Improved Ricci Cosmology

(1)

University of Amsterdam and the Free

University

Institute for Theoretical Physics

Bachelor Project Physics and Astronomy 15 EC

Conducted between 1-4-20 and 13-7-20

Improved Ricci Cosmology

Author:

Name Hendrik Huisman (11286113)

Supervisor: dr. J´_{acome Armas} Second examiner: dr. Jan Pieter van der Schaar

(2)

1 Abstract

Cosmology is the physics behind the evolution of the universe on the largest scales. It de-scribes how the universe will expand or contract over time. Shortly after the Big Bang there has been a period of inflation. Inflation is the process where the universe grew ex-ponentially for a very short period of time. The current cosmological model, the Lambda CDM model, explains this inflationary period by making use of exotic components such as dark matter and dark energy. The nature of these components is still mysterious. In or-der to find the cosmology of the universe the Einstein equation has to be solved. The most simple case of incorporating the matter contributions to the stress energy tensor in the Ein-stein equation is by making use of the stress energy tensor of a perfect fluid. Recently it has been shown that Ricci cosmology can explain an inflationary phase at early times with-out the need for any exotic components. Ricci cosmology makes use of corrections due to relativistic fluid dynamics to the stress energy tensor of a perfect fluid in the Einstein equa-tion. In this thesis a general overview of standard cosmology will be given and it will be shown how the inflationary phase of Ricci cosmology can be derived. The goal of this thesis is to see if another correction to the stress energy tensor will yield a new cosmology with an inflationary phase at early times. It turned out that this new correction doesn’t yield a new cosmology compared to Ricci cosmology, except for a special case. For this special case it has been shown that this cosmology doesn’t include an inflationary phase at early times and that overall it doesn’t describe our universe.

(3)

2 Samenvatting

De leidende kracht op de grote schaal, de schaal van het zonnestelsel is de zwaartekracht. Deze kracht zorgt ervoor dat wij op aarde blijven staan en deze zorgt er ook voor dat de aarde om de zon blijft draaien, maar hoe werkt dat op de allergroote schaal, de schaalg-roote van het gehele universum? Blijven de sterren in het universum bij elkaar of wordt de afstand tussen hen steeds groter? Je zou verwachten dat als er alleen zwaartekracht zou bestaan alles weer in elkaar zou storten. Er is immers niks dat de zwaartekracht zou kun-nen afremmen. Het tegengestelde is waar. De beroemde astronoom Edwin Hubble kwam er door metingen achter dat bijna alles in het universum van ons afbeweegt. Hij kwam met zijn beroemde wet van Hubble, die beschreef dat de snelheid van sterren steeds groter wordt naarmate ze verder van ons afstaan. Met de zwaartekrachtswetten van Einstein kon dat niet verklaard worden, dus er moest iets nieuws worden bedacht. Natuurkundigen denken dat donkere energie en ook donkere materie een rol spelen in deze uitdijing van het uni-versum. Dit zijn nog mysterieuze componenten van het universum die we niet kunnen zien maar alleen door indirecte metingen kunnen waarnemen. Een belangrijk bewijs voor het bestaan van deze donkere energie/materie is dat het de theorie van inflatie bewijst. Het idee van inflatie is dat aan het begin van het universum, dus net na de Big Bang, het uni-versum in een enorm kleine tijd uitbreidde van de grootte van een atoom tot de grootte van een grapefruit. We weten dat inflatie gebeurd zou moeten zijn, aangezien we dat uit waarnemingen kunnen zien. Zonder donkere energie en donkere materie zou dit onverklaar-baar zijn, dus dit is een voorbeeld van een aanwijzing van het bestaan van deze donkere substanties.

In deze scriptie laten we zien dat we door de toevoeging van een correctie op de zwaartekrachtswet van Einstein inflatie kunnen verklaren zonder deze mysterieuze donkere substanties. We

voegen daarna nog een extra correctie toe en we bekijken of dit ook inflatie teweeg brengt. Uit de resultaten bleek dat deze extra correctie niets nieuws oplevert. Wel liet het in een speciaal geval iets nieuws zien, maar dit bleek niet fysisch.

