The Energy Depletion Model of Light and Cosmology

1. Introduction

ΛCDM is currently the standard model of cosmology. It’s success, however, has been challenged recently by HST and JWST observations in the model’s early universe billions of light years away. Multiple mature galaxies were reported [1,2,3], which are not predicted by the standard model. The Hubble tension further highlights the difficulty facing ΛCDM if data measurement issues are not the culprit. These have prompted some to call for new physics, e.g. [4]. This paper presents a simple model to account for extinctions in vast intergalactic space. We discuss the proposed intergalactic extinction model in combination with the tired light model, which together had performed equally well as ΛCDM with Pantheon+ SN data and outperformed it with galaxy angular size data, in support of a static universe model over ΛCDM making the Hubble tension irrelevant and deep space mature structures explainable. In sections 2, we propose a simple model of light energy depletion through intergalactic extinction and energy redshift. The two components together are necessary to describe energy behaviors throughout the light’s travel path. Section 3 discusses previous estimation results and fitness of the proposed model vs. ΛCDM to the data. Conclusion is in section 4.

2. Materials and Methods

2.1. The Energy Depletion Model of Light

Without loss of generality, we decompose and write the flux received on earth from a beam of light with wavelength λ as

E = NE_λ

(1)

where  N = N₀/4πd²

(2)

is the number of photons per square area per time unit, d is the true distance (or the proper distance in an expanding universe) from the emitter, N₀ is the number of photons emitted by the emitter per time unit; and

E_λ = hc/λ

(3)

is the energy per photon, and h is the Planck constant. Also E_λ0=hc/λ₀ where λ₀ and E_λ0 are the wave length and energy per photon at source.

2.2. The Intergalactic Extinction Model

Interstellar and host (or internal, extragalactic) extinctions have been studied extensively close to earth on MW, LMC and SMC, and on host galaxies respectively. It has been essential to apply corrections to distance moduli to account for such extinctions when estimating distances or the intrinsic brightness of the source. Brout and Riess [5] give a comprehensive review on these extinctions. However, similar types of extinctions due to intergalactic media in between the host galaxy and MW/LMC/SMC along the vast distances of light travel has not been accounted for, which should increase with distance and thus become increasingly critical as our modern telescopes look into deeper space on the Gpc scale, resulting in overestimation of distances.

Intergalactic extinctions (IEs) may occur after the emitted light of a star leaves its host galaxy and before it enters the LMC/SMC/MW range. IE increases with distance, and may come from three sources:

• the light may encounter more galaxies or be “blocked” by them in our line-of-sight;
• it may encounter edges of galaxies or circumgalactic medium (CGM);
• it may encounter Warm-Hot intergalactic medium (WHIM) or Intracluster medium (ICM), and a great majority of baryonic matters in the universe too dimmed to be seen yet by existing telescopes (Massimo and Salucci [6]).

To gauge on the significance of intergalactic extinction source 1 above, Let’s do a rough estimation by answering the question: how likely would one expect to have a galaxy or galaxies in the line-of-sight between the light source and an earth observer? It’s a battle of two sheer numbers: the very large number of galaxies in the very large volume of the universe. Detailed in the Appendix A, the answer is: 6% probability per Gpc if the source is a galaxy, and 25% probability per Gpc if the source is a star (Cepheid, Supernova etc). These conservatively estimated probabilities are significant and must not be ignored.

Unlike interstellar and host extinctions, intergalactic extinction cannot be estimated with extinction curves since there is no photometric nor spectroscopic data available for unidentified extinction media. However, we can still model and estimate its effect statistically. We now apply the exponential decay function e^-2d/D to the photon rate in Eq. (2): N=(N₀/4πd²)*e^-2d/D = N₀/4π(de^d/D)² with the decay coefficient written as 2/D for subsequent convenience. Thus, the intergalactic extinction (IE) model can be written simply as

d_L(d) = de^d/D

(4)

where d_L is the indirectly measurable luminosity distance and D is a constant. D/2 is the distance by which 36.8% (1/e) photons remain alive from extinction, or the mean life-distance (analogue to mean lifetime for decays over time) of photons. (D was estimated to be 21.0 Gly with the Pantheon+ SN data as discussed in section 3 below.)

Figure 1 shows the behavior of d_L versus d under three different parameter values of D=13, 21, 29 Gly, and d_L=d as D->∞. The luminosity distance d_L approximates the true distance d well for small distance (d<<D) but deviates upwards exponentially.

