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Measuring the expansion of the universe

We draw a figure from where it is possible to measure the number of e-folds of expansion of the universe with a ruler. We find model independent bounds for the number of e-folds during inflation, reheating and radiation. We also give a lower bound to the size of the universe at the beginning of observable inflation. Finally, we show that consistency with a relevant diagram requires the existence of a new form of energy to drive the present expansion of the universe.


I. INTRODUCTION
The idea of the inflationary universe was proposed some 40 years ago [1], (for reviews see e.g., [2], [3], [4]). Throughout these years, a large number of articles has been published investigating different aspects of inflation as well as of the following stage of evolution of the universe, called reheating, where several ways of implementing constraints have been proposed. [5]- [18], (for reviews on reheating see e.g., [19], [20], [21]). One of the important parameters in the study of the evolution of the universe is the number of e-folds N ij ≡ ln a j a i which measures its expansion from the time the scale factor was a i to the time when the scalar factor was a j 1 . Sometimes N ij is even used in place of the inflaton field to describe observables such as the spectral index n s and the tensor-to-scalar ratio r. However, it is important to notice that to date there is no clear and direct way of obtaining bounds for N ij that are independent of a model of inflation.
In this article we present a simple geometrical approach which allows the determination of N ij directly by measuring lengths of segments in Fig. 1. The strategy is based in two simple observations: that it is possible to obtain an upper bound to the total number of e-folds of expansion of the universe from a k during inflation to the pivot scale with wavenumber mode k = k p = 0.05 M pc or to the present scale k = k 0 (see Eq. (6) below) and that the Hubble line specifying the radiation era with ω = 1/3 and slope m = −1 is fixed by determining the pivot scale factor a p at k = k p . The calibration for further measurements is simply obtained by dividing the total number of e-folds of expansion given by Eq. (6) over the length of the corresponding segment; this gives the number of e-folds per unit length.
The outline of the paper is as follows: in Section II we discuss the diagram shown in Fig. 1 and explain how can we possibly measure N ij directly from the figure. At the end of the section we give a lower bound to the size of the universe at the beginning of observable inflation. Section III contains our measurements for the −1/3 < ω < 1 and 0 < ω < 1/3 cases which are displayed in two pairs of columns in the Table I. Section IV presents a postdiction where consistency with a relevant diagram requires the existence of a new form of energy to drive the present expansion of the universe. Section V contains the main conclusions on the results presented in the article as well as a further discussion of possible extensions and applications of our approach. Finally, a collection of definitions for the various N ij ≡ ln a j a i used throughout the article is given in the Appendix. We know that the scale reenters at k p during the radiation era because at the scale k p the scale factor is a p < a eq , where a eq is the scale factor at matter-radiation equality. Lines for the ω = −1/3, ω = 0, ω = 1/3 and ω = 1 cases are also drawn. By measuring their projection on the ln(a) axis with a ruler we can determine bounds for the number of e-folds as shown in Table I. The reader can easily draw her/his own figure by remembering that the fixed main framework is given by the two big triangles tf d t and cf d c which have the same dimensions thus, tf =f d=f c.

