The four principal energy conditions (ECs) in general relativity prohibit negative energies, repulsive gravity and superluminal energy flows. One must invoke exotic matter to violate any one of these, yet $$\varLambda $$ CDM does so quite prominently during inflation and in the epoch of dark energy dominance. In this paper, we carry out model selection between the standard model and the $$R_{\textrm{h}}=ct$$ universe using a combination of H galaxy and cosmic chronometer measurements in the local Universe, and directly compare the results to the constraints imposed by the ECs. We find that the latter cosmology is not only strongly favored by these data, with a likelihood of $$\sim 92\%$$ versus only $$\sim 8\%$$ for the former, but that its optimized fit is fully compliant with all four ECs, while $$\varLambda $$ CDM’s best fit violates the so-called strong energy condition at $$z\lesssim 2$$ .

United States

Namit Chandak

Fulvio Melia

China

Jun-Jie Wei

The energy conditions and model selection in the local Universe

The conventional picture of supermassive black-hole growth in the standard model had already been seriously challenged by the emergence of $$\sim 10^9\;M_\odot $$ quasars at $$z\sim 7.5$$ , conflicting with the predicted formation of structure in the early $$\varLambda $$ CDM Universe. But the most recent discovery of a $$\sim 10^8\;M_\odot $$ source at $$z\sim 10.1$$ argues even more strongly against the possibility that these black holes were created in Pop II or III supernovae, followed by Eddington-limited accretion. Attempts at resolving this anomaly have largely focused on the formation of seeds via an exotic, direct collapse of primordial gas to an initial mass $$\sim 10^5\;M_\odot $$ – a process that has never been seen anywhere in the cosmos. Our goal in this is to demonstrate that the emergence of these black holes is instead fully consistent with standard astrophysics in the context of the alternative Friedmann–Lemaître–Robertson–Walker cosmology known as the $$R_\textrm{h}=ct$$ universe. We show that, while the predicted evolution in the standard model is overly compressed, the creation, growth and appearance of such high- quasars fall comfortably within the evolutionary history in this cosmology, thereby adding considerable observational support to the existing body of evidence favoring it over the standard scenario.

The cosmic timeline implied by the highest redshift quasars

The absence of large-angle correlations in the temperature of the cosmic microwave background (CMB), confirmed by three independent satellite missions, creates significant tension with the standard model of cosmology. Previous work has shown, however, that a truncation, <math><msub><mrow><mi>k</mi></mrow><mrow><mi>min</mi></mrow></msub></math>, of the primordial power spectrum comprehensively resolves the anomaly and the missing power at <math><mi>ℓ</mi><mo>≲</mo><mn>5</mn></math> (the low multipoles). Since this cutoff is consistent with the hypothesized delay of inflation well beyond the Planck time, we are strongly motivated to consider its possible impact on other observational signatures. In this Letter, we analyze and predict its influence on the most revealing probe awaiting measurement by upcoming missions—the B-mode polarization of the CMB, whose accurate determination should greatly impact the inflationary picture. We highlight the quantitative power of this discriminant by specifically considering the LiteBIRD mission, predicting the effect of <math><msub><mrow><mi>k</mi></mrow><mrow><mi>min</mi></mrow></msub></math> on both the angular power spectrum and the angular correlation function of the B-mode, for a range of tensor-to-scalar ratios, r. While its impact on the latter appears to be negligible, <math><msub><mrow><mi>k</mi></mrow><mrow><mi>min</mi></mrow></msub></math> should have a very pronounced effect on the former. We show that for <math><mi>r</mi><mo>=</mo><mn>0.036</mn></math>, <math><msub><mrow><mi>k</mi></mrow><mrow><mi>min</mi></mrow></msub></math>'s impact on <math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>B</mi><mi>B</mi></mrow></msubsup></math> at low ℓ's should be easily detectable by LiteBIRD, but will be largely hidden by the total uncertainty of the measurement if <math><mi>r</mi><mo>≲</mo><mn>0.02</mn></math>.

Jingwei Liu

A truncated primordial power spectrum and its impact on B-mode polarization

The quantum to classical transition of fluctuations in the early universe is still not completely understood. Some headway has been made incorporating the effects of decoherence and the squeezing of states, though the methods and procedures continue to be challenged. But new developments in the analysis of the most recent Planck data suggest that the primordial power spectrum has a cutoff associated with the very first quantum fluctuation to have emerged into the semi-classical universe from the Planck domain at about the Planck time. In this paper, we examine the implications of this result on the question of classicalization, and demonstrate that the birth of quantum fluctuations at the Planck scale would have been a ‘process’ supplanting the need for a ‘measurement’ in quantum mechanics. Emerging with a single wavenumber, these fluctuations would have avoided the interference between different degrees of freedom in a superposed state. Moreover, the implied scalar-field potential had an equation-of-state consistent with the zero active mass condition in general relativity, allowing the quantum fluctuations to emerge in their ground state with a time-independent frequency. They were therefore effectively quantum harmonic oscillators with classical correlations in phase space from the very beginning.

