Expanding wave solutions of the Einstein equations that induce an anomalous acceleration into the Standard Model of Cosmology

We consider the Standard Model of Cosmology during the pure radiation phase, after inflation, modeled by an FRW spacetime. In comoving coordinates this metric takes the form (2),where $\overline{r}=Rr$ measures arclength distance at fixed time t and R ≡ R(t) is the cosmological scale factor. Assuming a comoving perfect fluid with equation of state p = ρc²/3, the Einstein equations give (cf. ref. 1)where H is the Hubble constant. Note that H and $\overline{r}$ are scale-independent relative to the scaling law r → αr, $R\to \frac{1}{\alpha }R$ of the FRW metric Eq. 2.1; cf. ref. 1.

The next theorem gives a coordinate transformation that takes Eq. 2.1 to the SSC form,such that A and B depend only on $\overline{r}/\overline{t}$

Theorem 1. Assume $p=\frac{\rho c^2}{3}$ and k = 0. Then the FRW metricunder the change of coordinatestransforms to the SSC-metricwhereand v xis the SSC velocity given byHere $\overline{u}=({\overline{u}}^0,{\overline{u}}^1)$ gives the $(\overline{t}/\overline{r})$ components of the 4-velocity of the sources in SSC coordinates. [We include the constant ψ₀ to later account for the time rescaling freedom in Eq. 2.3 (compare with equation 2.18 on page 85 of ref. 1).]

We now assume p = ρc²/3 and that solutions depend only on ξ. In the next section we show how the Einstein equations for metrics taking the SSC form Eq. 2.3 reduce to a system of three ODEs. A subsequent lengthy calculation then shows that FRW is a special solution of these equations.

Putting the SSC metric ansatz into MAPLE the Einstein equations G = κT reduce to the four partial differential equations:whereOn smooth solutions, Eqs. 3.1–3.4 are equivalent to Eqs. 3.1–3.3 together with Div_jT^j1 = 0, where Div_jT^j1 = 0 can be written in the locally inertial form,where T_M denotes the Minkowski stress tensor (11). Assuming that A and B depend only on $\xi =\overline{r}/\overline{t}$, Eqs. 3.1–3.3 and 3.6 are equivalent to the three ODEs, Eqs. 3.1, 3.3, and 3.6, together with the one constraint, which represents the consistency condition obtained by equating A_ξ from Eqs. 3.1 and 3.2. Settinganda (long) calculation shows that the three ODEs can be written in the formwhere¶and the constraint becomesIn summary, we have the following theorem

Theorem 2. Assume that A(ξ), G(ξ) and v(ξ) solve the ODEs (Eqs.3.7–3.9), and use the constraint (Eq.3.12) to define the density byThen the metricsolves the Einstein equations with equation of state p = ρc²/3. Moreover, the transformation (Eq.2.4; Eq.2.5 of the FRW metric) leads to the following special SSC relations that hold on the Standard Model during the radiation phase:where ψ₀ is an arbitrary constant. Another (long) calculation verifies directly that Eq.3.14 indeed solves Eqs.3.7–3.9 with the constraint Eq.3.13.

We conclude that the Standard Model of cosmology during the radiation phase corresponds to a solution of the expanding wave equations Eqs. 3.7–3.9 and 3.13 with parameter ψ₀ accounting for the time-scaling freedom of the SSC metric (Eq. 2.3). That is, the time-scaling $\overline{t}\to {\psi }_0\overline{t}$ preserves solutions of Eqs. 3.7–3.9 and the constraint (Eq. 3.12). The next theorem states that modulo this scaling, distinct solutions of Eqs. 3.7–3.9 describe a two-parameter family of distinct spacetimes.

Theorem 3. The replacement $\overline{t}\to {\psi }_0\overline{t}$ takes A(ξ), G(ξ), and v(ξ) to A(ξ/ψ₀), G(ξ/ψ₀), and v(ξ/ψ₀), and this scaling preserves solutions of Eqs.3.7–3.9 and 3.12. Moreover, this is the only scaling law in the sense that any two solutions of Eqs.3.7–3.9 and 3.12 not related by the scaling $\overline{t}\to {\psi }_0\overline{t}$ describe distinct spacetimes.

Since Eqs. 3.7–3.9 admit three (initial value) parameters and one scaling law, it follows that Eqs. 3.7–3.9 and 3.12 describes a two parameter family of distinct spacetimes. In the next section we show that by imposing regularity at the center there results a further reduction to a continuous one-parameter family of expanding wave solutions, such that one value of the parameter corresponds to the FRW metric with pure radiation sources.

