Shape of Dark Matter Halos

I had used this post previously to help me write a paper a while back. The paper is out:

http://arxiv.org/abs/1106.1765

I am now struggling to finish my thesis for Monday and using the blog again to write up this particularly mentally blocked section :) will be back in writing action after it’s all done.

———below is the ooold part of this post——

ok so here’s the thing. i actually hate writing. or working maybe. and im having a terribly hard time trying to write a paper on what i’ve been working on these last few months. so im going to use you all to help me write it. hehe =) this way it doesnt feel like work. it feels like procrastination. and my thoughts would be more fluent, and i’d strive to be clearer. so i’ll explain it all. you dont have to read it or anything though hehe, but its nice if you want to. atleast the overall idea. in some parts im also going to pretend that you are interested in things like errors. i have made the post private, as i have heard horror stories of research being stolen, and im not sure if this is worthy of theft, but i dont want to risk it as i have spent a lot of time on it and it is very crucial to my career. i should also mention that some explanations are beyond the scope of this article, and you will have to look them up, for example the standard cosmological model.

UPDATE: I HAVE WRITTEN AN OK DRAFT NOW, THANK YOU FOR YOUR HELP. I WILL NOT TORTURE YOU ANY MORE WITH ALL THE CRAZY DETAILS, BUT I’LL LEAVE THE GENERAL BIT ON. FOR THOSE INTERESTED IN DETAILS, I’LL LINK YOU TO THE FINAL PAPER WHEN ITS DONE =)

general background (for non-physicists and people not interested in the subject) :

the \lambda CDM  model is the most widely accepted model of cosmology, the one that is used to understand how the universe as we know it today came to be. how the various structures in the universe like galaxies and clusters of galaxies formed. according to this theory, ordinary matter, known as baryonic matter contributes to only about 4%  of the energy of the universe. the rest of the matter in the universe – we dont know much about it – is given the name dark matter, as they do not emit any electromagnetic signals (x-rays, gamma rays, visible light etc). this dark matter contributes to 22% of the energy content of the universe, and is assumed to be cold, that is, these particles are not going too fast. so we have CDM – Cold Dark Matter. the thing about dark matter is that these particles only interact with other matter and with each other through gravity, as far as we know. they were initially proposed to account for the extra gravity that is inferred from the motion of stars. \lambda is the component which makes up the rest of energy density (~74%), and it is called dark energy. and we don’t need to go into that here. these days, there has been mounting evidence that cosmological structures form hierarchially - small galaxies merge to form bigger galaxies. small structures merge to form bigger structure. these structures are embedded in dark matter. so you can picture the universe as a neural web of dark matter – there are stars and galaxies embedded in dark matter and these are connected to each other by strands and filaments of dark matter. the large masses of dark matter are attracted to each other due to gravitation and so merge, thus causing the galaxies to collide and merge. here you can visualise all of this :

http://www.youtube.com/watch?v=X7APek5SQ2s&feature=related

http://www.cita.utoronto.ca/~dubinski/nbody/Mice_long.mpg – colliding galaxies

so you see, galaxies are immersed in dark matter, which is much more extended than the baryonic (ordinary) part of the galaxy. in the galaxy disk, where all the stars are, and lots of things are going on like supernovae, it is difficult to purely study the effect of dark matter. moreover, there’s more visible matter here than dark matter. so we have to look beyond the outer edges of the main stellar part of the galaxy and focus on the dark matter around it – the dark matter halo.

A study of the global features of the halo of a galaxy like its shape, extent, rotation etc are important in understanding the nature of  dark matter particles themselves. in this paper, we describe a method to study the shape of the halo, more specifically its flattening.

The density profile of a DM halo is generally considered to be a triaxial ellipsoid. The flatness of the distribution is given by the ratio of the polar to the equatorial axis (c/a) and the ovalness by the equatorial axis ratio (b/a). c/a and b/a vary with radius in general. Since we are interested in the flattening alone, let us assume that the equatorial axes are equal (b/a = 1).  Earlier studies have shown that this is a reasonable approximation for the halo of  a spiral galaxy, especially far from the galactic disk where it becomes spheroidal. (reference). Now the halo can be either spherical ( if c = a), oblate ( c < a), or prolate (c > a). These spheroids can be described by an axisymmetric potential :

\begin{equation}

\Phi_{halo} = \frac{1}{2}V_{0} ^2 ln\left(R_{c}^2+R^2 + \frac{z^2}{q^2}\right)

\end{equation}

in  cylindrical coordinates,

where $q = c/a$ is the flattening parameter or axial ratio, $V_{0}$ is the circular velocity, $R_{c}$ is the core radius. The galactic disk is in the XY (R) plane.

