text:
# Introduction

Your task is to generate the first 1000 terms in the continued fraction representation of digit-wise sum of square root of 2 and square root of 3.

In other words, produce exactly the following list (but the output format is flexible)

[2, 6, 1, 5, 7, 2, 4, 4, 1, 11, 68, 17, 1, 19, 5, 6, 1, 5, 3, 2, 1, 2, 3, 21, 1, 2, 1, 2, 2, 9, 8, 1, 1, 1, 1, 6, 2, 1, 4, 1, 1, 2, 3, 7, 1, 4, 1, 7, 1, 1, 4, 22, 1, 1, 3, 1, 2, 1, 1, 1, 7, 2, 7, 2, 1, 3, 14, 1, 4, 1, 1, 1, 15, 1, 91, 3, 1, 1, 1, 8, 6, 1, 1, 1, 1, 3, 1, 2, 58, 1, 8, 1, 5, 2, 5, 2, 1, 1, 7, 2, 3, 3, 22, 5, 3, 3, 1, 9, 1, 2, 2, 1, 7, 5, 2, 3, 10, 2, 3, 3, 4, 94, 211, 3, 2, 173, 2, 1, 2, 1, 14, 4, 1, 11, 6, 1, 4, 1, 1, 62330, 1, 17, 1, 5, 2, 5, 5, 1, 9, 3, 1, 2, 1, 5, 1, 1, 1, 11, 8, 5, 12, 3, 2, 1, 8, 6, 1, 3, 1, 3, 1, 2, 1, 78, 1, 3, 2, 442, 1, 7, 3, 1, 2, 3, 1, 3, 2, 9, 1, 6, 1, 2, 2, 2, 5, 2, 1, 1, 1, 6, 2, 3, 3, 2, 2, 5, 2, 2, 1, 2, 1, 1, 9, 4, 4, 1, 3, 1, 1, 1, 1, 5, 1, 1, 4, 12, 1, 1, 1, 4, 2, 15, 1, 2, 1, 3, 2, 2, 3, 2, 1, 1, 13, 11, 1, 23, 1, 1, 1, 13, 4, 1, 11, 1, 1, 2, 3, 14, 1, 774, 1, 3, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 8, 1, 3, 10, 2, 7, 2, 2, 1, 1, 1, 2, 2, 1, 11, 1, 2, 5, 1, 4, 1, 4, 1, 16, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 8, 1, 2, 1, 1, 22, 3, 1, 8, 1, 1, 1, 1, 1, 9, 1, 1, 4, 1, 2, 1, 2, 3, 5, 1, 3, 1, 77, 1, 7, 1, 1, 1, 1, 2, 1, 1, 27, 16, 2, 1, 10, 1, 1, 5, 1, 6, 2, 1, 4, 14, 33, 1, 2, 1, 1, 1, 2, 1, 1, 1, 29, 2, 5, 3, 7, 1, 471, 1, 50, 5, 3, 1, 1, 3, 1, 3, 36, 15, 1, 29, 2, 1, 2, 9, 5, 1, 2, 1, 1, 1, 1, 2, 15, 1, 22, 1, 1, 2, 7, 1, 5, 9, 3, 1, 3, 2, 2, 1, 8, 3, 1, 2, 4, 1, 2, 6, 1, 6, 1, 1, 1, 1, 1, 5, 7, 64, 2, 1, 1, 1, 1, 120, 1, 4, 2, 7, 3, 5, 1, 1, 7, 1, 3, 2, 3, 13, 2, 2, 2, 1, 43, 2, 3, 3, 1, 2, 4, 14, 2, 2, 1, 22, 4, 2, 12, 1, 9, 2, 6, 10, 4, 9, 1, 2, 6, 1, 1, 1, 14, 1, 22, 1, 2, 1, 1, 1, 1, 118, 1, 16, 1, 1, 14, 2, 24, 1, 1, 2, 11, 1, 6, 2, 1, 2, 1, 1, 3, 6, 1, 2, 2, 7, 1, 12, 71, 3, 2, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 3, 5, 5, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 2, 19, 1, 16, 2, 15, 1, 1, 3, 2, 3, 2, 4, 1, 3, 1, 1, 7, 1, 2, 2, 117, 2, 2, 8, 2, 1, 5, 1, 3, 12, 1, 10, 1, 4, 1, 1, 2, 1, 5, 2, 33, 1, 1, 1, 1, 1, 18, 1, 1, 1, 4, 236, 1, 11, 4, 1, 1, 11, 13, 1, 1, 5, 1, 3, 2, 2, 3, 3, 7, 1, 2, 8, 5, 14, 1, 1, 2, 6, 7, 1, 1, 6, 14, 22, 8, 38, 4, 6, 1, 1, 1, 1, 7, 1, 1, 20, 2, 28, 4, 1, 1, 4, 2, 2, 1, 1, 2, 3, 1, 13, 1, 2, 5, 1, 4, 1, 3, 1, 1, 2, 408, 1, 29, 1, 6, 67, 1, 6, 251, 1, 2, 1, 1, 1, 8, 13, 1, 1, 1, 15, 1, 16, 23, 12, 1, 3, 5, 20, 16, 4, 2, 1, 8, 1, 2, 2, 6, 1, 2, 4, 1, 9, 1, 7, 1, 1, 1, 64, 10, 1, 1, 2, 1, 8, 2, 1, 5, 4, 2, 5, 6, 7, 1, 2, 1, 2, 2, 1, 4, 11, 1, 1, 4, 1, 714, 6, 3, 10, 2, 1, 6, 36, 1, 1, 1, 1, 10, 2, 1, 1, 1, 3, 2, 1, 6, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 40, 1, 1, 1, 5, 1, 3, 24, 2, 1, 6, 2, 1, 1, 1, 7, 5, 2, 1, 2, 1, 6, 1, 1, 9, 1, 2, 7, 6, 2, 1, 1, 1, 2, 1, 12, 1, 20, 7, 3, 1, 10, 1, 8, 1, 3, 1, 1, 1, 1, 2, 1, 1, 6, 1, 2, 1, 5, 1, 1, 1, 5, 12, 1, 2, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 8, 2, 4, 1, 3, 1, 1, 1, 2, 1, 11, 3, 2, 1, 7, 18, 1, 1, 17, 1, 1, 7, 4, 6, 2, 5, 6, 4, 4, 2, 1, 6, 20, 1, 45, 5, 6, 1, 1, 3, 2, 3, 3, 19, 1, 1, 1, 1, 1, 1, 34, 1, 1, 3, 2, 1, 1, 1, 1, 1, 4, 1, 2, 1, 312, 2, 1, 1, 1, 3, 6, 6, 1, 2, 25, 14, 281, 4, 1, 37, 582, 3, 20, 2, 1, 1, 1, 2, 1, 3, 7, 8, 4, 1, 11, 2, 3, 183, 2, 23, 8, 72, 2, 2, 3, 8, 7, 1, 4, 1, 4, 1, 2, 2, 1, 2, 1, 8, 2, 4, 1, 2, 1, 2, 1, 1, 2, 1, 1, 10, 2, 1, 1, 5, 2, 1, 1, 1, 2, 1, 1, 2, 1, 3, 2, 9]

# Challenge

The following general introduction to continued fraction is taken from the challenge Simplify a Continued Fraction.

Continued fractions are expressions that describe fractions iteratively. They can be represented graphically:

Or they can be represented as a list of values: [a0, a1, a2, a3, ... an]

This challenge is to find out the continued fraction of the digit-wise sum of sqrt(2) and sqrt(3), the digit-wise sum is defined as follows,

Take the digits in the decimal representation of sqrt(2) and sqrt(3), and obtain the sum digit by digit:

    1.  4  1  4  2  1  3  5  6  2  3 ...
+   1.  7  3  2  0  5  0  8  0  7  5 ...
=   2. 11  4  6  2  6  3 13  6  9  8 ...

