Viable exact model universe without dark energy from primordial inflation David L. Wiltshire∗
arXiv:gr-qc/0503099 v4 6 Apr 2005
Department of Physics & Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand A new model of the observed universe, using solutions to the full Einstein equations, is developed on the basis of the suggestion of Kolb, Matarrese, Notari and Riotto [hep-th/0503117] that our observable universe is an underdense bubble in a spatially flat bulk universe. It is argued that on the basis of Mach’s principle, that true cosmic time is set by the bulk universe. With this understanding a systematic reanalysis of all observed quantities in cosmology is required. I provide such a reanalysis by giving an exact model of the universe depending on two measured parameters: the present density parameter, Ω0 , and the Hubble constant, H0 . The observable universe is not accelerating. Nonetheless, due to systematic factors in the luminosity distance relation the inferred luminosity distances will very closely mimic models with a cosmological constant, in accord with the evidence of distant type Ia supernovae. The measured Hubble constant is found to differ from the present physical Hubble parameter by a systematic offset. The predicted age of the universe agrees well with observation. For a universe with only baryonic matter, the expansion age can easily account for structure formation at large redshifts. It is also predicted that the low multipole (large angle) anomalies seen in the cosmic microwave background anisotropy spectrum might be resolved by the new model. PACS numbers: 98.80.-k, 98.80.Es, 98.80.Cq, 98.80.Bp
Observations in the past decade have been interpreted as suggesting that 70% of the matter–energy density in the universe at the present epoch is in the form of a smooth vacuum energy, or “dark energy”, which does not clump gravitationally. This is supported by two powerful independent lines of observation. Firstly, type Ia supernovae in distant galaxies [1] are dimmer than would be expected in standard Friedmann–Robertson–Walker (FRW) models, especially when it is noted that many independent dynamical estimates of the present clumped mass fraction, Ω0 , suggest values of order 20–30%. Secondly, observations of the power spectrum of primordial anisotropies in the cosmic microwave background radiation (CMBR), most recently by the WMAP satellite [2], indicate that the universe is spatially flat on the largest of scales, and also confirm the value of Ω0 . A cosmological constant, or alternatively dynamical dark energy, is most commonly invoked to explain the observed cosmological parameters, even though a fundamental origin for such a dark energy remains one of the profoundest mysteries of modern physics. However, even in the presence of dark energy at the present epoch, a number of problems remain. One of the most significant problems is that the epoch of reionization measured by WMAP [2] appears at a redshift of order z ∼ 20+10 −9 , indicating that the first stars formed much earlier than conventional models of structure formation would suggest. The detection of complex galaxies at relatively large redshifts [3] compounds the conundrum. In new work, Kolb, Matarrese, Notari and Riotto [4] have proposed a profoundly different resolution of the “cosmological constant problem”. They reason that primordial inflation will have produced density perturbations many times larger than the present horizon volume.
