Scale factor Definition and 62 Threads

A scale factor is usually a decimal which scales, or multiplies, some quantity. In the equation y = Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x. For example, doubling distances corresponds to a scale factor of two for distance, while cutting a cake in half results in pieces with a scale factor for volume of one half. The basic equation for it is image over preimage.
In the field of measurements, the scale factor of an instrument is sometimes referred to as sensitivity. The ratio of any two corresponding lengths in two similar geometric figures is also called a scale.

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  1. H

    I Boltzmann equation for annihilation

    In the Dodelson's textbook, the author introduce the Boltzmann equation for annihilation. ##a^{-3} \frac{d(n_1 a)}{dt} = \int \frac{d^3 p_1}{(2 \pi)^3 2E_1} \int \frac{d^3 p_2}{(2 \pi)^3 2E_2} \int \frac{d^3 p_3}{(2 \pi)^3 2E_3} \int \frac{d^3 p_4}{(2 \pi)^3 2E_4} \times (2 \pi)^4 \delta^3 (p_1...
  2. AHSAN MUJTABA

    I Dimensions of Cosmic Scale Factor ##a(t)##

    I know the formula for Hubble's parameter, ##\frac{\dot{a}}{a}##, but I cannot infer any dimension of ##a(t)## from it. Please guide me. Thanks.
  3. George Keeling

    I Is Scale Factor a Scalar? Sean Carroll Invitation

    Is the scale factor a scalar? I think that the answer is no but I want to check because god (or the universe) has been playing tricks on me... At Sean Carroll's invitation I wanted to check that the tensor$$ K_{\mu\nu}=a^2\left(g_{\mu\nu}+U_\mu U_\nu\right) $$was a Killing tensor...
  4. S

    MHB What is the ratio between sides of two triangles with lengths 6 and 15?

    Can someone please help me with these questions? I would be extremely thankful!
  5. Arman777

    I Is Scale Factor Continuity and Differentiability Necessary?

    Does scale factor must be continuous(Con.) and differentiable (Diff.) ? Or can it be one of them or neither ? Physically one expects it to be Con. and Diff. but is there a more rigorous proof. And as a separate question, if ##\dot{a}## is not continuous/differentiable in some case, does that...
  6. AHSAN MUJTABA

    Scale Factor & ##\Omega##: Finding the Relation

    I am trying to develop a relation between scale factor (a(t)) and ##\Omega##. The relation came out to be evolve as ##\Omega_i=\Omega_io * a^{-n}## but my graph isn't right it's giving values of ##a(t)## to higher extent. I consulted my instructor he only added that I should include ##H_o##...
  7. M

    MHB Find the scale factor of triangle ABC to triangle DEF

    Triangles ABC and DEF are similar. Triangle ABC has a perimeter of 16cm. Triangle DEF has side of 6cm, 8cm and 10cm. What is the scale factor of triangle ABC to triangle DEF? A. 1/2 B. 1/3 C. 2/3 D. 3/2 E. 2/1 I concluded the answer is D. Am I correct?
  8. Arman777

    What is the Derivative of the Scale Factor in Cosmology?

    In cosmology we have a scale factor that depends only on time ##a(t)##. Now how can I solve this thing $$\frac{d}{da}(\dot{a}(t)^{-2}) = ?$$ Is it 0 ? Or something else ?
  9. naushaan

    I Interpreting a scale factor vs. distance graph

    'Imagine that you live in a different universe, which may have a different cosmology to our own. You measure the distances to and redshifts of a large number of Type Ia supernovae, and you use the redshifts to calculate the scale-factor of the universe at the time when the supernova exploded...
  10. TheMercury79

    I Scale factor from Friedmann's equations

    If we take a flat universe dominated by radiation, the scale factor is ##a(t)=t^{1/2}## which can be derived from the first Friedmann Equation:$$(\dot a/a)^2 = \frac{8\pi G}{3c^2}\varepsilon(t)-\frac{kc^2}{R_0^2 a(t)^2}$$ But suppose I want to show this using the second Friedmann Equation (Also...
  11. TheMercury79

    I Hubble relation to Scale Factor

    Imagine a Universe where the Hubble parameter is truly a constant, in both space and time. How much smaller would such a Universe be 14 billion years ago compared to today? Using the Hubble parameter in terms of scale factor: ##H(t) = \frac{\dot{a}}{a}## leads to the differential equation...
  12. T

    I Question on the Derivation of Temperature/Scale Factor Relation

    In 'Introduction to Cosmology' by Barbara Ryden, there is an argument made using the first law of thermodynamics to derive the relation T(t) ∝ a(t)-1 on pages 29 and 30. MENTOR NOTE: removed copyrighted material. I've been able to work out all the omitted details up to 2.37, which gives the...
  13. Arman777

    I What Does Redshift and Scale Factor Tell Us About the Size of the Universe?