It is important to note that only d_L can be calculated with the distance modulus m-M=5logd_L-5. The true distance d in a static universe (or the proper distance in an expanding universe) can only be calculated after the parameter D is estimated. These properties of d_L and d are independent of cosmological models, expanding or static. In the case of expanding universe models such as ΛCDM, the distance modulus overestimates the proper distance d with d_L, unless corrections are made to the modulus formula; and further, Hubble’s law is the ratio of recession velocity to the measured luminosity distance, and not to the proper distance. Since v=H₀d_L, we can place velocity v on the vertical axis in Figure 1 which implies that the universe is decelerating instead of accelerating as the v-d curve is above the straight v=H₀d (or d_L=d). One may conclude an accelerating universe by using d_L but that is delusional as d_L is an overestimation.

2.3. The Redshift Model

Similar to the IE model, to model the light energy depletion due to redshift of each photon in the light, we apply the exponential decay function e^-d/Δ to Eq. (3): E_λ=E_λ0*e^-d/Δ, or λ= λ₀e^d/Δ. Since λ=λ₀(1+z),

z(d) = e^d/Δ-1

(5)

or d = Δln(1+z)

(5’)

where Δ is a constant. For d<<Δ and d<<D, with Taylor expansions, Eq. (4) and (5) reduce to d_L=d and z=d/Δ. Thus, we have Hubble’s law z=d_L/Δ in which Δ=c/H₀ and H₀ is the Hubble constant, and Δ$\approx$13.8 Gly is in fact the familiar Hubble distance. Given an observed value of z, the true distance to the light source can also be calculated simply with Eq. (5’). The observed d_L is very useful for estimation purposes but is not the true distance and is delusional as it does not take intergalactic extinction into account. Zhang [7] offered an additional and interesting explanation of the physics at play for photons to redshift over distance.

Under this model, for each Hubble distance Δ traveled, the photon’s wave length is redshifted by e-1 (~1.72) or the associated photon energy remaining is 1/e (~36.8%). Δ is also the expected value of distance weighted by the energy level as probability density during a photon's entire trip from distance zero to infinity. While Δ=Hubble distance, it does not have the same interpretation here as under the expanding universe hypothesis as there is no expanding space and recession velocity in a static universe.

2.4. The Combined Empirical Model

Figure 2 shows the functions d_L(d) and z(d) graphically. Substituting Eq. (5’) into Eq. (4), we obtain the empirical d_L-z relationship:

Figure 2. the functions d_L(d) at D=21Gly, and z(d).

Figure 2. the functions d_L(d) at D=21Gly, and z(d).

d_L = Δ(1+z)^Δ/Dln(1+z).

(6)

In terms of the distance modulus μ$\equiv$m-M=5logd_L-5,

μ = 5(logΔ-1)+5(log(ln(1+z))+log(1+z)^Δ/D).

(7)

Eq. (6) and (7) are equivalent and have two parameters, Δ (or H₀=c/Δ) and D (or Δ/D), which can be empirically estimated. (The total flux can also be written as E(d)=E₀e^-d(1/Δ+2/D) where E₀ is energy at source.) Figure 3 and Figure 4 shows the graphs of Eq. (6) and (7) respectively. The d_L-z relationship is a s-curve1 closely approximating the straight d_L=Δz or Hubble’s law, with deviations $\le$13.4% over 0$\le$z$\le$125 (for Δ=13.5 Gly and D=21 Gly taken from empirical estimation discussed in section 3 below). In other words, the luminosity distance per unit (100%) of redshift is roughly 13.5 Gly.

Note also the symmetry between Eq. (4) and (5) in terms of d_L/d and λ/λ₀(=1+z). We may call r_λ$\equiv$λ/λ₀ the redshift ratio, and r_d$\equiv$d_L/d the dimshift ratio (as the light is shifted dimer by extinction). We take first derivatives with respect to d in (4) and (5) and get

Dr_d^’/r_d = Δr_λ^’/r_λ = 1, and r_λr_d^’/r_dr_λ‘ = Δ/D

(8)

Thus, Δ/D ($\approx$0.64 per Pantheon+ SN sample) can be interpreted as the percentage change in the dimshift ratio as one percentage change in the redshift ratio or in wavelength λ.