Then only keep the last digit of the sum and compile them back to the decimal representation of a real number

    1.  4  1  4  2  1  3  5  6  2  3 ...
+   1.  7  3  2  0  5  0  8  0  7  5 ...
=   2. 11  4  6  2  6  3 13  6  9  8 ...
->  2.  1  4  6  2  6  3  3  6  9  8 ...

The digit-wise sum of sqrt(2) and sqrt(3) is therefore 2.1462633698..., and when it is expressed with continued fraction, the first 1000 values (i.e. a0 to a999) obtained are the ones listed in the introduction section.

# Specs

• You may write a function or a full program. Neither should take inputs. In other words, the function or the program should work properly with no inputs. It doesn't matter what the function or program does if non-empty input is provided.

• You should output to STDOUT. Only if your language does not support outputting to STDOUT should you use the closest equivalent in your language.

• You don't need to keep STDERR clean, and stopping the program by error is allowed as long as the required output is made in STDOUT or its equivalents.

• You can provide output through any standard form.

• This is , the lowest number of bytes wins.

• As usual, default loopholes apply here.

# Kotlin 1.1 script, 304 293 bytes

import java.math.BigDecimal as b
import java.math.*
val m=MathContext(1022)
var B=b(2)
var A=b((""+B.sqrt(m)).zip(""+b(3).sqrt(m)).joinToString(""){(a,b)->if(a=='.')".";else ""+(a-'0'+(b-'0'))%10})
val g=b(1).setScale(1022)
repeat(1000){println(B);A=g/(A-B);B=A.setScale(0,RoundingMode.FLOOR)}

A bit verbose unfortunately :/

Must be run with JDK 9, as sqrt was added to BigDecimal in this release. Interestingly, I couldn't find a TIO site with both Kotlin 1.1 and JDK 9 features (Ideone and repl.it both run Kotlin 1.0, which did not support destructuring in lambdas, and TIO complains that sqrt does not exist.)

Prints each element separated by a newline.

Edit (-11): moved println to the beginning of the loop body and added an additional iteration to avoid repeating the method call. An extra calculation is done, but it is not used for anything.

# Python 2, 193 ... 179 178 bytes

d=10
u=d**2000
v=u*u
def s(n,a=d,i=9):
while a-i:i,a=a,(a+n/a)/2
return a
p,q,r,t=s(2*v),s(3*v),1,0
while p:t+=(p+q)%d*r;p/=d;q/=d;r*=d
for i in range(1000):print t/u;t=v/(t%u)

Try it online!

Calculating sqrt(2) and sqrt(3) to such a precision with a short code is a tough job in Python and other languages.

2000 digits is needed to ensure the expansion is correct (1020 suffices, but I'm not going to modify it because no improvement), and lines 4-6 is the integer square root.

193 > 180 : The digit-wise modulo sum is now carried by a loop instead of array manipulation

180 > 179 : Replaced the 6 occurrences of 10 using d with the cost of defining with 5 bytes, cutting 1 byte in total

179 > 178 : Just realized that a!=i can be replaced by a-i

# Jelly, 32 bytes

ȷ*
%¢¢²¤:
2,3×Ñ×ÑÆ½DS%⁵ḌÇȷÐ¡Ṗ:Ñ

Try it online!

Basically use fixed-point arithmetic. M may work better here, but somehow floor(HUGE_NUMBER × sqrt(2) doesn't want to evaluate for too large HUGE_NUMBER. Anyway the fixed-point division is definitely better.

Explanation:

-------
ȷ*       Calculate the base for fixed-point arithmetic.
ȷ         Number 1000.
*        Raise to the power of...
       self. (so we have 1000 ** 1000 == 1e3000) Let B=1e3000.

-------
%¢¢²¤:    Given f × B, return a number approximately (1/frac(f)) × B.
Current value: f × B.
%¢        Modulo by B. Current value: frac(f) × B.
¢²¤     B² (that is, 1e6000)
:    integer-divide by. So we get B²/(frac(f)×B) ≃ 1/frac(f) × B.

-------
2,3                    The list [2,3].

Ñ                  This refers to the next link as a monad, which is the
×                   Multiply by. So we get [2,3]×1e3000 = [2e3000,3e3000]
×Ñ                Again. Current value = [2e6000,3e6000] = [2B²,3B²]

Æ½              Integer square root.
Current value ≃ [sqrt(2B²),sqrt(3B²)]
= [B sqrt(2),B sqrt(3)]

DS            Decimal digits, and sum together.
%⁵          Modulo 10.
Ḍ         Convert back from decimal digits to integer.

Ð¡     Repeatedly apply...

• Unfortunately ×⁺Ñ doesn't work. Alternatively ×Ñ²$. – user202729 Mar 6 '18 at 4:30 • Upvoted. Explanation would be much appreciated. – Weijun Zhou Mar 6 '18 at 4:34 • @WeijunZhou Done, tell me if you don't understand something. – user202729 Mar 6 '18 at 4:50 # Haskell 207 bytes I Couldn't find an easy way to compute the continued fraction lazilly, so I worked as well with 2000 digits. import Data.Ratio r#y|x<-[x|x<-[9,8..],r>(y+x)*x]!!0=x:(100*(r-(y+x)*x))#(10*y+20*x) c r|z<-floor r=z:c(1/(r-z%1)) main=print.take 1000.c$foldl1((+).(10*))(take 2000$(mod10)<$>zipWith(+)(3#0)(2#0))%10^1999
`


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# Recovery Tests

by Glenn M. Duffield, President, HydroSOLVE, Inc.

## What Is ARecovery Test?

A recovery test is a controlled field experiment performed at the end of a pumping test (constant-rate or step-drawdown) after pumping in the pumped (control) well has ended. Water-level response (residual drawdown) is measured after pumping has stopped in one or more surrounding observation wells and optionally in the control well itself.

In petroleum engineering, a buildup test refers to a recovery test after a production well is shut in (i.e., after pumping stops). Similarly, a falloff test is a recovery test performed after injection in a well ceases.

Figure 2 illustrates pumping rates recorded during and after a constant-rate pumping test in a confined aquifer (USBR 1995). A plot of drawdown and recovery recorded during the test is shown in Figure 3.

## Estimation OfAquifer Properties

The goal of a recovery test, as in any aquifer test, is to estimate hydraulic properties of an aquifer system such as transmissivity, hydraulic conductivity and storativity (storage coefficient). One typically measures recovery after the termination of a constant-rate or step-drawdown pumping test to provide additional data for estimating aquifer properties.

Typically, aquifer properties are estimated from a recovery test by fitting mathematical models (type curves) to residual drawdown data through a procedure known as curve matching.

Three methods available for estimating aquifer properties from recovery test data include the following:

One may use each of the three methods of recovery analysis to interpret tests performed in nonleaky confined, leaky confined, unconfined and fractured aquifers.

### Residual Drawdown Analysis

Theis (1935) was the first to devise a method for estimating aquifer properties from recovery data. The procedure involves fitting a straight line on a residual drawdown plot of s' (residual drawdown) versus log t/t' (ratio of time since pumping began to time since pumping stopped). In the petroleum engineering literature, a residual drawdown plot is known as a Horner plot (Horner 1951).

Figure 4 shows how to apply the Theis (1935) solution to residual drawdown data from the USBR (1995) recovery test (Figure 3). Note that the fitted line favors data from the end of the recovery period which plot closer to the origin of the graph (i.e., as t/t' approaches 1). The method estimates the transmissivity (T) of the aquifer and S/S', the ratio of storativity during pumping (S) to storativity during recovery (S'). For an infinite nonleaky confined aquifer, S/S' should be close to 1.

Residual drawdown plots are applicable to recovery data from variable-rate as well as constant-rate pumping tests (Horner 1951; Birsoy and Summers 1980). Furthermore, one may use solutions other than the Theis 1935 method to analyze recovery tests on a s' versus t/t' plot.

### Agarwal Method

Through a simple data transformation, Agarwal (1980) devised a procedure for the interpretation of recovery data that allows the application of standard curve-matching techniques, including diagnostic techniques like derivative analysis, routinely used for drawdown data. The Agarwal method is widely used by petroleum engineers (Horne 1995) and is steadily gaining recognition among groundwater scientists.

