Sparknotes – Urmila Chadayammuri

Primer: The Sunyaev-Zel’dovich Effect

April 16, 2018uc24 Leave a Comment

Galaxy cluster RX J1347.5-1145 observed with ALMA (blue) and HST (background)

The Sunyaev-Zel’dovich effect is this really really awesome process that allows us to see clusters of galaxies at all distances. It is going to be vastly important with all the fancy new submillimeter telescopes astronomers have built lately, such as the Atacama Large Millimeter Array (ALMA), the Sub-Millimeter Array (SMA) and the Atacama Cosmology Telescope (ACT). At its core, it’s a beautifully simple phenomenon, so let’s break it down.

About 10% of the mass of galaxy clusters is intra-cluster gas, not attached to any individual galaxy. To counter the strong gravitational potential of the dark matter with thermal pressure, the gas needs high temperatures of 1-100 million Kelvin. Gas this hot is best observed with X-ray telescopes.

Light from the cosmic microwave background (CMB) is coming at us from all directions. Since these photons started out ~13 billion light years away from us, when light and matter separated for the first time, they pass through plenty of galaxy clusters on their way to us. And when they hit the very hot electrons in the ionized intra-cluster gas, they get “upscattered” – that is to say, they get more energy, while the electrons cool down a tiny bit. This is called inverse Compton scattering; in regular Compton scattering, the photon would have been hotter than the electron, so the energy transfer would have gone the opposite way.

A photon Compton scatters against an electron, gaining some energy and sending the electron into a slight recoil.

The magnitude of cooling due to the SZ effect is:

$\frac{\Delta T}{T_{CMB}} = f(x)\int \sigma_T n_e \frac{kT_e}{m_ec^2}dl$

where x = $\frac{h\nu}{k_BT_{CMB}}$ describes how this cooling depends on the rest frame frequency $\nu$ that you observe. In words, it is the integral along the line of sight of the thermal pressure of the electrons normalised by the rest energy of electrons, all multiplied by the Thomson cross-section, which sets the scale of the photon-electron interaction. This quantity is called the SZ decrement.

Okay now are you ready for the coolest (IMHO) bit? Consider how the value of the SZ decrement varies with redshift/distance. Since the Universe was (1+z) times smaller in each dimension at a redshift z, the number density of a cluster with a given total number of electrons goes as:

$n_e = \frac{3 N_e}{4\pi r^3}$ $=\frac{3 N_e}{4\pi r_0^3} (1+z)^3$

Meanwhile, that line element dl that we’re integrating over can be written as:

$dl = \frac{dr^3}{d_A^2} = \frac{dx^3}{d_L^2}\frac{ (1+z)^2}{(1+z)^3}$

where, $d_A$ is the angular diameter distance and $d_L$ the luminosity distance. Following the same expansion argument as above, each of those dimensions was (1+z) times smaller, giving a total dependence of (1+z). So the intrinsic/rest frame SZ decrement increases with redshift as $(1+z)^4$ .

Now the energy density from a source at redshift z decreases as $(1+z)^3$ due to cosmological expansion, plus a $(1+z)$ fall-off in the frequency. This means that an object of a fixed luminosity L gets $(1+z)^{-4}$ times fainter with redshift.

The redshift dependences of the SZ decrement and fading due to distance exactly cancel out!

This means that given a mass of a cluster (and thus $N_e$ ), we can detect it with the SZ effect at every redshift ever. It’s a redshift-independent tracer of mass. It means that we can study the assembly history of massive objects in the Universe to as far out as we freaking want, as long as the temperature measurements are precise enough! And currently, they are enough to see anything upwards of about $10^{14}M_\odot$ .

#Sparknotes: AGN and Star Formation Feedback in Clusters

September 9, 2016uc24 Leave a Comment

Cooling, AGN feedback and star formation in cool-core clusters

Li et al, 2015

This is an adaptive-mesh simulation of an idealized, isolated, cool-core cluster modeled after the observed Perseus cluster. It studies the interplay between ICM cooling, AGN feedback and star formation.