(4)

3 Introduction

Physical Cosmology, or simply put Cosmology, is a branch of physics which studies the evo-lution of the universe on the largest time and length scales. It is concerned with the be-ginning and the ultimate fate of the universe, and everything in between. Modern cosmol-ogy started in 1915 with the publication of General Relativity by Albert Einstein, which showed that mass and energy can be described as curvature in 4 dimensional space time. Later, in 1922, Alexander Friedmann derived the Friedmann equation from General Rela-tivity. These equations describe the dynamics/expansion of an Universe which is isotropic and homogeneous. Solving these equations tells one how the Universe will expand or con-tract over time. These equations depend on certain parameters: the total energy density of matter m,the radiation energy density r and an energy density of dark energy arising from

a cosmological constant Λ. The current accepted model in Cosmology is called the Lambda

Cold Dark Matter model, or the Lambda CDM model. This model describes the universe where we live in today. This model includes exotic components such as dark matter and dark energy, whose nature still remains mysterious. The most basic way of describing these components in General Relativity is by treating them as a perfect fluid. This causes the stress energy tensor, a mathematical object in the Einstein equation, to take a simple form. This can then be used to derive the Friedmann equation, which can then be used to find how the universe will expand or contract over time.

Recent advances in relativistic fluid dynamics show that corrections to the stress energy tensor of a perfect fluid could cause a more natural explanation to inflation without the need for any exotic components such as dark matter and dark energy[1]. It is important to advance our knowledge in the field of cosmology so we can learn how the universe will evolve and what its eventual fate will be.

In this thesis an introduction will be given on what a metric is. Then an introduction will be given of the Friedmann-Robertson-Walker-Lemaitre metric for an expanding isotropic and homogeneous Universe. The Friedmann equations will be stated and it will be shown how these can be combined with the fluid equation and the equation of state to yield so-lutions for the scale factor. This will be done for simple single component universes. The Lambda CDM model, other models and the concept of inflation will be discussed. After-wards it will be shown how the Friedmann equation arises from the Einstein field equation. Some results from [1], where corrections from relativistic fluid dynamics to the stress energy tensor have been added, will be summarized. It will be shown how these corrections give rise to an inflationary phase at early times. Then other corrections to the effective pressure due to viscous terms as described in [4] will be added to the stress energy tensor. Various

(6)

solutions of the scale factor with this new correction will be shown. It will then be exam-ined if these new solutions can reproduce an inflationary phase in the early universe and if they are physical at all.

(7)

4 Theory

In order to get a grasp of cosmology a few features will be introduced in this section. First an explanation of a metric will be given by looking at line elements in flat 4 dimensional spacetime. Then the Cosmological principle and the FRW metric will be introduced. After-wards the Friedmann equation, the fluid equation and the equation of state will be stated. It will be shown how the Friedmann equation and the acceleration equation arise from the Einstein equation. These equations will be used to look at cosmologies of single component universes. A brief summary of the concept of inflation and various cosmological models and the Lambda CDM model will be given. Afterwards corrections to the effective pressure as described in [1] and [4] will be considered.

4.1 Metric

In a 2 dimensional Cartesian coordinate system the distance, S, between two points b and a, is given by

Sba= ((xb− xa)2+ (yb− ya)2)

1

2 (1)

Since the coordinates of the reference point with the coordinates xa and ya are arbitrary,

these could simply be put those to zero and look at an infinitesimal distance dx and dy to find

dS2 = dx2+ dy2 (2)

This quantity is also called the line element. The universe has 4 dimensions, 3 for space and 1 for time. This fabric of the universe is also called spacetime. The distance between two points in flat Minkowski spacetime is given by

dS2= −c2dt2+ dx2+ dy2+ dz2 (3)

where c is the speed of light. This can be written down more compactly by using the Ein-stein notation, which states that repeated upper and lower indices should be summed over. Now dS can be rewritten to

(8)

where µ and ν are 0,1,2,3 and gµν is called the metric. dx0, dx1, dx2, dx3 correspond to

cdt, dx, dy, dz respectively. Here gµν takes the form of a 4x4 matrix given by

gµν =        −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1        . (5)

