010]
Fan, Z., Shan, H., & Liu, J. 2010, , 719, 1408,

[Feroz & Hobson, 2008]
Feroz, F., & Hobson, M. P. 2008, , 384, 449,

[Feroz et al., 2009]
Feroz, F., Hobson, M. P., & Bridges, M. 2009, , 398, 1601,

[Feroz et al., 2019]
Feroz, F., Hobson, M. P., Cameron, E., & Pettitt, A. N. 2019, The Open
Journal of Astrophysics, 2, 10, 10.21105/astro.1306.2144

[Foreman-Mackey et al., 2013]
Foreman-Mackey, D., Hogg, D., Lang, D., & Goodman, J. 2013, , 125, 306,

[Foreman-Mackey et al., 2019]
Foreman-Mackey, D., Farr, W., Sinha, M., et al. 2019, The Journal of
Open Source Software, 4, 1864, 10.21105/joss.01864

[Fumagalli et al., 2024]
Fumagalli, A., Costanzi, M., Saro, A., Castro, T., & Borgani, S.
2024, , 682, A148, 10.1051/0004-6361/202348296

[Furusawa et al., 2018]
Furusawa, H., Koike, M., Takata, T., et al. 2018, , 70, S3,

[Gavazzi & Soucail, 2007]
Gavazzi, R., & Soucail, G. 2007, , 462, 459,

[Ghirardini et al., 2024]
Ghirardini, V., Bulbul, E., Artis, E., et al. 2024, arXiv e-prints,
arXiv:2402.08458, 10.48550/arXiv.2402.08458

[Gladders et al., 2007]
Gladders, M., Yee, H., Majumdar, S., et al. 2007, , 655, 128,

[Gladders & Yee, 2000]
Gladders, M. D., & Yee, H. K. C. 2000, , 120, 2148,

[Grandis et al., 2021]
Grandis, S., Bocquet, S., Mohr, J. J., Klein, M., & Dolag, K. 2021,
, 10.1093/mnras/stab2414

[Grandis et al., 2020]
Grandis, S., Klein, M., Mohr, J. J., et al. 2020, , 498, 771,

[Gruen et al., 2015]
Gruen, D., Seitz, S., Becker, M. R., Friedrich, O., & Mana, A. 2015,
, 449, 4264, 10.1093/mnras/stv532

[Gruen et al., 2014]
Gruen, D., Seitz, S., Brimioulle, F., et al. 2014, , 442, 1507,

[Haiman et al., 2001]
Haiman, Z., Mohr, J., & Holder, G. 2001, , 553, 545.

[Hamana et al., 2012]
Hamana, T., Oguri, M., Shirasaki, M., & Sato, M. 2012, , 425,
2287, 10.1111/j.1365-2966.2012.21582.x

[Hamana et al., 2015]
Hamana, T., Sakurai, J., Koike, M., & Miller, L. 2015, , 67, 34,

[Hamana et al., 2004]
Hamana, T., Takada, M., & Yoshida, N. 2004, , 350, 893,

[Hamana et al., 2020]
Hamana, T., Shirasaki, M., Miyazaki, S., et al. 2020, , 72, 16,

[Handley et al., 2015]
Handley, W. J., Hobson, M. P., & Lasenby, A. N. 2015, , 453, 4384,

[Hennawi & Spergel, 2005]
Hennawi, J. F., & Spergel, D. N. 2005, , 624, 59,

[Herbonnet et al., 2019]
Herbonnet, R., von der Linden, A., Allen, S. W., et al. 2019, ,
490, 4889, 10.1093/mnras/stz2913

[Herbonnet et al., 2022]
Herbonnet, R., Crawford, A., Avestruz, C., et al. 2022, , 513,
2178, 10.1093/mnras/stac997

[Hernández-Lang et al., 2023]
Hernández-Lang, D., Klein, M., Mohr, J. J., et al. 2023, ,
525, 24, 10.1093/mnras/stad2319

[Heymans et al., 2021]
Heymans, C., Tröster, T., Asgari, M., et al. 2021, , 646, A140,

[Hikage et al., 2019]
Hikage, C., Oguri, M., Hamana, T., et al. 2019, , 71, 43,

[Hildebrandt et al., 2020]
Hildebrandt, H., Köhlinger, F., van den Busch, J. L., et al. 2020,
, 633, A69, 10.1051/0004-6361/201834878

[Hilton et al., 2021]
Hilton, M., Sifón, C., Naess, S., et al. 2021, , 253, 3,

[Hinton, 2016]
Hinton, S. R. 2016, The Journal of Open Source Software, 1, 00045,

[Hoekstra, 2003]
Hoekstra, H. 2003, , 339, 1155,

[Hoekstra et al., 2011]
Hoekstra, H., Hartlap, J., Hilbert, S., & van Uitert, E. 2011, ,
412, 2095, 10.1111/j.1365-2966.2010.18053.x

[Hoekstra et al., 2015]
Hoekstra, H., Herbonnet, R., Muzzin, A., et al. 2015, , 449, 685,

[Hsieh & Yee, 2014]
Hsieh, B. C., & Yee, H. K. C. 2014, , 792, 102,

[Huang et al., 2018]
Huang, S., Leauthaud, A., Murata, R., et al. 2018, , 70, S6,

[Hunter, 2007]
Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90

[Huterer et al., 2015]
Huterer, D., Kirkby, D., Bean, R., et al. 2015, Astroparticle Physics,
63, 23, 10.1016/j.astropartphys.2014.07.004

[Ilbert et al., 2009]
Ilbert, O., Capak, P., Salvato, M., et al. 2009, , 690, 1236,

[Ishiyama et al., 2021]
Ishiyama, T., Prada, F., Klypin, A. A., et al. 2021, , 506, 4210,

[Ivezic et al., 2019]
Ivezic, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, , 873,
111, 10.3847/1538-4357/ab042c

[Jain & Van Waerbeke, 2000]
Jain, B., & Van Waerbeke, L. 2000, , 530, L1, 10.1086/312480

[Juric et al., 2017]
Juric, M., Kantor, J., Lim, K. T., et al. 2017, Astronomical Society of
the Pacific Conference Series, Vol. 512, The LSST Data Management System,
ed. N. P. F. Lorente, K. Shortridge, & R. Wayth, 279

[Kacprzak et al., 2016]
Kacprzak, T., Kirk, D., Friedrich, O., et al. 2016, , 463, 3653,

[Kaiser & Squires, 1993]
Kaiser, N., & Squires, G. 1993, , 404, 441.