The Agarwal method involves the following simple data transformations for recovery test data (assuming constant-rate pumping):

$sa=sp-s′$ $t a = t p t ′ t p + t ′$

where ${s}_{\text{a}}$ is recovery measured from ${s}_{\text{p}}$ [L], ${s}_{\text{p}}$ is total drawdown at the end of pumping [L], ${s}^{\prime }$ is residual drawdown during recovery [L], ${t}_{\text{a}}$ is Agarwal equivalent time [T], ${t}_{\text{p}}$ is total time of pumping [T], and ${t}^{\prime }$ is time since pumping stopped [T]. The method assumes that ${t}_{\text{p}}$ > ${t}^{\prime }$. For multi-rate tests, Agarwal (1980) proposed a different formula for ${t}_{\text{a}}$ that is slightly more complicated than the one given above.

Figure 5 shows results from the analysis of the USBR (1995) recovery data using the Agarwal method. After the Agarwal transformation, the familiar Theis (1935) type curve was matched to the recovery data to estimate transmissivity and storativity for the aquifer.

One may use other methods such as the Cooper and Jacob (1946) straight-line solution in conjunction with the Agarwal method of recovery analysis (Figure 6). The same derivative plateau used to identify infinite-acting radial flow in drawdown data is also evident on an Agarwal plot and guides the match of the Cooper-Jacob straight line (Figure 6).

Use of the Agarwal recovery method is not restricted to nonleaky confined aquifers. As a demonstration, consider the synthetic drawdown and recovery data from a constant-rate test in a leaky confined aquifer shown in Figure 7.

If we analyze only the recovery data from the synthetic test (Figure 6), the Agarwal method allows us to match a Hantush and Jacob (1955) drawdown curve to the transformed data very accurately (Figure 8).

### Combined Drawdown/Recovery Analysis

The residual drawdown and Agarwal methods only make use of data from the recovery period of a pumping test. You can combine the analysis of drawdown and recovery data using virtually any pumping test solution through the application of superposition in time.

Figure 9 shows results of analyzing combined drawdown and recovery data with the Theis (1935) solution.

## Catalog OfRecovery Plots

On the recovery plots presented below, drawdown responses are displayed as solid blue curves. For reference, the Theis solution is shown on selected plots by a dashed black curve. The following table provides well and aquifer parameters assumed for the plots (unless otherwise noted):

 Pumping (Control) Well constant discharge rate = 0.002 m3/min duration of pumping = 1440 min casing radius = 0.1 m well radius = 0.1 m depth to top of screen = 5 m screen length = 5 m Piezometer radial distance = 3.16 m depth = 0.75 m Aquifer thickness = 10 m vertical-to-horizontal anisotropy = 0.5

format: plain


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## Hensel’s lemma and a classification theoremSeptember 2, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
Tags: , ,
Let ${R}$ be a complete DVR with maximal ideal ${\mathfrak{m}}$ and quotient field ${F}$. We let ${k:=R/\mathfrak{m}}$; this is the residue field and is, e.g., the integers mod ${p}$ for the ${p}$-adic integers (I will discuss this more later).
Theorem 1 Suppose ${k}$ is of characteristic zero. Then ${R \simeq k[[X]]}$, the power series ring in one variable, with respect to the usual discrete valuation on ${k[[X]]}$. (more…)


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# Closed system piston cylinder device problem

1. Dec 2, 2009

### ricof

1. The problem statement, all variables and given/known data

A closed system comprising a cylinder and frictionless piston contains 1kg of a perfect gas of which molecular mass is 26. The piston is loaded so that the pressure is constant at 200kPa. Heat is supplied causing the gas to expand from 0.5m^3 to 1m^3. Calculate heat supplied

2. Relevant equations

w = int-(PdV)

PV = nRT

q = du - w

3. The attempt at a solution

I have worked out w as 100kJ by doing PdV = 200kPa x 0.5

Oh and Cp = 1.08 kJ/kg/K

Now am stuck.

2. Dec 2, 2009

### Andrew Mason

How did you get this? What is Oh?

You can determine the beginning and ending temperatures of the gas using the ideal gas equation.

Then determine the change in internal energy of the gas: $\Delta U = nC_v\Delta T$. Add that to the work done (which you have found) and that will give you the heat flow.

OR, as this is a process at constant pressure, you can simply use $$Q = nC_p\Delta T$$

Either way, however, you need to know the degrees of freedom of the gas. It is not monatomic as it would have to be iron. So it has either 2 or 3 degrees of freedom.

AM

3. Dec 2, 2009

### ricof

Sorry, 'Oh' as in the figure of speech.

So I use ideal gas law with the same P but different V each time to find the temperatures?

What does degree of freedom mean? I haven't heard of that before.

4. Dec 2, 2009

### Andrew Mason

Correct.

It is part of the kinetic theory of gases. See: http://en.wikipedia.org/wiki/Kinetic_theory

A monatomic gas has 3 degrees of freedom (of translational motion). A diatomic molecule has 5 degrees of freedom (because it can also spin on two perpendicular axes) and a polyatomic molecule has even more degrees of freedom (because it may be able to rotate on 3 axes and may also vibrate in different ways depending on its structure). Energy absorbed by a monatomic gas all goes into translational energy (which directly increases temperature and pressure). Energy absorbed by diatomic and polyatomic gases goes into rotational and vibrational energy, which does not affect temperature or pressure. So the number of degrees of freedom determines the heat capacity of the gas.

AM


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• ### Thermal Sunyaev-Zel'dovich effect in the intergalactic medium with primordial magnetic fields(1705.10054)

Dec. 24, 2018 hep-ph, astro-ph.CO, astro-ph.GA
The presence of ubiquitous magnetic fields in the universe is suggested from observations of radiation and cosmic ray from galaxies or the intergalactic medium (IGM). One possible origin of cosmic magnetic fields is the magnetogenesis in the primordial universe. Such magnetic fields are called primordial magnetic fields (PMFs), and are considered to affect the evolution of matter density fluctuations and the thermal history of the IGM gas. Hence the information of PMFs is expected to be imprinted on the anisotropies of the cosmic microwave background (CMB) through the thermal Sunyaev-Zel'dovich (tSZ) effect in the IGM. In this study, given an initial power spectrum of PMFs as $P(k)\propto B_{\rm 1Mpc}^2 k^{n_{B}}$, we calculate dynamical and thermal evolutions of the IGM under the influence of PMFs, and compute the resultant angular power spectrum of the Compton $y$-parameter on the sky. As a result, we find that two physical processes driven by PMFs dominantly determine the power spectrum of the Compton $y$-parameter; (i) the heating due to the ambipolar diffusion effectively works to increase the temperature and the ionization fraction, and (ii) the Lorentz force drastically enhances the density contrast just after the recombination epoch. These facts result in making the tSZ angular power spectrum induced by the PMFs more remarkable at $\ell >10^4$ than that by galaxy clusters even with $B_{\rm 1Mpc}=0.1$ nG and $n_{B}=-1.0$ because the contribution from galaxy clusters decreases with increasing $\ell$. The measurement of the tSZ angular power spectrum on high $\ell$ modes can provide the stringent constraint on PMFs.
• ### Planck 2015 constraints on the non-flat XCDM inflation model(1710.03271)