Equation 4 written out, ignoring all the zero terms, therefore looks like dS2 = g00dx0dx0+ g11dx1dx1+ g22dx2dx2+ g33dx3dx3

= −(dx0)2+ (dx1)2+ (dx2)2+ (dx3)2 = −c2dt2+ dx2+ dy2+ dz2

(6)

As one can see gµν, the metric, essentially contains all the information of how distances or

line elements are described in space. In this metric there is no specific spatial dependence on gµν, but in general one can have a time and spatial dependence. This can be seen by

looking at dS in spherical coordinates. This will yield a line element of the form

dS2 = −c2dt2+ dr2+ r2dθ2+ r2sin θ2dφ2. (7) The metric, gµν, is now given by

gµν =        −1 0 0 0 0 1 0 0 0 0 r2 0 0 0 0 r2sin θ2        (8)

Note that the metric in equation 8 is different from equation 5, but the space it describes is the same. Also note that the g22 component depends on r, so a metric can also be

coordi-nate dependent.

4.2 The Cosmological principle and FRW metric

The Cosmological principle states that the Universe is homogeneous and isotropic. Homo-geneous means that the universe, if looked at it at a sufficiently large scale, looks the same wherever you are. Isotropic means that the universe, again at large scales, looks the same in every direction. This is shown in the next figure.

The pattern on the left is homogeneous but anisotropic, because it’s not the same in every direction, whereas the pattern on the right is isotropic but inhomogenous, because it’s not the same everywhere.

(9)

Figure 1: Example of homogeneous and isotropic patterns [7]

In the previous section metrics were discussed which spatial parts weren’t time dependent, but one can take a look at metrics whose spatial parts are time dependent. The metric which includes the Cosmological principle, i.e. the universe is homogeneous and isotropic at all times, and a metric which includes a time dependent function which can expand or contract the spatial part of the given metric is given by the FRW metric, or the Friedmann-Robertson-Walker-Lemaitre metric. The FRW metric is given by [2]

dS2 = −c2dt2+ a(t)2(dr2+ Sκ(r)2dΩ2), (9) where dΩ2 ≡ dθ2+ sin θdφ2 (10) and Sκ(r) =          R sin(r/R) κ = 1 r κ = 0 R sinh(r/R) κ = −1. (11)

Here κ = 1 corresponds to a positively curved space, κ = 0 to a flat space and κ = -1 to a negatively curved space. The function a(t)2 is a function which depends on time and it is called the scale function or scale factor. The FRW metric, with κ = 0, written out in

(10)

matrix form reads gµν =        −1 0 0 0 0 a(t)2 0 0 0 0 r2a(t)2 0 0 0 0 r2sin θ2a(t)2        . (12)

4.3 The Friedmann equation

The equation that describes the expansion or contraction of the universe is called the Fried-mann equation. This equation links all features of general relativity, together with the FRW metric due the Cosmological Principle into one equation, which can be solved to find the scale factor, which describes how the universe evolves over time. The full derivation re-quires knowledge of General Relativity. It will be shown in the next section how the Fried-mann equation can be derived from the Einsten equation. The FriedFried-mann equation is given by [2] H2 = ˙a a 2 = 8πG 3c2 (t) − κc2 a(t)2, (13)

where H2 is called the Hubble parameter, (t) is the energy density of a perfect fluid and G is the gravitational constant. In order to solve the Friedmann equation an extra equation will be needed to specify how (t) depends on time. The equation needed is called the fluid equation, which is given by [2]

˙ + 3˙a

a( + P ) = 0. (14)

Here a perfect fluid is considered in an expanding or contracting universe, where is the energy density and P is the pressure. A derivation of equation 14 can be found in [2]. Equations 13 and 14 can be combined to find the acceleration equation, which is given by

¨ a

a =

4πG

3c2 ( + 3P ) (15)

Still a different equation is needed to eliminate the pressure out of equation 15. The equa-tion needed is called the equaequa-tion of state, which relates the pressure and energy density to each other. The equation of state is given by

P = w. (16)

Every component of the universe that contributes to the scale factor has a different value of w [2]. For example, radiation has wr = 1₃, matter has a value of wm = 0, and dark energy

has value of wde = −1. When considering a universe with multiple components those could

(11)

P =Xwii= wrr,0+ wmm,0+ wdede,0. (17)

Here the 0 specifies that these energy densities correspond to their present values. Equation 17 and equation 15 can be combined to find [2]

i(a) = i,0a−3(1+wi). (18)

This shows how the energy density of the components, which influence the scale factor, in-crease or dein-crease as the universe expands. This result can be used to take a look at simple single component flat universes to see how the scale factor evolves over time.