    We can define the relationship between ##z## and ##a(t_e)## as, $$1+z=\frac {a(t_0)=1} {a(t_e)}$$ When we assume ##z=2##, it means that ##a(t_e)=\frac {1} {3}## Is this means that universe was ##\frac {1} {3}## times smaller then now ? If its the case then let's suppose ##z=6## which means...
  14. G

    I Scale Factors: Is a 10x Bigger Earth Possible?

    Hello, Not really sure where this belongs (more sci fi) But, suppose the Earth were 10 times bigger. Would it be possible for everything on Earth to scale with this increase - Earth systems, land masses, trees etc (but also remain unchanged to us (except everything everything would be bigger)...
  15. J

    Cosmology - determining if a model universe would recollapse

    Homework Statement Show mathematically that a model with: Ω_M0 = 3 Ω_Λ0 = 0.01 Ω_R0 = 0 Ω_T0 = 3.01 is a model that re-collapses in the future. Be certain to indicate at what value of the scale factor 'a' the expansion reverses and becomes contraction. Homework Equations It's hinted pretty...
  16. Mictlantecuhtli

    Expressing surface charge density as volume charge density

    I'm working on this: When I consider a disc with radius ##a## and total charge ##Q## uniformly distributed (placed in the XY plane and centered at the origin) and determine the volume charge density in cylindrical coordinates, I have assumed is of the form ##\rho=A \delta (z) U(R-r)##, (##U## is...
  17. M

    I Object moving out of the Observable Universe due to Expansion

    With the LDCM, cosmological constant, model I understand that the scale factor of the Universe grows more rapidly than the Horizon. I believe the correct horizon I need to be considering is the Hubble Horizon and the point when objects recessional velocity hits the speed of light they disappear...
  18. C

    I Where the scale factor a(t) appears in the metric

    Hello, I was enjoying Zee's book on GR when I noticed the location of this "a(t)" thing in the metric sound quite disturbing to me. BTW: I experience the same annoyance and went down to the same conclusions, when I watched a related Theoretical Minimum lesson...Here's the setup, the flat...
  19. T

    MHB Solve Scale Factor Problem: 'ET Pizza' Pizzas

    Dear friends, I am unable to solve the following scale factor problem. Will appreciate your help here. Thanks in advance. 'ET Pizza' produces two pizzas that are similar in shape. The smaller pizza is 20 cm in diameter and costs $10. The larger pizza is 30 cm in diameter. What is a fair cost...
  20. Elnur Hajiyev

    A Hubble parameter vs Scale factor's derivative

    How does Hubble parameter and scale factor's derivative differ geometrically? I am reading S. Caroll's GR book. But I cannot get the full representation of these two parameters. On the book, it says How can \dot{H} and \ddot{a} be opposite of each other on the same instance if both are...
  21. S

    Scale factor of special conformal transformation

    Homework Statement (From Di Francesco et al, Conformal Field Theory, ex .2) Derive the scale factor Λ of a special conformal transformation. Homework Equations The special conformal transformation can be written as x'μ = (xμ-bμ x^2)/(1-2 b.x + b^2 x^2) and I need to show that the metric...
  22. S

    Density parameter against scale factor for perfect fluids

    Homework Statement This is a basic cosmology problem. The Friedmann equations are ##\Big( \frac{\dot{a}}{a}\Big)^{2}+\frac{k}{a^{2}}=\frac{8\pi}{3m_{Pl}^{2}}\rho## and ##\Big( \frac{\ddot{a}}{a} \Big) = - \frac{4\pi}{3m_{Pl}^{2}}(\rho + 3p)##. Using the density parameter ##\Omega \equiv...
  23. F

    I Hubble parameter in terms of the scale factor

    From Introduction to Cosmology by Matt Roos, he wanted to derive the Hubble parameter in terms of the scale factor. From the Friedmann's equation, ##\frac{R'^2 + kc^2}{R^2} = \frac{8πG}{3}ρ## The density parameter is ##~Ω(a) = \frac{8πG}{3H_o^2}ρ(a)~## and let ##~Ω_k = \frac{-kc^2}{H_o^2}##...
  24. P

    I Scale factor/redshift formula wrong at the end?