Firstly, it finds that all three quantities are tied to the ratio of the cooling time to free fall time, t_cool/t_ff. This is nice because that’s the quantity I’m studying in my simulations. Also because it is physically meaningful – if the gas can cool significantly before it can fall into the centre of the cluster, it will form filaments of cold clouds that rain onto the central black hole and build a reservoir of cold gas near the cluster centre.

Some of the cold gas is accreted onto the central SMBH, triggering mechanical outflows (modeled here are semi-thermalized bipolar jets). I don’t see mention of radiative feedback. In addition, the cold gas is also used for star formation. The three quantities (rows one, three and four above) are visibly correlated.

The two columns indicate runs with AGN feedback efficiencies of 1% and 0.1%, respectively. This was the only parameter in their model that significantly affected the qualitative results of a simulation. When AGN heating was lowered, there was little to pause star formation or accretion onto the central black hole; as a result, the AGN never “shuts off”, which is inconsistent with observations.

Another key point to note is that in the third column, which is the SMBH mass accretion rate, there is a large scatter corresponding to short-scale fluctuations. The black line instead shows the accretion rate averaged over a running window of 200Myr, and is much smoother. The average mass accretion rate is also much more tightly correlated with the star formation rate, as shown in this plot to the right. This emphasises the point that observations of AGN activity and star formation in a single galaxy can have a huge scatter because they measure only an instant in time.

The star formation rates in the simulation, as well as its relation to the ratio t_cool/t_ff, are consistent with observations, too, as shown in the two plots above. The key takeaway is that large sample of AGN galaxies is required to make a reasonable statement about the effect of AGN on star formation rates.

In summary, when you look at a cluster with a BCG, ICM with self-gravity, radiative cooling, AGN feedback in the form of jets, and star formation and feedback, you match observed measurements of star formation, t_cool/t_ff and the ratios between the two. I would like to also see a prediction for the X-ray surface brightness profile, and in particular how upcoming X-ray missions like E-Rosita could distinguish between different modes of accretion and feedback. Lastly, my simulation is looking at a zoom-in cluster from a cosmological box, with plenty of additional turbulence from the movement and mergers of galaxies within it. Since turbulence even in this simulation created filaments 15kpc long, I’d really like to see what happens when there’s more of it.

#Sparknotes: Black Holes as Chaotic Eaters

September 1, 2016uc24 Leave a Comment

Chaotic cold accretion onto black holes

Gaspari, Ruszowski and Oh, 2013

This paper describes a set of very high resolution, idealized simulations of a supermassive black hole (SMBH) in the central (cD) galaxy of a massive cluster. Since the goal is essentially to challenge the current near-universal use of the Bondi accretion model in simulations, let’s start with the key equation:

Setup Start with a simple NFW dark matter halo, a de Vaucouleurs stellar distribution, gas that traces the gravitational potential and a SMBH, all with masses similar to the observed galaxy group NGC 5044. All of these are modelled in an adaptive mesh grid overlaid on a box of side 52kpc for a total time of 40 Myr. With upto 44 levels of refinement, the simulation resolves sub-parsec scales around the central black hole.

Mass accretion onto the black hole, normalized by Bondi prediction.

Varying physics models

The initial setup thus consists of an ideal gas contracting under gravity. This results in adiabatic, isotropic, smooth accretion of warm gas, identical to the analytic model of Bondi (1952). Indeed, the (numerically) observed accretion in this case exactly matches Bondi’s prediction, since the solid line in Fig 1b is essentially = 1 throughout. What is the dashed line, you ask? Well, that is the accretion rate you would measure if you evaluated the parameters in the Bondi equation as an average over a cluster-centric radius of 1-2 kpc, instead of at the Bondi radius (85pc in this case). In other words, computing gas density and sound speed as an average over large cells overestimates the accretion rate.

The simulations sequentially complicate the physics.

