*Update 5 September 2022. *

A video covering much of the material in this post is available at:

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## Original post below

Although the Universe may be infinite in extent, in the generally accepted Big Bang cosmology we can only see a small fraction of it. This is known as the observable universe.

*Field of distant galaxies – Image Credit NASA*

There are three widely used definitions of cosmological horizons, which are limits imposed by cosmology to how far away we can see. The purpose of this post is to try and explain in a non-technical way these different definitions. I hope I succeed!

**The Particle Horizon** (boundary of the observable universe)

As discussed in a previous post , when we look at a distant galaxy its light will have taken millions (or even billions) of years to have reached us. if we could construct a cosmological-sized ruler between the galaxy and the Earth then we would measure a quantity known as the ** proper distance **to the galaxy

**.**This is the separation between the Earth and the galaxy at a given time. Because the Universe is expanding, the proper distance between two distant objects (which aren’t held together by gravity) increases with time. In the example below, light is emitted from a galaxy which 300 million years ago was at a proper distance of 294 million light years from Earth. However, the Earth is moving away from the emitted light photons all the time they are travelling towards us. So these photons actually travel 300 million light years to reach Earth. When they reach us the proper distance of the galaxy will be 306 million light years.

The ** particle horizon** is the theoretical maximum proper distance we can see to at the current time. It is a spherical shell approximately 46.5 billion light years in radius around the Earth. When we look at distant objects we are looking back in time and light from an object at the particle horizon will have been emitted at the beginning of the Universe and will have been travelling towards us for the entire age of the Universe.

*All the objects we observe today lie inside the particle horizon, which forms the boundary of the observable universe. If an object lies beyond the particle horizon, then the Universe is not old enough for its light to have had enough time to reach us.*

At the exact instant of the Big Bang the particle horizon would have been zero and as the Universe ages the particle horizon increases. This is for two reasons.

(1) As the age of the Universe increases, light can travel a greater distance before it reaches us.

(2) Because the particle horizon is the proper distance of the furthest object we can see, due to the expansion of the Universe, as the Universe ages the proper distance between two distant objects increases.

*The time axis shows the time since the Big Bang, the purple dashed line marks the current particle horizon.*

In reality, we cannot see all the way to the particle horizon. As readers of my earlier post will know, the early Universe was far too hot for atoms to exist. It contained a plasma of positively charged hydrogen and helium ions and negatively charged electrons. Electromagnetic radiation, of which light is an example, cannot pass through plasma. The oldest radiation we can detect is the cosmic microwave background (CMB) which was emitted when the Universe was only 400 000 years old, at which time it had cooled sufficiently for individual atoms to exist. The CMB radiation we observe today has been travelling towards us since this time and was emitted from a spherical shell of points, which lie at a proper distance of approximately 46 billion light years from Earth.

**The Event Horizon and Hubble Sphere**

As described previously the Universe is expanding. The further away an object is the faster it is receding from us.

There is a clear relationship between the recessional velocity and the distance of a galaxy. This relationship is known as Hubble’s Law and is written as

v = H_{o}D

where

- v is the velocity an object is moving away from us
- D is the object’s distance
- H
_{o}is a constant known as the Hubble constant. If v is measured in kilometres per second and D is in megaparsecs (Mpc) (1 Mpc =3.26 million light years) then H_{o}is approximately 70 km/s per Mpc. The Hubble constant measures how fast the Universe is expanding*. In reality, the Hubble constant changes over time (it is generally believed to be decreasing) and so is more correctly called the***Hubble parameter**H(t). The Hubble constant is the value of the Hubble parameter today. However, the current rate of change of the Hubble constant is very small. It will take hundreds of millions of years to fall by 1% from its current value.

Assuming that Hubble’s law is valid at all distances *(i.e at all times in the past)*, at a separation from us of more than 4,300 Mpc (or 14 billion light years) a galaxy will be receding at a velocity greater than 300 000 km/s which is the speed of light. In which case any light it emitted today could never reach us. The **Hubble sphere** is an imaginary sphere centred on the Earth of radius 4,300 Mpc. If the Hubble parameter didn’t change over time, we could only see objects which emitted light ** today** located inside the Hubble sphere.

However the Hubble parameter is changing over time, so we need to consider a further type of horizon, the **event horizon**. This is the largest proper distance from us from which light emitted * now* will reach us at some distance time in the far future.

- If an object lies closer than the event horizon then its light will reach us.
- If an object lies further away than the event horizon then it so far away that light emitted now will never reach us.

If the Hubble parameter didn’t vary over time, then the event horizon would be the radius of the Hubble sphere (14 billion light years). In most cosmological models, even though the Universe is expanding, the value of the Hubble constant falls over time. The net effect of this is that the event horizon is larger than the radius of the Hubble sphere and the difference between the event horizon and the Hubble sphere changes over time.

The graph below shows how the event horizon changes over time. In the current model of the Universe the event horizon will gradually increase with time but at a slower and slower rate reaching a maximum value of around 18 billion light years.

For more details on the event horizon see https://explainingscience.org/event-horizon-more-details/

**Technical Notes**

Since Hubble’s law v= H(t)D predicts superluminal recession at large distances (i.e. distances greater than c/H (t) ) it is sometimes wrongly stated that it needs some kind of* “special relativistic correction” *to prevent a galaxy moving away from us a velocity greater than the speed of light.

