Playing with Positive Feedback: External Pressure-triggering of a Starforming Disk Galaxy

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The Formation of Galaxies 

Galaxy formation and evolution is a highly non-linear process and poses significant chal-lenges to our current understanding of the Universe. The current paradigm of galaxy formation in a cosmological context is that galaxies form inside DM halos in a cold DM (CDM) Universe (White & Rees, 1978; Blumenthal et al., 1984) with a cosmological con-stant leading to a CDM model (Riess et al., 1998). DM comprises about 84% of the matter content of the Universe, is gravitationally dominant on large scales, and is, there-fore, crucial to the understanding of structure formation. The realisation that we live in a DM dominated Universe led to the development of the first comprehensive theory of galaxy formation (e.g., White & Rees, 1978; Fall & Efstathiou, 1980). These analytical models embedded simple gas physics within the hierarchical growth of structure forma-tion, whereby the DM halos evolve within the cosmic web and merged over time forming successively more massive halos (White & Rees, 1978).
Within these models the luminous matter composed of baryons (i.e., gas and stars) follows the DM into the halos, where they form gaseous halos and eventually cool to form cold, dense gas discs at their centre. Further cooling and fragmentation leads to the formation of dense molecular clouds and stars within.
The understanding of structure formation is, therefore, tightly connected with the for-mation and evolution of DM halos. Relevant for the theory of galaxy formation within a hierarchical cosmological growth are (i) the halo mass function of DM halos at a given time and (ii) the formation history of DM halos.
There are two ways to generate information about the abundance and evolution of DM halos: through Monte Carlo trees making use of the Press-Schechter (1974) theory and its extension to give progenitor distributions (Bond et al., 1991; White & Frenk, 1991; Lacey et al., 1993; Cole et al., 2000a; Parkinson et al., 2008) or through N-body simulations (Springel et al., 2005b; Diemand et al., 2008). The later has the advantage to track the non-linear collapse of structures. It has been shown that even with the simple underlying assumptions made in the Press-Schechter approach, their predictions agree remarkably well with N-body simulations (see Section 2.1.2 for more detail).
Despite the successes of the N-body simulations in understanding the scale of observed galaxy masses, it was soon realised that there were a number of problems. One particular issue is between the mass function of DM halos, provided by the N-body simulations, and the luminosity function of galaxies given by observations. By assuming that stellar mass follows halo mass we are left with a theoretical prediction that leads to too many small galaxies, too many big galaxies in the nearby Universe, and too many baryons within the galaxy halos.
To alleviate the problem of the excess baryons to form stars within galaxies, efficient outflows and feedback were introduced. Feedback both heats the gas in the halo, preventing it from cooling, and ejects gas from the galaxy and the halo, lowering the total gas content available to form stars.
While in low-mass galaxies, supernovae (SNe) and stellar winds can deplete cold-gas reservoirs and regulate star formation (e.g., Dekel & Silk, 1986), they are ineffective in the deeper gravitational potential well of massive galaxies. A more effective interaction for the high-mass galaxies may be provided by active galactic nuclei (AGN) feedback from supermassive black holes (SMBH) which are thought to be ubiquitous at the center of local galaxies (e.g., Magorrian et al., 1998; Hu, 2008; Kormendy et al., 2011). Rapid gas accretion onto BHs lead to energy release capable of driving high-velocity outflows (see e.g., Blandford & Payne, 1982; Begelman, 1985; Silk, 2005; Tortora et al., 2009), that are thought to have a high impact on the host galaxy by expelling and/or heating gas, and with this regulating the baryonic content and star formation of the galaxies (Silk & Rees, 1998).
Probably the strongest evidence supporting the existence of AGN feedback is provided by observations of X-ray lobes and radio cavities in galaxy groups and clusters (e.g., Mc-Namara & Nulsen, 2007, 2012). These have been interpreted as buoyantly rising bubbles of higher entropy material injected by an AGN. The inclusion of AGN feedback in numer-ical hydrodynamical simulations, but also semi-analytical models, lead, furthermore, to a better reproduction of the high-end tail of the galaxy mass function (e.g., Di Matteo et al., 2005; Croton et al., 2006; Bower et al., 2006; Sijacki et al., 2007; Di Matteo et al., 2008; Booth & Schaye, 2009; Dubois et al., 2010a), that additionally motivated the inclusion of AGN feedback in the galaxy formation theory.
However, we do not yet fully understand the physical mechanisms of the coupling between the AGN and the interstellar medium (ISM) of the host galaxy. The lack of understanding about the communication of the AGN feedback and the galaxy is partly due to the inability to capture the extremely wide dynamical range of the AGN cycle: from sub-pc scales where accretion discs form to galactic-size lobes of radio galaxies. Another complication arises because, to date, most of these studies have treated the ISM with relatively simple ‘sub-grid’ prescriptions that ignore the additional complications introduced by the multi-phase small-scale structure.
Interestingly, galaxy scale simulations show that properties of the ISM even determine whether AGN feedback is capable of driving out a large amount of gas out of the galaxy or whether it may actually foster star formation (e.g., Gaibler et al., 2012; Wagner & Bicknell, 2011; Wagner et al., 2012). The findings that AGN feedback may not only result in suppressed star formation but also may trigger the formation of stars (positive feedback) is also supported by recent observations (Cresci et al., 2015b,a; Salomé et al., 2015; Zinn et al., 2013).
This all motivates a deeper investigation of the precise interactions between AGN out-flows and the ISM, how feedback impacts the host galaxy, and the communication mech-anism of the AGN with the galaxy’s gas.
This Chapter will explain in more detail the relevant theories that lead to the comparison between the halo mass function and the galaxy luminosity function, which motivates the need for feedback processes as a regulator for the mass content in low and high mass galaxies. It will focus on AGN feedback and review observational evidence and numerical predictions for both negative and positive feedback from AGN.
Section 2.1 describes how the abundance of DM halos can be calculated using Press-Schechter theory. Section 2.2 explains the findings of N-body simulations. Section 2.3 discusses the inclusion of baryonic physics in the models, with Section 2.3.1 focusing on the physics of cooling and star formation within the ISM and Section 2.3.2 explaining the effect of feedback on the baryonic content of the galaxies. Section 2.3.3 discusses comparisons of hydrodynamical simulations with observations, mainly focusing on the stellar mass function. Finally, Section 2.4 focuses in more detail on AGN feedback and whether it only negatively or positively impacts the SFR within the galaxy.

