Shan–Chen interacting vacuum cosmology

Natalie Hogg

IPhT CEA Paris-Saclay

Shan–Chen interacting vacuum cosmology, N. B. Hogg and M. Bruni, MNRAS 511 3 2022, 2109.08676

Today's aim

To introduce a new class of interacting vacuum dark energy models and present the first observational constraints on them.

➡ This is motivated by the coincidence problem (relaxed in dynamical dark energy models) and the presence of cosmological tensions (can potentially be resolved in dynamical or interacting models).

The interacting vacuum (FLRW background)

Introduce an energy exchange $Q$ at the level of the conservation equations, \begin{align} \dot{V} &= Q, \\ \dot{\rho} + 3H\rho &= -Q. \end{align}

Covariant formalism ✅
Linear perturbations ✅

The cosmological behaviour of every interacting vacuum model thus boils down to the choice made for the coupling term $Q$. Is there a physically well-motivated choice?

Shan–Chen equation of state

The Shan–Chen equation of state comes from lattice kinetic theory (Shan & Chen 1993), \begin{align} P &= w \rho_* \left[\frac{\rho}{\rho_*} + \frac{g}{2} (1 - e^{- \alpha \frac{\rho}{\rho_*}})^2\right], \end{align}

where $P$ is the pressure and $\rho$ is the energy density of the fluid, $\rho_*$ is a characteristic energy scale and $g$ and $\alpha$ are free (dimensionless) parameters of the model.

➡ Phase transitions between two states naturally arise in fluids with this equation of state.

This was introduced to cosmology by Bini et al. in three works:

• Scalar field inflation and Shan–Chen fluid models, Bini et al., 2014a
• Dark energy from cosmological fluids obeying a Shan–Chen non-ideal equation of state, Bini et al., 2014b
• Late-time evolution of cosmological models with fluids obeying a Shan–Chen-like equation of state, Bini et al., 2016

Example behaviour of a cosmological fluid with the Shan–Chen equation of state: three possible scenarios.

Inspired by the Shan–Chen equation of state, we set

\begin{equation} Q = \dot{V} =-3H q\left[(1+\beta)V+ \frac{\beta g}{2} \rho_* \left(1 - e^{- \alpha \frac{V}{\rho_*}}\right)^2\right], \end{equation}

where $q$ is the coupling strength of the interaction and $\beta$ is a renaming of $w$ since we are no longer dealing with a true equation of state.

Parameter choice

• We're interested in late-time acceleration ➡ fix $\rho_*$ to the critical density at redshift zero.
• $\alpha$ only modifies the steepness of the peak/trough ➡ keep it fixed to Bini et al. best-fit value
• $g$ is degenerate with $\beta$ ➡ keep it fixed to Bini et al. best-fit value
• $q$ and $\beta$ left as the free parameters of the model

Constraining the models

• We implement the Shan–Chen interacting model in CAMB and CosmoMC and perform an MCMC analysis to constrain the free parameters of the model: $q$, $\beta$ and the cosmological parameters
• We firstly keep $\beta$ fixed and sample $q$; then vice versa
• We use the Planck 2018 CMB TTTEEE power spectra, SDSS BAO measurements and the Pantheon SNIa sample as our observational data.

In all of these cases, $\beta = 1/3$.

In all of these cases (except $\Lambda$CDM), $q=-0.1$.

Basic model comparison: $\chi^2$

• No model has a better $\chi^2$ than $\Lambda$CDM
• The best model apart from $\Lambda$CDM is that with $q=-0.1$ and $\beta=1/3$ (dark red)
• The worst model is that with $q=-0.1$ and $\beta=-1/3$ (orange)

Conclusions

• Some interesting behaviour in the Shan–Chen interacting vacuum scenario is possible
• Possibility for the model to be used as an early dark energy is yet to be explored
• Currently no resolution of the cosmological tensions possible

The future

• Is model building and testing (turning the crank) really the way to go?
• Care must be taken that extended models are consistent with more fundamental assumptions (see my talk tomorrow on the resilience of the Etherington–Hubble relation)
• An alternative approach is to attempt to increase the constraining power of current and future data by finding new observables
• One such example: line-of-sight shear in strong gravitational lensing – look out for forthcoming work with Pierre Fleury, Julien Larena and Matteo Martinelli

Thanks for listening!