In fact, there is no contradiction with special relativity when faster than light motion **occurs outside the observer’s inertial frame** and in any case general relatively not special relativity is needed to describe the Universe as whole.

Distant galaxies are receding from us superluminally. This means we will never be able to see their light emitted at the current time. However, they are at rest locally and** motion in their own local inertial frames remains is well described by special relativity. **For more details see Davis and Lineweaver (2003).

**Reference**

Davis, T A and Lineweaver, C H (2003) *Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe, *Available at: * https://arxiv.org/abs/astro-ph/0310808* (Accessed: 30 April 2021).

There is a short video on the history of the Universe after it was one second old on the Explaining Science YouTube channel.

For the fun of it, try this: Let a proton mass and an electron mass be in

stable circular gravitational orbit around their common center of mass.

Let the proton’s orbital angular moment be hbar. Set GMpMe/R^2 =

mu Vp^2 /R, where mu is the reduced mass, R is distance between Mp

& Me, Vp is Mp’s velocity, and rp is radius of Mp’s orbit.

NB: hbar = Mp*rp*Vp. Solve for 2*rp (the diameter of Mp’s orbit).

Convert this answer into light years and notice your jaw dropping.

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Yes it will be pretty big orbit! But then gravity is such a weak force

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Is there an equation with which you can enter a given age of the universe and calculate what the event horizon was at that time?

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Hi Jay,

Thanks for your comment. I have created a page which hopefully will answer your question

https://explainingscience.org/event-horizon-more-details/

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May I ask a question?

Can you tell how old the universe is when the particle horizon is 26.4 Gly?

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Firstly, as discussed in the post

The particle horizon is the theoretical maximum proper distance we can see to at the current time. It is a spherical shell approximately 46.5 billion light years in radius around the Earth. When we look at distant objects we are looking back in time and light from an object at the particle horizon will have been emitted at the beginning of the Universe and will have been travelling towards us for the entire age of the Universe.

Secondly,

The age of the Universe depends on the various parameters in our model of the Universe. Essentially what we do is measure its current density, composition and rate of expansion and work backwards to time when it had an extremely high density. The following link may prove useful

https://wmap.gsfc.nasa.gov/universe/uni_age.html

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I did not read the entire content, but how did the particle horizon reached 46.5 billion light years in radius IF the age of universe is only at 13.8 billion years since big bang? Should it not be only atmost 13.8 billion light years(the farthest the light would have traveled since big bang)?

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Hi Rodel, that is due to cosmic expansion of the fabric of space, so while the cumulative separation could be at well in excess of light speed, the objects themselves are just in their normal orbital peculiarities. The classic analogy example is leavening raisin dough. This holds true under both SCM-LCDM consensus and the competing SPIRAL cosmological redshift hypothesis and model.

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Nice presentation Steve.

Please advise approximate LY distance of the original departure point radius, of the CMB radiation we see here and now, that has travelled 13B rounded LY to get here.

was it at that distance x at the end of 400k years after the start of the big bang? or the end of cosmic inflation?

also

what is the nearest known departure point of any light we see here and now that has ever been subjected to any comic expansion?

by definition any light arriving here and now that has any degree of cosmological redshift (CR) has been subjected to cosmic expansion, is that LY distance closer than to the nearest stellar object whose light has any CR?

distance where it is now and how much closer it was when the light departed it, please.

TY,

rm

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Thanks for your comment. The photons we now observe as the cosmic microwave background were emitted (or be absolutely precise “last scattered”) when the universe was ~ 400 000 years old. The photons were emitted from a spherical shell which at the time was located at a proper distance of roughly 45 million light from us. But, during the 13.8 billion light years they have been travelling, the expansion of Universe means that the region of space they were emitted from now lies at a proper distance of roughly 46 billion light years from Earth.

As stated in my post https://explainingscience.org/2020/12/10/dark-energy-an-unexpected-finding/, the expansion of the Universe means that distance between our own Milky Way galaxy and any object not gravitationally bound to us increases over time. Our galaxy together with Andromeda galaxy and numerous smaller galaxies form a gravitational bound structure known as the Local Group. If you want to know more about the Local Group the link below gives a reasonable overview

https://imagine.gsfc.nasa.gov/features/cosmic/local_group_info.html

Galaxies outside the Local Group (i.e. lying at a distance of more than ~5 million light years) will, in general show a cosmological redshift

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Thank you Dr. Hurley, very helpful, have a great week, rm

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Very interesting. Could you please rewrite it without the assumption of a big bang, inflation and expansion. I really am curious about the outcome…

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Hi Hakan,

it depends on the model and assumptions.

For example if SPIRAL cosmological redshift hypothesis and model, the entire universe approximates the visible universe, that we (Earth-moon-sun ecliptic are at the approx. center of, that has a maximum radius of 4B LY.

The universe attained mature size and density after 4/365(5781) a fraction of history.

For now assume the radius is 4B LY.

So by the end of 4B years of the universe having attained mature size and density, no more light will reach us that has ever been subjected to cosmic expansion.

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