Classicalization of quantum fluctuations at the Planck scale in the = Universe

Today we have a solid, if incomplete, physical picture of how inertia is created in the standard model. We know that most of the visible baryonic ‘mass’ in the Universe is due to gluonic back-reaction on accelerated quarks, the latter of which attribute their own inertia to a coupling with the Higgs field – a process that elegantly and self-consistently also assigns inertia to several other particles. But we have never had a physically viable explanation for the origin of rest-mass energy, in spite of many attempts at understanding it towards the end of the nineteenth century, culminating with Einstein’s own landmark contribution in his Annus Mirabilis. Here, we introduce to this discussion some of the insights we have garnered from the latest cosmological observations and theoretical modeling to calculate our gravitational binding energy with that portion of the Universe to which we are causally connected, and demonstrate that this energy is indeed equal to  $$mc^2$$  <math> <mrow> <mi>m</mi> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </math>   when the inertia m is viewed as a surrogate for gravitational mass.

The origin of rest-mass energy

In the standard model of cosmology, the Universe began its expansion with an anomalously low entropy, which then grew dramatically to much larger values consistent with the physical conditions at decoupling, roughly 380,000 years after the Big Bang. There does not appear to be a viable explanation for this ‘unnatural’ history, other than via the generalized second law of thermodynamics (GSL), in which the entropy of the bulk,  $$S_\mathrm{bulk}$$  <math> <msub> <mi>S</mi> <mi>bulk</mi> </msub> </math>  , is combined with the entropy of the apparent (or gravitational) horizon,  $$S_{\mathrm{h}}$$  <math> <msub> <mi>S</mi> <mi>h</mi> </msub> </math>  . This is not completely satisfactory either, however, since this approach seems to require an inexplicable equilibrium between the bulk and horizon temperatures. In this paper, we explore the thermodynamics of an alternative cosmology known as the  $$R_{\mathrm{h}}=ct$$  <math> <mrow> <msub> <mi>R</mi> <mi>h</mi> </msub> <mo>=</mo> <mi>c</mi> <mi>t</mi> </mrow> </math>   universe, which has thus far been highly successful in resolving many other problems or inconsistencies in  $$\varLambda $$  <math> <mi>Λ</mi> </math>  CDM. We find that  $$S_{\mathrm{bulk}}$$  <math> <msub> <mi>S</mi> <mi>bulk</mi> </msub> </math>   is constant in this model, eliminating the so-called initial entropy problem simply and elegantly. The GSL may still be relevant, however, principally in selecting the arrow of time, given that  $$S_{\mathrm{h}}\propto t^2$$  <math> <mrow> <msub> <mi>S</mi> <mi>h</mi> </msub> <mo>∝</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </math>   in this model.

Thermodynamics of the  $$R_{\mathrm{h}}=ct$$  <math> <mrow> <msub> <mi>R</mi> <mi>h</mi> </msub> <mo>=</mo> <mi>c</mi> <mi>t</mi> </mrow> </math>   Universe: a simplification of cosmic entropy

