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I'm looking for some form of "rough and ready" formula to convert between redshift *z* value, years since BB, and distance, so that when I read an astronomy paper and it discusses an event that occurred at z=10+/-0.5, or a quasar at z=7, I can get a rough idea when the event occurred and how far away they are saying the quasar is, as context.

I'm aware that this is an inherently vague question, since the times and distances depend on the model and convention chosen, and also that I'm not specifying proper distances/comoving coordinates, etc.

My assumption is that most current models and their parameters, that are mainstream accepted and used, will be based on very similar parameters of very similar models (if not the same model), such as latest Planck parameters, or similar. They might differ more due to assumptions about very early times, but redshift is inherently >~380 k years, and if they did vary a lot for times beyond that timeframe, we'd have real issues. In other words, I suspect/hope that any differences in answers due to model variation, won't significantly change the answer to *this* question. So the only question will (hopefully) be, which convention or type of metric/distance will be appropriate.

If an assumption needs making (such as the meaning of 'distance' to be applied, or the zero point used for time: BB or end of inflation etc) please make an assumption most likely to help me, and I'll edit the question to clarify if needed, once I see what is too vague in the question as initially worded.

Thank you !

Use one of the "cosmology calculators". The conversions depend on what you assume for the cosmological parameters.

Here's one that will do what you want. http://home.fnal.gov/~gnedin/cc/

e.g. For the default parameters, $z=7$ corresponds to a look back time of 13.01 billion years, whilst $z=10.5$ corresponds to a look back time of 13.33 Gyr, with the current age of the universe being 13.78 Gyr. Choosing the "luminosity distance" option, we that the object has a luminosity distance of 112 Gpc.

If you want the proper distance you could use http://www.astro.ucla.edu/~wright/CosmoCalc.html According to this calculator the $z=10.5$ object is currently at a proper distance of 9.76 Gpc (31.8 billion light years).

I suspect that converting red-shift to distance, for any *particular* object, is going to have a lot of error. Especially for z > 1. Even for 0.05 < z < 1, the errors are often over 10%.

Half-life is defined as the amount of time it takes a given quantity to decrease to half of its initial value. The term is most commonly used in relation to atoms undergoing radioactive decay, but can be used to describe other types of decay, whether exponential or not. One of the most well-known applications of half-life is carbon-14 dating. The half-life of carbon-14 is approximately 5,730 years, and it can be reliably used to measure dates up to around 50,000 years ago. The process of carbon-14 dating was developed by William Libby, and is based on the fact that carbon-14 is constantly being made in the atmosphere. It is incorporated into plants through photosynthesis, and then into animals when they consume plants. The carbon-14 undergoes radioactive decay once the plant or animal dies, and measuring the amount of carbon-14 in a sample conveys information about when the plant or animal died.

Below are shown three equivalent formulas describing exponential decay:

- where

**N**is the initial quantity

_{0}**N**is the remaining quantity after time,

_{t}**t**

**t**is the half-life

_{1/2}**τ**is the mean lifetime

**λ**is the decay constant

If an archaeologist found a fossil sample that contained 25% carbon-14 in comparison to a living sample, the time of the fossil sample's death could be determined by rearranging equation 1, since **N _{t}**,

**N**, and

_{0}**t**are known.

_{1/2}This means that the fossil is 11,460 years old.

### Derivation of the Relationship Between Half-Life Constants

Using the above equations, it is also possible for a relationship to be be derived between **t _{1/2}**,

**τ**, and

**λ**. This relationship enables the determination of all values, as long as at least one is known.

## 2 Answers 2

The short answer is that, as you said, the redshift depends upon the scale factor at the time of transmission (as compared to the present). Since light travels at a finite speed, light from more distant sources was transmitted at a different time and hence scale factor.

You're redshift equation does NOT imply the same redshift for any distance, I think you were just interpreting, forgetting that light we're currently receiving from distant and near (relatively speaking) stars was released at VERY different times (and hence scale factors). The Hubble relation follows directly from the redshift equation for an expanding universe.

## Lecture 31: The Cosmic Microwave Background Radiation

### Solution

- Very early he Universe was radiation-dominated
**&rho**._{rad}> &rho_{m} - Transition took place when
**&rho**. i.e, when_{rad}= &rho_{m}**&rho**_{rad}(t_{0}) (1+z) 4 = &rho_{m}(t_{0}) (1+z) 3 - Therefore, the redshift of transition is given by the
**present day densities**of matter and radiation as

- The temperature of the Cosmic background Radiation changes at this redshift is
**T = T(t**_{0}) (1+z) &asymp 2.725 K x 5000 = 13600 K

### Can we compute the time of this or other events for which we now the redshift ?

## Redshift

Astronomers can learn about the motion of objects by looking at the way their color changes over time, or is different from the expected.

Redshift is an example of the the Doppler Effect. As an object moves away from us, the sound or light waves emitted by the object are stretched out, which makes them have a lower pitch and moves them towards the red end of the electromagnetic spectrum. In the case of light waves, this is called redshift. As an object moves towards us, sound and light waves are bunched up, so the pitch of the sound is higher, and light waves are moved towards the blue end of the electromagnetic spectrum. In the case of light waves, this is called blueshift.

The video below demonstrates the concepts of the Doppler Effect and redshift.

#### How Do Astronomers Measure Redshift?

The most accurate way to measure redshift is by using spectroscopy. By looking at the spectra of stars or galaxies, astronomers can compare the spectra they see for different elements with the spectra they would expect. If the absorption or emmission lines they see are shifted, they know the object is moving either towards us or away from us. For example, if the absorption lines are all shifted towards the red end of the spectrum, the object is redshifted. The object is moving away from us.

For far away objects such as quasars, some of which are too faint to be observed by spectrocopy, astronomers measure photometric redshifts. In this case they observe the peak brightness of the object through various filters. An object that is redshifted will have its peak brightness appear through filters towards the red end of the spectrum.

Astronomers talk about redshift in terms of the redshift parameter *z*. This is calculated with an equation:

*z* = (λobserved - λrest)/λrest

where λobserved is the observed wavelength of a spectral line, and λrest is the wavelength that line would have if its source was not in motion.

*z* is related to the distance of an object. This Cosmological Calculator lets you enter values of *z* and find the corresponding light travel time. This tells you the number of years the light from the object has traveled to reach us. This is not the distance to the object in light years, however, because the universe has been expanding as the light traveled and the object is now much farther away. The comoving radial distance takes this expansion into account and is the distance to the object now.

The highest known redshifts are from galaxies producing gamma ray bursts. The highest confirmed redshift is for a galaxy called UDFy-38135539 with a *z* value of 8.6, which corresponds to a light travel time of about 13.1 billion years. This means the light we see now left the galaxy about 600 million years after the Big Bang! The galaxy is now 30.384 billion light years away from us due to the expansion of the universe during the time the light from the galaxy traveled to us..

The table below gives light travel times and distances for some sample values of *z*:

z | Time the light has been traveling | Distance to the object now |
---|---|---|

0.0000715 | 1 million years | 1 million light years |

0.10 | 1.286 billion years | 1.349 billion light years |

0.25 | 2.916 billion years | 3.260 billion light years |

0.5 | 5.019 billion years | 5.936 billion light years |

1 | 7.731 billion years | 10.147 billion light years |

2 | 10.324 billion years | 15.424 billion light years |

3 | 11.476 billion years | 18.594 billion light years |

4 | 12.094 billion years | 20.745 billion light years |

5 | 12.469 billion years | 22.322 billion light years |

6 | 12.716 billion years | 23.542 billion light years |

7 | 12.888 billion years | 24.521 billion light years |

8 | 13.014 billion years | 25.329 billion light years |

9 | 13.110 billion years | 26.011 billion light years |

10 | 13.184 billion years | 26.596 billion light years |

#### What are Some Applications of Redshift?

Astronomers use redshift and blue shift (for nearby objects and measurements this technique is called the radial velocity method) to discover extrasolar planets. This method uses the fact that if a star has a planet (or planets) around it, it is not strictly correct to say that the planet orbits the star. Instead, the planet and the star orbit their common center of mass. Because the star is so much more massive than the planets, the center of mass is within the star and the star appears to wobble slightly as the planet travels around it. Astronomers can measure this wobble by using spectroscopy. If a star is traveling towards us, its light will appear blueshifted, and if it is traveling away the light will be redshifted. This shift in color will not change the apparent color of the star enough to be seen with the naked eye. Spectroscopy can be used to detect this change in color from a star as it moves towards and away from us, orbiting the center of mass of the star-planet system.

