Terminology in structure generating

[Discman]

I am very mixed-up with the terms perturbation, anisotropy and ripple.

What is the relation or difference between them?

 

[Mordred]

perturbation is a disturbance of motion, course, arrangement, or state of equilibrium; especially : a disturbance of the regular and usually elliptical course of motion of a celestial body that is produced by some force additional to that which causes its regular motion

Anisotropy is a property that is direction dependant.

a ripple is a capillary wave in fluid dynamics.

You can have a perturbation, that is evenly distributed such an example is the universe in thermal equilibrium that undergoes a phase transition where one particle drops out of thermal equilibrium. This is a change of state of equilibrium.

Anisotropy can take the form of a ripple ie wavelike, or it could simply be a direction of motion that is uniform. one example would be the movement of a gas cloud towards a black hole.

a ripple will usually have a preferred direction and is also be a perturbation. At least I can't think of any examples where that isn't the case

 

[abitslow]

I concur with Mordred. My take includes:
Perturbation implies a small change from a base state (could be an equilibrium, or a flow equation, or countless other things (including a set of equations describing a system)). see Perturbative Methods. Usually, if a change is large, I wouldn't call it a perturbation...but that's a moot point.
A ripple implies that a perturbation is continuous and wavelike. That is, it has temporal extent. It propagates in time (although there are other possibilities). It has continuity.
The meaning of the word anisotropy should be clear, but it is abstract. "an" means "not", "iso" means "same" and "tropy" means "shape". Something that has a different shape in different directions is anisotropic. The surface of a sphere is isotropic, no other shape in 3 dimensions is. Even a sphere, at the atomic scale, has anisotropy.

 

[bapowell]

Within the context of cosmology, these things have distinct meaning but they are not unrelated. "Perturbation" generally refers to the small fluctuations in the energy density of the universe. So, imagine a uniform distribution of energy (in the form, say, of photons and baryons) with small overdensities and underdensities superposed. These are density perturbations -- you might call them "ripples" if they look periodic, which some of them do (acoustic oscillations in the CMB), but that's colloquial. These perturbations in the energy density affect the temperature of the photons: on small scales, photons emitted from dense regions are hot, those from underdense are cold. Here on Earth, we receive these photons from a directions of the sky: because of the perturbations, photons coming from some directions are hotter than those coming from others. This difference amounts to an anisotropy in the temperature field of the CMB. So the anisotropies result from the density perturbations -- they are intimately related.

 

[Discman]

superhorizon anistropy

From the literature I understand that one speaks also about ripples in the super horizon areas. But according to me these areas are still virginal because gravity is not present. So why ripples?

 

[bapowell]

Inflation generates perturbations on superhorizon scales. The accelerated expansion causes comoving length scales to grow faster than the Hubble scale so that when inflation ends, fluctuations in the spatial curvature exist on scales surpassing what was the causal horizon during inflation.

 

[Discman]

perturbations

Thank you, I suppose that is the meaning of "primordial". So, a valley will become a potential gravitational valley only in the sub horizon position?

Still another question. Perturbations have different sizes (scales). I cannot cope that with the 1/100.000 distribution. All those different sizes and still that point-to-point difference? Strongly I have the idea there is a flaw in my thinking.

 

[bapowell]

Thank you, I suppose that is the meaning of "primordial". So, a valley will become a potential gravitational valley only in the sub horizon position?

The valleys are gravitational potential wells on all scales; they are just not causally "operative" on superhozion scales. It's not until perturbations fall back within the horizon that causal processes, like acoustic oscillations, commence.

Still another question. Perturbations have different sizes (scales). I cannot cope that with the 1/100.000 distribution. All those different sizes and still that point-to-point difference? Strongly I have the idea there is a flaw in my thinking.

The 10^-5 distribution is in reference to the amplitudes of the perturbations -- not their scales. For example, a scale invariant spectrum would have fluctuations of this amplitude on all scales.

 

[Discman]

wavelengths

It becomes clearer for me all the time. Now about wavelengths. Does a big perturbation have only one long wavelength or are all the different modes present in a perturbation?

And if there are different modes in a perturbation will they become active according to the progression of the Hubble radius or have they to wait till the whole perturbation is inside the horizon?

 

[the_pulp]

Inflation generates perturbations on superhorizon scales. The accelerated expansion causes comoving length scales to grow faster than the Hubble scale so that when inflation ends, fluctuations in the spatial curvature exist on scales surpassing what was the causal horizon during inflation.