4.4 Simple single component flat universes

In this section simple single component flat universes as described in [2] will be summa-rized. The scale factor of universes with matter only, radiation only and dark energy will be given. Because spatially flat are considered universes equation 13 reduces to

˙a2 = 8πG 3c2 i,0a

(−1−3wi)_, ₍₁₉₎

where equation 18 was plugged in. A general solution to equation 19 can be found. This general solution is given by

a(t) = ( t t0

)3+3w2 , (20)

with the restriction that w 6= −1. Here t0 corresponds to the age of the universe.

4.4.1 Matter only

For matter w = 0, so

a(t) = ( t t0

)23. (21)

This universe would be expanding forever. Here t0 is the age of this specific universe.

4.4.2 Radiation only

for radiation w = 1/3 so

a(t) = ( t t0

)12. (22)

This would also yield an expanding universe. Note that the t0 here is different compared to

(12)

4.4.3 Dark Energy only

For dark energy w = −1. This was the restriction of equation 20 so different solution than the general solution of equation 20 should be found. The solution is given by

a(t) = exp(H0(t−t0)) ₍₂₃₎ with H0 = ( 8πGde 3c2 ) 2 ₍₂₄₎

(13)

4.5 Lambda CDM model and inflation

The model that best describes our universe today is called the Lambda CDM model. This model incorporates a mixture of radiation, dark energy, and dark and normal matter. In the next figure the scale factor for different models is shown. Here Ωm,λ is the so called

Figure 2: The scale factor for universes with different components [5]

density parameter which is given by

Ωm,λ=

8πGρm,λ

3H2 . (25)

If this value is bigger than 1, the universe is positively curved, if it’s 1 it’s flat and if it’s smaller than 1 the universe is negatively curved. We can see in figure 2 that some cos-mologies allow the universe to return back to itself, whereas the Lambda CDM model pre-dicts accelerated growth. The Lambda CDM model does a great job at explaining the ex-istence of the Cosmic Microwave Background radiation (CMB) and the accelerated rate of the expansion of the universe and inflation, but it still needs exotic components, such as dark matter and dark energy, to explain these phenomena. Inflation is the phenomena that shortly after the big bang the universe expanded exponentially for a small period of time. This expansion was of the order of magnitude 1026 in 10−36s until 10−32 seconds after the big bang [6]. It will be shown that Ricci Cosmology as described in [1] will show a more natural explanation for inflation without the need for exotic elements such as dark matter and energy. First it will be shown how the Friedmann and acceleration equation can be de-rived from the Einstein equation.

(14)

4.6 The Friedmann and acceleration equation from the Einstein equation

The Einstein equation, where c=1 is used, without a cosmological constant is given by Rµν−

1

2gµνR = 8πGTµν, (26)

where Rµν is the Ricci tensor, R is the Ricci scalar, gµν the metric tensor and Tµν is the

stress-energy tensor. The stress-energy tensor for a perfect fluid is given by

Tµν = ρuµuν+ p(uµuν + gµν). (27)

Here ρ is the density, uµ and uν the four velocity of the fluid and p the pressure. When the metric tensor is set equal to equation 12, i.e. the FRW metric, and looked at in the comov-ing frame so that uµ= uν = (1,0,0,0) , the terms µ = ν = 0 yield

R00= −

1

2R + 8πGρ. (28)

In the comoving frame R and R00 are given by [3]

R00= −3 ¨ a a = −3( ˙H + H 2₎ R = 6(¨a a+ ˙ a2 a2) = 6( ˙H + 2H 2₎ (29)

Combining these equations yields back the Friedmann equation H2 = ˙a

a 2

= 8πG

3 ρ. (30)

The acceleration equation can be derived by looking at equation 26 and setting the terms µ = ν = 1, 2, 3. This essentially yields 3 identical equations given by

