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Quantum Phenomena in the Realm of Cosmology and Astrophysics

34 Part II. Dark Energy from **the** Vacuum Energy **of** **Quantum** Fields where ψ(n) is **the** Digamma function, **and** ψ ′ (n) **the** Trigamma function. The expansion **of** **the** Γ-function **the**n reads ( Γ −2 + ϵ ) ≃ 1 2 ϵ + 1 − γ [ ] π 2 2 + ϵ γ2 + 2(1 − γ) + ) + O(ϵ 2 ) . (4.12) 12 2 The mass term can be exp**and**ed for small ϵ as well, m −ϵ = µ −ϵ [ µ m ) ϵ ≃ µ −ϵ ( 1 + ϵ ln µ ] m + O(ϵ2 ) , (4.13) where µ is an auxiliary parameter with **the** dimension **of** a mass, **in**troduced to make **the** argument **of** **the** logarithm dimensionless. Truncat**in**g at l**in**ear order **in** ϵ, we end up with an expression for **the** effective action as S eff ≃ − iµ−ϵ (4π) 2 m4 [ 1 ϵ + ln µ m + 1 − γ 2 + O(ϵ) ] . (4.14) Thus, we see that **in** order to cancel **the** divergent terms proportional to 1/ϵ, we need to require that **the** sum over all quartic powers **of** **the** masses **of** **the** system vanish, ∑ m 4 b = ∑ f b m 4 f , (4.15) which will **the**n ensure **the** effective energy to be f**in**ite. This will also cancel **the** constant term 1 − γ/2 **in** **the** effective action. Thus for **the** convergent part, **the**re is only **the** term S eff,conv = − iµ−ϵ (4π) 2 m4 ln µ m (4.16) left, which should be tuned to result **in** **the** observed cosmological constant by requir**in**g **the** condition to be fulfilled. ∑ m 4 b ln µ − ∑ m b f b m 4 f ln µ m f = ρ Λ (4.17) In this argumentation, we ended up with one condition to cancel **the** divergences, **and** one to tune **the** convergent rema**in**der to a specific value ρ Λ . However, we failed to recover also **the** condition ∑ ν b = ∑ ν f . (4.18) b f to balance **the** degrees **of** freedom **of** **the** particles - a condition that was predicted by **the** naive estimations **of** **the** vacuum energy previously. The reason for this is that with**in** **the** formalism **of** dimensional regularisation as applied here **the** quadratically divergent contributions to **the** **in**tegral are lost due to **the** application **of** Veltman’s rule [54, 55].

4. Vacuum energy **in** flat spacetimes 35 Concretely, an **in**tegral like **the** one **in** Eq. (4.10) can always be reformulated as an **in**tegral over a fraction as ∫ Consider **the** identity ∫ d D p (2π) ln ( −p 2 + m 2) ∫ = D d D p (2π) D m 2 p 2 (p 2 + m 2 ) = ∫ d D ∫ p (2π) D dm 2 1 p 2 + m 2 . (4.19) [ ] d D p 1 (2π) D p − 1 . (4.20) 2 p 2 + m 2 By us**in**g **the** formula for Schw**in**ger’s proper time **in**tegral [56], ∫ i d D k (4πis) D/2 e−im2s +m = 2) (2π) D e−is(−k2 , (4.21) **and** tak**in**g **in**to account **the** **in**tegral representation **of** **the** Γ-function, we obta**in** ∫ d D p m 2 ∫ (2π) D p 2 (p 2 + m 2 ) ≡ d D p 1 Γ(1 − D/2) 1 = , (4.22) (2π) D p 2 + m2 (4π) D/2 (m 2 ) −D/2+1 which consequently implies ∫ d D p 1 (2π) D p = 0 , (4.23) 2 known as Veltman’s formula [52,54,55]. That shows that **in** us**in**g **the** technique **of** dimensional regularisation, **the** existence **of** an additional quadratic pole **in** **the** **in**tegr**and** is lost. On **the** contrary, us**in**g **the** formalism **of** a cut**of**f regularisation, **the** quadratic divergence is obta**in**ed correctly with an ultraviolet cut**of**f at p Λ : ∫ p Λ 0 d D p 1 (2π) D p 2 + m = − i ( ) p 2 2 (4π) 2 Λ − m 2 ln p2 Λ . (4.24) m 2 In **the** formalism **of** a dimensional regularisation, lost quadratic poles as **in** Eq. (4.23) can be recovered by consider**in**g **the** follow**in**g reparameterization **of** **the** masses: m 2 i → m 2 0 + m 2 i , (4.25) where **the** square **of** **the** masses are rewritten as **the** sum **of** a constant part m 2 0 **and** an **in**dividually different part m 2 i . m 2 0 is **the** same for each particle species, **and** **the** m 2 i characterise **the** differences between **the** particles. The effective action **in**tegral **the**n reads ∫ d D p (2π) ln ( ) −p 2 + m 2 D 0 + m 2 Γ(−D/2) i = −i (4π) D/2 (m2 0 + m 2 i ) D . (4.26) Do**in**g **the** same expansions as before, exp**and****in**g **the** Gamma function **and** **the** mass term, **and** neglect**in**g terms **of** l**in**ear order **in** ϵ, we obta**in** a similar expression as before, [ ] S eff ≃ − iµ−ϵ (m 4 i + 2m 2 0 m 2 i + m 4 0) 1 (4π) 2−ϵ/2 ϵ + 1 − γ 2 + ln µ √ + O(ϵ) . (4.27) m 2 i + m 2 0

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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 89 z y1 y 4z

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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 149 T [K] 0

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16. Numerical solution 151 Figure 1

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16. Numerical solution 153 lower ce

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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