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\(D_A = \frac{r_s}{\theta_s}\)

where the numerator is the known physical scale and the denominator is the measured angular scale. The angular diameter distance is a well-known function of the Hubble rate, and you can infer the Hubble rate from getting the angular diameter distance (assuming the only pertinent species of particles in the universe are dark matter, baryons, photons, neutrinos, and the cosmological constant). In particular, the equation is

\(\int_0^{z^*}\)

\(D_A = \int_0^{z^*}\)

\( D_A = \int_0^{z^*} \)

\( D_A = \frac{1}{1+z^*} \frac{dz}{H(z)} \)

\( D_A = \frac{1}{1+z^*} \int_0 \frac{dz}{H(z)} \) (works)

$D_A = \frac{1}{1+z^*} \int_0 \frac{dz}{H(z)}$ (works)

$D_A = \frac{1}{1+z^*} \int_0^z \frac{dz}{H(z)}$ (works) $D\_A = \frac{1}{1+z^*} \int_0^z \frac{dz}{H(z)}$

$D_A = \frac{1}{1+z^*} \int_0^{z^1} \frac{dz}{H(z)}$ (works) $D\_A = \frac{1}{1+z^1} \int_0^{z^1} \frac{dz}{H(z)}$

$D_A = \frac{1}{1+z^*} \int_0^{z*} \frac{dz}{H(z)}$ (works) $D\_A = \frac{1}{1+z^\*} \int_0^{z\*} \frac{dz}{H(z)}$

$D_A = \frac{1}{1+z^*} \int_0^{z^*} \frac{dz}{H(z)}$ (works) $D\_A = \frac{1}{1+z^\*} \int_0^{z^\*} \frac{dz}{H(z)}$

$D_A = \frac{1}{1+z^} \int_0^{z^} \frac{dz}{H(z)}$ (does not work) $D\_A = \frac{1}{1+z^*} \int_0^{z^*} \frac{dz}{H(z)}$

\( D_A = \frac{1}{1+z^} \int_0^{z^} \frac{dz}{H(z)} \) (does not work) \\( D\_A = \frac{1}{1+z^*} \int_0^{z^*} \frac{dz}{H(z)} \\)

$H(z) = H_0 \sqrt{\Omega_m (1+z)^3 + \Omega_{rad} (1+z)^4 + \Omega_{\Lambda}}$

with $z^*$ is the redshift of the CMB (~1100), and the $\Omega$ density parameters corresponding to the known total matter density, radiation density, and vacuum energy density today, respectively.

Tags: #CMB #Cosmology #Physics

ConnectedPapers.com

ACT Publications: https://act.princeton.edu/publications

Dillon Brout KITP talk July 2019, H0 Tension from the Viewpoint of the Dark Energy Survey (DES) CosmoComments | Hypothes.is cosmology comments | CosmoDiscussion on slack.com | CosmoCoffee board |

PyCosmo Hub | PyCosmo ETHzurich site | [2005.00543] Predicting Cosmological Observables with PyCosmo |

ncatlab Cosmology topics Cosmology at Home Astrophysics and Cosmology lecture notes | figures to accompany lecture notes |

ReadTheDocs: astronomy | astrophysics | cosmology | supernovae |

Sean Carroll's Spacetime and Geometry, an Introduction to General Relativity, html version freely available online until May 31, 2020

CalTech Ay 21 Galaxies and Cosmology, Winter 2018; class videos, slides, resources are online

Astrometry.net web service (David Hogg): http://nova.astrometry.net | http://astrometry.net/ | github | google groups |