˙ Ha2+ 3 ˙a2− 3a2(Ha˙ 2_{+ ˙a}2 a2 + ˙a2 a2) = 8πGpa 2 −2 ˙H = 8πGp + 3H2 ˙ H = −4πG(ρ + p) (31)

(15)

4.7 Corrections to the effective pressure due to relativistic fluid dynam-ics

In equation 27 the stress energy tensor of a perfect fluid was given. In [1] corrections up to second order in gradient expansion from [4] are used to rewrite the pressure to

p = pef f = p + ξ5R + ξ6uαuβRαβ. (32)

Here ξ5 and ξ6 are called transport coefficients which are not currently known but could

be calculated from a macroscopic theory, R is the Ricci scalar, Rαβ is the Ricci tensor and

uαuβ is the fluid four velocity. In [1] the new effective pressure has been added to the stress energy tensor. They found that for the case that c1 = 0 : ξ5 = 1₂ξ6, the Friedmann equation

and acceleration equation give rise to the implicit equation for the scale factor given by c2₀t

|c2|

+ 1 = c0ln(a) + ac0 (33)

where c0 = 3(1+w)₂ and c2= 3₂ξ6 For the case a(t) <<< 1 equation 33 becomes

a(t) = e

c2_0t

|c2| ₍₃₄₎

which shows exponential growth at the beginning of the universe, i.e. inflation, without the need for any exotic elements. At later times a(t) > 1

a(t) ≈ ( c 2 0 |c2| + 1) 1 c0 ₍₃₅₎

which is approximately the cosmology of single component universes as described in equa-tion 21 and 22.

In [4] it is shown that Tµν has an extra term, p∗, given by

p∗ = κR<µν>− 2(κ + κ∗)uλuρRλ<µν>ρ (36)

where κ and κ∗ are coefficients,uλuρthe fluid four velocity, R<µν> and Rλ<µν>ρ are the

Ricci tensors with the projection operator working on them. The projector operator for a 2nd rank tensor, A<µν> is given by

A<µν> = 1 2∆ µλ_∆νρ_(A λρ+ Aρλ) − 1 d − 1∆ µν_∆λρ_A λρ. (37)

Here d is the amount of dimensions, and ∆µλ _{is given by}

(16)

In order to find the contribution of equation 36 to the effective pressure as given in equa-tion 32, equaequa-tions 36, 37 and 38 were combined. Here the FRW metric as in equaequa-tion 12 was used and we were working in the comoving frame, such that uµ= uλ = (1,0,0,0). Equation 37 fully written out is given by

R<µν> = 1 2(g µλ_{+ u}µ_uλ_)(gνρ_{+ u}ν_uρ_)(R λρ+ Rρλ) − 1 3(g µν_{+ u}µ_uν_)(gλρ_{+ u}λ_uρ_)R λρ. (39)

Now 16 different terms have to be calculated because µ and ν range between 0 and 3. Luck-ily some terms turn out to be zero. The final result is given by

R<µν>= 1 a(t)4        0 0 0 0 0 2₃R11 R12 R13 0 R21 2₃R22 R23 0 R31 R32 2₃R33        . (40)

Since we are working in the comoving frame, all the off diagonal terms of the Ricci tensor are equal to zero[3], so Rµν can be written as

R<µν> = 2 3a(t)4        0 0 0 0 0 R11 0 0 0 0 R22 0 0 0 0 R33        . (41)

Now it’s still needed to compute Rλ<µν>ρ. This is done similarly as with R<µν>. Rλ<µν>ρ is given by Rλ<µν>ρ = 2 3a(t)4        0 0 0 0 0 Rρλ₁₁ 0 0 0 0 Rρλ₂₂ 0 0 0 0 Rρλ₃₃        (42)

Now equations 42 and 41 can be plugged in equation 36. Note that this is still the comov-ing frame, so that uλ = uρ= (1,0,0,0) and therefore

Rλ<µν>ρuλuρ= R0<µν>0u0u0= R<µν>.

