Lecture on Fuzzy Dark Matter Cosmology

Bodo Schwabe

Overview

Cold dark matter explains cosmological observations
But potential short comings on galactic scales
Fuzzy dark matter theory
Fuzzy dark matter observables

Successes of cold dark matter

( + cosmological constant )

Centripetal force = Gravitational force

\frac{v^2(r)}{r} = \frac{Gm(r)}{r^2}

v(r) = \sqrt{\frac{Gm(r)}{r}}

See also arXiv:1905.08103 for a jupyter notebook to reproduce this data

Potential shortcomings on "small" galactic scales

Title Text

The ΛCDM model encounters problems in describing structures at small scales. The main problems are/have been

The "cusp/core problem", designating the discrepancy between the flat density profiles of dwarf galaxies, Irregulars, and Low Surface Brightness galaxies, and the cuspy profile predicted by dissipationless N-body simulations, despite the fact that the observed galaxies are all of DM dominated types.

[arXiv:1606.07790 ]

[arXiv: 1011.2777]

Title Text

The ΛCDM model encounters problems in describing structures at small scales. The main problems are/have been

The “missing satellite problem”, coining the discrepancy between the number of predicted subhalos in N-body simulations and those actually observed.

[arXiv:1606.07790 ]

[arXiv:astro-ph/9901240]

Title Text

The ΛCDM model encounters problems in describing structures at small scales. The main problems are/have been

,“Too Big To Fail” problem, arising from the ΛCDM prediction of satellites that are too massive and too dense, compared to those observed, to hope for their destruction in the history of mass assembly up to today.

[arXiv:1606.07790 ]

[arXiv:1111.2048]

All of these problems have in common that N-body simulations modelling cold dark matter predict more substructure on galactic scales

[arXiv:1301.4498]

Fuzzy dark matter theory

The Cosmological Axion/FDM Field

\ddot{\phi}+3H\dot{\phi}-\nabla^2\phi+m_a^2\phi=0

Axion equation of motion:

Background energy density and pressure:

\overline{\rho}_a=\frac{1}{2}\dot{\phi}^2+\frac{1}{2}m_a^2\phi^2 \\ \overline{P}_a=\frac{1}{2}\dot{\phi}^2-\frac{1}{2}m_a^2\phi^2 \\ \overline{P}_a=\omega_a\overline{\rho}_a

\phi=a^{-3/2}(t/t_i)^{1/2}[C_1J_n(m_at)+C_2Y_n(m_at)]

Solution:

[arXiv:1510.07633]

The non-relatistic limit

[arXiv:2104.10128]

\phi = \frac{\hbar}{\sqrt{2}m}\overline{\psi} e^{-imt/\hbar}

i\hbar\partial_{t}\psi = -\frac{\hbar^{2}}{2a^{2}m}\nabla^{2}\psi+mV\psi

\left(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-\frac{m^{2}}{\hbar^{2}}\right)\phi = 0

T_{\mu\nu} = \nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(g^{\rho\sigma}\nabla_{\rho}\phi\nabla_{\sigma}\phi+\frac{m^{2}}{\hbar^{2}}\phi\right)

R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi GT_{\mu\nu}

\nabla^{2}V = 4\pi Ga^{-1}\delta\rho

\delta\rho=|\psi|^{2}

Madelung Transformation - Bohmian Quantum Mechanics

i\hbar\partial_{t}\psi = -\frac{\hbar^{2}}{2a^{2}m}\nabla^{2}\psi+mV\psi

\psi = \sqrt{\rho}e^{iS/\hbar}

\partial_{t}\sqrt{\rho} = -\frac{1}{2ma^{2}}\left[2(\nabla\sqrt{\rho})\cdot(\nabla S)+\sqrt{\rho}\nabla^{2}S\right]

imaginary part

\textbf{v}=\nabla\frac{S}{m}

\partial_{t}\rho+\frac{1}{a^{2}}\nabla\cdot(\rho\textbf{v}) = 0

real part

\partial_{t} S = -\left[\frac{(\nabla S)^{2}}{2a^{2}m}+mV+Q\right]

Q = -\frac{\hbar^{2}}{2ma^{2}}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}

\partial_{t}\textbf{v}+\frac{1}{a^{2}}\textbf{v}\cdot\nabla\textbf{v}+\nabla V+\nabla Q = 0

Euler equations of fluid dynamics + quantum corrections !

Schroedinger-Vlasov correspondence similarly to Ehrenfest theorem tells you that FDM behaves like CDM on large scales where gradient energy averages out !

[arXiv:1403.5567]

Linear growth of perturbations

a^{2}\partial_{t}\delta\rho+\rho_{0}\nabla\cdot\textbf{v} = 0

\rho_{0}\partial_{t}\textbf{v}+\rho_{0}\nabla V-\frac{\hbar^{2}}{4m^{2}a^{2}}\nabla\left(\nabla^{2}\delta \rho\right) = 0

Linearize Euler equations:

Take time derivative of the first equation and spatial derivative of the second one, combine and take Fourier transform

\partial_{t}^2\rho_{k}+2H\partial_{t}\rho_{k}+\left(\frac{\hbar^{2}k^{4}}{4m^{2}a^{4}}-4\pi G \rho_0\right)\rho_{k} = 0

Last term vanishes at Jeans scale

k_J=(16 \pi Ga \rho_{0})^{1/4}m^{1/2}

Jeans scale vs. deBroglie wavelength

\lambda_{\rm dB}=2\pi\hbar(mv)^{-1}

\simeq\hbar m^{-1}(G\rho)^{-1/2}r^{-1}

v\simeq r/t_{\text{ff}} = r\sqrt{G\rho}

\lambda_{\rm dB}\equiv r

\lambda_{\rm dB}\simeq r_J

\simeq 66.5 a^{1/4}\left(\frac{\Omega_a h^2}{0.12}\right)^{1/4}\left( \frac{m}{10^{-22}\text{ eV}}\right)^{1/2}\text{ Mpc}^{-1}

Linear growth of perturbations

\partial_{t}^2\rho_{k}+2H\partial_{t}\rho_{k}+\left(\frac{\hbar^{2}k^{4}}{4m^{2}a^{4}}-4\pi G \rho_0\right)\rho_{k} = 0

\delta_a=C_1 D_+(k,a)+C_2D_-(k,a)

D_+(k,a)= \frac{3\sqrt{a}}{\tilde{k}^2}\sin \left( \frac{\tilde{k}^2}{\sqrt{a}} \right) +\left[ \frac{3a}{\tilde{k}^4}-1 \right]\cos \left( \frac{\tilde{k}^2}{\sqrt{a}} \right)

D_-(k,a)= \left[ \frac{3a}{\tilde{k}^4}-1\right] \sin \left( \frac{\tilde{k}^2}{\sqrt{a}} \right) -\frac{3\sqrt{a}}{\tilde{k}^2}\cos \left( \frac{\tilde{k}^2}{\sqrt{a}} \right)

\tilde{k}=k/\sqrt{m_aH_0}\propto k/k_J