Shan–Chen interacting vacuum cosmology, N. B. Hogg and M. Bruni, MNRAS 511 3 2022, 2109.08676

📝 natalie.hogg@ipht.fr
🏡 nataliebhogg.com
🐦 @astronat

Back up slides

Covariant description

The energy–momentum tensors of the vacuum and cold dark matter are

\begin{align} \check{T}^\mu_\nu &= -V g^\mu_\nu, \\ T^\mu_\nu &= P g^\mu_\nu + (\rho + P)\, u^\mu u_\nu, \end{align}

where $V$ is the energy density of the vacuum, $P$ is the pressure of the cold dark matter and $\rho$ its energy density.

The coupling between the two is introduced as

\begin{align} \nabla_\mu \check{T}^\mu_\nu &= -\nabla_\nu V = Q_\nu, \\ \nabla_\mu T^\mu_\nu &= - Q_\nu. \end{align}

The geodesic condition

We project the energy–momentum flow 4-vector in two parts, one parallel and one orthogonal to the cold dark matter 4-velocity,

\begin{align} Q^\mu &= Q u^\mu + f^\mu, \end{align}

where $Q=-Q_\mu u^\mu$ represents the energy exchange and $f^\mu$ the momentum exchange between cold dark matter and the vacuum in the frame comoving with CDM, and $f_\mu u^\mu=0$.

Impose geodesic condition: $f^\mu = 0$, so that cold dark matter remains geodesic (no momentum transfer).

Linear perturbations in iVCDM

With linear scalar perturbations about the FLRW background the line element is \begin{align} ds^2 &= -(1+2\phi)dt^2 + 2a\partial_i B dx^i dt \\ &+ a^2[(1-2\psi)\delta_{ij} +2 \partial_i \partial_j E] dx^i dx^j. \end{align}

The perturbed energy conservation equations are (D. Wands et al., 2012) \begin{align} &\delta \dot{\rho}_c + 3H \delta \rho -3 \rho_c \dot{\psi} + \rho_c \frac{\nabla^2}{a^2}(\theta + a^2\dot{E} -aB) = \delta Q - Q \phi, \\ &\delta \dot{V} = \delta Q + Q \phi, \end{align} and the perturbed momentum conservation equations are \begin{align} &\dot{\theta} + \phi = 0, \\ &\delta V = -Q \theta. \end{align}

Gauge choice

Choose the synchronous comoving gauge, $\phi = v = B = 0$. Since \begin{align} \theta &= a(v + B), \\ -\delta V &= Q\theta,\\ \delta V &=0, \end{align} hence the vacuum is spatially homogeneous in this gauge.

Furthermore, since \begin{align} \phi &=0, \\ \delta \dot{V} &= \delta Q +Q \phi,\\ \delta Q &= 0, \end{align} the interaction is also spatially homogeneous in this gauge.

Effect of the interaction on structure growth

This means that the perturbed energy conservation equation reduces to the $\Lambda$CDM case, \begin{align} \delta \dot{\rho}_c + 3H\delta \rho_c - 3\rho_c \dot{\psi} + \rho_c \nabla^2 \dot{E} &=0. \end{align}

The background thus remains unperturbed in the synchronous comoving gauge (as used in CAMB), but the interaction term $Q$ does enter the evolution equation for the CDM density contrast,

\begin{equation} \dot{\delta_c} = \frac{Q}{\rho_c} + 3 \dot{\psi} - \nabla^2 E. \end{equation}

This implies that structure growth will still be affected even though the coupling does not explicitly appear in the perturbation equations.

Consequences of the geodesic condition

• Setting $f^\mu = 0$ means that dark matter is irrotational (H. Borges & D. Wands, 2017)
• However, purely irrotational CDM would lead to very rapid formation and growth of supermassive black holes (I. Sawicki et al., 2013)
• Assumption of pure energy exchange must break down somewhere

Choosing parameters

We can write down an effective equation of state for the interacting model,

\begin{equation} w_{\mathrm{int}} = q(1+\beta) + \frac{q\beta g}{2x} \left(1- e^{-\alpha x}\right)^2 -1, \end{equation}

where $x = V/\rho_*$, so that $Q = -3HV(1+w_{\rm int})$.

All $\chi^2$ values

Case	$\chi^2$	$\Delta \chi^2$
Ia ($\Lambda$CDM)	$3831.38$	$0.0$
Ib	$3831.87$	$0.49$
Ic	$3831.90$	$0.52$
Id	$3831.98$	$0.60$
IIa	$3833.95$	$2.57$
IIb	$3832.76$	$1.38$
III	$3832.20$	$0.82$