The Return of a Static Universe and the End of Cosmology
The Return of a Static Universe and the End of Cosmology
Lawrence M. Krauss1,2 and Robert J. Scherrer2
1Department of Physics, Case Western Reserve University,
Cleveland, OH 44106; email: krauss@cwru.edu and
2Department of Physics & Astronomy, Vanderbilt University,
Nashville, TN 37235; email: robert.scherrer@vanderbilt.edu
(Dated: February 1, 2008)
Abstract
We demonstrate that as we extrapolate the current ΛCDM universe forward in time, all evi-
dence of the Hubble expansion will disappear, so that observers in our “island universe” will be
fundamentally incapable of determining the true nature of the universe, including the existence
of the highly dominant vacuum energy, the existence of the CMB, and the primordial origin of
light elements. With these pillars of the modern Big Bang gone, this epoch will mark the end of
cosmology and the return of a static universe. In this sense, the coordinate system appropriate for
future observers will perhaps fittingly resemble the static coordinate system in which the de Sitter
universe was first presented.
http://arXiv.org/abs/0704.0221v3
Shortly after Einstein’s development of general relativity, the Dutch astronomer Willem
de Sitter proposed a static model of the universe containing no matter, which he thought
might be a reasonable approximation to our low density universe. One can define a coor-
dinate system in which the de Sitter metric takes a static form by defining de Sitter space-
time with a cosmological constant Λ as a four dimensional hyperboloid SΛ : ηABξ
AξB =
−R2, R2 = 3Λ−1 embedded in a 5d Minkowski spacetime with ds2 = ηABdξ
AdξB,
and (ηAB) = diag(1,−1,−1,−1,−1), A, B = 0, · · · , 4. The static form of the de Sitter
metric is then
ds2s = (1 − r
2)dt2s −
1 − r2s/R
− r2sdΩ
which can be obtained by setting ξ0 = (R2 − r2s)
1/2 sinh(ts/R), ξ
1 = rs sin θ cos ϕ, ξ
rs sin θ sin ϕ, ξ
3 = rs cos θ, ξ
4 = (R2 − r2s)
1/2 cosh(ts/R). In this case the metric only corre-
sponds to the section of de Sitter space within a cosmological horizon at R = r − s.
In fact de Sitter’s model wasn’t globally static, but eternally expanding, as can be seen by
a coordinate transformation which explicitly incorporates the time dependence of the scale
factor R(t) = exp(Ht). While spatially flat, it actually incorporated Einstein’s cosmological
term, which is of course now understood to be equivalent to a vacuum energy density, leading
to a redshift proportional to distance.
The de Sitter model languished for much of the last century, once the Hubble expansion
had been discovered, and the cosmological term abandoned. However, all present observa-
tional evidence is consistent with a ΛCDM flat universe consisting of roughly 30% matter
(both dark matter and baryonic matter) and 70% dark energy [1, 2, 3], with the latter
having a density that appears constant with time. All cosmological models with a non-zero
cosmological constant will approach a de Sitter universe in the far future, and many of the
implications of this fact have been explored in the literature [4, 5, 6, 7, 8, 9, 10, 11, 12, 13].
Here we re-examine the practical significance of the ultimate de Sitter expansion and
point out a new eschatological physical consequence: from the perspective of any observer
within a bound gravitational system in the far future, the static version of de Sitter space
outside of that system will eventually become the appropriate physical coordinate system.
Put more succinctly, in a time comparable to the age of the longest lived stars, observers
will not be able to perform any observation or experiment that infers either the existence of
an expanding universe dominated by a cosmological constant, or that there was a hot Big
Bang. Observers will be able to infer a finite age for their island universe, but beyond that
cosmology will effectively be over. The static universe, with which cosmology at the turn of
the last century began, will have returned with a vengeance.
Modern cosmology is built on integrating general relativity and three observational pillars:
the observed Hubble expansion, detection of the cosmic microwave background radiation,
and the determination of the abundance of elements produced in the early universe. We
describe next in detail how these observables will disappear for an observer in the far future,
and how this will be likely to affect the theoretical conclusions one might derive about the
universe.
