Re: A variable expansion speed theory of gravity
Sorry but this post will be a long one. It's a compilation of notes I took from the 2015 results of Plancks data. Which is naturally relevant to my opinion described in this discussion.
This Planck data release is both the first to include the data gathered over the full length of the mission and the first to contain polarization information.
Despite trying a wide range of extensions to the basic, 6-parameter ΛCDM model we find no significant evidence for a failure of the model. We continue to see tensions with some analyses of other astrophysical data sets. Planck Collaboration XIII (2016) shows that these tensions cannot be resolved with standard single parameter extensions of the base ΛCDM model. Resolving these discrepancies remains an area of active research.
XVIII. Background geometry and topology of the Universe
2. Previous results
It was concluded that a physically-motivated model was not favoured by the data.
For topology, we showed that a fundamental topological domain smaller than the Hubble volume is strongly disfavoured.
Using both frequentist and Bayesian methods applied for the first time to polarization data, we find no evidence for a multi-connected topology with a scale less than roughly the distance to the last-scattering surface.
Although the evidence thus far corroborates the conventional wisdom that we live in the simplest FLRW Universe, this is likely to be only an approximation vastly beyond the Hubble scale.
3.1. Background parameterizations
The conventional approach, that we adopt also here, is to choose a minimally-coupled scalar field model also known as quintessence, which corresponds to the choice of a rest-frame sound speed (i.e., equal to the speed of light) and σ = 0 (no scalar anisotropic stress). In this case the relativistic sound speed suppresses the dark energy perturbations on sub-horizon scales, preventing it from contributing significantly to clustering.
3.2.1. Modified gravity and effective field theory
The first approach starts from a Lagrangian, derived from an effective field theory (EFT) expansion in the context of DE. Specifically, EFT describes the space of scalar field theories, with a Lagrangian written in unitary gauge that preserves isotropy and homogeneity at the background level, assumes the weak equivalence principle, and has only one extra dynamical field besides the matter fields conventionally considered in cosmology.
3.2.2. MG and phenomenological parameterizations
The second approach adopted in this paper to test MG is more phenomenological and starts from the consideration that cosmological observations probe quantities related to the metric perturbations, in addition to the expansion rate.
The quest for Dark Energy and Modified Gravity is far from over. Our focus has been on the scales where linear theory is applicable, since these are the most theoretically robust. Overall, the constraints that we find are consistent with the simplest scenario, ΛCDM, with constraints on DE models and MG models that are significantly improved with respect to past analyses.
Information on DE, complementary to (w0,wa), comes from asking whether there can be any DE at early times.
The background is then forced to be very close to ΛCDM, unless the tight constraints on early DE can somehow be evaded in a realistic model by “counter balancing effects”.
6.2. Early-Universe physics
We found no evidence for a tensor component or running of the scalar spectral index, no strong evidence for isocurvature perturbations or features in the primordial power spectrum and no evidence for non-Gaussianity cosmic strings or other topological defects. On large angular scales, the Planck data showed some evidence for “anomalies” seen previously in the WMAP data.
Thus, at present there is no convincing evidence of a primordial B-mode signal. At these low values of r, there is no longer any tension with Planck temperature constraints. (Personal: Cosmologists predict two types of B-modes, the first generated during cosmic inflation shortly after the big bang (gravitational waves), and the second generated by gravitational lensing at later times)
6.2.2. Scale dependence of primordial fluctuations
In summary, the Planck data are consistent with zero running of the scalar spectral index. However, as illustrated in Fig. 23, the Planck data still allow running at roughly the 10-2 level, i.e., an order of magnitude higher than expected in simple inflationary models.
6.2.3. Isocurvature perturbations
Finally, neutrino velocity potential and vorticity modes are other possible consistent perturbations to the photon-neutrino fluid after neutrino decoupling. However, they are essentially impossible to excite, since they consist of photon and neutrino fluids coherently moving in opposite directions on super-horizon scales (although the relative velocity would have been zero before neutrino decoupling).
The simplifying assumptions of large-scale homogeneity and isotropy lead to the familiar Friedman-Lemaître-Robertson-Walker (FLRW) metric that appears to be an accurate description of our Universe.
Our Universe appears to be spatially flat to a 1σ accuracy of 0.25%. The red contours (on the graphic) tightly constrain the geometry of our Universe to be nearly flat.(Personal: which, in fact, means that no mesurements up to date has proven the universe being "curved")
6.3. Dark energy
The physical explanation for the observed accelerated expansion of the Universe is currently not known.