Oct. 27, 2018 hep-th, hep-ph, gr-qc, astro-ph.CO
We study Planck 2015 cosmic microwave background (CMB) anisotropy data using the energy density inhomogeneity power spectrum generated by quantum fluctuations during an early epoch of inflation in the non-flat XCDM model. Here dark energy is parameterized using a fluid with a negative equation of state parameter but with the speed of fluid acoustic inhomogeneities set to the speed of light. We use this simple parameterization of dynamical dark energy, that is relatively straightforward to use in a computation, in a first attempt to gain some insight into how dark energy dynamics and non-zero spatial curvature jointly affect the CMB anisotropy data constraints. Unlike earlier analyses of non-flat models, we use a physically consistent power spectrum for energy density inhomogeneities. We find that the Planck 2015 data in conjunction with baryon acoustic oscillation measurements are reasonably well fit by a closed XCDM model in which spatial curvature contributes a percent of the current cosmological energy density budget. In this model, the measured Hubble constant and non-relativistic matter density parameter are in good agreement with values determined using most other data. Depending on parameter values, the closed XCDM model has reduced power, relative to the tilted, spatially-flat $\Lambda$CDM case, and appears to partially alleviate the low multipole CMB temperature anisotropy deficit and can help partially reconcile the CMB anisotropy and weak lensing $\sigma_8$ constraints, at the expense of somewhat worsening the fit to higher multipole CMB temperature anisotropy data. However, the closed XCDM inflation model does not seem to improve the agreement much, if at all, compared to the closed $\Lambda$CDM inflation case, even though it has one more free parameter. Our results are interesting but tentative; a more thorough analysis is needed to properly gauge their significance.
• ### Planck 2015 constraints on the non-flat $\Lambda$CDM inflation model(1707.03452)

July 25, 2018 hep-th, hep-ph, gr-qc, astro-ph.CO
We study Planck 2015 cosmic microwave background (CMB) anisotropy data using the energy density inhomogeneity power spectrum generated by quantum fluctuations during an early epoch of inflation in the non-flat $\Lambda$CDM model. Unlike earlier analyses of non-flat models, which assumed an inconsistent power-law power spectrum of energy density inhomogeneities, we find that the Planck 2015 data alone, and also in conjunction with baryon acoustic oscillation measurements, are reasonably well fit by a closed $\Lambda$CDM model in which spatial curvature contributes a few percent of the current cosmological energy density budget. In this model, the measured Hubble constant and non-relativistic matter density parameter are in good agreement with values determined using most other data. Depending on parameter values, the closed $\Lambda$CDM model has reduced power, relative to the tilted, spatially-flat $\Lambda$CDM case, and can partially alleviate the low multipole CMB temperature anisotropy deficit and can help partially reconcile the CMB anisotropy and weak lensing $\sigma_8$ constraints, at the expense of somewhat worsening the fit to higher multipole CMB temperature anisotropy data. Our results are interesting but tentative; a more thorough analysis is needed to properly gauge their significance.
• ### Planck 2015 constraints on spatially-flat dynamical dark energy models(1802.05571)

Feb. 14, 2018 hep-th, hep-ph, gr-qc, astro-ph.CO
We determine constraints on spatially-flat tilted dynamical dark energy XCDM and $\phi$CDM inflation models by analyzing Planck 2015 cosmic microwave background (CMB) anisotropy data and baryon acoustic oscillation (BAO) distance measurements. XCDM is a simple and widely used but physically inconsistent parameterization of dynamical dark energy, while the $\phi$CDM model is a physically consistent one in which a scalar field $\phi$ with an inverse power-law potential energy density powers the currently accelerating cosmological expansion. Both these models have one additional parameter compared to standard $\Lambda$CDM and both better fit the TT + lowP + lensing + BAO data than does the standard tilted flat-$\Lambda$CDM model, with $\Delta \chi^2 = -1.26\ (-1.60)$ for the XCDM ($\phi$CDM) model relative to the $\Lambda$CDM model. While this is a 1.1$\sigma$ (1.3$\sigma$) improvement over standard $\Lambda$CDM and so not significant, dynamical dark energy models cannot be ruled out. In addition, both dynamical dark energy models reduce the tension between the Planck 2015 CMB anisotropy and the weak lensing $\sigma_8$ constraints.
• ### Updated observational constraints on quintessence dark energy models(1801.09446)

Jan. 29, 2018 astro-ph.CO
The recent GW170817 measurement favors the simplest dark energy models, such as a single scalar field. Quintessence models can be classified in two classes, freezing and thawing, depending on whether the equation of state decreases towards $-1$ or departs from it. In this paper we put observational constraints on the parameters governing the equations of state of tracking freezing, scaling freezing and thawing models using updated data, from the Planck 2015 release, joint light-curve analysis and baryonic acoustic oscillations. Because of the current tensions on the value of the Hubble parameter $H_0$, unlike previous authors, we let this parameter vary, which modifies significantly the results. Finally, we also derive constraints on neutrino masses in each of these scenarios.
• ### Planck 2015 constraints on the non-flat $\phi$CDM inflation model(1712.08617)

Dec. 4, 2017 hep-ph, gr-qc, astro-ph.CO
We perform Markov chain Monte Carlo analyses to put constraints on the non-flat $\phi$CDM inflation model using Planck 2015 cosmic microwave background (CMB) anisotropy data and baryon acoustic oscillation distance measurements. The $\phi$CDM model is a consistent dynamical dark energy model in which the currently accelerating cosmological expansion is powered by a scalar field $\phi$ slowly rolling down an inverse power-law potential energy density. We also use a physically consistent power spectrum for energy density inhomogeneities in this non-flat model. We find that, like the closed-$\Lambda$CDM and closed-XCDM models, the closed-$\phi$CDM model provides a better fit to the lower multipole region of the CMB temperature anisotropy data compared to that provided by the tilted flat-$\Lambda$CDM model. Also, like the other closed models, this model reduces the tension between the Planck and the weak lensing $\sigma_8$ constraints. However, the higher multipole region of the CMB temperature anisotropy data are better fit by the tilted flat-$\Lambda$ model than by the closed models.
• The Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) is a three-layered imaging survey aimed at addressing some of the most outstanding questions in astronomy today, including the nature of dark matter and dark energy. The survey has been awarded 300 nights of observing time at the Subaru Telescope and it started in March 2014. This paper presents the first public data release of HSC-SSP. This release includes data taken in the first 1.7 years of observations (61.5 nights) and each of the Wide, Deep, and UltraDeep layers covers about 108, 26, and 4 square degrees down to depths of i~26.4, ~26.5, and ~27.0 mag, respectively (5sigma for point sources). All the layers are observed in five broad bands (grizy), and the Deep and UltraDeep layers are observed in narrow bands as well. We achieve an impressive image quality of 0.6 arcsec in the i-band in the Wide layer. We show that we achieve 1-2 per cent PSF photometry (rms) both internally and externally (against Pan-STARRS1), and ~10 mas and 40 mas internal and external astrometric accuracy, respectively. Both the calibrated images and catalogs are made available to the community through dedicated user interfaces and database servers. In addition to the pipeline products, we also provide value-added products such as photometric redshifts and a collection of public spectroscopic redshifts. Detailed descriptions of all the data can be found online. The data release website is https://hsc-release.mtk.nao.ac.jp/.
• ### Subaru High-z Exploration of Low-Luminosity Quasars (SHELLQs). II. Discovery of 32 Quasars and Luminous Galaxies at 5.7 < z < 6.8(1704.05854)

June 9, 2017 astro-ph.CO, astro-ph.GA
We present spectroscopic identification of 32 new quasars and luminous galaxies discovered at 5.7 < z < 6.8. This is the second in a series of papers presenting the results of the Subaru High-z Exploration of Low-Luminosity Quasars (SHELLQs) project, which exploits the deep multi-band imaging data produced by the Hyper Suprime-Cam (HSC) Subaru Strategic Program survey. The photometric candidates were selected by a Bayesian probabilistic algorithm, and then observed with spectrographs on the Gran Telescopio Canarias and the Subaru Telescope. Combined with the sample presented in the previous paper, we have now identified 64 HSC sources over about 430 deg2, which include 33 high-z quasars, 14 high-z luminous galaxies, 2 [O III] emitters at z ~ 0.8, and 15 Galactic brown dwarfs. The new quasars have considerably lower luminosity (M1450 ~ -25 to -22 mag) than most of the previously known high-z quasars. Several of these quasars have luminous (> 10^(43) erg/s) and narrow (< 500 km/s) Ly alpha lines, and also a possible mini broad absorption line system of N V 1240 in the composite spectrum, which clearly separate them from typical quasars. On the other hand, the high-z galaxies have extremely high luminosity (M1450 ~ -24 to -22 mag) compared to other galaxies found at similar redshift. With the discovery of these new classes of objects, we are opening up new parameter spaces in the high-z Universe. Further survey observations and follow-up studies of the identified objects, including the construction of the quasar luminosity function at z ~ 6, are ongoing.
• ### Cosmological constraints on scalar-tensor gravity and the variation of the gravitational constant(1702.00742)

