Formalism for the Rapid Evolution of Primordial

Formalism for the Rapid Evolution of Primordial Black Holes

by Megan Russell

Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2017

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JUN 2 2 2017 LIBRARIES ARCHIVES Formalism for the Rapid Evolution of Primordial Black Holes by Megan Russell

Submitted to the Department of Physics on May 12, 2017, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics

Abstract This thesis presents two different formalisms which can be used together to evolve a density perturbation produced during inflation forwards in time in the radiation- dominated era of the early universe. The first formalism, based on early work by Misner and Sharp, is for the evolution of a perfect fluid in spherical symmetry under its own gravity. This allows us to efficiently determine if a black hole will form, but due to the formation of a singularity, it does not allow us to extract the black hole's mass. The second, new, formalism allows us to evolve the density perturbation past the formation of a singularity until a time where we can reasonably extract the mass of the black hole. This new formalism has been designed to do this computation as rapidly as possible, improving over our previous methods. The limits of this new formalism are investigated, details regarding a numerical implementation of our code are discussed, and a proof of concept example is presented.

Thesis Supervisor: Jolyon Bloomfield Title: Lecturer

2 Acknowledgments

First and foremost I would like to thank my thesis advisor, Jolyon Bloomfield. With- out his dedicated guidance and patient help over the last year, this thesis would never have happened. I would also like to thank the members of the Density Perturbation

Group, particularly Professor Alan Guth for his support and sharing his expertise, and Stephen Face for his help understanding our simulation. I am also grateful to MIT's UROP office and Professor Guth for providing me with funding that allowed me to do this research. I'd also like to thank my family for always encouraging me and listening to me when I start to talk about strange physics. Lastly, I'd like to thank my friends Mark,

Danielle and Jackie. For some reason they've stuck by my side for the last decade, and I can't imagine what my life would be like without them.

3 Contents

1 Introduction 7

1.1 M otivations ...... 7

1.2 Previous work ...... 9 1.3 N ew G oals ...... 10

2 Misner-Sharp Formalism 12

2.1 Defining a Metric and Stress Tensor ...... 12

2.2 Deriving the Equations of Motion ...... 14

2.3 Boundary and Initial Conditions ...... 17

2.4 Cosmological Equations of motion ...... 18 2.4.1 A FRW Background ...... 18

2.4.2 Rewriting the Equations of Motion for Numerical Efficiency 20

2.4.3 Outer boundary conditions ...... 22 2.4.4 Black hole formation ...... 23

3 A General Formalism 26

3.1 Defining the coordinates ...... 27

3.2 Equations of Motion ...... 27 3.2.1 Toy M odel ...... 27

3.2.2 Closing the set of equations with 5 ...... 28

3.2.3 New Equations of Motion ...... 29

3.2.4 Solving for 3 and U independently ...... 29

3.2.5 The general cosmological equations of motion ...... 31

4 3.3 Potential problems ...... 3 31

4 Implementation 34

4.1 Picking a convenient time coordinate ...... 34 4.2 Initial Conditions ...... 37

4.3 Numerical Evolution ...... 38 4.4 Initial Results ...... 39

4.5 Mass Extraction...... 43

5 Conclusions 45

5 List of Figures

1-1 Primordial Black Holes as a Dark Matter candidate ...... 8

2-1 Location of singularity, apparent horizon and event horizon in a black hole-forming spacetime ...... 24

4-1 Properties of f ...... 36 4-2 Example of new coordinates ...... 36 4-3 Derivatives of f with respect to r and r ...... 38 4-4 versus A for Misner Sharp and new coordinates ...... 40 4-5 Evolution of p ...... 41 4-6 Evolution of m ...... 42 4-7 Speed of Characteristic Wave ...... 43

6 Chapter 1

Introduction

The goal of this thesis is to build a formalism which we can use to evolve primordial density fluctuations, determine if they collapse to form a black hole, and if so, extract the mass of this black hole. Additionally, we aim to do all of this as computationally efficiently as possible, as ultimately, this formalism will be used to evolve millions of primordial black holes.

1.1 Motivations

The underlying motivation for this project is that theories of inflation produce a spectrum of perturbations in the density of the mostly homogeneous and isotropic plasma in the very early universe 1101. For a given model of inflation, there is a spectrum of density perturbations which are produced, and some of these perturbations may collapse into black holes. This idea was first proposed by Zel'dovic and Novikov in the 1960s 1191. Ultimately, we would like to be able to take a particular theory of inflation, and from that determine the number density and mass spectrum of black holes which are produced.

There are three broad ranges of black hole masses that are of particular interest.

How the super-massive black holes at the centers of galaxies were formed is an open question. They have masses on the order of millions to billions of sollar masses, and it is difficult to imagine they could have been formed by the normal process of stellar

7 10 -

10-2 MM" Capture by NSs in GCs, this work 104 Star Formation wking 10-6

10-8 1015 10l UPa10M 1000 SH mess, g

Figure 1-1: PrimordialBlack Holes as a Dark Matter candidate (figure from [7]) This shows the remaining phase space allowable for a given fraction of dark matter being black holes of a given mass. The only un-excluded region where all of dark matter is black holes is now limited to black holes masses near ~ 10 2 6g. collapse and mergers. An alternative formation mechanism is that the progenitors of these super-massive black holes were primordial black holes 18]. On the small end, primordial black holes on order of a lunar mass are a potential dark matter candidate 161. The remaining phase space for dark matter which is entirely low mass black holes has become increasingly narrow, as seen in Figure 1-1 171. Lastly, black holes of a more moderate mass are also of interest, as some suspect that the 10-30 solar mass black holes discovered by LIGO could have been formed primordially [12]. The process of going from a theory of inflation to a description of both the number density and mass spectrum of black holes it produces can be broken down into a few steps. The first step is to take a specific theory of inflation and calculate the power spectrum of density perturbations it creates. Roughly speaking, density profiles can then be drawn weighted by their probability, given by the power spectrum. Bardeen, Bond, Kaiser and Szalay [1 and Bloomfield, Face, Guth, Kalia, Lam and Moss 151 have done these probability calculations for two different field configurations relevant to various inflationary theories. The last step is to take millions of these density perturbations and numerically evolve them to determine which will collapse to produce

8 black holes, and what the mass of these black holes will be. A statistical analysis then yields the desired quantities. Our goal in this thesis is to address the numerical evolution step in this procedure.