[Kawanomoto et al., 2018]
Kawanomoto, S., Uraguchi, F., Komiyama, Y., et al. 2018, , 70, 66,

[Klein et al., 2023]
Klein, M., Hernández-Lang, D., Mohr, J. J., Bocquet, S., &
Singh, A. 2023, , 526, 3757, 10.1093/mnras/stad2729

[Klein et al., 2019]
Klein, M., Grandis, S., Mohr, J. J., et al. 2019, , 488, 739,

[Klein et al., 2022]
Klein, M., Oguri, M., Mohr, J. J., et al. 2022, , 661, A4,

[Klypin et al., 2016]
Klypin, A., Yepes, G., Gottlöber, S., Prada, F., & Heß, S.
2016, , 457, 4340, 10.1093/mnras/stw248

[Komiyama et al., 2018]
Komiyama, Y., Obuchi, Y., Nakaya, H., et al. 2018, , 70, S2,

[Laigle et al., 2016]
Laigle, C., McCracken, H. J., Ilbert, O., et al. 2016, , 224, 24,

[Lee et al., 2018]
Lee, B. E., Le Brun, A. M. C., Haq, M. E., et al. 2018, , 479,
890, 10.1093/mnras/sty1377

[Lee et al., 2023]
Lee, W., Cha, S., Jee, M. J., et al. 2023, , 945, 71,

[Lehmann et al., 2017]
Lehmann, B. V., Mao, Y.-Y., Becker, M. R., Skillman, S. W., &
Wechsler, R. H. 2017, , 834, 37, 10.3847/1538-4357/834/1/37

[Li et al., 2022]
Li, X., Miyatake, H., Luo, W., et al. 2022, , 74, 421,

[Li et al., 2023]
Li, X., Zhang, T., Sugiyama, S., et al. 2023, , 108, 123518,

[Liu et al., 2022]
Liu, A., Bulbul, E., Ghirardini, V., et al. 2022a, ,
661, A2, 10.1051/0004-6361/202141120

[Liu & Madhavacheril, 2019]
Liu, J., & Madhavacheril, M. S. 2019, , 99, 083508,

[Liu et al., 2015]
Liu, J., Petri, A., Haiman, Z., et al. 2015a, , 91,
063507, 10.1103/PhysRevD.91.063507

[Liu et al., 2022]
Liu, T., Merloni, A., Comparat, J., et al. 2022b, ,
661, A27, 10.1051/0004-6361/202141178

[Liu et al., 2023]
Liu, X., Yuan, S., Pan, C., et al. 2023, , 519, 594,

[Liu et al., 2015]
Liu, X., Pan, C., Li, R., et al. 2015b, , 450, 2888,

[Magnier et al., 2013]
Magnier, E. A., Schlafly, E., Finkbeiner, D., et al. 2013, , 205,
20, 10.1088/0067-0049/205/2/20

[Mandelbaum et al., 2006]
Mandelbaum, R., Seljak, U., Cool, R. J., et al. 2006, , 372, 758,

[Mandelbaum et al., 2018]
Mandelbaum, R., Miyatake, H., Hamana, T., et al. 2018a,
, 70, S25, 10.1093/pasj/psx130

[Mandelbaum et al., 2018]
—. 2018b, , 70, S25, 10.1093/pasj/psx130

[Mantz et al., 2015]
Mantz, A. B., von der Linden, A., Allen, S. W., et al. 2015, ,
446, 2205, 10.1093/mnras/stu2096

[Marques et al., 2024]
Marques, G. A., Liu, J., Shirasaki, M., et al. 2024, , 528, 4513,

[Martinet et al., 2018]
Martinet, N., Schneider, P., Hildebrandt, H., et al. 2018, , 474,
712, 10.1093/mnras/stx2793

[McClintock et al., 2019]
McClintock, T., Varga, T. N., Gruen, D., et al. 2019, , 482,
1352, 10.1093/mnras/sty2711

[McKerns et al., 2012]
McKerns, M. M., Strand, L., Sullivan, T., Fang, A., & Aivazis, M.
A. G. 2012, arXiv e-prints, arXiv:1202.1056.

[Medezinski et al., 2018]
Medezinski, E., Oguri, M., Nishizawa, A. J., et al. 2018a,
, 70, 30, 10.1093/pasj/psy009

[Medezinski et al., 2018]
Medezinski, E., Battaglia, N., Umetsu, K., et al. 2018b,
, 70, S28, 10.1093/pasj/psx128

[Miyatake et al., 2019]
Miyatake, H., Battaglia, N., Hilton, M., et al. 2019, , 875, 63,

[Miyatake et al., 2023]
Miyatake, H., Sugiyama, S., Takada, M., et al. 2023, , 108, 123517,

[Miyazaki, 2015]
Miyazaki, S. 2015, IAU General Assembly, 22, 2255916

[Miyazaki et al., 2007]
Miyazaki, S., Hamana, T., Ellis, R. S., et al. 2007, , 669, 714,

[Miyazaki et al., 2002]
Miyazaki, S., Hamana, T., Shimasaku, K., et al. 2002, , 580, L97,

[Miyazaki et al., 2018]
Miyazaki, S., Oguri, M., Hamana, T., et al. 2018a, ,
70, S27, 10.1093/pasj/psx120

[Miyazaki et al., 2018]
Miyazaki, S., Komiyama, Y., Kawanomoto, S., et al. 2018b,
, 70, S1, 10.1093/pasj/psx063

[More et al., 2023]
More, S., Sugiyama, S., Miyatake, H., et al. 2023, , 108, 123520,

[Murata et al., 2019]
Murata, R., Oguri, M., Nishimichi, T., et al. 2019, , 71, 107,