To obtain the leading order corrections to the FRW metric implied by Eqs. 3.7–3.9 and 3.12, we linearize the equations at ξ = 0 and then expand these equations to leading order in ξ. Modulo the scaling law, the resulting linearized equations admit one eigen-solution that tends to infinity as ξ → 0, and the other one satisfies A(ξ) → 1 and B(ξ) → 1 as ξ → 0. Removing the singular solution and including the scaling law leaves a two-parameter family that includes the FRW Standard Model. The analysis leads to the following theorem.

Theorem 4. The two-parameter family of bounded solutions of Eqs.3.7–3.9 and 3.12 that extends FRW of the Standard Model is given in SSC in terms of the two parameters ψ₀ and a, to leading order in ξ, bywhereand the SSC velocity v = v(ξ) is given byHere ψ₀ is the time-scaling parameter, a = 1 corresponds to FRW, v₁(ξ) now denotes the SSC velocity of the Standard Model given in Eqs.2.6–2.8, and a ≠ 1 introduces a new acceleration parameter, which gives the leading order perturbation of FRW. In particular Eq.2.7 givesso Eq.4.4 implies that the velocity v is independent of the parameter a up to second order in ξ.

In light of Eq. 3.14, when a = 1, Eq. 4.1 reduces exactly to the FRW metric

Note that the SSC representation of FRW depends only on H and $\overline{r}$, both of which are invariant under the scaling r → αr, R → R/α of the FRW metric (Eq. 2.1), so the SSC representation of FRW is independent of α and therefore independent of our choice of scale for R(t). Thus without loss of generality, we can take ψ₀ = 1, and we can also assume throughout that the FRW metric is scaled exactly so thatcompare Eq. 2.1 and ref. 1.

To get insight into the geometry of the spacetime metric (Eq. 5.4) when a≠1, consider the extension of the FRW (t,r) coordinate transformation (Eqs. 2.4 and 2.5) to a≠1 defined byA straightforward caculation shows that the metric of Eq. 4.1 transforms to (t,r)-coordinates asThe metric of Eq. 5.3 takes the form of a k = 0 Friedmann–Robertson–Walker metric with a small correction to the scale factor, (R_a(t) = t^a/2 instead of R(t) = t^1/2), and a small corrective mixed term. In particular, the time slices t = const. in Eq. 5.3 are all flat space R³ as in FRW, and the $\overline{r}$ = const. slices agree with the FRW metric modified by scale factor R_a(t). It follows that the t = const. surfaces given by Eqs. 5.1 and 5.2 define a foliation of spacetime into flat three dimensional spacelike slices. Thus, Eq. 5.3 exhibits many of the flat space properties characteristic of FRW.

The metric of Eq. 5.3 is not comoving, even at the leading order, when a≠1. To obtain (an approximate) comoving frame, note that Eq. 4.5 is independent of a up to order ξ³, so it follows that even when a≠1, the inverse of the transformation (Eqs. 2.4 and 2.5) gives, to leading order in ξ, a comoving coordinate system for Eq. 4.1 in which we can compare the Hubble constant and redshift vs. luminosity relations for Eq. 4.1 when a≠1 to the Hubble constant and redshift vs. luminosity relations for FRW as measured by Eqs. 2.1 and 2.2. Thus, from here on, we take the Standard Model coordinate map to FRW coordinates Eqs. 2.4 and 2.5, which corresponds to taking a = 1 in Eqs. 5.1 and 5.2.

Theorem 5. Take ψ₀ = 1, and set ζ = $\overline{r}$/t Then the inverse of the coordinate transformation, Eqs.2.4 and 2.5, maps Eq.4.1 over to (t,r)-coordinates aswhereand the SSC velocity v in Eq.4.5 maps to the (t,r)-velocityThe errors in Eqs.5.5 and 5.6 are written in terms of the comoving coordinate variable ζ, xwhich by Eqs.2.4 and 2.5 satisfies ζ = O(ξ) as ξ → 0, and $\overline{r}$ = R(t)r exactly measures arclength at t = const. in FRW when a = 1.

