hubble sphere - universe in problems

Hubble sphere

The Hubble radius is the proper distance . The sphere ofthis radius is called the Hubble sphere. From definition, the Hubble recession"velocity" of a comoving observer on the Hubble sphere is

and equal to the speed of light . This is true, ofcourse, at the same moment of time , for which is taken.

Problem 1All galaxies inside the Hubble sphere recede subluminally (slower than light)and all galaxies outside recede superluminally (faster than light). This is whythe Hubble sphere is sometimes called the "photon horizon". Does this meanthat galaxies and their events outside the photon horizon are permanentlyhidden from the observer's view? If that were so, the photon horizon would alsobe an event horizon. Is this correct?

At first glance, observation of galaxies beyond the Hubble sphere appears to be anunsolvable problem. As Eddington in 1933 wrote: ``Light is like a runner on anexpanding track with the winning-post receding faster than he can run. But nevergive up! In most models of the Universe the Hubble parameter is not constant.In a decelerating universe the Hubble radius increases with time. Moreover,the Hubble sphere expands faster than the Universe, so that the edge of theHubble sphere -- the photon horizon -- overtakes the receding galaxies. Light raysoutside the Hubble sphere moving toward us may therefore eventually beovertaken by the photon horizon. They will then be inside the Hubble sphere andwill at last start approaching us. Eddington's runner sees the winning-postreceding, but he must keep running and not give up; the expanding track isslowing down and eventually the winning-post will be reached.

Problem 2Show, by the example of static universe, that the Hubble sphere does notcoincide with the boundary of the observable Universe.

The observable Universe is limited by the particle horizon, and if the Hubblesphere and the observable Universe were the same, the latter in a static Universewould be infinitely large ( , so ). But static Universes of finite agehave particle horizons at finite distance. So, the Hubble sphere cannot be theboundary of the observable Universe.

Problem 3Estimate the ratio of the volume enclosed by the Hubble sphere to the fullvolume of a closed Universe.

(t) = c (t)RH H 1

v( ) =H = cRH RH ct H(t)

solution [hide]

HRH

solution [hide]

= 0H0 = RH

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Hubble sphere - Universe in Problems 3/5/15

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Problem 4Show that in a spatially flat Universe ( ), in which radiation is dominating,the particle horizon coincides with the Hubble radius.

Problem 5Find the dependence of comoving Hubble radius on scale factor in a flatUniverse filled with one component with the state equation .

Problem 6Express the comoving particle horizon through the comoving Hubble radius forthe case of domination of a matter component with state parameter .

Problem 7Show that

where

is the deceleration parameter.

As , the answer is obtained after one differentiation.

Problem 8Show that

Using the definition , we obtain this after differentiation.

Problem 9Show that in the Einstein-de Sitter Universe the relative velocity of the Hubblesphere and galaxies on it is equal to .

Problem 10Find in a universe with constant positive deceleration parameter .

Using

and the condition , we get

k = 0

/aRH = wp

w

= c(1 + q),dRHdt (1)

q = /aaH 2

solution [hide]= = a/RH H 1 a

= 1+ .dLpdtLpRH

solution [hide]= a dtLp t a1

c/2

a(t) qsolution [hide]

= ,H aaa2a2

q = const= 1 + q,ddt H

1

= (1 + q)t1



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thus and with

Also means .

Problem 11Show that in universes of constant positive deceleration , the the ratio ofdistances to the particle and photon horizons is .

Problem 12Show that the Hubble sphere becomes degenerate with the particle horizon at

and with the event horizon at .

Problem 13Show that if is not constant, comoving bodies can be inside and outside ofthe Hubble sphere at different times. But not so for the observable universe;once inside, always inside.

From ((1)) we see that recession velocity of the Hubble sphere can be lesser orgreater than , depending on the sign of . As recession velocity of matter on it isexactly , be definition, this means that matter can cross the Hubble sphere in bothdirections at different times. "Horizons are like membranes; the photon horizonacts as a two-way membrane (comoving bodies can cross in both directionsdepending on the value of ), and the particle horizon acts like a one-waymembrane (comoving bodies always move in and never out)." E. Harrison,Science of the Universe.

Category: Horizons

= (1 + q)tH 1 a tnn = .11 + q

q > 0 n < 1

q1/q

q = 1 q =1

q

solution [hide]

c qc

q



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