How can the Universe measure 93 billion light-years?

The Age of the Universe: A Temporal Measure

The question "how can the Universe be larger than its age in light-years?" arises from a common confusion between the age of the Universe expressed in light-years (which corresponds to a distance traveled by light in a certain time) and the actual size or observable extent of the Universe, which is much larger.

The current age of the Universe is estimated to be about 13.8 billion years. One might intuitively think that nothing could have traveled farther than 13.8 billion light-years since the light travels at a finite speed and nothing can exceed it (according to special relativity). But this intuition does not take into account the Expansion of Space.

The Expansion of Spacetime (General Relativity)

The Universe is not a static space in which galaxies simply move like projectiles. It is space itself that expands over time, according to general relativity. This phenomenon is modeled by the Friedmann–Lemaître–Robertson–Walker (FLRW) equations.

• Photons emitted by a distant galaxy 13.8 billion years ago have taken all this time to reach us.
• But while they were traveling, space expanded, increasing the distance between that galaxy and us.

Due to this expansion, the light we receive today, which has traveled for 13.8 billion years, comes from objects that are now much farther away than 13.8 billion light-years.

Calculations based on cosmological models give a radius of the observable Universe of about 46.5 billion light-years, or a diameter of 93 billion light-years. This does not mean that light has traveled faster than c, but rather that the fabric of space on which it moves has stretched during its journey.

Metaphor of an Elastic Balloon Inflating

Alice and Bob start from a point on the balloon, each walking at 1 km/h in opposite directions. The balloon stretches as they walk. After 1 hour, normally, they should be 2 km apart. But if the balloon has doubled in size during this time, they will actually be 4 km apart! Yet, neither has exceeded the speed of 1 km/h relative to the surface of the balloon.

How Many Times Has the Universe Inflated Since the Big Bang?

• Age of the Universe: 13.8 billion years.
• Current distance of galaxies whose light reaches us today: ~46.5 billion light-years ⇔ diameter = 93 billion light-years, as there are also galaxies on the other side.
• Expansion ratio (scale factor): The light has taken 13.8 billion years to reach us, but during this time, the initial distance has been multiplied by ~1,100 (according to cosmological models based on the Hubble constant and dark energy).

Concrete example: A galaxy located 1 million light-years away 13.8 billion years ago is today about 1.1 billion light-years away!

Calculating the Size of the Observable Universe

\[ds^2 = -c^2 \, dt^2 + a(t)^2 \left[ \frac{dr^2}{1 - k r^2} + r^2 \, d\Omega^2 \right]\]

• a(t) is the cosmological scale factor,
• t is the cosmological time,
• r is the comoving radial coordinate (fixed for a given object),
• dΩ2 is the angular element.

The proper distance at a given moment is then: \(D(t) = a(t) \cdot r\)

Example with Current Values

At the present moment \( t_0 \), we conventionally set \( a(t_0) = 1 \). The comoving radial coordinate of the cosmological horizon (limit of the observable Universe) is approximately:

\[ r \approx 14.4 \, \text{Gpc} = 14.4 \times 10^9 \, \text{pc} \]

We can then calculate the current proper distance:

\[ D(t_0) = a(t_0) \cdot r = 1 \times 14.4 \, \text{Gpc} = 14.4 \, \text{Gpc} \]

This corresponds to approximately \( 46.5 \) billion light-years (since \( 1 \, \text{Gpc} \approx 3.26 \) billion light-years).

Thus, although the Universe is only \( 13.8 \) billion years old, the size of the observable Universe is approximately:

\[ D(t_0) \approx 46.5 \, \text{billion light-years} \]

Why Does This Not Violate Special Relativity?

Special relativity forbids an object from moving through space faster than light. But the expansion of space is not a movement through space; it is the evolution of the very fabric of spacetime. This expansion can "separate" two galaxies at an apparent speed greater than c, without contradicting relativistic principles.

Summary in 3 Points

• Light travels at c, but space expands at the same time → cumulative effect.
• The Universe has inflated ~1,100 times since the emission of the oldest light (cosmic microwave background).
• No contradiction with Einstein: The limit c only applies to objects in space, not to space itself.
• It's like running on an escalator that lengthens faster: you progress at your max speed, but the final separation is much greater!