Cosmic acceleration driven by dark matter self-interactions: a phenomenological treatment

TL;DR

This paper proposes a model where dark matter self-interactions lead to negative pressure effects, potentially explaining cosmic acceleration and resolving quasar Hubble diagram tension, through a phenomenological approach inspired by structure formation.

Contribution

It introduces a novel scenario linking dark matter self-interactions to cosmic acceleration, supported by a phenomenological analysis fitting supernova and quasar data.

Findings

Model fits supernova and quasar data well

Resolves quasar Hubble diagram tension

Suggests dark matter self-interactions influence cosmic acceleration

Abstract

We explore the idea that cosmic acceleration may be a byproduct of late-time effects like structure formation in two steps. First, we consider the equation of state for an inhomogeneous cosmic fluid, which may lead to a Gedanken-model for cosmic evolution, where dark matter is strongly self-interacting and stays in a plasma state until late stages of the cosmic evolution. After decoupling, it condensates to super-structures with cosmic voids similar to the current picture of the universe, introducing a negative pressure term in relation to self-interaction strength. Secondly, we carry out a cosmological analysis inspired by this scenario via a phenomenological ansatz that exhibits a transient behavior. In this analysis, we use the recent Type Ia supernova compilation and high redshift quasar data and compare the results to that of $\Lambda$CDM. It turns out that proposed model can solve the quasar Hubble diagram tension.

Tables

Table 1. Table 1: Parameters for DM-interaction driven accelerated expansion model and Λ Λ \Lambda CDM. H 0 subscript 𝐻 0 H_{0} values are calculated for SNe Ia absolute magnitude interval of 19.2 < M < 19.3 19.2 𝑀 19.3 19.2<M<19.3 from [ 16 ]

	$\alpha$	$\beta$	${\Omega }_{M,0}$	$H_0$	${\chi }^2$/dof
interacting DM	0.03	600	0.35	72.4-76.1	1021.28/1044
$\Lambda$CDM	-	-	0.29	71.9-75.8	1024.35/1046

Equations

$$pV+\frac{2}{3}\sigma A=NkT$$

$$p=-\frac{2}{3}\frac{\sigma A}{V}.$$

$$q=\frac{1}{2}\left(1-2\frac{{\rho }_s}{{\rho }_c}\right)$$

$${\rho }_s=\frac{U}{V}\sim G{\sigma }_s^2$$

$${\sigma }_s{c}^2={\rho }_c\frac{V}{A}.$$

$$q=\frac{1}{2}\left(1-\frac{H^2{R}^2}{3\pi c^2}\right)$$

$${\rho }_{\mathrm{DE}}(z)={\rho }_0(z+1{)}^{\alpha }(10^3-z{)}^{\beta }10^{-3\beta }$$

$$d_L(z)=\frac{(z+1)c}{H_0}\int \frac{dz}{\sqrt{{\Omega }_{\mathrm{M}}+{\Omega }_{\mathrm{DE}}(z+1{)}^{\alpha }(10^3-z{)}^{\beta }10^{-3\beta }}}$$

Full text

1 Introduction

Despite being strongly favored by cosmological data, $\Lambda$CDM –the standard model of cosmology– still lacks convincing explanations to its two well-known setbacks: (i) the fine-tuning problem; the low but nonzero value of the observed vacuum energy density in comparison to the prediction coming from quantum field theory [1] and (ii) the coincidence problem; the surprisingly close present values of energy densities for matter and dark energy components in the cosmic fluid, a problem which implies that we live in a very special era in the cosmic lifetime [2]. One can argue whether those problems are relevant from a cosmological point of view or not [3]; however, it is still reasonable to invert this set of problems in an attempt to make sense of the cosmic puzzle of acceleration: It would be pleasing to come up with a cosmic scenario, where there is no dark energy and the acceleration of the universe is triggered by an event that took place in the recent cosmic history.

Cosmic-scale events that we can attribute to late time evolution are scarce, and they are mostly related to structure formation. The first stars are born around z$\sim$15 [4], causing a reionization period, an effect that we can single out from cosmological observations. A period of nonlinearization and cosmic structure growth, which can be regarded as a still ongoing process, follows reionization. A hierarchy of structures is pretty much observable to our instruments, starting from galaxies that form into clusters and further super-structures and voids of various sizes between them. Distribution of dark matter (DM) is not far from the visible one, according to the weak lensing observations that give large scale distribution of this mysterious component of cosmic fluid [5].

