1 00:00:03,000 --> 00:00:08,110 Let us now define some of the cosmological parameters which are used to 2 00:00:08,110 --> 00:00:11,163 characterize different cosmological models. 3 00:00:11,163 --> 00:00:16,628 The first of them is the Hubble Constant, or more accurately, Hubble Parameter. 4 00:00:16,628 --> 00:00:19,823 Because it's not constant, it changes in time. 5 00:00:19,823 --> 00:00:25,288 It is defined as the ratio of the time derivative of the scale factor by the 6 00:00:25,288 --> 00:00:29,380 scale factor itself. The normalized slope of R(t)t) curve. 7 00:00:29,380 --> 00:00:32,011 Today, its value is roughly 100 kilometers/MegaParsec. 8 00:00:32,011 --> 00:00:36,086 per second per megaparsec. Remember, this is the slope of the Hubble 9 00:00:36,086 --> 00:00:39,907 Diagram which plots velocity versus distance, 10 00:00:39,907 --> 00:00:44,576 thus the units. Since it's length over time in the 11 00:00:44,576 --> 00:00:49,245 numerator and length in denominator, its dimensions are one over time. 12 00:00:51,480 --> 00:00:55,955 Essentially, Hubble Constant gives the scale of the Universe. 13 00:00:55,955 --> 00:01:01,357 It's a slope of a given time and therefore when you draw a tangent on R0 14 00:01:01,357 --> 00:01:06,682 ticker, its intercepts on the time axis gives you inverse of the slope, 15 00:01:06,682 --> 00:01:12,315 which is one over Hubble Constant which is also known as the Hubble Time. 16 00:01:12,315 --> 00:01:17,794 And that's roughly some ten billion years or so in modern cosmological models. 17 00:01:17,794 --> 00:01:24,122 This is close to but does not have to be equal to the age of the Universe at that 18 00:01:24,122 --> 00:01:27,899 time, T0. If you multiply Hubble Time by the speed 19 00:01:27,899 --> 00:01:32,831 of light, which is length over time, then you get a Hubble Length. 20 00:01:32,831 --> 00:01:38,919 The next one is the density parameter. It characterizes the mean density of the 21 00:01:38,919 --> 00:01:43,539 Universe at any given time. If you look at the Friedmann Equation 22 00:01:43,539 --> 00:01:49,272 assuming that the logical constant lambda is equal to zero, Universe is flat if, 23 00:01:49,272 --> 00:01:54,280 has curvature constant to zero. And that can be also expressed as the 24 00:01:54,280 --> 00:01:59,360 critical density over three times H0^2 divided by eight pi G. 25 00:01:59,360 --> 00:02:04,840 Note that the critical density is determined just by the Hubble Constant of 26 00:02:04,840 --> 00:02:08,427 the time. The ratio of the actual density of the 27 00:02:08,427 --> 00:02:13,548 Universe to this value gives the Omega matter, the density parameter. 28 00:02:13,548 --> 00:02:19,498 If it's exact, if the density is exactly equal to critical, then Omega matter is 29 00:02:19,498 --> 00:02:24,705 one and the Universe is flat. The next would be the dark energy density 30 00:02:24,705 --> 00:02:28,007 parameter, just like mass has a mean density so 31 00:02:28,007 --> 00:02:31,659 would, energy. And we know that, from observations now, 32 00:02:31,659 --> 00:02:35,523 that actually Universe does have a mean energy density. 33 00:02:35,523 --> 00:02:40,230 It is defined from cosmological constant according to this formula. 34 00:02:40,230 --> 00:02:45,410 And the total matter energy density would be then simply a sum of the energy 35 00:02:45,410 --> 00:02:48,842 density of matter and energy density of the vacuum. 