Area metric cosmology How to explain the universe without dark energy or fine-tuning Bj ¨ orn Wehinger, Uppsala Universitet January 22, 2008 Abstract Using gener al relativit y and Ei nstein cosmology we have to intr odu ce some ki nd of dark matt er to ex pl ain the observed acceleration of our universe. A physical explanation of this dark energy does not exist until now. A cosmology based on area metric avoids this problem. If one take a general area measure insteadof a lengt h measure as a basis the cosmolo gical consta nt becomes unnecessa ry . The universe can be explained without dark matter or fine-tuning! 1. Introduction The idea of an area metric cosmology was first published by Raffaele Punzi, Frederic P. Schuller and Matti as N. R. Wohlfa rth on February 12, 2007 [1]. They trade space time as an area metric mani fold and show that this is a true generalizat ion of metric geometry in four (and more) dimensi ons. Based on generalized geometry in quantum string and gauge theory the universe is filled with bosonic and fermionic string radiation for the radiation dominated epoch [3] and filled with non-interacting string dust at large scales for the late universe [1] and [2]. For the radiation dominated epoch the area metric cosmol ogy is equi vale nt to the standard Einstei n cosmology . For the late time univ erse area metri c cosmology predicts a small acceleration. This theory does not require the introduction of a cosmological constant or any other not explainable factor. 2. The ory Area metric cosmo logy is a highl y theoretica l topic and require a good mathematical backg round. I present this theory in a way that foregoes mathematical proofs and explains you the most important background you have to know. 2.1. Are a metri c geome try The basic idea of area metric geometry is to promote area metrics to an own structure by measuring area instead of length. One define a metric manifold (M, g) and an area spacetime (M, G) with a four dimensional manifold Mand a four dimensional covariant tensor field G, see figure 1. 1
Transcript
8/3/2019 Bjorn Wehinger- Area metric cosmology: How to explain the universe without dark energy or fine-tuning
How to explain the universe without dark energy or fine-tuning
Bjorn Wehinger, Uppsala Universitet
January 22, 2008
Abstract
Using general relativity and Einstein cosmology we have to introduce some kind of dark matter to explain
the observed acceleration of our universe. A physical explanation of this dark energy does not exist untilnow. A cosmology based on area metric avoids this problem. If one take a general area measure instead
of a length measure as a basis the cosmological constant becomes unnecessary. The universe can be
explained without dark matter or fine-tuning!
1. Introduction
The idea of an area metric cosmology was first published by Raffaele Punzi, Frederic P. Schuller and
Mattias N. R. Wohlfarth on February 12, 2007 [1]. They trade space time as an area metric manifold
and show that this is a true generalization of metric geometry in four (and more) dimensions. Based
on generalized geometry in quantum string and gauge theory the universe is filled with bosonic and
fermionic string radiation for the radiation dominated epoch [3] and filled with non-interacting string
dust at large scales for the late universe [1] and [2]. For the radiation dominated epoch the area metric
cosmology is equivalent to the standard Einstein cosmology. For the late time universe area metric
cosmology predicts a small acceleration. This theory does not require the introduction of a cosmological
constant or any other not explainable factor.
2. Theory
Area metric cosmology is a highly theoretical topic and require a good mathematical background. I
present this theory in a way that foregoes mathematical proofs and explains you the most important
background you have to know.
2.1. Area metric geometry
The basic idea of area metric geometry is to promote area metrics to an own structure by measuring
area instead of length. One define a metric manifold (M, g) and an area spacetime (M, G) with a four
dimensional manifold M and a four dimensional covariant tensor field G, see figure 1.
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8/3/2019 Bjorn Wehinger- Area metric cosmology: How to explain the universe without dark energy or fine-tuning
Figure 1: Area measurement: Metric C g measures the squared area of the parallelogram spanned by
vectors (X, Y )
Remember that a manifold is an abstract mathematical space in which every point has a neighborhood
which resembles Euclidean space. A metric is a mathematical function referring two elements a non-
negative value. This value is usually interpreted as the distance of the two elements. The area metric C gis than defined as:
C g(X,Y,A,B) = g(X, A)g(Y, B)− g(X, B)g(Y, A) (1)
The assumption that spacetime can be interpreted as an area metric manifold is the central hypothesis of area metric cosmology and leads to area metric gravity.
2.2. Area metric gravity
The area metric geometry formalism allows us to derive a gravity theory different from Einsteins theory.
