This paper presents an analytical study of the behavior of radial free-geodesics in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime within the Lambda Cold Dark Matter ( $Λ$ CDM) model. Using the radial free motion solutions, we provide two methods for characterizing the geodesics and defines a general formula that encapsulates all possible solutions, determined by two initial conditions. We show that the past light cone, event horizon, and particle horizon, can be considered as special cases of this overarching formula. Furthermore, the paper explores the free geodesics within the currently accepted cosmological model based on the recent Planck results, thoroughly examining the various possible geodesic scenarios.

Keywords:

References

1. G. G. Smoot et al., Structure in the COBE differential microwave radiometer first-year maps, Astrophys. J. 396 (1992) 1–5. Crossref, Web of Science, Google Scholar
2. G. Hinshaw et al., Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological parameter results, Astrophys. J. Suppl. Ser. 208(2) (2013) 19, arXiv:1212.5226 [astro-ph.CO]. Crossref, Web of Science, Google Scholar
3. N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641(A6) (2000) 67, arXiv:1807.06209 [astro-ph.CO]. Google Scholar
4. P. J. E. Peebles , Principles of physical cosmology, (Princeton university press, 1993), Vol. 27. Google Scholar
5. A. Friedmann , Über die krümmung des raumes, Z. Phys. 10 (1922) 377–386. Crossref, Google Scholar
6. G. Lemaitre , A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae, in A Source Book in Astronomy and Astrophysics (Harvard University Press, 1979), pp. 844–848. Crossref, Google Scholar
7. H. P. Robertson , Kinematics and world-structure, Astrophys. J. 82 (1935) 284. Crossref, Google Scholar
8. A. G. Walker , On Milne’s theory of world-structure, Proc. London Math. Soc. 2(1) (1937) 90–127. Crossref, Google Scholar
9. D. Baumann , Cosmology, Part III Mathematical Tripos, University Lecture Notes, 56 (2014) 34. Google Scholar
10. A. B. Whiting, The expansion of space: Free particle motion and the cosmological redshift, preprint (2004), arXiv: astro-ph/0404095. Google Scholar
11. T. M. Davis, C. H. Lineweaver and J. K. Webb , Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects, Am. J. Phys. 71(4) (2003) 358–364, arXiv:astro-ph/0104349. Crossref, Web of Science, Google Scholar
12. O. Gron and O. Elgaroy , Is space expanding in the Friedmann universe models? Am. J. Phys. 75(2) (2007) 151–157, arXiv:astro-ph/0603162. Crossref, Web of Science, Google Scholar
13. L. A. Barnes, M. J. Francis, J. B. James and G. F. Lewis , Joining the Hubble flow: Implications for expanding space, Mon. Not. R. Astron. Soc. 373(1) (2006) 382–390, arXiv:astro-ph/0609271. Crossref, Web of Science, Google Scholar
14. M. Kerachian , Uniformly accelerated traveler in an FLRW universe, Phys. Rev. D 101(8) (2020) 083536, arXiv:1911.05436 [gr-qc]. Crossref, Web of Science, Google Scholar
15. G. Vachon, R. Vanderwee and V. Faraoni , Revisiting geodesic observers in cosmology, Eur. Phys. J. C 81 (2021) 1–9, arXiv:2108.01782 [gr-qc]. Crossref, Web of Science, Google Scholar
16. I. I. Cotaescu , Kinemaitcs in spatially flat FLRW spacetimes, Chin. Phys. C 45(10) (2021) 105101, arXiv:2102.03211 [gr-qc]. Crossref, Web of Science, Google Scholar
17. J. B. Hartle , Gravity: An Introduction to Einstein’s General Relativity, 1st edn. (American Association of Physics Teachers, 2003). Google Scholar
18. P. K. Townsend, Black holes, preprint (1997), arXiv:gr-qc/9707012 . Google Scholar
19. E. N. Saridakis et al., Modified gravity and cosmology, (Springer International Publishing, 2021), https://arxiv.org/abs/2105.12582arXiv:2105.12582. Google Scholar