Universe rate of expansion: speeding up or slowing down?

[CosmologyHobbyist]

A basic question... The further a galaxy is from ours, the more redshift, the faster it is moving away from us. This is taken as a sign the universe is expanding at ever increasing rate.
Yet... As we look farther out into space, we are also looking back in time. If the farther we look back in time, the faster the universe was expanding, doesn't that mean the more recent in time, the slower the expansion, and the expansion of the universe is slowing? What am I missing here?

 

[PeterDonis]

The further a galaxy is from ours, the more redshift, the faster it is moving away from us. This is taken as a sign the universe is expanding at ever increasing rate

No, that by itself only tells us that the universe is expanding. It doesn't tell us whether the expansion is accelerating or decelerating. For that we have to look at the details of how the redshift varies with distance over a large range of redshifts; the linear Hubble law (redshift equals a constant times distance) only holds for small redshifts/small distances ("small" on a cosmological scale). For larger redshifts the relationship is no longer linear, and the details of how the relationship varies are what we use to figure out things like the expansion of the universe decelerating until a few billion years ago, and then accelerating since then.

As we look farther out into space, we are also looking back in time. If the farther we look back in time, the faster the universe was expanding, doesn't that mean the more recent in time, the slower the expansion, and the expansion of the universe is slowing?

No, because it's not as simple as "the farther we look back in time, the faster the universe was expanding". See above.

 

[Vick]

The further a galaxy is from ours, the more redshift

If the farther we look back in time

There are two aspects you are talking about: 1) the looking back into the past and 2) redshift - meaning the objects are receding from our point of observation.

A simple example demonstrates the first aspect: Take the Sun which is about 150 million km from Earth. In terms of light speed, this is about 8.3 light minutes. Electromagnetic (EM) radiation including light and heat travels at the speed of light. Therefore when we look at the Sun now, it is a picture of what it looks like 8.3 minutes ago.

$T = D/S$

where T = time in seconds
D = distance in meters
S = speed in meters/second

So if for another object the distance is greater, the more in the past we are looking. Because it takes that much time for light (EM radiation) to cross that distance.
However, regarding the second aspect, the EM radiation that we observed from distant objects has longer wavelengths. The more distant the longer the wavelengths. That is called redshifted. This means that the object in question is receding from our point of observation.

$v = H_0 D$

where v is recessional velocity in km/sec
$H_0$ is the Hubble constant
and D is proper distance in Mpc

So an object that is, say, 100 Mpc from us, would have a receding velocity of 6780 km/sec (taking the Hubble constant to be 67.8 km/sec/Mpc).

The relationship between redshift and recessional velocity is as follows:

$z = \sqrt{\frac{1 + \frac{v}{c}} {1 - \frac{v}{c}}} - 1$

where c is the speed of light.

So the redshift (z) is 0.022877

 

[PeterDonis]

The relationship between redshift and recessional velocity is as follows

This is only true in flat spacetime. The spacetime of our universe is not flat. In a curved spacetime, there is no well-defined "recessional velocity" between spatially separated objects, only different coordinate-dependent quantities that don't have direct physical meaning.

For small enough distances and redshifts in our universe, spacetime curvature can be ignored and the relationship you give, as well as the linear velocity-distance Hubble relationship, can be taken as valid. However, at larger distances and redshifts in our universe, these relationships are no longer valid.

 

[Vick]

However, at larger distances and redshifts in our universe, these relationships are no longer valid.

Yes we will need General Relativity (GR) to calculate recessional velocity.

 

[PeterDonis]

we will need General Relativity (GR) to calculate recessional velocity

As I said, "recessional velocity" is coordinate-dependent and has no physical meaning for spatially separated objects in curved spacetime. It's unfortunate that cosmologists continue to use the term and convert directly observed redshifts into this physically meaningless quantity, apparently because they believe it will make more sense to lay people. IMO it only causes confusion.