Dynamics of neutrino lumps in growing neutrino quintessence

We investigate the formation and dissipation of large-scale neutrino structures in cosmologies where the time evolution of dynamical dark energy is stopped by a growing neutrino mass. In models where the coupling between neutrinos and dark energy grows with the value of the scalar cosmon field, the evolution of neutrino lumps depends on the neutrino mass. For small masses the lumps form and dissolve periodically, leaving only a small backreaction of the neutrino structures on the cosmic evolution. This process heats the neutrinos to temperatures far above the photon temperature, such that neutrinos acquire again an almost relativistic equation of state. The present equation of state of the combined cosmon-neutrino fluid is very close to $-1$. By contrast, for larger neutrino masses, the lumps become stable. The highly concentrated neutrino structures entail a large backreaction similar to the case of a constant neutrino-cosmon coupling. A present average neutrino mass of around 0.5 eV seems to be compatible with observations so far. For masses lower than this value, neutrino-induced gravitational potentials remain small, making the lumps difficult to detect.

Figure 1

Snapshots of the number density contrast of neutrinos $\delta n_{\nu }(\overrightarrow{x})\equiv n_{\nu }(\overrightarrow{x})/{\overline{n}}_{\nu }-1$ at different times. (Left panel) Model M2, at scale factors $a=0.45$, 0.7, 0.75, and 0.95, from top to bottom. The overdensity oscillates between values close to 1 (represented as light tones (yellow in the color version)) at early times, where there are no lumps, to values close to 10 (dark tones (blue and purple in the color version)), where several concentrated lumps form at intermediate times. At later times lumps dissolve and the overdensity decreases back to values close to unity. (Right panel) Model M4, at scale factors $a=0.35$, 0.42, 0.53, and 0.64, from top to bottom. The neutrino lumps start growing at early times and merge progressively into larger and more concentrated structures. At the end, almost all neutrinos are attracted to a single, very massive lump.

Figure 2

Snapshots of the neutrino overdensity field ${\delta }_{\nu }(\overrightarrow{x})$ for models (top left) M4 and (bottom left) M6 at scale factors of $a=0.64$, and 0.62, respectively. In these models, neutrino lumps cluster into large stable structures with a high concentration, starting from a bottom-up approach, as was shown for model M4 in Fig. 1. Neutrino structures occupy large parts of the simulation box, corresponding to scales of $\sim 50\mathrm{Mpc}$. At these scale factors, the forces introduced by the cosmon coupling are too strong to be resolved by our numerical approach, and our simulation breaks down.

Figure 3

Evolution of ${\mathrm{\Omega }}_{\varphi +\nu }$ (darker grey lines, blue in the color version) and ${\mathrm{\Omega }}_{\nu }$ (the lighter grey lines (orange in the color version)) for model M2, compared between the linear output from nuCAMB (the dashed lines) and the nonlinear calculation of the N-body simulation (the solid lines). The total cosmon-neutrino fluid has the same background evolution in the simulation as in the linear calculation. The neutrino energy density is somewhat larger in the simulation and shows a phase shift in its oscillations, as discussed in the text.

Figure 4

Equation of state of the combined neutrino-cosmon fluid $w_{\varphi +\nu }$ (darker lines (blue in the color version)) and equation of state of neutrinos $w_{\nu }$ (lighter grey lines (orange in the color version)). We compare the linear output (the dashed lines) to the nonlinear one obtained from the N-body simulation (the solid lines) for model M2. This model has a time averaged rms mass $⟨m_{\nu }⟩(a_l)=0.164$, where $a_l=0.9$ in the label denotes the center of the time interval $a=[0.8–1.0]$ used to take the average. For $w_{\nu }$, the linear output does not capture the oscillating equation of state of neutrinos due to the formation of structures, while, for $w_{\varphi +\nu }$, both codes agree relatively well. At late times, the equation of state predicted by the simulation has a somewhat higher value and is phase shifted due to the heating of the neutrino fluid.

Figure 5

Neutrino mass ${\overline{m}}_{\nu }$ (average over simulation volume) in model M2 (dotted line, blue in the color version) and model M1 (solid line, orange in the color version), as a function of the scale factor $a$, for the N-body simulation. The horizontal lines show the time averaged rms value at a late time $a_l=0.9$, denoting the center of the time interval of $[0.8:1.0]$ considered for taking the average. For model M2, the time averaged neutrino mass is $⟨m_{\nu }⟩(a_l)=0.164$ (dark grey dashed line, dark blue in the color version), while for model M1 it is somewhat smaller $⟨m_{\nu }⟩(a_l)=0.120$ (light grey dashed line, dark orange in the color version). One can observe that the oscillation frequency is higher for the smaller $⟨m_{\nu }⟩$ mass and the peaks are higher for the larger $⟨m_{\nu }⟩$. The present neutrino masses of the two models calculated in linear theory differ by an order of magnitude; on the contrary, the time averaged masses are very close to each other.

