There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper presents cosmological results based on full-mission Planck observations of temperature and polarization anisotropies of the cosmic microwave background (CMB) radiation. Our results are in very good agreement with the 2013 analysis of the Planck nominal-mission temperature data, but with increased precision. The temperature and polarization power spectra are consistent with the standard spatially-flat 6-parameter ΛCDM cosmology with a power-law spectrum of adiabatic scalar perturbations (denoted “base ΛCDM” in this paper). From the Planck temperature data combined with Planck lensing, for this cosmology we find a Hubble constant, H0 = (67.8 ± 0.9) km s-1Mpc-1, a matter density parameter Ωm = 0.308 ± 0.012, and a tilted scalar spectral index with ns = 0.968 ± 0.006, consistent with the 2013 analysis. Note that in this abstract we quote 68% confidence limits on measured parameters and 95% upper limits on other parameters. We present the first results of polarization measurements with the Low Frequency Instrument at large angular scales. Combined with the Planck temperature and lensing data, these measurements give a reionization optical depth of τ = 0.066 ± 0.016, corresponding to a reionization redshift of . These results are consistent with those from WMAP polarization measurements cleaned for dust emission using 353-GHz polarization maps from the High Frequency Instrument. We find no evidence for any departure from base ΛCDM in the neutrino sector of the theory; for example, combining Planck observations with other astrophysical data we find Neff = 3.15 ± 0.23 for the effective number of relativistic degrees of freedom, consistent with the value Neff = 3.046 of the Standard Model of particle physics. The sum of neutrino masses is constrained to ∑ mν < 0.23 eV. The spatial curvature of our Universe is found to be very close to zero, with | ΩK | < 0.005. Adding a tensor component as a single-parameter extension to base ΛCDM we find an upper limit on the tensor-to-scalar ratio of r0.002< 0.11, consistent with the Planck 2013 results and consistent with the B-mode polarization constraints from a joint analysis of BICEP2, Keck Array, and Planck (BKP) data. Adding the BKP B-mode data to our analysis leads to a tighter constraint of r0.002 < 0.09 and disfavours inflationarymodels with a V(φ) ∝ φ2 potential. The addition of Planck polarization data leads to strong constraints on deviations from a purely adiabatic spectrum of fluctuations. We find no evidence for any contribution from isocurvature perturbations or from cosmic defects. Combining Planck data with other astrophysical data, including Type Ia supernovae, the equation of state of dark energy is constrained to w = −1.006 ± 0.045, consistent with the expected value for a cosmological constant. The standard big bang nucleosynthesis predictions for the helium and deuterium abundances for the best-fit Planck base ΛCDM cosmology are in excellent agreement with observations. We also constraints on annihilating dark matter and on possible deviations from the standard recombination history. In neither case do we find no evidence for new physics. The Planck results for base ΛCDM are in good agreement with baryon acoustic oscillation data and with the JLA sample of Type Ia supernovae. However, as in the 2013 analysis, the amplitude of the fluctuation spectrum is found to be higher than inferred from some analyses of rich cluster counts and weak gravitational lensing. We show that these tensions cannot easily be resolved with simple modifications of the base ΛCDM cosmology. Apart from these tensions, the base ΛCDM cosmology provides an excellent description of the Planck CMB observations and many other astrophysical data sets.

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Planck 2015 results - XIII. Cosmological parameters

We present results based on full-mission Planck observations of temperature and polarization anisotropies of the CMB. These data are consistent with the six-parameter inflationary LCDM cosmology. From the Planck temperature and lensing data, for this cosmology we find a Hubble constant, H0= (67.8 +/- 0.9) km/s/Mpc, a matter density parameter Omega_m = 0.308 +/- 0.012 and a scalar spectral index with n_s = 0.968 +/- 0.006. (We quote 68% errors on measured parameters and 95% limits on other parameters.) Combined with Planck temperature and lensing data, Planck LFI polarization measurements lead to a reionization optical depth of tau = 0.066 +/- 0.016. Combining Planck with other astrophysical data we find N_ eff = 3.15 +/- 0.23 for the effective number of relativistic degrees of freedom and the sum of neutrino masses is constrained to < 0.23 eV. Spatial curvature is found to be |Omega_K| < 0.005. For LCDM we find a limit on the tensor-to-scalar ratio of r <0.11 consistent with the B-mode constraints from an analysis of BICEP2, Keck Array, and Planck (BKP) data. Adding the BKP data leads to a tighter constraint of r < 0.09. We find no evidence for isocurvature perturbations or cosmic defects. The equation of state of dark energy is constrained to w = -1.006 +/- 0.045. Standard big bang nucleosynthesis predictions for the Planck LCDM cosmology are in excellent agreement with observations. We investigate annihilating dark matter and deviations from standard recombination, finding no evidence for new physics. The Planck results for base LCDM are in agreement with BAO data and with the JLA SNe sample. However the amplitude of the fluctuations is found to be higher than inferred from rich cluster counts and weak gravitational lensing. Apart from these tensions, the base LCDM cosmology provides an excellent description of the Planck CMB observations and many other astrophysical data sets.