Note that ζ = $\overline{r}$/t is a natural dimensionless perturbation parameter that has a physical interpretation in (t,r)-coordinates because (assuming c = 1 or t ≡ ct), ζ ranges from 0 to 1 as $\overline{r}$ ranges from zero to the horizon distance in FRW (approximately the Hubble distance c/H), a measure of the furthest one can see from the center at time t units after the Big Bang (2); that is,Thus expanding in ζ gives an expansion in the fractional distance to the Hubble length (cf. ref. 1). Note also that when a = 1 we obtain the FRW metric (Eq. 2.1), where we have used $R(t)=\sqrt{t}$ (compare with Eq. 4.8).

Now for a first comparison of the relative expansion at a≠1 to the expansion of FRW, define the Hubble constant at parameter value a, bywhereequals the square root of the coefficient of dr² in Eq. 5.4. Then one can showWe conclude that the fractional change in the Hubble constant due to the perturbation induced by expanding waves a≠1 relative to the FRW of the Standard Model a = 1, is given by

In this section we obtain the first-order corrections to the redshift vs. luminosity relation of FRW, as measured by an observer positioned at the center ζ = 0 of the expanding wave spacetimes described by the metric of Eq. 4.1 when a≠1.∥ Recall that ζ ≡ $\overline{r}$/t measures the fractional distance to the horizon (compare with Eq. 5.7). The physically correct coordinate system in which to do the comparison with FRW (a = 1) should be comoving with respect to the sources. Thus we restrict to the coordinates (t,r) defined by Eqs. 2.4 and 2.5, in which our one-parameter family of expanding wave spacetimes are described, to leading order in ζ, by the metric of Eq. 5.4. Note that Eq. 5.4 reduces exactly to the FRW metric when a = 1 (compare Eq. 2.1 with Eq. 4.8). For our derivation of the redshift vs. luminosity relation for Eq. 5.4 we follow the development in ref. 12.

To start, assuming radiation is emitted by a source at time t_e at wave length λ_e and received at ζ = 0 at later time t₀ at wave length λ₀, defineand letThen using two serendipitous properties of the metric of Eq. 5.4, namely, the metric is diagonal in comoving coordinates, and there is no a-dependence on the spheres of symmetry, it follows that the arguments in ref. 12, section 11.8, can be modified to give the following result.

Theorem 6. The redshift vs. luminosity relation, as measured by an observer positioned at the center ζ = 0 of the spacetime described by the metric of Eq.4.1, is given to leading order in redshift factor z bywhere we used the fact that z and ζ are of the same order as ζ → 0.

Note that when a = 1, Eq. 6.5 reduces to the well-known FRW linear relation,correct for the radiation phase of the Standard Model (12). Thus the bracket in Eq. 6.5 gives the leading order quadratic correction to the redshift vs. luminosity relation implied by the change in the Hubble expansion law corresponding to expanding wave perturbations of the FRW spacetime when a≠1. Since (a² − 1) appears in front of the leading order correction in Eq. 6.5, it follows (by continuous dependence of solutions on parameters) that the leading order part of any anomalous correction to the redshift vs. luminosity relation of the Standard Model, observed at a time after the radiation phase, can be accounted for by suitable adjustment of parameter a. In particular, note that the leading order corrections in Eq. 6.5 imply a blue-shifting of radiation relative to the Standard Model, as observed in the supernova data, when a > 1, (12).

We have constructed a one-parameter family of general relativistic expansion waves, which, at a single parameter value, reduces to what we call the FRW spacetime, the Standard Model of Cosmology during the radiation epoch. The discovery of this family is made possible by a remarkable coordinate transformation that maps the FRW metric in standard comoving coordinates, over to SSC in such a way that all quantities depend only on the single self-similar variable $\xi =\overline{r}/\overline{t}$. Note that it is not evident from the FRW metric in standard comoving coordinates that self-similar variables even exist, and if they do exist, by what ansatz one should extend the metric in those variables to obtain nearby self-similar solutions that solve the Einstein equations exactly. The main point is that our coordinate mapping to SSC form explicitly identifies the self-similar variables as well as the metric ansatz that together accomplish such an extension of the metric.

The self-similarity of the FRW metric in SSC suggested the existence of a reduction of the SSC Einstein equations to a new set of ODEs in ξ. Deriving this system from first principles then establishes that the FRW spacetime does indeed extend to a three-parameter family of expanding wave solutions of the Einstein equations. This three-parameter family reduces to an (implicitly defined) one-parameter family by removing a scaling invariance and imposing regularity at the center. The remaining parameter a changes the expansion rate of the spacetimes in the family, and thus we call it the acceleration parameter. Transforming back to comoving coordinates, the resulting one-parameter family of metrics is amenable to the calculation of a redshift vs. luminosity relation, to second order in the redshift factor z, leading to the relation of Eq. 6.5. It follows by continuity that the leading order part of an anomalous correction to the redshift vs. luminosity relation of the Standard Model observed after the radiation phase, can be accounted for by suitable adjustment of parameter a.