In standard ΛCDM the acceleration is provided by a cosmological constant, i.e., an additional fluid satisfying an equation of state w ≡ pDE/ρDE = −1. However, there are many possible alternatives, typically described either in terms of extra degrees of freedom associated with scalar fields or modifications of general relativity on cosmological scales
6.4.1. Constraints on the total mass of active neutrinos
Detection of neutrino oscillations has proved that neutrinos have mass. The Planck base ΛCDM model assumes a normal mass hierarchy with ∑ mν ≈ 0.06 eV.
Here we give constraints assuming three species of degenerate massive neutrinos. At the level of sensitivity of Planck this is an accurate approximation, but note that it does not quite match continuously on to the base ΛCDM model (which assumes two massless and one massive neutrino with ∑ mν = 0.06 eV).
Masses well below 1 eV have only a mild effect on the shape of the CMB power spectra, since they became non-relativistic after recombination.
Masses below about 0.4 eV can provide an acceptable fit to the direct H0 measurements, and adding the BAO data helps to break the acoustic scale degeneracy and tightens the constraint on ∑ mνsubstantially.
Although the posterior has less weight at zero, the lensing data are incompatible with very large neutrino masses
6.4.2. Constraints on Neff
Dark radiation density in the early Universe is usually parameterized by Neff, defined so that the total relativistic energy density in neutrinos and any other dark radiation is given in terms of the photon density.
The numerical factors in this equation are included so that Neff = 3 for three standard model neutrinos that were thermalized in the early Universe and decoupled well before electron-positron annihilation.
In this section we focus on additional energy density from massless particles. In addition to massless sterile neutrinos, a variety of other particles could contribute to Neff. We assume that the additional massless particles are produced well before recombination, and neither interact nor decay, so that their energy density scales with the expansion exactly like massless neutrinos. (Personal: Yet all neutrinos have been observed with left-handed chirality, and all antineutrinos right-handed. So a massless neutrino would be an antineutrino. The question, thus, remains: can neutrinos and antineutrinos be differentiated only by chirality?).
For Neff> 3, the Planck data favour higher values of the Hubble parameter than the Planck base ΛCDM value, which may be in better agreement with some direct measurements of H0.
As a result, these models increase the tensions between the CMB measurements and astrophysical measurements of σ8. It therefore seems unlikely that additional radiation alone can help to resolve tensions with large-scale structure data.
Observations of both the primordial helium and deuterium abundance are compatible with the predictions of standard BBN for the Planck base ΛCDM value of the baryon density.
As discussed in the previous sections, neither a higher neutrino mass nor additional radiation density alone can resolve all of the tensions between Planck and other astrophysical data. However, the presence of additional massive particles, such as massive sterile neutrinos, could potentially improve the situation by introducing enough freedom to allow higher values of the Hubble constant and lower values of σ8.
In the case of massless radiation density, the cosmological predictions are independent of the actual form of the distribution function, since all particles travel at the speed of light.
Although Planck is perfectly consistent with no massive sterile neutrinos, a significant region of parameter space with fractional ΔNeff is allowed, where σ8 is lower than in the base ΛCDM model. This is also the case for massless sterile neutrinos combined with massive active neutrinos.
6.4.4. Neutrino models and tension with external data
In summary, modifications to the neutrino sector alone cannot easily explain the discrepancies between Planck and other astrophysical data described in Sect. 5.5, including the inference of a low value of σ8 from rich cluster counts.
6.4.5. Testing perturbations in the neutrino background
The Planck data provide evidence for a cosmic neutrino background at a very high significance level.
6.5. Primordial nucleosynthesis
The Planck base ΛCDM predictions of Eq. (74) lie within 1σ of the Cooke et al. (2014) result. This is a remarkable success for the standard theory of BBN.
The region preferred by CMB observations lies at the intersection between the helium and deuterium abundance 68% CL preferred regions and is compatible with the standard value of Neff = 3.046. This confirms the beautiful agreement between CMB and BBN physics. Figure 36 also shows that the Planck polarization data help in reducing the degeneracy between ωb and Neff.