[Nakajima et al., 2012]
Nakajima, R., Mandelbaum, R., Seljak, U., et al. 2012, , 420,
3240, 10.1111/j.1365-2966.2011.20249.x

[Navarro et al., 1997]
Navarro, J., Frenk, C., & White, S. 1997, , 490, 493,

[Nishizawa et al., 2020]
Nishizawa, A. J., Hsieh, B.-C., Tanaka, M., & Takata, T. 2020, arXiv
e-prints, arXiv:2003.01511, 10.48550/arXiv.2003.01511

[Oguri, 2014]
Oguri, M. 2014, , 444, 147, 10.1093/mnras/stu1446

[Oguri & Hamana, 2011]
Oguri, M., & Hamana, T. 2011, , 414, 1851,

[Oguri & Takada, 2011]
Oguri, M., & Takada, M. 2011, , 83, 023008,

[Oguri et al., 2005]
Oguri, M., Takada, M., Umetsu, K., & Broadhurst, T. 2005, , 632,
841, 10.1086/452629

[Oguri et al., 2018]
Oguri, M., Lin, Y.-T., Lin, S.-C., et al. 2018, , 70, S20,

[Oguri et al., 2021]
Oguri, M., Miyazaki, S., Li, X., et al. 2021, , 73, 817,

[Okabe & Umetsu, 2008]
Okabe, N., & Umetsu, K. 2008, , 60, 345,

[Okabe et al., 2010]
Okabe, N., Zhang, Y.-Y., Finoguenov, A., et al. 2010, , 721, 875,

[Pérez & Granger, 2007]
Pérez, F., & Granger, B. E. 2007, Computing in Science and Engineering, 9,
21, 10.1109/MCSE.2007.53

[Planck Collaboration et al., 2016]
Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, ,
594, A24, 10.1051/0004-6361/201525833

[Planck Collaboration et al., 2020]
Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, , 641,
A6, 10.1051/0004-6361/201833910

[Rau et al., 2023]
Rau, M. M., Dalal, R., Zhang, T., et al. 2023, , 524, 5109,

[Rykoff et al., 2014]
Rykoff, E. S., Rozo, E., Busha, M. T., et al. 2014, , 785, 104,

[Schirmer et al., 2007]
Schirmer, M., Erben, T., Hetterscheidt, M., & Schneider, P. 2007,
, 462, 875, 10.1051/0004-6361:20065955

[Schlafly et al., 2012]
Schlafly, E. F., Finkbeiner, D. P., Jurić, M., et al. 2012, ,
756, 158, 10.1088/0004-637X/756/2/158

[Schneider et al., 2020]
Schneider, A., Stoira, N., Refregier, A., et al. 2020, , 2020,
019, 10.1088/1475-7516/2020/04/019

[Schneider, 1996]
Schneider, P. 1996, , 283, 837

[Schrabback et al., 2010]
Schrabback, T., Hartlap, J., Joachimi, B., et al. 2010, , 516, A63,

[Schrabback et al., 2018]
Schrabback, T., Applegate, D., Dietrich, J. P., et al. 2018, ,
474, 2635, 10.1093/mnras/stx2666

[Schrabback et al., 2021]
Schrabback, T., Bocquet, S., Sommer, M., et al. 2021, , 505,
3923, 10.1093/mnras/stab1386

[Seppi et al., 2022]
Seppi, R., Comparat, J., Bulbul, E., et al. 2022, , 665, A78,

[Shan et al., 2018]
Shan, H., Liu, X., Hildebrandt, H., et al. 2018, , 474, 1116,

[Shirasaki et al., 2019]
Shirasaki, M., Hamana, T., Takada, M., Takahashi, R., & Miyatake, H.
2019, , 486, 52, 10.1093/mnras/stz791

[Simet et al., 2017]
Simet, M., McClintock, T., Mandelbaum, R., et al. 2017, , 466,
3103, 10.1093/mnras/stw3250

[Sommer et al., 2022]
Sommer, M. W., Schrabback, T., Applegate, D. E., et al.
2022a, , 509, 1127, 10.1093/mnras/stab3052

[Sommer et al., 2022]
—. 2022b, , 509, 1127, 10.1093/mnras/stab3052

[Sugiyama et al., 2023]
Sugiyama, S., Miyatake, H., More, S., et al. 2023, , 108, 123521,

[Sunayama et al., 2023]
Sunayama, T., Miyatake, H., Sugiyama, S., et al. 2023, arXiv e-prints,
arXiv:2309.13025, 10.48550/arXiv.2309.13025

[Sunyaev & Zel'dovich, 1972]
Sunyaev, R., & Zel'dovich, Y. 1972, Comments on Astrophysics and Space
Physics, 4, 173

[Tanaka et al., 2017]
Tanaka, M., Hasinger, G., Silverman, J. D., et al. 2017, arXiv
e-prints, arXiv:1706.00566.

[Tanaka et al., 2018]
Tanaka, M., Coupon, J., Hsieh, B.-C., et al. 2018, , 70, S9,

[Tasitsiomi et al., 2004]
Tasitsiomi, A., Kravtsov, A. V., Gottlöber, S., & Klypin, A. A.
2004, , 607, 125, 10.1086/383219

[Taylor, 2005]
Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series,
Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed.
P. Shopbell, M. Britton, & R. Ebert, 29

[Taylor, 2006]
Taylor, M. B. 2006, in Astronomical Society of the Pacific Conference Series,
Vol. 351, Astronomical Data Analysis Software and Systems XV, ed.
C. Gabriel, C. Arviset, D. Ponz, & S. Enrique, 666

[Tinker et al., 2008]
Tinker, J., Kravtsov, A., Klypin, A., et al. 2008, , 688, 709,

[Tonry et al., 2012]
Tonry, J. L., Stubbs, C. W., Lykke, K. R., et al. 2012, , 750, 99,

[Umetsu, 2020]
Umetsu, K. 2020, , 28, 7, 10.1007/s00159-020-00129-w

[Umetsu et al., 2014]
Umetsu, K., Medezinski, E., Nonino, M., et al. 2014, , 795, 163,

[Umetsu et al., 2020]
Umetsu, K., Sereno, M., Lieu, M., et al. 2020, , 890, 148,

[Van Der Walt et al., 2011]
Van Der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science
& Engineering, 13, 22

