Dark interactions beyond the Lambda-CDM model

MA van der Westhuizen and A Abebe, 'Dark coupling: cosmological implications of interacting dark energy and dark matter fluids', SA Inst. In this study, cosmological models are considered in which dark matter and dark energy are coupled and can exchange energy through non-gravitational interactions with each other.

A brief history of cosmology

At the time, Lemaˆıtre believed that if the universe was expanding, distant stars and galaxies should be moving away from us and that their electromagnetic radiation should be Doppler-shifted to longer wavelengths at the 'redder' end of the spectrum [12]. . It was generally accepted that the universe was expanding and was formed during the Big Bang.

Figure 1.1: Expansion of the Universe from the Big Bang [1]

The Einstein field equations

Derivation of the cosmological field equations

The Friedmann-Lemaˆıtre-Robertson-Walker metric
Calculating the Christoffel symbols
Calculating the Ricci curvature tensor
Calculating the scalar curvature
Calculating the stress-energy tensor
Obtaining the 1st Friedmann equation
Obtaining the 2nd Friedmann equation

To calculate the various components of the Einstein field equations for the FLRW metric (1.6), the Christoffel symbols for this metric will first be needed. This will be used to determine the components of the Ricci curvature tensorRαβ (1.10) as well as the scalar curvature R (1.12).

Geometry of the universe

This geometry can be measured directly from the CMB, since a light ray will follow a straight path (zero geodesic) on a curved surface such that lines converge on a sphere (closed universe), remain straight on a flat surface (flat universe) and diverge on a saddle (open universe) [8]. The geometry of the universe also dictates its evolution, as can be seen in Tables 2.1 and 2.2.

Figure 1.2: Curvature of the Universe [1]

The conservation equation

It can be seen from (1.46) that the equation of state of the liquid ωx determines the rate at which its energy density ρx decreases with increasing a as the universe expands. It is worth noting that in general relativity, the energy density ρ and the pressure P of these fluids have different energy conditions that must be considered.

Energy conditions

Components of the universe

Radiation

Taking the density dilution and the cosmological redshift into account, the energy density of the radiation has the dependence ρr = nE = n(hc/λ) ∝ a−3a−1 ∝ a−4 [10]. This coincides with the fact that the pressure P and the energy density ρ of radiation are related to an equation of state ω equal to P = 1/3ρ → ωr = 1/3, so that the energy evolves according to the conservation equation (1.46):.

Matter

Due to these limitations, a possible dark matter candidate has been proposed, called a Weakly Interacting Massive Particle (WIMP) [52]. For our purposes, we only need to assume that dark matter is cold (non-relativistic), as opposed to relativistic hot dark matter (HDM [61]).

Figure 2.1: Rotational velocities of hy- hy-drogen (HI regions) in NGC 3198 [2]

Dark energy

Einstein’s static universe

Hubble and deceleration parameters

Single-fluid cosmological models

Friedmann and Lemaˆıtre cosmological models
Solutions for radiation-dominated Friedmann models
Solutions for matter-dominated Friedmann models
Solutions for cosmological constant-dominated Lemaˆıtre models

The Friedmann equation has been analytically solved for radiation-dominated Friedmann models to obtain expressions for the evolution of the scale factor(t), the time span, and the age of the universe model t0. The Friedmann equation has been analytically solved for cosmological constant-dominated Lemaˆıtre models to obtain expressions for the evolution of the scale factor(t), the time setting, and either the universe model age0 or the time since the non-singular bouncetb.

Table 2.2: Distant past of Lemaˆıtre models (t → −∞)

The ΛCDM model

Cosmic acceleration
Cosmological parameters
Radiation, matter and dark energy dominated epochs
Origin, evolution and fate of ΛCDM model

Thus, it can be seen that the expansion of the ΛCDM model slows down between radiation and matter dominance before accelerating when dark energy becomes sufficiently dominant. Figure 2.11 shows the expansion history of the ΛCDM model (solid purple line), alongside the history of the de-Sitter universe (dashed purple line), the Einstein-de Sitter universe (dashed red line), and the closed matter-dominated universe (solid red line). and a flat, radiation-dominated universe (dashed blue line).

