# Get the expression of probed volume between 2 redshifts

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1) I can't manage to find/justify the relation (1) below, from the common relation (2) of a volume.

2) It seems the variable r is actually the comoving distance and not comoving coordinates (with scale factor R(t) between both).

The comoving volume of a region covering a solid angle $$Omega$$ between two redshifts $$z_{mathrm{i}}$$ and $$z_{mathrm{f}},$$is given by

$$Vleft(z_{mathrm{i}}, z_{mathrm{f}} ight)=Omega int_{z_{mathrm{i}}}^{z_{mathrm{f}}} frac{r^{2}(z)}{sqrt{1-kappa r^{2}(z)}} frac{c mathrm{d} z}{H(z)}quadquad(1)$$

for a spatially flat universe $$kappa=0)$$ the latter becomes

$$Vleft(z_{mathrm{i}}, z_{mathrm{f}} ight)=Omega int_{rleft(z_{mathrm{i}} ight)}^{rleft(z_{mathrm{f}} ight)} r^{2} mathrm{d} r=frac{Omega}{3}left[r^{3}left(z_{mathrm{f}} ight)-r^{3}left(z_{mathrm{i}} ight) ight] quadquad(2)$$

I would like to demonstrate it from the comoving distance with :

$$D_{mathrm{A}}(z)=left{egin{array}{ll} {(1+z)^{-1} frac{c}{H_{0}} frac{1}{sqrt{left|Omega_{mathrm{K}, 0} ight|}} sin left[sqrt{left|Omega_{mathrm{K}, 0} ight|} frac{H_{0}}{c} r(z) ight],} & { ext { if } Omega_{mathrm{K}, 0}<0} {(1+z)^{-1} r(z),} & { ext { if } Omega_{mathrm{K}, 0}=0} {(1+z)^{-1} frac{c}{H_{0}} frac{1}{sqrt{Omega_{mathrm{K}, 0}}} sinh left[sqrt{Omega_{mathrm{K}, 0}} frac{H_{0}}{c} r(z) ight]} & { ext { if } Omega_{mathrm{K}, 0}>0} end{array} ight.$$

UPDATE 1: It's been a long time that I posted this question.

I recently took over this isssue and I have done little progress, at least I think.

The FLRW metric can be expressed under following (0,2) tensor form :

$$left[egin{array}{cccc}1 & 0 & 0 & 0 0 & -frac{R^{2}(t)}{1-k r^{2}} & 0 & 0 0 & 0 & -R^{2}(t) r^{2} & 0 0 & 0 & 0 & -R^{2}(t) r^{2} sin ^{2} hetaend{array} ight]$$

If I consider only slice times constant, my goal is to compute the volume probeb by a satellite between 2 redshifts.

1. We can easily find that :

$$int_{0}^{z_0}frac{cdz}{H(z)} = int_{0}^{t_0}frac{cdt}{R(t)}$$

1. Then, If I consider a volume with $$ext{d}r$$, $$ext{d} heta$$ and $$ext{d}phi$$ coordinates, I have the following expression for determinant :

$$g= ext{det}[g_{ij}] = -dfrac{R(t)^6}{1-kr^2},r^4, ext{sin}^2 heta$$

Which means that I have :

$$ext{d}V=sqrt{-g} ext{d}^3x = dfrac{R(t)^3}{sqrt{1-kr^2}},r^2, ext{d}r ext{sin} heta, ext{d} heta, ext{d}phi$$

$$V=int ext{d}V= int dfrac{R(t)^3}{sqrt{1-kr^2}},r^2, ext{d}r ext{sin} heta, ext{d} heta, ext{d}phi$$

$$V = int ext{d}Omega int dfrac{R(t)^3}{sqrt{1-kr^2}},r^2, ext{d}r$$

$$ightarrowquad V = Omega int dfrac{R(t)^3}{sqrt{1-kr^2}},r^2, ext{d}r$$

with $$Omega$$ the solid angle considered.

But as you can see, I am far away from the expression $$(1)$$ that I would like to find, i.e :

$$Vleft(z_{mathrm{i}}, z_{mathrm{f}} ight)=Omega int_{z_{mathrm{i}}}^{z_{mathrm{f}}} frac{r^{2}(z)}{sqrt{1-kappa r^{2}(z)}} frac{c mathrm{d} z}{H(z)}quadquad(1)$$

EDIT : Maybe I have found a partial explanation to my issue to determine the expresion of this volume between 2 redshifts. Here below a formula :

The main expression to keep in mind is :

$$d V_{C}=D_{H} frac{(1+z)^{2} D_{A}^{2}}{E(z)} d Omega d zquad(3)$$

1. Could anyone explain me please the different justifications to introduce all the factors implied in this expression ?

2. I have not yet with this expression the same expression (1) at the beginning of my post, so could anyone manage to find (1) from (3) ?

Any help would be fine, I am stucked for the moment.

## A Lyman-α protocluster at redshift 6.9

Protoclusters, the progenitors of the most massive structures in the Universe, have been identified at redshifts of up to 6.6 (refs. 1,2,3,4,5,6 ). Besides exploring early structure formation, searching for protoclusters at even higher redshifts is particularly useful to probe the reionization. Here we report the discovery of the protocluster LAGER-z7OD1 at a redshift of 6.93, when the Universe was only 770 million years old and could be experiencing rapid evolution of the neutral hydrogen fraction in the intergalactic medium 7,8 . The protocluster is identified by an overdensity of 6 times the average galaxy density, and with 21 narrowband selected Lyman-α galaxies, among which 16 have been spectroscopically confirmed. At redshifts similar to or above this record, smaller protogroups with fewer members have been reported 9,10 . LAGER-z7OD1 shows an elongated shape and consists of two subprotoclusters, which would have merged into one massive cluster with a present-day mass of 3.7 × 10 15 solar masses. The total volume of the ionized bubbles generated by its member galaxies is found to be comparable to the volume of the protocluster itself, indicating that we are witnessing the merging of the individual bubbles and that the intergalactic medium within the protocluster is almost fully ionized. LAGER-z7OD1 thus provides a unique natural laboratory to investigate the reionization process.

