Slideshow summarizing the findings of a project to investigate the impact of dark matter and dark energy on large scale structures in the universe.

1. Impact of Dark Matter
and Dark Energy on
Large Scale Structures
Carole-Anne Collins, Ruike Li, Joe Santrock, Rohan Srivastava, Carson West

2. Introduction to project
● Time evolving a Model Universe to show the impacts of dark energy and dark matter on
the evolution of large scale structures
○ Focussing specifically on the expansion rate of the universe
○ Comparing the Bulk Viscosity theory vs the Cosmological Constant Theory
● Simulation consists of 256³ (~16.7 million) bodies to lower the computational power
needed
● Simulation scale still allows the cosmological principle (>100 Mpc) to apply
○ Findings can be generalized to that of an entire universe

3. Why our topic is important/why we chose this
topic
● Understanding how large scale structures form
○ Cosmology has a basis in Relativity, which can draw support/proof from
simulations of how these structures form
○ The cause of the expansion of the universe is still largely unknown. Since we can’t
experiment with a universe, simulations are the next best tool
● Models can predict the age and position of astrophysical objects
● A manipulatable model gives the ability to answer “what ifs”
○ Gives a visual representation of how much contribution certain parameters have
on the universe’s motion

4. Observational Basis
● High-Z Supernova Search Team
○ Observations of Type 1a supernovae prove that these supernovae were moving away from each other, and at an
accelerating rate
○ Gave the first observational data that the universe was expanding
● Wilkinson Microwave Anisotropy Probe
○ Imaged the Cosmic Microwave Background
○ There must be some negative pressure or anti-gravity force in order for clusters we observe today to have
formed
○ Approximately flat universe (~Ω =1.002)
● Planck 2018
○ Measured parameters to greatest accuracy

5. Initial conditions
● Amplitude of the (linear) Power Spectrum on a scale of 8 h-1 Mpc
○ σ8 = 0.8159 ±0.0060
○ Power Spectrum:
● Scalar value tied to comoving distance and spatial curvature
○ ns= 0.9671±0.0038
● Spatial Curvature
○ κ ≈ 0
● Hubble Constant
○ H0 =
● Hubble Parameter
○ h = 0.6774 ± 0.0046
● Initial simulation size
○ L0 = 100 Mpc
● N-body mesh size
○ 256 x 256 x 256

6. Numerical Methods
Method Purpose Numerical methods used
ModelUniverse.simulate Simulates N-body system
with cosmological factors
FFT, Gaussian integration,
interpolation, parallel
computing
utils.scale_factor Solves first order ODEs to
find a(t)
ODE, integration, derivative
utils.animation Animates 3D scatter plots,
density plots, and scale
factor plots
Time evolved animations,
matplotlib

7. Simulated Universes
Theory Universe Parameters
Dark Energy Our Universe (Cosmological
Constant)
Ωr = 8.490*10^5, ΩM = 0.3103, ΩDE = 0.6897, ω = -1, κ = 0
Strong Dark Energy (Phantom
Energy)
Ωr = 8.490*10^5, ΩM = 0.3103 , ΩDE = 0.6897, ω = -1.5, κ = 0
Weak Dark Energy
(Quintessence)
Ωr = 8.490*10^5, ΩM = 0.3103, ΩDE = 0.6897, ω = -0.5, κ = 0
Bulk
Viscosity
Our Universe 𝜁0=4.389, 𝜁1=-2.166
No Big Bang 𝜁0=8.000, 𝜁1=2.000
Constant Pressure 𝜁0=1.000, 𝜁1=0
Logarithmic Expansion Ωr = 8.490*10^5, ΩM = 1 , ΩDE = 0, k= 0
Analytical
Models
Overdense Matter Ωr = 0, ΩM = 2, ΩDE = 0, κ = -1, 𝜁0=0.000, 𝜁1=0.000
de Sitter ΩDE = 1, ω = -1
Theory Universe Parameters
Dark Energy Our Universe (Cosmological
Constant)
Ωr = 8.490*10^-5, ΩM = 0.3103, ΩDE = 0.6897, ω = -1, κ = 0
Strong Dark Energy (Phantom
Energy)
Ωr = 8.490*10^-5, ΩM = 0.3103 , ΩDE = 0.6897, ω = -1.5, κ = 0
Weak Dark Energy
(Quintessence)
Ωr = 8.490*10^-5, ΩM = 0.3103, ΩDE = 0.6897, ω = -0.5, κ = 0
Bulk
Viscosity
Our Universe Ωr = 8.490*10^-5, ΩM = 1, 𝜁0=4.389, 𝜁1=-2.166
No Big Bang Ωr = 8.490*10^-5, ΩM = 1, 𝜁0=8.000, 𝜁1=2.000
Constant Pressure Ωr = 8.490*10^-5, ΩM = 1, 𝜁0=1.000, 𝜁1=0
Logarithmic Expansion Ωr = 8.490*10^5, ΩM = 1, 𝜁0<<0.000 and/or 𝜁1<<0.000
Analytical
Models
Overdense Matter Ωr = 0, ΩM = 2, ΩDE = 0, κ = -1, 𝜁0=0.000, 𝜁1=0.000
de Sitter ΩDE = 1, ωDE = -1

