Structure formation and the matter power-spectrum in the $R_{\rm h}=ct$ universe

Our current view of cosmic evolution holds that large-scale structure originated from primordial quantum fluctuations, believed to have been seeded during inflation in the early Universe [1, 2, 3, 4, 5, 6]. According to this theory, quantum fluctuations classicalized and grew due to self-gravity to form the inhomogeneous Universe we see today. The existence of these inhomogeneities in the CMB and matter power spectrum [7, 8, 9] has become quite evident with the advent of high precision measurements. Moreover, the observation of galaxy rotation curves [10, 11, 12] and weak lensing measurements [13, 14, 15, 16, 17] have indicated a necessity for the existence of dark matter. Hence primordial fluctuations must include dark and baryonic matter, and radiation. Dark matter is assumed to be collisionless and non-interacting, so its fluctuations grew due to self-gravity only, without the suppression from radiation pressure. Once dark matter perturbations grew beyond a critical limit, they formed bound objects [18]. In contrast, baryons could not form bound objects in the early Universe because their growth was suppressed by the radiation to which they were coupled. As the baryons gradually decoupled from the radiation, however, they accreted into the potential wells created by dark matter to form the large-scale structure we see in the Universe today. This scenario—in which dark matter participated in the formation of structure in the early Universe—is indispensable to $\Lambda$CDM. Without the dark matter, structure formation would have been delayed by the baryon-radiation coupling, creating an inconsistency with the observations.

But the high precision data coming down the pipeline over the past decade have created a conflict with the timeline of structure formation in $\Lambda$CDM, even with the contribution due to dark matter. In particular, the discovery of SDSS010013.02+280225.8, an ultraluminous quasar at redshift $z=6.3$ [19], has exacerbated the problem of supermassive black-hole growth and evolution in the early Universe [20]. Almost all of the 122 previously discovered quasars at redshifts $z\approx 6$ [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] have a black hole with mass $\sim 10^{9}\;M_{\odot}$, challenging the standard model’s predicted timeline, which would have afforded them less than 900 Myr to grow after the big bang, but likely even less than $\sim$ 500 Myr since the onset of Population II star formation.

In the context of $\Lambda$CDM, the formation of Pop III stars might have occurred by redshift $z\approx 20$ [38, 39, 40, 41], corresponding to a time of 200 Myrs. Then the subsequent transition to Pop II stars would have taken at least 100 Myrs because the gas expelled by the first generation stars had to cool and recollapse [42, 43]. These Pop II stars would have acted as seeds for black holes. If the early black holes grew according to the Eddington rate, it would have taken at least 930 Myrs for them to reach their observed mass. The observed quasar at redshift $z=6.3$, however, would have been 880 Myrs old in $\Lambda$CDM, which is not consistent with how these objects grew.

Two possibilities have been proposed to reconcile this problem: 1) the black holes might have grown with highly anomalous accretion rates, but this phenomenon is not observed anywhere in the Universe; 2) Pop III stars might have formed earlier than expected, thus partially reconciling the problem. But this too doesn’t appear to be reasonable according to various simulations [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. In addition, the structure formation timeline in $\Lambda$CDM predicts that at redshifts $z=4-8$ most massive galaxies should have been transitioning from an initial halo assembly to baryonic evolution. But the recent Cosmic Assembly Near-infrared Deep Extragalactic Survey (CANDELS)[56, 57] and Spitzer Large Area Survey with Hyper-Suprime-Cam (SPLASH) surveys have found that these massive halos formed much earlier than predicted by $\Lambda$CDM, giving to rise to what some have called ‘The Impossibly Early Galaxy Problem’ [58, 59, 60]. Attempts to reconcile this problem have indicated that a $\sim 0.8$ dex change is required in the halo to stellar mass ratio over this redshift range, though such a drastic change could come about only with a complete absence of dark matter at redshift 8 or with essentially $100\%$ of the baryons condensing into stars at higher redshifts [58]. Both of these scenarios constitute implausible physics, such as the need to convert all of the baryons into stars instantly upon halo virialization (see ref. [58] and references cited therein). Other attempted remedies, such as an evolution of the halo mass to light ratio, could reconcile the problem, but could only happen if the initial mass function were top-heavy. Studies have shown that the ratio of halo mass to light should remain the same, at least within the redshift range ($4\leq z\leq 8$).