1.2 Previous work

Our first tool in constructing a method of evolving black holes is the Misner Sharp formalism 1151. This formalism was originally formulated to model stellar collapse in a vacuum in 1964, and it assumes that our matter is both spherically symmetric and a perfect fluid. The Misner Sharp formalism is a good first step in our ability to numerically model the evolution of primordial fluctuations, and is enough to tell us if a black hole forms or not. However it has a major problem: it allows the singularity to form at the center of the black hole, causing numerical code to crash as the black hole forms. We want to extract the black hole's mass, and therefore need to evolve the black hole further in time than just the early moments of its formation. In particular, when a black hole forms, there is typically mass nearby that will fall into the black hole shortly thereafter. To obtain an accurate mass estimate we must evolve the black hole until this has happened. To do this, we could move to a coordinate system with null time slicing as originally formulated by Hernandez and Misner 1111. This null time slicing is a physically intuitive way of formulating a coordinate system where the singularity of the black hole never forms. In this system, the time of an event is said to occur at the time that light emitted from the event reaches an observer at a distant radius. Therefore it takes infinite time for the black hole's horizon to form, while the outer regions of the black hole can continue to evolve. Once we allow our black hole to settle, and any mass that is going to fall in has been allowed to fall in, we can extract an accurate estimate of our black hole's mass. Previous work in this area has focused on obtaining a heuristic for what magnitude of an overdensity would allow primordial black holes to form. Niemeyer and Jedamzik in 1999 1161 were the first to try and obtain this heuristic using numerical methods.

9 They used Misner Sharp for initial evolution and Hernandez Misner to reach a point where the mass could be extracted. Musco et al. [171 corrected the initial conditions, and repeated Niemeyer and Jedamzik's simulation. Bloomfield, Bulhosa, and Face

141 continued to build on the previous work, developing a cosmological formalism for Misner Sharp and Hernandez Misner.

Modern evolutions of black holes typically utilize more complicated formalisms than Misner Sharp and Hernandez Misner. They take full advantage of the much more powerful computers which have become available in the last sixty years, and do their evolutions in three dimensional space. These techniques are more precise, and work well if you only need to evolve one black hole and have a month of time on a supercomputer to do it. As an example, in Lehner and Pretorius' Numerical

Relativity review [14], numerical simulations of the last orbits of an in-spiraling black hole binary are cited as having taken tens to hundreds of thousands of CPU hours.

We are interested in evolving enough black holes for a Monte Carlo simulation, meaning we will need to evolve on the order of millions of black holes to obtain an accurate number density and mass spectrum. Therefore we will stick to evolutions in

1 + 1 dimensions, which allows for much more rapid evolutions than simulations in more dimensions. Even if the techniques are older, they are much faster and therefore much better suited for our needs.

1.3 New Goals

The major issue with the previous technique of using Misner Sharp and then Hernan- dez Misner is that this is still not numerically efficient enough for our purposes. The

Misner Sharp formalism is very efficient for evolving primordial fluctuations, and a modern laptop takes approximately a second to evolve a perturbation to the point where a black hole either forms or it becomes obvious one will not form. Unfortunately the Hernandez Misner formalism is not as fast. Although Hernandez Misner does a good job preventing the singularity from forming, it also slows down the evolution of the outer regions of the black hole. This means that we need to take many time

10 steps to reach a point where the black hole is stable enough to extract a final mass. The goal of this thesis is to formulate a new coordinate system where the formation of the singularity is suppressed, but the outer regions of the black hole can complete their evolution relatively rapidly. In Chapter 2, we discuss the Misner Sharp formalism in detail, derive the equations of motion, and then modify these equations of motion to be better adapted for our cosmological background. This chapter will also discuss how to determine if a black hole has formed. Chapter 3 formulates a new coordinate system to evolve forming black holes in a numerically efficient way and investigates the limits of the equations of motion. Chapter 4 details the implementation of these formalism in code, our initial conditions, extracting the mass, and some initial results. Lastly Chapter 5 will summarize and give conclusions.

11 Chapter 2

Misner-Sharp Formalism

Our first step is to follow the work of Misner and Sharp 115] to construct the equations of motion of a spherically symmetric perfect fluid. This was a topic of interest to them for its application to stellar collapse in a vacuum. We make the assumption that our collapsing mass will be nearly spherically symmetric based on numerical 12] and analytic [91 results of other studies which found that the non-symmetric modes in gravitational collapse are oscillatory, and have little impact on whether or not a black hole forms, or its final mass.

We then consider the effect that a cosmological background will have on these equations of motion. We care about a spherically symmetric mass in a Friedmann-

Robertson-Walker background, where pressure and energy density do not go to zero far from our collapsing mass. Misner and Sharp's work assumed that the background was a Schwarzschild metric where pressure and density vanish at some radius. This section largely follows Bloomfield et al. 141.

