Show that in an exponentially expanding Universe (q=-1) the...

Show that in an exponentially expanding Universe $(q=-1)$ the Hubble sphere is stationary (see section 5.18). Show that it constitutes an event horizon in the sense that events beyond it will never be observable. Show that in this Universe there is no particle horizon. (In a Universe which has undergone a period of normal expansion with $q>0$, which then enters a period of exponential expansion, particles which have come within the horizon prior to the onset of inflation are swept beyond the Hubble sphere and contact with them is lost. This contact can then be re-established later once a period of normal expansion is resumed.)

Key Concepts

Exponential Expansion

Exponential expansion refers to a phase in the evolution of the Universe during which the scale factor grows exponentially with time. This rapid increase in the scale factor is characteristic of an inflationary period, where distances between points in space double in constant time intervals, leading to dramatic effects on observable regions and causal contact between different parts of the Universe.

Hubble Sphere

The Hubble sphere is defined as the spherical region around an observer beyond which objects recede faster than the speed of light due to the expansion of the Universe. In an exponentially expanding Universe, this sphere becomes stationary because the expansion rate (characterized by a constant Hubble parameter) balances the recession of galaxies at a fixed proper distance, creating a fixed boundary in proper coordinates.

Event Horizon

In cosmology, an event horizon is a boundary delineating the region of spacetime from which light signals can eventually reach an observer. In an exponentially expanding Universe, the Hubble sphere acts as an event horizon since any event occurring beyond this fixed boundary will never be observable by an observer, no matter how long one waits, due to the continuously accelerated separation driven by exponential expansion.

Particle Horizon

The particle horizon represents the maximum distance from which light has had time to reach an observer since the beginning of the Universe. In an exponentially expanding Universe, the usual notion of a particle horizon does not apply because the rapid expansion prevents a fixed maximum observable region from being established; instead, earlier causal contact may be lost during the exponential phase, and later resumed, highlighting a different causal structure than in universes undergoing slower, decelerated expansion.

Transcript

00:01 Hello, and in this question here, we want to find the hubble constant as a function of the radius of curvature, which is itself a function of time.

00:12 Now, we're told in the question that 1 plus the redshift, which we denote by z, is equal to the wavelength we receive divided by the wavelength emitted.

00:22 Now, we have, let's say we have two galaxies, and in one galaxy, a wave, a photon is emitted, and we'll see a different wavelength as to what was emitted.

00:36 Now, the reason for this is that the space in between the two galaxies, so the two galaxies are at rest, but the space in between them is actually expanding.

00:48 And we can denote that we can say that this ratio between the wavelength that was received divided by the wavelength emitted is equal to the radius of recar curvature at the time when we received it divided by the radius of curvature at the time we admitted it.

01:09 We're also told in the question that the red shift is also equal to the hubble's constant times the distance between the two galaxies, divided by the speed of light.

01:21 Sorry, that should be c.

01:22 Okay? so we want to use these formulas to write an expression for h, sorry.

01:31 So what we can do is we can say that z from the first expression is equal to the radius of curvature at the time when we receive the signal divided by the time when the signal was emitted minus 1.

01:50 And that is what equal to the redshift.

01:52 And this is equal to h times the distance between the galaxies divided by the speed of light.

01:59 Now, we can say that the radius of curvature, at the time when was emitted, we can tailor expand this in terms of the time when we received the signal.

02:15 Okay? and that we get by this that the radius of curvature at the time it when it was emitted is equal to the radius of curvature as the time when it was received.

02:33 Plus we take, because we're tailor expanding, we take the first derivative with respect to time multiplied by time it was emitted minus the time it was received.

02:45 Okay.

02:46 And we have also of higher order terms, order t squared and stuff like that, but we're just going to ignore them for now.

02:53 So we can sub in this expression in here to guess z is equal to the radius of curvature at the time the signal was received, divided by the radius of curvature at the time the signal was received, multiplied by 1 plus the derivative of the radius of curvature with respect to t.

03:17 Divided by the radius of curvature when the signal was received multiplied by the time emitted minus the time received okay and this here is equal to h times delta s times c okay well we can just cancel these here and get that we can get that one over 1 plus d a d t divided by a at time received times times emitted minus the time received okay now what is the time emitted minus the time received well if we were to say the distance traveled okay the distance between the two galaxies that is equal to the speed of light so it's the speed of the photon multiplied by the time the signal was received at minus the time that the signal was emitted at okay and that there is the distance that the photon will have traveled so we can sub this in here to get this is equal to one minus one divided by one minus d a divided by a at the time the signal was received okay multiplied by by delta s over c okay and this is equal to h times delta s over c okay so let's look at this here again now we're going to assume that delta s is very very large okay so we can once again tailor expand this for sorry we're assuming that this is we're going to assume sorry that this quantity here is small...

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