More generally, astronomers use redshift and blueshift or radial velocity to study objects that are moving, such as binary stars orbiting each other, the rotation of galaxies, the movement of galaxies in clusters, and even the movement of stars within our galaxy.

#### Cosmological Redshift

Astronomers also use redshift to measure approximate distances to very distant galaxies. The more distant an object, the more it will be redshifted. Some very distant objects may emit energy in the ultraviolet or even higher energy wavelengths. As the light travels great distances and is redshifted, its wavelength may be shifted by a factor of 10. So light that starts out as ultraviolet may be become infrared by the time it gets to us!

As the universe expands, the space between galaxies is expanding. The more distance between us and a galaxy, the more quickly the galaxy will appear to be moving away from us. It is important to remember that although such distant galaxies can appear to be moving away from us at near the speed of light, the galaxy itself is not traveling so fast. Its motion away from us is due to the expansion of the space between us.

#### Example to try:

Use the equation for the *z* parameter and the table above to answer the following:

Suppose light with a wavelength of 400 nm (violet) leaves a galaxy, and by the time it reaches us, its wavelength has been redshifted to 2000 nm in the infrared.

## Answers and Replies

Hi Suede - welcome to Physics Forums!

That's an interesting list of references (some of which weren't working just now but seem to be improving). It appears that they are mostly concerned with the quantization of intrinsic redshift (for galaxies within a cluster as well as quasars) rather than its existence, and I think that whole idea is on somewhat dodgy ground as it seems that the statistical strength of the arguments is getting weaker as the amount of data is getting larger.

If quasars aren't black holes, then it would certainly be possible that some the evolution of new quasars could proceed in a stepwise way (for example regularly blowing off layers on reaching certain critical energy density levels). However, I'd prefer to consider the evidence for or against decreasing intrinsic redshift without getting into the issue of quantization.

Do you have a similar list of references which claim to prove that there is no intrinsic redshift, or which criticize the articles you've listed? I'd like to see a more balanced view.

Hi Suede - welcome to Physics Forums!

That's an interesting list of references (some of which weren't working just now but seem to be improving). It appears that they are mostly concerned with the quantization of intrinsic redshift (for galaxies within a cluster as well as quasars) rather than its existence, and I think that whole idea is on somewhat dodgy ground as it seems that the statistical strength of the arguments is getting weaker as the amount of data is getting larger.

If quasars aren't black holes, then it would certainly be possible that some the evolution of new quasars could proceed in a stepwise way (for example regularly blowing off layers on reaching certain critical energy density levels). However, I'd prefer to consider the evidence for or against decreasing intrinsic redshift without getting into the issue of quantization.

Do you have a similar list of references which claim to prove that there is no intrinsic redshift, or which criticize the articles you've listed? I'd like to see a more balanced view.

Yeah I just fixed a bunch of those links.

They got hosed up when I collected them.

As for the counter arguements, you can look up intrinsic redshift on wiki which is dominated by people opposed to the idea. You're not going to find many published papers refuting it though, just a lot of ad hom attacks and pontification.

I haven't seen any published papers refuting the findings to date.

1) http://arxiv.org/PS_cache/arxiv/pdf/. 712.3833v2.pdf [Broken]

Fourier spectral analysis has been carried out on the quasar number count as a function of redshift calculated from the quasar data of the Sloan Digital Sky Survey DR6 data release. The results indicate that **quasars have preferred periodic redshifts** with redshift intervals of 0.258, 0.312, 0.44, 0.63, and 1.1. Within their standard errors these intervals are integer multiples 4, 5, 7, 10 and 20 of 0.062. Could this be indicative of an intrinsic redshift for quasars as has been suggested by some?

The redshift distribution of all 46,400 quasars in the Sloan Digital Sky Survey (SDSS) Quasar Catalog, Third Data Release (DR3), is examined. Six peaks that fall within the redshift window below z=4 are visible. **Their positions agree with the preferred redshift values predicted by the decreasing intrinsic redshift (DIR) model.**

Evidence is presented for redshift quantization and variability as detected in global studies done in the rest frame of the cosmic background radiation. **Quantization is strong and consistent** with predictions derived from concepts associated with multidimensional time. Nine families of periods are possible but not equally likely. The most basic family contains previously known periods of 73 and 36 km s–1 and shorter harmonics at 18.3 and 9.15 km s–1.

Using new data for unassociated galaxies with wide H I profiles and values of period and solar motion predicted by Tifft and Cocke (1984), **a periodicity has been found which is significant at the conventional 5 percent level.** Together with Tifft's work on galaxy pairs and small groups, this result appears to provide evidence in favor of the hypothesis that measured galaxy redshifts occur in steps of a little more than 72 km/s or a simple multiple of this period.

Power spectrum analyses of the corrected redshifts are used to search for a significant periodicity in the prescribed range 70-75 km/s. No such periodicity is found for the dwarf irregulars, but there is a possible **periodicity of about 71.1 km/s for the bright spirals**. In a further exploratory study, the sample of 112 spirals is divided up according to environment. The spirals in high-density regions of the cluster show no quantization, whereas those in low-density regions appear to be partially quantized in intervals of about 71.0 km/s.

The present study investigates the notion that extragalactic redshifts are periodic in ranges around 24.2, 36.3, or 72.5 km/s for an independent sample of 89 nearby spirals, in the general field, with accurately determined heliocentric redshifts. A **strong periodicity of about 37.2 km/s is found**, against a white noise background, for an assumed solar vector coincidental, within the uncertainties, with that corresponding to the sun's probable motion around the Galactic Center. Comparison with sets of synthetic data simulating the overall characteristics of the real data show the periodicity to be present at a high confidence level.

Published observational data on galaxies of redshift z less than about 1000 km/s are compiled in extensive tables and diagrams and analyzed, searching for additional Local Group members among fainter higher-redshift galaxies. A concentration toward the center of the Local Group and a concentration associated with NGC 55, NGC 300, and NGC 253 are identified in the south Galactic hemisphere and characterized in detail. The **galaxies near the centers of the concentrations are found to obey a quantization interval of Delta-cz0 = 72.4 km/s, as for the Local Group** (Tifft, 1977) the accuracy of this finding is shown to be to within + or - 8.2 km/s (for galaxies with redshifts known to + or - 8 km/s) and to within 3-4 km/s (for a subset of galaxies with more accurately measured redshifts).

Samples of 97 and 117 high-precision 21 cm redshifts of spiral galaxies within the Local Supercluster were obtained in order to test claims that extragalactic redshifts are periodic (P36 km s–1) when referred to the centre of the Galaxy. **The power spectral density of the redshifts, when so referred, exhibits an extremely strong peak at 37.5 km s–1.** The signal is seen independently with seven major radio telescopes. Its significance was assessed by comparison with the spectral power distributions of synthetic datasets constructed so as to closely mimic the overall properties of the real datasets employed it was found to be real rather than due to chance at an extremely high confidence level.

Persistent claims have been made over the last

15yr that extragalactic redshifts, when corrected for the Sun's motion around the Galactic centre, occur in multiples of

36km/s. A recent investigation by us of 40 spiral galaxies out to 1000km/s, with accurately measured redshifts, gave evidence of a periodicity

37.2-37.7km/s. Here we extend our enquiry out to the edge of the Local Supercluster (

2600km/s), applying a simple and robust procedure to a total of 97 accurately determined redshifts. We find that, when corrected for related vectors close to recent estimates of the Sun's galactocentric motion, the **redshifts of spirals are strongly periodic (P**

** **

**37.6km/s).** The formal confidence level of the result is extremely high, and the signal is seen independently with different radio telescopes. We also examine a further sample of 117 spirals observed with the 300-foot Green Bank telescope alone. The periodicity phenomenon appears strongest for the galaxies linked by group membership, but phase coherence probably holds over large regions of the Local Supercluster.