2.1 Defining a Metric and Stress Tensor

Before we construct the metric, we should define our matter. We are analyzing our black holes during a time period where the universe is radiation dominated. We can treat this radiation dominated universe as a perfect fluid because there is enough non- radiation matter also present so that the radiation scatters frequently and therefore

12 maintains thermal equilibrium. We can treat our matter as a perfect fluid with the stress energy tensor

T" = (p + P)uuv + PgV (2.1) where ut' is the fluid's velocity four-vector, g" is the inverse metric tensor, P is pressure, and p is density. We are dealing with radiation, and have an equation of state where P = wp, where w = 1/3. We decided to use co-moving coordinates, which requires that only the time component of u will be nonzero. This component is then fixed by demanding ulu, = -1. For a metric to have spherical symmetry, we need it to be true that three dimensional rotations of the metric will produce two dimensional spheres. The metric of a two dimensional sphere in three dimensional space is

dQ 2 = dO 2 + sin2 d 2 . (2.2)

Our metric needs to have this same symmetry, so that it is also invariant under three dimensional rotations, meaning it must have the form

ds 2 = 9ABdxA dB + R 2 dQ 2 (2.3) where xA and xB are our two remaining coordinates and gAB is a two dimensional metric that is only depends on x0 and x1. Next we fix our gauge so that our metric is diagonal, while respecting our co-moving coordinate condition. Labeling our time and radial coordinates as t and A respectively, we can then write

ds2 = e2<(t'^)d2 + eA(t,A)dA 2 + R2 (t, A)dQ 2 (2.4) with free functions #(t, A), A(t, A) and R(t, A). We can still make changes in how we define t and A, we will do this in Chapter 3. Beyond this, the behavior of our free functions are determined by employing the Einstein field equations.

13 Now, with our metric we can construct the fluid four-velocity vector as

Ut = e -, ur = u0 = uO = 0. (2.5)

2.2 Deriving the Equations of Motion

Now we can calculate equations of motion for this system using the Einstein Field

Equations,

Gtt = 8,rT,, (2.6) where we have set G = c = 1. Because our metric is diagonal, the calculations can be simplified using the equations in the appendix of 1181. The resulting equations are as follows.

2 2 2 2 2 Gt = 1 (e2 + 12 - e o,-(R') ) + 1 (e + A +e 0 R'A - 2e OR") - 8,re 0P R2 (2.7a)

Grr = 1 ((R' )2 + 2RR'0' - eA-20(e 20 + p2 - 2R &5+ 2RR)) = 8,reAP (2.7b) R2 1 Gtr, = -(iR' + 2Rq$' - 2k') = 0. (2.7c) R

Although G09 and Goo are also non-zero, it is simpler to obtain the remaining necessary equations by using the conservation of the stress energy tensor,

VAT" = 0. (2.8)

Calculating VTpA and VATut give us

21R + (P + p)(4N + AR) = 0 (2.9a) P' + (p + P)' = 0. (2.9b)

14 Now we are going to write Equations (2.7) and (2.9) in a more useful form. The simplest is to rewrite Equation (2.9b) as

P' (2.10) < P'= P*.

To rewrite the rest, we define a few useful quantities. First, we can express a fluid element's coordinate velocity as

U = e-o. (2.11)

Next we can rewrite A in a more convenient form,

F = e-A/2R' (2.12)

and by applying &a to Equation (2.12),

' e-/ 2 R" - -PFA'. (2.13) 2

Rewriting (2.7c) gives us

2eOU' (2.14) R'

Using Equations (2.11), (2.12), (2.13), and (2.14), we can simplify Equation (2.7a) to

d 8irpR2 R' = dA (R(1 + U2 _ 1-2)). (2.15)

Next we define a measure of the gravitational mass contained in the comoving radius A,

m A R (2.16)

Now integrating Equation (2.15), using the definition of m from Equation (2.16), and

15 enforcing a boundary condition that R(t, 0) = 0, we obtain

2 F2= I+ U2- m (2.17) R

Next, we take Equation (2.9a), and multiply it by a factor of 27rRR' to obtain

0 = 4ir R2 R'+ 27rRR'(P + p)( 4J? + AR). (2.18)

If we substitute Equation (2.14) into this we get

0 = 4irR 2 R'6+ 8irRR'Rp + 4eOU'wrR2p + 8irRR'RP + 41reOU'1R2 P. (2.19)

From here we can use Equation (2.11) to write

0 = -(4rpR2R') + dA( 4 rPR2). (2.20) dt d A

Again integrating from 0 to A, using the boundary condition R(t, 0) = 0, and using Equations (2.11) and (2.16), we find

2 Th = -47reOR PU. (2.21)

The only equation left to rewrite is Equation (2.7b), which can be done by using

Equations (2.11), (2.12), and the following expression calculated by applying at to Equation (2.11).

e-oj = U +Up (2.22)

Altogether this gives us

2e-ORU = F2 - 1 - U2 + 2F2 R- - 87rPR2 (2.23) R'

2 which can be further simplified using our equations for 0' and 1 in Equations (2.10)

16 and (2.17) to give

U = -e( + m + 4rPR). (2.24) (T (R'(p + P)

Altogether this results in the following, the Misner-Sharp equations of motion.

R = Ueo (2.25a)

mh = -e047rR 2PU (2.25b)

= -e + 47RP)r (2.25c) R'( p + P) R2

p = (2.25d) 47r R2R/ R' - P(2.25e) p+P

]F2 = + U2 _2m (2.25f) R

Here primes indicate derivatives with respect to A, and overdots indicate derivatives with respect to t. The equation of state P = wp closes the system of equations.

2.3 Boundary and Initial Conditions

The system of equations still needs boundary conditions and initial conditions, and so far only one has been specified. In our derivation of the Misner-Sharp equations of motion we chose that R(t, 0) = 0 while integrating several quantities. From the definition of m we can see that m(t, 0) = 0, because we only integrate from A = 0 to 0. We will also require that the regularity of the metric is maintained at A = 0, which requires that p'(t, 0) = 0 and P'(t, 0) = 0. Equation (2.25e) tells us that if P'(t, 0) = 0 then #'(t, 0) = 0. We also see that R(t, 0) = 0, and therefore U(t, 0) = 0. Lastly, when trying to calculate F2 (t, 0) because both m and R are 0 at A = 0, we need to use L'H6pital's Rule.