Chapter 1: [1909.13740] The Fundamentals of the 21-cm Line Chapter 2: [1909.12595] Astrophysics from the 21-cm background Chapter 3: [1909.12430] Physical Cosmology From the 21-cm Line Chapter 4: [1909.13860] Inference from the 21cm signal Chapter 5: [1909.11938] 21 cm observations: calibration, strategies, observables Chapter 6: [1909.12369] Foregrounds and their mitigation Chapter 7: [1909.12491] The status of 21cm interferometric experiments Chapter 8: ? Chapter 9: [1909.12797] Future prospects To appear as a book chapter in The Cosmic 21-cm Revolution: Charting the first billion years of our Universe, Ed Andrei Mesinger (Bristol: IOP Publishing Ltd) AAS-IOP ebooks; expected publication Feb. 2020 Andrei Mesinger's home page | reddit post | -—

See this wp cmb primary anisotropy section for interpretations of the power spectrum peaks, and adiabatic density perturbations vs isocurvature density perturbations. The CMB can distinguish between the two: “adiabatic density perturbations produce peaks whose locations are in the ratio 1:2:3:...[56] Observations are consistent with the primordial density perturbations being entirely adiabatic, providing key support for inflation, and ruling out many models of structure formation involving, for example, cosmic strings.” adiabatic vs isocurvature perturbations | primordial isocurvature perturbations |

The Hubble tension might be pointing towards new physics, a possibility which has been the subject of significant study in the literature: see e.g. [88–137] for a selection of works examining this possibility. However, as pointed out in a number of recent works (e.g. [138–140]), it is important to check that proposed solutions are consistent with BAO distance measurements Soundness of Dark Energy Properties, pg 8.

Some of the numerous examples proposing to alleviate Hubble constant tension [1908.03663] The Hubble Hunter's Guide, reviews “a variety of types of departures from ΛCDM that could, in principle, restore concordance among these datasets, and we explain why we find almost all of them unlikely to be successful.” [2004.09487] Relieving the Hubble tension with primordial magnetic fields [2003.03602] Reconciling Hubble Constant Discrepancy from Holographic Dark Energy [2002.10831] Solving the curvature and Hubble parameter inconsistencies through structure formation-induced curvature [2002.06782] Reducing the \(H_{0}\) tension with generalized Proca theory [2002.06127] Tensions in the dark: shedding light on Dark Matter-Dark Energy Interactions [2002.05602] Phenomenological model explaining Hubble Tension origin [2001.07536] Resolving the \(H_0\) tension with diffusion [1912.00242] \(H_0\) tension and the String Swampland [1912.00190] Can Non-standard Recombination Resolve the Hubble Tension? [1911.11760] Early dark energy from massive neutrinos — a natural resolution of the Hubble tension, by Sakstein and Trodden [1911.06281] Thermal Friction as a Solution to the Hubble Tension; Early Dark Energy, EDE [1910.00459] Quintessence Axion Dark Energy and a Solution to the Hubble Tension [1908.06995] Oscillating scalar fields and the Hubble tension: a resolution with novel signatures; by Tristan Smith, Poulin, Amin; early dark energy, EDE [1908.04281] Interacting dark energy after the latest Planck, DES, and \(H_0\) measurements: an excellent solution to the \(H_0\) and cosmic shear tensions [1908.03324] Resolving Hubble Tension with Quintom Dark Energy Model [1908.02401] The Hubble-Lemaître constant and sound horizon from low-redshift probes [1907.12639] Can redshift errors bias measurements of the Hubble Constant? [1905.10198] Implications of a transition in the dark energy equation of state for the \(H_0\) and σ8 tensions [1907.07569] New physics in light of the \(H_0\) tension: an alternative view, by Sunny Vagnozzi | response1 | response2 | [1907.05608] Evaporating primordial black holes as varying dark energy; alleviates \(H_0\) tension [1906.11628] Baryon Acoustic Oscillations and the Hubble Constant: Past, Present and Future [1905.02278] Super-CMB fluctuations can resolve the Hubble tension [1904.09689] A possible solution to the Hubble constant discrepancy — Cosmology where the local volume expansion is driven by the domain average density [1904.01016] Rock 'n' Roll Solutions to the Hubble Tension [1903.07603] Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics Beyond LambdaCDM [1903.06220] Late universe decaying dark matter can relieve the \(H_0\) tension; Avi Loeb et al [1902.07081] The local and distant Universe: stellar ages and H0 [1811.04083] Early Dark Energy Can Resolve The Hubble Tension [1809.02340] Can the \(H_0\) tension be resolved in extensions to ΛCDM cosmology? [1802.03404] Prospects for resolving the Hubble constant tension with standard sirens [1712.02967] Emerging spatial curvature can resolve the tension between high-redshift CMB and low-redshift distance ladder measurements of the Hubble constant [1410.0960] Stable FLRW solutions in Generalized Massive Gravity | [1401.4173] Massive Gravity | wp article | Massive Gravity article | [1907.11594]The BAO+BBN Take on the Hubble Tension

