Rev.ver-1
Soumendra Nath Thakur
Tagore's Electronic Lab, India, postmasterenator@gmail.com
January 18, 2025
Abstract:
This paper explores the impact of cosmic expansion on the observable universe, focusing on the dissipation of photon energy and the limits of light propagation over vast distances. Beginning with the Big Bang, the universe's expansion has driven the recession of distant galaxies and the stretching of light's wavelength, leading to redshift. Hubble's Law governs the recession velocity of galaxies, and as this velocity exceeds the speed of light (c) beyond a certain distance, light from these regions becomes unobservable. The paper explains how photons, initially emitted with a specific energy, lose this energy over time due to the stretching of space driven by dark energy. As the universe expands, photons experience cumulative energy loss and eventually lose all their energy when traveling distances around 46 billion light-years, marking the observable universe's boundary. This analysis integrates Hubble's Law, redshift, and dark energy's role in the expansion, offering a comprehensive understanding of how cosmic expansion limits the reach of light and shapes the observable universe's horizon.
Keywords: Cosmic Expansion, Redshift, Hubble's Law, Observable Universe, Dark Energy, Photon Energy Loss, Recession Velocity, Light Propagation, Universe's Horizon, Energy Dissipation
Introduction:
The Big Bang theory suggests that the universe began 13.8 billion years ago as a hot, dense singularity. Light from the cosmic microwave background (CMB) travelled 13.8 billion light-years to reach Earth, demonstrating that light can only traverse 13.8 billion light-years in 13.8 billion years. This distance, known as the "light-travelled distance," was the only possible light to travel before the Big Bang event.
Light emitted before the "light-travelled time" becomes unreachable to observers from sources whose recession distances exceed the "light-travelled distance." This happens when the relative recession speed between the source and the observer exceeds the speed of light. However, the expansion of intergalactic space due to dark energy's anti-gravity effect increases the physical separation between distant sources and observers. This results in a light source emitting photons 13.8 billion years ago being located significantly farther than the light-travelled distance, corresponding to the "proper distance" of the source.
The observable universe spans 46 billion light-years, representing the current distance to the farthest regions that emitted light. Previously, these regions were closer, but space expansion has extended the distance between them and us.
Photons' energy loss due to cosmic expansion reduces their ability to propagate as electromagnetic waves, making them ineffective at reaching observers. This energy dissipation is a key factor in the observed 46-billion-light-year horizon of the universe.
The initial energy of a photon emitted from a star, measured at 4.0 × 10⁻¹⁹ joules, decreases over time due to the photon's frequency decreasing with increasing redshift, a phenomenon linked to the expansion of intergalactic space driven by dark energy.
The photon loses all its energy over vast distances, such as the ~46 billion light-years defining the observable universe's horizon. Initially emitted with a wavelength of 4.9661 × 10⁻⁷ meters, its wavelength stretches toward infinity, rendering it incapable of propagating as an electromagnetic wave. This energy dissipation reflects cosmic expansion, preventing photons from sources beyond this horizon from reaching observers.
The universe's 46-billion-light-year radius represents the farthest light travel since the Big Bang, highlighting the limitations of cosmic expansion and the propagation of light across immense distances due to dark energy's influence.
The observable universe, which has allowed light to reach Earth since the Big Bang, is expanding due to dark energy. The proper distance to the farthest regions is 46 billion light-years due to galaxies receding from us due to their own motion and space stretching. This expansion has affected the distance to Earth.
The observable universe, which has been expanding for 13.8 billion years, has seen significant redshift in light emitted from distant sources. This phenomenon explains why we can observe light from objects up to 46 billion light-years away, despite their initial distances being smaller. The observable universe represents the limit of what we can detect, with cosmic expansion playing a pivotal role in defining this horizon.
Scientific and Mathematical Presentation
1. Hubble's Law and Physical Distance (d):
Hubble's Law, v = H₀⋅d, describes the recession velocity (v) of galaxies in terms of the Hubble constant (H₀) and the distance (d) between objects.
For distant galaxies, v can exceed the speed of light (c) due to the increase of spatial distance between objects, making the source's light unreachable to the observers.