2 2 liF= - im2m' 8irpR R' lim F2 = I - lim = 1 - limr = 1 (2.26) A-4O A-+O R' A-*O R'

17 However we still require outer boundary conditions, and a further gauge condition is also required in the choice of a reference value for 0. We will return to these issues after constructing the equations of motion in terms of cosmological variables. For initial conditions we need a slice of data for the evolution variables. In other words, we need to know R(to, A), the initial gauge condition, m(to, A), the initial energy density, and U(to, A), the initial fluid velocity, where to is the start time for the evolution.

2.4 Cosmological Equations of motion

We now take into account the fact that we have a cosmological Friedmann-Robertson- Walker (FRW) background, not an asymptotically flat one. To improve the numerical precision of our code it would also be beneficial to rescale our equations of motion by the relevant constants.

2.4.1 A FRW Background

We begin by investigating how our equations of motion work for an FRW background. Starting with the fact that in FRW space-time the density of energy is spatially constant we can say p' = 0, where the b subscript indicates the value of the FRW background. Combining this with Equation (2.25e) we can show that q's = 0.

To obtain a value for mb, we need to be able to integrate Equation (2.16). Because

Pb is a constant we can write

mb=PbR (2.27)

From there we can use the Misner Sharp evolution equations (2.25a), and (2.25c), to obtain

Nb=Ub (2.28a) 4ir U = 4-pbRb(1 + 3w) (2.28b) 3

18 where we have used the equation of state P = wp. Combining these two equations

results in

-- = -- PO( + 3w). (2.29) Rb 3

At t = to, we can exploit reparametrization invariance to write Rb(to, A) = A. The

lack of spatial derivatives in the equations of motion then imply that Rb(t, A) = a(t)A

with a(to) = 1, where a(t) contains all of the time dependence in Rb, and will turn

out to be the FRW scale factor. Substituting this into Equation (2.29) yields

5i 4ir 1 a- = Pb( + 3w) (2.30) a 3

which is recognizable as the acceleration equation for FRW. Next we can take the

Misner Sharp equation for p from Equation (2.25d) and when combined with Rb a(t)A, obtain

pb = -3-pb(1 + w) (2.31) a

which is the cosmological continuity equation. When combined with the FRW accel-

eration equation we obtain

= -- Pba2p2 - k (2.32) 3

where k is an arbitrary constant of integration. We can rearrange this to get an

expression for the Hubble Parameter, H = &/a,

Pb-k =(2.33)- H2=87r k

which is recognizable as the Friedman equation, where our integration constant has

become the curvature of spacetime. As we are working in the era after inflation, we

will set k = 0 from now on. From here, using Ub = A&, along with Equations (2.28a),

19 (2.27), and (2.25f), we obtain

F2= 1-kA2 = 1. (2.34)

We can now solve the background cosmology. The cosmological continuity equation

(2.31) can be integrated to obtain

Pb = poa-3(1+w) (2.35) where pb(to) = Po. Using this, the Friedmann equation can be integrated to find

a = - C and Hand =- (2.36)(2a6 (to) t where we define a 2 and relate 3(1+w)

3 to = a (2.37) 87rpo

2.4.2 Rewriting the Equations of Motion for Numerical Effi- ciency

From here we can define variables that have been normalized by reasonable quantities so that numbers remain small, and our variables are unitless.

We also choose to use a dimensionless logarithmic time = ln(t/to), and rescale our radial coordinate like A = RHA, where RH = to/a, the horizon radius at the beginning of the evolution. This results in dimensionless equations of motion. The

20 equations dealing with our new coordinates are

a = e (2.38a)

H = -e _ 1 -4 (2.38b) to RH

=- -e- Ce a (2.38c) = . (2.38d) RH

Defining rescaled versions of our remaining variables,

R = aRH (2.39a)

P = PbP (2.39b) 47r m = mb= -PbR m~n (2.39c) 3 U = HRU = Ne(c-') Q (2.39d)

P = pbP = PobWf (2.39e)

F = aHRHr (2.39f) where they have now been writen in terms of background FRW values. We can additionally take Equation (2.25e) and rewrite it as

P w (2.40) +P 1+w p which integrates to obtain

= -3aw/2 (2.41) where a (purely gauge) constant of integration has been chosen to set the coefficient of this expression to unity. We can now write the cosmological Misner Sharp equations

21 of motion as follows.

R = aR(Ueo - 1) (2.42a)

m = 2f~ - 3ae*(w + in) (2.42b)

=U - ae ~~ + 2(2U2 + ~n + 3wi) (2.42c)

i = f~ + i (2.42d) 3R'

Co = i-3wc/2 (2.42e)

2 r2 = e1-aM + R(U - 7~n) (2.42f)

Here primes indicate derivatives with respect to A, and overdots indicate derivatives with respect to . Revisiting our boundary conditions, many of our variables will be even functions of

A, meaning that they are symmetric about the origin: p'((, 0) = P'((, 0) = ~'((, 0) =

U'( , 0) = 0. R on the other hand is an odd function, therefore R( , 0) = 0. Again, our initial conditions are described with slices of data over all A at our initial time, = 0, for our three evolution variables, fTf, U, and R.

2.4.3 Outer boundary conditions

In the system of equations as Misner and Sharp originally used them, the outer boundary was simple. Their collapsing star was embedded in a vacuum, and so the outer bound was specified by the Schwarzschild metric. However our collapsing black hole has a FRW background, and we can not simply set the boundary to the values of R, p, etc, to what it would be in a plain FRW metric. Doing this would cause any outflowing matter to reflect off of our boundary to maintain perfect external FRW. Instead, we use an appropriately-constructed "wave" boundary condition which allows outgoing waves to exit the domain. The exact details of this can be found in [4].