Voyage through the hidden physics of the cosmic web UBVRI filters

Will the CMB ever recede outside our visibility: SE1 | SE2 | R1 | H1 | Doug Scott's answer to CMB questions | Ned Wright | S1 | [0704.0221] The Return of a Static Universe and the End of Cosmology |

[0907.2887] Cosmic Neutrino Last Scattering Surface, Dodelson [1904.10544] Developments in Cosmic Growth and Gravitation Wayne Hu PhD thesis Observing the Big Bang, Luca Amendola, May 2019, a nice 100+ pages on cosmology Lots of pages available for Longair's Galaxy Formation [1501.03822] Real time cosmology – A direct measure of the expansion rate of the Universe with SKA

Emission spectra of the elements: Atomic Emission Spectra of the Periodic Table of Elements Atomic Absorption and Emission Spectra | The Atomic Spectrum of Hydrogen |

Planck 2013 results. XVI. Cosmological parameters

Quantum Fluctuations in Cosmology and How They Lead to a Multiverse, by Alan Guth

Cosmological parameter inference with Bayesian statistics [1903.11127] The main aim of this work is to provide an introduction of Bayesian parameter inference and its applications to cosmology. We assume the reader is familiarized with the basic concepts of statistics, but not necessarily with Bayesian statistics. Then, we provide a general introduction to this subject, enough to work out some examples

Constraining the physics of the early universe [1903.11472] PhD thesis

The Degree of Fine-Tuning in our Universe — and Others [1902.03928], Fred C. Adams

Speaker Deck

HashTags: #cosmology

https://arxiv.org/abs/1807.06211 Planck 2018: Constraints on Inflation in relation to https://arxiv.org/abs/1810.05216 https://old.reddit.com/r/cosmology/comments/9ocvsv/bicep2_keck_array_x_constraints_on_primordial/

MUSE reddit | 1810.00843 |

Subaru Hyper Suprime-Cam Y1 survey Sept '18: Cosmology from cosmic shear power spectra with Subaru Hyper Suprime-Cam first-year data [1809.09148] Hyper Suprime-Cam: https://hsc.mtk.nao.ac.jp/ssp/instrument/ Survey status: https://hsc.mtk.nao.ac.jp/ssp/survey/#survey_status Public Data Release 1: page https://hsc-release.mtk.nao.ac.jp/doc/ Phys.org article reddit thread tw9916

Hashtags: #cosmology

GitHub project: Tools for plotting CMB polarization power spectra

Spherical CMB map or 2D CMB map

Wayne Hu: graphic taking apart features in the power spectrum, slide 9 | CMB fluid approximation, slide 9+ |.

https://en.wikipedia.org/wiki/Cosmic_microwave_background (the refs, esp to Wayne Hu's work, are good)

Will the CMB ever recede outside our visibility: SE1 | SE2 | R1 | H1 | Doug Scott's answer to CMB questions | Ned Wright | S1 |

https://www.ast.cam.ac.uk/~pettini/Intro%20Cosmology/Lecture09.pdf Recombination was not an instantaneous process but proceeded relatively quickly nevertheless, with the fractional ionisation decreasing from X = 0.9 to X = 0.1 over a time interval ∆t ∼ 70,000 yr. With the number density of free electrons dropping rapidly, the time when photons and baryons decoupled follows soon, once the rate for Thomson scattering falls below the expansion rate H. Section 9.3, Photon Decoupling, pg 6 (These excellent notes cover the following epochs: Radiation-Matter Equality, Recombination, Photon Decoupling, Last Scattering.)