## Periodicity in the distribution of quasar redshifts and density perturbation in the early universe

We made a power spectrum analysis on the quasar emission redshift distribution, and further confirmed the existence of periodicity in respect of the quantity x = F(z,qo) defined at (8).

The existence of this periodicity does not mean that the quasar redshift is non-cosmological, for it can be interpreted as a remnant of density (acoustic) perturbations in the early big-bang universe. For this model, we made a number of tests. We found: 1) the ratio of periodic to non-periodic components falls as the sample size increases 2) the periodicity should be more marked for quasars in one region of the sky than for all quasars, and 3) the Jeans wavelength before the recombination epoch determines the length of the period. Using this model we also found that qo > 0.5, lending further support to the conclusion reached by other means that the universe may be closed.

## The Cosmic Tug of War

We might summarize our discussion so far by saying that a “tug of war” is going on in the universe between the forces that push everything apart and the gravitational attraction of matter, which pulls everything together. If we can determine who will win this tug of war, we will learn the ultimate fate of the universe.

The first thing we need to know is the density of the universe. Is it greater than, less than, or equal to the critical density? The critical density today depends on the value of the expansion rate today, H0. If the Hubble constant is around 20 kilometers/second per million light-years, the critical density is about 10 –26 kg/m 3 . Let’s see how this value compares with the actual density of the universe.

Critical Density of the Universe
As we discussed, the critical density is that combination of matter and energy that brings the universe coasting to a stop at time infinity. Einstein’s equations lead to the following expression for the critical density (ρcrit):

where H is the Hubble constant and G is the universal constant of gravity (6.67 × 10 –11 Nm 2 /kg 2 ).

Solution
Let’s substitute our values and see what we get. Take an H = 22 km/s per million light-years. We need to convert both km and light-years into meters for consistency. A million light-years = 10 6 × 9.5 × 10 15 m = 9.5 × 10 21 m. And 22 km/s = 2.2 × 10 4 m/s. That makes H = 2.3 ×10 –18 /s and H 2 = 5.36 × 10 –36 /s 2 . So,

which we can round off to the 10 –26 kg/m 3 . (To make the units work out, you have to know that N, the unit of force, is the same as kg × m/s 2 .)

Now we can compare densities we measure in the universe to this critical value. Note that density is mass per unit volume, but energy has an equivalent mass of m = E/c 2 (from Einstein’s equation E = mc 2 ).

1. A single grain of dust has a mass of about 1.1 × 10 –13 kg. If the average mass-energy density of space is equal to the critical density on average, how much space would be required to produce a total mass-energy equal to a dust grain?
2. If the Hubble constant were twice what it actually is, how much would the critical density be?

a. In this case, the average mass-energy in a volume V of space is E = ρcritV. Thus, for space with critical density, we require that

Thus, the sides of a cube of space with mass-energy density averaging that of the critical density would need to be slightly greater than 10 km to contain the total energy equal to a single grain of dust!

b. Since the critical density goes as the square of the Hubble constant, by doubling the Hubble parameter, the critical density would increase by a factor a four. So if the Hubble constant was 44 km/s per million light-years instead of 22 km/s per million light-years, the critical density would be

We can start our survey of how dense the cosmos is by ignoring the dark energy and just estimating the density of all matter in the universe, including ordinary matter and dark matter. Here is where the cosmological principle really comes in handy. Since the universe is the same all over (at least on large scales), we only need to measure how much matter exists in a (large) representative sample of it. This is similar to the way a representative survey of a few thousand people can tell us whom the millions of residents of the US prefer for president.

There are several methods by which we can try to determine the average density of matter in space. One way is to count all the galaxies out to a given distance and use estimates of their masses, including dark matter, to calculate the average density. Such estimates indicate a density of about 1 to 2 × 10 –27 kg/m 3 (10 to 20% of critical), which by itself is too small to stop the expansion.

A lot of the dark matter lies outside the boundaries of galaxies, so this inventory is not yet complete. But even if we add an estimate of the dark matter outside galaxies, our total won’t rise beyond about 30% of the critical density. We’ll pin these numbers down more precisely later in this chapter, where we will also include the effects of dark energy.

In any case, even if we ignore dark energy, the evidence is that the universe will continue to expand forever. The discovery of dark energy that is causing the rate of expansion to speed up only strengthens this conclusion. Things definitely do not look good for fans of the closed universe (big crunch) model.

Some say the world will end in fire,

From what I’ve tasted of desire

I hold with those who favor fire.

—From the poem “Fire and Ice” by Robert Frost (1923)

Given the destructive power of impacting asteroids, expanding red giants, and nearby supernovae, our species may not be around in the remote future. Nevertheless, you might enjoy speculating about what it would be like to live in a much, much older universe.

The observed acceleration makes it likely that we will have continued expansion into the indefinite future. If the universe expands forever (R increases without limit), the clusters of galaxies will spread ever farther apart with time. As eons pass, the universe will get thinner, colder, and darker.

Within each galaxy, stars will continue to go through their lives, eventually becoming white dwarfs, neutron stars, and black holes. Low-mass stars might take a long time to finish their evolution, but in this model, we would literally have all the time in the world. Ultimately, even the white dwarfs will cool down to be black dwarfs, any neutron stars that reveal themselves as pulsars will slowly stop spinning, and black holes with accretion disks will one day complete their “meals.” The remains of stars will all be dark and difficult to observe.