10) Halton Arp, Quasars, Redshifts and Controversies

http://books.google.com/books?id=_JY. result#PPP1,M1 [Broken]

I seem to see a recurring theme here.

73 km/s periodicities show up over and over and over again in any cogent study of redshifts.

Jonathan, I'm glad to see you've read some of Arp's work.

(I've added numbers, to help readers with my comments, below)

It looks quite an impressive list, doesn't it?

And as many of the papers on your list have been around for a long time, I'm sure you won't be at all surprised to learn that Tifft et al.'s claims (3, 4, 5, 6, 7, 8, 9)* have been given a pretty thorough working over. And some curious things emerge from these papers - and the couple of dozen or so that you don't cite:

a) despite the apparent similarity in findings, between papers, a closer read of them shows that most are, in fact, inconsistent - would you like to go through a sample in detail?

b) there is a paper which points out that the statistical methods used, in most of the early papers, is wrong, rendering the stated conclusions invalid (I'll see if I can dig it up, if anyone is interested)

c) the later the paper, generally, the weaker or more limited the 'redshift periodicity' reported. For example, http://arxiv.org/abs/astro-ph/0511260" [Broken] (2005):

IOW, a more careful analysis, using a larger set of data (a superset of DR3), found no signal.

1) is in its second version, and AFAICS is not yet published, despite going up in arXiv over a year ago. Maybe we should wait until it appears in a relevant peer-reviewed journal before commenting?

Oh, and 10), the book by Arp? Well, anyone can write a book, can't they? No peer-review required wrt any claims made, is there?

* not all these have Tifft as an author (Guthrie and Napier are an independent pair, for example), but they all address the

I've starting to learn about Hubble's law and I have a very simple question. How are the velocities to distant objects calculated from a redshift? I understand the basic principle, that faster objects have longer wavelengths, but I'm not sure about the formula which links the two.

The wikipedia page has some formula for redshift, but the cosmological formula doesn't seem to have a term in velocity.

How are the velocities [of] distant objects calculated from a redshift? I understand the basic principle, that **faster objects have longer wavelengths,** but I'm not sure about the formula which links the two.

The wikipedia page has some formula for redshift, but the cosmological formula doesn't seem to have a term in velocity.

For small local velocities a handy rule of thumb is simply that a Doppler shift of 1/1000 corresponds to radial (towards or away) speed of c/1000.

That is, about 300 km per second.

We distinguish between small DOPPLER shifts caused by small local motions, and COSMOLOGICAL REDSHIFTS caused by the expansion of the universe's geometry---the distance expansion rates---the rates we see large scale distances increasing without anybody actually getting anywhere.

Hubble law distance expansion rates are a different story from Doppler. You should probably get familiar with the convenient online calculators. For example, google "wright calculator" and put in a redshift, like 3, and press calculate.

It will give you a distance. Unfortunately it does not give a distance expansion rate, but you can calculate that yourself using Hubble law, if you want.

If you are interested in expansion rates, a handy shortcut is to use an online calculator with more features like where it says ". ocalc.2010.htm" in my signature. That one gives you the distance expansion rate too, as well as the distance itself.

Put in 3 for the redshift and it will tell you that the current recession rate is some multiple of the speed of light.

I just checked. The rate it gives is 1.53 c. About 53% faster than the speed of light.

Hubble law distance expansion rates should really not be called "velocities". It confuses people because it makes them think that geometry expansion is like ordinary motion (where you get somewhere).

In geometry expansion nobody gets anywhere---distances between everybody just get larger. Typically at rates faster than the speed of light. (The recession rate is proportional to distance and most objects we observe are far away enough that the distances to them are expanding faster than c.)

## More Observations of GRB 090423, the Most Distant Known Object in the Universe

On April 23, 2009 the Swift satellite detected a gamma ray burst and as we reported back in April, scientists soon realized that it was more than 13 billion light-years from Earth. GRB 090423 occurred 630 million years after the Big Bang, when the Universe was only four percent of its current age of 13.7 billion years. Now, continued observations of the GRB by astronomers around the world have yielded more information about this dramatic and ancient event: the GRB didn’t come from a monster star, but it produced a fairly sizable explosion.

Several of the world’s largest telescopes turned to the region of the sky within the next minutes and hours after Swift’s announcement of the GRB’s detection, and were able to locate the faint, fading afterglow of the GRB. Detailed analysis revealed that the afterglow was seen only in infrared light and not in the normal optical. This was the clue that the burst came from very great distance.

The Very Large Array radio telescope first looked for the object the day after the discovery, detected the first radio waves from the blast a week later, then recorded changes in the object until it faded from view more than two months later.

Images of the afterglow of GRB 090423 taken (left to right) with the Y, J, H and K filters. The absence of any flux in the Y filter is a strong indication that the GRB is very high redshift (Credit: A. J. Levan & N. R. Tanvir)

Astronomers have thought that the very first stars in the Universe might be very different — brighter, hotter, and more massive — from those that formed later.

“This explosion provides an unprecedented look at an era when the Universe was very young and also was undergoing drastic changes. The primal cosmic darkness was being pierced by the light of the first stars and the first galaxies were beginning to form. The star that exploded in this event was a member of one of these earliest generations of stars,” said Dale Frail of the National Radio Astronomy Observatory.

Universe Today spoke with Edo Berger with the Gemini Telescope shortly after the GRB was detected, and he said the burst itself was not all that unusual. But even that can convey a lot of information. “That might mean that even these early generations of stars are very similar to stars in the local universe, that when they die they seem to produce similar types of gamma ray bursts, but it might be a little early to speculate.”

“This happened a little more than 13 billion years ago,” said Berger. “We’ve essentially been able to find gamma ray bursts throughout the Universe. The nearest ones are only about 100 million light years away, and this most distant one is 13 billion light years away, so it seems that they populate the entire universe. This most distant one demonstrates for the first time that massive stars exist at those very high red shifts. This is something people have suspected for a long time, but there was no direct observational proof. So that is one of the cool results from this observation.”

The scientists concluded the explosion was more energetic than most GRBs, but was certainly not the most energetic ever detected. The blast was nearly spherical that expanded into a tenuous and relatively uniform gaseous medium surrounding the star.

Antennas of the Very Large Array CREDIT: NRAO/AUI/NSF

“It’s important to study these explosions with many kinds of telescopes. Our research team combined data from the VLA with data from X-ray and infrared telescopes to piece together some of the physical conditions of the blast,” said Derek Fox of Pennsylvania State University. “The result is a unique look into the very early Universe that we couldn’t have gotten any other way,” he added.

## 2. Data

### 2.1. Horizon Run 4

The Horizon Run 4 (HR4) simulation (Kim et al. 2015) is a massive cosmological simulation that evolved *N*_{p} = 6300 3 particles in a cubic box of a side length of *L*_{box} =3150 *h* −1 Mpc. It uses a flat ΛCDM cosmological model in concordance with a *Wilkinson Microwave Anisotropy Probe* (*WMAP*) 5 yr observation (Dunkley et al. 2009), where the matter density fraction, dark energy density fraction, and dark energy equation of state at *z* = 0 are . The volume of the HR4 is big enough to simulate the formation of large-scale structures, and at the same time its force and mass resolutions are high enough to simulate the formation of individual galaxies down to a relatively small mass scale. Thanks to these unique features, the HR4 has been extensively used for cosmological model tests and study of galaxy formation under the influence of large-scale structures in the universe (Kim et al. 2015 Hwang et al. 2016 Li et al. 2016, 2017 Appleby et al. 2017, 2018b Einasto et al. 2018 Uhlemann et al. 2018b, 2018a).

