The cosmological constant of emergent spacetime in the Newtonian approximation

[EN] We present a simple quantum-mechanical estimate of the cosmological constant of a Newtonian Universe. We first mimic the dynamics of a Newtonian spacetime by means of a nonrelativistic quantum mechanics for the matter contents of the Universe (baryonic and dark) within a fixed (i.e. nondynamical) Euclidean spacetime. Then we identify an operator that plays, on the matter states, a role analogous to that played by the cosmological constant. Finally, we prove that there exists a quantum state for the matter fields, in which the above-mentioned operator has an expectation value equal to the cosmological constant of the given Newtonian Universe.This research was supported by grant no. RTI2018-102256-B-I00 (Spain).Castro-Palacio, JC.; Fernández De Córdoba, P.; Isidro, J. (2020). The cosmological constant of emergent spacetime in the Newtonian approximation. International Journal of Modern Physics D. 29(13):1-11. https://doi.org/10.1142/S0218271820500935S1112913Susskind, L. (1995). The world as a hologram. Journal of Mathematical Physics, 36(11), 6377-6396. doi:10.1063/1.531249Padmanabhan, T. (2016). The atoms of space, gravity and the cosmological constant. International Journal of Modern Physics D, 25(07), 1630020. doi:10.1142/s0218271816300202Padmanabhan, T. (2017). The atoms of spacetime and the cosmological constant. Journal of Physics: Conference Series, 880, 012008. doi:10.1088/1742-6596/880/1/012008Cabrera, D., & Isidro, J. M. (2017). Boltzmann Entropy of a Newtonian Universe. Entropy, 19(5), 212. doi:10.3390/e19050212Isidro, J. (2018). On the Holographic Bound in Newtonian Cosmology. Entropy, 20(2), 83. doi:10.3390/e20020083Verlinde, E. (2011). On the origin of gravity and the laws of Newton. Journal of High Energy Physics, 2011(4). doi:10.1007/jhep04(2011)029Astashenok, A. V., Elizalde, E., & Yurov, A. V. (2013). The cosmological constant as an eigenvalue of a Sturm-Liouville problem. Astrophysics and Space Science, 349(1), 25-32. doi:10.1007/s10509-013-1606-zBarrow, J. D., & Shaw, D. J. (2011). The value of the cosmological constant. General Relativity and Gravitation, 43(10), 2555-2560. doi:10.1007/s10714-011-1199-1Riess, A. G., Filippenko, A. V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P. M., … Tonry, J. (1998). Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. The Astronomical Journal, 116(3), 1009-1038. doi:10.1086/300499Hubble, E. (1929). A relation between distance and radial velocity among extra-galactic nebulae. Proceedings of the National Academy of Sciences, 15(3), 168-173. doi:10.1073/pnas.15.3.168Madelung, E. (1927). Quantentheorie in hydrodynamischer Form. Zeitschrift f�r Physik, 40(3-4), 322-326. doi:10.1007/bf01400372Cadoni, M., Casadio, R., Giusti, A., Mück, W., & Tuveri, M. (2018). Effective fluid description of the dark universe. Physics Letters B, 776, 242-248. doi:10.1016/j.physletb.2017.11.058Cadoni, M., Casadio, R., Giusti, A., & Tuveri, M. (2018). Emergence of a dark force in corpuscular gravity. Physical Review D, 97(4). doi:10.1103/physrevd.97.044047Elizalde, E., Odintsov, S. D., Romeo, A., Bytsenko, A. A., & Zerbini, S. (1994). Zeta Regularization Techniques with Applications. doi:10.1142/2065Blanchet, L., & Faye, G. (2000). Hadamard regularization. Journal of Mathematical Physics, 41(11), 7675-7714. doi:10.1063/1.1308506Finster, F., & Isidro, J. M. (2017). Lp-spectrum of the Schrödinger operator with inverted harmonic oscillator potential. Journal of Mathematical Physics, 58(9), 092104. doi:10.1063/1.4997418Rajeev, K., Chakraborty, S., & Padmanabhan, T. (2018). Inverting a normal harmonic oscillator: physical interpretation and applications. General Relativity and Gravitation, 50(9). doi:10.1007/s10714-018-2438-