Dark Matter Statistics

The abundance of DM halos is an important factor for theories of structure formation. To first order, the abundance of a halo of mass M is a function of mass only. The Press-Schechter theory (Press & Schechter, 1974), which uses the spherical top-hat collapse model and linear growth theory, gives an intuitive and useful analytic description of this mass function. This section will briefly review the spherical top-hat collapse model and ex-plain the halo mass function that later will become important when comparing theoretical predictions with observations. It will summarise how the Press-Schechter theory can be extended to get some information about the evolution of the DM halos.

Spherical Top-Hat Collapse Model

The spherical top-hat collapse model is a simple way to develop an understanding of the formation of structures from the evolution of non-linear perturbations within the density field. It assumes a spherical symmetric region with radius R with a uniform overdensity at initial time t in an Einstein de Sitter Universe with = 0 (see Chapter 1). It does not make any assumption on the dynamics of the system.
Using the equations above, especially equation 2.8, a few key events in the history of the density perturbations can be quantified. At the beginning, overdense regions grow with the Hubble expansion. At a specific time, this expansion stops under its own self-gravity, reaching a maximum expansion at = . Substituting this solution into equation 2.8 reveals that this turnaround occurs when the linear density contrast is turnaround 1:06. Hence, shortly after a perturbation’s overdensity exceeds unity, it turns around and begins to contract. The final collapse occurs when = 2 . The overdensity at this point is collapse 1:69.
It is at the turnaround point where the linear spherical collapse model starts to be-come invalid. Clearly, the assumption of a perfectly spherically symmetric, pressureless overdensity is ideal and the collapse is not expected to continue to a singularity. In fact, the overdense regions virialise rather than collapse to a point. Because of the collisionless nature of dark matter, it cannot dissipate its energy and the collapse of the DM halts. The DM particles obtain virial equilibrium through dynamical friction which results in a pressure-supported virialised collapsed structure, the dark matter halo.
The average density within the virialised object is usually estimated by assuming a virialised radius of half of the turnaround radius. The density of the spherical region will, therefore, increase by a factor of eight until virilisation whereas the density of the non virialised region increases only by a factor of four. This occurs when the density reaches This number is important for defining a bound object in N-body simulations. Typically an overdensity of vir = 200 is assumed.