This means that the light that now reveals galaxies to us will eventually go out. Even if a small pocket of raw material were left in one unsung corner of a galaxy, ready to be turned into a fresh cluster of stars, we will only have to wait until the time that their evolution is also complete. And time is one thing this model of the universe has plenty of. There will surely come a time when all the stars are out, galaxies are as dark as space, and no source of heat remains to help living things survive. Then the lifeless galaxies will just continue to move apart in their lightless realm.

If this view of the future seems discouraging (from a human perspective), keep in mind that we fundamentally do not understand why the expansion rate is currently accelerating. Thus, our speculations about the future are just that: speculations. You might take heart in the knowledge that science is always a progress report. The most advanced ideas about the universe from a hundred years ago now strike us as rather primitive. It may well be that our best models of today will in a hundred or a thousand years also seem rather simplistic and that there are other factors determining the ultimate fate of the universe of which we are still completely unaware.

## 2. Estimates of Z and E of Known FRBs

The observed DM of an FRB can be broken down to

is the external DM contribution outside the Milky Way galaxy, and DMhost and DMsrc are the DM contributions from the FRB host galaxy and source environment, respectively, in the cosmological rest frame of the FRB. The measured values of both are smaller by a factor of (1 + z) (Ioka 2003 Deng & Zhang 2014). The intergalactic medium (IGM) portion of DM is related to the distance (redshift) of the source through (Deng & Zhang 2014)

in the flat ΛCDM universe (i.e., the dark energy equation of state parameter w = −1), where Ωb is the baryon density, H0 is Hubble constant, fIGM

0.83 is the fraction of baryons in the IGM (Fukugita et al. 1998), 3

denotes the free electron number per baryon in the universe, with χe,H and χe,He denoting the ionization fraction of hydrogen and helium, respectively, and y1

1 denoting the possible slight deviation from the 3/4-1/4 split of hydrogen and helium abundance in the universe. If both hydrogen and helium are fully ionized (valid below z

3), one has χ(z) 7/8. Adopting the latest Planck results (Planck Collaboration et al. 2016) for the ΛCDM cosmological parameters, i.e., H0 = 67.74 ± 0.46 km s −1 kpc −1 , Ωb = 0.0486 ± 0.0010, Ωm = 0.3089 ± 0.0062, ΩΛ = 0.6911 ± 0.0062, Equation (3) has the numerical value

which lies in the range 1–1.12 for z < 3. If one adopts an average value F(z)

DME/(1200 pc cm −3 ) has been adopted (Caleb et al. 2016 Petroff et al. 2016) to estimate the upper limit of the FRB redshifts based on the earlier calculations by Ioka (2003) and Inoue (2004). These calculations have assumed that essentially all baryons are in the IGM (fIGM

1) and that the universe is composed of hydrogen only (χ = 1), which significantly underestimates the redshift upper limit z for a given DME (by a factor of

0.73). According to our results, a rough estimate

is recommended for z < 3, which has a

6% error. Notice that this relation is valid on average. Due to the existence of large-scale structures, different lines of sight may give different DMIGM values for the same z (McQuinn 2014). The variation is redshift-dependent, and can be up to

1 and drops at higher redshifts. If one adopts the

40% variation, the conversion factor 855 would be in the range

In order to derive the DMIGM of an FRB, one needs to know DMhost + DMsrc. This is difficult to derive from an individual FRB, but may be derived statistically using a sample of FRBs (Yang & Zhang 2016 Yang et al. 2017). The observations of FRB 121102 (Chatterjee et al. 2017 Marcote et al. 2017 Tendulkar et al. 2017) and a statistical study (Yang et al. 2017) suggest that this sum is not small, which is comparable to DMIGM for FRB 121102 (if the true DMIGM of the source is close to the average value derived in Equation (3)). In any case, DME can be used to derive an average upper limit of DMIGM, and hence an average upper limit of z, of a particular FRB (again noting the fluctuations of DMIGM along different lines of sight McQuinn 2014). As DM increases, this average upper limit gets closer to the true value due to the (1 + z) suppression factor of DMhost + DMscr. The average z upper limits of the published FRBs (extracted from the FRB catalog, Petroff et al. 2016) are presented in Table 1. The external DME values are directly taken from the FRB catalog, which were presented in the original papers that reported the discovery of each FRB (Petroff et al. 2016, and references therein). In those original papers, some authors have used the Galactic electron density model NE2001 (Cordes & Lazio 2002) while some others used YMW17 (Yao et al. 2017). The DMMW values derived from the two models usually agree with each other, but could be very different for some FRBs. In any case, because DMMW is usually a small portion of the total DM, the derived DME from the two Galactic electron density models would not differ significantly, and the conclusions presented in this Letter are essentially not affected. In the derivations of DME of these original papers, the DM contribution from the Galactic halo (e.g., Dolag et al. 2015) was not deducted.

Table 1. Observational Properties of a Sample of FRBs (Including "All Events" in the FRB Catalog as of 2018 August 15, http://www.frbcat.org Petroff et al. 2016) and Their Estimated Average Upper Limits of Redshift (z), Isotropic Peak Luminosity (Lp), and Isotropic Energy (E)