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Planck 2015 results. XIII. Cosmological parameters

This paper presents the first cosmological results based on Planck measurements of the cosmic microwave background (CMB) temperature and lensing-potential power spectra. We find that the Planck spectra at high multipoles (l ≳ 40) are extremely well described by the standard spatially-flat six-parameter ΛCDM cosmology with a power-law spectrum of adiabatic scalar perturbations. Within the context of this cosmology, the Planck data determine the cosmological parameters to high precision: the angular size of the sound horizon at recombination, the physical densities of baryons and cold dark matter, and the scalar spectral index are estimated to be θ∗ = (1.04147 ± 0.00062) × 10-2, Ωbh2 = 0.02205 ± 0.00028, Ωch2 = 0.1199 ± 0.0027, and ns = 0.9603 ± 0.0073, respectively(note that in this abstract we quote 68% errors on measured parameters and 95% upper limits on other parameters). For this cosmology, we find a low value of the Hubble constant, H0 = (67.3 ± 1.2) km s-1 Mpc-1, and a high value of the matter density parameter, Ωm = 0.315 ± 0.017. These values are in tension with recent direct measurements of H0 and the magnitude-redshift relation for Type Ia supernovae, but are in excellent agreement with geometrical constraints from baryon acoustic oscillation (BAO) surveys. Including curvature, we find that the Universe is consistent with spatial flatness to percent level precision using Planck CMB data alone. We use high-resolution CMB data together with Planck to provide greater control on extragalactic foreground components in an investigation of extensions to the six-parameter ΛCDM model. We present selected results from a large grid of cosmological models, using a range of additional astrophysical data sets in addition to Planck and high-resolution CMB data. None of these models are favoured over the standard six-parameter ΛCDM cosmology. The deviation of the scalar spectral index from unity isinsensitive to the addition of tensor modes and to changes in the matter content of the Universe. We find an upper limit of r0.002< 0.11 on the tensor-to-scalar ratio. There is no evidence for additional neutrino-like relativistic particles beyond the three families of neutrinos in the standard model. Using BAO and CMB data, we find Neff = 3.30 ± 0.27 for the effective number of relativistic degrees of freedom, and an upper limit of 0.23 eV for the sum of neutrino masses. Our results are in excellent agreement with big bang nucleosynthesis and the standard value of Neff = 3.046. We find no evidence for dynamical dark energy; using BAO and CMB data, the dark energy equation of state parameter is constrained to be w = -1.13-0.10+0.13. We also use the Planck data to set limits on a possible variation of the fine-structure constant, dark matter annihilation and primordial magnetic fields. Despite the success of the six-parameter ΛCDM model in describing the Planck data at high multipoles, we note that this cosmology does not provide a good fit to the temperature power spectrum at low multipoles. The unusual shape of the spectrum in the multipole range 20 ≲ l ≲ 40 was seen previously in the WMAP data and is a real feature of the primordial CMB anisotropies. The poor fit to the spectrum at low multipoles is not of decisive significance, but is an “anomaly” in an otherwise self-consistent analysis of the Planck temperature data.

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Planck 2013 results. XVI. Cosmological parameters