[Varga et al., 2019]
Varga, T. N., DeRose, J., Gruen, D., et al. 2019, , 489, 2511,

[Vikhlinin et al., 2009]
Vikhlinin, A., Burenin, R., Ebeling, H., et al. 2009, , 692, 1033,

[Virtanen et al., 2020]
Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature
Methods, 17, 261, https://doi.org/10.1038/s41592-019-0686-2

[von der Linden et al., 2014]
von der Linden, A., Allen, M. T., Applegate, D. E., et al. 2014,
, 439, 2, 10.1093/mnras/stt1945

[von der Linden et al., 2014]
von der Linden, A., Mantz, A., Allen, S., et al. 2014, , 443, 1973,

[Wang et al., 2020]
Wang, J., Bose, S., Frenk, C. S., et al. 2020, , 585, 39,

[Weinberg et al., 2013]
Weinberg, D. H., Mortonson, M. J., Eisenstein, D. J., et al. 2013,
, 530, 87, 10.1016/j.physrep.2013.05.001

[White et al., 2002]
White, M., van Waerbeke, L., & Mackey, J. 2002, , 575, 640,

[Wittman et al., 2001]
Wittman, D., Tyson, J. A., Margoniner, V. E., Cohen, J. G., &
Dell'Antonio, I. P. 2001, , 557, L89, 10.1086/323173

[Wu et al., 2022]
Wu, H.-Y., Costanzi, M., To, C.-H., et al. 2022, , 515, 4471,

[Zhang et al., 2019]
Zhang, Y., Jeltema, T., Hollowood, D. L., et al. 2019, , 487,
2578, 10.1093/mnras/stz1361

[Zhang et al., 2023]
Zhang, Z., Wu, H.-Y., Zhang, Y., et al. 2023, ,

[Zuntz et al., 2015]
Zuntz, J., Paterno, M., Jennings, E., et al. 2015, Astronomy and
Computing, 12, 45, 10.1016/j.ascom.2015.05.005

§ THE CONSTRAINTS OF ALL PARAMETERS

In Figure <ref>, we show the fully marginalized posteriors and covariance of all the free parameters used in the default analysis.
The fully marginalized posteriors of the cosmological parameters (, , and ) against various systematic uncertainties are plotted in Figure <ref>.

The posteriors of and covariances between all free parameters in the default analysis.
The same plotting configuration used in generating Figure <ref> is also adopted here.

The systematic uncertainty in the marginalized posterior of  (left),  (middle) and,  (right).
These plots are made using the same configuration as in Figure <ref>.


# Black string solutions in Lifshitz spacetime

L. A. Lessa<EMAIL_ADDRESS>Universidade Federal do Ceará,
Departamento de Física, 60455-760, Fortaleza, CE, Brazil  J. E. G. Silva
<EMAIL_ADDRESS>Universidade Federal do Ceará, Departamento de Física,
60455-760, Fortaleza, CE, Brazil  J. Furtado<EMAIL_ADDRESS>Universidade Federal do Cariri, Centro de Ciências e Tecnologia, 63048-080,
Juazeiro do Norte, CE, Brazil  Department of Physics, Faculty of Science, Gazi
University, 06500 Ankara, Turkey

###### Abstract

In this paper we study black string solutions considering the Lifshitz
anisotropic scaling. We have shown that a new class of asymptotically Lifshitz
solutions can be generated by an Einstein-Maxwell-Dilaton theory with a
cosmological constant. In the limit where we recover conformal scale
invariance, we retrieve the usual black string solution. Furthermore, we
demonstrated that to incorporate the effects of electric charge in the black
string, at least two independent gauge fields coupled to the dilaton field are
necessary. The charged black string solution exhibits new horizons that depend
on the potential in Lifshitz exponent $z$. The stability of these new
solutions is investigated through the thermodynamic analysis of the charged
black string. The temperature, entropy, and heat capacity indicate that these
modified black strings are thermodynamically stable.

##  I Introduction

Gauge/gravity duality studies have been proving to be a very promising area in
theoretical physics in recent years. This is mainly due to the operational
ease that such a tool offers to obtain weakly coupled and computable dual
descriptions of strongly coupled conformal theories maldacena1999large . The
strongly-coupled systems exhibits a scaling symmetry near critical points.
When the critical fixed point is not dynamic, the more familiar scale
invariance which arises in the conformal group is given by

$t\rightarrow\lambda t,\ \ x_{i}\rightarrow\lambda x_{i}$ (1)

where $\lambda$ is a real constant, the $t$ is time coordinate and $x_{i}$ are
spatial coordinates. In the context of asymptotically anti-de Sitter (AdS)
spacetimes that satisfy the invariant property described by (1), the AdS/CFT
correspondence proves to be a valuable tool. This correspondence highlights
the significance of such invariances in theoretical physics. Over time, the
AdS/CFT correspondence has gained substantial traction, as evidenced by
numerous studies applying this duality to various physical scenarios co1 ; co2
; co3 ; co4 .

Our primary interest, however, lies in the development of gravitational dual
descriptions for models that exhibit anisotropic scale invariancei
kachru2008gravity , i.e.,

$t\rightarrow\lambda^{z}t,\ \ x_{i}\rightarrow\lambda x_{i}\ \
r\rightarrow\lambda^{-1}r,$ (2)

where $z$ is called the Lifshitz exponent. The aforementioned scale symmetry
is referred to as the Lifshitz scale symmetry balasubramanian2008gravity . For
$z=1$, the scaling is isotropic, which corresponds to relativistic invariance.
It has been proposed in kachru2008gravity ; Mann:2009yx  the gravity duals of
field theories with Lifshitz scaling should possess metric solutions that
exhibit asymptotic behavior of the following form:

$ds^{2}=-\frac{r^{2z}}{l^{2z}}dt^{2}+\frac{l^{2}dr^{2}}{r^{2}}+r^{2}d\Omega^{2}_{d-1}.$
(3)

where $d\Omega^{2}_{d-1}$ the metric of a unit-radius $S^{d-1}$.