Rich information on structure formation is contained in the merger trees of dark matter (DM) halos forming in the big simulation box of HR4, constructed at 75 time steps between *z* = 12 and 0 with the time interval of

0.1 Gyr. In each snapshot, DM halos are found with the Friends-of-friends (FoF) algorithm with the linking length of Mpc. The minimum number of DM particles constructing DM halos is set to 30, which corresponds to the minimum DM halo mass of . The mock galaxy catalogs of HR4 were modeled by applying the most bound halo particle (MBP)-galaxy abundance matching to its DM halo merger tree (Hong et al. 2016). For each DM halo at each snapshot, we found the most gravitationally bound member particle (MBP). The particle is marked as the center of a "galaxy" if the given halo is isolated or if it is the most massive member halo (namely the central halo) in the merger events. On the other hand, for less massive member halos (satellites), we trace their "galaxies" from the time when they were isolated ones just before merger until they are completely disrupted. The time between the infall and the complete disruption of satellite galaxies is estimated by adopting the modified merger timescale model of Jiang et al. (2008):

where are the circularity of the satellite's orbit, mass of central and satellite halos, and the orbital period of virialized objects, respectively. We set *α* = 1.5, which makes the 2pCF of our mock galaxies match that of the SDSS Main galaxies down to scales below 1 *h* −1 Mpc (Zehavi et al. 2011).

For our analysis, we divide the HR4 simulation box into 6 pieces in each dimension, thereby creating 216 sub-cube mock samples that are 525 *h* −1 Mpc long on a side. This choice is made to have a sufficient number of samples for likelihood analysis. Galaxy surveys like the SDSS cover a larger volume at the redshift of our interest (*z*

1). Thus, we plan to analyze larger sample volumes with larger simulations in future studies.

We adopt 10 −3 galaxy per (*h* −1 Mpc) 3 for the galaxy number density in the mock sample, which corresponds to 0.145 million in each sub-cube mock. This number density roughly correspond to the *r*-band magnitude at *z* = 0 (Choi et al. 2010) and it is also similar to the expected number density galaxies to be observed by the PFS survey. We will also show some results with 10 times more galaxies for comparison. We note that these mass cuts are rather arbitrary. The actual value to be used in the analysis of a given observational data should be determined by the survey data.

### 2.2. Multiverse Simulations

The multiverse simulations are a set of cosmological *N*-body simulations designed to see illustrate the effects of cosmological parameters on the clustering and evolution of cosmic structures. We changed the cosmological parameters around those of the standard concordance model with Ω_{m} = 0.26, Ω_{de} = 0.74, and *w* = −1. We used exactly the same set of random numbers to generate the initial density fluctuations of all the simulations, which allow us to make a proper comparison between the models with the effects of the cosmic variance compensated.

Five multiverse simulations we use in this paper are listed in Table 1. Two models have the matter density parameter shifted by 0.05 from the fiducial model, while the dark energy equation of state is fixed to *w* = −1. The other two quintessence models (Sefusatti & Vernizzi 2011) have *w* shifted by 0.5 from the fiducial value of −1, while Ω_{m} is fixed to 0.26. These parameters are chosen so that they are reasonably large enough to cover the area in the Ω_{m}–*w* space constrained by many existing studies at the time *WMAP* 5 yr results have been announced (Spergel et al. 2003).

**Table 1.** Multiverse Simulation Parameter

Label | w | Ω_{m} | Ω_{de} |
---|---|---|---|

Low-w | −1.5 | 0.26 | 0.74 |

Low-Ω_{m} | −1 | 0.21 | 0.79 |

Fiducial | −1 | 0.26 | 0.74 |

High-Ω_{m} | −1 | 0.31 | 0.69 |

High-w | −0.5 | 0.26 | 0.74 |

The power spectrum of each model is normalized in such a way that the rms of the matter fluctuation linearly evolved to *z* = 0 has *σ*_{8} = 0.794 when smoothed with a spherical top hat with *R* = 8 *h* −1 Mpc.

The number of particles evolved is *N*_{p} = 2048 3 and the comoving size of the simulation box is 1024 *h* −1 Mpc. The starting redshift is *z*_{init} = 99 and the number of global time steps is 1980 with equal step size in the expansion parameter, *a*. We have used the CAMB package to calculate the power spectrum at *z*_{init}. We have extended the original GOTPM code (Dubinski et al. 2004) to gravitationally evolve particles according to the modified Poisson equation of

where *D*_{de} and *D*_{m} are the linear growth factors of the dark energy and matter, respectively (see Sefusatti & Vernizzi 2011 for details).

## APPENDIX: DESCRIPTIONS AND EXAMPLES OF DISTANCE INDICATORS IN NED-D

Descriptions of distance indicators that follow are brief. The references were chosen randomly from uses in NED-D, and are provided only as illustrative examples. For in-depth reviews of specific indicators or to obtain references giving the original, first uses of indicators, follow the references given and the references therein. For in-depth reviews on primary indicators see Ferrarese et al. (2000), Freedman & Madore (2010), de Grijs et al. (2014) and de Grijs & Bono (2015, 2014, and references therein), and for secondary indicators see Tully et al. (2009, 2013, 2016, and references therein).

Descriptions of standard candle indicators are given in Section A.1, followed by standard ruler indicators in Section A.2, and secondary indicators in Section A.3. Additional information on applying Cepheids in particular, and applicable to standard candle-based indicators in general, is given in Section A.4. Brief descriptions of luminosity relations, apparent versus reddening-corrected distance, and corrections related to age or metallicity, as well as others are provided.

Researchers are cautioned that at least three indicators have considerable overlap with others. Asymptotic Giant Branch (AGB) stars are a particular type of brightest stars indicator. The Subdwarf Fitting indicator makes use of the CMD indicator, but is applied specifically to globular clusters. The Dwarf Elliptical indicator makes use of the better-known fundamental plane relation for elliptical galaxies, but is applied specifically to dwarf elliptical galaxies. The indicators mentioned are considered distinct empirically, because they pertain to different stellar populations. They are treated as distinct in the references provided for the indicators, and in the literature in general. Further, distinguishing indicators based on the stellar populations targeted is in keeping with recognition of the TRGB, Horizontal Branch, and Red Clump indicators as distinct indicators, though all are related to the CMD indicator.

### A.1. Standard Candles

*Active galactic nucleus (AGN) timelag*

Based on the time lag between variations in magnitude observed at short wavelengths compared to those observed at longer wavelengths in AGNs. For example, using a quantitative physical model that relates the time lag to the absolute luminosity of an AGN, Yoshii et al. (2014) obtain a distance to the AGN host galaxy MRK 0335 of 146 Mpc.

Based on the maximum absolute visual magnitude for these stars of *M*_{V} = −2.8 (Davidge & Pritchet 1990). Thus, the brightest AGB stars in the galaxy NGC 0253, with a maximum apparent visual magnitude of *m*_{V} = 24.0, have a distance modulus of (m-M)_{V} = 26.8, for a distance of 2.3 Mpc.

Based on the relation between absolute magnitude and beta-index in these stars, where beta-index measures the strength of the star's emission at the wavelength of hydrogen Balmer or H-beta emission. Applied to the LMC by Shobbrook & Visvanathan (1987), to obtain a distance modulus of (m-M) = 18.30, for a distance of 46 kpc, with a statistical error of 0.20 mag or 4 kpc (10%).

*BL Lac object luminosity (BL Lac luminosity)*

Based on the mean absolute magnitude of the giant elliptical host galaxies of these AGNs. Applied to BL Lacertae host galaxy MS 0122.1+0903 by Sbarufatti et al. (2005), to obtain a distance of 1530 Mpc.

Based on super-Eddington accreting massive black holes, as found the host galaxies of certain AGNs at high redshift, and a unique relationship between their bolometric luminosity and central black hole mass. Based on a method to estimate black hole masses (Wang et al. 2014), the black hole mass–luminosity relation is used to estimate the distance to 16 AGN host galaxies, including for example galaxy MRK 0335, to obtain a distance of 85.9 Mpc, with a statistical error of 26.3 Mpc (31%).

Based on the absolute magnitude and the equivalent widths of the hydrogen Balmer lines of these stars. Applied to the SMC by Bresolin (2003) to obtain a distance modulus of (m-M) = 19.00, for a distance of 64 kpc, with a statistical error of 0.50 mag or 16 kpc (25%).