22 2.4.4 Black hole formation

We now discuss what occurs during these numerical evolutions for the sake of physical intuition. There are three cases that will be discussed. In broad terms what happens when: a black hole forms, a black hole fails to form, and when our evolved region is just FRW spacetime. Evolving just FRW is the simplest case: in our new cosmological variables rin, U, and p will stay equal to one at all locations, and ft = A. When some overdensity is placed into our region of numerical evolution, the matter falls inwards and the density of matter correspondingly increases at small radii. Both the formation and lack of formation of a black hole look relatively similar in the beginning, as matter collapses inwards. At some time, if the black hole does not form, the matter will cease to infall and instead create an outgoing density wave. If a black hole is forming, the matter will continue to infall and an apparent horizon will form, rapidly followed by the formation of the singularity. A depiction of this can be found in Figure 2-1. This is where our current formalism will fail due to the formation of the singularity, and we will need to move to a new formalism. To stop the Misner Sharp evolution before a singularity forms, we will detect the presence of an apparent horizon, which will prompt us to move to the formalism discussed in Chapter 3. Following 7 in [31, we will begin by defining timelike, orthogonal to t, and spacelike, orthogonal to A, unit vectors, n' and s' respectively. This means our outgoing null vector is k = (n' + s')/v'2 = (e-0, e-A/ 2, 0, 0)/V. If we further define m'ab = gab + nanb - sasb, a metric defined on a surface of constant t and A, we can say that the expansion of outgoing null geodesics is described by

kaVa(41rR2 ) S= .iR (2.43) 47r R2

Substituting in our metric, we obtain

8=- (U +I). (2.44) R

23 * t

Figure 2-1: Location of singularity, apparent horizon and event horizon in a black hole- forming spacetime This is a schematic representation of the location of the singularity (denoted by a star), the apparent horizon (dashed blue line), and the event horizon (solid red line). At some very large t, which we will not reach in our simulations, the apparent and event horizon will be at the same radius.

24 We can say that an apparent horizon forms when 8 < 0, or when the null geodesics

are beginning to head inwards, which corresponds to the condition that F + U < 0.

We can rewrite this using Equation (2.25f) to get

2m - > 1 U < 0 (2.45) R -

where the second condition comes from the fact that F > 0. We can rewrite this in

cosmological variables as

rn 2(ae-) ;> 1 U < 0. (2.46)

These two equations together are what allows us to detect when an apparent horizon

has formed. We are using this a stand in for detecting an event horizon, and therefore the formation of a black hole. This is because to determine the location of an event

horizon we need to know the entire history and future of our spacetime, which is not computationally efficient. We do know an inner bound on the location of the event

horizon though: the apparent horizon is always inside the event horizon.

Altogether, we now have the ability to evolve a primordial density perturbation forwards in time until a singularity forms, and can halt the evolution before this happens by detecting the apparent horizon. Practically this part of the evolution takes

about one second to do on current laptops. Now we must evolve our perturbation until the surrounding mass has fallen into it, and the final mass of our primordial black hole can be extracted.

25 Chapter 3

A General Formalism

So far, we have the ability to evolve black holes right up until a singularity is about to form. Although this is enough to tell us if a black hole forms, it is not enough to tell what the mass of the black hole will be. At this time, not all the mass that will settle into the black hole has settled into the black hole. A new coordinate system is necessary in order to compute the mass accurately.

In Section 1.2 the Hernandez Misner formalism was discussed. It uses a null coordinate, which works to evolve black holes past the formation of a singularity by preventing the singularity (and even the event horizon) from forming. This approach has been used in many previous studies of primordial black hole formation, however we will not use this formalism because it is very computationally intensive. This process took about two orders of magnitude longer than the portion of the evolution done in Misner Sharp 14]. For the purposes of many black hole evolution projects, it is fine for the evolution of a single black hole to be time consuming. However we would ultimately like to evolve millions of black holes in order to compute statistics from inflationary models, and we desire a formalism that is more numerically efficient.

Our goal is to construct a new formalism where we prevent the formation of the singularity, while allowing the regions outside the black hole to evolve in a computationally efficient manner. We will now perform an arbitrary time coordinate redefinition, for which we can specify something convenient later.

26 3.1 Defining the coordinates

Our present coordinates are and A (9 and q don't appear in any of our equations, and so we will ignore them). We define new coordinates T and q, which are related

to c and A by

((T, q) = f(T, TI) (3.1) A(r) =T. (3.2)

T will be our timelike coordinate, and TI will be our spacelike coordinate. For now we will not specify f and we will allow it to remain an arbitrary function. From here we can relate O and 9A to 0,, and (9, as follows.

Of aA a, = IA + a (3.3) Of (3.4) aA & OA = Tg

3.2 Equations of Motion

However we are not going to be able to create a complete set of equations of motion

just by performing a coordinate transform. We will demonstrate why with a toy case

before moving on to calculate a complete set of equations of motion.

3.2.1 Toy Model

We will consider a single quantity, m(t, a), with an equation of motion

atm = f(m, 0am). (3.5)

27 From here we are going to try a coordinate transformation from (t, a) to (T, A) with dT = Tdt + T'da and dA = da. The chain rule will give us the following relations.

& _ t 0 (36 - -= ot (3.6) A aTT Ata

& _Ot & &a 0 -AT = A-a+ =t'at+0a a'A T =A aot Ara (3.7)

In this toy case we try to calculate our new equation of motion for m in the (T, A) coordinate system.

&Tm=if m, Am-- - m) (3.8)

Except for special cases of f, there is no way to solve this for aTm. On the other hand, letting p = &Am, and promoting Dam to an evolution variable allows us to close the system. We can get both &Tm and aTp in the new coordinate system. In this case we can recover

aTm = if (m, p). (3.9)

Unfortunately we now need an evolution equation for p in the original coordinate system, which we can then transform to our new coordinate system.

3.2.2 Closing the set of equations with P

To obtain an evolution equation for , we begin with Equation (2.42d), which we can rearrange to write

(aAR),3 = rinaR + 3RaAffi. (3.10)

Using the chain rule in reverse, we can write this as

aA(R)3 = ~A(,hR 3 ). (3.11)

28 From here if we apply &9 to the equation, and use Equations (3.13b), (3.13c), (2.40)

along with a = 3(1 2 w)'wcawrt, we can write oO+

~ &0A)) p=2P( 1 ~? . (3.12)

3.2.3 New Equations of Motion

Using the relations in Equation (3.3) and (3.4), we can transform the Cosmologi-

cal Misner Sharp equations of motion to obtain the following. We also transform Equation (3.12).