The EDGES collaboration has reported the detection of a global 21-cm signal with a plateau centered at 76 MHz (i.e., redshift 17.2), with an amplitude of  $$500^{+200}_{-500}$$  <math> <msubsup> <mn>500</mn> <mrow> <mo>-</mo> <mn>500</mn> </mrow> <mrow> <mo>+</mo> <mn>200</mn> </mrow> </msubsup> </math>   mK. This anomalous measurement does not comport with standard cosmology, which can only accommodate an amplitude  $$\lesssim 230$$  <math> <mrow> <mo>≲</mo> <mn>230</mn> </mrow> </math>   mK. Nevertheless, the line profile’s redshift range ( $$15\lesssim z\lesssim 20$$  <math> <mrow> <mn>15</mn> <mo>≲</mo> <mi>z</mi> <mo>≲</mo> <mn>20</mn> </mrow> </math>  ) suggests a possible link to Pop III star formation and an implied evolution out of the ‘dark ages.’ Given this tension with the standard model, we here examine whether the observed 21-cm signal is instead consistent with the results of recent modeling based on the alternative Friedmann–Lemaître–Robertson–Walker cosmology known as the  $$R_{\mathrm{h}}=ct$$  <math> <mrow> <msub> <mi>R</mi> <mi>h</mi> </msub> <mo>=</mo> <mi>c</mi> <mi>t</mi> </mrow> </math>   universe, showing that – in this model – the CMB radiation might have been rethermalized by dust ejected into the IGM by the first-generation stars at redshift  $$z\sim 16$$  <math> <mrow> <mi>z</mi> <mo>∼</mo> <mn>16</mn> </mrow> </math>  . We find that the requirements for this process to have occurred would have self-consistently established an equilibrium spin temperature  $$T_{\mathrm{s}}\approx 3.4$$  <math> <mrow> <msub> <mi>T</mi> <mi>s</mi> </msub> <mo>≈</mo> <mn>3.4</mn> </mrow> </math>   K in the neutral hydrogen, via the irradiation of the IGM by deep penetrating X-rays emitted at the termination shocks of Pop III supernova remnants. Such a dust scenario has been strongly ruled out for the standard model, so the spin temperature ( $$\sim 3.3$$  <math> <mrow> <mo>∼</mo> <mn>3.3</mn> </mrow> </math>   K) inferred from the 21-cm absorption feature appears to be much more consistent with the  $$R_{\mathrm{h}}=ct$$  <math> <mrow> <msub> <mi>R</mi> <mi>h</mi> </msub> <mo>=</mo> <mi>c</mi> <mi>t</mi> </mrow> </math>   profile than that implied by  $$\Lambda $$  <math> <mi>Λ</mi> </math>  CDM, for which adiabatic cooling would have established a spin temperature  $$T_\mathrm{s}(z=17.2)\sim 6$$  <math> <mrow> <msub> <mi>T</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>=</mo> <mn>17.2</mn> <mo>)</mo> </mrow> <mo>∼</mo> <mn>6</mn> </mrow> </math>   K.

The anomalous 21-cm absorption at high redshifts

The growth of structure may be traced via the redshift-dependent halo mass function. This quantity probes the re-ionization history and quasar abundance in the Universe, constituting an important probe of the cosmological predictions. Halos are not directly observable, however, so their mass and evolution must be inferred indirectly. The most common approach is to presume a relationship with galaxies and halos. Studies based on the assumption of a constant halo to stellar mass ratio $$M_h/M_*$$ <math><mrow><msub><mi>M</mi><mi>h</mi></msub><mo>/</mo><msub><mi>M</mi><mrow><mrow></mrow><mo>∗</mo></mrow></msub></mrow></math>  (extrapolated from $$z \lesssim 4$$ <math><mrow><mi>z</mi><mo>≲</mo><mn>4</mn></mrow></math> ) reveal significant tension with $$\varLambda $$ <math><mi>Λ</mi></math> CDM – a failure known as “The Impossibly Early Galaxy Problem”. But whether this ratio evolves or remains constant through redshift $$4 \lesssim z \lesssim 10$$ <math><mrow><mn>4</mn><mo>≲</mo><mi>z</mi><mo>≲</mo><mn>10</mn></mrow></math>  is still being debated. To eliminate the tension with $$\varLambda $$ <math><mi>Λ</mi></math> CDM, it would have to change by about 0.8 dex over this range, an issue that may be settled by upcoming observations with the James Webb Space Telescope. In this paper, we explore the possibility that this major inconsistency may instead be an indication that the cosmological model is not completely correct. We study this problem in the context of another Friedmann–Lemaître–Robertson–Walker (FLRW) model known as the $$R_{\mathrm{h}}=ct$$ <math><mrow><msub><mi>R</mi><mi>h</mi></msub><mo>=</mo><mi>c</mi><mi>t</mi></mrow></math>  universe, and use our previous measurement of $$\sigma _8$$ <math><msub><mi>σ</mi><mn>8</mn></msub></math>  from the cosmological growth rate, together with new solutions to the Einstein–Boltzmann equations, to interpret these recent halo measurements. We demonstrate that the predicted mass and redshift dependence of the halo distribution in $$R_{\mathrm{h}}=ct$$ <math><mrow><msub><mi>R</mi><mi>h</mi></msub><mo>=</mo><mi>c</mi><mi>t</mi></mrow></math>  is consistent with the data, even assuming a constant $$M_h/M_*$$ <math><mrow><msub><mi>M</mi><mi>h</mi></msub><mo>/</mo><msub><mi>M</mi><mrow><mrow></mrow><mo>∗</mo></mrow></msub></mrow></math>  throughout the observed redshift range ($$4\lesssim z\lesssim 10$$ <math><mrow><mn>4</mn><mo>≲</mo><mi>z</mi><mo>≲</mo><mn>10</mn></mrow></math> ), contrasting sharply with the tension in $$\varLambda $$ <math><mi>Λ</mi></math> CDM. We conclude that – if $$M_h/M_*$$ <math><mrow><msub><mi>M</mi><mi>h</mi></msub><mo>/</mo><msub><mi>M</mi><mrow><mrow></mrow><mo>∗</mo></mrow></msub></mrow></math>  turns out to be constant – the massive galaxies and their halos must have formed earlier than is possible in $$\varLambda $$ <math><mi>Λ</mi></math> CDM.

