Abstract
Although bulk peculiar motions are commonplace in the universe, most theoretical studies either bypass them, or take the viewpoint of the idealised Hubble-flow observers. As a result, the role of these peculiar flows remains largely unaccounted for, despite the fact that relative-motion effects have led to the misinterpretation of the observations in a number of occasions. Here, we examine the implications of large-scale peculiar flows for the interpretation of the deceleration parameter. We compare, in particular, the deceleration parameters measured by the Hubble-flow observers and by their bulk-flow counterparts. In so doing, we use Newtonian theory and general relativity and employ closely analogous theoretical tools, which allows for the direct and transparent comparison of the two studies. We find that the Newtonian relative-motion effects are generally too weak to make a difference between the two measurements. In relativity, however, the deceleration parameters measured in the two frames differ considerably, even at the linear level. This could deceive the unsuspecting observers to a potentially serious misinterpretation of the universe’s global kinematic status. We also discuss the implications and the observational viability of the relativistic study.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Hereafter, we will use the terms Hubble frame and CMB frame interchangeably.
Throughout this manuscript Greek indices run between 1 and 3, while their Latin counterparts take values from 0 to 3. Also, round brackets imply symmetrisation, square antisymmetrisation and angled indicate the symmetric, trace-free component of second-rank tensors (e.g. \(\sigma _{\alpha \beta }= \partial _{( \beta }u_{\alpha )} -(\partial ^{\mu }u_{\mu }/3)h_{\alpha \beta }\)).
Newtonian studies of peculiar flows typically use physical (\(r^{\alpha }\)) and comoving (\(x^{\alpha }\)) coordinates, with \(r^{\alpha }=ax^{\alpha }\). The time-derivative of the latter gives \(v_{t}=v_{H}+v_{p}\), where \(v_{t}=\dot{r}^{\alpha }\), \(v_{H}=Hr^{\alpha }\) and \(v_{p}=a\dot{x}^{\alpha }\) are the total, the Hubble and the peculiar velocities respectively. On an FRW background, the above relation coincides with (9).
The two 4-velocity fields seen in Eq. (11) form the hyperbolic angle \(\beta \), with \(\cosh \beta =-\tilde{u}_{a}u^{a}= \tilde{\gamma }\geq 1\) (see King and Ellis (1973) and also Fig. 1 here). The latter determines the “tilt” between the two timelike directions and also explains why we use the term “titled” when referring to these cosmological models.
The volume scalar is related to the Hubble parameter. In fact, \(\Theta =3H\) and \(\tilde{\Theta }=3\tilde{H}\), with \(H\) and \(\tilde{H}\) being the Hubble parameters in the CMB and the tilted frames respectively. Then, Eq. (15) reads \(\tilde{H}=H+\tilde{\vartheta }/3\).
Taken at face value, relation (24) seems to suggest that \(\tilde{q}\) could take negative values in low-density domains that expand faster than the background universe, namely in voids with \(\delta \rightarrow -1\) and \(\tilde{\vartheta }>0\). In that case \(\tilde{q}\) can become marginally negative, even when \(q\simeq 1/2\). Having said that, one should be very cautious before applying linear results to nonlinear structures, like the large-scale voids. An alternative (also unlikely) possibility occurs when \(0< q\ll 1\), in which case \(\tilde{q}\) turns negative when \(\delta <0\) and \(|\delta |>q\).
We have employed the familiar harmonic decomposition for the perturbations (e.g. \(\tilde{\vartheta }= \sum _{n} \tilde{\vartheta }_{(n)}\mathcal{Q}_{(n)}\)), where \(\tilde{\mathrm{D}}_{a}\tilde{\vartheta }_{(n)}=0=\mathcal{Q}^{\prime }_{(n)}\) and \(\tilde{\mathrm{D}}^{2}\mathcal{Q}_{(n)}=-(n/a)^{2}\mathcal{Q}_{(n)}\) by construction. Note that \(n\) represents the comoving wavenumber of the harmonic mode (with \(n>0\)), which makes \(\lambda _{(n)}=a/n\) its physical wavelength.
Following definition (33), once the background values of \(H\) and \(q\) are fixed, the peculiar Jeans length (\(\lambda _{P}\)) associated with any given bulk flow depends on the latter’s local expansion/contraction rate (\(\tilde{\vartheta }\)). Also note that, the lower the Hubble parameter, the stronger the relative-motion effects and the larger the value of \(\lambda _{P}\).
The same is also true for the apparent over-deceleration (with \(\tilde{q}^{(+)}>1\)) measured on scales smaller than \(\lambda _{P}\) by observers inside locally expanding bulk flows. Globally, the universe is decelerating with \(q=0.5\).
The considerable differences between the Newtonian and the relativistic treatments of peculiar velocities, as well as the underlying theoretical reasons responsible for them, hold irrespective of the current Hubble value and the age of the universe.
Recall that the standard Jeans length marks the scale below which pressure-gradient perturbations dominate over the background gravitational pull and determine the linear evolution of density inhomogeneities.