Based on the mean absolute visual magnitude for red supergiant stars, *M*_{V} = −8.0, Davidge et al. (1991) present an application to NGC 0253 where red supergiant stars have apparent visual magnitude *m*_{V} = 19.0, leading to a distance modulus of (m-M)_{V} = 27.0, for a distance of 2.5 Mpc.

Based on the mean absolute near-infrared magnitude of these stars *M*_{I} = −4.75 (Pritchet et al. 1987). Thus, carbon stars in galaxy NGC 0055 with a maximum apparent infrared magnitude *m*_{I} = 21.02, including a correction of −0.11 mag for reddening, have a distance modulus of (m-M)_{I} = 25.66, for a distance of 1.34 Mpc, with a statistical error of 0.13 mag or 0.08 Mpc (6%).

Based on the mean luminosity of Cepheid variable stars, which depends on their pulsation period, *P*. For example, a Cepheid with a period of *P* = 54.4 days has an absolute mean visual magnitude of *M*_{V} = −6.25, based on the period–luminosity (PL) relation adopted by the *HST* Key Project on the Extragalactic Distance Scale (Freedman et al. 2001). Thus, a Cepheid with a period of *P* = 54.4 days in the galaxy NGC 1637 (Leonard et al. 2003) with an apparent mean visual magnitude *m*_{V} = 24.19, has an apparent visual distance modulus of (m-M)_{V} = 30.44, for a distance of 12.2 Mpc. Averaging the apparent visual distance moduli for the 18 Cepheids known in this galaxy (including corrections of 0.10 mag for reddening and metallicity) gives a corrected distance modulus of (m-M)_{V} = 30.34, for a distance of 11.7 Mpc, with a statistical error of 0.07 mag or 0.4 Mpc (3.5%).

Based on the absolute magnitude of a galaxy's various stellar populations, discernable in a CMD. Applied to the LMC by Andersen et al. (1984), to obtain a distance modulus of (m-M) = 18.40, for a distance of 47.9 kpc.

Based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with Cepheid and Mira variables, a PL relation gives their absolute magnitude. Applied to the LMC by McNamara et al. (2007), to obtain a distance modulus of (m-M) = 19.46, for a distance of 49 kpc, with a statistical error of 0.19 mag or 4.5 kpc (9%).

*Flux-weighted gravity–luminosity relation (FGLR)*

Based on the absolute bolometric magnitude of A-type supergiant stars, determined by the FGLR (Kudritzki et al. 2008). Applied to galaxy Messier 31, to obtain a distance of 0.783 Mpc.

Based on six correlations of observed properties of GRBs with their luminosities or collimation-corrected energies. A Bayesian fitting procedure then leads to the best combination of these correlations for a given data set and cosmological model. Applied to GRB 021004 by Cardone et al. (2009), to obtain a luminosity distance modulus of (m-M) = 46.60 for a luminosity distance of 20,900 Mpc. With the GRB's redshift of *z* = 2.3, this leads to a linear distance of 6330 Mpc, with a statistical error of 0.48 mag or 1570 Mpc (25%).

*Globular cluster luminosity function (GCLF)*

Based on an absolute visual magnitude of *M*_{V} = −7.6, which is the location of the peak in the luminosity function of old, blue, low-metallicity globular clusters (Larsen et al. 2001). So, for example, the galaxy NGC 0524 with an apparent visual magnitude *m*_{V} = 24.36 for the peak in the luminosity function of its globular clusters, has a distance modulus of (m-M)_{V} = 31.99, for a distance of 25 Mpc, with a statistical error of 0.14 mag or 1.8 Mpc (7%).

*Globular cluster surface brightness fluctuations (GC SBF)*

Based on the fluctuations in surface brightness arising from the mottling of the otherwise smooth light of the cluster due to individual stars (Ajhar et al. 1996). Thus, the implied apparent magnitude of the stars leading to these fluctuations gives the distance modulus in magnitudes. Applied to galaxy Messier 31, to obtain a distance modulus of (m-M) = 24.56, for a distance of 0.817 Mpc, with a statistical error of 0.12 mag or 0.046 Mpc (6%).

*H ii luminosity function (H ii LF)*

Based on a relation between velocity dispersion, metallicity, and the luminosity of the H-beta line in H ii regions and H ii galaxies (e.g., Siegel et al. 2005, and references therein). Applied to high-redshift galaxy CDFa C01, to obtain a luminosity distance modulus of (m-M) = 45.77, for a luminosity distance of 14,260 Mpc. With a redshift for the galaxy of *z* = 3.11, this leads to a linear distance of 3470 Mpc, with a statistical error of 1.58 mag or 3,710 Mpc (93%).

Based on the absolute visual magnitude of horizontal branch stars, which is close to *M*_{V} = +0.50, but depends on metallicity (Da Costa et al. 2002). Thus, horizontal branch stars in the galaxy Andromeda III with an apparent visual magnitude *m*_{V} = 25.06, including a reddening correction of −0.18 mag, have a distance modulus of (m-M)_{V} = 24.38, for a distance of 750 kpc, with a statistical error of 0.06 mag or 20 kpc (3%).

*M stars luminosity (M stars)*

Based on the relationship between absolute magnitude and temperature-independent spectral index for normal M Stars. Applied to the LMC by Schmidt-Kaler & Oestreicher (1998), to obtain a distance modulus of (m-M) = 18.34, for a distance of 46.6 kpc, with a statistical error of 0.09 mag or 2.0 kpc (4%).

Based on the mean absolute magnitude of Mira variable stars, which depends on their pulsation period. As with Cepheid variables, a PL relation gives their absolute magnitude. Applied to the LMC by Feast et al. (2002), to obtain a distance modulus of (m-M) = 18.60, for a distance of 52.5 kpc, with a statistical error of 0.10 mag or 2.5 kpc (5%).

Based on the maximum absolute visual magnitude reached by these explosions, which is *M*_{V} = −8.77 (Ferrarese et al. 1996). So, a nova in galaxy Messier 100 with a maximum apparent visual magnitude of *m*_{V} = 22.27, has a distance modulus of (m-M)_{V} = 31.0, for a distance of 15.8 Mpc, with a statistical error of 0.3 mag or 2.4 Mpc (15%).

*O- and B-type supergiants (OB stars)*

Based on the relationship between spectral type, luminosity class, and absolute magnitude for these stars. Applied to 30 Doradus in the LMC by Walborn & Blades (1997), to obtain a distance of 53 kpc.

*Planetary nebula luminosity function (PNLF)*

Based on the maximum absolute visual magnitude for planetary nebulae of *M*_{V} = −4.48 (Ciardullo et al. 2002). So, planetary nebulae in the galaxy NGC 2403 with a maximum apparent visual magnitude of *m*_{V} = 23.17 have a distance modulus of (m-M)_{V} = 27.65, for a distance of 3.4 Mpc, with a statistical error of 0.17 mag or 0.29 Mpc (8.5%).

*Post-asymptotic giant branch stars (PAGB Stars)*

Based on the maximum absolute visual magnitude for these stars of *M*_{V} = −3.3 (Bond & Alves 2001). Thus, PAGB stars in Messier 31 with a maximum apparent visual magnitude of *m*_{V} = 20.88 have a distance modulus of (m-M)_{V} = 24.2, for a distance of 690 kpc, with a statistical error of 0.06 mag or 20 kpc (3%).

Based on the observed apparent spectrum of a quasar, compared with the absolute spectrum of comparable quasars as determined based on *HST* spectra taken of 101 quasars. Applied to 11 quasars by de Bruijne et al. (2002), including quasar [HB89] 0000–263, to obtain a distance of 3.97 Gpc.

Based on the mean absolute visual magnitude of these variable stars, which depends on metallicity: *M*_{V} = F/H **×** 0.17 + 0.82 mag (Pritzl et al. 2005). So, RR Lyrae stars with metallicity F/H = −1.88 in the galaxy Andromeda III have an apparent mean visual magnitude of *m*_{V} = 24.84, including a 0.17 mag correction for reddening. Thus, they have a distance modulus of (m-M)_{V} = 24.34, for a distance of 740 kpc, with a statistical error of 0.06 mag or 22 kpc (3.0%).

