Content uploaded by Hans-Otto Carmesin

Author content

All content in this area was uploaded by Hans-Otto Carmesin on Jan 22, 2019

Content may be subject to copyright.

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

61

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).

62

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).

63

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.

5th edition. Springer Berlin Heidelberg New

York

Kiefer, Claus and Sandhöfer, Barbara (2008): Quan-

tum Cosmology. Beyond the Big Bang. Vaas,

Rüdiger (Herausgeber). Berlin: Springer, 1-29.

Lemaitre, Georges (1927): Un Univers homogene de

masse constante et de rayon croissant rendant

compte de la vitesse radiale des nebuleuses ex-

tra-galactiques. Annales de la Societe Scien-

tifique de Bruxelles. A47, 49-59.

64