First they add cooling, which occurs due to atomic transitions in the ICM. Observed cluster ICMs tend to be quite enriched in metals, mostly due to ejecta from supernovae. They assume the metallicity of the cluster gas to equal that of the sun, which I thought was generous but is actually supported by Chandra observations (e.g. Vikhlinin et al 2005). The accretion is now boosted by over two orders of magnitude. Of course, the simulation doesn’t model star formation; if it did, a lot of this centrally accreted gas could actually be converted into stars, so we would observe very large star formation rates in over very short time scales in the central galaxies of clusters. We d not.

Next, they add turbulence by “stirring” the gas on large scales (lol 4kpc; this is just about the resolution of our cosmological simulation. It’s so relieving to see that someone is actually probing the smaller regions so that our sub-grid models aren’t full of hot air.)(I’m sorry I can’t help these things.) In reality turbulence can be induced by galaxy motions through the viscous ICM, galaxy-galaxy mergers, AGN and stellar feedback, etc. And this is where things start to look really different.

You now see a slice of the temperature profile of the gas. Adding the metal cooling, as mentioned above, decreases the temperature in the core by over 4 orders of magnitude, but retains the spherical symmetry and isotropy. Adding turbulence creates very cold filaments on very large scales. The accretion is on average as high as in the cooling-only scenario, but with more fluctuations.

Lastly, they consider global heating. In a real cluster, this could come from cosmic rays, AGN feedback, massive stellar feedback, etc. This suppresses the star formation somewhat from the previous case, but the “boost factor” with respect to the Bondi prediction is just under 100 by the end of the simulation. The filaments induced by turbulence are not broken up or significantly heated up.

In summary: accretion of gas onto the supermassive black holes at the centres of galaxy clusters is cold, chaotic and filamentary. Averaged over tens of megayears, the boost factor with respect to the Bondi model is just under 100, compared to the prevalent norm of 100-400.

#Sparknotes: AGN

August 15, 2016uc24 Leave a Comment

So excited about this fall! I get to implement a new subgrid model for accretion onto, and feedback from, active galactic nuclei (AGN) at the centres of clusters in a cosmological simulation we develop here at Yale.

Developing subgrid models (and numerical approximations in general) is an art as much as a science. On the one hand, you need to reproduce large-scale, observable phenomena, like star formation histories, stellar masses, stellar and metallicities, and dark and visible substructure. On the other, you want your input parameters to be motivated by plausible underlying physics.

Fig 1 from Urry and Padovani, 1995

AGN are a pesky beast. They consist of a supermassive black hole (SMBH) at the core, or nucleus, of a galaxy, surrounded by a hot disk of infalling matter. Since the material in the disk is unlikely to have falling in on a radial trajectory, it settles in a rotating disk around the black hole. Friction between layers of the disk heat it up to tens of millions of Kelvin, so that if you catch an AGN at the correct angle, it is bright in the X-ray. More often than not, due to geometric reasons, you’ll end up seeing dust-obscured AGN, which are bright in the radio. That’s a two-line summary of the Unified Model of AGN.

This obscuration, combined with the final parsec problem, is the main reason AGN are so pesky. Stuff falls onto black hole, aggravates it, isothermal heat ejections and mechanical jets arise! But what sort of stuff can fall into the black hole? Why does that cause it to spit stuff out? How exactly does it eject this energy? How far does the energy travel, and how does it interact with gas on its way (more specifically, the Intra-Cluster Medium or the ICM)?

There has been a sea of observations and theoretical models on this topic in the last few decades, and I’m just starting to dip my toes in it. Here’s a summary of the papers I’ll review in the next months.