8. No Scale Factor: N-Body Animation

9. Math/Theory: Kinematics of Cosmology
● Relativistic, time independent radius:
Newtonian
Relativistic
● Fluid equation is a direct result of a thermodynamic system that is allowed to expand and
contract
● By taking the time derivative, and substituting the fluid equation, the acceleration equation
pops out
A sphere of radius Rs(t), and
mass Ms, expanding or
contracting under its own
gravity.
Rs(t) - Newtonian time
dependent
rs - Newtonian, time
independent
R0 - Relativistic, time
independent

10. Math/Theory: Kinematics of Cosmology
● Usually, a more useful form of the Friedmann equation is used where it is parameterized using density
parameters known as ‘ingredients’:
● Friedmann equation becomes
● Where index i sums over all ‘ingredients’ and exponent ki is the exponential dependence of that ingredient.
Here we also introduce the Hubble value H(a). It describes the relative expansion of the universe, where H(a0)
= H0

11. Math/Theory: Bulk
● A bulk viscous matter-dominated Universe
○ No dark energy, or cosmological constant
● Agrees with observations
● Scale factor dependence is known for ‘normal’ matter and
radiation. For BV, we solve fluid equation with pressure derived
from the conservation equation of a viscous fluid

12. Math/Theory: Bulk
● Solving the systems of equations of the fluid, acceleration, and
Friedmann equation yields a bulk viscosity scale factor dependence
● Using this in the parameterized version of the Friedmann equation
does not have an analytical solution

13. Math/Theory: Cosmological Constant
● One particularly interesting ingredient is dark energy
○ It explains the expansion we observe that no other ingredient does
● The general from of pressure is
○ Dark energy is a negative pressure ingredient where ⍵DE < 0
● Plugging this into the fluid equation and solving for energy density,

14. Cosmological Constant theory
● Now that we know the scale factor dependence, we can construct an ODE from the Friedmann
equation:

15. Simulated Universes Revisited
Theory Universe Parameters
Dark Energy Our Universe (Cosmological
Constant)
Ωr = 8.490*10^-5, ΩM = 0.3103, ΩDE = 0.6897, ω = -1, κ = 0
Strong Dark Energy (Phantom
Energy)
Ωr = 8.490*10^-5, ΩM = 0.3103 , ΩDE = 0.6897, ω = -1.5, κ = 0
Weak Dark Energy
(Quintessence)
Ωr = 8.490*10^-5, ΩM = 0.3103, ΩDE = 0.6897, ω = -0.5, κ = 0
Bulk
Viscosity
Our Universe Ωr = 8.490*10^-5, ΩM = 1, 𝜁0=4.389, 𝜁1=-2.166
No Big Bang Ωr = 8.490*10^-5, ΩM = 1, 𝜁0=8.000, 𝜁1=2.000
Constant Pressure Ωr = 8.490*10^-5, ΩM = 1, 𝜁0=1.000, 𝜁1=0
Logarithmic Expansion Ωr = 8.490*10^5, ΩM = 1, 𝜁0<<0.000 and/or 𝜁1<<0.000
Analytical
Models
Overdense Matter Ωr = 0, ΩM = 2, ΩDE = 0, κ = -1, 𝜁0=0.000, 𝜁1=0.000
de Sitter ΩDE = 1, ωDE = -1

16. ● Scale factor vs time animation
○ 𝜁1 = 0
○ Evolving over the range: -5 < 𝜁0 < 5
● As 𝜁0 increases, the scale factor increases
more rapidly
○ 𝜁0 plays a big role in universes that
are not expanding quickly
● Increasing 𝜁0 increases the age of the
universe
○ Lower left corner of graph
Bulk Viscosity: Scale Factor Plot (𝜁0 )

17. ● Scale factor vs time animation
○ 𝜁0 = -1
○ Evolving over the range: -5 < 𝜁1 < 5
● As 𝜁1 increases, the scale factor increases
more rapidly
○ As ȧ increases, 𝜁1 has a stronger
effect; thereby increasing ȧ even
more
● Increasing 𝜁1 increases the age of the
universe
○ Lower left corner of graph
Bulk Viscosity: Scale Factor Plot (𝜁1)

18. Method: death_of_universe
● Big Rip: da/dt > c
● Big Crunch da/dt << 0
● Heat/Big Freeze death: No thermodynamic processes can happen
○ Based on density fluctuations
○ Rate of expansion cannot exceed the speed of light, but it needs to be great
enough to break apart gravitationally bound systems