The Einstein-Hilbert action is interpreted as an action for an area manifold. The Einstein-Hilbert action
is a mathematical object (an action) that is used to derive Einstein’s field equations of general relativity.
In analogy to general relativity theory Raffaele Punzi et al. [1] derive the full equations of motion for
area metric gravity:
K C abcd = κT C abcd (2)
K Φ = κT Φ (3)
With gravitational tensor K and the energy-momentum tensor T with cyclic contributions K C , T C and
scalar contributions K Φ and T Φ.
The question of most interest is of course how Einstein gravity fits into the more general area metric
theory. It turns out that area metric with cosmological symmetries are ”almost metric”: G−1 = (GΦG)−1.
Inserting this ansatz into the equations of motion one gets precisely Einstein’s equation coupled to a
massless scalar field. Any vacuum solution (M, g) of Einsteins gravity is a vacuum solution of area
metric gravity. Including matter, however, it is no longer true that all solutions of Einstein gravity are
solutions of area metric gravity. But it is compatible for a so called string fluid discussed in the following
section.
2.3. String fluids
Area metric space time is based on area measure instead of length measure. Formulating space time
dynamics for massive point particles in general relativity one gets so called worldlines for minimal
dynamics. However for area measure based dynamics one obtains string worldsheets for a perfect fluid
matter instead, as illustrated in figure 2.
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With the generalized density ρ the generalized momentum ˜ p and the generalized coordinate q. Thisrepresents the sum of three components of non-interacting strings. This equation can be solved for
different cosmological constraints and leads us directly to area metric cosmology and the explanation of
different states of the universe.
3. Area metric cosmology
The equation of motion for string fluids can be solved for different cosmological constraints. The so-
lutions can be compared with Einstein cosmology and the Benchmark model. Since equation 6 holds
only for non-interacting strings, it can’t be used for the matter dominated epoch of the universe. But
when matter interaction can be neglected, as in the late epoch and in the radiation dominated epoch, this
equation describes our universe very well and without an additional cosmological constant.
Combining the equations of motion for area metric gravity (equation 2 and 3) and the equation of
motion for non-interactive string fluids (equation 6) one can derive a solution for the universe. In analogy
to the Friedmann equation [4] one obtains a parameter of state w for the string fluid. The equation of
state is defined as
w =p
q=
x + y
3(x− y), x = −H
˙ΦΦ−1 + 4κ(ρ + q)Φ2, y = 4κq. (7)
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3of an effective radiation fluid. So the vacuum cosmology
in area metric geometry is equivalent to Einstein cosmology filled with a radiation fluid. One obtains
w = −1
3for the limit x
y→ 0 describing a universe with zero acceleration. Since we may obtain any value
of w the string fluid should be able to describe any physical universe. Generally area metric cosmology
combined with the string fluid can be interpreted as Einstein cosmology with a perfect fluid. This is
illustrated in figure 3. We will now discuss the case of the late universe characterized by a negligiblematter interaction.
Figure 3: Comparison of area metric geometry and Einstein gravity
3.1. String dust cosmology - the accelerating universe
In the late epoch the matter on average has spread out so much that the gravitational interactions are
no longer important. The universe can be described by string dust which is non interacting matter. The
equations of motion for area metric filled with non interactive matter can be solved exactly. Energy
conservation on the generalized continuity equation yields
˜ p = 0, q = −ρ. (8)
The area metric cosmology equations become:
Φ = λa, ρ = ζa−2, a +a2
a+
k − 4κζ
a(9)
with curvature k, gravitational constant κ and two positive integration constants ζ and λ. Substituting
k − 4κζ by ξ the equations can be solved by the scale factor a
a(t) =
c(t− t0) ξ = 0 cξ−1 − ξ(t− t0)2 ξ = 0
(10)
with integration constants c and t0. We will now discuss three possibilities: ξ = 0, ξ > 0 and ξ < 0assuming ζ > 0 (ζ < 0 is not consistent for a flat universe).
ξ = 0: This solution requires a positively curved universe k = +1 since k = 4κζ in this case, c > 0and t− t0 is also required. For the parameter of state turns out: w > −1
3, the late time limit is −1
3. This
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8/3/2019 Bjorn Wehinger- Area metric cosmology: How to explain the universe without dark energy or fine-tuning
solution describes an open universe with eternal deceleration and initial singularity. The acceleration
tends to zero for late times. It would require some fine tuning in order to agree with observations, such
as a positive curvature.