Deviating from cosmological principle and taking this inhomogeneity into account to see if it can be an explanation to observed cosmic acceleration is not a new idea among cosmologists. A fair amount of work argues that backreaction of matter inhomogeneity may be the reason for the observed acceleration [6]. Einstein’s field equations can be solved in a perturbed background as well, and the deceleration parameter that is also weakly dependent on space in addition to its usual time dependence can be served as an alternative [7].

Nevertheless, the fact that the universe is not exactly homogeneous or isotropic does not disclaim the idea that the universe is still at least statistically homogeneous and isotropic at large scales; the probability of deviating from average density is the same for the whole space. It is fair to assume that the cosmic structure/fluid follows a similar void-filament pattern everywhere in the universe. At this point, it is also fair to ask the following question: Is it possible to propose a cosmic fluid whose inhomogeneous nature is implicitly given in its equation of state; and to solve Einstein’s equations implementing such fluid in a Friedman-Robertson-Walker (FRW) setting?

Fortunately, such an equation of state was proposed previously in the context of fluid dynamics for dry foam (bubbles with ideal gas in them); a fluid consisting of walls and voids [8]:

[TABLE]

Here $p$ and $V$ are the total pressure and volume of the system, $\sigma$ is the surface tension on the bubble surfaces, $A$ is the total area of the interfaces between bubbles, $N$ is the number of ideal gas particles, $T$ is the temperature, and $k$ is the Boltzmann constant. If we adapt this model to cosmology, we may assume that almost all matter is concentrated in thin walls of structure and $T\sim 0$. So, it is possible to come up with a negative pressure term in the form of tension in structure walls,

[TABLE]

We can think of this tension as the repelling part of gravity since pressure counterintuitively contributes to attraction in general relativity. The term “$\sigma A/V$” can be treated as the surface energy per volume and will be denoted by $\rho_{s}$ from now on.

If we solve Einstein’s equations with (2) for the spatially flat case of the FRW metric, the deceleration parameter takes the form,

[TABLE]

where $\rho_{c}$ is the critical density. One can easily see that if $\rho_{s}=\rho_{c}$, i.e., all energy density in the universe is in the form of surface energy, we recover $q=-\frac{1}{2}$, the value for a universe dominated by cosmic branes.

If we move on without introducing any exotic components like dark energy or cosmic branes, it is convenient to interpret this tension energy as Newtonian gravitational potential within the DM structure. Assuming that we have sheets with uniform mass density, we make the estimation,

[TABLE]

where $U$ is the gravitational potential energy, $\sigma_{s}$ is the surface mass density, and $G$ is the gravitational constant. We assume that the voids are almost empty, so

[TABLE]

The important parameter, $V/A$, is the typical volume-to-surface area ratio for a cosmic void. Assuming spherical voids, this ratio is given by $2R/3$, in terms of void radius $R$. A factor of 2 was introduced to avoid the double-counting of interfaces. Rearranging terms, we get the following equation for the deceleration parameter:

[TABLE]

We can make an estimation for the term with the negative sign to see if it can sustain any acceleration. The Hubble parameter can be estimated as $H\sim 70$ km/s/Mpc [9], and the average void radius from surveys is $R\sim 100$ Mpc [10]. It turns out that the introduced contribution is only about $10^{-6}$. We can also calculate the necessary void size for acceleration (e.g., $q=-1/2$), which is about $R\sim 10^{5}$ Mpc, bigger than the Hubble horizon itself.

It would be naive to expect a gravity-only tension within the cosmic structures to drive the cosmic acceleration. But we are well aware that gravity is not the only long-range interaction in the universe, and it is actually the weakest by far. To assume that DM particles are not interacting with each other is still part of the benchmark cosmology, but this assumption is being heavily argued lately [11]. Actually, it is natural to think that DM particles should be interacting with each other in a yet unknown non-standard model mechanism, like every other particle in the universe does through some interaction other than gravity.

