Pavle I. Premović
Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,University of Niš,
pavleipremovic@yahoo.com, 18000 Niš, Serbia
Light emitted by all nearby and distant galaxies [1] shows the cosmological redshift. This shift is defined as
z = (λ – λ 0)/λ0
where λ0 is the wavelength of the emitted light and λ is the wavelength measured by the Earth’s observer. Modern cosmology, which is based on the General theory of relativity, explains the cosmological redshift by the expansion of the Universe.
Denote now with D the distance between the Earth and a nearby or distant galaxy (hereinafter galaxy). It is rational to propose that the photon’s wavelength λ cannot be larger than this distance or λ ≤ D. Thus, the maximum wavelength of a galaxy’s photon can be about the distance D or λmax = D.
We know that the photon’s wavelength is inversely proportional to its frequency. The relation between D (= λmax) and the corresponding frequency νmin is given by the formula D = c/νmin where c (= 299792 km sec-1) is the speed of light. Multiplying both sides of this inequality with Planck’s constant h (= 6.63 × 10-24 J sec) and after a bit of algebra we find
Emin = hc/D
where Emin = hνmin. Introducing into this equation the above-given values for c and h we have
Emin = 2 × 10-25/D.
In Hubble terminology, the Hubble distance is defined as
D H = c/H0
where H0 is the Hubble constant. (Note the reciprocal of H0 is known as the Hubble time tH ≈ 14 Gy). This distance represents, roughly speaking, the radius of the observable Universe. For H0 = 72 km sec-1 (Mpc)-1, we can take very roughly DH ≈ 14 Gly. This is the limit of the observable Universe or our observable cosmic horizon, the maximum wavelength of the galaxy’s light, λmax, coming from the horizon to the Earth would be about 14 Gly. Of course, a photon with a wavelength of 14 Gly cannot be detected. The oldest photon that can be detected is that of the cosmic microwave background (CMB) radiation. Its wavelength of 1.9 mm is in the microwave regime.
We also know that the photon’s wavelength is inversely proportional to its energy or in our case
Emin = hc/λmax = hc/DH.
Introducing the above data for h, c and λmax or DH in the appropriate part of this expression, we find that Emin (= hνmin) = 1.5 × 10-51 J. This energy is previously denoted as ε. (For further details see references 3 and 4). Very obviously, it represents the minimum quantum of energy of the observable Universe. This was the main aim of this piece of work.
Moreover, for H0 = 72 km sec-1 (Mpc)-1 = 2.3 × 10-18 sec-1, νmin = H0 = 2.3 × 10-18 sec-1, the minimum photon momentum pmin = ε/c = hH0/c (= 1.5 × 10-51 J/3 × 108 m) = 5 × 10-60 kg m sec-1 and its minimum rest mass mmin = ε/c2 (= 1.5 × 10-51 J/9 × 1016 m2 sec-2) = 1.7 × 10-68 kg. These values are listed in Table 1 of Reference [3] but marked with the subscript ε instead mmin.
Heisenberg's Uncertainty Principle for Energy and time is related to simultaneous measurements of energy and time. In expression form
ΔEΔt ≥ h
where ∆E is the uncertainty in energy and ∆t is the uncertainty in time. The maximum uncertainty in time Δtmax is found using the equals sign in this equation. In our case, after a bit of algebra,
Δtmax = h/Emin.
Putting the above value of Emin (= 1.5 × 10-51 J) in this expression we get Δtmax ≈ 14 Gy. It is not unreasonable in this case to estimate the maximum uncertainty of time here as simply the maximum time itself. (It will at least give the correct order of magnitude). In other words, the Hubble time tH (= 14 Gy) represents the maximum time of the observable Universe.
There is a lower limit for the photon energy in the observable Universe, and it comes from the above Uncertainty principle expression. This minimal photon energy is of the order of Emin = h/AU, where AU (= 1/H0) is the age of the Universe. So, Emin = hH0 = ε.
References
[1]
We define nearby galaxies as
those whose redshift z is from 0.001 to 0.1 (or 0.001 ≤ z ≤ 0.1) and distant galaxies with z > 0.1 [1, 2]. Of course,
there is no sharp line between nearby and distant galaxies.
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