Why is it said that inflation generates perturbations on superhorizon scales? I mean, every field generates quantum perturbations just because of its existence, right? Why are the inflation quantum perturbations needed? Cant you generate the anisotropies in CMB and such with just the Standard Model + GR fields? (Ive seen the equation related to perturbations and I managed to follow them at least in a high level, but I couldn't see, conceptually, why the inflation quantum perturbations are needed)

Thanks in advance!

 

[bapowell]

It becomes clearer for me all the time. Now about wavelengths. Does a big perturbation have only one long wavelength or are all the different modes present in a perturbation?

And if there are different modes in a perturbation will they become active according to the progression of the Hubble radius or have they to wait till the whole perturbation is inside the horizon?

It's all about the Fourier modes. A density perturbation in space can be resolved into its Fourier modes -- it is these modes that we track and evolve during inflation.

 

[bapowell]

Why is it said that inflation generates perturbations on superhorizon scales? I mean, every field generates quantum perturbations just because of its existence, right? Why are the inflation quantum perturbations needed? Cant you generate the anisotropies in CMB and such with just the Standard Model + GR fields? (Ive seen the equation related to perturbations and I managed to follow them at least in a high level, but I couldn't see, conceptually, why the inflation quantum perturbations are needed)

You do indeed have fluctuations in all the SM fields, and they can in important circumstances contribute to the overall density perturbations. What sets the inflaton apart is that it is the dominant component of the energy density. Fluctuations in the inflaton field cause different parts of the universe to undergo reheating at different times, transferring the inhomogeneities in the inflaton field to the matter sector after inflation.

 

[the_pulp]

You do indeed have fluctuations in all the SM fields, and they can in important circumstances contribute to the overall density perturbations. What sets the inflaton apart is that it is the dominant component of the energy density. Fluctuations in the inflaton field cause different parts of the universe to undergo reheating at different times, transferring the inhomogeneities in the inflaton field to the matter sector after inflation.

Ok thanks, you are confirming what I was guessing. Nevertheless I have 2 questions:

1) Why is it the more dominant? (my guess is that it should have something to do with the energy at which this field should be excited)
2) This explanation must surely have some counterpart in the maths. There should be some parameter in the inflaton part of the lagrangian that should be very large in order to make it the more important in the quantum fluctuations. Just to use an easy example, in fi^2 chaotic inflation, what part of the lagrangian is big enough to produce this effect in quantum fluctuations?

Thank you very much!

 

[phsopher]

Why is it said that inflation generates perturbations on superhorizon scales? I mean, every field generates quantum perturbations just because of its existence, right? Why are the inflation quantum perturbations needed? Cant you generate the anisotropies in CMB and such with just the Standard Model + GR fields? (Ive seen the equation related to perturbations and I managed to follow them at least in a high level, but I couldn't see, conceptually, why the inflation quantum perturbations are needed)

Thanks in advance!

You can't do this with vector fields (photons, gluons, gauge bosons of the weak interaction) because they are conformally invariant. What this means is basically that they are insensitive to length scales, so they don't feel the expansion of the universe and microscopic quantum fluctuations can't be stretched to superhorizon scales.

I must say I can't think of what would be the problem with fermions but I assume there are some given that they aren't considered much in this context in the literature as far as I'm aware. Perhaps other people have more insight.

That leaves scalar fields. The only scalar field in the Standard Model is the Higgs, and like any light scalar field it will acquire a spectrum of superhorizon perturbations. However, as bapowell pointed out, if inflation is driven by another scalar field - the inlfaton - the perturbations in the Higgs field will be subdominant compared to the perturbations of the inflaton. You can't realize inflation with the Higgs itself in the vanilla Standard Model because the potential is not sufficiently flat. However, if the Higgs is coupled non-minimally to gravity then this can be done. This scenario (Higgs inflation) is however disfavored if the the recent measurement from BICEP is a signature of primordial gravitational waves.

Maybe that's a lot of information, but what to take away is that is none of the fields in the vanilla Standard Model + gravity possesses all the necessary properties for successful inflation and generation of primordial perturbation.

 

[bapowell]

You can't do this with vector fields (photons, gluons, gauge bosons of the weak interaction) because they are conformally invariant. What this means is basically that they are insensitive to length scales, so they don't feel the expansion of the universe and microscopic quantum fluctuations can't be stretched to superhorizon scales.

Theories with massive bosons, like the weak interaction, are not scale invariant.

 

[the_pulp]

You can't do this with vector fields (photons, gluons, gauge bosons of the weak interaction) because they are conformally invariant. What this means is basically that they are insensitive to length scales, so they don't feel the expansion of the universe and microscopic quantum fluctuations can't be stretched to superhorizon scales.