The proposal presented herein entails a metric characterized by cylindrical
and static symmetry with Lifshitz scaling. So that the metric ansatz suitable
for this purpose is formulated as follows:

$ds^{2}=-\frac{r^{2z}}{l^{2z}}dt^{2}+\frac{l^{2}dr^{2}}{r^{2}}+r^{2}d\phi^{2}+\frac{r^{2}}{l^{2}}dy^{2}.$
(4)

where $t\in(-\infty,\infty)$, the radial coordinate $r\in(0,\infty)$, the
axial coordinate $y\in(-\infty,\infty)$ and angular coordinate
$\phi\in[0,2\pi)$. Note that the metric above is invariant under
transformations (2). Additionally, we have three Killing vectors
$\frac{\partial}{\partial
t},\frac{\partial}{\partial\phi},\frac{\partial}{\partial y}$, indicating that
indeed we have a space-time with cylindrical-static symmetry Stephani:2003tm .
In general, the black string solutions of Einstein gravity have been
extensively analyzed in the many literatures, see the Refs. Horne:1991gn ;
Lemos:1994xp ; Darlla:2023qgf . However, the literature still lacks a black
string solution that asymptotically approaches the Lifshitz solution with
cylindrical symmetry (4).

The objective of this work is to generate a black string solution that
accounts for the Lifshitz anisotropic scaling (2). As we will demonstrate
later, this exact solution can be derived within the framework of Einstein-
Maxwell-dilaton theories (EMDT). These theories are well-suited for generating
asymptotically Lifshitz solutions, as Lifshitz spacetime cannot be supported
by vacuum solutions and requires matter fields for its generation. We find
black hole and black brane solutions in Lifshitz spacetime in EMDT in Refs.
Lim:2019hci ; Tarrio:2011de . In Ref. Lessa:2023dbd , Lifshitz black holes
with Lorentz violation are presented. Significant solutions incorporating
dilation, but lacking Lifshitz symmetry, are discussed in Refs. d1 ; d2 .
Moreover, the dilaton field can naturally arise through the Kaluza-Klein (KK)
reduction over the extra dimensions in higher-dimensional theories, see
Ref.Lessa:2023yvw .

This paper is organised as follows. In section II, we obtain an asymptotically
Lifshitz black string solution, as well as its charged version. Subsequently,
we analyze some geometric properties of the solution. In section III, we study
the thermodynamics of the solution found and then we analyze its stability.
The paper concludes in section IV. We shall also take the Lorentzian signature
for the spacetime metric to be $(-,+,+,+)$.

##  II black string geometry in Lifshitz gravity

In this section, our aim is to seek for a (3+1)-dimensional black string
solutions that exhibit asymptotically Lifshitz behavior.

Let us consider a static and cylindrically symmetric line element in the form

$ds^{2}=-\frac{r^{2z}}{l^{2z}}f(r)dt^{2}+\frac{l^{2}dr^{2}}{r^{2}f(r)}+r^{2}d\varphi^{2}+\frac{r^{2}}{l^{2}}dy^{2}.$
(5)

where the $f(r)$ is the blackening function. In addition, we impose the
condition that the function $f(r)$ satisfies the following requirement:

$\lim_{r\rightarrow\infty}f(r)=1,$ (6)

ensuring that the metric asymptotically approaches the form given by (4).

The action considered to govern the black string dynamics is given by an
Einstein-dilaton-Mawxell model in the form Lim:2019hci ; Tarrio:2011de

$\displaystyle S=\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\bigg{[}R-\frac{1}{2}(\partial\phi)^{2}-2\Lambda_{0}e^{-2\xi_{0}\phi}-\frac{1}{4}F^{2}e^{-2\xi_{1}\phi}\bigg{]}.$
(7)

where $G$ is the Newton gravitational constant and $\Lambda_{0}$ is the
cosmological constant. The first term of the above action is the Einstein-
Hilbert term. The scalar field $\phi$ is called the dilaton field that couples
to the matter fields of the theory, where $\xi$ are the coupling constants
that measure this interaction. Further, the strength field of gauge field is
defined as $F_{\mu\nu}=\partial_{[\mu}A_{\nu]}$.

Through the of action (7), we can derive the equations of motion by varying
the action with respect to the metric and the fields. This leads us to the
following equations, respectively:

$\displaystyle
R_{\mu\nu}=\Lambda_{0}e^{-2\xi_{0}\phi}g_{\mu\nu}+\frac{e^{-2\xi_{1}\phi}}{2}T^{EM}_{\mu\nu}+\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi,$
(8) $D_{\mu}(e^{-2\xi_{1}\phi}F^{\mu\nu})=0,$ (9)
$\displaystyle\Box\phi+4\xi_{0}\Lambda_{0}e^{-2\xi_{0}\phi}+\frac{\xi_{1}}{2}F^{2}e^{-2\xi_{1}\phi}=0.$
(10)

where $\Box=g^{\mu\nu}D_{\nu}D_{\mu}$ and

$T^{EM}_{\mu\nu}=F_{\mu}\ ^{\sigma}F_{\nu\sigma}-\frac{1}{4}g_{\mu\nu}F^{2}.$
(11)

Note that we used in Eq. (8) the fact that the gauge field above have
traceless in 4-dimensions.

Let us assume an electric field configuration, i.e., $F_{rt}\neq 0$. From the
Maxwell equation (9), in the ansatz metric (5), we obtain:

$F_{rt}=\frac{qe^{2\xi_{1}\phi}}{r^{3-z}},$ (12)

where $q$ is a integration constant. The constant $q$ is related to the total
charge through further consideration Tarrio:2011de

$Q=\frac{1}{16\pi G}\int e^{-2\xi_{1}\phi}\star F.$ (13)

where $\star$ is Hodge star operator.