How is energy transported within an accretion disk? How do the viscosity, density and temperature of the gas in the accretion disk determine whether energy transport is dominated by radiation, advection or convection? What does each of these processes look like? Advection-Dominated Accretion around Black Holes – Narayan, Mahadevan and Quataert, 1998
How exactly does the gas fall onto the black hole? How cold does it have to be? Does this depend on whether the gas is in filaments or clouds, and how those may be oriented? Growing supermassive black holes by chaotic accretion – Gaspari et al, 2013
The simulation I work with extracts clusters of galaxies from a cosmological box. This captures things idealized/isolated cluster simulations cannot, like smooth accretion of gas from filaments and mergers of clusters. Accretion during the merger of supermassive black holes – Armitage and Natarajan, 2002
Several self-regulating mechanisms have resulted in tight relations between galaxies and the supermassive black holes that live in their centres. Accreting supermassive black holes in the COSMOS field and the connection to their host galaxies – Bongiorno et al, 2012.
- Also, Self-regulated Growth of Supermassive Black Holes in Galaxies as the Origin of the Optical and X-Ray Luminosity Functions of Quasars – Wyithe and Loeb, 2003

Simulating the First Dwarf Galaxies and Globular Clusters

July 20, 2016uc24 Leave a Comment

A Common Origin for Globular Clusters and Ultra-faint Dwarfs in Simulations of the First Galaxies

Massimo Ricotti, Owen H. Parry and Nickolay Y. Gnedin

This paper presents the results of simulations of four cosmological boxes, each 1 Mpc/h a side, using the Adaptive Refinement Tree (ART) technique. The adaptive refinement scheme creates finer spatial and temporal resolutions in regions of high density, where the physics is more interesting. In the highest-resolution run, the authors resolve individual star particles as small at 40 $M_\odot$ , at sub-parsec sizes. The numerical simulation ends at z ~ 9, sufficient to make predictions about what the James Webb Space Telescope would see. Afterwards, an analytical prescription extrapolates what the galaxies found at z=9 would look like today, and comparisons are made to dwarf galaxies and globular clusters in the Local Group.

The physics implementation here is very neatly explained and physically motivated. Stars form whenever a gas cell meets certain criteria of metallicity, number density and molecular hydrogen fraction. There are two sets of prescriptions, corresponding to metallicity requirements for Pop III and Pop II stars. Unless the gas cell is of the minimum mass, i.e. 40 $M_\odot$ , it is converted into a stellar particle, which is understood as a population of stars following the Chabrier IMF. Feedback occurs in the form of supernova (SNe) explosions 3 Myr after star formation has occurred in a given cell – this timescale corresponds to the main sequence lifetime of an 8 $M_\odot$ star. This releases $10^{51}$ ergs of thermal energy into the neighbouring cells, on time scales that range from 0 (for Pop III hypernovae) to 35 Myr (for Pop II supernovae). The feedback also serves to enrich the gas with metals.

Finally, substructure is identified at every time step using the friend-of-friends (FoF) algorithm and refined using SubFind. The high-res simulation produced galaxies as small as 2.8 $\times 10^5 M_\odot$ , comparable to ultra-faint dwarfs today! Btw, I was elated that someone finally explained linking length, which is key to FoF! Particles are considered linked if their separation is less than

(linking length)*(mean separation between particles in the box)

The group catalogues are then used to construct merger trees.

At z = 9, they find that many galaxies have gas disks, but stars still form spheroids –

This makes sense if a spheroidal gas cloud was cool enough for star formation before dissipative processes turned it into a disk.

What do the orbits of the stars look like? Defining circularity as the angular momentum of a star particle in the direction of the mean angular momentum of the galaxy, divided by that of a particle of that mass moving on a circular orbit. The star particles in the simulated galaxies are consistent with non-rotating spheroids, i.e. symmetric distributions of circularity peaking at zero. In some galaxies, the mean circularity is positive, indicating that at least some stars are undergoing some rotation. No difference between metal rich and metal poor stars, divided at [Fe/H] = -1.5.

How big are the galaxies? Half-light radius $r_h$ computed at 100 different viewing angles, error bars represent range between 10th and 90th percentile of measured values for each. Unlike observations in Local Group, where $r_h$ scales with luminosity/stellar mass, in the simulation there is a large spread in $r_h$ at fixed stellar mass. That said, a lot can happen between z = 9 and the present day. If tidal stripping, for example, occurs at a rate inversely proportional to the density of the dwarf galaxy, more extended low-mass galaxies at high z would be smaller by z=0. Patience – this comes a couple sections later!