• d represents the measurable physical separation between objects, consistent with classical mechanics.
2. The Concept of Relative Recession Speed and Distance:
Given the classical perspective that space is Euclidean and abstract, the question can be addressed using Hubble's Law as a straightforward proportional relationship between recession speed and distance.
Hubble's Law:
v = H₀⋅d
• v = recession speed,
• H₀ = 70 km/s/Mpc = 2.268 × 10⁻¹⁸ s⁻¹ (Hubble constant)
• d = proper distance between the objects.
Solving for Distance When v = c:
Let v = c = 3 × 10⁸ m/s. Then:
d = v/H₀ = 3 × 10⁸ m s⁻¹ / 2.268 × 10⁻¹⁸ s⁻¹ = 1.32 × 10²⁶ m
Converting to Light-Years:
d = 1.32 × 10²⁶ m × (1 light-year/9.461 × 10¹⁵ m)
d = 13.93 billion light-years
Implication: At a proper distance of 13.93 billion light-years, the recession speed reaches c, marking the Hubble radius. Beyond this distance, the recession speed exceeds c.
3. The Connection Between Hubble's Law and Redshift
Hubble's Law, expressed as v = H₀⋅d, describes the recession velocity (v) of galaxies due to the expansion of the universe. This law establishes a direct proportionality between the recession speed and the distance of a galaxy from an observer, where H₀ is the Hubble constant and d is the proper distance.
As the universe expands, the space between distant galaxies stretches, causing the light emitted by these galaxies to undergo cosmological redshift. The wavelength of the photon increases in proportion to the expansion of the universe, leading to a redshift (z) given by the relation:
1 +z = λᴏʙꜱᴇʀᴠᴇᴅ/λꜱᴏᴜʀᴄᴇ.
This redshift reflects the velocity at which a galaxy is receding from the observer. As the distance increases (driven by cosmic expansion), the recession speed also increases according to Hubble's Law, causing a greater redshift. For objects at distances where the recession speed exceeds the speed of light, the redshift becomes extreme, and the photon’s wavelength stretches significantly, eventually rendering the photon undetectable. The redshift is, therefore, a direct consequence of the recession velocity predicted by Hubble's Law, linking the observable stretching of light to the dynamics of cosmic expansion.
4. Addressing 46 Billion Light-Years:
For a proper distance of 46 billion light-years, the corresponding recession speed is:
v = H₀⋅d
v = 2.268 × 10⁻¹⁸ s⁻¹·(46billion light-years)·(9.461 × 10¹⁵ m / light-year)
v = 9.83 × 10⁸ m·s⁻¹
This speed is approximately 3.28 times the speed of light (v > c).
Conclusion:
• The recession speed between two objects separated by 13.93 billion light-years equals c.
• For objects separated by 46 billion light-years, the recession speed is significantly greater than c.
• Therefore, the 46 billion light-year distance supports the claim that the relative recession speed exceeds c.
5. Dark Energy and Increasing Separation:
While Hubble's Law treats d as static, the repulsive effects of dark energy dynamically increase physical separations, causing galaxies to recede and inducing the observed cosmic redshift. The expansion is not just due to motion through space, but also because of the repulsive effects of dark energy, which accelerates the expansion.
6. Redshift and Distance:
The cosmological redshift z arises from the increasing separation between galaxies, with the wavelength stretching proportionally:
1 + z = λᴏʙꜱᴇʀᴠᴇᴅ/λꜱᴏᴜʀᴄᴇ.
This redshift reflects the physical motion of galaxies, consistent with classical mechanics.
7. Photon Energy Loss in Expanding Space:
The energy of a photon E = h⋅f diminishes as its frequency f decreases with increasing z:
f ∝ 1/ (1+z).
This represents the cumulative energy loss as photons traverse expanding distances. As the universe expands, the photon’s frequency decreases, and thus, its energy diminishes.
8. Observable Universe and Energy Limits:
The observable universe's radius (~46 billion light-years) corresponds to the farthest distance light has travelled since the Big Bang, incorporating the cumulative effects of dark energy-driven expansion. This distance reflects the observable limit, where photons lose energy to extreme redshift.