Figure 2.6: Temperature anisotropies of the CMB [1]

Problems with the ΛCDM model

The cosmological constant problem

The largest energy fluctuation must therefore occur within the Planck time and can be defined as Planck energyEPlanck. The predicted vacuum energy density ρΛ is thus obtained by dividing the Planck energy EPlanck (2.36) by the Planck volume vPlanck.

The cosmic coincidence problem

The magnitude of the deviation from ζ = 0, indicates the magnitude of the randomness problem (used from here on to denote the cosmic randomness problem) [82-84]. This implies that models where ωde>−1 alleviate, while ωde<−1 exacerbate the randomness problem.

The Hubble tension

Therefore, any model with ζ > 3 will exacerbate the coincidence problem, while ζ < 3 alleviates the problem (relative to the ΛCDM model). Therefore, the randomness problem motivates research into dark matter and dark energy models that maintain a small or constant ratio r in both past and future expansion.

Quintessence and phantom dark energy

Particle horizon and Hubble distance

This has the physical consequence that the number of visible galaxies will grow as more galaxies enter the horizon [9,95]. In contrast, during dark energy dominance (in the quintessence regime −1 < ω < lt; < 1 and for a cosmological constantω =−1→= 1) the horizon either remains constant or grows more slowly than the scale factor, which causing distant galaxies to disappear beyond the cosmic horizon over time and become imperceptible to us.

Big rip

From the left panel in Figure 2.12 it can be seen that while a cosmological constant has a constant density with expansion, quintessence and phantom dark energy are rather dynamic. Quintessence and phantom dark energy models open up a whole new world of possibilities for what dark energy could be, beyond the cosmological constant Λ.

Interactions between dark matter and dark energy

This creates a mechanism where the components can exchange energy (or momentum) in a non-gravitational way, so that dark matter and dark energy have the following special conservation equations. Here Q is an arbitrary coupling function whose sign determines how energy (or momentum) is transferred between dark energy and dark matter.

Properties of interacting dark energy models

Effective equations of state
Addressing the coincidence problem
Cosmological implications of a dark coupling
Instabilities and the doom factor
Evolution of energy densities and phase portraits
Phase portrait primer: ΛCDM model

The coupling between the dark sectors will affect the evolution of dark matter and dark energy disturbances. For our purposes, we are interested in the parameter space for how dark matter and dark energy evolve in relation to each other. These will be used to see if the relationship between dark matter and dark energy becomes fixed in the past or the present, thus addressing the potential of the model to solve the randomness problem.

Table 3.1: Consequences of interacting dark energy models (relative to uncoupled models)

Phase portraits - Q 1 = δHρ dm
Background analytical equations - Q 1 = δHρ dm
Positive energy conditions - Q 1 = δHρ dm
Cosmic coincidence problem - Q 1 = δHρ dm
Evolution of energy densities and cosmic equalities - Q 1 = δHρ dm
Evolution of deceleration parameter - Q 1 = δHρ dm
Hubble parameter and age of the universe - Q 1 = δHρ dm
Doom factor and big rip - Q 1 = δHρ dm
Concluding remarks on IDE model Q 1 = δHρ dm

The effective equations of state for dark matter and dark energy are therefore the same in the distant past (ωeffdm = ωeffde). This is due to the dark energy density ρde approaching zero and then becoming negative in the past at redshift z(de=0) (indicated by red dashed line) from (3.43). This difference in the slope between how ρdm and ρth redshift is indicative of the coincidence problem.

Figure 3.2: Phase portraits for Ω dm and Ω de (Q 1 = δHρ dm )

Phase portraits - Q 2 = δHρ de
Background analytical equations - Q 2 = δHρ de
Positive energy density conditions - Q 2 = δHρ de
Cosmic coincidence problem - Q 2 = δHρ de
Evolution of energy densities and cosmic equalities - Q 2 = δHρ de
Evolution of deceleration parameter - Q 2 = δHρ de
Hubble parameter and age of the universe - Q 2 = δHρ de
Doom factor and big rip - Q 2 = δHρ de
Concluding remarks on IDE model Q 2 = δHρ de

For this model, the randomness problem is not solved in the past, but instead in the future. The effective equations of state for dark matter and dark energy are therefore the same in the distant future (ωeffdm=ωdeeff). For this model, dark energy never completely dominates in the future, but instead dark matter and dark energy have the density parameters (Ωdm+bm,Ωde)+ = −3ωδ ,1 +3ωδ.