[ arXiv: gr-qc/0503099 ]
We should not view the observable universe as typical of the universe on scales larger than our particle horizon, and the values of cosmological parameters na¨ıvely inferred should not necessarily be taken as typical of the whole. Furthermore, they argue that our present observations are compatible with the observed universe being an underdense bubble in an otherwise spatially flat k = 0 FRW universe with an energy density Ω TOT = 1 in ordinary clumped matter. In this Letter I give an alternative model to that of ref. [4], at the level of the full nonlinear Einstein equations. I will demonstrate that the fundamental premises of ref. [4] are sound, though my quantitative results differ. In particular, I will develop a cosmology which passes observational tests with no cosmic acceleration. The new model is based solely on two cosmological parameters which are already measured: the present density parameter, Ω0 , and the Hubble constant, H0 . All other cosmological parameters are completely determined. It is a consequence of the inflationary paradigm that an initial spectrum of density perturbations is stretched to all observable scales within our past light cone, and also to scales beyond our particle horizon. The fact that this is true for the past light cone is well supported by the CMBR. The hypothesis that such perturbations should extend to super–horizon scales is a feature of most inflationary models, independent of their details. One important realisation is the fact that since primordial inflation ended at a finite very early time, the scale of super–horizon sized modes, although huge, must have a cut–off at an upper bound. Beyond that scale we assume following ref. [4] that there is no dark energy, and
2 the universe is described by a spatially flat bulk metric ds2 = −dt2 + a ¯2 (t)(dx2 + dy 2 + dz 2 ),
(1)
where a ¯(t) = a ¯i (t/ti )2/3 , and we use units with c = 1. Although the observable universe will undoubtedly be embedded in many regions of under- and over-density, like the smallest figure inside a Russian doll, it is nonetheless reasonable to assume that provided the density perturbation immediately containing our observed universe extends sufficiently beyond our horizon then there is a super-horizon sized underdense bubble containing the observable universe, with matter density equal to the average matter density we measure locally, which we can model as a super–horizon sized underdense sphere, S. We can therefore treat the spacetime inside S by the spatially open FRW geometry dr2 2 2 2 2 2 2 2 2 ds = −dτ + α a (τ ) + r (dθ + sin θ dφ ) , 1 + r2 (2) where parametrically in terms of conformal time, η,
To define a global cosmic time in reference to regions beyond the horizon may seem puzzling at first. However, prior to inflation, such regions were in causal contact with the observed universe. Thus a variant of Mach’s principle, in terms of defining true surfaces of homogeneity, and with that a notion of inertial frames of isotropic observers, can be understood conceptually as a logical consequence of inflation.
We also note that regardless of what is happening in the overdense and underdense regions beyond the bubble, S, which contains our universe, there are only two relevant geometries to consider. One is the local geometry (2)–(4), which is relevant because general relativity is a local theory and we can use it as a valid solution of Einstein’s equations. The other is the bulk geometry (1), which dominates on the largest scales, and by Mach’s principle defines a cosmic time throughout the universe. In fact, the observed universe is not particularly homogeneous: we see structures of clusters of galaxies, super– clusters, voids, and streaming motions. While one might ultimately drop the assumption of local homogeneity, it is reasonable to assume that on average the geometry a i Ωi (cosh η − 1) , a = within S is described by the geometry (2)–(4). 2(1 − Ωi ) We are making conservative assumptions which agree Ωi (sinh η − η) . (3) Hi τ = with general relativity. The one new ingredient is that 2(1 − Ωi )3/2 in accord with Mach’s principle we assume that physical quantities are referred to the parameter t of (4), rather where Ωi is the initial density contrast of the bubble, than parameter τ of (3). This means that the parameter S, α its inverse Gaussian curvature magnitude, and ai a τ , which has been previously used as a cosmic time paconstant to be set. rameter, would actually refer to the time of a clock of an It is important to note that there is nothing in general observer boosted with respect to true isotropic observers. relativity which demands that the parameter τ of (2), (3) To correct this anomalous boost we must now systematcorresponds to the time measured on the clock of an ideically reanalyse all observed quantities in cosmology. alised isotropic observer – defined to be an observer who In order to perform the required reanalysis we will conmeasures no dipole anisotropy in the CMBR. Indeed we struct a “spherical expansion model” of the underdense are perfectly free to reparameterize the time parameter in region S, in parallel to the “spherical collapse model” (2), (3) in terms of some arbitrary parameter, t, provided dt [5], keeping careful track of what the observable quanti, so we introduce a non-trivial lapse function γ(t) ≡ dτ ties are. Despite being a very standard approach for the that (2) becomes early stages of structure formation, such an approach is dr2 not widely used for the analogous problem of the growth 2 2 2 2 2 2 2 2 2 ds = −γ (t)dt +α a (t) + r (dθ + sin θ dφ ) , 1 + r2 of voids in our observable universe [6], since in such (4) situations spatial gradients would no doubt render the In other words, the Hubble parameter is not an invariant, model inaccurate. On super–horizon scales considered as it depends on a choice of coordinate time to define the here, however, this is not an issue. surfaces of homogeneity. Following the standard approach, we assume the initial We now make our second crucial assumption. We asdensity parameter is set sufficiently early that it is very sert that in accordance with Mach’s principle, physical close to unity: Ωi = 1 − δi , 0 < δi ≪ 1. Furthermore clocks of idealised isotropic observers within S, or indeed since the universe has Ωi ≃ 1 initially we require the anywhere in the bulk universe, match the time parameter comoving scales to match at that epoch. This means we t of (1). Thus in order to make contact with observation set a ¯i ≃ ai , γi ≃ 1, τi ≃ ti and Hi ≃ 2/(3ti ). the geometry within S must be referred to the form (4) of There are three possible Hubble parameters to take the FRW metric, whenever measurements involving time ¯ = into account: (i) the background Hubble parameter H are performed. Furthermore, the cosmic time parameter, 2/(3t) of the metric (1) which cannot be measured in S; t, for idealised isotropic observers in (4) exactly matches the parameter t of the bulk geometry (1).