In this paper, we present the formation of structure in $R_{\rm h}=ct$ [61, 62, 63, 64, 65, 66], an alternative Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology with zero active mass [65, 66]. The linear expansion in this model afforded sufficient time for supermassive objects to form, thus solving the ‘too early’ appearance of massive quasars [20]. Even more importantly, we recently showed that the growth rate in this model accounts very well for the (otherwise too) early appearance of massive halos and galaxies [59, 60].

In § 2, we summarize the basis for the two cosmological models we consider here (i.e., $\Lambda$CDM and $R_{\rm h}=ct$). Then, in §§ 3–5, we shall derive the necessary mathematical formalism for the formation of structure in the $R_{\rm h}=ct$ universe, and describe the evolution of the matter power spectrum in §§ 6–8. Our conclusions will be presented in § 9.

We begin by deriving the perturbed Einstein equations for an arbitrary FLRW metric and then use them in the $R_{\rm h}=ct$ universe to obtain the growth rate and power spectrum at various epochs. All the modes remain sub-horizon in $R_{\rm h}=ct$, but one has to use relativistic perturbation theory rather than Newtonian theory because the cosmic fluid in $R_{\rm h}=ct$ is always dominated by dark energy. We proceed by perturbing the general FLRW metric [102, 103, 104, 105, 106, 107, 108, 109, 110, 111], which we write in the form

where $a(\eta)$ is the scale factor in terms of the conformal time $\eta$. The components of the tensor $\bar{h}$ are the perturbations to the homogeneous FLRW metric. The indices $\alpha$ and $\beta$ run from 1 to 3. Using the conventional scalar-vector-tensor decomposition, we write $\bar{h}_{00}$, $\bar{h}_{0\alpha}$ and $\bar{h}_{\alpha\beta}$ as follows

and

with $\psi={h^{\alpha}}_{\alpha}$ and $h_{\alpha}^{\perp\perp{\alpha}}=0$. All these metric perturbations are highly coordinate dependent, so one cannot automatically attribute any physical meaning to the perturbed components of the metric.

For infinitesimal coordinate transformations, such as $\bar{x}^{i}=x^{i}+\xi^{i}$, the decomposed perturbed metric components transform as follows

and

To use these perturbed components, one may either choose a particular observer (i.e., a particular set of coordinates), resort to a particular gauge, or work with Bardeen’s four gauge independent variables [112]. It is evident from Equations (5)–(11) that there are seven independent metric perturbations, implying seven degrees of freedom, but there are only four gauge-invariant variables. So when choosing a particular gauge, one must fix 3 degrees of freedom. In this work, we follow the first approach by choosing the Newtonian gauge and fix three independent metric perturbations as follows (with $\phi^{\parallel}=\Phi_{1}=U_{\alpha}^{\perp}=0$):

and

After setting three metric perturbations to zero (in the Newtonian gauge), one ends up with

and

It is quite evident from the above equations that, in the Newtonian gauge, the independent metric perturbations are in fact the Bardeen variables. This is the primary reason for choosing this gauge. Then the perturbed metric is given as

It is well known that the vector perturbations die away as the Universe expands, and since our primary motivation in this paper is to compute the power spectrum and evolution of the perturbed dark-matter density field, we shall ignore both the vector and tensor perturbations for this work. The metric with only the scalar perturbations is then given as

which may be used to compute the Ricci tensor with the following components:

the Ricci scalar,

and also the Einstein tensor,

In these expressions, ${\mathcal{H}}$ is the Hubble parameter written as ${a^{{}^{\prime}}}/{a}$.