The Halo Mass Function

The so-called Press-Schechter (Press & Schechter, 1974) theory provides an analytic for-malism for the process of structure formation once the overdensities have collapsed into a halo. It relies entirely on the linear theory and provides insight into the evolution of the halo mass function, discussed below, and succeeds in describing the hierarchical clustering seen in N-body simulations discussed in Section 3.1.
The idea of the Press-Schechter theory is that a halo forms at peaks of matter density fluctuations in the early Universe. Those fluctuations are described by spheres of mass M and the initial density distribution is assumed to be Gaussian. It therefore makes the assumption that the probability distribution function of relative overdensity can be described by a Gaussian function with zero mean
where is defined as in equation 2.6 and is the overdensity associated with M. Finally, (M) is the standard deviation.
Further, it assumes that within a sphere of radius R is monotonically decreasing as a function of mass. Large scales, therefore, correspond to a smaller standard deviation and hence are smaller in amplitude than those on small scales. This results in a bottom up picture of structure formation as small structures are believed to form before large structures.
The Press-Schechter theory assumes the linear theory of the spherical collapse model (described above in Section 2.1.1) to be valid until the density reaches the threshold of c and collapses immediately afterwards, where c is derived above It is assumed that at any given time, all regions that have a density over c collapse and form halos. This approximation can somehow be justified by the fact that gravitational instability operates very quickly.
The fraction of a sphere of radius R that exceeds a threshold at a given time t is given by with erfc(x) being the complementary error function that describes the probability that the error of a measurement drawn from a standard Gaussian distribution lies outside the region c and c. It is defined as the integral of the Gaussian function.
By, mistakenly, assuming that this fraction can be identified with the fraction of parti-cles which are part of collapsed lumps with masses greater than M we run into a problem. For small masses M, the mean square function goes to infinity and, therefore, the fraction of points, f, converges to 1=2. Hence, the formula above predicts that only half of the particles are part of lumps of any mass. In other words, the negative part of the Gaus-sian distribution has been left out as it corresponds to underdense regions. The so-called ‘swindle’ in the Press-Schechter approach is to multiply the mass fraction by an arbitrary factor of 2.
Generally, the fraction of mass that is in halos between mass M and M + dM is given by then differentiating the function (Eq. 2.12) with respect to M. To get the number density of collapsed objects, the fraction of mass in halos in the range M ! M + dM is multiplied by the number density of all halos m=M assuming that all of the mass is composed of such halos. The number density then becomes where c(t) = c=D(t) is the critical overdensity linearly extrapolated to the present time.
Note that, here, equation 2.12 is here multiplied by a factor of 2 as discussed above.
The normal procedure to find the number density is by calculating and its derivative from the linear theory matter power spectrum.
Despite the simple underlying assumptions made above, the Press-Schechter formula for dn=d ln M agrees remarkably well with N-body simulations. The formula is known to deviate in detail at both high and low mass ends as it tends to systematically under-predict large-mass halos and overestimates the abundance of small-mass halos in comparison with simulations (see Hu & Kravtsov, 2003 for a study of the relative contributions of each source of error as a function of halo mass).
Refinements to this theory have since been made. In particular, a better fit to the mass function in N-body simulations has been proposed by assuming that the halos are elliptical instead of spherical (e.g., Lee & Shandarin, 1998; Sheth et al., 2001). Sheth et al. (2001) were able to show that this replacement plausibly leads to a mass function almost identical to that which Sheth & Tormen (1999) had earlier fitted to a subset of numerical data. Similar results have been found by Jenkins et al. (2001).
The Press-Schechter theory can be extended to allow predictions for merger histories and merger rates of DM halos (e.g., Bower, 1991). Bond et al. (1991) proposed an alternative to the Press-Schechter theory based on excursion sets to statistically estimate how many small halos would be subsumed into larger ones. This approach reduced the number of low-mass halos compared to the Press-Schechter theory and increased the number of large-mass halos and is therefore in better agreement with simulations.