3 (ii) the boosted Hubble parameter Hb (τ ) ≡
1 da a dτ
;
(5)
(iii) the physical Hubble parameter H(t) ≡
1 da a dt
.
(6)
By our assumptions, while all three parameters coincide at early times, as the universe expands all three diverge. As isotropic observers within S, we measure the local spatial geometry (2)–(4), referred to global cosmic time, t, not to the parameter τ . We simply need to determine τ (t) to fix all cosmological quantities. 2/3 Using the bulk universe scale factor, a ¯ = ai 32 Hi t and (2)–(4) the density parameter is determined to be Ω=
a ¯ 3 a
=
18Hi 2 (1 − Ωi )3 t2 . Ωi 3 (cosh η − 1)3
(7)
Comparing this with the standard parametric expression Ω(η) = 2(cosh η − 1)/ sinh2 η derived from (2), (3), we obtain a parametric relation for t(η), 3/2
Hi t =
Ωi (cosh η − 1)2 . 3(1 − Ωi )3/2 sinh η
(8)
The physical Hubble parameter is found to be H(η) =
3Hi (1 − Ωi )3/2 sinh3 η 3/2 Ωi (cosh η
= =
−
1)2 (sinh2
η + cosh η − 1)
where Ω0 = 2/(1 + cosh η0 ) is the density parameter at the present epoch. The lapse function is given by dτ 3(cosh η + 1) H = = , dt Hb 2(cosh η + 2)
(10)
where we have used Ωi ≃ 1. Observe that at early times, η ∼ 0, H ∼ Hb as expected but at late times, H ∼ 23 Hb . At the present epoch γ0 = 3/(2 + Ω0 ). For future reference we also define a quantity w(η) ≡
H (Ω + 2)(1 − Ω0 )3/2 sinh η 1 dγ = 0 0 . γ dt (cosh η + 2)2 (cosh η − 1)
Furthermore, the fact that H → 32 Hb as η → ∞ will also systematically modify the luminosity distance relation. Recall that the supernovae measurements [1] involve distances, and that the interpretation as cosmic acceleration depends on model–dependent assumptions. We now show that the effects we have found can conspire to mimic the effects usually attributed to dark energy. To compare with observations (9)–(13) need to be expressed in terms of the observed cosmological redshift, z. Great care is needed at this point in identifying physical quantities. It proves simplest to always use the line element (4), remembering that t corresponds to clock time for true isotropic observers. It follows that
1+z =
H0 (1 − Ω0 )3/2 (Ω0 + 2)(cosh η + 1)3/2 (9) Ω0 (cosh η − 1)3/2 (cosh η + 2)
γ(η) =
It is equal to the bulk deceleration parameter q¯ = 21 at early times η = 0, but at late times as η → ∞, q → 0. The late time value of q(t) differs from that of ref. [4], where apparent cosmic acceleration was obtained. This discrepancy is not surprising given that Kolb et al. worked with a number of approximations, whereas the present analysis is exact. Although we have no cosmic acceleration, the ratio of q to the na¨ıve deceleration parameter deduced from (2), (3) is 3/(2 + cosh η), which vanishes at late times. Thus deceleration is dramatically reduced as compared to a standard open universe. At the present epoch q0 = 3Ω20 /[2(Ω0 + 2)]. For Ω0 ∼ 0.2–0.3 we have q0 ∼ 0.025–0.06; and for Ω0 ∼ 0.05, q0 ∼ 0.002.