Note that we have a set of 5 differential equations (8),(9) and (10) and 3
functions $f,\phi$, and $F_{rt}$ to be determined, in addition to 4 parameters
$\xi_{1,2}$, $q$, and $\Lambda$ to be fixed. In this regard, we can observe
that the system is potentially integrable. Initially, we adopt the standard
procedure, as outlined in some instances Chan:1995fr; d1 ; Cai:1997ii , to
determine the dilaton field. Thus, we shall substitute the solution (12) in
the components $tt$ and $rr$ of Eq.(8), we can find from the combination
$R^{t}\ _{t}-R^{r}\ _{r}$ that

$\phi=\phi_{0}+\phi_{1}\text{ln}r,$ (14)

where $\phi_{0}$ is a integration constant and $\phi_{1}$ is given by

$\phi_{1}=2\sqrt{z-1}.$ (15)

From the expression for $\phi_{1}$ presented above, it is evident that we
require $z\geq 1$. Once the expression for the scalar field is obtained, we
can substitute it into one of the equations of the components $tt$ and $rr$ of
Eq.(8), then utilize it to eliminate the second derivative of the function $f$
in the component $\varphi\varphi$ or $yy$ of Eq.(8). Consequently, we are left
with solving only a first-order equation for $f$. In conclusion, we thus find
that

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{m}{r^{z+2}}+\frac{q^{2}e^{2\xi_{1}\phi_{0}}(\xi_{1}\phi_{1}-1)l^{2z}r^{2\xi_{1}\phi_{1}-4}}{4z(2\xi_{1}\phi_{1}+z-2)}$
(16)
$\displaystyle-\frac{\Lambda_{0}l^{2}e^{-2\xi_{0}\phi_{0}}(\xi_{0}\phi_{1}+1)r^{-2\xi_{0}\phi_{1}}}{z(-2\xi_{0}\phi_{1}+z+2)}$

where $m$ is a integration constant. To ensure that the limit (6) is obeyed,
we adopt

$\xi_{0}=0.$ (17)

Therefore, we need to fix the cosmological constant as

$\Lambda_{0}=-\frac{(z+1)(z+2)}{2l^{2}}.$ (18)

In this case, as we will see later, we can recover a solution that is indeed
asymptotically Lifshitz. Also note that for this choice, we find that the
dilatonic field decouples from the cosmological constant. After doing so, we
need to satisfy Eq. (10). Substituting (16), (18), and (17) into Eq. (10), we
obtain the following algebraic equation

$4(1+z)(2+z)\phi_{1}-e^{2\xi_{1}\phi_{0}}l^{2z}q^{2}(4\xi_{1}+\phi_{1})r^{2\xi_{1}\phi_{1}-4}=0$
(19)

In order for this equation to be satisfied, we must assume that

$\xi_{1}=\frac{2}{\phi_{1}}.$ (20)

Additionally, we need to fix the charge as follows:

$q^{2}=2(z-1)(z+2)l^{-2z}e^{-\frac{4\phi_{0}}{\phi_{1}}}$ (21)

Hence, the electromagnetic field introduced in the action (7) is, in fact,
just an auxiliary field. A similar conclusion was obtained in Refs.
Lim:2019hci ; Tarrio:2011de . Finally, the ultimate solution obtained after
substituting (17), (18), (20), and (33) into (16) is given by

$f(r)=1-\frac{m}{r^{z+2}}.$ (22)

where the constant $m$ is related to mass, as we will see later.

Considering all of the aforementioned considerations, we can conclude that the
solution is characterized by only three free parameters: $m$, $z$, and
$\phi_{0}$. Summarizing, the solution we found to action (7) is

$ds^{2}=-\frac{r^{2z}}{l^{2z}}\bigg{[}1-\frac{m}{r^{z+2}}\bigg{]}dt^{2}+\frac{dr^{2}}{\frac{r^{2}}{l^{2}}\bigg{[}1-\frac{m}{r^{z+2}}\bigg{]}}+r^{2}d\varphi^{2}+\frac{r^{2}}{l^{2}}dy^{2}.$
(23)

Note that in the limit $z=1$ and assuming that $l^{2}=\alpha^{-2}$ and
$m=\frac{b}{\alpha^{-3}}$, we then recover the well-known solution by Lemos
Lemos:1994xp .

###  II.1 Lifshitz charged black string

Figure 1: Black string’s Hawking temperature. We have considered zero electric
charge and the first three values of $z$ (left panel), $q_{2}=0.5$ and the
first three values of $z$ (middle panel) and $z=2$ and three values of charge
(right panel). We set $l=1$.

As seen above, the gauge charge ends up not affecting the charge of the black
string, since the electric charge is fixed by the geometry (33). However, it
is possible to incorporate a contribution from the gauge field into the
solution. In this case, one just needs to introduce new gauge fields in (7)
that are coupled to the dilaton field, i.e.,
$-\frac{1}{4}F_{i}^{2}e^{-2\xi_{i}\phi}$ ($i=2,3,...$). For simplicity, let’s
consider only one additional gauge field. In this case, the new electric field
can be given by

$F_{rt,2}=\frac{q_{2}e^{2\xi_{2}\phi}}{r^{3-z}},$ (24)

where the total charge $U(1)$ is given by

$Q_{2}=\frac{1}{16\pi G}\int e^{-2\xi_{2}\phi}\star F_{2}.$ (25)

It is straightforward to show that the addition of the charge $q_{2}$ only
change Eq. (16) and Eq. (19), as, respectively

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{m}{r^{z+2}}+\frac{q^{2}e^{2\xi_{1}\phi_{0}}(\xi_{1}\phi_{1}-1)l^{2z}r^{2\xi_{1}\phi_{1}-4}}{4z(2\xi_{1}\phi_{1}+z-2)}+\frac{q_{2}^{2}e^{2\xi_{2}\phi_{0}}(\xi_{2}\phi_{1}-1)l^{2z}r^{2\xi_{2}\phi_{1}-4}}{4z(2\xi_{2}\phi_{1}+z-2)}-\frac{\Lambda_{0}l^{2}e^{-2\xi_{0}\phi_{0}}(\xi_{0}\phi_{1}+1)r^{-2\xi_{0}\phi_{1}}}{z(-2\xi_{0}\phi_{1}+z+2)}$
(26)

and

$\displaystyle
4(1+z)(2+z)\phi_{1}-e^{2\xi_{1}\phi_{0}}l^{2z}q^{2}(4\xi_{1}+\phi_{1})r^{2\xi_{1}\phi_{1}-4}$
$\displaystyle-e^{2\xi_{2}\phi_{0}}l^{2z}q_{2}^{2}(4\xi_{2}+\phi_{1})r^{2\xi_{2}\phi_{1}-4}=0.$
(27)