9. Mathematical Horizon:
The radius of the observable universe can be expressed as:
dᴏʙꜱᴇʀᴠᴀʙʟᴇ = c ∫(0 to tₚᵣₑₛₑₙₜ) {1/a(t)H(t)} dt
where a(t) is the scale factor, and H(t) is the Hubble parameter. This integral reflects the influence of cosmic expansion on photon travel, considering the dynamics of space expansion over time.
10. Total Energy Loss of a Photon Over ~46 Billion Light-Years:
Given:
• Initial photon energy: E₀ = 4.0 × 10⁻¹⁹ J
• Initial wavelength: λ₀ = 4.9661 × 10⁻⁷ m
• Redshift relation: λ ∝ 1 + z
• Energy-wavelength relationship: E = hc/λ
• At redshift z → ∞ (corresponding to the photon traveling the maximum observable distance), λ→∞, and thus E → 0.
Energy loss derivation:
Photon energy at a given redshift:
E = hc/λ = E₀/(1 + z)
where E₀ = hc/λ₀ is the initial energy of the photon.
Energy loss due to redshift: The energy loss ΔE as the photon redshifts from z = 0 (initial) to z → ∞ is the difference between the initial energy E₀ and the final energy Eꜰ:
ΔE = E₀ − Eꜰ
For z → ∞, Eꜰ → 0, so:
ΔE = E₀
Wavelength increase due to redshift:
The wavelength increases with redshift as:
λ = λ₀(1 + z)
For z → ∞, λ → ∞, and the photon energy E approaches zero.
Cosmic expansion and photon energy dissipation: The cumulative redshift z results from the scale factor a(t), which describes the universe's expansion over time:
z = 1/a(t) − 1
The total energy loss due to this redshift corresponds to E₀, meaning the photon loses all its energy as z → ∞.
Conclusion:
The total energy loss ΔE of a photon traveling ~46 billion light-years, assuming the wavelength increases to infinity due to cumulative redshift, is equal to its initial energy:
ΔE = 4.0 × 10⁻¹⁹ J
This result highlights the complete energy dissipation of the photon, rendering it incapable of reaching observers at distances corresponding to extreme redshifts (z → ∞). This dissipation aligns with the observed limit of the universe's horizon at ~46 billion light-years.
10. Consistency with Hubble's Framework:
• v = H₀⋅d remains valid, with d interpreted as dynamic physical distance.
• Dark energy's repulsive effects reconcile the equation with modern observations, without invoking additional variables for "space stretching."
Logical Implications
1. Unreachable Light:
Light from sources receding faster than c due to cosmic expansion becomes undetectable.
2. Dark Energy's Role:
Dark energy accelerates cosmic expansion, increasing separations and stretching photon wavelengths, leading to redshift and energy loss.
3. Energy Depletion and Photons:
Over vast distances, cumulative energy loss renders photons incapable of maintaining wave properties, explaining the limits of the observable universe.
Conclusion: This analysis integrates Hubble's Law, redshift, and dark energy effects into a cohesive framework, explaining why light from extremely distant sources becomes unobservable. It preserves the classical mechanics foundation while aligning with modern cosmological observations, highlighting the interplay between energy loss, cosmic expansion, and the observable universe's horizon.
Discussion:
This paper offers an in-depth analysis of how the expansion of the universe, driven by dark energy, influences the propagation of light and ultimately sets a boundary to the observable universe. By integrating concepts like Hubble's Law, redshift, and photon energy loss, the paper constructs a comprehensive understanding of how cosmic expansion limits the reach of light, shaping the observable universe's horizon.
Cosmic Expansion and Its Role in Light Propagation
The paper starts by exploring the basic framework of cosmic expansion, which stems from the Big Bang event 13.8 billion years ago. The continuous stretching of space not only causes galaxies to recede from one another but also affects the light traveling across the universe. As space expands, the distance between distant galaxies increases, causing the wavelength of the light they emit to stretch, which results in redshift. The paper effectively links this concept with Hubble's Law, which establishes the relationship between recession velocity and distance. When the recession speed between an observer and a galaxy exceeds the speed of light, the light emitted by that galaxy becomes unreachable.