Once taking self-interacting DM models into account, we would like to rewind the cosmic evolution to identify a past DM plasma stage where the universe was small and too hot for DM to sustain any structure. Such a cosmic dark plasma scenario was considered in the literature [12], but there is no reason to expect such an era to take place before the photon-baryon decoupling. On the contrary, considering that DM is five times denser than the baryonic matter in the universe, it is possible that DM-plasma would decouple much later than photon-baryon plasma, depending on the type and strength of the DM self-interaction itself. Recent observations of early galaxies with no DM can be regarded as hints of yet uncoupled DM-plasma at that time [13].

2 Analysis

Is it possible to construct a model for DM interactions as a function of redshift so that we can compare it to cosmological datasets? We assert that those interactions would start to affect cosmic expansion after a hypothetical DM-plasma decoupling and start to increase as cosmic structures evolve (such an increasing energy density would be phantom by definition). It is also plausible to think that they will start to lose their strength after pinching of DM walls and filaments when the universe is diluted enough.

However complicated to construct a cosmological model exactly from DM-interactions may seem, we can still come up with a semi-phenomenological model for this DM-interaction-motivated “dark energy” density (inspired by the well-known Beta distribution function):

[TABLE]

Our assumption in this model is that that type of energy content will have zero effect at around $z=1000$ when the universe was significantly homogeneous and at $z=-1$ when the expansion goes to infinity. The last factor is introduced as a “$z=0$” correction (There are also different phenomenological models for dark energy density in literature with different motivations [14]). The luminosity distance function for this model, assuming zero spatial curvature, would be given as,

[TABLE]

We neglected the radiation component because we are only interested in late-time cosmological data, i. e. Type Ia supernovae (SNe Ia).

We use the recent Pantheon compilation [15] to estimate the model parameters via standard $\chi^{2}$ minimization. Best values are given in Table 1.

The obtained value 600 for $\beta$ may seem unusual for a parameter that should also be estimated from fundamental physics; however, this is only a byproduct of standard redshift parametrization. For example, using $y$-redshift parametrization introduced in [17], $\beta$ would be around 1 for the same ansatz.

It is yet difficult to distinguish two models statistically for the redshift range and precision of SNe Ia data. Two energy densities differ more drastically for higher redshifts (Figure 1). Maybe more precise gamma-ray burst observations up to redshifts $\sim$10 can break the tie.

Matter-dark energy equality is shifted towards more recent times for the introduced model with respect to $\Lambda$CDM, as expected, and phantom behavior continues past $z=0$. Parameter $\alpha$ (“future” side of the curve) is lightly constrained by data. However, we see that DM-interactions dominate the cosmic expansion up to the point where the universe becomes 20 times its current size and then lose their effect.

3 Conclusion

Works that relate self-interaction of DM to the late acceleration of the universe exist in the literature [18]; however, with no emphasis on structure formation as a triggering mechanism. There are also works that suggest an acceleration driven by collapse and tension in the emerging structures [19], but the idea is limited to the gravitational interaction.

It is too early to speculate on the type or strength of the DM self-interactions; we are still far from telling that they even exist (for a discussion of astrophysical effects of DM-interactions see [20]). But we can lay down a framework for our cosmic scenario assuming that DM is self-interacting:

First of all, the DM self-interaction should be strong enough –maybe on the electromagnetic scale– to support high tensions that can cause negative deceleration. Secondly, formed DM structures should not be neutral, unlike structures bound under electromagnetism, or they should at least expose strong van der Waals type leaks, to reach intergalactic scales, as gravity does. Additionally, DM should stay in a plasma state up until late times, maybe a couple of redshifts late, in accordance with the beginning of the acceleration epoch. Lastly, a fast condensation reminiscent of a phase transition or a more complex chemical interaction picture that results in strong bonds between DM-particles and substructures may be needed to avoid early pinching of the cosmic DM-filaments/walls.

An increased number of constraints, in this case, does not necessarily mean that we are dealing with a more complex and fine-tuned model. One should keep in mind that models that include dark energy still include dark matter, maybe an already self-interacting one. We have argued that, if this interaction has certain properties, the apparent acceleration may be explained without the need for dark energy.

We depend on future observations to see if this mechanism is viable within a reasonable interaction picture. In the meantime, computer N-body simulations, running on different types of DM self-interaction models, would be the way to get the most out of this scenario and to see if a strong enough mechanism can be found to drop dark energy from the cosmic picture; to be replaced with particle interactions, a more familiar and natural, less exotic concept.