Of the ten most compact objects, 5 are DM dominated and 5 baryon-dominated. In fig 6, this is seen as a wide range in pseudo mass-to-light ratios at a fixed dynamical mass. There is, nevertheless, an apparently bimodal distribution – one where mass/light hovers around 10, and another around 10,000.

The money plot:

The three panels correspond to three different values adopted for star formation efficiency, ranging from 1-100%. To quote the authors, “luminosity of dwarfs increases by about a factor of two whenis increased by a factor of 100”; i.e. the qualitative results are depend very weakly on the assumed $\eta$ (which is excellent, since this quantity is still poorly constrained by observations)! The grayscale is the log of the stellar mass to dark matter mass.

Takeaway: the simulation self-consistently forms several objects with half-light radii ranging from 1-150 pc and stellar-to-dark matter ratios ranging from 1:1,000 to 10,000:1! The latter systems live in the top left corner of each plot, and are consistent with the mass-to-light ratios observed in globular clusters today. The more dark-matter dominated systems, on the other hand, would be the progenitors of dwarf galaxies.

[The plots in this paper are so damn nice. Like, they really know the physics point they’re trying to get across.]

Finally, the paper analytically calculates what these high-redshift compact objects would look like in the Local Universe, and compare it to observations.

Overall, it seems to me that the dwarf galaxies in the simulation don’t fit observations nearly as well as the globular clusters do – the simulation+analytic evolution produce dwarfs that are more compact, have smaller velocity dispersions, and with a smaller range of masses than observed. That said, the 13 billion years between the end of the simulation and the present day are very complex to model, and the fact that the predictions are so close to the observations is pretty impressive!

Ultimately the conclusion is that globular clusters and dwarf galaxies form through the same processes in the very early universe. How do you form very compact, baryon-dominated systems at high redshifts, though? I might have to re-read the paper to get this one.

Gravitational Waves are HERE!

February 12, 2016uc24 Leave a Comment

If everyone spent their time thinking about how black holes collide and send out bunches of energy as ripples in the fabric of spacetime that can change the shape of things in their way just enough that one billion light years away four kilometres of metal on Earth could get stretched by one thousandth the width of a proton, which means a packet of light (that is also a wave) traveling along that piece of metal takes just a little longer to get back to where it started and collides with another wave that was not expecting it to take that much longer and therefore produces a signal on a photographic plate that HUMANS CAN SEE
we’d have a lot less time for bigotry and war and other ways of being mean.

Also a bunch of LIGo data is publicly accessible at http://havewedetectedgravitationalwavesyet.com. Yes yes that is the official website name.

Good day for science!!

#Shareable: Einstein’s Riddle

January 12, 2016uc24 Leave a Comment

A lot of times you’ll have a problem with multiple unknowns and known correlations. In elliptical galaxies, for example, you know that:

Size is correlated with velocity dispersion and brightness.
But velocity dispersion is related to total mass
And baryonic mass, which is bright, follows dark mass..

So you try to figure out how stellar mass is correlated with total mass (velocity dispersion) on average in different environments so you can feed into into your sub-grid model in a large-scale simulation. It gets messy, but by making sure all your known constraints match up, your answer can be maximally likely under the circumstances.

#Shareable: The Hat Riddle

January 6, 2016uc24 Leave a Comment

Math at its finest is not about arithmetic, but about modelling patterns using some symbolic language. This colourfully visualized riddle from Ted-Ed is a fun example!

Reading List

January 4, 2016uc24 Leave a Comment

Papers

Shakura & Sunyaev 1973. Black holes in binary systems: Observational appearance.

McCulloch & Pitts 1970. A Logical Calculus of Ideas Immanent in Nervous Activity.