N-body simulations of gravitational collapse of density peaks show that a flattened halo implies that the DM particles are cold, whereas a spherical one implies that they are hot. A massless spherical halo component would mean that the ‘missing mass’ is due to a modification of gravity rather than due to undetectable matter. The study of the distribution of the halo could also tell us about the local orientation of the cosmic web, since the smallest axis tends to be normal to the sheet in which it lies. A survey of galaxies and their halos could tell us how the shape of the halo relates to the main stellar component, as well as recent mergers.

Various methods have been developed to study the properties of dark halos, but some of the results obtained are inconclusive and some of them seem to contradict each other, and there is no real consensus on several of these properties. For example, efforts to constrain the flattening of the NGC 5907 halo by N-body simulations of its tidal stream  show that the similar streams are produced for different values of q. Nevertheless, these tidal streams are important probes of dark halos and in this paper, we improve on the existing methods based on them. These stellar streams are left behind by a satellite galaxy as it falls into the host galaxy and trace out the orbit of its progenitor. We use maximum likelihood estimation (MLE) to constrain the parameters of the halo potential using the guiding center orbit of the stream.

Besides tidal streams, gravitational lensing has proved to be a powerful tool in constraining the flattening parameter as it is sensitive to the ellipticity of the lens, but the values obtained by strong lensing are also sensitive on  how well the substructures in the halo have been accounted for. Weak lensing, being much less sensitive to the substructure, provides a better way to measure the halo profile. Other methods that have been used involve HI, X-ray gas, polar rings and velocity distributions of halo stars. Fitting the warps in the gas in spiral galaxies have also been used to study the DM halo profiles.

One of the main aspects of the problem we would like to explorein this project is: what is the minimum information required to uniquely estimate the parameters? For most stellar streams, only projected positional data are available, so we attempt to constrain the parameters with only projected positions. We also try to see if the parameters are better constrained if we use more than one stream.The method can be easily extended to account for line of sight velocity information for those streams where this is available. We also assume that the system is viewed edge-on, such as in NGC 5907.

A study of the global features of the halo of a galaxy like its shape, extent, rotation etc are important in understanding the nature of  dark matter particles themselves. The DM halo is in general ellipsoidal, but we will assume its spheroidal (valid assumption for spiral galaxies). We want to find its flatness q, that is the ratio between the vertical axis and equatorial axis. The halo can be either spherical ( if c = a), oblate ( c < a), or prolate (c > a).

N-body simulations of gravitational collapse of density peaks show that a flattened halo implies that the DM particles are cold, whereas a spherical one implies that they are hot. A massless spherical halo component would mean that the ‘missing mass’ is due to a modification of gravity rather than due to undetectable matter. The study of the distribution of the halo could also tell us about the local orientation of the cosmic web, since the smallest axis tends to be normal to the sheet in which it lies. A survey of galaxies and their halos could tell us how the shape of the halo relates to the main stellar component, as well as recent mergers.

Various methods have been developed to study the properties of dark halos, but some of the results obtained are inconclusive and some of them seem to contradict each other, and there is no real consensus on several of these properties. For example, efforts to constrain the flattening of the NGC 5907 halo by N-body simulations of its tidal stream (image)  show that similar streams are produced for different values of q. Nevertheless, these tidal streams are important probes of dark halos and in this paper, we improve on the existing methods based on them. These stellar streams are left behind by a satellite galaxy as it falls into the host galaxy and trace out the orbit of its progenitor. We use maximum likelihood estimation (MLE) to constrain the parameters of the halo potential using the guiding center orbit of the stream. 

Besides tidal streams, gravitational lensing has proved to be a powerful tool in constraining the flattening parameter as it is sensitive to the ellipticity of the lens, but the values obtained by strong lensing are also sensitive on  how well the substructures in the halo have been accounted for. Weak lensing, being much less sensitive to the substructure, provides a better way to measure the halo profile. Other methods that have been used involve HI, X-ray gas, polar rings and velocity distributions of halo stars. Fitting the warps in the gas in spiral galaxies have also been used to study the DM halo profiles.

TO CUT A LONG STORY SHORT :

We find the combination of parameters that generate the best fitting orbit to the stream using Monte Carlo Markov Chains. And it works! We can get back the flatness parameter q of the potential that we make the fake data with.