We present the first results based on Planck measurements of the CMB temperature and lensing-potential power spectra. The Planck spectra at high multipoles are extremely well described by the standard spatially-flat six-parameter LCDM cosmology. In this model Planck data determine the cosmological parameters to high precision. We find a low value of the Hubble constant, H0=67.3+/-1.2 km/s/Mpc and a high value of the matter density parameter, Omega_m=0.315+/-0.017 (+/-1 sigma errors) in excellent agreement with constraints from baryon acoustic oscillation (BAO) surveys. Including curvature, we find that the Universe is consistent with spatial flatness to percent-level precision using Planck CMB data alone. We present results from an analysis of extensions to the standard cosmology, using astrophysical data sets in addition to Planck and high-resolution CMB data. None of these models are favoured significantly over standard LCDM. The deviation of the scalar spectral index from unity is insensitive to the addition of tensor modes and to changes in the matter content of the Universe. We find a 95% upper limit of r<0.11 on the tensor-to-scalar ratio. There is no evidence for additional neutrino-like relativistic particles. Using BAO and CMB data, we find N_eff=3.30+/-0.27 for the effective number of relativistic degrees of freedom, and an upper limit of 0.23 eV for the summed neutrino mass. Our results are in excellent agreement with big bang nucleosynthesis and the standard value of N_eff=3.046. We find no evidence for dynamical dark energy. Despite the success of the standard LCDM model, this cosmology does not provide a good fit to the CMB power spectrum at low multipoles, as noted previously by the WMAP team. While not of decisive significance, this is an anomaly in an otherwise self-consistent analysis of the Planck temperature data.

We show that dissipationless CDM simulations predict that the majority of themost massive subhalos of the Milky Way are too dense to host any of its brightsatellites (L V > 10 5 L ). These dark subhalos have circular velocities at infall ofV infall = 30 1070kms 1 and infall masses of [0:2 4] 10 M . Unless the Milky Way isa statistical anomaly, this implies that galaxy formation becomes e ectively stochasticat these masses. This is in marked contrast to the well-established monotonic relationbetween galaxy luminosity and halo circular velocity (or halo mass) for more massivehalos. We show that at least two (and typically four) of these massive dark subhalosare expected to produce a larger dark matter annihilation ux than Draco. It maybe possible to circumvent these conclusions if baryonic feedback in dwarf satellites ordi erent dark matter physics can reduce the central densities of massive subhalos byorder unity on a scale of 0.3 { 1 kpc.Key words: Galaxy: halo { galaxies: abundances { dark matter { cosmology: theory

Too big to fail? The puzzling darkness of massive Milky Way subhaloes

We use cosmological simulations to study the effects of self-interacting dark matter (SIDM) on the density profiles and substructure counts of dark matte r halos from the scales of spiral galaxies to galaxy clusters, focusing explicitly on mod els with cross sections over dark matter particle mass σ/m = 1 and 0.1 cm 2 /g. Our simulations rely on a new SIDM N-body algorithm that is derived self-consistently from the Boltz mann equation and that reproduces analytic expectations in controlled numerical experiments. We find that well-resolved SIDM halos have constant-density cores, with significantly lowe r central densities than their CDM counterparts. In contrast, the subhalo content of SIDM halos is only modestly reduced compared to CDM, with the suppression greatest for large hosts and small halo-centric distances. Moreover, the large-scale clustering and halo circular vel ocity functions in SIDM are effectively identical to CDM, meaning that all of the large-scale successes of CDM are equally well matched by SIDM. From our largest cross section runs we are able to extract scaling relations for core sizes and central densities over a range o f halo sizes and find a strong correlation between the core radius of an SIDM halo and the NFW scale radius of its CDM counterpart. We construct a simple analytic model, based on CDM scaling relations, that captures all aspects of the scaling relations for SIDM halos. Our results show that halo core densities in σ/m = 1 cm 2 /g models are too low to match observations of galaxy clusters, low surface brightness spirals (LSBs), and dwarf spheroidal galaxies. However, SIDM with σ/m ≃ 0.1 cm 2 /g appears capable of reproducing reported core sizes and central densities of dwarfs, LSBs, and galaxy clusters without the need for velocity dependence. Higher resolution simulations over a wider range of masses will be required to confirm this expectation. We discuss constraints arising from the Bullet cluster observ ations, measurements of dark matter density on small-scales and subhalo survival requirements, and show that SIDM models with σ/m ≃ 0.1 cm 2 /g ≃ 0.2 barn/GeV are consistent with all observational constraints.

Cosmological Simulations with Self-Interacting Dark Matter I: Constant Density Cores and Substructure

