lPlanck and the timetPlanck are thus mathematically defined as [8,10]:
lPlanck= r
~G
c3 = 1.616×10−35 m, (2.34)
tPlanck= lPlanck
c =
r
~G
c5 = 5.319×10−44 s. (2.35)
This Planck time acts as a temporal limit to the Heisenberg uncertainty principle. The largest energy fluctuation must therefore occur within the Planck time and can be defined as the Planck energyEPlanck. This is obtained by substituting the Planck time (2.35) into the Heisenberg uncer- tainty principle (2.33:
EPlanck∼ ~
∆t =~ r c5
~G = r
~c5
G = 1.96×109 Joule. (2.36) This may be used to determine the maximum energy density due to vacuum fluctuations. This should occur at the minimum volume, which is just the cube of the Planck length and is known as the Planck volume vPlanck:
vPlanck= (lPlanck)3 = r
~G c3
!3
= r
~3G3
c9 = 4.222×10−105 m3. (2.37) The predicted vacuum energy density ρΛ is thus obtained by dividing the Planck energy EPlanck (2.36) by the Planck volumevPlanck(2.37):
ρ(Λ,predicted)= EPlanck vPlanck
= r
~c5 G
c9
~3G3 =
r c14
G4~2 = c7 G2~
∼10114 Joule.m−3. (2.38) This is an enormously large energy density. Conversely, the current measured value of the vacuum energy is much smaller and given by:
ρ(Λ,measured)= 3H02
8πGΩ(Λ,0) ∼10−10 Joule.m−3. (2.39) The magnitude of the difference between the theoretical predicted vacuum energy ρ(Λ,predictedand the measured vacuum energy density is (ρ(Λ,measured) is:
ρ(Λ,predicted)
ρ(Λ,measured)
= 10114 Joule.m−3
10−10 Joule.m−3 ∼10−10 Joule.m−3 ∼10124. (2.40)
Thus, the predicted energy density of the vacuum from quantum mechanics is over 120 orders of magnitude larger than the cosmologically measured value. This is also known as the vacuum catastrophe and has been referred to as the worst prediction in the history of physics [9]. The cosmological constant problem, therefore, casts doubt on dark energy being a cosmological constant, motivating research into alternative dark energy models [82].
2.6.2 The cosmic coincidence problem
The cosmic coincidence problem states that the dark matter and dark energy densities have the same order of magnitude at the present moment of cosmic history while differing with many orders of magnitude in the past and predicted future [83]. This is due to dark matter scaling asρdm∝a−3 (2.5), while cosmological constant scales asρΛ∝a0. This causesρdmto vary greatly with time while ρΛstays constant (as shown in Figure 2.7). Therefore, the distant past is completely dominated by dark matter (ρdmρΛ), while the distant future is dominated by dark energy (ρΛρdm), but we happen to be observing the universe when the dark matter to dark energy ratio is approximately one to two. This coincidence seems unlikely, and a possible explanation to this problem would add more viability to any cosmological model [84].
The magnitude of this problem is indicated by the ratio r = ρρdm
de. For conserved fluids (where ρdm/de evolves according to (1.46)), as well as for the special ΛCDM case, this ratio is:
r= ρdm
ρde = ρ(dm,0)a−3(1+ωdm)
ρ(de,0)a−3(1+ωde) =r0a−3(ωdm−ωde) ; rΛCDM = ρdm
ρΛ = ρ(dm,0)a−3
ρ(Λ,0) =r0a−3, (2.41) where r0 = ρ(dm,0)/ρ(de,0)
= Ω(dm,0)/Ω(de,0)
is the present ratio of dark matter to dark energy.
To understand the order of magnitude of this problem, it is informative to calculater at the Planck scale as an indicator of the amount of fine-tuning the ΛCDM model needs from initial conditions [82]. This may be done by considering that the cosmic microwave background (CMB) temperature scales as T =T0a−1 →a= TT0 [9]. This ratio can then give the scale factor at the Planck scale:
aPlanck= T0
TPlanck. (2.42)
The current temperature of the CMB is T0 = 2.725K [72] and the temperature at the Planck scale is TPlanck =
q
~c5
GkB = 1.416×1032K [85]. Substituting (2.42) into (2.41) for the ΛCDM case (ω=−1; Ωdm= 0.266; Ωde= 0.685), gives the size of the coincidence problem at the Planck scale:
rPlanck=r0a−3Planck=r0
T0
TPlanck −3
= 0.266 0.685
2.725K
1.416×1032K −3
= 5.1×1094≈1095. (2.43)
This indicates that the initial conditions of dark matter and dark energy should be fine-tuned to about 95 orders of magnitude to produce a universe where the two densities nearly coincide today, approximately 14 billion years later [86]. The magnitude of this problem for any specific model can be denoted by the constantζ, which is defined from the relation:
r ∝r0a−ζ. (2.44)
If ζ = 0, thenr is a constant throughout cosmic evolution, and the coincidence problem is solved.
The magnitude of the deviation from ζ = 0, indicates the magnitude of the coincidence problem (used from here on forward to denote cosmic coincidence problem) [82–84]. Thus for the ΛCDM case (2.41), the value of this constant is ζΛCDM = 3. Therefore, any model which has ζ > 3 will worsen the coincidence problem, whileζ <3 alleviates the problem (relative to the ΛCDM model).
ζΛCDM = 3→
ζ >3 worsens the coincidence problem ζ <3 alleviates the coincidence problem ζ = 0 solves the coincidence problem
(2.45)
Furthmore, from the first equation in (2.41), we see that for the general case ζ = 3 (ωdm−ωde).
This shows that the magnitude of the coinicidence problem is determined by the difference in the equation of state of dark matter ωdm and dark energyωde, which causes the fluids to redshift and dilute at different rates. The coincidence problem would also be solved if ζ = 0, which happens when ωdm = ωde. This may also be obtained if some exotic interaction causes dark matter and dark energy to redshift at the same rate, causing similar Effective equations of state, such that ωdmeff = ωeffde. Finally, if dark matter is pressureless ωdm = 0, then ζ = −3ωde. This implies that models where ωde>−1 alleviate, while ωde<−1 worsen the coincidence problem. These models are known as Quintessence and phantom dark energyrespectively and will be discussed shortly.
Therefore, the coincidence problem motivates research into dark matter and dark energy mod- els that maintain a small or constant ratio r in either the past or future expansion.
2.6.3 The Hubble tension
The Hubble tension concerns the 4.4σ level discrepancy between values of the Hubble constant H0, that the ΛCDM model predicts using data either from early time probes (mainly the Cosmic Microwave Background) and late time determinations using redshift and distances (mostly from Type Ia Supernovae using a calibrated local distance ladder) [87–89].
As we have mentioned, the Planck collaboration has measured the Hubble constant to be ap- proximately H0 = 67.4±0.5km s−1Mpcs−1 [4]. Conversely, recent measurements of Type Ia su- pernovae (calibrated with Cepheid variables) have obtained a best estimate for Hubble constant of H0= 74.03±1.42km s−1Mpcs−1 [90] in 2019 and more recently H0 = 73.0±1.4km s−1Mpcs−1 in 2021 [91]. As measurements have become more accurate, this tension has only become greater.
This discrepancy has hinted at new physics beyond the standard model. This problem has become one of the most relevant issues in all of cosmology. This may be seen from a recent review article on the problem and its possible solutions [88], which include over 1000 references while concluding that no specific proposal makes a strong case for solving this tension.