=jaN(Teo - 1) (3.13a)

S=f(2fi - 3a eo(w + in)) (3.13b)

U O - e +2U2 + ?n + 3w,) (3.13c) j-f')(1 + w) 2 \I,

I = e(--e" + RU 2 - 7 (3.13d)

eo = -3aw/2 (3.13e)

p= 2f/( 1 - eo U&+ -fUf.) (3.13f) 3 fr - f'R

Here overdots represent derivatives with respect to T, and primes represent derivatives

with respect to rq. While complete, the equations of motion for p- and U depend on

each other and need to be isolated.

3.2.4 Solving for p- and U independently

To simplify the algebra of this we are going to redefine U and p as

U = A + B(' - pf'/f) (3.14) p= C + D(U' - Uf'/f) (3.15)

29 where

A = f(U - iaeO(2U2 + n + 3w)) (3.16a) 2 -jae*I2W B =(3.16b)B= R(R' - f'aR(UeO - 1))fi(1 + w) C = 2fi(1 - e0U) (3.16c)

D = -2f~e 3 (3.16d) R' - f'aR(UeO - 1)

From there it is simple to solve for

~ -Af - BCf' - BDf'U' + Bfp' U= f f-Bf2(3.17) j- BDf' 2

. .C- ADf' - BDf'p'+ Df ' P= f . (3.18) j 2 -BDf' 2

30 3.2.5 The general cosmological equations of motion

Finally we will restate our equations of motion in one place.

= jf n(UeO - 1) (3.19a)

nh f(2f~n - 3aUeO(wp + i)) (3.19b)

e =-3aw/2 (3.19c)

r2 2- a)f + A 2 ((2 - ~n (3.19d)

.Af - BCf' - BDf'U' + Bfp'( U =U~f f ~ 2 -BDf' 2/ (3.19e) .Cf - ADf' - BDf'p' + Df U' p = f -Bf2(3.19f) f 2 -BDf' 2

A = f( - ae0(2U2 + in- + 3w,5)) (3.19g) 2

B = _jerW(3.19h) R(R' - f'cd?(UeO - 1)),(1 + w)

C = 2M,(1 - eC"() (3.19i)

D = (3.19j) R' - f'aR(UeO - 1)

Again, overdots represent derivatives with respect to T, and primes represent derivatives with respect to q.

3.3 Potential problems

Looking our equations of motion there are two types problems that can beset our coordinate system. The first is that eventually, our time slice will stop being a spacelike curve, while the second is that there are some denominators in our evolution equations that could possibly vanish, leading to a divergence.

We can investigate when our coordinates will become timelike and spacelike by deriving a metric for our coordinate system. We can transform the original Mis- ner Sharp metric, in Equation (2.4), to a cosmological metric by using our previous

31 definitions of A, ft and .

2 ( 2 2 2 2 ds2 = R 2 + eAdA + e a R d ) (3.20)

We can then transform this to our general coordinate system by noting that

< = f'd + jdr (3.21)

dA = d (3.22) which gives us

2 ds =RH( - -Z2cCedf+-2f'jdqdT (3.23)

+ (eA - oe 2 e 2(f+O)f'2 )dn2 + e2 af R 2 dQ2). (3.24)

By inspection, if f 0, T will not become spacelike, and this is always true (for our eventual choice of f, this will be true except at the origin, which we wish to freeze anyway). However, if eA/2 = +aef+'f'then q will become timelike. Using Equations (2.12) and (2.38c), we can compute

eA/2 _ - (3.25) ef = [a5 - arf~f'/j] (3.26)

Hence, our spatial coordinate will become timlike if

=-faeOP ?' - f'ac(UeO - 1). (3.27)

This essentially corresponds to f becoming too steep. However, this turns out to be unimportant, as a separate divergence kicks in before this one can be met. Looking at the denominators of the equations of motion there are two different terms that could potentially go to zero and cause our equations to become divergent.

2 f2 - BDf' , which appears in the denominator of U and , and R'- f'aR(Ue5 - 1) =

32 OAR, contained in the denominator of both B and C. The second is trivial, it corresponds to OAR, which must always be positive, and cannot be zero. It is possible for f2 - BDf'2 to become zero however. Solving for when this quantity vanishes, we obtain the condition

ifaeoirv5= ' - f'aON(Ueq - 1). (3.28)

This is almost identical to the condition in Equation (3.27), except for a factor of

Vw-, which arises from the speed of sound in the fluid. At the linear level, the Misner Sharp equations possess wave-like solutions that travel inwards and outwards at the same speed. When redefining our time coordinate as we have done, the speed of one direction increases, while the speed of the other decreases. The condition in Equation (3.27) corresponds to ensuring that the speed of light in one direction doesn't become zero, while the condition in Equation (3.28) corresponds to ensuring that the speed of sound in one direction doesn't vanish in that coordinate system. The speed of sound will always vanish before the speed of light does, and so this condition provides a stringent upper-limit on how far forwards in time we can evolve our system of equations. Hence, the only condition which we need to be wary of is 2 = BDf'2 . As we only need to evolve the system long enough to extract the black hole mass, we hope that this limit is inconsequential. This limit can also be mitigated by increasing 0, which produces a shallower gradient in f.

33 Chapter 4

Implementation

Now that the two formalisms necessary to fully evolve a primordial black hole are in place, we can discuss the details of how to numerically implement them. To do this we will need to first pick an appropriate f. Then we will determine what we need for initial conditions, and how we will use our Misner Sharp equations of motion to step these conditions forwards in time. Then we must determine how to evolve our black hole according to the new general equations of motion once we have detected a black hole, particularly what our exact form of f should be. Lastly, we discuss how to extract our black hole's mass once we have evolved it for long enough.