May 8, 2017 hep-th, gr-qc, astro-ph.CO
We present cosmological constraints on the scalar-tensor theory of gravity by analyzing the angular power spectrum data of the cosmic microwave background obtained from the Planck 2015 results together with the baryon acoustic oscillations (BAO) data. We find that the inclusion of the BAO data improves the constraints on the time variation of the effective gravitational constant by more than $10\%$, that is, the time variation of the effective gravitational constant between the recombination and the present epochs is constrained as $G_{\rm rec}/G_0-1 <1.9\times 10^{-3}\ (95.45\%\ {\rm C.L.})$ and $G_{\rm rec}/G_0-1 <5.5\times 10^{-3}\ (99.99 \%\ {\rm C.L.})$. We also discuss the dependence of the constraints on the choice of the prior.
• ### The FMOS-COSMOS survey of star-forming galaxies at $z\sim1.6$. V: Properties of dark matter halos containing H$\alpha$ emitting galaxies(1703.08326)

March 24, 2017 astro-ph.GA
We study the properties of dark matter halos that contain star-forming galaxies at $1.43 \le z \le 1.74$ using the FMOS-COSMOS survey. The sample consists of 516 objects with a detection of the H$\alpha$ emission line, that represent the star-forming population at this epoch having a stellar mass range of $10^{9.57}\le M_\ast/M_\odot \lesssim 10^{11.4}$ and a star formation rate range of $15\lesssim \mathrm{SFR}/(M_\odot \mathrm{yr^{-1}}) \lesssim 600$. We measure the projected two-point correlation function while carefully taking into account observational biases, and find a significant clustering amplitude at scales of $0.04$-$10~h^{-1}~\mathrm{cMpc}$, with a correlation length $r_0 = 5.21^{+0.70}_{-0.67}~h^{-1}~\mathrm{cMpc}$ and a bias $b=2.59^{+0.41}_{-0.34}$. We interpret our clustering measurement using a halo occupation distribution model. The sample galaxies appear to reside in halos with mass $M_\mathrm{h} = 4.6^{+1.1}_{-1.6}\times10^{12}~h^{-1}M_\odot$ on average that will likely become present-day halos of mass $M_\mathrm{h} (z=0) \sim2\times10^{13}~h^{-1}M_\odot$, equivalent to the typical halo mass scale of galaxy groups. We then confirm the decline of the stellar-to-halo mass ratio at $M_\mathrm{h}<10^{12}~M_\odot$, finding $M_\ast/M_\mathrm{h} \approx 5\times10^{-3}$ at $M_\mathrm{h}=10^{11.86}~M_\odot$, which is lower by a factor of 2-4 than those measured at higher masses. Finally, we use our results to illustrate the future capabilities of Subaru's Prime-Focus Spectrograph, a next-generation instrument that will provide strong constraints on the galaxy-formation scenario by obtaining precise measurements of galaxy clustering at $z>1$.
• ### Subaru high-z exploration of low-luminosity quasars (SHELLQs). I. Discovery of 15 quasars and bright galaxies at 5.7 < z < 6.9(1603.02281)

Sept. 5, 2016 astro-ph.CO, astro-ph.GA
We report the discovery of 15 quasars and bright galaxies at 5.7 < z < 6.9. This is the initial result from the Subaru High-z Exploration of Low-Luminosity Quasars (SHELLQs) project, which exploits the exquisite multiband imaging data produced by the Subaru Hyper Suprime-Cam (HSC) Strategic Program survey. The candidate selection is performed by combining several photometric approaches including a Bayesian probabilistic algorithm to reject stars and dwarfs. The spectroscopic identification was carried out with the Gran Telescopio Canarias and the Subaru Telescope for the first 80 deg2 of the survey footprint. The success rate of our photometric selection is quite high, approaching 100 % at the brighter magnitudes (zAB < 23.5 mag). Our selection also recovered all the known high-z quasars on the HSC images. Among the 15 discovered objects, six are likely quasars, while the other six with interstellar absorption lines and in some cases narrow emission lines are likely bright Lyman-break galaxies. The remaining three objects have weak continua and very strong and narrow Ly alpha lines, which may be excited by ultraviolet light from both young stars and quasars. These results indicate that we are starting to see the steep rise of the luminosity function of z > 6 galaxies, compared with that of quasars, at magnitudes fainter than M1450 ~ -22 mag or zAB ~24 mag. Follow-up studies of the discovered objects as well as further survey observations are ongoing.
• ### Planck constraints on scalar-tensor cosmology and the variation of the gravitational constant(1602.00809)

May 30, 2016 hep-th, gr-qc, astro-ph.CO
Cosmological constraints on the scalar-tensor theory of gravity by analyzing the angular power spectrum data of the cosmic microwave background (CMB) obtained from the Planck 2015 results are presented. We consider the harmonic attractor model, in which the scalar field has a harmonic potential with curvature ($\beta$) in the Einstein frame and the theory relaxes toward the Einstein gravity with time. Analyzing the {\it TT}, {\it EE}, {\it TE} and lensing CMB data from Planck by the Markov chain Monte Carlo method, we find that the present-day deviation from the Einstein gravity (${\alpha_0}^2$) is constrained as ${\alpha_0}^2<2.5\times10^{-4-4.5\beta^2}\ (95.45\% {\rm\ C.L.})$ and ${\alpha_0}^2<6.3\times10^{-4-4.5\beta^2}\ (99.99\%\ {\rm C.L.})$ for $0<\beta<0.4$. The time variation of the effective gravitational constant between the recombination and the present epochs is constrained as $G_{\rm rec}/G_0<1.0056\ (95.45\% {\rm\ C.L.})$ and $G_{\rm rec}/G_0<1.0115\ (99.99 \%{\rm\ C.L.})$. We also find that the constraints are little affected by extending to nonflat cosmological models because the diffusion damping effect revealed by Planck breaks the degeneracy of the projection effect.
• ### Reconstruction of CMB Temperature Anisotropies with Primordial CMB Induced Polarization in Galaxy Clusters(1603.06166)

March 20, 2016 astro-ph.CO
Scattering of cosmic microwave background (CMB) radiation in galaxy clusters induces polarization signals determined by the quadrupole anisotropy in the photon distribution at the location of clusters. This "remote quadrupole" derived from the measurements of the induced polarization in galaxy clusters provides an opportunity of reconstruction of local CMB temperature anisotropies. In this {\em Letter} we develop an algorithm of the reconstruction through the estimation of the underlying primordial gravitational potential, which is the origin of the CMB temperature and polarization fluctuations and CMB induced polarization in galaxy clusters. We found a nice reconstruction for the quadrupole and octopole components of the CMB temperature anisotropies with the assistance of the CMB induced polarization signals. The reconstruction can be an important consistency test on the puzzles of CMB anomaly, especially for the low quadrupole and axis of evil problems reported in WMAP and Planck data.
• ### Primordial magnetic fields from the string network(1601.01059)

