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Didaktik der Physik
Frühjahrstagung – Würzburg 2018
A Computer Simulation of Cosmic Inflation
Lennert Sprenger1, Hans-Otto Carmesin1,2,3
1Gymnasium Athenaeum Stade, Harsefelder Straße 40, 21680 Stade
2Studienseminar Stade, Bahnhofstraße 5, 21682 Stade
3Fachbereich Physik, Universität Bremen, 28334 Bremen
Abstract
From the Cosmic Microwave Background CMB, the flatness problem and the horizon problem
arose. An extraordinarily increase of distances in the early universe, the Cosmic Inflation, was
proposed as a possible solution, whereby suggested mechanisms for such an increase have been
criticized (Steinhard, 2011). We apply a theory that explains the Cosmic Inflation by an extended
Friedmann-Lemaitre model combined with an energy term (Carmesin, 2017). We investigate vari-
ous questions by performing computer simulations. We observe a sequence of phase transitions
that cause an extraordinarily fast increase in distances. Our findings are in excellent quantitative
agreement with observations of the CMB. Thereby the theory depends only on first principles and
the fundamental constants G, c and h and we apply no fit in particular. We present the develop-
ment of the project in the framework of a Jugend forscht club.
1. Introduction
From the Cosmic Microwave Background CMB, the
flatness problem and the horizon problem arose. To
describe the expansion of the universe, a Friedmann-
Lemaitre model has been frequently used until now
(see Karttunen 2007). The Friedmann-Lemaitre
differential equation
{1}
does not solve the two problems arising from the
Cosmic Microwave Background. One additional
problem, which is not solved by the Friedmann-
Lemaitre equation is the singularity problem (see
Kiefer 2008), which considers a singularity as non-
physical. In the Friedmann-Lemaitre equation is
the matter density and is the vacuum density. The
vacuum density is constant. The solution of the
Friedmann-Lemaitre model provides the evolution
of the scale factor a(t).
2. Problems
The singularity problem is visible in the solved
Friedmann-Lemaitre equation, where the density
tends to infinity, when a(t) and t tends to zero (see
figure 1).
Figure 1: Symbolic evolution of the Friedmann-Lemaitre
equation, numerically solved. With , and
.
The horizon problem and the flatness problem arise
from the CMB (see figure 2).
Figure 2: Cosmic Microwave Background (figure: NASA
WMAP Science Team, 9 year WMAP image of back-
ground radiation).
3. The horizon problem
The horizon problem arises from the temperature
fluctuations which amount to
. So the
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Sprenger, Carmesin
full solid angle covered by the Cosmic Micro-
wave Background has a homogenous temperature
distribution. The full solid angle is so large that on
the basis of the Friedmann-Lamaitre equation it
would be causally disconnected and it would not be
possible for radiation to distribute the energy homo-
geneously in the time since the big bang.
4. The flatness problem
The second problem arising from the CMB is the
flatness problem. The space-time curvature is meas-
ured and indicates a flat space on a large scale.
Modelled with the Friedmann-Lemaitre equation it
would have been even more flat in the early universe
which is highly unlikely.
5. Possible solution for the problems
To solve the three problems a rapid increase in dis-
tances in the early universe, the Cosmic Inflation,
was proposed by Allan Guth in 1981 (see Guth
1981). In many approaches to the problems an in-
flaton field is used, which requires very sensitive fit
parameters to fit to observations. In this paper, we
propose a new model without any fit parameters, an
extended Friedmann-Lemaitre equation, to describe
the Cosmic Inflation and solve these problems.
6. The model
The standard Friedmann-Lemaitre equation can be
derived from a spherical model with a scale factor a,
a probing mass m and the density of the sphere.
Figure 3: Model from which the Friedmann-Lemaitre
equation can be derived.
From this spherical model, we can derive the Fried-
mann-Lemaitre equation (see Carmesin 2017, Car-
mesin 2018 a,d):
{2}
We extend the model by adding a radius for our
probing mass in order to describe the density of the
probing mass, which was not possible in the old
model based on a point-like mass.
Figure 4: Model from which the extended Friedmann-
Lemaitre equation can be derived.
