Simplicity in cosmology: add virialisation, remove Λ , keep classical GR
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Simplicity in cosmology: add virialisation,remove Λ, keep classical GR
Boudewijn F. RoukemaToru´n Centre for AstronomyFaculty of Physics, Astronomy and InformaticsNicolaus Copernicus Universityul. Gagarina 1187-100 Toru´n, Polandle 14 juillet 2014 chapter to appear in“Mathematical structures of the Universe”
Abstract
Present-day extragalactic observations are mostly rather well-modelled by a general-relativistic model, the ΛCDM model. Themodel appears to surpass the limits of known physics by requiringthat the Universe be dominated by “dark energy”. However, themodel sacrifices physical simplicity in favour of applied mathematicalsimplicity. A physically simpler, general-relativistic alternative to theΛCDM model is described here, along with preliminary observationalchecks. Thus, it will be argued that extragalactic observations suchas the distance-modulus–redshift relation of type Ia supernovae arewell-modelled within classical general relativity, without the additionof “new physics”.
Within the family of locally homogeneous and isotropic solutions ofthe Einstein equation, i.e. the Friedmann–Lemaˆıtre–Robertson–Walker(FLRW) model (de Sitter, 1917; Friedmann, 1923, 1924; Lemaˆıtre, 1927;Robertson, 1935), observations since the early 1990’s—faint galaxy counts1nd correlation functions (e.g. Fukugita et al., 1990; Roukema & Yoshii,1993; Yoshii & Peterson, 1995), gravitational lensing (e.g. Chiba & Yoshii,1997; Fort et al., 1997), supernovae type Ia magnitude–redshift relations(e.g. Perlmutter et al., 1999; Schmidt et al., 1998), and the cosmic mi-crowave background (CMB) from the Wilkinson Microwave AnisotropyProbe (WMAP) (Spergel et al., 2003) and Planck Surveyor (Ade et al.,2013)—have required the addition of a parameter that does not correspondto any empirically detected physical phenomenon: the cosmological constantor dark energy parameter Λ (Ostriker & Steinhardt, 1995). This “discovery”has stimulated much theoretical interest, including hypotheses of new phys-ical components of the Universe and theories of gravity that extend beyondclassical general relativity.However, the FLRW solutions (of which the ΛCDM model is a specialcase) do not take into account the fact that we live during the virialisa-tion epoch, i.e. the epoch during which dense structures—galaxies, galaxyclusters, and the cosmic web (e.g. de Lapparent et al., 1986) in general—and underdense regions—voids on scales of many h − Mpc—have recentlyformed. Thus, at small scales and recent epochs, it should be expected thatinterpretations of observational data inferred by assuming the FLRW familyof models may fail. Wrong assumptions tend to imply wrong conclusions.Defining the fraction (by mass) of non-relativistic matter (baryonic anddark) contained in virialised objects of typical galaxy scales and above f vir ( z )at any given redshift, it was shown in Fig. 1 of Roukema et al. (2013) thatthe evolution of the dark energy parameter Ω Λ ( z ) follows that of f vir ( z ) towithin a factor of a few. Thus, as the Universe evolves from high redshift(early epochs) to low redshift (recent epochs), the virialisation fraction growsfrom very little to a high fraction of unity, and the dark energy parameterinferred from observations by assuming homogeneity despite its increasinginvalidity grows similarly from very