These results suggest an interpretation that we might call a Conservation Law Scenario of the Big Bang. That is, it is well-known that highly oscillatory interactive solutions of genuinely nonlinear conservation laws decay in time to noninteracting waves (shock waves and expanding waves), by the mechanism of shock wave dissipation. The subtle point is that even though dissipation terms are neglected in the formulation of the equations, there is a canonical dissipation and consequent loss of information due to the nonlinearities, and this can be modeled by shock wave interactions that drive solutions to noninteracting wave patterns. [This viewpoint is well-expressed in celebrated works (3, 5, 6)]. Since the one fact most certain about the Standard Model is that our universe arose from an earlier hot dense epoch in which all sources of energy were in the form of radiation, and since it is approximately uniform on the largest scale but highly oscillatory on smaller scales,** one might reasonably conjecture that decay to a noninteracting expanding wave occurred during the radiation phase of the Standard Model, via the highly nonlinear evolution driven by the large sound speed, and correspondingly large modulus of GN. Our analysis has shown that FRW is just one point in a family of noninteracting expanding waves, and as a result we conclude that some further explanation is required as to why, on some length scale, decay during the radiation phase of the Standard Model would not proceed to a member of the family satisfying a≠1. If decay to a≠1 did occur, then the galaxies that formed from matter at the end of the radiation phase (some 379,000 years after the Big Bang), would be displaced from their anticipated positions in the Standard Model at present time, and this displacement would lead to a modification of the observed redshift vs. luminosity relation. In principle such a mechanism could account for the anomalous acceleration of the galaxies as observed in the supernova data. Of course, if a≠1, then the spacetime has a center, and this would violate the so-called Copernican Principle, a simplifying assumption generally accepted in cosmology (compare with the discussions in ‡ and refs. 8 and 13). As a consequence, if the earth did not lie within some threshold of the center of expansion, the expanding wave theory would imply large angular variations in the observed expansion rate.†† In any case, the expanding wave theory presented here can in principle be tested. For such a test of Eq. 6.5, one must first evolve the relation, valid at the end of the radiation phase, up through the p ≈ 0 stage to present time in the Standard Model, thereby obtaining (an approximation to) the special value of a that gives the leading order correction to the redshift vs. luminosity relation observed in the supernova data. Then a derivation of the next order correction to (6.5) during the radiation phase, at that special a -value, evolved up through the p ≈ 0 stage to present time, would make a prediction of the next order correction to redshift vs. luminosity at present time, and this could be compared with an accurate plot of the supernove data.

To summarize, the expanding wave theory could in principle give an explanation for the observed anomalous acceleration of the galaxies within classical general relativity, with classical sources. In the expanding wave theory, the so-called anomalous acceleration is not an acceleration at all, but is a correction to the Standard Model due to the fact that we are looking outward into an expansion wave. The one parameter family of noninteracting general relativistic expansion waves derived here, are all equally possible end-states that could result after dissipation by nonlinear wave interaction during the radiation phase of the Standard Model, is done; and when a≠1 they introduce an anomalous acceleration into the Standard Model of cosmology. Unlike the theory of dark energy, this provides a possible explanation for the anomalous acceleration of the galaxies that is not ad hoc in the sense that it is derivable exactly from physical principles and a mathematically rigorous theory of expansion waves. That is, this explanation does not require the ad hoc assumption of a universe filled with an as yet unobserved form of energy (dark energy) with antigravitational properties in order to fit the data. The idea that the anomalous acceleration might be accounted for by a local under-density in a neighborhood of our galaxy was expounded in a recent article (8). Our results here might then give an accounting for the source of such an under-density.

In conclusion, these expanding wave solutions of the Einstein equations provide a paradigm to test against the Standard Model. Moreover, even if these general relativistic expansion waves do not in the end explain the anomalous acceleration of the galaxies, their presence represents an instability in the Standard Model in the sense that an explanation is required as to why small-scale oscillations have to settle down to large-scale a = 1 expansions instead of a≠1 expansions (either locally or globally), during the radiation phase of the Big Bang.