    In this video: The professor at the end (at about 7:28), used the formula for scale factor and redshift as a(t) = 1/z, instead of the actual one a(t) = 1/1+z. And when we apply both of them, they give very different results. So, how could the professor use the first formula, which we were...
  25. thecourtholio

    Angular diameter distance to surface of last scattering

    Homework Statement 1) Calculate the angular diameter distance to the last scattering surface in the following cosmological models: i) Open universe, ΩΛ= 0.65, Ωm = 0.30 ii) Closed universe, ΩΛ = 0.75, Ωm = 0.30 ii) Flat universe, ΩΛ = 0.75, Ωm = 0.25 Describe how the CMB power spectrum...
  26. T

    Time dependence of the Ratio Hubble length to scale factor

    Let's assume a universe like ours which after inflation expands decelerated and accelerated thereafter. How will the ratio Hubble length ##1/H## to scalefactor ##a## evolve over time? And how could one calculate the time dependence of this ratio. Any help appreciated.
  27. S

    Negative scale factor RW metric with scalar field

    Homework Statement The aim is to find a solution for the scale factor in a Robertson Walker Metric with a scalar field and a Lagrange multiplier. Homework Equations I have this action S=-\frac{1}{2}\int...
  28. M

    MHB Calculate % Increase from Scale Factor: Stumped? Help Here!

    Hi guys, This question has me a bit stumped and I can't seem to find any methods on Google that would help me solve this: The last part of the question asks me to find the value of the scale factor, which in this equation I believe is 1.8. I know how to calculate percentage increases the...
  29. U

    How do I find the scale factor of cosmological constant?

    Homework Statement (a)Sketch how the contributions change with time (b)For no cosmological constant, how long will this universe exist? (c)How far would a photon travel in this metric? (d)Find particular density ##\rho_E## and scale factor (e)How would this universe evolve?[/B] Homework...
  30. M

    How Is the Scale Factor R(t)/R(0) Calculated in Cosmology?

    Homework Statement The energy density of the universe for radiation, matter and cosmological constant have changed over the years and there was a time (t), when it was equal for matter and radiation. Assuming the universe is 13.7 billion years old, estimate R(t) / R(0) where R(0) is the...
  31. ognik

    MHB General orthogonal scale factor identity

    Please be patient as I struggle with latex here ... Part 1 of the problem says to start with: $ \frac{\partial\bar{r}}{\partial{q}_{1}} ={h}_{1} \hat{q}_{1} $ and then to find an expression for $ {h}_{1} $ that agrees with $ {g}_{ij}=\sum_{l}...
  32. P

    Calibration cosmic distance scale on age globular clusters

    Hi, Could someone enlighten me in this matter? . how the calibration of cosmic distance scale affects the determination of the age of globular star clusters? thanks a lot :)
  33. P

    Velocity of receding galaxies. Hubble <-> scale factor

    Hi there, This is my first post but I've been a spectator for a long time now. So I've been working on some of the basics of cosmic expansion and there is one contradiction that I came upon that I can't seem to resolve. I've looked around some of the similar threads but I couldn't find anything...
  34. binbagsss

    GR: FRW Metric relation between the scale factor & curvature

    Mod note: OP warned about not using the homework template. I have read that 'a(t) determines the value of the constant spatial curvature'.. Where a(t) is the scale factor, and we must have constant spatial curvature - this can be deduced from the isotropic at every point assumption. I'm trying...
  35. TrickyDicky

    The scale factor in flat FRW model

    In the FRW model with Euclidean 3-space, k=0 in the first Friedmann equation makes the density(times a constant) directly related to the Hubble parameter. The Hubble parameter is the time derivative of the scale factor divided by the scale factor itself. My question is: does a Euclidean...
  36. tartaneto

    Comparing FLRW and Scale Factor Metrics

    The generic FLRW metric is dS^2 = a^2.(dx^2 + dy^2) - (c.dt)^2. Is it equivalent to the metric dS^2 = (dx^2 + dy^2) - (c.dt/a)^2 with the scale factor in the denominator of the time dimension? (I suppressed one dimension just for simplicity). Thanks for the help.
  37. marcus

    Switch to scale factor time (universe its own clock)

    Switch to scale factor "time" (universe its own clock) In a sense the universe is its own clock and the pointer-hand is the scalefactor, as in: "when distances were 0.1 what they are today" "when distances were 0.3 what they are today" "when distances were 0.9 what they are today" For some...
  38. H