little to nearly unity. In other words,the more that the Universe becomes inhomogeneous, the more that the as-sumption of homogeneity leads to the sudden emergence of dark energy. Thisquantitative similarity, Ω Λ ( z ) ∼ f vir ( z ), reverses the onus of proof for darkenergy: unless or until relativistically acceptable cosmological models in-cluding structure formation show that dark energy is needed in order to fitobservational data, the simplest explanation for dark energy is that it is anartefact of inhomogeneity.Exact relativistic inhomogeneous cosmological solutions of the Einsteinequation have been known since the 1930’s (Lemaˆıtre, 1933; Tolman, 1934;2 assive E mpty D omain Figure 1: Examples of M assive (virialised), E mpty (un-virialised), andfull averaged D omains in a 60 h − Mpc × h − Mpc region showing galaxiesobserved in the two-degree–field galaxy redshift survey (Colless et al., 2003,2dFGRS, ) with the observer at the left.The slice is about 10 great circle degrees in thickness.Bondi, 1947) and reviewed during the last few decades (Krasinski, 1997;Krasi´nski, 2006). While not directly applicable as cosmological models with-out sacrificing the Copernican principle, these solutions provide qualitativeunderstanding of more realistic alternatives to the FLRW model, and canbe used to model the “holes” in “Swiss cheese” cosmological models (e.g.Lavinto et al., 2013, and refs therein).A relativistic approach to inhomogeneous cosmology that allows the in-clusion of standard statistical representations of structure in the Universe isthe scalar averaging approach to cosmology (Buchert et al., 2000; Buchert,2008; Buchert & Carfora, 2008; Buchert & Ostermann, 2012; Buchert et al.,2013; Kolb et al., 2005b,a; R¨as¨anen, 2006a,b; Kolb, 2011; Wiltshire, 2007a,b,2009; Duley et al., 2013). As in the FLRW approach, a spacetime foliation is3hosen, but instead of assuming homogeneity on the slices, volume-weightedmeans (using the metric to measure volume) are calculated within spatialdomains of interest.Here we briefly describe the virialisation approximation presented infull in Roukema et al. (2013). This implementation of multi-scale scalaraveraging uses the definitions, derivations and terminology introduced inBuchert & Carfora (2008) and Wiegand & Buchert (2010). Figure 1 illus-trates the terminology, with complementary M assive (virialised) and E mpty(un-virialised) domains, and the full averaged D omains, labelled genericallyas F . Let us define the virialisation volume fraction λ M := |M||D| , (1)which is related to f vir ( z ) by λ M = f vir / ∆ vir , (2)where ∆ vir ∼
100 to 200 is the overdensity ratio after collapse, e.g. estimatedfor a top-hat initial overdensity using the scalar virial theorem for an isolatedsystem (e.g. Lacey & Cole, 1993). The homogeneous Friedmann equationΩ m + Ω Λ + Ω k = 1 (3)can then be generalised toΩ F m + Ω F Λ + Ω FR + Ω FQ = H F H D , (4)where the F superscript (or subscript) indicates which domain is being usedfor averaging, the scalar-averaged expansion rate H F is defined H F := ˙ a F a F , (5)the rigid curvature parameter Ω k has been replaced by a spatially vary-ing (domain-averaged) 3-Ricci curvature parameter Ω FR , and the kinematicalbackreaction parameter Ω FQ arises, representing statistics (variance, shear,and vorticity) of the extrinsic curvature tensor (in Newtonian thinking, thevelocity gradient). See Roukema et al. (2013) for detailed