Figure 3.11: Phase portraits for Ω dm and Ω de (Q 2 = δHρ de )

Standard candles and type Ia supernovae

Distance modulus

Distance modulus ΛCDM

The distance modulus for the ΛCDM model is then obtained by substituting the normalized Hubble parameter from (4.7) into (4.6), which gives:

Distance modulus Q 1 = δHρ dm

Distance modulus Q 2 = δHρ de

Cosmological Parameters

Cosmological parameters - ΛCDM
Cosmological parameters - Q 1 = δHρ dm
Cosmological parameters - Q 2 = δHρ de
Implications of observational parameters

Using the parameters from Figure 4.1 along with the distance modulus (4.8), the ΛCDM model can be fit to the supernova data. These values are very small and show how well the ΛCDM model with the parameters from Figure 4.1 fits the data. In Figure 4.4 we can see that this model fits the supernova data well, as the predicted distance modulus corresponds to the data points.

Figure 4.1: MCMC simulation results - (ΛCDM)

Statistical analysis

AIC and BIC values

This result, δ > 0, physically implies energy flow from dark energy to dark matter (iDEDM regime). These values tell you whether your model can be considered valid against the "real" model. It should be noted that this is "supported" relative to the "true" (ΛCDM) model.

Results from the statistical analysis

This encourages research into dark matter and dark energy models that maintain a small or constant ratio in the past or future expansion. This would mean that the currently observed relationship between dark matter and dark energy is not a coincidence, but a general property of the universe, which solves the problem of coincidence. Since negative energy densities are unphysical (even those predicted to appear in the future), the iDMDE regime should not be taken seriously as a potential candidate for dark energy.

Calculations for a radiation-dominated flat model

Calculations for a radiation-dominated closed model

From the solution, it can be seen that this model has a cyclic nature with a big bang at θ= 0 and a big crunch at θ =π (forming a semi-circle). The amount of time until the big contraction tc can therefore be calculated by the difference between the lifetime of the universe tl (A.14) and the current age t0 (A.55):.

Calculations for a radiation-dominated open model

Calculations for matter-dominated Friedmann models

Calculations for a matter-dominated flat model

Calculations for a matter-dominated closed model

From the solution it can be seen that this solution is a cycloid as shown in Figure 2.4. So this universe model has an infinite age, consisting of successive big bounces and big crunches. The amount of time until the next bounceN b can therefore be calculated by the difference in time between one bounce cycleb (A.42) and the time since the previous bounce tP b (A.41): A.43) It can also be noted that the scale factor reaches a maximum size amaxatθ=π, so from (A.37) we have:.

Calculations for a matter-dominated open model

Calculations for cosmological constant dominated Lemaˆıtre models

Calculations for a cosmological constant-dominated flat model

Because of this exponential nature of the solution, it can be inferred that the size of the universe will decrease in the past, but will have no big bang singularity.

Calculations for a cosmological constant-dominated closed model

From this solution it is clear that the expansion has a parabolic nature and therefore will not have a big bang, but will have a non-singular big rebound at some finite time in the past. The time since the bounce tb will be the time elapsed for the universe to grow from size ab to size a0 = 1.

Calculations for a cosmological constant-dominated open model

Solving the Friedmann Equation with 4th-order Runge-Kutta Method

In this appendix, the equilibrium points of the phase portraits (Figures are calculated for ΛCDM models, Q1=δHρdmandQ2 =δHρde. We do this by setting the derivative of the energy densities to zero in (3.22) and obtaining the coordinates for the equilibrium points.

Calculations for dark matter energy density ρ dm

Calculations for dark energy density ρ de

This can also be written in terms of the effective dark matter equation of state ωdmeff =−δ3 (3.36), so that: C.4), which reduces back to the uncoupled case if δ= 0 or ωdmeff =ωdm. To solve the linear differential equation of the first order, we invent two new functions u and v, so that y = uv and therefore dxdy = udvdx+vdudv.