Jan. 6, 2016 astro-ph.CO
Cosmic strings are a type of cosmic defect formed by a symmetry-breaking phase transition in the early universe. Individual strings would have gathered to build a network, and their dynamical motion would induce scalar--, vector-- and tensor--type perturbations. In this paper, we focus on the vector mode perturbations arising from the string network based on the one scale model and calculate the time evolution and the power spectrum of the associated magnetic fields. We show that the relative velocity between photon and baryon fluids induced by the string network can generate magnetic fields over a wide range of scales based on standard cosmology. We obtain the magnetic field spectrum before recombination as $a^2B(k,z)\sim4\times10^{-16}G\mu/((1+z)/1000)^{4.25}(k/{\rm Mpc}^{-1})^{3.5}$ Gauss on super-horizon scales, and $a^2B(k,z)\sim2.4\times10^{-17}G\mu/((1+z)/1000)^{3.5}(k/{\rm Mpc}^{-1})^{2.5}$ Gauss on sub-horizon scales in co-moving coordinates. This magnetic field grows up to the end of recombination, and has a final amplitude of approximately $B\sim10^{-17\sim -18} G\mu$ Gauss at the $k\sim1\ {\rm Mpc}^{-1}$ scale today. This field might serve as a seed for cosmological magnetic fields.
• ### Magnetic field spectrum at cosmological recombination revisited(1504.03790)

June 3, 2015 gr-qc, astro-ph.CO
If vector type perturbations are present in the primordial plasma before recombination, the generation of magnetic fields is known to be inevitable through the Harrison mechanism. In the context of the standard cosmological perturbation theory, non-linear couplings of first-order scalar perturbations create second-order vector perturbations, which generate magnetic fields. Here we reinvestigate the generation of magnetic fields at second-order in cosmological perturbations on the basis of our previous study, and extend it by newly taking into account the time evolution of purely second-order vector perturbations with a newly developed second-order Boltzmann code. We confirm that the amplitude of magnetic fields from the product-terms of the first-order scalar modes is consistent with the result in our previous study. However, we find, both numerically and analytically, that the magnetic fields from the purely second-order vector perturbations partially cancel out the magnetic fields from one of the product-terms of the first-order scalar modes, in the tight coupling regime in the radiation dominated era. Therefore, the amplitude of the magnetic fields on small scales, $k \gtrsim 10~h{\rm Mpc}^{-1}$, is smaller than the previous estimates. The amplitude of the generated magnetic fields at cosmological recombination is about $B_{\rm rec} =5.0\times 10^{-24}~{\rm Gauss}$ on $k = 5.0 \times 10^{-1}~h{\rm Mpc}^{-1}$. Finally, we discuss the reason of the discrepancies that exist in estimates of the amplitude of magnetic fields among other authors.
• ### Primordial magnetic fields from self-ordering scalar fields(1501.06304)

Jan. 26, 2015 astro-ph.CO
A symmetry-breaking phase transition in the early universe could have led to the formation of cosmic defects. Because these defects dynamically excite not only scalar and tensor type cosmological perturbations but also vector type ones, they may serve as a source of primordial magnetic fields. In this study, we calculate the time evolution and the spectrum of magnetic fields that are generated by a type of cosmic defects, called global textures, using the non-linear sigma (NLSM) model. Based on the standard cosmological perturbation theory, we show, both analytically and numerically, that a vector-mode relative velocity between photon and baryon fluids is induced by textures, which inevitably leads to the generation of magnetic fields over a wide range of scales. We find that the amplitude of the magnetic fields is given by $B\sim{10^{-9}}{((1+z)/10^3)^{-2.5}}({v}/{m_{\rm pl}})^2({k}/{\rm Mpc^{-1}})^{3.5}/{\sqrt{N}}$ Gauss in the radiation dominated era for $k\lesssim 1$ Mpc$^{-1}$, with $v$ being the vacuum expectation value of the O(N) symmetric scalar fields. By extrapolating our numerical result toward smaller scales, we expect that $B\sim {10^{-17}}((1+z)/1000)^{-1/2}({v}/{m_{\rm pl}})^2({k}/{\rm Mpc^{-1}})^{1/2}/{\sqrt{N}}$ Gauss on scales of $k\gtrsim 1$ Mpc$^{-1}$ at redshift $z\gtrsim 1100$. This might be a seed of the magnetic fields observed on large scales today.
• ### Impact of anisotropic stress of free-streaming particles on gravitational waves induced by cosmological density perturbations(1412.1081)

Jan. 8, 2015 gr-qc, astro-ph.CO
Gravitational waves (GWs) are inevitably induced at second-order in cosmological perturbations through non-linear couplings with first order scalar perturbations, whose existence is well established by recent cosmological observations. So far, the evolution and the spectrum of the secondary induced GWs have been derived by taking into account the sources of GWs only from the product of first order scalar perturbations. Here we newly investigate the effects of purely second-order anisotropic stresses of photons and neutrinos on the evolution of GWs, which have been omitted in the literature. We present a full treatment of the Einstein-Boltzmann system to calculate the spectrum of GWs with anisotropic stress based on the formalism of the cosmological perturbation theory. We find that photon anisotropic stress amplifies the amplitude of GWs by about $150 %$ whereas neutrino anisotropic stress suppress that of GWs by about $30 %$ on small scales $k\gtrsim 1.0 h{\rm Mpc}^{-1}$ compared to the case without anisotropic stress. The second order anisotropic stress does not affect GWs with wavenumbers $k\lesssim 1.0 h{\rm Mpc}^{-1}$. The result is in marked contrast with the case at linear order, where the effect of anisotropic stress is damping in amplitude of GWs.
• ### Implications of the B-mode Polarization Measurement for Direct Detection of Inflationary Gravitational Waves(1406.1369)

Oct. 9, 2014 hep-th, hep-ph, gr-qc, astro-ph.CO
The prospects for direct measurements of inflationary gravitational waves by next generation interferometric detectors inferred from the possible detection of B-mode polarization of the cosmic microwave background are studied. We compute the spectra of the gravitational wave background and the signal-to-noise ratios by two interferometric detectors (DECIGO and BBO) for large-field inflationary models in which the tensor-to-scalar ratio is greater than the order of 0.01. If the reheating temperature $T_{\rm RH}$ of chaotic inflation with the quadratic potential is high ($T_{\rm RH}>7.9\times10^6$ GeV for upgraded DECIGO and $T_{\rm RH}> 1.8\times 10^{6}$ GeV for BBO), it will be possible to reach the sensitivity of the gravitational background in future experiments at $3\sigma$ confidence level. The direct detection is also possible for natural inflation with the potential $V(\phi)=\Lambda^4 [1-\cos(\phi/f)]$, provided that $f>4.2 M_{\rm pl}$ (upgraded DECIGO) and $f>3.6 M_{\rm pl}$ (BBO) with $T_{\rm RH}$ higher than $10^8$ GeV. The quartic potential $V(\phi)=\lambda \phi^4/4$ with a non-minimal coupling $\xi$ between the inflaton field $\phi$ and the Ricci scalar $R$ gives rise to a detectable level of gravitational waves for $|\xi|$ smaller than the order of 0.01, irrespective of the reheating temperature.
• ### Probing large scale filaments with HI and $^3$HeII(1408.2450)

Aug. 11, 2014 astro-ph.CO, astro-ph.GA
We explore the observability of the neutral hydrogen (HI) and the singly-ionized isotope helium-3 ($^3$HeII) in the intergalactic medium (IGM) from the Epoch of Reionization down to the local Universe. The hyperfine transition of $^3$HeII, which is not as well known as the HI transition, has energy splitting corresponding to 8 cm. It also has a larger spontaneous decay rate than that of neutral hydrogen, whereas its primordial abundance is much smaller. Although both species are mostly ionized in the IGM, the balance between ionization and recombination in moderately high density regions renders them abundant enough to be observed. We estimate the emission signal of both hyperfine transitions from large scale filamentary structures and discuss the prospects for observing them with current and future radio telescopes. We conclude that HI in filaments is possibly observable even with current telescopes after 100 hours of observation. On the other hand, $^3$HeII is only detectable with future telescopes, such as SKA, after the same amount of time.
• ### Evolution of perturbations and cosmological constraints in decaying dark matter models with arbitrary decay mass products(1402.2972)