4.1 Picking a convenient time coordinate

Now that we have equations of motion for an arbitrary change of coordinate described by some function f, we should decide what function would work well for us. Again, we want a time coordinate that prevents the formation of the black hole's singularity but which also allows the outer parts of the black hole to continue to evolve rapidly so that our code is computationally efficient. We know that Misner Sharp allows rapid evolution, so we will design our time coordinate such that the outer regions are still

34 in fact Misner Sharp. So, we are going to define , such that

6o if r7 < qo = f(7, 7) if7h < 7 < '1 (4.1) 7- if q ;> /1 which allows us to have our outer region, 7 ;> qj, remain as the Misner Sharp evolution, while the middle region's time slicing is defined by the still-to-be-determined f, and the innermost region is completely frozen. Now we need to define an f that smoothly connects to Misner Sharp at n1 and to o at qO. The following function fulfills both of those requirements,

f(q, 7) + T 2 tanh (tan r( 77 7 ) +1. (4.2)

A plot of this function at a time T is in Figure 4-1. Here f(0,r) = o and f(771 , ) = , where we have picked o to be the time at which an apparent horizon is first detected, m1 the radius at which there is an apparent horizon, and mO to be some distance away from the center. In other words, the center of the time slice stays fixed at a time before the formation of the singularity. At a radius outside of the apparent horizon, where we are a safe distance from regions which would cause numerical problems in

Misner Sharp, our function smoothly matches up to Misner Sharp. Altogether this formalism will artificially stop the evolution of the center of the black hole, allowing us to evolve for longer. A graphical depiction of this can be found in Figure 4-2.

From Figure 4-2, the potential problem with our new coordinate system discussed in in Section 3.3 is visible. This system will break down after a certain time span, because once it has been evolved for too long, the region that interpolates between

A = 0 and A = ql at the current time will become increasingly vertical, and our time slice will violate our speed of sound condition.

From here we can also calculate more explicit forms of our partial derivatives,

35 2

1.8

1.6 -

.4-'

1.4 -

1.2 -

0 0.2 0.4 0.6 0.8 I

Figure 4-1: Properties of f A plot of what our function for f looks like at a given r and for O = 971 = 1 and qo = 0.

2.5 -I, ------

-- / -- -1------2

,,,-oI,, ------'U-P 1.5 * k ,e ------At ------1

0.5

VI' 0 0.5 1 1.5 2 2.5 3 A

Figure 4-2: Example of new coordinates Here we plot an example progression of time slices of what we should expect when evolving a black hole. The solid blue curves are time slices in the Misner Sharp evolution. At = 1, an apparent horizon is detected at A = 1, and we transition to our new coordinates with T11 = 1 and 7o = 0. We then switch to our new coordinate system with the dashed red time slices. The black star represents the formation of a singularity, which the new coordinate system avoids.

36 f = tanh (tan(7r(, -- + " (4.3) 27 211 2 and

f'=7r T - sec 2r( - ) sech 2(tan(7r(" 1)) (4.4)

=7r T GSC2(7r( - io)/91)sec 2 (ot(7r( - 70)/91)). (4.5) 21

Plots of these derivatives can be found in Figure 4-3. Note that f'(r, 0) = 0, which means that the first time slice in the new coordinates is the same as the Misner Sharp time slice. This means that we can use our data from Misner Sharp as our intial conditions in the new coordinates. Also, f'(A = 1 + qo, T) = 0 which smoothly matches with the Misner Sharp condition that (' 0. One reason that this was function was chosen is that f(")(A - 771 + YO,T) = 0, meaning that every spatial derivative matches to Misner Sharp. Lastly, near the center f'( T1, T) = f(q <

77, T) = 0, and so the center is frozen in time.

4.2 Initial Conditions

Our equations of motion in Misner Sharp consist of evolution equations for f~n, U and R, with constraint equations for F2, , and ed. Therefore we know that we need to specify i, U and f? for all A at some initial time. These initial conditions correspond to the initial energy density, the initial velocity and an initial gauge choice. These initial conditions come from selecting the growing mode from inflation of a given perturbation. The derivation to determine the initial conditions based on a growing mode is rather long, and can be found in Section IV of [4], although it is worth noting that the discrepancies between the conditions listed here and those in the paper are due

37 1.6

1.4 -

1.2 -

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 7) Figure 4-3: Derivatives of f with respect to r and rj The blue solid line is f and the red dashed line is f'. to differing definitions of R. They are

ffn(A, = 0) = 1 + 6mo (4.6)

(A, = 0) = 1 - a6mo (4.7) 2 R(A,(= )= 1 -- mo+ A. (4.8) 2 mW+1 )

4.3 Numerical Evolution

Once we have our initial data calculated from 6mo we can begin to evolve our system using the Misner Sharp formalism. The first step will be to use our initial data to calculate the auxiliary variables, and 1', and then e1 . Additionally many of our equations of motion have spatial derivatives that need to be evaluated. To do this we fit a polynomial of order 5 to the data whose derivative is being taken, and the derivative of the polynomial is evaluated. Once we have those we can use our evolution equations to take a time step for-

38 wards in time for I?, in, and . To do this we use a 4-5 Runge-Kutta-Fehlberg method, which takes ffn(A, o), R(A, co), U(A, o) and rfi, R, U and computes a numerical approximation for ri(A, o + 6 ), R(A, o + 6 ), and U(A, o + 6k). We continue this process until either an apparent horizon is detected, or our maximum evolution time for Misner Sharp is reached. This maximum time was chosen such that if no apparent horizon had yet been detected, it is very unlikely that a black hole will form. If an apparent horizon was detected we move to the general equations of motion detailed in Chapter 3, and continue to evolve our black hole using the same numerical methods as before, now with the additional L equation.