3. Results and Discussions

The two simple equations (4) and (5) are all we need to describe a static universe consistent with Hubble’s Law without the complicated ΛCDM model assuming expansion of space accelerated by dark energy. The two equations combine to produce Eq. (6) or (7) which can be empirically applied with z directly measured and d_L indirectly via the distance modulus. The true distance, d, is not measurable outside the model and is a confounding exogeneous variable that affect both d_L and z through Eq. (4) and (5) respectively. In this static universe, there is no space expansion and acceleration, no dark energy to search for (which one will never find if it doesn’t exist outside the hypothetical world of the standard model), not even age of the universe, beginning of time, nor the big bang. We will not be surprised to see JWST and other future telescopes continue to find more mature galaxies and structures in unbound distances.

Gupta [8] discussed the ΛCDM model and several alternative models including a TL+ model where an arbitrary or unknown flux loss term 5log(1+z)^β was added to the distance modulus formula. This TL+ model is indeed the exact same model mathematically as the one proposed here (Eq. (7)) in that the TL+ parameter β=Δ/D. So, thanks to Gupta, the proposed model was already tested with the Pantheon+ SN data which showed an excellent fit as good as the ΛCDM model with estimated H₀=72.46 Km s^-1 Mpc^-1 and β=0.6418, and thus Δ=c/H₀=13.5 Gly and D=Δ/β=21.0 Gly. Gupta further tested the models with galaxy angular size data and found that the TL+ or the proposed model’s fit was acceptable for high redshift while the standard ΛCDM model was not, which may be a smoking gun in favor of the proposed model over ΛCDM. Although Gupta included the TL+ model for discussion purpose only, it in fact turned out to explain the data equally well as or better than ΛCDM. For detailed results of the proposed model fitted with the data, please refer to the TL+ model in Gupta. Note that Zhang’s model is also the same but the β parameter there was given a value of 1.2 before fitting the SCP Union SNIa data. Both papers added the β-term somewhat arbitrarily to tired light models or otherwise the models were inadequate. The missing piece to previous redshift or tired light models is in fact the extinction equation (4). No arbitrary term is needed. Again in cosmology, the dust tricked us.

From Eq. (4), the relative bias of d_L as a measure of d is given by d_L/d-1=e^d/D-1. For d<<D, this bias is approximately d/D, and thus is linear and small, For example, at d=1%D$\approx$210 Mly, the bias is only 0.005%. When d_L approaches D, the bias grows exponentially and dominate. At d=D$\approx$21 Gly, the bias is a whopping e-1=172%! Thus, any focus on achieving a small measurement uncertainty as in the case of estimating the Hubble constant using the distance ladder method for Hubble tension discussions, is irrelevant if one ignores the biggest bias of all – intergalactic extinction. It is surreal to claim that the total measurement errors can be so small when measuring vast distance of space still full of many unknowns and measuring it indirectly through source brightness.

4. Conclusions

The proposed IE model is independent of cosmology models and may coexist with ΛCDM. However, the IE model also implies that the Hubble tension is an artifact in that the luminosity distance d_L in Hubble’s law used for estimation is subject to intergalactic extinction, and is not the true or proper distance d. It thus identifies a major bias for distance and H₀ local measurements using standard candles and other luminosity-based methods. It implies both H₀ measurement errors and (not so) new physics in the context of Hubble tension discussions. Additionally, given ΛCDM together with the IE model, the universe is not accelerating and is decelerating instead.

The IE model also provides the missing piece of the puzzle to tired light theories and explains the deep space data equally well as or better than ΛCDM. Why marginalize a simple static model in favor a complex expanding-universe model, particularly in light of increasing observations in deep space favoring the former over the latter as well as the addition of the proposed IE model here?

It is not a sound argument to refute the tired light model on the ground that there is no knowledge in physics for photons to redshift over distance. The evidence of such knowledge is unobtainable independently in lab or in nature as it would require distances in millions of light years to conduct such an experiment, which is the exact observations that the proposed model here and others do predict. This can be seen easily from Hubble’s law z=d_L/13.8 Gly: a distance of 138 Mly (or 138 million years in scientists’ life to observe) is required to detect z=0.01, 13.8 Mly to detect z=0.001, etc. The “decay” rate of energy redshift, i.e., the Hubble distance Δ, is just too large.

In my opinion as an outsider, it is just a matter of time that the cosmology community must reconsider a better alternative to the FLRW metrics and the ΛCDM model. The standard model of cosmology today strikes me as very puzzling to say the least. The claims of high precision in distance measurements are also surreal. The standard model has encountered and will very likely continue to encounter more contradictory evidence in deeper space, and I don’t believe that bandages to patch it are the solution.