July 16, 2014 astro-ph.CO
Decaying dark matter (DDM) is a candidate which can solve the discrepancies between predictions of the concordance $\Lambda$CDM model and observations at small scales such as the number counts of companion galaxies of the Milky Way and the density profile at the center of galaxies. Previous studies are limited to the cases where the decay particles are massless and/or have almost degenerate masses with that of mother particles. Here we expand the DDM models so that one can consider the DDM with arbitrary lifetime and the decay products with arbitrary masses. We calculate the time evolutions of perturbed phase-space distribution functions of decay products for the first time and study effects of DDM on the temperature anisotropy in the cosmic microwave background and the matter power spectrum at present. From a recent observational estimate of $\sigma_{8}$, we derive constraints on the lifetime of DDM and the mass ratio between the decay products and DDM. We also discuss implications of the DDM model for the discrepancy in the measurements of $\sigma_8$ recently claimed by the Planck satellite collaboration.
• ### Cosmological signatures of tilted isocurvature perturbations: reionization and 21cm fluctuations(1311.3294)

Nov. 13, 2013 hep-ph, astro-ph.CO
We investigate cosmological signatures of uncorrelated isocurvature perturbations whose power spectrum is blue-tilted with spectral index 2<p<4. Such an isocurvature power spectrum can promote early formation of small-scale structure, notably dark matter halos and galaxies, and may thereby resolve the shortage of ionizing photons suggested by observations of galaxies at high redshifts (z~7-8) but that are required to reionize the universe at z~10. We mainly focus on how the formation of dark matter halos can be modified, and explore the connection between the spectral shape of CMB anisotropies and the reionization optical depth as a powerful probe of a highly blue-tilted isocurvature primordial power spectrum. We also study the consequences for 21cm line fluctuations due to neutral hydrogens in minihalos. Combination of measurements of the reionization optical depth and 21cm line fluctuations will provide complementary probes of a highly blue-tilted isocurvature power spectrum.
• ### The effect of primordial black holes on 21 cm fluctuations(1207.6405)

Oct. 16, 2013 astro-ph.CO
The 21 cm signal produced by non-evaporating primordial black holes (PBHs) is investigated. X-ray photons emitted by accretion of matter onto a PBH ionize and heat the intergalactic medium (IGM) gas near the PBH. Using a simple analytic model, we show that this X-ray heating can produce an observable differential 21 cm brightness temperature. The region of the observable 21 cm brightness temperature can extend to 1-10 Mpc comoving distance from a PBH depending on the PBH mass. The angular power spectrum of 21 cm fluctuations due to PBHs is also calculated. The peak position of the angular spectrum depends on PBH mass, while the amplitude is independent of PBH mass. Comparing this angular power spectrum with the angular power spectrum caused by primordial density fluctuations, it is found that both of them become comparable if $\Omega_{{\rm PBH}} = 10^{-11} (M/10^{3} M_\odot)^{-0.2}$ at $z=30$ and $10^{-12} (M/10^{3} M_\odot)^{-0.2}$ at $z=20$ for the PBH mass from $10 M_\odot$ to $10^8 M_\odot$. Finally we find that the Square Kilometer Array can detect the signal due to PBHs up to $\Omega_{\rm PBH}=10^{-5} (M/10^{3} M_\odot)^{-0.2}$ at $z=30$ and $10^{-7} (M/10^{3} M_\odot)^{-0.2}$ at $z=20$ for PBHs with mass from $10^2 M_\odot$ to $10^8 M_\odot$.
• ### Optimal constraint on g_NL from CMB(1303.4626)

Sept. 2, 2013 hep-ph, astro-ph.CO
An optimal method to constrain the non-linearity parameter g_NL of the local-type non-Gaussianity from CMB data is proposed. Our optimal estimator for g_NL is separable and can be efficiently computed in real space. Combining the exact filtering of CMB maps with the full covariance matrix, our method allows us to extract cosmological information from observed data as much as possible and obtain a tighter constraint on g_NL than previous studies. Applying our method to the WMAP 9-year data, we obtain the constraint g_NL = (-3.3 +- 2.2) 10^5, which is a few times tighter than previous ones. We also make a forecast for PLANCK data by using the Fisher matrix analysis.
• ### Generation of magnetic fields in Einstein-Aether gravity(1302.4189)

June 14, 2013 hep-th, hep-ph, gr-qc, astro-ph.CO
Recently the lower bounds of the intergalactic magnetic fields $10^{-16} \sim 10^{-20}$ Gauss are set by gamma-ray observations while it is unlikely to generate such large scale magnetic fields through astrophysical processes. It is known that large scale magnetic fields could be generated if there exist cosmological vector mode perturbations in the primordial plasma. The vector mode, however, has only a decaying solution in General Relativity if the plasma consists of perfect fluids. In order to investigate a possible mechanism of magnetogenesis in the primordial plasma, here we consider cosmological perturbations in the Einstein-Aether gravity model, in which the aether field can act as a new source of vector metric perturbations and thus of magnetic fields. We estimate the angular power spectra of temperature and B-mode polarization of the Cosmic Microwave Background (CMB) Anisotropies in this model and put a rough constraint on the aether field parameters from latest observations. We then estimate the power spectrum of associated magnetic fields around the recombination epoch within this limit. It is found that the spectrum has a characteristic peak at $k=0.1 h{\rm Mpc^{-1}}$, and at that scale the amplitude can be as large as $B\sim 10^{-22}$ Gauss where the upper bound comes from CMB temperature anisotropies. The magnetic fields with this amplitude can be seeds of large scale magnetic fields observed today if the sufficient dynamo mechanism takes place. Analytic interpretation for the power spectra is also given.
• ### Exploring the origin of the fine structures in the CMB temperature angular power spectrum(1211.3097)

Nov. 13, 2012 astro-ph.CO
The angular power spectrum of the cosmic microwave background (CMB) temperature anisotropies is a good probe to look into the primordial density fluctuations at large scales in the universe. Here we re-examine the angular power spectrum of the Wilkinson Microwave Anisotropy Probe data, paying particular attention to the fine structures (oscillations) at $\ell=100 \sim 150$ reported by several authors. Using Monte-Carlo simulations, we confirm that the gap from the simple power law spectrum is a rare event, about 2.5--3$\sigma$, if these fine structures are generated by experimental noise and the cosmic variance. Next, in order to investigate the origin of the structures, we examine frequency and direction dependencies of the fine structures by dividing the observed QUV frequency maps into four sky regions. We find that the structures around $\ell \sim 120$ do not have significant dependences either on frequencies or directions. For the structure around $\ell \sim 140$, however, we find that the characteristic signature found in the all sky power spectrum is attributed to the anomaly only in the South East region.


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## Friday, 13 June 2014

### Metaphysics as Über-theory and Metaphysics as Meta-theory, II

Though it is common for logicians to be a bit negative about metaphysics, I am very fond of metaphysics. I can trace the reason: I purchased a scruffy copy of W.V. Quine's From a Logical Point of View (2nd ed., 1961) from a second-hand shop in Hay-on-Wye, around 1987, containing Quine's essays -- "On what there is" and other essays on related themes, such as modality, reference, opacity, etc. I found "On what there is" so engrossing that I numbered each paragraph and learnt it by heart. A few years ago, I lent this copy to a close friend, but it was never returned (aleha hashalom).

In an older post, and to some extent tongue-in-cheek responding to some criticisms of analytic metaphysics, I listed a number of achievements in analytic metaphysics. Analytic metaphysics is so closely related to mathematics that one might simply confuse the two, but this is an error. There is massive overlap between analytic metaphysics and mathematics. This is why some responded to the list of achievements of metaphysics by saying "is this not just mathematics?". Well, they overlap, and when X and Y overlap, then saying (truly) something is X, one does not establish that it is not Y. Sometimes, the criticism of "analytic" metaphysics, as opposed to "naturalized" metaphysics, is ad hominem, directed not so much of analytic metaphysics, but rather of analytic metaphysicians; they are theorists and not experimentalists, and they are bad theorists, because their knowledge of (empirical) science does not go beyond "A Level Chemistry". The important criticisms I see are: an epistemological criticism (how might knowledge of the relevant kind even be possible, entirely by a priori "armchair" reasoning?); a competence criticism ("A-Level chemistry"); an irrelevance criticism ("what a waste of time"). I don't know the answer to the first, but then no one knows how mathematical knowledge is possible, and yet mathematical knowledge exists.