(11)
(1 − Ω0 )(2 + Ω0 )(cosh η + 1) a0 γ = aγ0 Ω0 (cosh η + 2)(cosh η − 1)
We must be careful to note that when (14) is expanded as a Taylor series for small distances then we find
z = ℓr (H0 − w0 ) = ℓr H0
The observed deceleration parameter q(t) = −H −2 a ¨/a = 2 ˙ −1 − H/H (with overdot denoting a t derivative), is given by q(η) = −1 +
cosh2 η + 3 cosh η + 5 (cosh η + 2)(cosh η + 1)
(13)
!
,
(15)
t0
(1 − Ω0 )Ω0 /(Ω0 + 2). The measured Hubble constant is therefore not the present value of the physical Hubble parameter (6), but is given by
Hm = (12)
2 + Ω0 2 2 + Ω0
in terms of proper radial distance, ℓr . Thus the Hub ble relation contains an additional term, w0 = γ1 dγ dt =
The expansion age (8) is usefully rewritten as Ω0 (cosh η − 1)3/2 t(η) = H0 (1 − Ω0 )3/2 (Ω0 + 2)(cosh η + 1)1/2
(14)
2 + Ω0 2 2 + Ω0
!
H0 .
(16)
For small values of Ω0 there will be a systematic offset, which is close to unity, but nonetheless large enough to have significance for all measured quantities in cosmology. E.g., for Ω0 ∼ 0.05, Hm ≃ 0.977H0 . The physical solution to eqn. (14) is
4
−1 (1 − Ω0 )(2 + Ω0 ) + + cosh η = 2
q Ω0 z[9Ω0 z − 2Ω0 2 + 16Ω0 + 4] + (Ω0 2 + 2)2 2Ω0 (z + 1)
(17)
We are now ready for our first cosmological tests. The expansion age (12) is given by substituting (17) in (12) and is plotted in Fig. 1 for two values of Ω0 in comparison to some standard Friedmann–Lemaˆıtre models. It gives a present day age of the universe of t0 = 2H0−1 /(Ω0 + 2). The luminosity distance is readily computed in the standard p fashion and is found to be dL = (1 + z)γ0 −1 αa0 sinh(η − η0 ), and since 1 − Ω0 = γ0 /(a0 αH0 ) it follows that ) ! ( q 2 + Ω0 2 (1 + z)(2 + Ω0 2 ) (2 − Ω0 ) 2 Hm dL = cosh η − 1 . (18) H0 dL = 2 cosh η − p 2 + Ω0 Ω0 (2 + Ω0 ) 1 − Ω0
10.0 1.0
5.0 Hm dL
Age: t (Gyr)
1.0
0.5
0.1
0.5
0.05 2
4 6 Redshift: z
8
10
FIG. 1: The expansion age for a typical example new model universes with Ω0 = 0.05 (solid line), Ω0 = 0.23 (dashed line), as compared to a vacuum energy model with Ω0 = 0.23, ΩΛ0 = 0.77 (dotted line), and open FRW model with Ω0 = 0.23 (lowest dash–dotted line). H0 = 70 km sec−1 Mpc−1 in both cases.