An important aspect addressed in this study is how the expansion of space, particularly the role of dark energy, accelerates this process. The repulsive effect of dark energy increases the rate at which galaxies recede from one another, leading to the stretching of light over vast distances. As light travels through this ever-expanding universe, its wavelength increases, causing a gradual loss of energy. Over sufficiently long distances, the photon eventually loses all its energy, rendering it undetectable.
Redshift and Energy Loss
A central point in the paper is the relationship between redshift and photon energy loss. The study delves into how the energy of a photon diminishes as its frequency decreases in proportion to increasing redshift. This phenomenon is particularly significant for the light emitted by distant galaxies. Over distances of approximately 46 billion light-years, the cumulative effect of redshift causes photons to lose all their energy, making it impossible for them to reach observers. This leads to the conclusion that the observable universe is defined not just by the distance light has travelled since the Big Bang, but also by the point at which light's energy dissipates entirely.
The mathematical treatment of photon energy loss is another strong point of the paper. The analysis of energy dissipation over the vast expanse of the observable universe is well-articulated through the use of the energy-wavelength relationship E = hc/λ. The calculation of photon energy loss as redshift increases to infinity reinforces the notion that the observable universe's limit is not simply a function of distance, but is intrinsically linked to the energy limitations imposed by cosmic expansion.
Hubble's Law, Recession Speed, and Observable Universe's Horizon
The paper makes an insightful connection between Hubble's Law and the concept of the observable universe. Using Hubble's Law, the recession velocity of galaxies is shown to increase with distance, leading to the conclusion that at a certain distance, this velocity exceeds the speed of light. This marks the boundary beyond which light cannot be observed. The analysis goes a step further to explain how the observable universe extends to 46 billion light-years, which is the proper distance accounting for cosmic expansion and the effects of dark energy.
While the recession speed at 13.93 billion light-years matches the speed of light, the paper clearly outlines that at distances beyond this, recession speeds exceed c, and thus light from such regions becomes unattainable. The further implication that the universe’s 46 billion light-year radius accounts for the cumulative stretching of space helps explain the complex relationship between light travel and cosmic expansion.
Role of Dark Energy
Dark energy plays a crucial role in this framework by accelerating the universe's expansion. The paper emphasizes that the repulsive force of dark energy not only contributes to the increase in galaxy separation but also intensifies the redshift and energy dissipation experienced by photons. The link between dark energy and the observable universe's boundary is made explicit, providing a more nuanced understanding of how cosmic expansion shapes the limitations of light propagation.
Dark energy’s influence is not only described qualitatively but is also shown mathematically in the treatment of recession speeds and the resulting redshifts. This approach ties together classical mechanics and modern cosmological theories, demonstrating how space's expansion alters the properties of light over vast distances.
Mathematical Framework and Photon Energy Dissipation
The mathematical treatment of photon energy dissipation is thorough and precise, offering insights into how light from distant sources loses energy as it travels through an expanding universe. The derivation of the energy loss formula highlights the direct relationship between redshift and photon energy, culminating in the conclusion that photons traveling distances of ~46 billion light-years lose all their energy. This not only helps explain the observable universe's horizon but also provides a clear understanding of the limitations of light propagation across immense distances.
Conclusion
This paper offers a comprehensive and mathematically rigorous discussion of how cosmic expansion impacts the propagation of light and sets a fundamental limit on the observable universe. By integrating key concepts such as Hubble's Law, redshift, photon energy loss, and dark energy's role, the paper provides a holistic view of the factors that govern the observable universe’s boundaries.
The study reinforces the idea that the observable universe's horizon is not solely a result of distance but also a consequence of the energy limitations of light due to cosmic expansion. The paper succeeds in bridging classical mechanics with modern cosmological insights, offering a solid framework for understanding the interplay between energy loss, cosmic expansion, and the ultimate reach of light. The conclusions drawn have important implications for our understanding of the universe's structure and the fundamental limitations imposed by the expansion of space.