FRB Name DM DME z Sν,p tobs νc a Lp E Telescope S/N
(yymmdd) (pc cm −3 ) (pc cm −3 ) (Jy) (ms) (MHz) (10 43 erg/s) (10 40 erg)
FRB 010125 790 ± 3 680 <0.76 0.3 9.4 1372.5 <1.16 <6.22 Parkes 17
FRB 010621 b 745 ± 10 222 <0.26 0.41 7 1374 <0.124 <0.691 Parkes 16.3
FRB 010724 375 330.42 <0.38 30 5 1374 <21.9 <79.3 Parkes 23
FRB 090625 899.55 ± 0.01 867.86 <0.97 1.14 1.92 1352 <11.7 <11.4 Parkes 30
FRB 110220 944.38 ± 0.05 909.61 <1.01 1.3 5.6 1352 <18.6 <51.8 Parkes 49
FRB 110523 623.3 ± 0.06 579.78 <0.65 0.6 1.73 800 <0.928 <0.972 GBT 42
FRB 110626 723 ± 0.3 675.54 <0.76 0.4 1.4 1352 <1.53 <1.22 Parkes 11
FRB 110703 1103.6 ± 0.7 1071.27 <1.19 0.5 4.3 1352 <5.74 <11.3 Parkes 16
FRB 120127 553.3 ± 0.3 521.48 <0.59 0.5 1.1 1352 <1.03 <0.711 Parkes 11
FRB 121002 1629.18 ± 0.02 1554.91 <1.75 0.43 5.44 1352 <12.7 <25.1 Parkes 16
FRB 121102 557 ± 2 369 <0.42 0.4 3 1375 <0.370 <0.782 Arecibo 14
FRB 130626 952.4 ± 0.1 885.53 <0.99 0.74 1.98 1352 <5.39 <5.36 Parkes 21
FRB 130628 469.88 ± 0.01 417.3 <0.48 1.91 0.64 1352 <2.38 <1.03 Parkes 29
FRB 130729 861 ± 2 830 <0.92 0.22 15.61 1352 <1.34 <10.9 Parkes 14
FRB 131104 779 ± 1 707.9 <0.79 1.12 2.08 1352 <4.69 <5.45 Parkes 30
FRB 140514 562.7 ± 0.6 527.8 <0.60 0.471 2.8 1352 <1.00 <1.76 Parkes 16
FRB 150215 1105.6 ± 0.8 678.4 <0.76 0.7 2.88 1352 <2.68 <4.38 Parkes 19
FRB 150418 776.2 ± 0.5 587.7 <0.66 2.2 0.8 1352 <5.93 <2.85 Parkes 39
FRB 150610 1593.9 ± 0.6 1471.9 <1.65 0.7 2 1352 <17.9 <13.5 Parkes 18
FRB 150807 266.5 ± 0.1 229.6 <0.27 128 0.35 1352 <41.7 <11.5 Parkes 0 c
FRB 151206 1909.8 ± 0.6 1749.8 <1.99 0.3 3 1352 <12.1 <12.2 Parkes 10
FRB 151230 960.4 ± 0.5 922.4 <1.03 0.42 4.4 1352 <3.36 <7.28 Parkes 17
FRB 160102 2596.1 ± 0.3 2583.1 <3.10 0.5 3.4 1352 <59.2 <49.1 Parkes 16
FRB 160317 1165 ± 11 845.4 <0.94 3 21 843 <12.0 <129 UTMOST 13
FRB 160410 278 ± 3 220.3 <0.26 7 4 843 <1.30 <4.13 UTMOST 13
FRB 160608 682 ± 7 443.7 <0.50 4.3 9 843 <3.69 <22.1 UTMOST 12
FRB 170107 609.5 ± 0.5 574.5 <0.65 22.3 2.6 1320 <56.9 <89.6 ASKAP 16
FRB 170827 176.4 ± 0 139.4 <0.17 50.3 0.4 835 <3.57 <1.22 UTMOST 90
FRB 170922 1111 1066 <1.19 2.3 26 835 <16.3 <194 UTMOST 22
FRB 171209 1458 1445 <1.62 0.92 2.5 1352 <22.6 <21.5 Parkes 40
FRB 180301 520 365 <0.42 0.5 3 1352 <0.455 <0.962 Parkes 16
FRB 180309 263.47 218.78 <0.26 20.8 0.576 1352 <6.20 <2.84 Parkes 411
FRB 180311 1575.6 1530.4 <1.72 0.2 12 1352 <5.68 <25.1 Parkes 11.5
FRB 180528 899 830 <0.92 13.8 1.3 835 <51.7 <35.0 UTMOST 14
FRB 180714 1469.873 1212.873 <1.35 5 1 1352 <78.0 <33.2 Parkes 20
FRB 180725A d 716.6 647.6 <0.73 2 600 CHIME 20.6

a Notice that νc can be different for the same telescope. The values presented are the ones reported in the original discovery papers. b This FRB reached saturation so that the peak flux and S/N reported (Lorimer et al. 2007) was greatly underestimated. c No S/N was reported in the original paper (Ravi et al. 2016). d No flux was reported in the original ATel (Boyle 2018).

With the z upper limit, one can derive the upper limit of the isotropic peak luminosity and isotropic energy of an FRB within the observed bandwidth, which read

where Sν,p is the specific peak flux (in units of erg s −1 cm −2 Hz −1 or Jy), and is the specific fluence (in units of erg cm −2 Hz −1 , or Jy ms). Notice that Equation (9) is different from the formula used in some previous influential papers including the FRB catalog paper (Caleb et al. 2016 Petroff et al. 2016) in two aspects. First, we use the central frequency νc, rather than the bandwidth B of the telescope, to derive Lp and E. We believe that this is more appropriate. Bandwidth B makes a connection between the detected energy and fluence, but for estimating the source energy, one should use the central frequency νc. Let us consider the same FRB detected by two telescopes with the same νc but different bandwidths B. The telescope with a wider band receives more energy than the one with a narrower band, but their derived specific flux (energy per unit frequency per unit time per unit area) should be the same. When one estimates the luminosity and energy of the source, the formula of Petroff et al. (2016), Caleb et al. (2016) would give two different values for the same source, which is apparently incorrect. One may also consider two telescopes with the same B but operating at two different νc values. If these two telescopes each detected an FRB with the same specific flux/fluence, using the formula of Petroff et al. (2016), Caleb et al. (2016) would give rise to the same Lp and E for the two FRBs, while in reality the burst detected in the higher frequency band should have higher Lp and E than the other one. Therefore, using νc to calculate Lp and E is more reasonable. Second, the factor (1 + z) was applied incorrectly in those papers when connecting specific fluence with the FRB energy. 4 The definition of luminosity distance DL is such that the luminosity L (in units of erg s −1 ) and flux S (in units of erg s −1 cm −2 or Jy Hz) are connected through . When this is multiplied by the burst-frame intrinsic time τ = τobs/(1 + z) the result is energy, which is Equation (9) note , where is the fluence (in units of erg cm −2 or Jy ms Hz).