Based on the maximum absolute infrared magnitude for red clump stars of *M*_{I} = −0.67 (Dolphin et al. 2003). So, red clump stars in the galaxy Sextans A with a maximum apparent infrared magnitude of *m*_{I} = 24.84, including a 0.07 mag correction for reddening, have a distance modulus of (m-M)_{I} = 25.51, for a distance of 1.26 Mpc, with a statistical error of 0.15 mag or 0.09 Mpc (7.5%).

*Red supergiant variables (RSV stars)*

Based on the mean absolute magnitude of these variable stars, which depends on their pulsation period (Jurcevic 1998). As with Cepheid and Mira variables, a PL relation gives their absolute magnitude. Applied to galaxy NGC 2366, to obtain a distance modulus of (m-M) = 27.86, for a distance of 3.73 Mpc, with a statistical error of 0.20 mag or 0.36 Mpc (10%).

*Red variable stars (RV stars)*

Based on the mean absolute magnitude of RV stars, which depends on their pulsation period (Kiss & Bedding 2004). As with Cepheid variables, a PL relation gives their absolute magnitude. Applied to the SMC to obtain a distance modulus of (m-M) = 18.94, for a distance of 61.4 kpc, with a statistical error of 0.05 mag or 1.4 kpc (2.3%)

Based on the mean absolute magnitude of these stars, which is derived based on their amplitude-luminosity relation. Applied to galaxy Messier 31 by Wolf (1989), to obtain a distance modulus of (m-M) = 24.40, for a distance of 0.759 Mpc.

Based on SNIa (Type Ia supernovae). It is distinguished from normal SNIa however, because it has been applied to candidate SNIa obtained in the SDSS Supernova Survey that have not yet been confirmed as bona fide SNIa (Sako et al. 2014). Applied to Type Ia supernova SDSS-II SN 13651, to obtain a luminosity distance modulus of (m-M) = 41.64 for a luminosity distance of 2130 Mpc. With a redshift for the supernova of *z* = 0.25, this leads to a linear distance of 1700 Mpc.

Based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with Cepheid and Mira variables, a PL relation gives their absolute magnitude (e.g., McNamara 1995). Applied to the Carina Dwarf Spheroidal galaxy, to obtain a distance modulus of (m-M) = 20.01, for a distance of 0.100 Mpc, with a statistical error of 0.05 mag or 0.002 Mpc (2.3%).

*Short gamma-ray bursts (SGRBs)*

Similar to but distinct from the GRB standard candle, because it employs only GRBs of short, less than 2 s duration (Rhoads 2010). SGRBs are conjectured to be a distinct subclass of GRBs, differing from the majority of normal or "long" GRBs, which have durations of greater than 2 s. Applied to SGRB GRB 070724A, to obtain a linear distance of 557 Mpc.

Based on the mean distance obtained from multiple distance estimates, based on at least several to as many as a dozen or more different standard candle indicators, although standard ruler indicators may also be included. For example, Freedman & Madore (2010) analyzed 180 estimates of the distance to the LMC, based on two dozen indicators not including Cepheids, to obtain a mean distance modulus of (m-M) = 18.44, for a distance of 48.8 kpc, with a statistical error of 0.18 mag or 4.2 kpc (9%).

Gives an improved calibration of the distances and ages of globular clusters. Applied to the LMC by Carretta et al. (2000), to obtain a distance modulus of (m-M) = 18.64, for a linear distance of 53.5 kpc, with a statistical error of 0.12 mag or 3.0 kpc (6%).

*Sunyaev–Zeldovich effect (SZ effect)*

Based on the predicted Compton scattering between the photons of the cosmic microwave background radiation and electrons in galaxy clusters, and the observed scattering, giving an estimate of the distance. For galaxy cluster CL 0016+1609, Bonamente et al. (2006) obtain a linear distance of 1300 Mpc, assuming an isothermal distribution.

Based on the fluctuations in surface brightness arising from the mottling of the otherwise smooth light of the galaxy due to individual stars, primarily red giants with maximum absolute K-band magnitudes of *M*_{K} = −5.6 (Jensen et al. 1998). So, the galaxy NGC 1399, for example, with brightest stars at an implied maximum apparent K-band magnitude *m*_{K} = 25.98, has a distance modulus of (m-M)_{K} = 31.59, for a distance of 20.8 Mpc, with a statistical error of 0.16 mag or 1.7 Mpc (8%).

Based on the maximum absolute infrared magnitude for TRGB stars of *M*_{I} = −4.1 (Sakai et al. 2000). So, the LMC, with a maximum apparent infrared magnitude for these stars of *m*_{I} = 14.54, has a distance modulus of (m-M)_{I} = 18.59, for a distance of 52 kpc, with a statistical error of 0.09 mag or 2 kpc (4.5%).

Based on the mean absolute magnitude of these variable stars, which depends on their pulsation period. As with normal Cepheids and Miras, a PL relation gives their absolute magnitude. Applied to galaxy NGC 4603 by Majaess et al. (2009), to obtain a distance modulus of (m-M) = 32.46, for a linear distance of 31.0 Mpc, with a statistical error of 0.44 mag or 7.0 Mpc (22%).

*Type II supernovae, radio (SNII radio)*

Based on the maximum absolute radio magnitude reached by these explosions, which is 5.5 **×** 10 23 ergs s −1 Hz −1 (Clocchiatti et al. 1995). So, the type-II SN 1993J in galaxy Messier 81 (NGC 3031), based on its maximum apparent radio magnitude, has a distance of 2.4 Mpc.

Based on the maximum absolute blue magnitude reached by these explosions, which is *M*_{B} = −19.3 (Astier et al. 2006). Thus, for example, SN 1990O (in the galaxy MCG +03-44-003) with a maximum apparent blue magnitude of *m*_{B} = 16.20, has a luminosity distance modulus of (m-M)_{B} = 35.54 (including a 0.03 mag correction for color and redshift), or a luminosity distance of 128 Mpc. With a redshift for the galaxy of z = 0.0307, this leads to a linear distance of 124 Mpc, with a statistical error of 0.09 mag or 6 Mpc (4.5%).

Based on the absolute magnitudes of white dwarf stars, which depends on their age. Applied to the LMC by Carretta et al. (2000), to obtain a distance modulus of (m-M) = 18.40, for a linear distance of 47.9 kpc, with a statistical error of 0.15 mag or 3.4 kpc (7%).

Based on the mean absolute magnitude of these massive stars. Applied to galaxy IC 0010, by Massey & Armandroff (1995), to obtain a distance of 0.95 Mpc.

### A.2. Standard Rulers

Based on the mean absolute radius of a galaxy's inner carbon monoxide (CO) ring, with compact rings of *r* =

200 pc and broad rings of *r* =

750 pc. So, a CO compact ring in the galaxy Messier 82 with an apparent radius of 130 arcsec, has a distance of 3.2 Mpc (Sofue 1991).

Based on the absolute radii of certain kinds of dwarf galaxies surrounding giant elliptical galaxies such as Messier 87. Specifically, dwarf elliptical (dE) and dwarf spheroidal (dSph) galaxies have an effective absolute radius of

1.0 kpc that barely varies in such galaxies over several orders of magnitude in mass. So, the apparent angular radii of these dwarf galaxies around Messier 87 at 11.46 arcsec, gives a distance for the main galaxy of 18.0 ± 3.1 Mpc (Misgeld & Hilker 2011).

A hybrid method between standard rulers and standard candles, using stellar pairs orbiting one another fortuitously such that their individual masses and radii can be measured, allowing the system's absolute magnitude to be derived. Thus, the absolute visual magnitude of an eclipsing binary in the galaxy Messier 31 is *M*_{V} = −5.77 (Ribas et al. 2005). So, this eclipsing binary, with an apparent visual magnitude of *m*_{V} = 18.67, has a distance modulus of (m-M)_{V} = 24.44, for a distance of 772 kpc, with a statistical error of 0.12 mag or 44 kpc (6%).