It's fair to say that there is more weight in the "competence cricitism" of some modern metaphysicians, as one might call it. By and large, David Lewis tends to have a very classical picture of the (actual!!) world, with "lumps of stuff" at spacetime points (and regions), and perhaps the criticisms made against this is fair. However, one must be careful about stones and glass houses. There is some physics in Every Thing Must Go but not much: for example, no detailed computation of the electronic orbitals of a hydrogen atom using separation of variables in the Schroedinger equation, or of the Schwarzschild metric in GR, or of the properties of gases, or calculations of Clebsch-Gordan coefficients, etc. And what there is there includes a mistaken formulation of Ehrenfest's Theorem, as explained here: in the book, the equation given (twice) has the quantum inner product brackets (viz., expressions of the form $\langle \psi \mid \hat{O} \mid \psi \rangle$) misplaced in the equation.

But the basic point here is still unfair to those criticized by the "competence criticism", even if there's some legitimacy to the criticism. It is extremely difficult for someone whose specialization is metaphysics - but has not studied, say, theoretical physics or mathematics to graduate level - to acquire a detailed understanding of what can, and cannot, be said fruitfully about, say, the (alleged) implications of QM or GR. As an example of this, there's an (unpublished) article on Leibniz equivalence, and I gave it as a talk perhaps six times now, with physicists and philosophers of physics; audience response is this; 50% say it's obviously wrong and 50% say it's obviously right.

Since Frege, Russell, Wittgenstein, Carnap et al. were not experimentalists, it must be that whatever progress they made, if any at all, they must have made as theorists, and yes, in their armchair (or deckchair, for Wittgenstein). It is hard to see how an experiment might help me understand, for example, the semantics of sentences about fictional objects or possible worlds or transfinite cardinals. I may analyse the semantic content of, say, "Scott is the author of Waverley" or "I buttered the toast with a knife"; or I may try to analyse the Dirac equation. Or I may analyse, "You are almost as interesting as Sherlock Holmes is", in which a real-life person is compared with a fictional character. Consequently, "analytic" refers to a method, not to any specific content. The content is unconstrained: it may be possible words, fictional objects, moral values, topological field theories, transfinite sets, the unit of selection debate, etc., etc.

In an older M-Phi post, Metaphysics as Über-theory and Metaphysics as Meta-theory I suggested one could think of metaphysics in two ways:
• Metaphysics as über-theory (think - Plato).
• Metaphysics as meta-theory (think - Aristotle).
One could probably run through any piece of work classifiable as "metaphysics" and identify which bits are über-theoretic and which bits are meta-theoretic. It is Pythagorean über-theory that "all things are numbers" and it is Aristotelian meta-theory to say that "to say of what is, that it is, is true". So, as I want to use this word, in über-theory, one attempts an overall picture of "how things ... hang together". That is,
"The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term” (Sellars, 1962, "Philosophy and the Scientific Image of Man")
Sellars says this of philosophy in general, but I think it is an overestimate. Philosophers should be able to work on small problems without feeling the intellectual burden, a somewhat pretentious one too, of trying to understand how "things hang together". For example, I don't think Russell's "On Denoting" (1905) fits this picture at all, but I do think his Principles of Mathematics (1903) or "Philosophy of Logical Atomism" (1918-19) do.

For example, when Prof. Max Tegmark suggests that physics is ultimately mathematics, then that claim is an example of über-theory. I think this somehow neglects the modal contingency of concrete entities, that "how the concreta are" changes from world to word; and that rather it is mathematics that is ultimately physics -- the physics of modally invariant objects -- then that's über-theory too. If you like, mathematics is the physics of modality. The idea is that purely abstract entities, like $\pi$, $\aleph_0$ and $SU(3)$, don't modally change their relationships to one another as we let the worlds vary. There is no world "in" which $3 < 2$, $(\omega, <)$ is not wellordered, or $e^{i \pi} + 1 \neq 0$. Purely abstract objects are not even "in" possible worlds at all. The distinction between physics and mathematics is not, I think, connected to how knowledge of their objects is acquired, but is connected to the fact that relations amongst purely abstract entities (e.g., how $\omega$ is related to any $n \in \omega$) are fixed and invariant ("Being"in Plato's terminology), whereas the relations of concreta, such as e.g., Blackpool Tower and the Eiffel Tower, are a matter of change ("Becoming", in Plato's terminology). For example, at the moment, the Eiffel Tower (a concretum) is higher than the Blackpool Tower (a concretum), but this temporary French advantage over the British could, of course, be remedied by an "accident" (Team America; World Police, 4:15).

Russell's Principles of Mathematics contains a great deal of über-theory and meta-theory, but his "On Denoting" is a classic of meta-theory. Meta-theory was the central focus of Rudolf Carnap's Der logische Aufbau (1928). The logical apparatus for doing meta-theory had blossomed with the publication of Frege's Begriffsschrift (1879) and then was amplified in his later writings on semantics and applied to the case of the foundations of arithmetic in Die Grundlagen der Arithmetik (1884). Bertrand Russell joined this revolution against German idealism in 1899, after attending a conference at which Peano was present. Russell's friend, G.E. Moore, was part of this rebellion too, although not a logician. A decade later, on the advice of Frege, a young Austrian, Ludwig Wittgenstein, spent a year and half visiting Russell at Trinity College. In general, and in practice, all published work in metaphysics does both. Meta-theoretic work in metaphysics has no serious objection to it, aside perhaps from mild "competence" accusations of insufficient expertise in difficult parts of, let's say, mathematical logic or theoretical physics. This work appears alongside the work of other logicians, mathematics and computer scientists (and sometimes cognitive scientists), often in the same journals. Aside from the usual internecine waffle and squabbles -- e.g., about one's favourite "logic", etc. -- there is no deep disagreement as to methods and also as to the genuine progress that is made. For example, Michael Clark once showed me, on a blackboard in 2000, a paradox, involving an infinite list of sentences, and I was struck by how one might make it precise. When I went home and did that, I discovered that the infinite set (simplifying notation quite bit),
$\{Y(n) \leftrightarrow \forall x>n \neg T(Y(x)) \mid n \in \mathbb{N} \} \cup \{T(\phi) \leftrightarrow \phi \mid \phi \in L\}$
actually had a model: a non-standard model. That's progress and I did no "experiment".

With über-theory, it is different. For how can a priori reflection, from the armchair, tell us "how everything hangs together". Surely, that task is for the empirical scientist, and, in the end, for the physicist. In other words, it seems utterly pretentious for a metaphysician to even insinuate some ability to discover "how things hang together". I agree. Well, sort of. There are two main lines of response. The first is that while it is true that the armchair metaphysician is not performing experiments on neutrinos or gravitational waves (and neither of course is the mathematician or theoretical scientist), the armchair metaphysician is going to have, and should be expected to have, some degree of knowledge and acquaintance with science - with mathematics, with formal parts of linguistics and computer science, with parts of physics, chemistry, biology and psychology (cognitive science, more broadly). But this is material for analysis. It is not therefore a direct attempt to find out how "everything hangs together", but an attempt to see how our best scientific theories (or even how our discourse in general) depict "how things hang together". A second response focuses on what these "things" might be in "how things hang together"? The scientist may be interested in galaxies or ganglia; for the metaphysician, there also is a more or less canonical list of the kinds of things one are interested in: properties, relations, quantities, abstract entities and structures, formal systems, moral values, propositionspieces of discourse, possible worlds and fictional entities.


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### Home > AC > Chapter 14 > Lesson 14.3.1.2 > Problem3-29

3-29.

Factor each expression completely.

1. $x^{2} − 49$

$\left(x + 7\right)\left(x − 7\right)$

1. $4x^{2} − 1$

1. $x^{2}y^{2 }− 81z^{2}$

$\left(xy + 9z\right)\left(xy − 9z\right)$

1. $2x^{3} − 8x$

Remove the common factor and then factor the quadratic.