Books

Frank Wilczek 2015. A Beautiful Question: Finding Nature’s deep design

Cedric Villani 2015. Birth of a Theorem: A mathematical adventure

Jim Al-Khalili 2012. Paradox: The nine greatest enigmas in physics

Richard Feynman 1985. Surely you’re joking, Mr Feynman!

Buckminster Fuller 1969. Utopia or Oblivion: The prospects for humanity

#Toolbox: Identifying Cluster Members

July 15, 2015uc24 Leave a Comment

When trying to identify structure in the sky, astronomers are faced with the problem of distance; we see the sky as two-dimensional, and can’t tell between a faint object nearby and a bright one further away. Sure, there are a few objects whose special properties make their distance uniquely calculable – Type Ia Supernovae, for example, always have the same intrinsic luminosity, and Cepheid Variable stars pulsate slower if they are intrinsically brighter. These are called Standard Candles, and they are very rare. Moreover, they cannot tell us whether objects that appear close to them in the sky are in fact nearby in space.

The most accurate solution, of course, lies in spectroscopy. Take light from any source, disperse it through a prism, note absorption lines, see how much they are offset from the same absorption lines in light in a laboratory on Earth, and plug that shift into the redshift equation:

The catch, as usual, is that spectroscopy is expensive. The signal-to-noise of a spectrometer decreases with increasing spectral resolution, and you want high resolution to correctly identify spectral lines. If you’re looking at a galaxy with millions of (resolved) stars, or a cluster with hundreds of galaxies, you can only hope to have spectroscopy for a handful of the members.

HR Diagram. Image Courtesy Chandra team.

Sample Hertzsprung-Russell diagram, courtesy Chandra team.

Thankfully, we know a couple things about the structures we are interested in. In the case of galaxies, we know that if you were to plot the luminosity against the temperature – and temperature is related to colour, thanks to Wien’s law – we always end up with a scatterplot that looks more or less the same. The pattern is called a Hertzsprung-Russell (HR) diagram.

This works because star populations within a galaxy evolve together. So if you plot the magnitudes and colours of everything in a region of sky, those belonging to the same galaxy with form an HR diagram. You decide how close a given star needs to be to the most clear lines to be considered part of the galaxy, and chuck out the rest. Sounds shoddy, but it works.

The galaxy CMD features are shown here for nine clusters. Image from Li et al, ApJ 749 150.

You can plot a similar colour-magnitude diagram for a set of galaxies. That ends up giving you a red sequence of old, quiescent, ellipticals, a green valley of quiescent spirals and a blue cloud of star-forming spirals. The red sequence, as it turns out, will consist mostly of galaxies within the same cluster because, like stars in a single galaxy, these guys have lived in a similar environment and have similar star-formation and merger histories that leave them with similar shapes and colours. Again, there will be some scatter in the relation. How much scatter you think is acceptable.. is pretty arbitrary. And there could still be outliers.

The way to check for contaminants from one method is to approach the same problem from another angle. And a pretty cool other angle out there merges the principles of spectroscopy and CMD selection, and that is photometric redshift.

Photo-z takes in photometry of your object in multiple bands; more is usually better, but the improvement is usually limited after 7-10 bands. It compares these points to various Spectral Energy Distribution (SED) templates, each corresponding to a model of galaxy evolution. Based on the agreement between the observed and template SED, each template – and the redshift corresponding to it – is given a probability. The output of a photo-z code is then a probability distribution function (PDF) of redshifts for the galaxy based on its input colours. You can choose to deal with either the best-fit redshift value, or the marginalized value, which is the sum of the products of each possible redshift and the probability of the galaxy being at that redshift.

The key value-add over spectroscopy is that the spectral resolution is very low – you measure an entire band of wavelengths in a single filter – and the redshift estimate comes from fitting a few points on the SED to those of a model. There are many, many ways of coming up with the template SEDs, of deciding which one(s) best describe the observed photometry, modeling errors, etc. I’ll share notes on EAZY, which I’ve been working with, in a future post.