We use the Aquarius simulations to show that the most massive subhaloes in galaxy-mass dark matter (DM) haloes in Λ cold dark matter (ΛCDM) are grossly inconsistent with the dynamics of the brightest Milky Way dwarf spheroidal galaxies. While the best-fitting hosts of the dwarf spheroidals all have , ΛCDM simulations predict at least 10 subhaloes with Vmax > 25 km s−1. These subhaloes are also among the most massive at earlier times, and significantly exceed the reionization suppression mass back to z∼ 10. No ΛCDM-based model of the satellite population of the Milky Way explains this result. The problem lies in the satellites’ densities: it is straightforward to match the observed Milky Way luminosity function, but doing so requires the dwarf spheroidals to have DM haloes that are a factor of ∼5 more massive than is observed. Independent of the difficulty in explaining the absence of these dense, massive subhaloes, there is a basic tension between the derived properties of the bright Milky Way dwarf spheroidals and ΛCDM expectations. The inferred infall masses of these galaxies are all approximately equal and are much lower than standard ΛCDM predictions for systems with their luminosities. Consequently, their implied star formation efficiencies span over two orders of magnitude, from 0.2 to 20 per cent of baryons converted into stars, in stark contrast with expectations gleaned from more massive galaxies. We explore possible solutions to these problems within the context of ΛCDM and find them to be unconvincing. In particular, we use controlled simulations to demonstrate that the small stellar masses of the bright dwarf spheroidals make supernova feedback an unlikely explanation for their low inferred densities.

The Milky Way’s bright satellites as an apparent failure of ΛCDM

We derive an accurate mass estimator for dispersion-supported stellar systems and demonstrate its validity by analysing resolved line-of-sight velocity data for globular clusters, dwarf galaxies and elliptical galaxies. Specifically, by manipulating the spherical Jeans equation we show that the mass enclosed within the 3D deprojected half-light radius r 1/2 can be determined with only mild assumptions about the spatial variation of the stellar velocity dispersion anisotropy as long as the projected velocity dispersion profile is fairly flat near the half-light radius, as is typically observed. We find M 1/2 = 3G -1 (σ 2 los )r 1/2 ≃ 4G -1 (σ 2 los )R e , where (σ 2 los ) is the luminosity-weighted square of the line-of-sight velocity dispersion and R e is the 2D projected half-light radius. While deceptively familiar in form, this formula is not the virial theorem, which cannot be used to determine accurate masses unless the radial profile of the total mass is known a priori. We utilize this finding to show that all of the Milky Way dwarf spheroidal galaxies (MW dSphs) are consistent with having formed within a halo of a mass of approximately 3 x 10 9 M ⊙ . assuming a A cold dark matter cosmology. The faintest MW dSphs seem to have formed in dark matter haloes that are at least as massive as those of the brightest MW dSphs, despite the almost five orders of magnitude spread in luminosity between them. We expand our analysis to the full range of observed dispersion-supported stellar systems and examine their dynamical I-band mass-to-light ratios Υ I 1/2 . The Υ I 1/2 versus M 1/2 relation for dispersion-supported galaxies follows a U shape, with a broad minimum near Υ I 1/2 ≃ 3 that spans dwarf elliptical galaxies to normal ellipticals, a steep rise to Υ I 1/2 ≃ 3200 for ultra-faint dSphs and a more shallow rise to Υ I 1/2 ≃ 800 for galaxy cluster spheroids.

Accurate masses for dispersion-supported galaxies

We use the Aquarius simulations to show that the most massive subhalos in galaxy-mass dark matter halos in LCDM are grossly inconsistent with the dynamics of the brightest Milky Way dwarf spheroidal galaxies. While the best-fitting hosts of the dwarf spheroidals all have 12 25 km/s. These subhalos are also among the most massive at earlier times, and significantly exceed the UV suppression mass back to z ~ 10. No LCDM-based model of the satellite population of the Milky Way explains this result. The problem lies in the satellites' densities: it is straightforward to match the observed Milky Way luminosity function, but doing so requires the dwarf spheroidals to have dark matter halos that are a factor of ~5 more massive than is observed. Independent of the difficulty in explaining the absence of these dense, massive subhalos, there is a basic tension between the derived properties of the bright Milky Way dwarf spheroidals and LCDM expectations. The inferred infall masses of these galaxies are all approximately equal and are much lower than standard LCDM predictions for systems with their luminosities. Consequently, their implied star formation efficiencies span over two orders of magnitude, from 0.2% to 20% of baryons converted into stars, in stark contrast with expectations gleaned from more massive galaxies. We explore possible solutions to these problems within the context of LCDM and find them to be unconvincing. In particular, we use controlled simulations to demonstrate that the small stellar masses of the bright dwarf spheroidals make supernova feedback an unlikely explanation for their low inferred densities.

Manoj Kaplinghat

Papers

Too big to fail? The puzzling darkness of massive Milky Way subhaloes

Cosmological Simulations with Self-Interacting Dark Matter I: Constant Density Cores and Substructure

The Milky Way’s bright satellites as an apparent failure of ΛCDM

Accurate masses for dispersion-supported galaxies

The Milky Way's bright satellites as an apparent failure of LCDM