Physics of the cosmic microwave background anisotropy [1501.04288], by Martin Bucher | Anisotropy of the Cosmic Background Radiation and Cosmological Parameters, Bartelmann | Determining Cosmological Parameters from Anisotropies in the CMB |

Wayne Hu's writings: CMB, power spectrum, BAO, etc | Great graphic | BAO section | Lecture notes from introductory cosmology course: e.g., cosmic geometry |

https://www.reddit.com/r/AskPhysics/comments/8i79wf/why_do_we_believe_there_is_dark_matter/

The CMB, J. Kenney class 22 slides, very good

Halverson – Measuring the CMB Angular Power Spectrum with DASI.pdf (local PDF) | [astro-ph/0104489] DASI First Results: A Measurement of the Cosmic Microwave Background Angular Power Spectrum, 8 pages |

The simplicity of CMB physics – due to linearity – is mirrored in analysis by the apparent Gaussianity of both the signal and many sources of noise... The Gaussianity of the CMB is not shared by other cosmological systems since gravitational non-linearities turn an initially Gaussian distribution into a non-Gaussian one. http://background.uchicago.edu/~whu/araa/node32.html

most of the secondary anisotropies are not linear in nature and hence produce non-Gaussian signatures. Non-Gaussianity in the lensing and SZ signals will be important for their isolation. The same is true for contaminants such as galactic and extragalactic foregrounds. Finally the lack of an initial non-Gaussianity in the fluctuations is a testable prediction of the simplest inflationary models [Guth & Pi, 1985,Bardeen et al, 1983]. Consequently, non-Gaussianity in the CMB is currently a very active field of research. http://background.uchicago.edu/~whu/araa/node31.html

arXiv listing with abstracts of all Planck 2013, 2015, 2018 papers

Astro useful info

Hashtags: #CMB #Cosmology #physics

Planck 2013 results. XVI. Cosmological parameters | Nice wp article on measuring SNe distances |

WikiProject Astronomy/Cosmology | Cosmology Task Force | Physics | Science Directory | reddit Q&A |

Discovery of the hot Big Bang: What Happened in 1948 Past, Present and Future of Cosmic Microwave Background Observations: Implications for Cosmology

Astro2020 Science White Paper: A proposal to exploit galaxy-21cm synergies to shed light on the Epoch of Reionization https://arxiv.org/abs/1903.03628

3 cosmology lectures by Celine Boehm at Sept. 2017 European School of High Energy Physics (slides) https://t.co/bbl9reJMU3 https://t.co/dtn0oy08gi https://t.co/XcEIqpkksQ All ESHEP 2017 lectures: https://t.co/dHsZMwuOn5

Sound Waves from the Beginning of Time, PBS SpaceTime video on Baryon Acoustic Oscillation (BAO) Secrets of the Cosmic Microwave Background, PBS SpaceTime video | reddit post |

The Distance Scale of the Universe; excellent! [astro-ph/0310571] A Map of the Universe This explains the “future visability limit”, which is also refered to in this Ethan Siegel article

[1812.06841] Inflation in Loop Quantum Cosmology, Ed Copeland et al

Megamaser Cosmology Project

Las Cumbres Observatory: Measuring the Age of the Universe | Magnitudes and Distance Measurement | Star in a Box: interactive app on lifecycle of stars |

John C. Brown, Astronomer Royal for Scotland | on wp | U Glasgow | what being an astronomer is like | known for Cold Thick-Target model |