A. The disappearance of the Hubble Expansion
The most basic component of modern cosmology is the expansion of the universe, firmly
established by Hubble in 1929. Currently, galaxies and galaxy clusters are gravitationally
bound and have dropped out of the Hubble flow, but structures on larger length scales are
observed to obey the Hubble expansion law. Now consider what happens in the far future
of the universe. Both analytic [7] and numerical [10] calculations indicate that the Local
Group remains gravitationally bound in the face of the accelerated Hubble expansion. All
more distant structures will be driven outside of the de Sitter event horizon in a timescale on
the order of 100 billion years ([4], see also Refs. [8, 9]). While objects will not be observed
to cross the event horizon, light from them will be exponentially redshifted, so that within
a time frame comparable to the longest lived main sequence stars all objects outside of our
local cluster will truly become invisible [4].
Since the only remaining visible objects will in fact be gravitationally bound and decou-
pled from the underlying Hubble expansion, any local observer in the far future will see a
single galaxy (the merger product of the Milky Way and Andromeda and other remnants of
the Local Group) and will have no observational evidence of the Hubble expansion. Lacking
such evidence, one may wonder whether such an observer will postulate the correct cosmo-
logical model. We would argue that in fact, such an observer will conclude the existence of
a static “island universe,” precisely the standard model of the universe c. 1900.
This will be true in spite of the fact that the dominant energy in this universe will not
be due to matter, but due to dark energy, with ρM/ρΛ ∼ 10
−12 inside the horizon volume
[9]. The irony, of course, is that the denizens of this static universe will have no idea of
the existence of the dark energy, much less of its magnitude, since they will have no probes
of the length scales over which Λ dominates gravitational dynamics. It appears that dark
energy is undetectable not only in the limit where ρΛ ≪ ρM , but also when ρΛ ≫ ρM .
Even if there were no direct evidence of the Hubble expansion, we might expect three
other bits of evidence, two observational and one theoretical, to lead physicists in the future
to ascertain the underlying nature of cosmology. However, we next describe how this is
unlikely to be the case.
B. Vanishing CMB
The existence of a Cosmic Microwave Background was the key observation that convinced
most physicists and astronomers that there was in fact a hot big bang, which essentially
implies a Hubble expansion today. But even if skeptical observers in the future were inclined
to undertake a search for this afterglow of the Big Bang, they would come up empty-
handed. At t ≈ 100 Gyr, the peak wavelength of the cosmic microwave background will be
redshifted to roughly λ ≈ 1 m, or a frequency of roughly 300 MHz. While a uniform radio
background at this frequency would in principle be observable, the intensity of the CMB will
also be redshifted by about 12 orders of magnitude. At much later times, the CMB becomes
unobservable even in principle, as the peak wavelength is driven to a length larger than the
horizon [4]. Well before then, however, the microwave background peak will redshift below
the plasma frequency of the interstellar medium, and so will be screened from any observer
within the galaxy. Recall that the plasma frequency is given by
where ne and me are the electron number density and mass, respectively. Observations
of dispersion in pulsar signals give [14] ne ≈ 0.03 cm
−3 in the interstellar medium, which
corresponds to a plasma frequency of νp ≈ 1 kHz, or a wavelength of λp ≈ 3 × 10
7 cm.
This corresponds to an expansion factor ∼ 108 relative to the present-day peak of the CMB.
Assuming an exponential expansion, dominated by dark energy, this expansion factor will
be reached when the universe is less than 50 times its present age, well below the lifetime
of the longest-lived main sequence stars.
After this time, even if future residents of our island universe set out to measure a
universal radiation background, they would be unable to do so. The wealth of information
about early universe cosmology that can be derived from fluctuations in the CMB would be
even further out of reach.
C. General Relativity Gives No Assistance
We may assume that theoretical physicists in the future will infer that gravitation is
described by general relativity, using observations of planetary dynamics, and ground-based
tests of such phenomena as gravitational time dilation. Will they then not be led to a Big
Bang expansion, and a beginning in a Big Bang singularity, independent of data, as Lemaitre
was? Indeed, is not a static universe incompatible with general relativity?