Praying

My belief in God, but not so much my faith, just my belief in the existence of a personal God..has been fluctuating a lot this past year. I even put away my tiny Jesus Christ picture that I have always had under my pillow since I was eight or so, only to retrieve it every once in a while when I felt weak or scared. It has been the constant in my life and I am emotionally dependant on it..especially during periods of crises, which is almost every now and then of late. In the first pangs of doubt, I reasoned that even if a personal God did not exist, I have the need to believe in one (for strength or whatever), and so I’m just gonna go right ahead and believe. Once I accepted that, however, my need to believe reduced. In fact, the idea of no God began to appeal to me. And then I reasoned that I should learn to depend more on myself and not some higher power, and whatever goes wrong or right that is out of my hands, I can put down to just sheer luck and the ups and downs of life. It’s a more responsible way of approaching life. Blah blah blah. Anyway, it’s safe to say that I’m basically agnostic now. Though at times, I feel strongly atheistic and at others, theistic and at yet others, something else…spiritual in a different way..I guess it just depends on which part of my brain is active (or inactive) at the time.

Anyway, taking an average of all these moments and those in which I don’t give a damn at all, I could call myself agnostic for now. That’s not what I wanted to talk about though. What I wanted to talk about and still want to is praying. Which I had conveniently stopped doing, in the way I used to always do it. But a relic remained. The fossil of my prayers include:-

1. a certain amount of faith that everything is okay and will continue to remain so

2. pleading tearfully to God during times of desperation.

3. a sense of gratitude

4. praying for others if they are having a tough time or starting on something new or in a risky situation or if they just happen to be my brother.

I don’t do that consciously all the time, but after years of doing that it’s there in my head somewhere. I only do it consciously when it resurfaces or for special occasions when I make a mental note of what I need to pray for. What I have come to realise and what I wanted to say is that I still should make it a point to pray consciously, daily, God or no God. Especially for #4.,,,,#1,2,3 I am bound to do more easily, but they serve only myself. #4 is what reminds me that there are others, and that they are going about life like me too, that I can do something for them even if it’s just wishing them well, that I am capable of kind thoughts and empathy. It always snaps me out of my own bubble. Leading a more or less individual life can increase more of this alone in my world feeling. Maybe for others who live with a lot of people, with children and family maybe, #1,2,3 help to gain some alone-time. #4 is still great for perspective. And cultivating kindness. I mean, there are other ways to cultivate kindness of course, for example, by actually doing something kind. But kind thoughts lead to kind actions, and even if they don’t..sometimes you can’t do anything else…it is good I think to remember that we are kind of kind. Kindness is one of those rare things that anybody can have, at any point. Just like that. It’s free! And freeing! :D

——

OK so I thought about it some more…. it’s a good way to like yourself but I don’t think in an ego-stroking way. It’s this fraternal feeling. I always feel calm when I pray for others…I feel strong, like a giver, not always taking or asking. But I’m asking for someone else…and they are asking for me, and I get this sense of we are all in it together…hmm…ok..I’m off again.

The Perfect Couple

Like Tea and Biscuit,

made for each other,

complimenting and enhancing

and so much fun :)

Delicious and refreshing and comforting.

Kind of half-assed without the other,

even though each is so full on their own,

giving away no clues of a need of a partner.

True love. Good love. Accidental soul mates.

Softening one, and doing something inexplicable to the other.

Anyway, a whole a whole lot greater than the sum of its parts.

……………………………………………………………………………….

And then when Biscuit is gone, its still there in Tea,

who by now is a bit cold, but richer in a way and quiet.

Not the hot bubbly thing she used to be.

Solemnly, fondly longing for Biscuit …but its okay she tells herself ,

she’s going to be gone soon too.

Looking for a hug

Sometimes you just need someone to give you a tight hug and say ‘Everything’s going to be okay, you’re going to be fine’. And if there’s no one around to give you that, when you need it, you need to step up and tell your self that. Very much like massages. Yeah, reach your hand back and press that magic spot. It may not be as satisfying as someone else telling you, touching you…but it’s still comforting. This may also lead to a very mild form of split personality, being the you and the I at the same time.

Do you also get those voices in your head…like when you mumble to yourself at night when you’re tired…and I catch myself saying ‘I love you’ ‘I love you too sweetie’ ‘really?’ ‘yeah :)’. It’s not creepy really, it’s just one of those automatic compulsive conversations in my head. Do you also get those? (-I do. -really? – yeah…lol)

Anyway I’m drifting off the original intent of this post.. see there are these two quotes that I like a lot:

1. Falling in love with your self is the beginning of a lifelong romance  – Oscar Wilde

2. Love me the most when I deserve it the least, because that’s when I really need it – Dunno who

I always have #2 in mind, actually more with respect to others than to myself, that is #2 being a plea to me rather than a plea of mine. Anyway, combining 1 and 2 brings me to why I need to love myself the most now. Coz I really feel like I don’t deserve it. But I think I’m being nice to myself and more patient than I would have been earlier. I’m learning something. That’s for sure. Because I think I’m in one of the worst self created situations in my life, I haven’t felt so pressured or guilty ever, and yet I am actually also in a happy place. Not entirely of course, but I’m pretty happy in many ways. Actually given the circumstances, I don’t think I could be any happier. Still though I’m not doing very well. I’m anxious and very negative. And want to run away from it all. But I won’t.

