Gravitational field inside a void

[Bill McKeeman]

Sci Am August 2016 discusses a supervoid detected in the direction of the CMB cold spot. The analysis assumes the gravitational potential is less in the center of the void than near its edges (thus near its surrounding galaxies). On the other hand the gravitational field inside a spherical shell of matter is constant zero everywhere (e.g. http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/sphshell2.html). Why isn't the field inside a void also constant zero? What am I missing here?

 

[Chalnoth]

Sci Am August 2016 discusses a supervoid detected in the direction of the CMB cold spot. The analysis assumes the gravitational potential is less in the center of the void than near its edges (thus near its surrounding galaxies). On the other hand the gravitational field inside a spherical shell of matter is constant zero everywhere (e.g. http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/sphshell2.html). Why isn't the field inside a void also constant zero? What am I missing here?

The void isn't spherical and it isn't empty. It just has lower matter density than its surroundings.

 

[timmdeeg]

The void isn't spherical and it isn't empty. It just has lower matter density than its surroundings.

Just for my understanding. Would a photon loose energy after entering a void, in contrast to a photon which "falls" into a supercluster and gains energy thereby?

 

[Chronos]

Just for my understanding. Would a photon loose energy after entering a void, in contrast to a photon which "falls" into a supercluster and gains energy thereby?

Correct. The integrated Sachs-Wolfe effect causes photons to be slightly blue shifted upon passing through a large overdense region of the universe and slightly red shifted upon passing through a large underdense region. This is due to expansion of the universe. A photon gains energy [blueshifts] as it approaches an overdense region, but, it does not have to pay back all the energy gained when it exits the overdense region because the gravitational well is slightly shallower by the time a photon exits an overdens region. The opposite effect occurs when a photon passes through a large void. The density of mass behind a photon entering a void is slightly greater than the desity of mass in the direction opposite the void which causes a slight energy loss [redshift], but, the photon does not recover all the lost energy when it exits because the mass in the direction opposite the void is slightly diluted by expansion by the time the photon exits.

 

[Chalnoth]

Correct. The integrated Sachs-Wolfe effect causes photons to be slightly blue shifted upon passing through a large overdense region of the universe and slightly red shifted upon passing through a large underdense region. This is due to expansion of the universe. A photon gains energy [blueshifts] as it approaches an overdense region, but, it does not have to pay back all the energy gained when it exits the overdense region because the gravitational well is slightly shallower by the time a photon exits an overdens region. The opposite effect occurs when a photon passes through a large void. The density of mass behind a photon entering a void is slightly greater than the desity of mass in the direction opposite the void which causes a slight energy loss [redshift], but, the photon does not recover all the lost energy when it exits because the mass in the direction opposite the void is slightly diluted by expansion by the time the photon exits.

This explanation isn't quite correct.

If there were no dark energy, there'd be no ISW effect. Dark energy causes gravitational potentials to decay slowly over time, so that a large potential (whether an overdensity or void) will be a little bit more shallow by the time the photon exits. Thus it keeps some of the energy it gained or lost as it entered the region. If we had a matter-dominated universe with no dark energy, then the potentials would be essentially constant (at least in linear theory, meaning on very large scales), so that the photon would revert to its exact energy before entering the region.

 

[Chronos]

The explanation I offered is essentially the same as the one given here, https://briankoberlein.com/2015/04/22/stoking-the-fire/

 

[Chalnoth]

The explanation I offered is essentially the same as the one given here, https://briankoberlein.com/2015/04/22/stoking-the-fire/

Yeah, it's not correct that it's just the expansion. In a flat, matter-dominated universe, the Integrated Sachs-Wolfe Effect cancels entirely (because at least in linear theory, gravitational potentials do not change over time in such a universe). I'm sure that non-linear effects change this on smaller scales, but the ISW effect largely cancels on smaller scales anyway (the effect is most significant at large scales).

The Wikipedia write-up goes into a little bit of detail on this point:
https://en.wikipedia.org/wiki/Sachs–Wolfe_effect#Late-time_integrated_Sachs.E2.80.93Wolfe_effect

If you want a more technical write-up:
https://arxiv.org/pdf/1404.5102v1.pdf

Apparently the ISW effect is also apparent in a universe with significant spatial curvature. But given that our universe is quite flat, the ISW effect is currently the most direct evidence we have of dark energy.

 

[timmdeeg]

The opposite effect occurs when a photon passes through a large void.

Ok, thanks.

 

[timmdeeg]

But given that our universe is quite flat, the ISW effect is currently the most direct evidence we have of dark energy.

Without dark energy the universe would still expand, albeit decelerated. Would large structures not participate because of $\ddot{a}=0$ in this case?

 

[Chalnoth]

Without dark energy the universe would still expand, albeit decelerated. Would large structures not participate because of $\ddot{a}=0$ in this case?

This effect is a feature of dealing with the inhomogeneities, so FRW doesn't really apply.

When you use linear perturbation theory to estimate how the underdense and overdense regions evolve over time, you find that the gravitational potential is constant in the flat, matter-dominated universe.

I'm not completely sure of the physical interpretation of this fact, but it might be related to the fact that once gravitationally-bound systems form, they are quite stable. I think that this means that in such a universe, overdense regions don't collapse so much as they just stop expanding once the expansion slows to the point that they can remain gravitationally-bound. It's been a number of years since I studied this subject in detail, unfortunately.

There are certainly some non-linear effects that can also cause gravitational potentials to change over time, but those effects are very small as they're only really significant at small scales and the ISW effect cancels more efficiently at small scales (because if you're looking at smaller scales, chances are that there are so many voids/overdensities of those scales between us and the CMB that they average out to zero).

 

[timmdeeg]

Thanks for your answer.

This effect is a feature of dealing with the inhomogeneities, so FRW doesn't really apply.

So it seems it isn't as simple as I thought. My reasoning was that large structures like superclusters are just enough loosely bound gravitationally in order to "feel" tidal forces. Hopefully it is at least correct that tidal forces require $\ddot{a}<>0$.