References
Ade, P.A.R., et al.: Astron. Astrophys. 561, A97 (2014)
Al Mamon, A., Das, S.: Eur. Phys. J. C 77, 495 (2017)
Antoniou, I., Perivolaropoulos, L.: J. Cosmol. Astropart. Phys. 12, 012 (2010)
Bengaly, C.A.P. Jr., Bernui, A., Alcaniz, J.S.: Astrophys. J. 808, 39 (2015)
Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, Princeton, New Yersey (1987)
Cai, R.-G., Tuo, Z.-L.: J. Cosmol. Astropart. Phys. 02, 004 (2012)
Chang, Z., Lin, H.-N.: Mon. Not. R. Astron. Soc. 446, 2952 (2015)
Coles, P., Lucchin, F.: Cosmology: The Origin and Evolution of Cosmic Structure. Wiley, Chichester (2002)
Colin, J., Mohayaee, R., Sarkar, S., Shafieloo, A.: Mon. Not. R. Astron. Soc. 414, 264 (2011)
Colin, J., Mohayaee, R., Rameez, M., Sarkar, S.: Astron. Astrophys. 631, L13 (2019)
Colin, J., Mohayaee, R., Rameez, M., Sarkar, S.: (2020). arXiv:1912.04257
Collins, C.B., Ellis, G.F.R.: Phys. Rep. 56, 65 (1979)
Cooke, R., Lynden-Bell, D.: Mon. Not. R. Astron. Soc. 401, 1409 (2010)
Ehlers, J.: Abh. Maiz. Akad. Wiss. Lit. 11, 1 (1961)
Ellis, G.F.R.: In: Sachs, R.K. (ed.) General Relativity and Cosmology, p. 104. Academic Press, New York (1971)
Ellis, G.F.R.: In: Schatzman, E. (ed.) Cargese Lectures in Physics, vol. 6, p. 1. Gordon and Breach, New York (1973)
Ellis, G.F.R.: Mon. Not. R. Astron. Soc. 243, 509 (1990)
Ellis, G.F.R., Maartens, R., MacCallum, M.H.A.: Relativistic Cosmology. Cambridge University Press, Cambridge (2012)
Filippou, K., Tsagas, C.G.: Astrophys. Space Sci. 366, 4 (2021)
Gong, Y., Wang, A.: Phys. Rev. D 75, 043520 (2007)
Heckmann, O., Schücking, E.: Z. Astrophys. 38, 95 (1955)
Heinesen, A.: J. Cosmol. Astropart. Phys. 05, 008 (2021)
Herrera, L., Di Prisco, A., Ibánẽz, J.: Phys. Rev. D 84, 064036 (2011)
Hewitt, C.G., Uggla, C., Wainwrihgt, J.: In: Wainwrihgt, J., Ellis, G.F.R. (eds.) Dynamical Systems in Cosmology. Cambridge University Press, Cambridge (1997)
King, A.R., Ellis, G.F.R.: Commun. Math. Phys. 31, 209 (1973)
Lavaux, G., Afshordi, N., Hudson, M.J.: Mon. Not. R. Astron. Soc. 430, 1617 (2013)
Ma, Y.-Z., Pan, J.: Mon. Not. R. Astron. Soc. 437, 1996 (2014)
Maartens, R.: Phys. Rev. D 58, 124006 (1998)
Macpherson, H.J., Heinesen, A.: (2021). arXiv:2103.11918
Migkas, K., Schellenberger, G., Reibrich, T.H., Pacaud, F., Ramos-Ceja, M.E., Lovisari, L.: Astron. Astrophys. 636, A15 (2020)
Migkas, K., Pacaud, F., Schellenberger, G., Erler, J., Nguyen-Dang, N.T., Reibrich, T.H., Ramos-Ceja, M.E., Lovisari, L.: Astron. Astrophys. 649, A151 (2021)
Nair, R., Jhingan, S., Jain, D.: J. Cosmol. Astropart. Phys. 01, 018 (2012)
Nájera, S., Sussman, R.A.: Eur. Phys. J. C 81, 374 (2021)
Nusser, A., Dekel, A., Bertschinger, E., Blumenthal, G.R.: Astrophys. J. 379, 6 (1991)
Padmanabhan, T.: Structure Formation in the Universe. Cambridge University Press, Cambridge (1993)
Peebles, P.J.E.: Astrophys. J. 205, 318 (1976)
Peebles, P.J.E.: The Large-Scale Structure of the Universe. Princeton University Press, Princeton, New Yersey (1980)
Perivolaropoulos, L., Skara, F.: (2021). arXiv:2105.05208
Raychaudhuri, A.K.: Ann. Astrophys. 43, 161 (1957)
Rubin, D., Heitlauf, J.: Astrophys. J. 894, 68 (2020)
Salehi, A., Yarahmadi, M., Fathi, S., Bamba, K.: Mon. Not. R. Astron. Soc. 504, 1304 (2021)
Schwarz, D.J., Weinhorst, B.: Astron. Astrophys. 474, 717 (2007)
Scrimgeour, M.I., et al.: Mon. Not. R. Astron. Soc. 455, 386 (2016)
Tsagas, C.G.: Mon. Not. R. Astron. Soc. 405, 503 (2010)
Tsagas, C.G.: Phys. Rev. D 84, 063503 (2011)
Tsagas, C.G.: Eur. Phys. J. C 81, 753 (2021)
Tsagas, C.G., Kadiltzoglou, M.I.: Phys. Rev. D 92, 043515 (2015)
Tsagas, C.G., Challinor, A., Maartens, R.: Phys. Rep. 465, 61 (2008)
Tsaprazi, E., Tsagas, C.G.: Eur. Phys. J. C 80, 757 (2020)
Valcin, D., Bernal, J.L., Jimenez, R., Verde, L., Wandelt, B.D.: J. Cosmol. Astropart. Phys. 12, 002 (2020)
Acknowledgements
We would like to thank Kostas Migkas, Roya Mohayaee, Mohamed Rameez and Subir Sarkar for helpful discussions and comments. This work was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.), under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant” (Project Number: 789).
Funding
This study was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I. – Project Number: 789).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
No conflict of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tsagas, C.G., Kadiltzoglou, M.I. & Asvesta, K. The deceleration parameter in “tilted” Friedmann universes: Newtonian vs relativistic treatment. Astrophys Space Sci 366, 90 (2021). https://doi.org/10.1007/s10509-021-03995-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10509-021-03995-7