Equation (37) describes the evolution of the perturbed gravitational potential $\Phi$ with a source term $\delta\rho$ driving its growth. To solve it, one must determine the evolution equation for $\delta\rho$ as well. In principle $\delta\rho$ represents the total perturbed energy density of the universe, comprised of perturbations of individual species that clump above the smooth background. In this section, we shall obtain the evolution equations for individual species by perturbing their corresponding Boltzmann equations.

The Boltzmann equation for a given species (i.e., dark matter, baryons, radiation, etc.) describes the evolution of its distribution function in the 8-dimensional phase space with 4-dimensions representing the four space-time coordinates, $x^{\,i}$, and the other four comprising the 4-momentum, $p^{\,i}$. Using the additional constraint $g_{ij}p^{\,i}p^{\,j}=-m^{2}$, this 8-dimensional phase space is reduced to 7-dimensions. In this work, we choose the four spacetime coordinates, $x^{\,i}$, the magnitude of the 3-momentum, $p\equiv|\vec{p}|$, and its direction, $\hat{p}^{\,\mu}$, as the independent variables. From Liouville’s theorem, one obtains the following equation for the evolution of the distribution function $f_{s}(x^{\,i},p,\hat{p}^{\,\mu})$ (for species ‘s’):

where $\lambda$ and $C[f_{s}]$ are the affine parameter and the source/collision term for this species, respectively. We define $P^{\,i}={dx^{\,i}}/{d\lambda}$, so that ${dx^{\,0}}/{d\lambda}=P^{\,0}$. Thus, dividing the above equation by $P^{\,0}$, and neglecting the fourth term that is of second order, one gets

where $\eta$ is the conformal time, in terms of the expansion factor $a(t)$ and cosmic time $t$. Using ${P^{\,\mu}}/{P^{\,0}}=({p}/{E})\hat{p}^{\,\mu}$ (where $E$ is the energy) and geodesic equation, we may write the above equation as follows:

where $h_{\mu\nu}$ are the perturbed metric coefficients. Then substituting Equation (40) into Equation (39), we get

In arriving at this equation, we have substituted $P_{0}=\frac{E}{a}(1+\Phi)$, in terms of the perturbed gravitational potential, $\Phi$.

Next, we partition the distribution function into its unperturbed, $\bar{f}_{s}$, and perturbed, ${\mathcal{F}}_{s}$, components:

Thus, multiplying Equation (41) by the energy and integrating over momentum space, and collecting the zeroth-order terms, one gets

In the context of $\Lambda$CDM, the quantity $C[f_{s}]$ is zero because the particle number is conserved during the phase of fluctuation growth. Therefore, integrating the second term on the left-hand side by parts and neglecting the boundary term, we arrive at the expression

This equation may be reduced further by using the following definitions of density and pressure:

and

If we now apply Equations (44-46) to the dark matter fluctuation, we arrive at the expression

We multiply Equation (41) by $E(p)$ and again integrate over momentum space using the definition of $\rho$ and $\mathcal{P}$, but now collecting the first order terms:

We define $\delta_{\rm dm}\equiv{\delta\rho_{\rm dm}}/{\rho_{\rm dm}}$ and set both ${\mathcal{P}}_{\rm dm}$ and $\delta{\mathcal{P}}_{\rm dm}$ equal to zero. Then,

Substituting this expression into Equation (48) and decomposing them into Fourier modes, we find that

In arriving at this equation, we have also used $\partial_{\,\mu}v^{\,\mu}_{k}=ku_{dm,k}$, where $u_{dm,k}$ is the velocity perturbation of dark matter.