Calculations for dark energy density ρ de
Calculations for dark matter energy density ρ dm
Calculations for the radiation-matter equality
Calculations for the matter-dark energy equality
Calculations for the cosmic jerk

In this appendix, explicit equations will be derived for the scale factor a and redshift z where important events in cosmic history took place. These events are the radiation-matter equation (initiating the period of matter dominance), the matter-dark energy equation (starting the dark energy-dominant epoch), and the cosmic jerk; when the cosmic acceleration starts. In the ΛCDM model, the universe consists of radiations, matter (which is the sum of baryonic and dark matter, since both evolve as ρbm∝ρdm∝a−3) m=dm+bm and dark energy de.

Calculations for the radiation-matter equality

Calculations for the matter-dark energy equality

Calculations for the cosmic jerk

Calculations for the radiation-matter equality

Calculations for the matter-dark energy equality

Calculations for the cosmic jerk

Expansion of the Universe from the Big Bang [1]
Curvature of the Universe [1]
Rotational velocities of hydrogen (HI regions) in NGC 3198 [2]
Strong gravitational lensing from dark matter around galaxy cluster CL0024+17 [3]. 17
Evolution of matter-dominated Friedmann models
Evolution of cosmological constant-dominated Lemaˆıtre models
Temperature anisotropies of the CMB [1]
Evolution of energy densities ρ with redshift (1 + z) (ΛCDM)
Evolution of density parameters Ω with redshift (1 + z) (ΛCDM)
Evolution of the effective equation of state ω eff with redshift (1 + z) (ΛCDM)
Evolution of deceleration parameter q with redshift (1 + z) (ΛCDM)
Origin, evolution and fate of ΛCDM model
Evolution of energy density and scale factor for quintessence and phantom dark
Phase portraits for Ω dm and Ω de (ΛCDM)
Phase portraits for Ω dm and Ω de (Q 1 = δHρ dm )
Coincidence problem and effective equations of state (Q 1 = δHρ dm )
Energy densities ρ vs redshift - (Q 1 = δHρ dm )
Density parameters vs redshift - (Q 1 = δHρ dm )
Evolution of effective equation of state ω eff with redshift (Q 1 = δHρ dm )
Evolution of deceleration parameter q with redshift (1 + z) (Q 1 = δHρ dm )
Relative Hubble parameter (H/H δ=0 ) vs redshift (Q 1 = δHρ dm )
Evolution of scale factor with time (Q 1 = δHρ dm )
Evolution of energy density, scale factor and the big rip for phantom (ω = −1.15)
Phase portraits for Ω dm and Ω de (Q 2 = δHρ de )
Coincidence problem and effective equations of state (Q 2 = δHρ de )
Energy densities ρ vs redshift - (Q 2 = δHρ de )
Density parameters vs redshift - (Q 2 = δHρ de )
Evolution of effective equation of state ω eff with redshift (Q 2 = δHρ de )
Evolution of deceleration parameter q with redshift (1 + z) (Q 2 = δHρ de )
Relative Hubble parameter (H/H δ=0 ) vs redshift (Q 2 = δHρ de )
Evolution of scale factor with time (Q 2 = δHρ de )
Evolution of energy density, scale factor and the big rip for phantom (ω = −1.15)
Evolution of energy density, scale factor and the big rip for vacuum (ω = −1),
MCMC simulation results - (ΛCDM)
ΛCDM model fitted with supernovae data
MCMC simulation results - (Q 1 = δHρ dm )
Q 1 = δHρ dm model fitted with supernovae data
MCMC simulation results - (Q 2 = δHρ de )

This is because this model solves the randomness problem for the future expansion, thereby adjusting the ratio of dark matter to dark energy. For this to hold, we may need the effective equation of state for dark energy to be less than −1. which is the predicted major recovery time for the IDE model Q2 = δHρde. Newhubble Space Telescope discoveries of the type ia supernova atz 1: Tight constraints on the early behavior of dark energy.