ξ > 0: The solution holds for c > 0 and is a half ellipsoid. But it is inconsistent since it turns out that
the derivative of a diverges. This closed universe is therefore no valid solution.
ξ < 0: This leads to k < 4κζ so negative curved k = −1 and flat k = 0 universe are allowed. For4ζ > 1
κpositive curvature can also be realized. The time dependence of the scale factor can as well be
calculated exactly but the solution depends as well from the sign of the integration constant c: c > 0describes an open universe with eternal deceleration and initial singularity. If c < 0 the universe is open
and eternally accelerating. It has no singularity and its acceleration is tending to zero. We obtain the
same parameter of state w → −1
3as the late time limit for both solutions. All the solutions are shown in
figure 4.
Figure 4: Solutions for the string dust filled universe : From top to bottom: ξ < 0 with c < 0, ξ < 0with c > 0, ξ = 0 and ξ > 0 for the dashed line. From [2].
3.2. Radiation dominated area metric cosmology
Area metric filled with bosonic and fermionic string radiation will now be compared to the radiation
dominated Einstein universe. Including the spins of bosonic and fermionic massless strings the fluid
equation 6 leads to the following characteristics describing a radiation fluid [3]:
q = 0, ρ + ˜ p− 2ρΦ2 = 0 (11)
This equation of state implies a really interesting result. Using the definition w = p
ρit turns out that
p = 1
3ρ equal to w = 1
3. This shows immediately the equality of area metric cosmology filled with
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string radiation to Einstein cosmology filled with a perfect radiation fluid! Therefore the area metric
cosmology of the early, radiation-dominated universe is completely unchanged with respect to Einstein
cosmology.
4. Comparasion with observed data
In principle it is quite easy to determine the expansion history of the universe. One just have to find a
standard light source with known absolute magnitude which is observable over a wide distance range.
Type Ia supernovae are good candidates for such measurement, they have a defined brightness. The
distance and consequencly the time of emittance can be determined quite precisely by measuring the
apparent magnitude. Redshift observations give us information about the velocity the object is moving
away from us. The recorded redshift and brightness of each Ia supernova allows a measurement of the
total integrated expansion of the universe since the time the light was emitted. The observed expansion
shows a late time accelerating, see figure 5.
Figure 5: Observed magnitude for Ia supernovae plotted versus the redshift from [5].
Figure 6 finally shows the expansion (scale factor) of the universe as a function of time. The standard
Einstein cosmology fits only by introducing a cosmological constant. Since the universe seems to haveno large scale curvature as indicated by the microwave background measurement the amount of dark
energy is ΩΛ ≈ 0.70 in the standard model [6]. The observed data points don’t exclude the new solu-
tion for the dust filled universe. Rather fits an open and eternally accelerating universe without initial
singularity as well.
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Figure 6: Scale factor as a function of time. The data points are the results of measured high redshifted
Ia supernovae from [5]. The best fit in the standard model contains dark energy. All fits assume Einstein
cosmology but the new solution for the string dust filled universe fits as well, compare figure 4.
5. Conclusions
The single hypothesis ”space time is an area metric manifold” turns Einsteins gravity into a consistent
alternative theory. Based on a four dimensional area metric the universe can be described without a
cosmological constant or fine tuning. For the late time fate one observe a new result from area metriccosmology which cannot be obtained by Einstein cosmology: The late time acceleration tending to zero.
The main pillar is that every vacuum solution of Einstein’s equation also solves the vacuum equation of
area metric gravity. The consequence of area metric cosmology is that motion of minimal mechanical
objects are no longer based on worldlines drawn by point particles but on worldsheets drawn by strings.
5.1. Open questions
It is not clear yet if the proved vacuum solution turns into a consistent solution with matter. Including
matter it is no longer true that all solutions of Einstein gravity leads to solutions of area metric gravity. To
make area metric cosmology experimentally provable one has to derive predictions for the solar system.
This is complicated by the fact that spherical symmetric area metric is not of almost metric form. A overall theory should include the matter - matter interaction.
References
[1] Raffaele Punzi, Frederic P. Schuller and Mattias N. R. Wohlfarth, “Area metric gravity and accelerating cos-
mology,” J. High Energy Phys., 02 (2007) 030.
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