Latest cosmology results page

Amplitude of Matter Fluctuations, σ8. The σ8 parameter is a measure of matter clumping in the LCDM model The amplitude of the primordial matter power spectrum is parameterized by σ8. This provides an important constraint on clustering of matter (CDM dominated) and the formation of structure.
Determining the cosmological parameters Ωm and σ8 from peculiar velocity and density-contrast data Another, albeit somewhat weaker, tension (compared to H0) exists between the values of the amplitude of galaxy clustering S8 σ8 and the matter fraction Ωm (S₈ − Ωm tension) in the Planck best fit model and those inferred from the latest surveys of large scale structure. [2004.09487] Adam Riess interview with Sean Carroll: this is not a quest to measure a number, this is a quest to do an end-to-end test of the universe and understand why it’s failing or we’re failing...there is this problem with the lumpiness of the universe as well...there’s another parameter, which often goes by the catchy name of Sigma 8 and it measures the clumpiness of matter. And so likewise in the early universe, because we know the state of the universe shortly after the Big Bang, and then we know how it dilutes due to expansion, we should be able to predict how clumpy it should be today. And then we have techniques which are also hard but are powerful in measuring how clumpy is, either by measuring this phenomenon called loop lensing, where we measure how clumps of matter distort the shapes, the apparent shapes of galaxies, or by measuring the motions of galaxies, extra motions due to not the expansion of space but matter, the clumpiness of matter, and those two at this point disagree with each other at… I guess the most recent number that I’ve seen is sort of 3 sigma from each of those, the lensing method and the peculiar velocity method. And in the same direction, that the late universe measurements are in a different direction in terms of size to the sigma 8 number as multiple early universe measurement.

IPython notebook tutorials covering NumPy, Matplotlib, SciPy. etc. and overview of the Python language itself. Matplotlib plotting | Precision data plotting in Python with Matplotlib | Do you often find a figure you made and have no idea what script you used to generate it? I just discovered the metadata keyword in matplotlib (plt.savefig). You can pass a dictionary including the “Creator” and the name of the script! [tweet]

ICRAR cosmology calculator | iCosmos | Ned Wright cosmology calculator |

Lambda-Tools | wp cosmology software | astrobites software refs | ASCL web resources and tools | LightCone7 Combo Tutorial – calculator for expansion history of universe |

Computing the Expansion History of the Universe, pdf |

CosmoloPy | nbinderkit |

Colossus html exported from ipynb notebooks | Colossus cosmology python astrophysics toolkit | Overview docs | LSS and DM Halo modules | Diemer 1712.04512 paper | -The easiest way to learn how to use Colossus is by following the tutorial notebooks. -You can learn more about Jupyter notebooks on the Jupyter website. * Cosmology <static/tutorialcosmology.html> * LSS: peaks in Gaussian random fields <static/tutoriallss_peaks.html> -The editable notebooks are located in the colossus/tutorials directory, the links below point to their html output: https://bitbucket.org/bdiemer/colossus/commits/f73536addcb083173bbbe3adb9021da45cac1bbe https://pypi.org/project/colossus/ from the WinPython command interpreter: pip install colossus

>python 
>>>from colossus.tests run import run_tests
Ran 87 tests in 17.344s

Astropy | Astrophysics source code library | astropy tutorials, Jupyter notebooks on github |

Cosmology Collaboration Wikis: https://cmb-s4.org/wiki/index.php/UChicago-2020:_Cosmology_with_CMB-S4 (MediaWiki) Planck Legacy Archive: https://wiki.cosmos.esa.int/planck-legacy-archive/index.php/Planck_Collaboration https://wiki.cosmos.esa.int/planck-legacy-archive/index.php/Main_Page (MediaWiki) NRAO: https://safe.nrao.edu/wiki/bin/view/Main/WebHome (public wiki, separate staff wiki) MCP: https://safe.nrao.edu/wiki/bin/view/Main/MegamaserCosmologyProject TASI: https://sites.google.com/a/colorado.edu/tasi-2018-wiki/home

Hashtags: #cosmology #physics