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N-body Simulations

The evolution of cosmic structures beyond the linear regime can only be addressed in limiting cases, notably the large-scale limit, with a simple extension of the linear theory to higher order and under a prohibitive amount of simplifying assumptions.
To better understand the non-linear evolution of the DM density field on a long timescale, as well as to get an understanding of the distribution of the DM halos within the Universe, the help of numerical simulations is needed. Numerical simulations are extremely important in the study of structure formation in a cosmological context as they provide accurate numerical predictions to be tested against theories and observations.
Gravity is the dominant force that drives structure formation. Unless an accurate de-scription of the very early Universe, or the behaviour of objects near a massive black hole is needed, gravity can be well described by the Newtonian theory. DM is thought to be made of collisionless particles that interact only through gravity and can be well represented by a set of point particles, N-bodies (see Section 3.1 for a more detailed discussion).
N-body simulations follow the evolution of the DM component from the very early stages, where small density fluctuations perturb an otherwise homogeneous distribution, to the current time with its highly non-linear structures. Hence, the features introduced by the specific cosmological model are imprinted on the initial conditions of the simulation, i.e. on the primordial perturbation field. Usually, the initial conditions are derived analytically and start a certain time after the Big Bang (e.g., Zaldarriaga & Seljak, 2000; Lewis & Bridle, 2002; Lewis, 2013). This can be done because about the first 100 million years (Myr) of the evolution of the Universe after the Big Bang are still within the quasi-linear regime and, hence, can be derived analytically.
Including a statistical distribution of initial positions and velocities of the particles, the system is evolved forward in time according to the gravitational forces acting on the DM density field within an expanding Universe.
Such N-body simulations of large-scale structures have met great success. There are various simulations (e.g., Springel et al., 2005b; Diemand et al., 2008; Klypin et al., 2011) that successfully simulated the evolution of the DM content of the Universe and reproduce the observed luminous cosmic web (de Lapparent et al., 1986). They all reveal a picture where over the course of time the DM collapses into large walls, filaments, and haloes building up the cosmic web structure that is surrounded by immense voids, with densities as low as one tenth of the cosmological mean.
A visual representation of an evolution of such a cosmological simulation (here the Mil-lenium simulation performed by Springel et al., 2005b) is shown in Figure 2.1, where it visualises that as the simulation proceeds, the matter distribution evolves from a state of homogeneity to a more and more pronounced filamentary structure. A comparison of the N-body simulations with the observations such as the Sloan Digital Sky Survey (SDSS) reveals that the general structure of the cosmic web matches the cosmic structure revealed by deep galaxy studies, shown in Figure 2.2, where the galaxy distribution predicted by the N-body (Bolshoi) simulation of Klypin et al. (2011) is shown in comparison of the galaxy structure observed by SDSS.
Such a close match between the predictions of the N-body simulations and the observa-tions, provides evidence that the underlying theoretical assumptions of structure formation, coming from the standard CDM model, provide, at least on large scales, a fairly accu-rate picture of how the Universe actually evolved. However, while the simulations appear to agree broadly with observations, the details in the highly non-linear regime exhibit important differences.

N-body Simulations

Comparing N-body Simulations with Observations

First of all, in order to compare the statistical properties of the N-body simulations with observations, the number of galaxies have to be counted within the simulation. By as-suming that each DM halo hosts a galaxy of proportional mass, the problem is reduced to finding the number of halos within the N-body simulations. The essence of finding the halos is to convert a discrete representation of a continuous density field into a countable set of ’objects’, the halos. In general, this conversion is affected both by the degree of discreteness in the realisation (i.e., the particle mass within the simulation) as well as by the detailed characteristics of the object definition algorithm. Additionally, high resolution simulations, such as e.g., Springel et al., 2005b; Diemand et al., 2008; Klypin et al., 2011, reveal that within the virialised region of a large halo there can also be various smaller, bound ‘sub-halos’. The distinction between the larger halo and its substructures compli-cates comparisons between theory, simulation, and observation. It is beyond the subject of this thesis to go into detail of the different ways halos can be defined and the reader may be referred to Hoffmann et al., 2014; Lee et al., 2014; Behroozi et al., 2015 for a detailed discussion.
Having found the number of DM halos, and by assuming a constant DM mass-to-light ratio, the theoretical predictions can be compared with observations. Such comparisons are however not without complications. For instance, attempts to reproduce the observed number of Milky-Way satellites using N-body simulations, and by assuming that each simulated DM satellite halo contains a galaxy, overproduced the number of low mass satellites by several orders of magnitude (Moore et al., 1999). Additionally, Moore et al. (1999) found that simulations with cold DM fails to reproduce the rotation curves of DM dominated galaxies, one of the key problems that it was designed to resolve. While it is almost certain that the existing sample of Milky-Way satellite galaxies is incomplete (Koposov et al., 2008; Tollerud et al., 2008) and new satellite galaxies are still discovered (e.g., Belokurov et al., 2009, 2010), it appears unlikely that new observations will uncover as many galaxies as DM simulations predict to exist around the Milky-Way.
For another comparison, one can compare the mass function of DM halos with the luminosity function of galaxies obtained from wide redshift surveys.