Manoj Yennapureddy

A comparison of the $$R_{\mathrm{h}}=ct$$ <math><mrow><msub><mi>R</mi><mi>h</mi></msub><mo>=</mo><mi>c</mi><mi>t</mi></mrow></math>  and $$\varLambda $$ <math><mi>Λ</mi></math> CDM cosmologies based on the observed halo mass function

The recently measured cutoff, $$k_\mathrm{min}=4.34\pm 0.50/r_{\mathrm{cmb}}$$ <math><mrow><msub><mi>k</mi><mi>min</mi></msub><mo>=</mo><mn>4.34</mn><mo>±</mo><mn>0.50</mn><mo>/</mo><msub><mi>r</mi><mi>cmb</mi></msub></mrow></math>  (with $$r_{\mathrm{cmb}}$$ <math><msub><mi>r</mi><mi>cmb</mi></msub></math>  the comoving distance to the last scattering surface), in the fluctuation spectrum of the cosmic microwave background, appears to disfavor slow-roll inflation and the associated transition of modes across the horizon. We show in this Letter that $$k_{\mathrm{min}}$$ <math><msub><mi>k</mi><mi>min</mi></msub></math>  instead corresponds to the first mode emerging out of the Planck domain into the semi-classical universe. The required scalar-field potential is exponential, though not inflationary, and satisfies the zero active mass condition, $$\rho _\phi +3p_\phi =0$$ <math><mrow><msub><mi>ρ</mi><mi>ϕ</mi></msub><mo>+</mo><mn>3</mn><msub><mi>p</mi><mi>ϕ</mi></msub><mo>=</mo><mn>0</mn></mrow></math> . Quite revealingly, the observed amplitude of the temperature anisotropies requires the quantum fluctuations in $$\phi $$ <math><mi>ϕ</mi></math>  to have classicalized at $$\sim 3.5\times 10^{15}$$ <math><mrow><mo>∼</mo><mn>3.5</mn><mo>×</mo><msup><mn>10</mn><mn>15</mn></msup></mrow></math>  GeV, consistent with the energy scale in grand unified theories. Such scalar-field potentials are often associated with Kaluza–Klein cosmologies, string theory and even supergravity.

Quantum fluctuations at the Planck scale

Particle physics suggests that the Universe may have undergone several phase transitions, including the well-known inflationary event associated with the separation of the strong and electroweak forces in grand unified theories. The accelerated cosmic expansion during this transition, at cosmic time $$t\sim 10^{-36}-10^{-33}$$ <math><mrow><mi>t</mi><mo>∼</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>36</mn></mrow></msup><mo>-</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>33</mn></mrow></msup></mrow></math>  s, is often viewed as an explanation for the uniformity of the CMB temperature, T, which would otherwise have required inexplicable initial conditions. With the discovery of the Higgs particle, it is now quite likely that the Universe underwent another (electroweak) phase transition, at $$T=159.5\pm 1.5$$ <math><mrow><mi>T</mi><mo>=</mo><mn>159.5</mn><mo>±</mo><mn>1.5</mn></mrow></math>  GeV – roughly $$\sim 10^{-11}$$ <math><mrow><mo>∼</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>11</mn></mrow></msup></mrow></math>  s after the big bang. During this event, the fermions gained mass and the electric force separated from the weak force. There is currently no established explanation, however, for the apparent uniformity of the vacuum expectation value of the Higgs field which, like the uniformity in T, gives rise to its own horizon problem in standard $$\varLambda $$ <math><mi>Λ</mi></math> CDM cosmology. We show in this paper that a solution to the electroweak horizon problem may be found in the choice of cosmological model, and demonstrate that this issue does not exist in the alternative Friedmann–Robertson–Walker cosmology known as the $$R_\mathrm{h}=ct$$ <math><mrow><msub><mi>R</mi><mi>h</mi></msub><mo>=</mo><mi>c</mi><mi>t</mi></mrow></math>  universe.

A solution to the electroweak horizon problem in the $$R_\mathrm{h}=ct$$ <math><mrow><msub><mi>R</mi><mi>h</mi></msub><mo>=</mo><mi>c</mi><mi>t</mi></mrow></math>  universe