Although models with values Ω0 ∼ 0.2–0.3 fit the supernova data reasonably well, given that the relation (15) demands a reanalysis of all cosmological quantities, including the rotation curves of galaxies, we should be open to the possibility that this model may allow us to dispense with dark matter as well as dark energy. We could then take Ω0 ∼ 0.025–0.075, corresponding to ordinary baryonic matter only, with the density set by primordial nucleosynthesis bounds. This is the reason for the two distinct values of Ω0 plotted in Figs. 1 and 2. Statistical comparison of the model with the supernovae data indicates a good fit for the lower values of Ω0 [7]. We note that with H0 = 70 ± 5 km sec−1 Mpc−1 the age of the universe is t0 = 12.5+1.3 −1.1 Gyr for Ω0 = 0.23 ± 0.05, or alternatively t0 = 13.6+1.3 −1.1 Gyr for Ω0 = 0.05 ± 0.025. Both values agree well with age estimates from old globular clusters and with WMAP results [2], especially when it is noted that the values quoted in ref. [2] are model–dependent best–fit values, and tighter bounds are expected once the sky maps are recomputed for the new
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Redshift: z
FIG. 2: The luminosity distance, Hm dL , for a typical example new model universes with Ω0 = 0.05 (solid line), Ω0 = 0.23 (dashed line) as compared to a vacuum energy model with Ω0 = 0.23, ΩΛ0 = 0.77 (uppermost dotted line), and an open FRW model with Ω0 = 0.23 (lowest dash–dotted line).
model. If it is possible to take Ω0 to be the density of baryonic matter, then we have the added advantage of a dramatic difference in the expansion age as seen in Fig. 1. For Ω0 ∼ 0.05 the expansion age is significantly larger at large z: t ∼ 4 Gyr at z ∼ 2, t ∼ 1.5 Gyr at z ∼ 2, and t ∼ 0.3 Gyr at z ∼ 20. This would buy precious time for structure formation early on, and easily explain the redshift of the reionization epoch [2], and observations of “early” formation of structure such as those of Refs. [3]. As a model universe, the present model is remarkable in that it depends only on two already observed parameters. With fewer parameters than many competing cosmological models with dark energy, it is readily testable, and numerous observable quantities can be directly computed. Indeed, if this model is correct, then all observed quantities in cosmology must be systematically reanalysed. Since the required changes would in some sense correct an anomalous boost, it is expected that anomalies
5 seen in the low multipoles (large angles) of the CMBR anisotropy data [8], are precisely of the sort that this model would resolve. Reanalysis of the WMAP data with the new cosmological model may be its first real test. It would be ironic that Einstein’s idea concerning the irrelevance of the cosmological constant, and his idea of the importance of Mach’s principle, may prove to both be right in understanding the universe. This work was supported by the Mardsen Fund of the Royal Society of New Zealand. I thank Roy Kerr, Tam Nguyen Phan, Alex Nielsen, Benedict Carter, Ben Leith, Marni Sheppeard, Jorma Louko, Ishwaree Neupane, John Barrow and Paul Davies for helpful comments.
∗
Electronic address:
[email protected]; URL: http://www2.phys.canterbury.ac.nz/~dlw24/
[1] S. Perlmutter et al., Astrophys. J. 483, 565 (1997); A.G. Riess et al., Astron. J. 116, 1009 (1998); S. Perlmutter et al., Astrophys. J. 517, 565 (1999). [2] C. L. Bennett et al., Astrophys. J. Suppl. 148, 1 (2003). [3] See, e.g., K. Glazebrook et al., Nature 430, 181 (2004); A. Cimatti et al., Nature 430, 184 (2004). [4] E. W. Kolb, S. Matarrese, A. Notari and A. Riotto, arXiv:hep-th/0503117. [5] See, e.g., E. W. Kolb and M. S. Turner, “The Early Universe”, (Addison–Wesley, Reading, Mass., 1990), §9.2.1. [6] E. Bertschinger, Astrophys. J. Suppl. 58, 1 (1985). [7] B. M. N. Carter, B. M. Leith, S. C. C. Ng, A. B. Nielsen and D. L. Wiltshire, in preparation. [8] K. Land and J. Magueijo, arXiv:astro-ph/0502237.