*Globular cluster radii (GC radius)*

Based on the mean absolute radii of globular clusters, *r* = 2.7 pc (Jordan et al. 2005). So, globular clusters in the galaxy Messier 87 with a mean apparent radius of *r* = 0.032 arcsec, have a distance of 16.4 Mpc.

Based on the absolute diameter at which a galaxy reaches the critical density for gravitational stability of the gaseous disk (Zasov & Bizyaev 1996). A distance to galaxy Messier 74 is obtained of 9.40 Mpc.

*Gravitational lenses (G Lens)*

Based on the absolute distance between the multiple images of a single background galaxy that surround a gravitational lens galaxy, determined by time-delays measured between images. Thus, the apparent distance between images gives the lensing galaxy's distance. Applied to the galaxy 87GB[BWE91] 1600+4325 ABS01 by Burud et al. (2000), to obtain a distance of 1,920 Mpc.

*H ii region diameters (H ii )*

Based on the mean absolute diameter of H ii regions, *d* = 14.9 pc (Ismail et al. 2005). So, H ii regions in the galaxy Messier 101 with a mean apparent diameter of *r* = 4.45 arcsec, have a distance of 6.9 Mpc.

Based on the apparent motion of individual components in parsec-scale radio jets, obtained by observation, compared with their absolute motion, obtained by Doppler measurements and corrected for the jet's angle to the line of sight. Applied to the quasar 3C 279 by Homan & Wardle (2000), to obtain an angular size distance of 1.8 ± 0.5 Gpc.

Based on the absolute motion of masers orbiting at great speeds within parsecs of supermassive black holes in galaxy cores, relative to their apparent or proper motion. The absolute motion of masers orbiting within the galaxy NGC 4258 is *V*_{t} = 1,075 km s −1 , or 0.001100 pc yr −1 (Humphreys et al. 2004). So, the maser's apparent proper motion of 31.5 **×** 10 −6 arcsec yr −1 , gives a distance of 7.2 Mpc, with a statistical error of 0.2 Mpc (3.0%).

*Orbital mechanics (Orbital mech.)*

Based on the predicted orbital or absolute motion of a galaxy around another galaxy, and its observed apparent motion, giving a measure of distance. Applied by Howley et al. (2008) to the Messier 31 satellite galaxy Messier 110, to obtain a linear distance of 0.794 Mpc.

Based on the absolute motion of a galaxy, relative to its apparent or proper motion. Applied to galaxy Leo B by Lepine et al. (2011), to obtain a linear distance of 0.215 Mpc.

Based on the apparent angular ring diameter of certain spiral galaxies with inner rings, compared to their absolute ring diameter, as determined based on other apparent properties, including morphological stage and luminosity class (Pedreros & Madore 1981). For galaxy UGC 12914, a distance modulus is obtained of (m-M) = 32.30, for a linear distance of 29.0 Mpc, with a statistical error of 0.84 mag or 13.6 Mpc (47%), assuming *H* = 100 km s −1 Mpc −1 .

*Type II supernovae, optical (SNII optical)*

Based on the absolute motion of the explosion's outward velocity, in units of intrinsic transverse velocity, *V*_{t} (usually km s −1 ), relative to the explosion's apparent or proper motion (usually arcseconds year −1 ) (e.g., Eastman et al. 1996). So, the absolute motion of Type II SN 1979C observed in the galaxy Messier 100, based on the Expanding Photosphere Method (EPM), gives a distance of 15 Mpc, with a statistical error of 4.3 Mpc (29%). An alternative SNII Optical indicator uses the Standardized Candle Method (SCM) of Hamuy & Pinto (2002). Applied to Type II SN 2003gd in galaxy Messier 74, by Hendry et al. (2005), to obtain a distance of 9.6 Mpc, with a statistical error of 2.8 Mpc (29%).

### A.3. Secondary Methods

*Brightest cluster galaxy (BCG)*

Based on the fairly uniform absolute visual magnitudes of *M*_{V} = −22.68 ± 0.35 found among the brightest galaxies in galaxy clusters (see Hoessel 1980). So, for example, for the brightest galaxy in the galaxy cluster Abell 0021, which is the galaxy 2MASX J00203715+2839334 and which has an apparent visual magnitude of *m*_{V} = 15.13, the luminosity distance modulus can be calculated, as done by Hoessel et al. (1980). The result is a luminosity distance modulus of (m-M) = 37.81, or a luminosity distance of 365 Mpc. With a redshift for the BCG in Abell 0021 of *z* = 0.0945, this leads to a linear distance of 333 Mpc, with a statistical error of 0.35 mag or 59 Mpc (18%).

Provides standard candles based on the absolute magnitudes of elliptical and early-type galaxies, determined from the relation between the galaxy's apparent magnitude and apparent diameter (e.g., Willick et al. 1997). Applied to galaxy ESO 409- G 012, to obtain a distance modulus of (m-M) = 33.9, for a linear distance of 61 Mpc, with a statistical error of 0.40 mag or 12 Mpc (20%).

Certain galaxy's major diameters may provide secondary standard rulers based on the absolute diameter for example of only the largest, or "supergiant" spiral galaxies, estimated to be

52 kpc (van der Kruit 1986). So, from the mean apparent diameter found for supergiant spiral galaxies in the Virgo cluster of

9 arcmin, the Virgo cluster distance is estimated to be 20 Mpc, with a statistical error of 3 Mpc (15%).

Based on the absolute magnitude of dwarf elliptical galaxies, derived from a surface-brightness/luminosity relation, and the observed apparent magnitude of these galaxies (Caldwell & Bothun 1987). Applied to dwarf elliptical galaxies around galaxy NGC 1316 in the Fornax galaxy cluster, to obtain a distance of 12 Mpc.

Based on the absolute magnitudes of elliptical and early-type galaxies, determined from a relation between a galaxy's apparent magnitude and velocity dispersion (Lucey 1986). Applied to galaxy NGC 4874, to obtain a distance modulus of (m-M) = 34.76, for a linear distance of 89.5 Mpc, with a statistical error of 0.12 mag or 5.1 Mpc (6%).

Based on the absolute magnitudes of early-type galaxies, which depend on effective visual radius *r*_{e}, velocity dispersion sigma, and mean surface brightness within the effective radius *I*_{e}: log *D* = log *r*_{e}–1.24 log sigma + 0.82 log *I*_{e} + 0.173 (e.g., Kelson et al. 2000). The galaxy NGC 1399 has an effective radius *r*_{e} = 55.4 arcsec, a rotational velocity sigma = 301 km s −1 , and surface brightness, *I*_{e} = 428.5 *L*_{Sun} pc −2 . So, from the FP relation, its distance is 20.6 Mpc.

The globular cluster *K*-band magnitude versus *J*-band minus *K*-band CMD secondary standard candle is similar to the CMD standard candle, but applied specifically to globular clusters within a galaxy, rather than entire galaxies (Sitko 1984). Applied to galaxy Messier 31, to obtain a linear distance of 0.689 Mpc.

Based on the absolute magnitude at which this ratio equals one, which compares energy emitted at two wavelengths, giga-electron volt and tera-electron volt (Prandini et al. 2010). Applied to galaxy 3C66A, to obtain a linear distance of 794 Mpc.

*Globular cluster fundamental plane (GC FP)*

Based on the relationship among velocity dispersion, radius, and mean surface brightness for globular clusters, similar to the fundamental plane for early-type galaxies (Strader et al. 2009). Applied to globular clusters in galaxy Messier 31, to obtain a distance modulus of (m-M) = 24.57, for a linear distance of 0.820 Mpc, with a statistical error of 0.05 mag or 0.019 Mpc (2.3%).

*H I* + *optical distribution*

Based on neutral hydrogen I mass versus optical distribution or virial mass provides a secondary standard ruler that applies to extreme H I-rich galaxies, such as Michigan 160, based on the assumption that the distance-dependent ratio of neutral gas to total (virial) mass should equal one (Staveley-Smith et al. 1990). Applied to galaxy UGC 12578, to obtain a distance modulus of (m-M) = 33.11, for a linear distance of 41.8 Mpc, with a statistical error of 0.20 mag or 4.0 Mpc (10%).