The answer is no. The inference that the universe must be expanding or contracting is
dependent upon the cosmological hypothesis that we live in an isotropic and homogeneous
universe. For future observers, this will manifestly not be the case. Outside of our local
cluster, the universe will appear to be empty and static. Nothing is inconsistent with
the temporary existence of a non-singular isolated self-gravitating object in such a universe,
governed by general relativity. Physicists will infer that this system must ultimately collapse
into a future singularity, but only as we presently conclude our galaxy must ultimately
coalesce into a large black hole. Outside of this region, an empty static universe can prevail.
While physicists in the island universe will therefore conclude that their island has a finite
future, the question will naturally arise as to whether it had a finite beginning. As we next
describe, observers will in fact be able to determine the age of their local cluster, but not
the nature of the beginning.
D. Polluted Elemental Abundances
The theory of Big Bang Nucleosynthesis reached a fully-developed state [15] only after
the discovery of the CMB (despite early abortive attempts by Gamow and his collaborators
[16]). Thus, it is unlikely that the residents of the static universe would have any motivation
to explore the possibility of primordial nucleosynthesis. However, even if they did, the
evidence for BBN rests crucially on the fact that relic abundances of deuterium remain
observable at the present day, while helium-4 has been enhanced by only a few percent
since it was produced in the early universe. Extrapolating forward by 100 Gyr, we expect
significantly more contamination of the helium-4 abundance, and concomitant destruction
of the relic deuterium. It has been argued [17] that the ultimate extrapolation of light
elemental abundances, following many generations of stellar evolution, is a mass fraction of
helium given by Y = 0.6. The primordial helium mass fraction of Y = 0.25 will be a relatively
small fraction of this abundance. It is unlikely that much deuterium could survive this degree
of processing. Of course, the current “smoking gun” deuterium abundance is provided by
Lyman-α absorption systems, back-lit by QSOs (see, e.g., Ref. [18]). Such systems will be
unavailable to our observers of the future, as both the QSOs and the Lyman-α systems will
have redshifted outside of the horizon.
Astute observers will be able to determine a lower limit on the age of their system,
however, using standard stellar evolution analyses of their own local stars. They will be able
to examine the locus of all stars and extrapolate to the oldest such stars to estimate a lower
bound on the age of the galaxy. They will be able to determine an upper limit as well, by
determining how long it would take for all of the observed helium to be generated by stellar
nucleosynthesis. However, without any way to detect primordial elemental abundances, such
as the aforementioned possibility of measuring deuterium in distant intergalactic clouds that
currently absorb radiation from distant quasars and allow a determination of the deuterium
abundance in these pre-stellar systems, and with the primordial helium abundance dwarfed
by that produced in stars, inferring the original BBN abundances will be difficult, and
probably not well motivated.
Thus, while physicists of the future will be able to infer that their island universe has not
been eternal, it is unlikely that they will be able to infer that the beginning involved a Big
Bang.
E. Conclusion
The remarkable cosmic coincidence that we happen to live at the only time in the history
of the universe when the magnitude of dark energy and dark matter densities are comparable
has been a source of great current speculation, leading to a resurgence of interest in possible
anthropic arguments limiting the value of the vacuum energy (see, e.g., Ref. [19]). But
this coincidence endows our current epoch with another special feature, namely that we can
actually infer both the existence of the cosmological expansion, and the existence of dark
energy. Thus, we live in a very special time in the evolution of the universe: the time at
which we can observationally verify that we live in a very special time in the evolution of
the universe!
Observers when the universe was an order of magnitude younger would not have been
able to discern any effects of dark energy on the expansion, and observers when the universe
is more than an order of magnitude older will be hard pressed to know that they live in an
expanding universe at all, or that the expansion is dominated by dark energy. By the time
the longest lived main sequence stars are nearing the end of their lives, for all intents and
purposes, the universe will appear static, and all evidence that now forms the basis of our
current understanding of cosmology will have disappeared.
Note added in proof: After this paper was submitted we learned of a prescient 1987
paper [20], written before the discovery of dark energy and other cosmological observables
that are central to our analysis, which nevertheless raised the general question of whether
there would be epochs in the Universe when observational cosmology, as we now understand
it, would not be possible.
Acknowledgments
L.M.K. and R.J.S. were supported in part by the Department of Energy.
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	The disappearance of the Hubble Expansion
	Vanishing CMB
	General Relativity Gives No Assistance
	Polluted Elemental Abundances
	Conclusion
	Acknowledgments
	References