The results are presented in Table 1. Without knowing DMhost and DMsrc and their distributions, one can only present the upper limits of z, Lp and E. As there are line-of-sight fluctuations (McQuinn 2014), one can only present the average values.

For the FRB sample published in the FRB Catalogue (FRBCAT) so far, the average z upper limit ranges from 0.17 (FRB 170827 Farah et al. 2017b) to 3.10 (FRB 160102 Bhandari et al. 2018). The average isotropic peak luminosity Lp upper limit ranges from 1.24 × 10 42 erg s −1 (FRB 010621 Keane et al. 2012) to 7.80 × 10 44 erg s −1 (FRB 180714 Oslowski et al. 2018) with a spread of 2.80 dex. The average isotropic energy E upper limit ranges from 6.91 × 10 39 erg (FRB 010621) to 1.94 × 10 42 erg (FRB 170922 Farah et al. 2017a) with a spread of 2.45 dex.

## Comparing Ages

We now have one estimate for the age of the universe from its expansion. Is this estimate consistent with other observations? For example, are the oldest stars or other astronomical objects younger than 13.8 billion years? After all, the universe has to be at least as old as the oldest objects in it.

In our Galaxy and others, the oldest stars are found in the globular clusters (Figure 4) which can be dated using the models of stellar evolution described in the chapter Stars from Adolescence to Old Age.

Figure 4. Globular Cluster 47 Tucanae: This NASA/ESA Hubble Space Telescope image shows a globular cluster known as 47 Tucanae, since it is in the constellation of Tucana (The Toucan) in the southern sky. The second-brightest globular cluster in the night sky, it includes hundreds of thousands of stars. Globular clusters are among the oldest objects in our Galaxy and can be used to estimate its age. (credit: NASA, ESA, and the Hubble Heritage (STScI/AURA)-ESA/Hubble Collaboration)

The accuracy of the age estimates of the globular clusters has improved markedly in recent years for two reasons. First, models of interiors of globular cluster stars have been improved, mainly through better information about how atoms absorb radiation as they make their way from the center of a star out into space. Second, observations from satellites have improved the accuracy of our measurements of the distances to these clusters. The conclusion is that the oldest stars formed about 12–13 billion years ago.

This age estimate has recently been confirmed by the study of the spectrum of uranium in the stars. The isotope uranium-238 is radioactive and decays (changes into another element) over time. (Uranium-238 gets its designation because it has 92 protons and 146 neutrons.) We know (from how stars and supernovae make elements) how much uranium-238 is generally made compared to other elements. Suppose we measure the amount of uranium relative to nonradioactive elements in a very old star and in our own Sun, and compare the abundances. With those pieces of information, we can estimate how much longer the uranium has been decaying in the very old star because we know from our own Sun how much uranium decays in 4.5 billion years.

The line of uranium is very weak and hard to make out even in the Sun, but it has now been measured in one extremely old star using the European Very Large Telescope (Figure 5). Comparing the abundance with that in the solar system, whose age we know, astronomers estimate the star is 12.5 billion years old, with an uncertainty of about 3 billion years. While the uncertainty is large, this work is important confirmation of the ages estimated by studies of the globular cluster stars. Note that the uranium age estimate is completely independent it does not depend on either the measurement of distances or on models of the interiors of stars.

Figure 5. European Extremely Large Telescope, European Very Large Telescope, and the Colosseum: The European Extremely Large Telescope (E-ELT) is currently under construction in Chile. This image compares the size of the E-ELT (left) with the four 8-meter telescopes of the European Very Large Telescope (center) and with the Colosseum in Rome (right). The mirror of the E-ELT will be 39 meters in diameter. Astronomers are building a new generation of giant telescopes in order to observe very distant galaxies and understand what they were like when they were newly formed and the universe was young. (credit: modification of work by ESO)

As we shall see later in this chapter, the globular cluster stars probably did not form until the expansion of the universe had been underway for at least a few hundred million years. Accordingly, their ages are consistent with the 13.8 billion-year age estimated from the expansion rate.

### Key Concepts and Summary

Cosmology is the study of the organization and evolution of the universe. The universe is expanding, and this is one of the key observational starting points for modern cosmological theories. Modern observations show that the rate of expansion has not been constant throughout the life of the universe. Initially, when galaxies were close together, the effects of gravity were stronger than the effects of dark energy, and the expansion rate gradually slowed. As galaxies moved farther apart, the influence of gravity on the expansion rate weakened. Measurements of distant supernovae show that when the universe was about half its current age, dark energy began to dominate the rate of expansion and caused it to speed up. In order to estimate the age of the universe, we must allow for changes in the rate of expansion. After allowing for these effects, astronomers estimate that all of the matter within the observable universe was concentrated in an extremely small volume 13.8 billion years ago, a time we call the Big Bang.

### Glossary

the theory of cosmology in which the expansion of the universe began with a primeval explosion (of space, time, matter, and energy)

cosmological constant:

the term in the equations of general relativity that represents a repulsive force in the universe

the study of the organization and evolution of the universe

dark energy:

the energy that is causing the expansion of the universe to accelerate its existence is inferred from observations of distant supernovae

The New Enlightenment: Cosmo-Transcendental Positioning of the Living Being in the Universe

On the First Principle of Biology and the Foundation of the Universal Science

The Relation Between Man and World

Observers, Freedom, and the Cosmos

The Cosmological Circumstances and Results of the Anno Domini Invention: Anno Mundi 6000, Great Year, Precession, and End of the World Calculation

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Mind in the Quantum Universe

Why is the Universe Just Right for Life?