I must say I can't think of what would be the problem with fermions but I assume there are some given that they aren't considered much in this context in the literature as far as I'm aware. Perhaps other people have more insight.

That leaves scalar fields. The only scalar field in the Standard Model is the Higgs, and like any light scalar field it will acquire a spectrum of superhorizon perturbations. However, as bapowell pointed out, if inflation is driven by another scalar field - the inlfaton - the perturbations in the Higgs field will be subdominant compared to the perturbations of the inflaton. You can't realize inflation with the Higgs itself in the vanilla Standard Model because the potential is not sufficiently flat. However, if the Higgs is coupled non-minimally to gravity then this can be done. This scenario (Higgs inflation) is however disfavored if the the recent measurement from BICEP is a signature of primordial gravitational waves.

Maybe that's a lot of information, but what to take away is that is none of the fields in the vanilla Standard Model + gravity possesses all the necessary properties for successful inflation and generation of primordial perturbation.

It was not a lot of information, you were very clear. Are not fermions invariant to scales too? (Im having in mind one lecture of susskind on renormalization -the 1st of the supersymmetry course-)

 

[bapowell]

1) Why is it the more dominant? (my guess is that it should have something to do with the energy at which this field should be excited)

The vacuum energy of the inflaton field must dominate in order for the universe to accelerate!

2) This explanation must surely have some counterpart in the maths. There should be some parameter in the inflaton part of the lagrangian that should be very large in order to make it the more important in the quantum fluctuations. Just to use an easy example, in fi^2 chaotic inflation, what part of the lagrangian is big enough to produce this effect in quantum fluctuations?

Generally, the field value must be larger than $M_{\rm Pl}$ to support the required energy density.

 

[phsopher]

Theories with massive bosons, like the weak interaction, are not scale invariant.

That's true, thank you for clarifying. However, what is important in this context is that the kinetic term is conformally invariant which is also true of massive gauge bosons. For this reason there will be no Hubble friction term and therefore no production of superhorizon perturbations. Perhaps what I should have said is that the kinetic term is insensitive to scales.

 

[phsopher]

It was not a lot of information, you were very clear. Are not fermions invariant to scales too? (Im having in mind one lecture of susskind on renormalization -the 1st of the supersymmetry course-)

Could be. I confess I'm not very familiar with what happens to fermions in a curverd spacetime. You need to mess around with spin connections and I never ended up looking into it. I should do at some point.

 

[Discman]

S-W effect

I still have a problem with the S-W effect. This effect is typical, I thought, for the super horizon areas. But gravity is not active there. So why is there red shifting en blue shifting? Have potential valleys and hills……Oh, my goodness, it's a question of energies, of course, during questioning it enters my mind. Am I right?

 

[bapowell]

I still have a problem with the S-W effect. This effect is typical, I thought, for the super horizon areas. But gravity is not active there.

Why isn't gravity active there? Are there not curvature perturbations on superhorizon scales?

 

[Discman]

I interfere again with a simple question (I hope). What is the relation between amplitude and density. I mean the contrast between densities on super-horizon scales are becoming stronger during expansion because of lagging behind of the denser regions. I thought all perturbations had in the beginning the same density (amplitude?). What is my flaw?

 

[bapowell]

The amplitudes of perturbations on different scales is determined by the expansion history, given in terms of the power spectrum. I don't know what you mean by superhorizon perturbations growing because they "lag behind" denser regions. Can you clarify?

 

[Discman]

Dear bapowell,

This is a quote from Mark Whittle in Big Bang Acoustics. Perhaps I interpret it wrong.

We'll come to these sub-horizon changes in a moment, but first let's ask whether large super-horizon regions can change. Although they can't change internally, they do participate in the cosmic expansion, and because of this the entire population of super-horizon clumps does evolve. Here's why. Emerging from creation, each region expands at a rate that depends on its density — denser regions expand a bit more slowly, while sparser regions expand a bit more rapidly — it's as if the denser/sparser regions act like miniature closed/open universes, each with their own expansion history. As time passes, therefore, the denser regions "lag" further behind the sparser regions, and the density contrast steadily increases. Thus, overall, the super-horizon "landscape" gradually gets rougher.

 

[bapowell]

Yes, I agree with this.

 

[Discman]

A very late reaction. What is meant (by Mark Whittle) with "the denser/sparser regions act like miniature closed/open universes"? Because super horizon perturbations are able to curve space-time? My simple layman thinking tells me there is no any gravitational action behind the horizon. Or is it in a potentially way possible? Then still I don't understand the lagging behind during expansion, that sounds like real active gravitational effects on the surroundings of the perturbations. Last question: does this roughening of the super horizon landscape mean a transition to non-lineairity?