6.6. Dark matter annihilation
Note that to produce the observed dark matter density from thermal DM relics requires an annihilation cross-section of ⟨ σν ⟩ ≈ 3 × 10-26 cm3 s-1 (assuming s-wave annihilation) at the time of freeze-out (see, e.g., the review by Profumo 2013).
The dark grey dots indicate the best-fit dark matter models described in that paper. The favoured value of the cross-section is about two orders of magnitude higher than the thermal relic cross-section (≈3 × 10-26 cm3 s-1). Attempts to reconcile such a high cross-section with the relic abundance of DM include a Sommerfeld enhanced cross-section (that may saturate at ⟨ σν ⟩ ≈ 10-24 cm3 s-1) or non-thermal production of DM. ). Both of these possibilities are strongly disfavoured by the Planck data. ). Since the relative velocity of DM particles at recombination is many orders of magnitude smaller than in the Galactic halo, such a model cannot be constrained using CMB data.
6.7. Testing recombination physics with Planck
The cosmological recombination process determines how CMB photons decoupled from baryons around redshift z ≈ 103, when the Universe was about 400 000 years old.
For the base ΛCDM model, we find that the largest bias is on ns, at the level of 0.15σ (≈0.0006) for Planck TT,TE,EE+lowP+BAO. Although this is about 5 times larger than the difference in ns between CosmoRec and HyRec, this bias is nevertheless unimportant (really?) at the current level of precision.
a) The 2015 PlanckTT, TE, EE, and lensing spectra are consistent with each other under the assumption of the base ΛCDM cosmology. However, when comparing the TE and EE spectra computed for different frequency combinations, we find evidence for systematic effects caused by temperature-to-polarization leakage.
b) The PlanckTT, TE, and EE spectra are accurately described by a purely adiabatic spectrum of fluctuations with a spectral tilt ns = 0.968 ± 0.006, consistent with the predictions of single-field inflationary models. Combining Planck data with BAO, we find tight limits on the spatial curvature of the Universe, | ΩK | < 0.005, again consistent with the inflationary prediction of a spatially-flat Universe.
c) The Planck data show no evidence for tensor modes. Adding a tensor amplitude as a one-parameter extension to base ΛCDM, we derive a 95% upper limit of r0.002< 0.11. This is consistent with the B-mode polarization analysis reported in BKP, resolving the apparent discrepancy between the Planck constraints on r and the BICEP2 results reported by BICEP2 Collaboration (2014). In fact, by combining the Planck and BKP likelihoods, we find an even tighter constraint, r0.002< 0.09, strongly disfavouring inflationary models with a V(φ) ∝ φ2potential.
d) The Planck data show no evidence for any significant running of the spectral index. We also set strong limits on a possible departure from a purely adiabatic spectrum, either through an admixture of fully-correlated isocurvature modes or from cosmic defects.
e) The Planck best-fit base ΛCDM cosmology (we quote numbers for Planck TT+lowP+lensing here) is in good agreement with results from BAO surveys, and with the recent JLA sample of Type Ia SNe. The Hubble constant in this cosmology is H0 = (67.8 ± 0.9) km s-1Mpc-1, consistent with the direct measurement of H0 of Eq. (30) used as an H0 prior in this paper.
The amplitude of the present-day fluctuation spectrum, σ8, of the Planck base ΛCDM cosmology is higher than inferred from weak lensing measurements from the CFHTLenS survey and, possibly, from counts of rich clusters of galaxies.
The Planck base ΛCDM cosmology is also discordant with Lyα BAO measurements at z ≈ 2.35 At present, the reasons for these tensions are unclear.
f) The Planck data strongly disfavour fully thermalized sterile neutrinos with msterile ≈ 1 eV that have been proposed as a solution to reactor neutrino oscillation anomalies. FromPlanck, we find no evidence for new neutrino physics. Standard neutrinos with masses larger than those in the minimal mass hierarchy are still allowed.
g) The standard theory of big bang nucleosynthesis, with Neff = 3.046 and negligible leptonic asymmetry in the electron neutrino sector, is in excellent agreement with Planck data and observations of primordial light element abundances.
h) We have investigated the temperature and polarization signatures associated with annihilating dark matter and possible deviations from the standard recombination history. Again, we find no evidence for new physics from the Planck data.
i) The Planck results offer powerful evidence in favour of simple inflationary models, which provide an attractive mechanism for generating the slightly tilted spectrum of (nearly) Gaussian adiabatic perturbations that match our data to such high precision.
j) the Planck data show that the neutrino sector of the theory is consistent with the assumptions of the base ΛCDM model and that the dark energy is compatible with a cosmological constant. If there is new physics beyond base ΛCDM, then the corresponding observational signatures in the CMB are weak and difficult to detect.