definitions.4onsidering the virialised and un-virialised regions together, i.e. F = D ,the right-hand side of Eq. (4) simplifies to unity. In addition, approxi-mating | Ω FQ | ≪ | Ω FR | , as found with the relativistic Zel’dovich approxima-tion (Buchert & Ostermann, 2012; Buchert et al., 2013), the resemblance ofEq. (4) to the rigid comoving space version, Eq. (3), is obvious. The param-eter required by the FLRW model for fitting observations is now removed,i.e. we set Ω F Λ = 0. For future observations, it might eventually be necessaryto allow a dark energy parameter again, though the history of relativisticcosmology suggests that Λ is a parameter that falls in or out of cosmologicalfashion every few decades. Thus, Eq. (4) becomesΩ F m + Ω FR = H F H D , (6)or Ω D m + Ω DR = 1 (7)for the full domain.To evaluate these equations, we assume the stable clustering hypothesisPeebles (1980), i.e. H M ≈ , (8)virialised regions ( M ) are stable, we assume an Einstein–de Sitter (EdS)model at early epochs (high redshift), and we extrapolate this as a “back-ground” model (with parameters labelled “bg”) to recent epochs (low red-shift). Simplifying this set of equations leads to Eq. (2.22) of Roukema et al.(2013), i.e. Ω effm ( z ) := Ω D m ≈ Ω bgm ( H eff /H bg ) , (9)where the label D has been replaced by “eff” for convenience.Since virialised regions are assumed to be stable [Eq. (8)], and since voidshave to expand faster than the extrapolated background model in order to beable to become (nearly) empty, it is already clear that at low redshifts, H eff will increase to become greater than H bg . In Eq. (2.23) of Roukema et al.(2013), we model this with a smooth transition from the pre-virialisationepoch to the present epoch, we require that the effective expansion rateat the present epoch matches the observed Hubble constant [Eq. (2.32),Roukema et al. (2013)], and we estimate the peculiar expansion rate of voidsusing void and cluster surveys of approximately similar sizes [Eq. (2.36),5oukema et al. (2013)]. An effective metric can then be defined, allowing cal-culation of effective luminosity distances [Sect. 2.4, Roukema et al. (2013)].Thus, making minimal assumptions beyond the homogeneous model apartfrom allowing inhomogeneous structure on spatial slices and correspondingcurvature, and defining averages on spatial domains rather than forcing uni-formity, the early-time Einstein–de Sitter model evolves to a model with a lowmatter density at the present, Ω effm (0) ≈ . References
Ade, P. A. R., Aghanim, N., Armitage-Caplan, C., Arnaud, M., Ashdown,M., Atrio-Barandela, F., Aumont, J., Baccigalupi, C., Banday, A. J., &et al. 2013, ArXiv e-prints, [arXiv:1303.5076]
Bondi, H. 1947, MNRAS, 107, 410Buchert, T. 2008, Gen. Rev. Grav., 40, 467, [arXiv:0707.2153]
Buchert, T., & Carfora, M. 2008, CQG, 25, 195001, [arXiv:0803.1401]
Buchert, T., Kerscher, M., & Sicka, C. 2000, Phys. Rev. D, 62, 043525, [arXiv:astro-ph/9912347]
Buchert, T., Nayet, C., & Wiegand, A. 2013, Phys. Rev. D, 87, 123503, [arXiv:1303.6193]
Buchert, T., & Ostermann, M. 2012, Phys. Rev. D, 86, 023520, [arXiv:1203.6263]
Chiba, M., & Yoshii, Y. 1997, ApJ, 489, 485Colless, M., Peterson, B. A., Jackson, C., Peacock, J. A., Cole, S., Norberg,P., Baldry, I. K., Baugh, C. M., Bland-Hawthorn, J., Bridges, T., Cannon,R., Collins, C., Couch, W., Cross, N., Dalton, G., De Propris, R., Driver,S. P., Efstathiou, G., Ellis, R. S., Frenk, C. S., Glazebrook, K., Lahav,6., Lewis, I., Lumsden, S., Maddox, S., Madgwick, D., Sutherland, W., &Taylor, K. 2003, (astro-ph/0306581)de Lapparent, V., Geller, M. J., & Huchra, J. P. 1986, ApJL, 302, L1de Sitter, W. 1917, MNRAS, 78, 3Duley, J. A. G., Ahsan Nazer, M., & Wiltshire, D. L. 2013, Classical andQuantum Gravity, 30, 175006, [arXiv:1306.3208]