So that the Eq. (II.1) above continues to be satisfied for (20) and (33) if

$\xi_{2}=-\frac{\phi_{1}}{4}.$ (28)

Finally, if we substitute (20), (33), and (28) into Eq. (26), we find the
solution for the asymptotically Lifshitz charged black string given by

$ds^{2}=-\frac{r^{2z}}{l^{2z}}f(r)dt^{2}+\frac{dr^{2}}{\frac{r^{2}}{l^{2}}f(r)}+r^{2}d\varphi^{2}+\frac{r^{2}}{l^{2}}dy^{2},$
(29)

where $f(r)$ is given by

$f(r)=1-\frac{m}{r^{z+2}}+\frac{q_{2}^{2}l^{2z}e^{-\sqrt{z-1}\phi_{0}}}{4zr^{2(z+1)}}$
(30)

Now, let us analyze some geometric properties of the solution (29). First, we
can determine the position of the horizon $r_{h}$ by setting $f=0$. This
results in the following algebraic equation for $r_{h}$:

$\displaystyle
1-\frac{m}{r_{h}^{z+2}}+\frac{q_{2}^{2}l^{2z}e^{-\sqrt{z-1}\phi_{0}}}{4zr_{h}^{2(z+1)}}=0.$
(31)

In this case, we assume that $r_{h}$ represents the largest positive real root
of the equation $f=0$. Unfortunately, an exact solution for
$r_{h}=r_{h}(m,q_{2},\phi_{0})$ is not readily available. However, as noted by
ref.Tarrio:2011de , the mass parameter $m$ is not a fundamental parameter of
the theory. Motivated by this, we can solve Eq. (31) with respect to the mass,
yielding

$\displaystyle
m=\frac{q_{2}^{2}l^{2z}r_{h}^{-z}e^{-\sqrt{z-1}\phi_{0}}}{4z}+r_{h}^{z+2}.$
(32)

Additionally, we can compute the Kretschmann scalar near the origin, with the
leading term being proportional to $K\sim r^{4(1+z)}$. This indicates that our
solution is singular at the origin.

##  III Thermodynamics

Let us now investigate the thermodynamics of the charged black string in
Lifshitz spacetime. It is crucial to study the thermodynamic properties of the
solution (29) and verify the validity of the first law. But first, we set
$\phi_{0}=0$, as this choice does not affect any subsequent thermodynamic
analysis since it only pertains to a phase. To proceed, we must formally
define the relationship between the parameter $m$ and the mass of the black
string. First, let us explicitly define the charge from Eq. (25) as

$Q=\frac{q_{2}l^{z-2}}{8G_{N}}.$ (33)

Additionally, the potential associated with this charge in the thermodynamic
relations, measured at infinity with respect to the horizon, is defined by:

$\Phi(r)=A_{\mu}\chi^{\mu}\big{|}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\big{|}_{r=r_{h}},$
(34)

where $\chi=\partial_{t}$ is the null generator of the horizon. Considering
Eq. (24) and the equation above, we find that:

$\Phi(r)=-\frac{q_{2}}{z}(r^{-z}-r_{h}^{-z}).$ (35)

Using the modified Brown and York formalism york , we can calculate the mass
of the solution as:

$M=\frac{ml^{-z-2}}{4G_{N}}.$ (36)

Figure 2: Heat capacity. We adopt the same Hawking temperature choices in Fig.
(1). We set $l=1$.

The black string possesses a Hawking temperature $T_{H}=\kappa/2\pi$, which
can be derived from the surface gravity provided by

$\kappa=-\frac{1}{2}(D_{\mu}\chi_{\nu})(D^{\mu}\chi^{\nu})$ (37)

where $\chi=\partial_{t}$ is the Killing vector. Hence, the computed Hawking
temperature can be written as

$\displaystyle
T_{H}=\frac{l^{-z-1}r_{h}^{-z}\left(4(z+2)r_{h}^{2z}-\frac{q_{2}^{2}l^{2z}}{r_{h}^{2}}\right)}{16\pi}.$
(38)

From the above expression, we can calculate the charge value at which we reach
the extremal limit ($T=0$). This occurs when

$q_{ext}=2l^{-z}r_{ext}^{z+1}\sqrt{(z+2)}$ (39)

where $r_{ext}$ is the position of the horizon at extremality, defined by the
conditions $f(r_{ext})=f^{\prime}(r_{ext})=0$. With the expression from (39),
we can use Eq. (32) to obtain the extremal mass given by

$m_{ext}=\frac{2(1+z)}{z}r_{ext}^{2+z}.$ (40)

The behaviour of the Hawking temperature is depicted in Fig. (1) for three
values of the electric charge and the Lifshitz exponent $z$, in order to
investigate the influence of the charge and the Lifshitz scaling in the black
string’s Hawking temperature. In the left panel of Fig. (1) we have considered
$q_{2}=0$ in order to study the pure influence of the anisotropic scaling. As
we can see, when $z=1$ we recover the linear behaviour of the black string’s
Hawking temperature, as expected. By increasing the value of $z$ the behaviour
of the Hawking temperature changes by a simple modification in the power of
$r_{h}$. In the middle and right panels of Fig. (1) we have considered non-
vanishing $q_{2}$. Notice that the non-vanishing electric charge forces the
black string to exhibit a phase transition of zeroth order in which the
temperature vanishes and the black string evaporation halts at finite horizon
radii. Therefore, remnant masses appear as a result of such phase transitions.

Furthermore, we can use the temperature expression (38) to find the critical
points in the canonical ensemble. These points indicate first-order
transitions in the system and can be obtained by solving the system of
equations generated by the conditions $\frac{\partial T}{\partial
r_{h}}=\frac{\partial^{2}T}{\partial r_{h}^{2}}=0$. Directly, we can show that
there are no solutions to these conditions; therefore, there are no first-
order transitions for asymptotically Lifshitz charged black strings.