Now, taking the second moment of Equation (41) by multiplying it with $p\hat{p}^{\,\mu}$ and contracting with $i\hat{k}_{\,\mu}$, and then integrating it over momentum, we get

where $u_{{\rm dm},\,k}$ is the $k^{\rm th}$ velocity perturbation of dark matter. Again substituting for ${d\rho_{\rm dm}}/{d\eta}$ in this equation, we get

The early $R_{\rm h}=ct$ universe consisted of $\approx 80\%$ dark energy and $\approx 20\%$ radiation, with a small contamination of matter [101]. At late times, these fractions change to $\approx 67\%$ dark energy and $33\%$ matter. A coupling between dark matter and dark energy is therefore unavoidable in this model, since particle number is not conserved. The Boltzmann equations in this model will therefore be considerably different from those in $\Lambda$CDM. The pertinent equations were derived earlier in [60], but we summarize the procedure here for consistency and convenience.

Starting with Equation (41), an integration over momentum space yields

where the principal difference between this result and Equation (44) is that $C[f_{s}]$ on the righthand side is zero for $\Lambda$CDM, but not for $R_{\rm h}=ct$. The interaction term cannot be zero in the latter due to the coupling between dark energy and dark matter. Then, using the definition of density and pressure for dark matter (from Equations 45 and 46), one gets

The aforementioned transition in densities, e.g., $\rho_{\rm de}\approx 0.8\rho_{\rm c}$ after the Big Bang to $\approx 2\rho_{\rm c}/3$ more recently, and analogously for radiation and matter, may be modeled simply using a straightforward empirical expression,

to reflect the fact that, in $R_{\rm h}=ct$, the Universe transitioned from a radiation-dark-energy dominated early phase to a matter/dark-energy dominated one at late times. In the above expression, $\rho_{\rm c}$ is the critical density today, and $a_{*}$ represents the scale factor at matter radiation equality. Using this expression, one may easily evaluate the collision/source term,

which describes the interaction between dark energy and dark matter. A fraction of dark energy must be transformed (possibly via decay) into dark matter. Thus, the corresponding collision term for dark energy is

Returning now to Equation (41), with the use of Equation (56), we find for dark matter that

We take the first moment of this equation by multiplying it with $E(p)$ and integrating over momentum space. Collecting the first order terms yields

We also have, from the definition of $\delta_{\rm dm}$ and ${\mathcal{P}}_{\rm dm}=\delta{\mathcal{P}}_{\rm dm}=0$ for dark matter, that

And substituting for $d(\delta\rho_{\rm dm})/d\eta$ in this equation, and isolating the Fourier mode $k$, we find that

Then, taking the second moment of Equation (58) by multiplying it with $p\hat{p}^{\,\mu}$, contracting it with $i\hat{k}_{\,\mu}$, and integrating over momentum space, we find

Substituting for ${d\rho_{\rm dm}}/{d\eta}$ we thus get

A similar procedure allows to derive the analogous equation for dark energy. Using the interaction term in Equation (57), we may write

where $f_{\rm de}$ is the distribution function for dark energy. Then, partitioning $f_{\rm de}$ into its unperturbed ($\bar{f_{de}}$) and perturbed (${\mathcal{F}}_{de}$) components, we find

Taking the first momentum of this equation and isolating them into Fourier modes gives

From the definition of $\delta_{\rm de}={\delta\rho_{\rm de}}/{\rho_{\rm de}}$ and the inferred equation-of-state ${\mathcal{P}}_{\rm de}=-\rho_{\rm de}/2$ [77] for dark energy, we may write

so that, combining Equations (66) and (67), we get

We do not yet know the sound speed for the coupled dark matter/dark energy fluid, so we characterize it as follows:

and, assuming adiabatic fluctuations, we shall put

For the sake of simplicity, we assume the sound speed to be a constant limited to the range $0<(c_{s}/c)^{2}<1$. We have found that the actual value of this constant has a negligible impact on the solutions to the above equations since the ratio of dark matter density to dark energy is always much smaller than $1$ in the $R_{\rm h}=ct$ universe, and we therefore adopt the simple fraction $c_{s}^{2}=c^{2}/2$ throughout this work. Thus, using Equations (68) and (70), we get

Finally, taking the second moment of Equation (65), we find that