(43)

Equation 36 is now given by

(17)

Now equation 27 can be rewritten to include the terms pef f and p∗ and plug this into 26. This yields Rµν− 1 2gµνR = 8πGTµν Rµν− 1 2gµνR = 8πG(ρu µ_uν _{+ p} ef f(uµuν + gµν) − (κ + 2κ∗)R<µν>). (45)

If µ and ν are set to zero one finds

R00+ 1 2R = 8πGρ −3( ˙H + H2) + 3( ˙H + 2H2) = 8πGρ H2 = 8πG 3 ρ (46)

Which is the same as the Friedmann equation as in 30. For the terms µ = ν = 1, 2, 3 3 equations which are identical are found. Let’s consider µ = ν = 1 given by

R11− 1 2g11R = 8πG(ρu 1_u1_{+ p} ef f(u1u1+ g11) − (κ + 2κ∗)R<11>) R11(1 + 2 3(2κ ∗_{+ κ)B) = Ba}2_p ef f + 1 2Ra 2 (a2H + 3 ˙a˙ 2)A = Bpef fa2+ 3a2( ˙H + 2H2) ˙ H = B(ρ(2 − A) + pef f) A − 3 (47)

Where B and A are given by B = 8πG and A = (1 +2₃B(2κ∗+ κ)). Equation 47 can be combined with 29 to find

˙ H((A − 3) − 2Hc1) − 2H2( 3 2 − 3 2A + c0) − 2H 3_c 2 = 0 (48)

where c0 = 3(1+w)₂ , c1 = 3ξ5−3₂ξ6, c2 = 6ξ5−3₂ξ6 and G was set to _8π1 . Equation 48 can be

rewritten to ˙ H(1 −Hc6 c3 ) − H2c4 c3 − H3c5 = 0, (49) where c0= − c4 c3 c1 = − c6 c3 c2= − c5 c3 c3 = A − 3. (50)

The unmodified version of equation 49 given in [1] is given by ˙

H(1 + c1H) + c0H2+ c2H3= 0 (51)

The coefficients in equation 49 depend on κ, κ∗, ξ5, w and ξ6 and since the value of these

(18)

not going to yield any new solutions in comparison to equation 51 for c3 6= 0. Therefore it

will suffice to look only at equation 49 for the case that c3 = 0 so that A = 3. The relation

between κ and κ∗ is then given by

κ = 3 − 2κ∗. (52)

If this relation holds in the real world, equation 49 can then be written to the form ˙

H

Hc6+ c4+ Hc5 = 0, (53)

where

c6= 2c1 c4 = c0− 3 c5 = 2c2. (54)

This can be solved analytically for a(t), where a(0) = H(0) = 1, yielding

a(t) = e−c4t+c6 ln (e

3+c4t

c6 +ec4+c6 c3t ₅₎

c5 _. ₍₅₅₎

In the next section the behavior of the scale factor will be considered when ξ5 and ξ6 are

varied and it will be examined if it is possible to reproduce inflation at early times.

4.8 Scale factor for the case A = 3

The behavior of the scale function for the cases where w ≥ 0, normal matter or radiation only, and ξ5 = ξ6 and the case where ξ5 6= ξ6 will be considered. For the case w ≥ 0 and

ξ5= ξ6 the solution of the scale factor is given by

a(t) = e −t(1+w) 6ξ6 _(e 3+t(1+w) 2ξ6 _{+ 9e} t ξ6+ 3(1+w) 2 _ξ 6) 1 3. (56)

When ξ6 > 0 and w = 1₃ the result is an exponential function.

(19)

This graph will get steeper when the value of w is decreased. The case ξ6 < 0 is shown in the next figure.

Figure 4: Scale factor for the case ξ5= ξ6= −2.5, w = 1₃

The fact that this scale factor only begins at later times is caused by the fact that at those time a root is taken of a negative number. It can be seen that this scale function asymp-totically rises to e. When w is decreased the behavior remains the same. Now for the case w ≥ 0 and ξ5 6= ξ6 it should be noted that the behavior of the scale factor is sensitive to

small changes in the choice of the coefficients ξ5, ξ6. The behavior will be either

exponen-tial growth forever or a logistic curve. A plot of the scale factor in a logistic curve form is shown in in figure 5.