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Astrobiology: From Extremophiles in the Solar System to Extraterrestrial Civilizations

Rationality and Wonder: From Scientific Cosmology to Philosophy and Theology

Positive Contribution of Religion to Cosmology

Principle of Greatest Happiness

Astronomy: Brightest and Most Fascinating Shining Path for Mankind Future

## Fourier transform and Cosmic variance - a few precisions

If we are interested in power spectrum, we want to estimate the
variance of the amplitude of the modes ##k## of our Fourier
decomposition. If one observes the whole observable Universe and we
do the Fourier transformation we get a cube whose center is the mode
## |vec| = 0## which corresponds to the mean value of the observed
field.

This mode has only one pixel. How do we measure the variance of the
process at ## |vec| = 0 ## ? We can not.

In fact we can but the value of doesn't mean anything because the
error is infinite. In other words, we have an intrinsic (statistical)
error which depends on the number of achievements to which we have
access.

one will be able to consider spheres of sizes ## dk ## between ## [k: k +
dk] ## which will contain a number of pixels ## N_ = V_ /(dk)^<3>## where ## V_ = 4 pi k^ <2>dk ## is the volume of the sphere
and ## (dk)^ <3>## is the volume of a pixel in our Fourier transform
cube.

So one can estimate how many values one can use to calculate our
power spectrum for each value of ## k ##. The greater the ## k ##, the
greater the number of accessible values and therefore the statistical
error decreases. The power spectrum is a variance estimator so the
statistical error is basically a relative error:

So we can see that for the case ##|vec| = 0## we have an infinite error because ##N_ = 1##.

1) I can't manage to proove that the statistical error is formulated like :

and why it is considered like a relative error ?

2) Which are the conditions to assimilate a statistical error (standard deviation) to a relative error (##dfrac##) ?

## Get the expression of probed volume between 2 redshifts - Astronomy

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#### Images

###### Figure 1

(a) Schematic illustration of the device: a diamond nanobeam containing a single NV spin is positioned near a hybrid Pt/YIG structure. (b) Scanning electron micrograph of the device (before positioning the diamond nanobeam). (c) NV optical (green) and microwave (red) spin-echo sequence for stray-field magnetometry of spin-wave resonances. A spin-wave (FMR) drive is applied during the central free-precession period. A change in the YIG stray field Δ B | | imparts a phase on the NV spin state and is read out via spin-dependent photoluminescence. A free precession time τ ≈ 5.5 μ s is used for such stray-field magnetometry. (d) Example of YIG spin-wave resonances measured with the pulse sequence in (c), at applied static magnetic field B ext = 337 G aligned with the NV axis. Plotted is the NV-measured stray static magnetic field along the NV axis, Δ B | | , as a function of the spin-wave drive frequency. The signal is normalized by the square of the drive field b 1 2 , which is proportional to the spin-wave drive power and is independently measured on-chip using the same NV sensor [20]. Blue dots: data. Red line: double Gaussian fit, yielding FWHM = 8.5 ( 6 ) MHz for the dominant mode attributed to the spatially homogeneous ( n = 1 ) ferromagnetic resonance (FMR) of the YIG bar. (e) Green dots: Magnetic-field ( B ext ) dependence of the fundamental spin-wave resonance frequency extracted from fits to measurements such as shown in (d). Blue line: fit reveals characteristic Kittel-like behavior of FMR. Black lines: NV transition frequencies corresponding to the m s = 0 ↔ ± 1 transitions. NV-spin manipulation pulses are applied on the m s = 0 ↔ + 1 transition.

###### Figure 2

(a) Sketch of the magnetization dynamics of the Pt/YIG device under the influence of a spin current. The electrical current ( J e ) injects a spin current ( J s ) into the YIG, leading to a spin-orbit torque (labeled SOT) that either reduces or enhances magnetic damping depending on the relative orientation between the injected spins and the magnetization M. DT denotes the (intrinsic) damping torque. (b) NV-measured, microwave-driven spin-wave resonance spectra in the YIG as a function of the DC current I dc through the Pt. [Measurement sequence shown in Fig. 1]. Blue traces: normalized change in YIG stray field Δ B | | / b 1 2 as a function of microwave driving frequency for I dc = 5 , 4, 3, 2, 1, 0, −2, −5 mA. Red lines: double Gaussian fit to data. Green dots: center frequency of the fundamental spin-wave mode vs. I dc . Black curve: parabolic fit to green dots. (c) On-resonance Δ B | | as a function of microwave driving power b 1 2 for different values of I dc . Black, red, and pink dots correspond to I dc = 4.5 , 4.75, and 5 mA, for which the initial slopes ( d Δ B | | / d b 1 2 ) have no discernable difference. Blue and green squares correspond to I dc = 3 and 3.5 mA, for which initial slopes are significantly smaller. (d) Plot of the inverse of the initial slopes, i.e., d b 1 2 / d Δ B | | , as a function of I dc . Diagonal and horizontal dashed lines serve as eye guide to illustrate that there exists a current threshold as onset of an auto-oscillating spin torque oscillator (STO).

###### Figure 3

(a) Blue dots show temperature, measured with NV, changes with I dc . (b) Green dots represent center frequencies of fundamental SW mode at different I dc measured by NV stray-field magnetometry measurements [extracted from Fig. 2]. (c) Normalized effective saturation magnetization ( M e / M s ) as a function of I dc : data shown as blue dots are derived from temperature measurements. Data shown as solid green dots are from FMR center frequency measurements without correction to remove impact from Oersted field, while data shown as open green circles are corrected.