36 00:02:48,842 --> 00:02:53,619 We also defined the deceleration parameter, often designated with lower 37 00:02:53,619 --> 00:02:59,540 case q or q0 at the present, which in the absence of cosmological constant reduces 38 00:02:59,540 --> 00:03:04,350 to one half of the density parameter. But, if a cosmological constant is 39 00:03:04,350 --> 00:03:09,196 present, its value is different. So, this is what cosmological parameters 40 00:03:09,196 --> 00:03:12,268 do. Hubble's Constant gives the overall scale 41 00:03:12,268 --> 00:03:14,793 of the Universe, temporal and spatial. 42 00:03:14,793 --> 00:03:18,206 It does not depend on the density of the Universe, 43 00:03:18,206 --> 00:03:23,393 cosmological constant or anything else. The other parameters Omega of matter, 44 00:03:23,393 --> 00:03:29,058 Omega vacuum, and so on, determine the shape of the R(t) curves and do not 45 00:03:29,058 --> 00:03:33,243 depend on Hubble's constant. Therefore, their measurements will be 46 00:03:33,243 --> 00:03:36,622 really done a different way and completely independent. 47 00:03:36,622 --> 00:03:41,475 You can determine scale of the Universe without knowing what its density or, or, 48 00:03:41,475 --> 00:03:45,837 or if it has dark energy or not. Conversely, you can find out the mix of 49 00:03:45,837 --> 00:03:50,076 the densities in the Universe without knowing accurately solar scale. 50 00:03:50,076 --> 00:03:54,315 This is done through cosmological tests which we will describe later. 51 00:03:54,315 --> 00:03:59,989 So, let's have a few notes here. Hubble Parameter is often, or always 52 00:03:59,989 --> 00:04:05,950 called Hubble constant, and is sometimes written in the following normalized form. 53 00:04:05,950 --> 00:04:12,555 We divide actual value of Hubble Constant which is usually expressed in kilometers 54 00:04:12,555 --> 00:04:18,418 per second per megaparsec by 100, and that is denoted with lower case h, 55 00:04:18,418 --> 00:04:23,215 normalized Hubble Constant. Then, you can scale for other values of 56 00:04:23,215 --> 00:04:28,413 the Hubble's constant as you wish. Sometimes, it's divided by 70, which is 57 00:04:28,413 --> 00:04:31,445 actually very close to it's present value. 58 00:04:31,445 --> 00:04:35,272 And that's denoted as little h with the subscript 70. 59 00:04:35,272 --> 00:04:41,120 And that is Hubble Constant in units of 70 kilometers per second per megaparsec. 60 00:04:43,240 --> 00:04:48,548 The current value of the critical density, which is determined entirely by 61 00:04:48,548 --> 00:04:54,718 the value of the Hubble Constant is approximately 10^-29 grams per cubic 62 00:04:54,718 --> 00:04:58,377 centimeter, including bulk matter, matter and energy. 63 00:04:58,377 --> 00:05:03,614 So, we can write density parameters in the following form, the Omega of the 64 00:05:03,614 --> 00:05:07,130 matter, the Omega of the energy density or vacuum, 65 00:05:07,130 --> 00:05:13,347 plus Omega curvature all add up to one. Omega curvature really is defined simply 66 00:05:13,347 --> 00:05:19,410 as a deviation of the other two from unit, but it makes a more elegant formula 67 00:05:19,410 --> 00:05:22,829 this way. So now, when we look at the definitions 68 00:05:22,829 --> 00:05:27,522 of these parameters as shown here, we can plug those in Friedmann's 69 00:05:27,522 --> 00:05:30,618 Equation. And here is the Friedmann Equation 70 00:05:30,618 --> 00:05:33,573 expressed through cosmological parameters. 71 00:05:33,573 --> 00:05:39,273 As we can see, as we can see on the left, its value of Hubble Constant and other 72 00:05:39,273 --> 00:05:42,580 parameters participate in the equation. Alright. 73 00:05:42,580 --> 00:05:47,928 Now, next time, we will actually try to solve Friedmann Equation and actually 74 00:05:47,928 --> 00:05:50,040 reach the cosmological models.