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Quantum Phenomena in the Realm of Cosmology and Astrophysics

32 Part II. Dark Energy from **the** Vacuum Energy **of** **Quantum** Fields That means, **the** effective action **and** thus **the** vacuum energy is proportional to **the** expression S eff ∝ = ∫ ∞ p c [ p 4 p 2 s**in** θ dφ dθ dp (2π) 3 4 + m2 p 3 6 − m4 8 p √ 1 + m2 p 2 ln p − m6 48p 2 − ... ] ∞ p c . (4.5) This expression diverges **in** **the** ultraviolet limit **of** **the** momentum for **the** first three terms; **and** is moreover dependent on an **in**frared cut**of**f parameter p c . The cut**of**f p c has been **in**troduced because **the** expansion (4.4) is only valid for large momenta, **and** we are not **in**terested **in** **the** behaviour for small momenta. The ultraviolet divergences pose a serious problem s**in**ce **the**re is no more elegant argumentation for **the** **the**ir renormalisation than **the** assumption **of** **the** **in**validity **of** conventional quantum field **the**ories at scales beyond **the** Planck scale. The usual way to deal with those **in**f**in**ities is thus to **in**troduce ano**the**r cut**of**f [50] **in** phase space, for example at **the** Planck scale, **and** disregard particles with momenta higher than that cut**of**f, lead**in**g to **the** **in**famous energy density **of** ρ Λ,th ≃ 10 76 GeV 4 mentioned before. There are alternative renormalisation methods, such as dimensional regularisation [51], which however all serve **the** purpose **of** peel**in**g out **the** s**in**gularities explicitly only to **the**n disregard those terms or elim**in**ate **the**m by appropriate counterterms. The concept we would like to put forward **in** this work to deal with **the**se divergences is to make **the** divergent contributions from bosons **and** fermions cancel exactly, so that only **the** f**in**ite parts **of** **the** **in**tegral rema**in**, which **the**n determ**in**es **the** vacuum energy. The way to achieve this is by consider**in**g **the** one o**the**r quantity occurr**in**g **in** **the** effective action above, i.e. **the** mass **of** **the** particle. If we would like to cancel **the** divergences between two particles, **and** tak**in**g **in**to account that bosons **and** fermions give opposite-sign contributions to **the** effective action, it is clear that by f**in**e-tun**in**g **the** masses **of** **the** particles **in** **the** system accord**in**gly, it is possible to get rid **of** **the** divergent parts. In order to have **the** first three terms **in** **the** above expression for bosons **and** fermions cancel each o**the**r out **the**re are some relations that have to be fulfilled. The total expression for **the** vacuum energy for i particle species is S eff ∝ ∑ [ ν i p 4 + m2 i p 3 4 6 i − m4 i 8 ln p − ... ] . (4.6) To cancel **the** first term **in** **the** sums, we dem**and** to have **the** same number **of** degrees **of** freedom on **the** bosonic **and** fermionic sector **of** **the** system, denoted by b **and** f, ∑ ν b = ∑ ν f . (4.7) b f

4. Vacuum energy **in** flat spacetimes 33 To cancel **the** o**the**r two divergent terms **in** **the** sum, **the** masses **of** **the** bosons **and** fermions must obey **the** relations ∑ m 2 b = ∑ m 2 f , b f ∑ m 4 b = ∑ m 4 f . (4.8) b f Given **the** assumption that **the** masses **of** bosons **and** fermions can be chosen to fulfil **the**se relations, this is **the** basic schedule that has to be obeyed **in** order to achieve **the** cancellation **of** divergences. These simple estimations have shown **the** pr**in**ciple that we employ. In **the** follow**in**g sections, we will calculate **the** vacuum energy exactly, peel out **the** divergences with slightly different methods, **and** try to f**in**d **the** exact cancellation conditions that have to be fulfilled. We expect however that **the**y will be **in** pr**in**ciple **the** same as what we obta**in**ed from **the**se sketchy considerations. 4.1. Exact calculation **of** **the** vacuum energy Go**in**g back to **the** **in**tegral **in** question, ∫ d D p (2π) D ln ( −p 2 + m 2) , (4.9) we will apply **the** technique **of** dimensional regularisation, which means that **the** **in**tegral will be rewritten **in** terms **of** Γ-functions as a function **of** **the** number **of** spacetime dimensions D, which will **the**n be set to a natural number plus an **in**f**in**itesimal part, i.e. D = 4 − ϵ. The result**in**g expression will be exp**and**ed **in** terms **of** **the** small quantity ϵ, **and** **in** **the** end **the** limit ϵ → 0 will be taken. From **the** above expression, we first carry out a Wick rotation, **and** **the**n calculate **the** **in**tegral as (see e.g. [52]) ∫ d D p (2π) ln ( −p 2 + m 2) = −i Γ(−D/2) 1 . (4.10) D (4π) D/2 (m 2 ) −D/2 This **in**tegral is divergent because **the** Γ-function diverges for negative even values without any possibility **of** analytic cont**in**uation or such tricks, **and** for D = 4, we have Γ(−2) **in** **the** numerator. The expression needs to be processed fur**the**r us**in**g dimensional regularisation, i.e. we substitute D by D = 4 − ϵ, which will make it possible to explicitly isolate **the** **in**f**in**ities **in** terms which diverge **in** **the** limit ϵ → 0. The result **of** **the** **in**tegral conta**in**s ϵ twice: with**in** **the** Γ-function, **and** as power **of** **the** mass. It is possible to exp**and** **the** Γ-function as [53] Γ(−n + ϵ) ≃ (−1)n n! [ 1 ϵ + ψ(n + 1) + ϵ ( )] π 2 2 3 + ψ2 (n + 1) − ψ ′ (n + 1) , (4.11)

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10. Numerical analyses 85 10. Numer

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10. Numerical analyses 87 priors to

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10. Numerical analyses 91 Table 10.

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10. Numerical analyses 93 Model A M

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10. Numerical analyses 95 Table 10.

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10. Numerical analyses 97 Figure 10

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10. Numerical analyses 99 Table 10.

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10. Numerical analyses 101 Figure 1

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10. Numerical analyses 103 Table 10

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106 Part III. Dark Energy from an O

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12. Bose-Einstein condensates in As

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122 Part IV. Bose-Einstein condensa

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124 Part IV. Bose-Einstein condensa

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126 Part IV. Bose-Einstein condensa

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16. Numerical solution 145 16. Nume

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16. Numerical solution 147 16.3. Ou

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16. Numerical solution 155 Figure 1

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16. Numerical solution 157 The best

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16. Numerical solution 159 Figure 1

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16. Numerical solution 161 The zero

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164 Part IV. Bose-Einstein condensa

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Bibliography 167 Bibliography [1] A

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Bibliography 169 [27] V. Vitagliano

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Bibliography 171 [58] L. E. Parker

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Bibliography 173 [87] R. R. Caldwel

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Bibliography 175 [116] R. C. Tolman

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Bibliography 177 [143] P. Oehberg a

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Acknowledgments 181 Acknowledgments