Sachs-Wolfe Effect on power spectrum

Question

From my understanding, there are two main types of Sachs-Wolfe effect:

1)Non-intergrated Sachs-Wolfe effect happens when the photon escapes from a potential well/peak at the last scattering, therefore appears red/blue shifted

2)Intergrated Sachs-Wolfe effect is when the photon, while on its way to reach us, fall into potential well(gains energy) and then come out of it(loses energy). Now normally the gain and loss will balance out, but since the universe is expanding, the potential well becomes shallower while the photon is in the well, therefore as it emerges, it loses less energy than it initially gained, hence emerging with a surplus of energy

What I do not understand are:

1)how these two effects result in the flat shape of the power spectrum and

2)why it only affects lumps with large angular size

If there are mistakes in my understandings, please do point out too. Any help is greatly appreciated!

Answer 1

I will answer your question 2 in full, and give a pointer to the answer to question 1. Someone else may like to answer question 1 more fully.

You do often see somewhat sloppy language in which the spectrum at low $l$ is said to be "caused by" or "owing to" the Sachs-Wolfe effect. This is not quite right. The fluctuations, whether at low $l$ or high $l$, are caused by whatever gave rise to them in the very early universe, followed by the gravitational collapse in an expanding background going on before last scattering.

The Sachs-Wolfe effect is the name for the process which comes in and says the temperature fluctuations observed by ourselves, long after and far away from the surface of last scattering, are not simply a map of the temperature on that surface. Rather, the combined effect of redshift on leaving a gravitational well, and also the slight change in the local expansion rate of the universe (imprinted on the metric) associated with a local density increase or decrease, make the $\Delta T$ observed by us partly a reflection of the original $\Delta T$, and partly an indication of the size of this further gravitational effect. For example, if the original temperature had been completely smooth, but there were metric perturbations of size $\Delta \Phi$, then we would observe temperature fluctuations of size $$ \frac{\Delta T}{T} = - \frac{\Delta \Phi}{3 c^2} $$ owing to the way the frequency of light is affected by gravity, which we here call the Sachs-Wolfe effect in order to remember that it is the full dynamic effect we want, not just redshift from a static potential well.

Now for adiabatic perturbations the local temperature (at the scattering surface) will be related to the matter perturbation $\delta$ by $$ \frac{\Delta T}{T} = \frac{1}{3} \delta_m. $$ So we have two ways for our observed temperature changes to come about, and in practice both ways occur. But their relative sizes depend on the distance scale of the perturbation, and hence on $l$. To see this, note that what I have called a metric perturbation $\Delta \Phi$ could equally well be called the gravitational potential. Newtonian gravity will suffice to make the main point we need (and Newtonian gravity is sufficient for this part of the argument, except at the largest distance scales). In Newtonian gravity the gravitational potential is related to the matter plus radiation fluctuation $\delta$ through Poisson's equation, so we find, for perturbations with wave vector $k$, $$ k^2 \Delta \Phi = 4 \pi G \rho a^2 \delta. $$ Therefore $\Delta \Phi$ is small compared to $\delta$ when $k$ is large. This is why the Sachs-Wolfe effect is only significant at low $k$, which corresponds to low $l$ (in practice, $l < 50$ or so). This answers your question (2).

For question (1) the only way to answer is to have a model of the origin of the fluctuations themselves. This is where inflationary cosmology has something to offer. Broadly speaking, if the fluctuations originate in the kind of quantum effects that occur in an expanding vacuum then one expects this part of the spectrum to be pretty flat after the Sachs-Wolfe effect is accounted for. But someone else may wish to say more about that.