In order to obtain the corresponding dynamics, we
derive the quantum physical expectation values
and and denote these by and in the following
(for details see Carmesin 2018 a,d). In the resulting
extended Friedmann-Lemaitre equation we intro-
duce the scaled density
which is the matter
density divided by the maximal density, the Planck
density . From this new spherical model (see
figure 4), we get the following extended Friedmann-
Lemaitre equation for three dimensions (see equa-
tion {3}). Hereby we introduce the Planck time .
{3}
Moreover we generalize this model to spatial dimen-
sions and obtain the following extended
Friedmann-Lemaitre equation (see Carmesin 2018
a,d, see {4})
{4}
with the corresponding scaled energy term (see
Carmesin, 2018 a,d, see {5}):
{5}
Thereby the scaled energy term is the quantum
physical expectation value at the ground state
divided by .
7. Dimensional transitions
The scaled energy is calculated for each scaled
density and for any Dimension (see figure
5).
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A Computer Simulation of Cosmic Inflation
Figure 5: Scaled energy as a function of the spatial
dimension at the scaled density .
From the table in figure 5 we get the following
graph.
Figure 6: Scaled energy as a function of the spatial di-
mension D at the scaled density . Here the
ground state energy is at .
In order to apply the D-dimensional version of the
extended Friedmann-Lemaitre equation (see equa-
tion {4}), we calculate a critical density at
which there occurs a dimensional transition of the
ground state from to for each dimension.
For it we minimize the scaled energy term (see equa-
tion {5}). By applying the variational principle, we
obtain the following values for the critical scaled
densities (see figure 7).
Figure 7: Critical scaled densities for dimensional transi-
tions.
8. Solving the extended Friedmann-Lemaitre
equation
The extended Friedmann-Lemaitre equation can be
solved with numerical integration. For it we used the
Runge-Kutta method of fourth order.
Figure 8: Solved extended Friedmann-Lemaitre equation:
Discontinuities arise at dimensional transitions.
The dimensional transitions take place where the
graph is not continuous (see figure 8). The graph in
figure 8 shows symbolically the dimensional transi-
tions with the corresponding increase in the scale
factor. The exact values can be found in Model for
the Dynamics of Space (see Carmesin 2018 a,d).
Our critical density can be utilized in order to
calculate the observed density of of the universe.
Our result is in excellent accordance with observa-
tions. Thereby no fit must be applied (see Carmesin
2017, see Carmesin 2018 a,d).
9. Summary
We solve the extended Friedmann-Lemaitre equa-
tion generalized for spatial dimensions nu-
merically (see section 8). So we obtain the scaling
radius as a function of the time including dimen-
sional transitions at critical densities (see section
7). Based on this solution the singularity problem
and the flatness problem can be solved when the
durations of the dimensional transitions are calculat-
ed with help of Fermi’s golden rule (see Carmesin
2018 a,d). Furthermore these durations show in full
detail how the singularity problem is solved by the
dimensional transistions (see Carmesin 2018 a,d).
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Sprenger, Carmesin
10. Literatur
Bennett, Charles L. and others (2013): Nine-year
Wilkinson microwave anisotropy probe
(WMAP) Observations: final maps and results.
The Astrophysical Journal Supplement Series.
208. 1-54.
Carmesin, Hans-Otto (2017): Vom Big Bang bis
heute mit Gravitation – Model for the Dynam-
ics of Space. Berlin: Verlag Dr. Köster.
Carmesin, Hans-Otto (May 2018a): Entstehung
dunkler Materie durch Gravitation - Model for
the Dynamics of Space and the Emergence of
Dark Matter. Berlin: Verlag Dr. Köster.
Carmesin, Hans-Otto Carmesin and Carmesin, Mat-
thias (2018b): Quantum Gravity Model for
Cosmic Inflation. To be publ.
Carmesin, Hans-Otto (July 2018c): Entstehung
dunkler Energie durch Quantengravitation -
Universal Model for the Dynamics of Space,
Dark Matter and Dark Energy. Berlin: Verlag
Dr. Köster.
Carmesin, Hans-Otto (2018d): A model for the Dy-
namics of Space. PhyDid B.
Friedmann, Alexander (1922): Über die Krümmung
des Raumes. Z. f. Physik, 10, 377-386.
Guth, Alan (1981): Inflationary Universe: A possible
to the horizon and flatness problem. Phys. Rev.
D 23, 347-356.
Karttunen et al. (2007): Fundamental Astronomy.
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Kiefer, Claus and Sandhöfer, Barbara (2008): Quan-
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