    Calculating the scale factor (cosmology) for a universe starting at the Big Bang

    Homework Statement Show that by using the Friedmann-equation you can determine the scale factor for a Universe in it's early stages (starting with the Big Bang) to: Homework Equations The equation for the scale factor: Where r(t) is the distance from us to a given galaxy to the time t...
  39. H

    Trying to understand the normalisation of the scale factor to be 1 today

    Hello all! I'm trying to understand the standard normalisation of the scale factor to be set to 1 at today's time. Looking at the first Friedmann Equation for a spatially flat Robertson Walker metric with no cosmological constant gives \frac{\dot{a}^2}{a^2} = \frac{8\pi G}{3}\rho If we...
  40. R

    Understanding the Scale Factor in the Friedman Equation | Cosmology Explained

    I'm having trouble understanding the part in the Friedman equation where a'/a. I can't figure out why it's the derivative of a over a itself. If someone could give me some sort of analogy to something I already understand that would help. I'm pretty sure a is the rate at which the universe...
  41. E

    How to find scale factor at recombination?

    If we know that the temperature of photons was apprx. 3000 K at recombination and the temperature of the CMB is apprx. 2.725 K today, how can we extrapolate the value of the scale factor at recombination? I know that recombination happens at a matter-dominated era, such that the density goes...
  42. P

    Does Friedmann equation allow for complex scale factor?

    Looking at the Friedmann equation H^2=\left[\frac{\dot{a}}{a}\right]^2=\frac{8\pi G\rho}{3}-\frac{kc^2}{a^2} and considering positive curvature, then for the limit where the second term dominates, we're left with \left[\frac{\dot{a}}{a}\right]^2=-\frac{kc^2}{a^2} This implies a complex...
  43. S

    Origins of Scale Factor of FRW Metric and Misc Questions of GR Equations

    In the context of Friedmann's time, 1922, how did he know to make the metric scale factor, a, a function of time when Hubble's redshifts were not yet published? I understand that he took the assumption that the universe is homogenous and isotropic, but does that naturally imply that the universe...
  44. R

    Deriving a scale factor expression for FLRW model

    Hello, I am reading this paper about quantum gravity, trying and failing to follow along in the derivation of this, eq (204) p. 72, expression for the scale factor: a(t) = a_1 \left( 1 + \frac 3 2 ( 1 + \omega ) H_1 (t - t_1) \right)^{\frac 2 {3(1+\omega)}} (I'm not sure what the...
  45. J

    Universe expansion + light speed limit -> scale factor linear in time?

    The Hubble parameter is defined by: H(t) = a'(t) / a(t) where a is the scale factor which is a function of cosmological time t. This definition is equivalent to the Hubble relation: v(t) = H(t) r(t) where v(t) and r(t) are the proper velocity and distance of an object at...
  46. Z

    The Scale Factor & Universe Entropy: Examining Heat Death

    Empirical evidence supports that the scale factor is proportional to the following. a(t) = e^(HT) Where the distance between any two objects are D(t) = a(t)Δx Where x does not measure physical distance, but a conventional coordinate distance. This means that eventually any physical distance...
  47. A

    Does This Cosmological Model Re-Collapse?

    Hello. I have been working through some questions and answers to do with cosmology. One of them asks you to consider a model where: \Omega_{MO}=3 \Omega_{\Lambda O}=0.01 \Omega_{RO}=0 and asks you to show mathematically that the model re-collapses. Following through the math, I get three...
  48. R

    Integrating an equation with scale factor

    Homework Statement Ok, the problem is simple enough, I think. I just think I'm missing something obvious. I have an equation involving the scale factor R(t) and need to integrate it. I am at the first equation and need to get to the second by integrating (with respect of R, I suppose)...
  49. G

    What Is the Age of the Universe According to Different Models of Cosmology?

    Homework Statement From CMB measurements, we know the total density of the Universe matches the critical density to 2 percent. From observations of Supernova it seems that 70 percent of that is in the form of dark energy. But the supernova observations are not as solid as the CMB result...
  50. A

    Elementary level question on Accelerometer scale factor

    Hi, I've received a new accelerometer with a scale factor declared in datasheet as: 800 mV/g @ 1.5g. I googled and searched the forum but still can't fully be sure I understand the accelerometer and its functioning, so please correct me where I'm wrong: One thing that very helped me to get...
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