19. Method: death_of_universe (continued)
● If neither of the two aforementioned criteria are not
satisfied throughout the iteration, we check for a
possible Heat Death at the last time step
○ Calls on average_finder to determine average
spacing between bodies in our universe
■ Calls on simulate function to get
position arrays
○ Calculate expected average for an evenly
spaced universe
○ Compare these two values within a buffer of
2% to determine if the Model Universe is
showing signs of trending to an evenly spaced
universe

20. Bulk Viscosity: Death of Universe
Universe Theory Numerical
Result
Our Universe Big Rip Big Rip
No Big Bang Big Rip Big Rip
Constant Pressure Expand to infinity Expand to infinity
Logarithmic Expansion Heat/Big Freeze Expand to infinity
Overdense Matter Big Crunch Big Crunch

21. Overdense Matter (Big Crunch)
Bulk Viscosity: N-Body Animations
No Big Bang

22. Bulk Viscosity: N-Body and Density Animations
Our Universe Our Universe Heat Map

23. ● Evolving -1.5 < ⍵DE < -0.5
● The smaller ⍵DE the stronger dark energy’s
effect is
● Stronger effects also slow expansion at early
times
Cosmological constant: scale factor plot

24. Cosmological Constant: Death of Universe
Universe Theory Numerical
Result
Our Universe (Cosmological Constant) Big Rip Big Rip
Strong Dark Energy (Phantom Energy) Big Rip Big Rip
Weak Dark Energy (Quintessence) Expand to infinity Expand to
infinity
de Sitter Big Rip Big Rip

25. de Sitter
Cosmological Constant: N-Body Animations
Quintessence

26. Cosmological Constant: N-Body and Density Animations
Our Universe Our Universe Heat Map

27. Conclusions
● Our code agrees to the theory at small scales (256^3 bodies), so generalizations to the entire universe
should be accurate.
● Similar to how Classical Mechanics models a pendulum, given a set of initial conditions, our code uses
Cosmology to model universes and agrees with other researcher’s results
●
● Youtube with all simulations: https://www.youtube.com/channel/UC55UoPo_8jY-iLoBns5QGPw
● Gitlab repository: https://gitlab.com/phys3266/darksim

28. Comparison to Millennium Simulation Project

29. Code Improvement
● ModelUniverse.simulate is not fully parallelized, though ~90% is using Tensorflow
○ Runtime for 256 mesh (~16.7 million bodies) ~4 minutes per time step
○ ~60% of the runtime is only using 1 CPU core (doing operations such as np.copy)
○ RAM usage peaks at 27 GB, with continuous usage at ~17 GB
● 3D animations are bulky
○ ~50 minutes to run
○ RAM usage peaks at ~16-30 GB
● Finish making it a package that is installable using pip (and documentation!)
● Larger Projects (CosmoFlow, ~134 million bodies)
○ More advanced usage of Tensorflow, namely 3D convolutional neural networks to cut down dataset size and calculation
time
○ Their goal was finding σ8, Ωm, and ns using 12,632 simulation boxes of 5123 particles (1.7 trillion total)

30. References
Aghanim, N. et al. “Cosmological parameters.” Planck Collaboration, vol. VI, no. ms, July 16, 2018, pp. 1-71. Astronomy &
Astrophysics, URL: https://www.cosmos.esa.int/documents/387566/387653/Planck_2018_results_L06.pdf/38659860-
210c-ffac-3921-e5eac3ae4101.
Avelino, Arturo, and Ulises Nucamendi. “Exploring a Matter-Dominated Model with Bulk Viscosity to Drive the Accelerated
Expansion of the Universe.” Journal of Cosmology and Astroparticle Physics, vol. 2010, no. 08, 2010, pp. 009–009.,
doi:10.1088/1475-7516/2010/08/009.
Barbara Sue Ryden.Introduction to cosmology. Cambridge University Press, 2017.
Bolotinet, Yu. L. al. “Cosmology In Terms Of The Deceleration Parameter. Part II.” arXiv, arXiv:1506.08918vl [gr-qc] 30 June 2015.
https://arxiv.org/pdf/1506.08918.pdf
Mathuriya, Amrita et al. “CosmoFlow: Using Deep Learning to Learn the Universe at Scale.” arXiv, Publisher, Publication Date,
arXiv:1808.04728v2 [astro-ph.CO] 9 Nov 2018. https://arxiv.org/pdf/1808.04728.pdf.
S Pfalzner, M B Davies, M Gounelle, A Hohansen, C M ̈unker, P Lacerda, S PortegiesZwart, L Testi, M Trieloff, D Veras, and et al.
The formation of the solar system. Royal Swedish Academy of Sciences, Apr 2015.

31. Questions