Table of contents :

1 Introduction 
2 The Formation of Galaxies 
2.1 Dark Matter Statistics
2.1.1 Spherical Top-Hat Collapse Model
2.1.2 The Halo Mass Function
2.2 N-body Simulations
2.2.1 Comparing N-body Simulations with Observations
2.3 Including the Baryons into the Picture
2.3.1 Physics of the Interstellar Medium
2.3.1.1 Radiative Cooling and Heating
2.3.1.2 Star Formation
2.3.2 Feedback Processes
2.3.3 Comparing Hydrodynamical Simulations with Observations
2.4 Active Galactic Nuclei
2.4.1 Classification of AGN
2.4.2 Correlating the AGN power to the Accretion Rate and AGN mode
2.4.3 Radio Galaxies and Jets
2.4.4 Quasars
2.4.5 Positive or Negative Feedback
2.4.5.1 Observations
2.4.5.2 Theoretical Work
3 Numerical Modeling of Galaxies 
3.1 Collisionless N-body systems
3.1.1 Particle-Mesh Method
3.2 Collisional Systems
3.2.1 Deriving the Fluid Equations
3.2.2 Different Approaches to solve the Fluid Equations
3.2.3 Discretising the Fluid to solve it on a Grid
3.3 ramses: A numerical N-body and HD code using adaptive mesh refinement (AMR)
3.3.1 Adaptive Mesh Refinement structure
3.3.2 Time-stepping Scheme
3.4 Sub-grid physics to study galaxy formation and evolution
3.4.1 Radiative Cooling and Heating
3.4.2 Polytropic Equation of State
3.4.3 Star Formation
3.4.4 Supernova Feedback
3.4.5 Black Hole Feedback
3.5 Radiation-Hydrodynamics (RHD)
3.5.1 The Radiation-Hydrodynamics equations
3.5.2 Moments of the RT equation
3.5.3 The RHD equations
3.5.4 Closing the Moment Equations
3.6 Ramses-RT: An RHD extension to ramses to model propagation of photons
4 External pressure-triggering of star formation 
4.1 Jet Propagation
4.2 Playing with Positive Feedback: External Pressure-triggering of a Starforming Disk Galaxy
4.2.1 Introduction
4.2.2 Simulation Set-up
4.2.3 Results
4.2.3.1 Disc fragmentation and star formation history
4.2.3.2 Mass Flow Rate
4.2.4 Conclusions
4.3 External pressure-triggering of star formation in a disc galaxy: a template for positive feedback
4.3.1 Introduction
4.3.2 Simulation Set-up
4.3.2.1 Basic simulation scheme
4.3.2.2 Application of external pressure
4.3.3 Results
4.3.3.1 Qualitative differences
4.3.3.2 Disc fragmentation
4.3.3.3 Star formation history
4.3.3.4 Clump properties
4.3.3.5 The galaxy’s mass budget
4.3.3.6 The star formation rate
4.3.3.7 The Kennicutt-Schmidt relation
4.3.4 Conclusions
4.3.5 Appendix
4.3.5.1 Bipolar pressure increase
4.3.5.2 Effects of supernova feedback
4.3.5.3 Convergence Studies
5 Feedback from Radiatively-driven AGN Winds 
5.1 Setting up the initial two-phase density distribution
5.2 Outflows Driven by Quasars in High-Redshift Galaxies with Radiation Hydrodynamics
5.2.1 Introduction
5.2.2 Methods
5.2.2.1 Initial Gas Density Distribution
5.2.3 Radiation Hydrodynamics
5.2.3.1 Modeling the Quasar
5.2.4 Results
5.2.4.1 Effects of Different Cloud Sizes
5.2.4.2 Qualitative Effects of Cloud Sizes
5.2.4.3 Effects of Different Photon Groups on the Cloud Evolution
5.2.4.4 Efficiency of the Photon-Gas Coupling
5.2.5 Evolution of the Optical Depth
5.2.5.1 Effects of the Quasar Position
5.2.5.2 Comparison between Different Luminosities
5.2.6 Discussion
5.2.7 Conclusions
5.2.8 Appendix
6 Conclusions and Perspectives
6.1 Conclusions
6.2 Future Prospects

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