—-

Addendum: That conversation I have is a relic, one I used to have on a nightly basis with my mother, and still do when I get to be home. I guess I just miss having someone say that to me, with that much truth in it. It’s comforting, it makes me feel safe.

You, me, them and us

Opposing perspectives. In the world’s eye, I’m so tiny, just an insignificant drop in the ocean. But in mine, I’m nearly everything, I’m the center, so big. Sometimes, it seems to me like that is the whole purpose of life, so that the universe can sample itself locally, magnifying things at different scales. Every scale matters.

These opposing perspectives can be confusing, do I behave like I am insignificant or do I behave like I am the boss of it all? Maybe an in-between take…I guess that’s what most of us resort to.. a middle stance between mattering and non-mattering. Things matter, but then nothing matters. It’s a good thing we die, otherwise the insignificance would rot us. Yes, better to keep making fresher batches of life that feel like the bomb themselves.

Still, what surprises me sometimes, is the finiteness of our world as we know it. Sometimes it feels infinite, but it isn’t. 7 billion people. Not infinite. 150 million square kilometers of land. 24 hours in a day. Michael Jackson, Gandhi, Obama, Osama, Newton, Einstein, Jesus, Mohammed, Pamela Anderson. We all know them. There may have been a time when the world was more apart and scattered and news travelled slower, but now it seems like a really really small place. It seems to be, effectively, just one big ghetto. And we all go about doing our own things, our own nothings, creating and trying to solve our problems, others’ problems, fighting with one another, bonding with one another. Helping, hating, gossiping. Feeling important and unimportant and helpless and instrumental and massively confused all at once.

Cutesy world

The West’s fascination with the East and the East’s fascination with the West reminds me so much of boys’ fascination with girls and girls’ fascination with boys, it’s cute :)

Un

J’ai eu une bonne idee hier soir – Je vais ecrire en Francais! Mon francais..c’est merde, mais ca va, je peut apprendre comme ca. Je voudrais developer concepts et engage plus que le simple conversation en francais. Je lire rien en francais sauf les phrase sur cartons et a la supermarche et par fois les emails a travaill, donc ces posts vont etre plein d’erreurs. Je veux pas verifier mes phrases avec google traduction toujours, alors pardonnez moi et n’hesitez pas me signe si tu (tu peux me tutoyer), si tu note quelque erreurs ou tu connais un meilleur facon a dire le meme chose.

Mais t’inquiete pas! Je suis amusant en francais aussi =D Mais un peut plus simple, moin complex, moin de trucs au hazards. Par example, je vais essayer un rigolo en francais….err.. non, peut etre la prochaine fois. Ma tete me fait mal quand je pense en francais. C’est pas normal pour ma tete les sonds rrrrrrrrr et jjjj et aaannn etc et les silence de lettres quand ils plait. C’est comme une mer calme, les sonds, il me fait seasick…malade de mer? voila..

C’est suffit pour maintentant, mais je vais revenir!! Bisous :)

how’s life?

humbling. very very humbling..

Being Cool

Some people are just so fucking cool, and then some others are just fucking.

Tension

I love it when the guy is too scared to make a wrong move, when he is so still lest he blows it, I love the helpless nervousness on his face, and the underlying excitement in his eyes, his tense muscles, the way he is silently frustrated he can’t be his usual self, an anxious mess in place of the confident, almost arrogant, man he is..

Getting somewhere

Jul 16 2011 16:30

I was going somewhere, but I can’t remember where. I need to go back to being very basic and boring, in the sense that I don’t

take an interest in all the distractions that surround me. This is the only way I know to get something substantial done. This is the only way

I seem to be able to focus, unfortunately.

This thought is under construction, going back to trying to focus.

————————————–

Jul 16 2011 22:00

Really need to simplify my life. It’s hard to say no. Especially to going out and being social.

I need to be home more. And not on the computer (unless it’s work related). And less daydreaming. And less going out for coffee on my own.

Keep it simple. The fewer procrastination options I allow myself, the more I’ll work.

So I’m staying home tonight even though it’s a Saturday and a bunch of friends and others are going out.

I’m having soup and toast and green peas for dinner. Will study for a bit and wind up for the day with a movie in bed.

Sounds pretty good actually :)