10.3. Dark energy and modified gravity
Observations have long shown that only a small fraction of the total energy density in the Universe (around 5%) is in the form of baryonic matter, with the dark matter needed for structure formation accounting for about another 26%. In one scenario, the dominant component, generically referred to as dark energy (DE), brings the total close to the critical density and is responsible for the recent phase of accelerated expansion. In another scenario, the accelerated expansion arises, partly or fully, owing to a modification of gravity on cosmological scales.
As for Planck Collaboration XIII (2016), the results are consistent with the simplest scenario, ΛCDM, though all constraints on dark energy models… and modified gravity models…) are considerably improved with respect to past analyses. In particular, we improve significantly the constraint on the density of dark energy at early times, finding that it has to be below 2%... of the critical density (and an even tighter bound results if high-ℓ polarization is included).
These results imply that V(φ) ∝ φ2 and natural inflation are now disfavoured compared to models predicting a smaller tensor-to-scalar ratio, such as R2inflation. We search for several physically motivated deviations from a simple power-law spectrum of curvature perturbations, including those motivated by a reconstruction of the inflaton potential not relying on the slow-roll approximation. We find that such models are not preferred, either according to a Bayesian model comparison or according to a frequentist simulation-based analysis.
These results are consistent with the Planck 2013 analysis based on the nominal mission data and further constrain slow-roll single-field inflationary models as expected from the increased precision of Planck data using the full set of observations.
The addition of polarization data has significantly improved the limits on any isocurvature modes, which are now constrained at the percent level. Despite a detailed search, and study of several models, we see no statistically significant evidence for departures from a power law.
Single-field inflationary models with a standard kinetic term were also found to be compatible with the new tight upper bounds on the primordial non-Gaussianity parameters fNL No evidence of isocurvature perturbations as generated in multi-field inflationary models or by cosmic strings or topological defects was found.
The Planck 2013 results overall favoured the simplest inflationary models. However, we noted an amplitude deficit for multipoles ℓ ≲ 40 whose statistical significance relative to the six-parameter base Λ cold dark matter (ΛCDM) model is only about 2σ, as well as other anomalies on large angular scales.
10.6. Primordial non-Gaussianity
The global picture that emerges is one of consistency with the premises of ΛCDM cosmology, namely that the structure we observe today is the consequence of the passive evolution of adiabatic, Gaussian, nearly scale-invariant, primordial seed perturbations.
10.7. Isotropy and statistics
A large number of statistical tests indicate consistency with Gaussianity, while a power deficit at large angular scales is manifested in several ways.
10.8. The ISW effect (integrated Sachs-Wolfe effect)
Our analysis, using specially-constructed CMB temperature maps that are correlated and uncorrelated with E-modes, cannot rule out the ISW effect as the cause of these anomalies.
10.9. Cosmology from clusters
We confirm the 2013 results with the larger 2015 catalogue.
12. Summary and conclusions
Specifically, we can estimate the optical depth of reionization, τ, independently of other experiments. The value of τ is smaller than found in previous determination, implying later reionization.
There is no compelling evidence for any extensions to the 6-parameter model, or any need for new physics; five of the six parameters are now measured to better than 1% precision.
Using only Planck data, we find that the Universe is flat to 0.7% (1σ). Including BAO data, the constraint tightens to a remarkable 0.25%.
Models of inflation are more tightly constrained than ever before, with the simplest φn models being ruled out for n ≥ 2.
The Planck full mission temperature and polarization data are consistent with the spatially flat base ΛCDM model whose perturbations are Gaussian and adiabatic with a spectrum described by a simple power law, as predicted by the simplest inflationary models.
Among the models considered using this approach, the R2 inflationary model proposed by Starobinsky (1980) is the most favoured. Due to its high tensor-to-scalar ratio, the quadratic model is now strongly disfavoured and natural inflation is also disfavoured.
Finally we examined the connection between inflation and statistical isotropy. We found that a modulated curvaton model proposed to explain the observed large-scale dipolar power asymmetry cannot account for all of the asymmetry, and hence is not preferred over statistically isotropic base ΛCDM.
The rest of the work should be done by Occam's razor.