4.4 Initial Results

We implemented a preliminary version of these equations in Python and performed some trial runs, the initial results of which will be presented here. We began our evolution with Gaussian initial data for 6m which was slightly above the threshold for collapse. From there we hand picked qo and ql for the values which allowed us to evolve the furthest in time. This initial set up has allowed us to successfully evolve a black hole past the formation of the singularity. In Figure 4-4, versus A is plotted for two different runs. The first is an evolution attempt where only Misner Sharp was used, and it is plotted until Misner Sharp crashes due to the formation of a singularity. The other data set is our new implementation where we switch to our new equations of motion, and they are run until our code must stop. We can see that first, we can now evolve past the formation of the singularity. We can also see that the outer region of our second evolution, which should still be Misner Sharp matches up with our first evolution which only used Misner Sharp. Additionally we can see where the apparent horizon is. Next, we can look at what our black hole looks like by plotting p versus A, shown in Figure 4-5. It behaves as expected, the matter in the outer regions is falling inwards and a large central overdensity is forming. The difference is that the density plots in

39 6.2

5.8

5.6

5.4

I 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 A

Figure 4-4: versus A for Misner Sharp and new coordinates We can here see that we are now able to evolve our black holes past the formation of a singularity. The black points are those points which meet the criterion to be on or inside the apparent horizon. We can additionally see that in the region of our new formalism where it is Misner Sharp, our time slices match to the evolution where it is just Misner Sharp.

40 105.

100

10-5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 A

Figure 4-5: Evolution of 3 Here we have timeslices of 3 as our black hole is forming from two different simulations. The timeslices in green show the evolution using only Misner Sharp. Here it is apparent that a singularity was forming as the density at the center of the last time slice becomes divergent. The pink and blue lines are from an evolution with our new coordinate system, with blue lines representing early slice and pink lines representing later ones." End of the caption - "continues to fall inwards in agreement with Misner Sharp. We can see that it behaves as expected, the inner region varies from frozen to slowed while the material outside continues to fall inwards. the simulation that only used Misner Sharp was clearly beginning to diverge, with the final timeslice reaching a peak density of 10', while in the evolution that employs the new coordinate system it is possible to see that the innermost region is frozen in time after a certain timeslice. What we really care about however is if we have reached a place where we can extract an accurate black hole mass. To determine this we will examine Figure 4-6, a plot of timeslices of ~nh. As the lines change from blue to pink, later and later timeslices are being plotted. We can see that at the last timeslice the region around A = 15 is nearly flat. This has an impact on our ability to extract the mass of the black hole, which will be discussed in Section 4.5. Unfortunately we encounter the condition discussed in Section 3.3. f 2 _ f' 2BD is plotted in Figure 4-7, and we can see that we are in fact reaching the condition where

41 3

2.5

1.5

0.5

0 0 2 4 6 8 10 12 14 16 18 20 A

Figure 4-6: Evolution of m Here we have timeslices of m as our black hole is forming, as the lines change from blue to pink later and later timeslices are being plotted. The pink lines show that matter is rapidly moving towards A = 0, which represents the matter falling into the singularity. We can see that at the latest timeslice our enclosed mass is nearly unchanging for A, but is still changing significantly.

42 0.9 -

0.8 -

0.7

0.6 - ~.0.5-

0.4

0.3

0.2

0.1

0- 2.2 2.3 24 2.5 2.6 2.7 2.8 2.9 3 A

Figure 4-7: Speed of CharacteristicWave Here we have timeslices of f 2 _ f 2'BD as our black hole is forming, as the lines change from blue to pink later and later timeslices are being plotted. We can see that near A = 2.7 this is approaching 0, the condition under which our equations will break down. this is 0 and our equations of motion break down. This was the concern that we had about these equations at the end of Section 3.3. Without modifying our choice of f this means that we have evolved our black hole as far as we can.

4.5 Mass Extraction

Now that we have a final state of a primordial black hole, we need to extract the most accurate mass that we can. A black hole's mass is typically defined as the mass that is enclosed by the event horizon. However to determine the location of the event horizon we would need to know the current state of the black hole, its entire history, and its entire future. Therefore trying to determine the event horizon's location is not reasonable, and we will have to determine the best practical alternative.

Looking at Figure 4-6, we can see that there is a region of very little change in the enclosed mass surrounding the black hole. The event horizon is located somewhere in this region, but again we do not know where. Ideally we would like this region to

43 be constant enough that our different A in this region would produce a change of less than one percent in m by our last timeslice. However this is not the case, although we are very close. This means that we still can not extract the black hole mass quite as accurately as we would like.

44 Chapter 5

Conclusions

The goal of this thesis was to rapidly numerically evolve a primordial density perturbation forwards in time until a black hole has formed, and accurately extract its mass.

To do this we derived a cosmological Misner Sharp formalism following the work of

[15, 41 which is capable of evolving our density perturbation until a singularity forms.

We use the formation of an apparent horizon to detect that a black hole has formed, and an indicator that a singularity would form shortly thereafter.

We then performed an arbitrary redefinition of the time coordinate and derived cosmological equations of motion in this new coordinate system. We presented an appropriate choice for a new time coordinate that would prevent the formation of a singularity while still allowing the outer regions of the simulation to be evolved as rapidly as in the Misner Sharp formalism. Potential numerical problems with our new coordinate system were then explored. We presented an overview of a proof-of- concept numerical implementation of our new formalism, including initial conditions, evolution details, and suggestions for mass extraction.

The preliminary numerical simulations using this new formalism effectively evolves our black holes until the speed of one of our characteristics goes to zero. When this happens we are nearly able to extract an accurate mass, but we would need a few more time steps to get a truly accurate mass extraction.

For future work we would like to trace out the null rays of our characteristic waves for our new time slices. This can also be thought of as using the null of sound as

45 our time coordinate, similar to how Hernandez Misner used light nulls as their time coordinate. Because some sound nulls will travel inwards to the center of the black hole and some will escape, there must be a sound null which allows us to reach a state where we can accurately extract the mass of the black hole.

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