Assuming that the black string solution (29) obeys the area law, we have that
the Bekenstein-Hawking entropy is given by

$S=\frac{\pi r_{h}^{2}}{2Gl}.$ (41)

Given these thermodynamic quantities, it is straightforward to demonstrate
that the first law of thermodynamics holds, namely,

$dM=TdS+\Phi(\infty)dQ,$ (42)

where $\Phi(\infty)=\frac{q_{2}}{z}r_{h}^{-z}$

To conclude this part of the thermodynamic analysis, let us calculate the heat
capacity at constant charge $C_{q}$ of the charged black string, corresponding
to the canonical ensemble, to determine its thermodynamic stability. We know
that this stability can be associated with microscopic fluctuations of the
system. To analyze this, we need to examine the interval of positivity of the
$C_{q}$, namely,

$\displaystyle C_{q}=T\bigg{(}\frac{\partial S}{\partial T}\bigg{)}_{q}\geq 0$
(43)

Substituting Eqs. (38) and (41) in the above expression considering $q_{2}$
fixed, we obtain the thermal capacity of the asymptotically Lifshitz charged
black string given by

$C_{q}=\frac{\pi
r_{h}^{2}\left(4(z+2)r_{h}^{2z+2}-q_{2}^{2}l^{2z}\right)}{Gl(z+2)\left(q_{2}^{2}l^{2z}+4zr_{h}^{2z+2}\right)}$
(44)

With the above expression, we can show that thermodynamic stability, i.e.,
$C_{q}\geq 0$ is achieved when $q_{2}\leq q_{ext}$ (39). Further, the heat
capacity (44) does not exhibit singular points. Thus, we indeed do not have
first-order transitions, as can be verified in the Fig. (2).

For completeness, let us calculate the free energy associated with the
canonical ensemble, the Helmholtz free energy. This quantity is given by

$\displaystyle\mathcal{F}$ $\displaystyle=\Delta M-TS$ (45)
$\displaystyle=\frac{[(2+z)q_{2}^{2}l^{2z}-4z^{2}r_{h}^{2(1+z)}]}{32zGl^{2+z}r_{h}^{z}}.$

Indeed, as shown in the Fig. (3), we can observe the absence of the
swallowtail structure, which is characteristic of first-order transitions

Figure 3:  Helmholtz free energy. We set $l=1$.

##  IV Final remarks

We have shown that it is possible to generate asymptotically Lifshitz black
string solutions in the Einstein-Maxwell-Dilaton theory with a cosmological
constant. Furthermore, we demonstrated that it is necessary to assume at least
two gauge fields to electrically charge the black string.

The geometric properties of the new solution found depend on the value of the
Lifshitz exponent $z$, which must be $z\geq 1$ to ensure that the dilaton
field is real. However, this field still diverges at the boundary. We found
that our solution, which is singular at the origin, has possible horizons that
depend on the value of $z$. The new solution recovers Lemos’s black string
solution when $z=1$. On the other hand, for $z\neq 1$, we find new
modifications in the horizons and thermodynamic quantities.

To conclude the paper, we investigated the thermodynamic properties of the
asymptotically Lifshitz charged black string solution. We calculated the
associated Hawking temperature and showed that it increases with $r_{h}$ as a
power of $z$ in the chargeless case. On the other hand, when we consider the
effects of electric charge on the black string, we find a zero-order phase
transition, resulting in the formation of remnant masses. Additionally, we
showed that the solution (29) maintains the validity of the first law of
thermodynamics.

Finally, we explored the thermodynamic stability of the solution (29) with
microscopic fluctuations in the canonical ensemble. We calculated the heat
capacity of the charged black string and showed that it can be positive, and
thus thermodynamically stable, if $q_{2}\leq q_{ext}$, where $q_{ext}$ is the
extremal charge (39) responsible for $T=0$. Additionally, we did not find any
first-order transitions in our solution.

## Acknowledgments

The authors thank the Fundação Cearense de Apoio ao Desenvolvimento Científico
e Tecnológico (FUNCAP), the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES), and the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Grant no. 200879/2022-7 (RVM).

## References

  * (1)

## References

* (2) J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998), 231-252
* (3) E. Witten, Adv. Theor. Math. Phys. 2 (1998), 505-532
* (4) A. Karch and E. Katz, JHEP 06 (2002), 043
* (5) D. T. Son and A. O. Starinets, JHEP 09 (2002), 042
* (6) S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, JHEP 12 (2008), 015
* (7) S. Kachru, X. Liu and M. Mulligan, Phys. Rev. D 78 (2008), 106005
* (8) K. Balasubramanian and J. McGreevy, Phys. Rev. Lett. 101 (2008), 061601
* (9) R. B. Mann, JHEP 06 (2009), 075
* (10) D. T. Son, Phys. Rev. D 78 (2008), 046003
* (11) M. Guica, K. Skenderis, M. Taylor and B. C. van Rees, JHEP 02 (2011), 056
* (12) H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt, Cambridge Univ. Press, 2003, ISBN 978-0-521-46702-5, 978-0-511-05917-9 doi:10.1017/CBO9780511535185
* (13) J. H. Horne and G. T. Horowitz, Nucl. Phys. B 368, 444-462 (1992)
* (14) R. G. Cai, J. Y. Ji and K. S. Soh, Phys. Rev. D 57 (1998), 6547-6550
* (15) J. P. S. Lemos, Phys. Lett. B 353 (1995), 46-51
* (16) R. Dárlla, F. A. Brito and J. Furtado, Universe 9, no.6, 297 (2023)
* (17) Y. K. Lim, JHEP 03, 134 (2020)
* (18) J. Tarrio and S. Vandoren, JHEP 09, 017 (2011)
* (19) L. A. Lessa and J. E. G. Silva, Eur. Phys. J. C 84 (2024) no.2, 194
* (20) M. Alishahiha, E. O Colgain and H. Yavartanoo, JHEP 11 (2012), 137
* (21) L. Li, Phys. Lett. B 767 (2017), 278-284
* (22) L. A. Lessa and J. E. G. Silva, Eur. Phys. J. C 83 (2023) no.11, 1035
* (23) J. D. Brown and J. W. York, Jr., Phys. Rev. D 47 (1993), 1407-1419