*Infra-Red Astronomical Satellite* (*IRAS*)

Based on a reconstruction of the local galaxy density field using a model derived from the 1.2 Jy *IRAS* survey with peculiar velocities accounted for using linear theory (e.g., Willick et al. 1997). Applied to galaxy UGC 12897, to obtain a distance modulus of (m-M) = 35.30, for a linear distance of 115 Mpc, with a statistical error of 0.80 mag or 51 Mpc (44%).

Based on the SBF standard candle, which is based on the fluctuations in surface brightness arising from the mottling of the otherwise smooth light of a galaxy due to individual stars, but applied specifically to low surface brightness (LSB) galaxies (Bothun et al. 1991). Applied to LSB galaxies around galaxy NGC 1316 in the Fornax galaxy cluster, to obtain a distance modulus of (m-M) = 31.25, for a linear distance of 17.8 Mpc, with a statistical error of 0.28 mag or 2.4 Mpc (14%).

Based on an extragalactic object's magnetic energy and particle energy, and calculations assuming certain relations between the two. It has been applied so far to only one gamma-ray source, HESS J1507-622 (Domainko 2014). Depending on which theoretical possibilities are assumed, the distance is estimated to range from 0.18 Mpc to 100 Mpc, indicating that HESS J1507-622 is extragalactic.

Based on the apparent magnitudes of certain galaxies, which may provide a secondary standard candle based on the mean absolute magnitude determined from a sample of similar galaxies with known distances. Assuming a mean absolute blue magnitude for dwarf galaxies of *M*_{B} = −10.70, the dwarf galaxy DDO 155 with an apparent blue magnitude of *m*_{B} = 14.5, has a distance modulus of (m-M)_{B} = 25.2, for a distance of 1.1 Mpc (Moss & de Vaucouleurs 1986).

Based on the absolute radii of galaxy halos, estimated from the galaxy plus halo mass as derived from rotation curves and from the expected mass density derived theoretically (Gentile et al. 2010). Applied to galaxy NGC 1560, to obtain a linear distance of 3.16 Mpc.

Based on the absolute radio brightness assumed versus the apparent radio brightness observed in a galaxy (Wiklind & Henkel 1990). Applied to galaxy NGC 0404, to obtain a distance of 10 Mpc.

"Look Alike," or in French "Sosies," galaxies provide standard candles based on a mean absolute visual magnitude of *M*_{V} = −21.3 found for spiral galaxies with similar Hubble stages, inclination angle, and light concentrations (Terry et al. 2002). So, the galaxy NGC 1365, with an apparent visual magnitude of *m*_{V} = 9.63, has a distance modulus of (m-M)_{V} = 30.96, for a distance of 15.6 Mpc. Galaxy NGC 1024, with an apparent visual magnitude of *m*_{V} = 12.07 that is 2.44 mag fainter and apparently farther than NGC 1365, is also estimated to be 0.06 mag less luminous than NGC 1365, leading to a distance modulus of (m-M)_{V} = 33.34, for a distance of 46.6 Mpc.

A catch-all term for various distance indicators employed by de Vaucouleurs et al. in the 1970s and 1980s, including galaxy luminosity index and rotational velocity (e.g., McCall 1989). Applied to galaxy IC 0342, to obtain a distance modulus of (m-M) = 26.32, for a linear distance of 1.84 Mpc, with a statistical error of 0.15 mag or 0.13 Mpc (7%).

Based on various parameters, including galaxy magnitudes, diameters, and group membership (Tully, NGC, 1988). For galaxy ESO 012- G 014, the estimated distance is 23.4 Mpc.

Introduced by Tully & Fisher (1977), based on the absolute blue magnitudes of spiral galaxies, which depend on their apparent blue magnitude, *m*_{B}, and their maximum rotational velocity, sigma: *M*_{B} = −7.0 log sigma—1.8 (e.g., Karachentsev et al. 2003). So, the galaxy NGC 0247 has an absolute blue magnitude of *M*_{B} = −18.2, based on its rotational velocity, sigma = 222 km s −1 . With an apparent blue magnitude of *m*_{B} = 9.86, NGC 0247 has a distance modulus of (m-M)_{B} = 28.1, for a distance of 4.1 Mpc.

### A.4. Additional Information on Indicators

Here are some notes relating to Cepheids distances in particular, and to standard candle indicators in general, regarding different luminosity relations, apparent versus reddening-corrected distance, and corrections related to age or metallicity.

### A.4.1. Period–Luminosity Relation

Cepheid variable stars have absolute visual magnitudes related to the log of their periods in days

This is the PL relation adopted by NASA's *HST* Key Project On the Extragalactic Distance Scale (Freedman et al. 2001).

In the galaxy NGC 1637, the longest-period Cepheid of 18 observed has a period of 54.42 days, yielding a mean absolute visual magnitude of *M*_{V} = −6.25 (Leonard et al. 2003). With the star's apparent mean visual magnitude of *m*_{V} = 24.19, its apparent visual distance modulus of is (m-M)_{V} = 30.44, corresponding to a distance of 12.2 Mpc.

NGC 1637's shortest-period Cepheid, with a period of 23.15 days, has a mean absolute visual magnitude of *M*_{V} = −5.23. The shorter period variable's mean apparent visual magnitude is *m*_{V} = 25.22, giving an apparent visual distance modulus of (m-M)_{V} = 30.45, for a distance of 12.3 Mpc. This is in excellent agreement with the distance found from the longest-period Cepheid in the same galaxy.

### A.4.2. Apparent Distance

Nevertheless, there is in practice a significant scatter in the individual Cepheid distance moduli within a single galaxy. In the galaxy NGC 1637, for example, the average of the apparent distance moduli for all 18 Cepheids is (m-M)_{V} = 30.76, corresponding to a distance of 14.2 Mpc. This is

0.3 mag fainter than the distance moduli obtained from either the longest- or shortest-period Cepheids, and corresponds to a 15% greater distance.

### A.4.3. Reddening-corrected Distance

Scatter in individual Cepheid distance moduli is caused primarily by differential "reddening" or dimming due to differing patches of dust within target galaxies, and to a lesser extent by reddening due to foreground dust within the Milky Way, as well as differences in the intervening intergalactic medium. Because reddening is wavelength-dependent (greater at shorter wavelengths) the difference between distance moduli measured at two or more wavelengths can be used to estimate the extinction at any wavelength, *E*_{V-I} = (m-M)_{V} - (m-M)_{I}. For NGC 1637, with (m-M)_{V-I} = 30.76–30.54, the extinction between V and I is *E*_{V-I} = 0.22. Extinction, when multiplied by the ratio of total-to-selective absorption and assuming that ratio to be *R*_{V} = 2.45, equals the total absorption, or dimming in magnitudes of the visual distance modulus due to dust, *A*_{V} = *R*_{V} **×** *E*_{V-I} = 0.54 in the case of NGC 1637. Note different total-to-selective absorption ratios are assumed by different authors. The correction for dimming due to dust obtained by Leonard et al. (2003) is deducted from the apparent visual distance modulus of (m-M)_{V} = 30.76 to obtain the true, reddening-corrected, "Wesenheit" distance modulus of (m-M)_{W} = 30.23, corresponding to a distance of 11.1 Mpc.

### A.4.4. Metallicity-corrected Distance

Cepheids formed in galaxies with higher "metal" abundance ratios (represented here by measured oxygen/hydrogen ratios), are comparatively less luminous than Cepheids formed in "younger" less evolved galaxies.

Leonard et al. (2003) apply a metallicity correction of *Z* = 0.12 mag, based on the difference in metal abundance between galaxy NGC 1637 and the LMC. Their final, metallicity- and reddening-corrected distance modulus is (m-M)_{Z} = 30.34, corresponding to a distance of 11.7 Mpc.

Different corrections for reddening and age or metallicity are applied by different authors. For a review see Freedman & Madore (2010).

### A.4.5. Distance Precision

Differences affecting distance estimates, whether based on Cepheid variables or other methods, include corrections for: