Dark Future for Dark Matter

The prevailing cosmological constant and cold dark matter (ΛCDM) cosmic concordance model accounts for the radial expansion of the universe after the Big Bang. The model appears to be authoritative because it is based on the Einstein gravitational field equation. However, a thorough scrutiny of the underlying theory calls into question the suitability of the field equation, which states that the Einstein tensor µν G is a constant multiple of the stress-energy tensor µν T when they both are evaluated at the same 4D space-time point: , where k is the gravitational constant. Not-withstanding its venerable provenance, this equation is incorrect unless the cosmic pressure is p = 0; but then all that remains of the Einstein equation is the Poisson equation which models the Newtonian gravity field. This short-coming is not resolved by adding the cosmological constant term to the field equation, T , as in the ΛCDM model, because then p = Λ, so the pressure is a universal constant, not a variable. Numerous studies sup-port the concept of a linearly expanding universe in which gravitational forces and accelerations are negligible because the baryonic mass density of the universe is far below its critical density. We show that such a coasting universe model agrees with SNe Ia luminosity vs. redshift distances just as well or even better than the ΛCDM model, and that it does so without having to invoke dark matter or dark energy. Occam’s razor favors a coasting universe over the ΛCDM model.


Introduction
The ΛCDM model is used to represent the expanding universe for the interpretation 1) of type Ia supernovae (SNe Ia) brightness and redshift data, and 2), of measurements of anisotropies in the cosmic microwave background (CMB) spectrum of the early universe. The model conjectures the existence of dark matter and dark energy-a lot of both. With a felicitous balance of the two, an ostensibly credible model is derived. However, a ΛCDM model has drawbacks.
Firstly, the existence of dark matter is an unverified hypothesis. 1 There have been numerous experiments dedicated to finding dark matter here on Earth, in underground experiments [2] [3] and satellite detectors [4], as well as in the Large Hadron Collider [5]. Yet, there has been no unambiguous dark matter detection [6], which, by induction [7], is evidence of absence. Moreover, aside from the ΛCDM model, there is no theoretical call for dark energy; nor has there even been an observational hint of its existence. Kang et al. [8] have recently challenged a key premise in the derivation of the ΛCDM model, and by addressing the problem with a different approach, we show that the model is utterly untenable. We expand, revise and interpret the Einstein field equation to find a simple viable alternative to the ΛCDM model.

Einstein Field Equation
We use the standard nomenclature of GR (e.g., see Kenyon [9]), except k is the According to Einstein ([11], p. 84), µν G "must be a differential tensor in the µν g that is completely determined by the following three conditions: 1) It may contain no differential coefficients of the µν g higher than the second. 1 Dark matter had a sullied reputation from the nineteenth century, when astronomers searched fruitlessly for another sort of invisible matter-an intramercurial planet, Vulcan [1], that was thought to be responsible for otherwise unexplainable orbital perturbations of Mercury. As it turned out, what had been missing was not matter, but an authoritative theory: GR. So, is it now missing matter, or missing theory? Journal of Modern Physics 2) It must be linear in these second differential coefficients.
3) Its divergence must vanish identically." For the nonce, let ∂ represent any partial derivative operator i x ∂ , and let g represent any component of µν g . According to the first two conditions, each component of µν G has the form therefore 11T ρ , and 11 0 G ≠ implies that p ρ . With such an equation of state, the fluid is relativistic, which is to say that the characteristic thermal speed of the fluid is close to c, which it generally is not. The only viable alternative is to set G 11 and all other terms of µν G except G 00 equal to zero. But this requires that 0 p = , which is not generally true. We overcome this dilemma by adding  (4) remain generally covariant and, with this essential correction, Equation (4) be- . This is the full Poisson equation, which applies in all coordinate systems. It notably takes into account the gravitational charge equivalence of mass density ρ and energy density p [12]; 2 adapted to this nicety, φ is the Newtonian potential.
In general, the density does not need to be uniform. We can approximate the

The Expanding Universe
For a universe with vanishingly small  , the line element 3 reduces to Then 0 µν = G because this metric is independent of φ ; GR obviously does not apply. But what if this nearly empty universe expands according to Hubble's law? Then But b is merely an invention that allows us to broaden the scope of the field equation by allowing for Hubble expansion; it has no significance of its own. In fact, the Hubble expansion is real, but it is independent of GR.
Refashioned to try to accommodate Hubble expansion, Equation (5) becomes where 2 0 3H b Λ = + , which is zero according to Equation (9). There is no real change from Equation (5), but we use this form of the equation in Section 5, where Λ-there designated as the cosmological constant-is regarded as a proxy for dark energy.

The Coasting Universe
Evidence for the existence of dark matter is not at all conclusive. It is not needed to explain the flat rotation curves of spiral galaxies nor the CMB simulations.  Kolb [27] proposed such a linearly coasting universe, which Sethi et al. [28] found to agree "surprisingly" well with SNe Ia observations. Subsequently many others [29]- [58] have confirmed the validity of a coasting fit, which we now corroborate. The coasting universe expands without acceleration, For redshift z, the distance between a source (then) and observer at 0 The distance modulus depends on the area, For an open universe [59], 3 We derive this expression using L. Parker's Mathematica [13] notebook, Curvature and the Einstein Equation 1 is the luminosity distance.
In Figure 1, we plot a set of high-confidence Union 2.1 data [60]. We used these data to estimate This is gratifyingly close to the 14.46 ± 0.31 Gyr age of HD 140283, the oldest star for which a reliable age has been determined [62].

The ΛCDM Model
Equating the 00 µν = element pairs on both sides of Equation (10) gives whereas equating any of the 11, 22 µν = The ΛCDM model is based on Equation (14) (but while also implicitly ignoring Equation (15)). The first integral of Equation (14) is which, by Newton's second law of motion, is ( ) With this choice, the ΛCDM scale factor for an expanding universe decelerates at first when 8 k π > Λ  , then later accelerates after 8 k Λ > π  ; the orthodox ΛCDM model does not expand at the same rate as the well substantiated linearly coasting universe that are discussed in Section 4. There is no way of formulating a linearly coasting universe with the ΛCDM model, so purported ΛCDM scale accelerations [60] [63]- [68] are then merely expected artifacts of the Equation (17) procrustean template. Anyway, according to Equation (9) of Section 3, and to the conventionally ignored Equation (15), the cosmological constant must be zero: the ΛCDM model should be rejected because it does not expand at all.

Conclusions
GR can be reconciled with a static universe by starting with the Einstein field equation (Equation (1)) if 0 p = , or with the modified field equation (Equation (4)) if 0 p ≥ . GR can also be reconciled with a uniformly expanding universe by replacing the Einstein tensor with its Lovelock form b µν µν µν → + G G g . GR then turns out to be in full accord with the linear coasting universe determined from the collation of type Ia supernovae (SNe Ia) brightness and redshift data-without recourse to the existence of dark matter or dark energy.
The ΛCDM model requires the existence of vast quantities of (attractive) dark matter and (repulsive) dark energy. They are the sources of fanciful opposing forces that mostly cancel each other out. We have shown why this model must be wrong: according to Equation (15), 0 Λ = . There can be no dark energy in the context of the ΛCDM approach, so this broadly accepted model is incorrect. Whereas dark energy is implausible and unnecessary, dark matter is not implausible and its need is unresolved. We recommend a linearly coasting universe-one that is consistent both with GR and with SNe Ia observations-as a viable ΛCDM alternative.