Figure 5: Scale factor for the case ξ5= 1, ξ6= 5, w = 0

(20)

Figure 6: Scale factor for the case ξ5= 3, ξ6= 5, w = 0

5 Discussion

As seen from the previous section, the addition of κ and κ∗ term to the stress energy ten-sor won’t yield new solutions if A 6= 3. When A = 3, it can be seen from the plots that the scale factor will only yield an ever growing exponential function or a logistic curve. It’s therefore not possible to reproduce inflation at early times with the addition of κ and κ∗ at the special case A = 3.

Figure 3 shows that the scale factor will exponentially increase for the case w = 1/3 i.e. ra-diation only. It is shown in equation 22 that in an universe with only rara-diation the scale factor follows a t2/3 function. Since this solution is exponential and it doesn’t drop off in some limit, this solution can be deemed non compatible with standard simple cosmology. Figure 4 shows that the scale factor in that case will rise quickly towards e. This solution is not physical because it starts at t>0 and it’s not compatible with our universe since it doesn’t allow that universe to grow to a reasonable size. Figure 5 shows a scale function which follows a logistic curve. This solution also rises quite quickly towards it’s maximum. Figure 6 shows exponential growth for the case w = 0 i.e. matter only. The same argument can be applied as for the radiation case. Since all these solutions are weird, it can be as-sumed that the case A=3 is not physical and that equation 52 doesn’t hold.

Equation 34 describes an inflationary scale factor at early times. Dimensional analysis shows that there should be a factor of

√ ξ5G

c in front of the c2

0

|c2|t. Since ξ5 is not known, we can

rewrite this factor to its numerical value as 2.723110−14√ξ5. When this is compared to the

currently accepted value of inflation in the inflationary epoch of the universe[6], which is an expansion of the order of magnitude 1026 in 10−36s until 10−32seconds after the big bang, a calculation shows that in order to explain inflation ξ5 should be at least of the order of

(21)

magnitude 5 ∗ 1099. In [1] it is shown that ξ5 has dimensions of energy density, so at early

times this could have been the energy density of the universe. Further analysis should be done to check if this is a possible value.

(22)

6 Conclusion

The main aim of this thesis was the addition of new terms to the effective pressure as de-scribed in [4] to see if this addition gives rise to new solutions of the scale factor and if this addition can reproduce an inflationary phase, without the need of exotic elements such as dark matter and dark energy, as described in [1]. First a summary of standard cosmology was given. An explanation of what a metric is was given and how the cosmological princi-ple leads to the FRW metric. This metric was combined with the Einstein equation to find the Friedmann equation and the acceleration equation. The Friedmann equation was solved for simple single component universes as described in [2]. Then a summary was given of various models and the current accepted model in cosmology, the Lambda CDM model. A summary was given of a result as described in [1], where corrections to the stress energy tensor of a perfect fluid lead to a scale factor with an inflationary early period without the need of any exotic elements such as dark matter and dark energy. Finally a new term was added to the stress energy tensor of a perfect fluid as described in [4]. It has been shown that this addition doesn’t lead to any new solutions for the scale factor for the case A 6= 3, but for the case A = 3 new solutions could be found. In order for these solutions to have any physical implications, equation 52 should hold. It turned out that these new solutions didn’t include an inflationary phase at early times. These solutions also showed some non-physical behavior. Therefore it can be stated that the case A=3 doesn’t describe our uni-verse.

(23)

7 References

[1] Ricci cosmology, Rudolf Baier, Sayantani Lahiri, Paul Romatschke, 2019 [2] Introduction to Cosmology, B. Ryden, 2017

[3] http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll8.html

[4] Relativistic Fluid Dynamics In and Out of Equilibrium – Ten Years of Progress in The-ory and Numerical Simulations of Nuclear Collisions, Paul Romatschke, Ulrike Romatschke, 2017

[5] https://en.wikipedia.org/wiki/Age of the universe [6] https://en.wikipedia.org/wiki/Inflation (cosmology)

Improved Ricci Cosmology

University of Amsterdam and the Free

University

Institute for Theoretical Physics

Improved Ricci Cosmology

1

Abstract

2

Samenvatting

Contents

3

Introduction

4

Theory

5

Discussion

6

Conclusion

7

References