###### Figure 4

(a) NV spin-relaxometry measurement sequence. The NV spin is initialized into m s = 0 by a green laser pulse and let to relax for a time τ, after which the spin population is characterized via the spin-dependent PL during a laser readout pulse. Noise that is resonant with an NV transition frequency causes NV spin relaxation. (b) NV spin-relaxometry measurement at I dc = 5.8 mA and τ = 5 μ s . By tuning the magnetic field B ext , the frequency of the m s = 0 ↔ − 1 transition is swept over three spin-wave (SW) modes in the YIG, whose field-noise causes strong NV spin relaxation and thus dips in the normalized PL signal. (c) Performing the measurement shown in panel (b) for different I dc yields a 2D plot of PL vs I dc and B ext that displays the presence and dispersion of spin-torque oscillators (STOs) in the system. Different delay times τ of 150, 50, 15, 5, and 3 µs are used for the different I dc ranges of [−5 mA:0 mA], [0.2 mA:1.8 mA], [2 mA:3 mA], [3.2 mA:5 mA], and [5.2 mA:6 mA], respectively. Top horizontal axis shows the m s = 0 ↔ − 1 transition frequency at corresponding B ext . Blue stars indicate fits of peak centers for the first resonance on the right-hand-side ( ST O 1 ), while red stars are fits of peak centers for the second ( ST O 2 ). These two STOs are also indicated in panel (b). Note that an additional oscillator [data points are orange in color and designated as STO* in panel (b)] appears when I dc = 5.8 mA and persists for higher current. Inset illustrates mode spatial distribution of ST O 1 and ST O 2 along width of Pt/YIG microstructure (W). (d) and (e). Zoomed-in, high-resolution views of (c), where spin-wave modes are observed to approach each other.

###### Figure 5

(a) NV spin relaxation rate (Γ) is measured at the current and magnetic field values indicated by the blue and red stars in Fig. 4, where the spin-torque oscillators (STOs) are resonant with the m s = 0 ↔ − 1 transition frequency. At each of these current and magnetic field values, we sweep τ, perform an NV spin-relaxometry measurement sequence [Fig. 4], and extract the exponential decay time constant Γ. Γ STO 1 (red) and Γ STO 2 (blue) are plotted as a function of I dc . The dramatic order-of-magnitude increase of the relaxation rate above I dc ∼ 3 mA indicates spin-torque induced auto-oscillation of the STOs. Inset shows 1 / Γ = T 1 vs I dc for both STOs. Linear fits at low current ( I dc < 4 mA) intersect with T 1 = 0 at I t h 1 = 3.5 mA and I t h 2 = 4.4 mA , which we define as the auto-oscillation threshold currents. (b) Measured STO linewidth Δ B as a function of I dc for ST O 1 (blue dots) and ST O 2 (red dots). The vertical axis on the right gives the linewidth in frequency (MHz), calculated from Δ B using the Kittel relation at B ext ∼ 250 G and I dc = 0 .

###### Figure 6

(a) NV spin-relaxometry measurement sequence as in Fig. 4, with added MW drive. For synchronization measurement, B ext is tuned such that the NV m s = 0 ↔ − 1 transition coincides with STO resonance. (b) Measured NV photoluminescence (PL) as a function of the MW drive frequency f M W , at I dc = 5.2 mA , B ext = 292 G , and MW drive amplitude b 1 = 1.5 G . When the MW drive is resonant with the NV transition frequency, a dip in the PL is observed because the driving deplete m s = 0 population. Over a frequency interval Δ f s the STO can be locked to the MW drive and thus detuned from the NV transition, thereby decreasing the NV spin relaxation and correspondingly increasing the measured PL. When the MW drive frequency is detuned beyond the locking interval (i.e., synchronization bandwidth), the STO remains resonant with the NV transition frequency, leading to strong NV-spin relaxation and a corresponding reduced PL. (Appendix pp6 for detailed data analysis.) (c) 2D map of PL vs f M W and MW drive amplitude b 1 . The synchronization bandwidth increases linearly with b 1 . b 1 th is the threshold amplitude at which synchronization begins to occur. (d) Synchronization bandwidth vs b 1 at different I dc (4, 5.2, and 5.6 mA).

###### Figure 7

Pt/YIG hybrid device fabrication processes. (a) A r + plasma cleaning inside sputter chamber. (b) DC sputtering of 10 nm Pt. Note: this procedure has successfully generated Pt/YIG interfaces showing excellent spin mixing conductance [43]. (c) Coating substrate with PMMA/HSQ/FOX resist stack. (d) E-beam lithography and developing in TMAH to form etching mask. (e) O 2 reactive ion etch to remove excessive PMMA and to induce proper undercut. (f) A r + milling to transfer pattern from the etching mask onto the substrate. (g) Lift-off to remove the resist stack. (h) E-beam patterning of electrical leads and MW driving lines with proper alignment with respect to existing Pt/YIG microstructures.

###### Figure 8

(a) SEM image of Pt/YIG hybrid shows smooth edge. Throughout the fabrication process, extra care is taken to preserve material quality, e.g., only mild solvents are used. (b) SEM image of an example YIG disc (∼400 nm in diameter), fabricated with the use of high-resolution e-beam resist.

###### Figure 9

SOT and its dependence on the relative orientation between injected spins and bias field. (a) Assuming the DC current J e is along the long direction of the device ( x axis), a spin current is sent along the vertical, z direction. The injected spins are necessarily oriented along the short direction ( y axis) of the device. Therefore, the ideal orientation of the bias field B ext is along the y direction, as then the YIG magnetization M (in the absence of excitation) is parallel to the orientation of the injected spins s . This leads to the maximal spin-orbit torque (SOT) available. The SOT can act either along or against the damping torque (DT), depending on the orientation of injected spins, which is set by the direction of the DC current. (b) Orientation of the NV, and hence B ext , employed in single-spin sensing in this work ( N V a ). The projection of B ext on the y axis is ∼ 0.72 | B ext | . (c) Orientation of another NV ( N V b ) present in the nanobeam. The projection of the corresponding B ext on the y axis is ∼ 0.08 | B ext | . (d) Influence of I dc on SW linewidth measured via NV spin-relaxometry [similar to those shown in main text Fig. 5] but now with N V b .