Figure 6

Distribution of the momenta of the neutrino particles in the simulation for two different times, $a=0.75$ (the dark shade (purple in the color version)) and $a=1.0$ (the lighter shade (orange in the color version)), compared to a Fermi-Dirac distribution with a temperature given by the mean of the distribution (the dashed lines). (Left panel) For model M1, the Fermi-Dirac fits very well for temperatures of $\overline{T}=0.018\mathrm{eV}$ and $\overline{T}=0.077\mathrm{eV}$, respectively, for each scale factor. (Right panel) For model M2, the fit is also good, with the corresponding temperatures being $\overline{T}=0.018\mathrm{eV}$ and $\overline{T}=0.065\mathrm{eV}$. The CMB photon temperature is $2.35\times {10}^{-4}\mathrm{eV}$, this means that the nonlinear cosmon-neutrino interactions heat the neutrino background by more than a factor of 100.

Figure 7

Power spectra of the total gravitational potential ${\mathrm{\Phi }}_t$ (the darker grey lines (blue in the color version)) and of the neutrino contribution ${\mathrm{\Phi }}_{\nu }$ (the lighter grey lines (orange in the color version)) for models M2 and M5 at the scale factors $a=0.40$ (the solid lines) and $a=0.65$ (the dashed lines), as a function of scale. Model M2 has a rms time averaged neutrino mass $⟨m_{\nu }⟩(a_e)=0.07$, where $a_e=0.5$ stands for the central time in the interval $a=[0.4–0.6]$ used to take the average. Model M5 has in the same interval a higher rms mass of $⟨m_{\nu }⟩(a_e)=0.40$. In the first model, ${\mathrm{\Phi }}_t$ at large scales is of the order of ${10}^{-5}$, while ${\mathrm{\Phi }}_{\nu }$ is 3 to 4 orders of magnitude smaller at both cosmological times. For model M5, in which neutrino lumps are stable and growing, one sees that, at large scales, the total ${\mathrm{\Phi }}_t$ starts with a value of ${10}^{-5}$ at $a=0.4$, but it reaches ${10}^{-4}$ at later times. At $a=0.65$, the neutrino contribution is dominant and neutrino structures have migrated from small scales to large scales, as can be seen from the dip in ${\mathrm{\Phi }}_{\nu }$ at modes where $k=0.2-1.0\mathrm{h}/\mathrm{Mpc}$.

Figure 8

Power spectra of the matter gravitational potential ${\mathrm{\Phi }}_m$ (the darker grey lines (blue in the color version)) and of the neutrino contribution ${\mathrm{\Phi }}_{\nu }$ (the lighter grey lines (orange in the color version)) at two different scales, $k=0.016$ (the solid lines) and $k=0.116$ (the dashed lines) as a function of scale factor $a$ with different scales on the top and bottom axis. For model M2, the matter gravitational potential ${\mathrm{\Phi }}_m$ is, at most, ${10}^{-5}$ at all times, while the neutrino contribution is 2 to 3 orders of magnitude smaller and displays time oscillations. In clear contrast, the neutrino contribution from model M5 for very large scales, reaches and dominates over the matter contribution for $a\gtrsim 0.5$ and pushes the total $\mathrm{\Phi }$ to high values that would be ruled out by observations; see also Fig. 7. The rms neutrino mass has been taken in the same interval range as for Fig. 7, where $a_e=0.5$.

Figure 9

(Left panel) Line plot through the main diagonal of the simulation box for model M2 at $a=0.75$. The negative of the neutrino contribution to the gravitational potential ${\mathrm{\Phi }}_{\nu }$ (the dot-dashed lighter lines (blue in the color version)) oscillates in the range $\pm 1.0\times {10}^{-7}$. This is correlated to the neutrino density contrast (the black solid lines) and the number density contrast $\delta n_{\nu }=n_{\nu }(\overrightarrow{x})/{\overline{n}}_{\nu }-1$ (the darker dashed lines (orange in the color version)), reaching values of up to 1.5. (Right panel) Snapshot of the same simulation, showing a equipotential contour of the gravitational potential, for ${\mathrm{\Phi }}_{\nu }=+1.0\times {10}^{-7}$ in lighter areas (yellow in the color version), and the small but dense neutrino lumps in darker areas (blue, purple and red in the color version), corresponding to density contrasts $\delta n_{\nu }$ of 1.5, 2.0, and 4.0, respectively.

Figure 10

Different $w(z)$ curves in the case of model M1 computed with the N-body simulation and compared to linear theory. The dark grey dashed lines (red in the color version) represent $w_{\mathrm{DE}}$ computed from the Hubble function which is an output of the linear code nuCAMB. The lighter solid line (orange in the color version), representing $w_{\nu +\varphi }$ is computed using the standard pressure and density outputs from the simulation. The dark dots (blue in the color version) stand for some set of computed values of $w_{\mathrm{DE}}(z)$ obtained from the Hubble function that is calculated entirely within the N-body simulation. Because of numerical noise in the oscillating derivatives of $E(z)$, there is some scatter in the EOS obtained in this case.