Fort, B., Mellier, Y., & Dantel-Fort, M. 1997, A&A, 321, 353Friedmann, A. 1923,
Mir kak prostranstvo i vremya (The Universe as Spaceand Time) (Petrograd: Academia)Friedmann, A. 1924, Zeitschr. f¨ur Phys., 21, 326Fukugita, M., Yamashita, K., Takahara, F., & Yoshii, Y. 1990, ApJL, 361,L1Kolb, E. W. 2011, CQG, 28, 164009Kolb, E. W., Matarrese, S., Notari, A., & Riotto, A. 2005a, Phys. Rev. D,71, 023524, [arXiv:hep-ph/0409038]
Kolb, E. W., Matarrese, S., & Riotto, A. 2005b, ArXiv Astrophysics e-prints, [arXiv:astro-ph/0506534]
Krasinski, A. 1997, Inhomogeneous Cosmological Models (Cambridge, UK:Cambridge University Press), 1–333, ISBN: 0521481805Krasi´nski, A. 2006, Inhomogeneous Cosmological Models (Cambridge, UK:Cambridge University Press)Lacey, C., & Cole, S. 1993, MNRAS, 262, 627Lavinto, M., R¨as¨anen, S., & Szybka, S. J. 2013, JCAP, 12, 51, [arXiv:1308.6731]
Lemaˆıtre, G. 1927, Annales de la Soci´et´e Scientifique de Bruxelles, 47, 49Lemaˆıtre, G. 1933, Annales de la Soci´et´e Scientifique de Bruxelles, 53, 517striker, J. P., & Steinhardt, P. J. 1995, ArXiv Astrophysics e-prints, [arXiv:astro-ph/9505066]
Peebles, P. J. E. 1980, Large-Scale Structure of the Universe (Princeton Uni-versity Press)Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R. A., Nugent, P., Castro,P. G., Deustua, S., Fabbro, S., Goobar, A., Groom, D. E., Hook, I. M.,Kim, A. G., Kim, M. Y., Lee, J. C., Nunes, N. J., Pain, R., Pennypacker,C. R., Quimby, R., Lidman, C., Ellis, R. S., Irwin, M., McMahon, R. G.,Ruiz-Lapuente, P., Walton, N., Schaefer, B., Boyle, B. J., Filippenko,A. V., Matheson, T., Fruchter, A. S., Panagia, N., Newberg, H. J. M.,Couch, W. J., & The Supernova Cosmology Project. 1999, ApJ, 517, 565, [arXiv:astro-ph/9812133]
R¨as¨anen, S. 2006a, CQG, 23, 1823, [arXiv:astro-ph/0504005]
R¨as¨anen, S. 2006b, International Journal of Modern Physics D, 15, 2141, [arXiv:astro-ph/0605632]
Robertson, H. P. 1935, ApJ, 82, 284Roukema, B. F., Ostrowski, J. J., & Buchert, T. 2013, JCAP, 10, 043, [arXiv:1303.4444]
Roukema, B. F., & Yoshii, Y. 1993, ApJL, 418, L1Schmidt, B. P., Suntzeff, N. B., Phillips, M. M., Schommer, R. A., Cloc-chiatti, A., Kirshner, R. P., Garnavich, P., Challis, P., Leibundgut, B.,Spyromilio, J., Riess, A. G., Filippenko, A. V., Hamuy, M., Smith, R. C.,Hogan, C., Stubbs, C., Diercks, A., Reiss, D., Gilliland, R., Tonry, J.,Maza, J., Dressler, A., Walsh, J., & Ciardullo, R. 1998, ApJ, 507, 46, [arXiv:astro-ph/9805200]
Spergel, D. N., Verde, L., Peiris, H. V., Komatsu, E., Nolta, M. R., Bennett,C. L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., Limon, M., Meyer,S. S., Page, L., Tucker, G. S., Weiland, J. L., Wollack, E., & Wright, E. L.2003, ApJSupp, 148, 175, [arXiv:astro-ph/0302209]
Tolman, R. C. 1934, Proceedings of the National Academy of Science, 20,169 8iegand, A., & Buchert, T. 2010, Phys. Rev. D, 82, 023523, [arXiv:1002.3912]
Wiltshire, D. L. 2007a, New Journal of Physics, 9, 377, [arXiv:gr-qc/0702082]
Wiltshire, D. L. 2007b, Physical Review Letters, 99, 251101, [arXiv:0709.0732]
Wiltshire, D. L. 2009, Phys. Rev. D, 80, 123512, [arXiv:0909.0749][arXiv:0909.0749]