###### Figure 10

Determination of saturation magnetization ( M s ). (a) NV spin-relaxometry measurements at I dc = 0 on m s = 0 ↔ − 1 transition over a wide range of B ext (red dots on gray vertical plane). The bottom plane of the figure including green dots showing center frequency of the fundamental SW mode, a blue trace showing a fit to the green dots, and black line indicating the NV m s = 0 ↔ − 1 transition, is reproduced from main text Fig. 1. The strongest NV spin-relaxometry resonance dip at I dc = 0 corresponds to the crossing point of the NV m s = 0 ↔ − 1 transition and the fundamental mode of the Pt/YIG microstructure at B ext = 245 Gauss and f = 2.18 GHz. (b) To validate our analytical formalism to calculate M s , a micromagnetic simulation (OOMMF) is performed at B ext = 245 Gauss with different values of M s . Simulation results (black dots) match very well with analytical calculation (blue line). Red dashed line corresponds to the frequency of the fundamental mode ( f = 2.18 GHz) and the crossing point between this red dashed line and blue line determines the saturation magnetization M s used throughout this work.

###### Figure 11

SW mode designation and micromagnetic simulation (OOMMF). (a) This figure is reproduced from main text Fig. 4 with SW mode labeled as M 1 , M 2 ,…, M 5 . (b) OOMMF simulation along with NV spin-relaxometry data [line cut from (a)] at I dc = 0 . Vertical plane (gray color shaded) can be projected onto the NV m s = 0 ↔ − 1 transition frequency (black diagonal line on the bottom plane). Relaxometry data (red points) and fit (blue line) are in this vertical plane and the resonance dip can thus be projected onto the NV transition frequency connected via color-coded dashed lines (with the same color as the label of the corresponding mode). Green solid line is OOMMF simulation at bias field of 245 G (along NV axis). Resonance peaks (bulk mode n = 1 , 2, 3 edge modes n ′ = 1 , 2) are visible in simulation. Solid color-coded lines on the bottom plane show well-recognized correspondence between simulated resonance peaks and resonance dips measured with NV spin-relaxometry.

###### Figure 12

Micromagnetic simulation (OOMMF) of SW spectrum and spatial magnetization profiles for different eigenmodes. (a) OOMMF simulated spectrum (performed at 245 G bias field along NV axis). Vertical axis represents spatially averaged magnetic susceptibility (imaginary part), 〈 χ ″ 〉 . (b) and (c) At f = 2.08 and 2.14 GHz, map of simulated local susceptibility (imaginary part), χ ″ , shows an edge mode localized at the corner of YIG microstructure. (d)–(f) At f = 2.18 , 2.26, and 2.36 GHz, map of simulated local susceptibility (imaginary part), χ ″ , shows bulk mode sustained over entire YIG microstructur(e) with mode order n = 1 , 2, 3 respectively. All spatial mode profile maps use the same color scale (right bottom color bar).

###### Figure 13

Estimate of mode hybridization coupling strength. (a) Measured NV upper transition frequencies at 337 G as a function of DC current. The data is fit to a quadratic function (as well as smaller linear and constant terms). (b) NV spin-relaxometry data [from Fig. 4] plotted at the plane of f NV ( B , I dc ) . (c) Frequency value associated with each mode center ( B 1 , 2 , I dc ) from (b). (d) Dispersion at a fixed field ( B = 284.54 G ) as a function of the DC current.

###### Figure 14

ST O 1 synchronization test to understand I dc range of auto-oscillation and extract synchronization frequency interval. (a) NV PL vs field sweep [similar measurements as in main text Fig. 4] at I dc = 4 mA to find mode resonance as indicated by blue arrow. (b) At I dc = 4 mA and B ext = 270 G (mode resonance matching NV transition frequency), we perform 2D sweeping of the MW driving frequency and amplitude and simultaneously record the corresponding NV PL signal. This measurement is the same as that performed in Fig. 6. Here the 2D plot color bar corresponds to the PL level shaded by gray color shown in (a) and a dashed arrow is used to illustrate such correspondence. (c) and (e) show the same field sweep as in a but at different values of I dc : 5.2 and 5.6 mA, respectively. (d) and (f) show a similar 2D plot but with different values of both I dc : 5.2 and 5.6 mA and B ext = 292 and 302.6 G.

###### Figure 15

Characterization of ST O 2 by testing synchronization to external MW signals. (a) NV PL vs field sweep at I dc = 5.1 mA to find center frequency of free running ST O 2 , as indicated by red arrow. (b) At I dc = 5.1 mA and B ext = 273.4 G , we perform 2D sweeping of the MW driving frequency and amplitude, and simultaneously record the corresponding NV PL signal. 2D plot color bar corresponds to the PL level covered by the gray shaded region shown in (a) and a dashed arrow is used to illustrate such correspondence. (c) Similar measurement as in (a) but at I dc = 5.8 mA , and mode resonance takes place at B ext = 294 G . (d) At I dc = 5.8 mA , and B ext = 294 G , we perform synchronization test measurements. No discernible signal contrast is observed.

###### Figure 16

Coexistence of STOs and their competing synchronization to external MW source. (a) At I dc = 5.2 mA , NV PL vs field sweep shows two resonance peaks, which correspond to ST O 2 (lower field) hence higher frequency) and ST O 1 . (b) At I dc = 5.2 mA and B ext = 292 G , synchronization test shows a parameter space area with an interesting signal feature surrounded by dashed yellow triangle.

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