II. THE DIAGRAM
Let us briefly explain the general strategy for obtaining bounds to the number of e-folds.
To do this it is necessary to draw a fixed framework containing and constraining the epochs of reheating and of radiation (see between the scales k p and k 0 (see Fig. 1) is and is clearly the same in both sides of the figure. This does not occur, however, with the horizontal "distances". While in the l.h.s triangle it is also equal to 5.4 e-folds the corresponding r.h.s distance is ln(a 0 /a p ), larger than 5.4. To calculate a p we write the Friedmann equation in the form from where it follows that where Ω md,0 = 0.315, Ω rd,0 = 5.443 × 10 −5 , and Ω de = 0.685. The resulting solution to Eq. (4) is a p = 3.6512 × 10 −5 and the number of e-folds from a p to a 0 is N p0 = ln(a 0 /a p ) = 10.2. Since we find that a p < a eq where a eq is the scale factor at matter-radiation equality, the radiation line is fixed. To fix the inflation line we have to fix a k , to do this we use an equation recently proposed by the author [22] N ke where N ke ≡ ln( ae a k ) is the number of e-folds from a k to the end of inflation at a e and N ep ≡ ln( ap ae ) is the number of e-folds from the end of inflation to the pivot scale at a p . The Hubble function can be written as H = π rA s /2 and using the bound r < 0.063, [23], [24] at k p it follows that where A s = 2.1 × 10 −9 is the scalar power spectrum amplitude. Fixing To calculate the bounds it is necessary to know the minimum number of e-folds during radiation. The number of e-folds during the radiation dominated era is given by N rd = ln Tr where T r is the temperature of the universe at the beginning of radiation, T 0 = 2.725K and a eq = 2.97 × 10 −4 . Also, the number of e-folds during the radiation dominated era from the beginning of radiation at a r to the pivot scale factor a p is given by ap ). We see that both numbers depend on the temperature T r however their difference N peq ≡ ln( aeq ap ) = N rd − N rp does not. We find that N peq = 2.1 e-folds (the very small triangle just below k p in the r.h.s of the diagram, amplified in Fig. 2).
The important point is that a p < a eq and k p is inside the radiation dominated era. Thus, the radiation line is represented in the diagram of  Before proceeding with our measurements, just a small comment in relation with the size of the universe at the beginning of observable inflation. From the discussion above we see that there is an upper bound for the total number of e-folds for the observable universe given by measuring the bottom k 0 line or by adding the extra e-folds to the k p line: 112.5+5.4+10.2=128.1 e-folds where 5.4 comes from the expansion between ln(k 0 ) to ln(k p ) during inflation (l.h.s. triangle with lower vertex at ln(k 0 ) in Fig. 1) and 10.2 comes from the number of e-folds from a p to a 0 (r.h.s. composition of triangles with lower vertex at ln(k 0 ) in Fig. 1). From here we find that the scale factor at the beginning of the inflation giving rise to our observable universe is a k 0 = e −128.1 a 0 = 2.3 × 10 −56 . For our present universe of diameter 8.8 × 10 26 m we find that the size of the early universe at the beginning of observable inflation at a k 0 was at least 2.05 × 10 −29 m in diameter, that is, at least 1.27 × 10 6 times larger than the Planck length. Finally we would like to remark that all results presented in this article are model independent, in the sense that no model of inflation has been used to obtain them. We display in the first row bounds for the total number of e-folds from inflation plus post inflation evolution for the pivot scale wavenumber k = k p = 0.05 M pc as well as for the present scale at k = k 0 = a 0 H 0 . These bounds come from Eq. (6) where a bound r < 0.063 to the tensor-to-scalar ratio has been imposed. Columns first and second display bounds for the cases −1/3 < ω < 1, both for k = k p and k = k 0 . Columns third and fourth show bounds for 0 < ω < 1/3, for the k = k p and k = k 0 cases. The number of e-folds during the radiation era is N req ≡ ln aeq ar is bounded as 16.7 < N req < 58.3 . In our definition of N req just given, a r is the scale factor at the beginning of the radiation era and a eq is the matter-radiation equality scale factor, First we will determine bounds when ω = −1/3 (line of slope m = 0 in Fig. 1). This line (segment ra), when composed with the smallest number of e-folds from radiation (segment bc), gives the most post inf lationary e-folds. This means a lower bound for the number of e-folds during inflation N ke , from a k at k = k p to the end of inflation at a r (segment ts). The task is then very simple: we take our ruler and measure e.g., in millimeters the length of the segment ts, multiply it by the calibration obtained before (after Eq. (6)) and the resulting number is the minimum number of e-folds during inflation: 14.6 < N ke (see first column of Table I Table I. In particular we find that the number of e-folds during inflation is bounded as 20 < N ke < 75.5 at k 0 , in excellent agreement with old estimates [25], [26] ranging from 18 to 77 e-folds. Since this is really an article on experimental physics, uncertainties should be given. Thus we convert millimeters into e-folds with the result that all the numbers quoted in Table I should carry a ± 0.2 e-folds but it is not written there for clarity of the presentation.
B. The case 0 < ω < 1/3 Proceeding in a similar way to the previous case we now determine bounds when 0 < ω < 1/3. First we pay attention to the line with ω = 0 and slope m = −1/2 (segment ae) in Results for 0 < ω < 1/3 at k = k p are given in the third column of Table I with those corresponding to the present scale wavenumber k 0 in the fourth column.
Note that for a given ω < 1/3 the maximum N ke occurs when we displace the segment e.g., ea in the ω = 0 case parallel to itself along the radiation line i.e., we slide the vertex at a towards d with b of the segment c b approaching f . In the limit a reaching d we get 0 e-folds for reheating and a maximum number of e-folds for the radiation era (instantaneous reheating and maximum temperature). Thus the maximum number of e-folds for N ke from a k to a d (segment tf ) is 56.2 at k = k p for any ω < 1/3 (see [22] for a concrete example using Starobinsky model of inflation).

IV. THE POSTDICTION
While the previous description is useful for obtaining bounds on the number of e-folds of inflation, reheating and radiation, it also allows us to study the regimen between the pivot scale and the present as illustrated in Fig. 2. This diagram corresponds to the lower r.h.s corner of the main diagram (Fig. 1) and comprises the last bit of radiation ac, the epoch of matter domination c d and the present dark energy dominated era df (see Fig. 2).
In this section we would like to show the usefulness of the diagrammatic approach by elaborating on a postdiction related to the necessity of a new form of energy between matter domination and the present. We will show that consistency with the relevant diagram requires the existence of a new form of energy to drive the present expansion of the universe. Let us assume for a moment that we have no knowledge of the existence of dark energy as we know it today. We can assume that it is the year 1997 or some time before the first supernovae observations indicating a new form of energy driving the expantion of the universe where reported [27], [28]. Thus we focus in the r.h.s lower section of Fig. 1 ending in the present scale k 0 . This section of the figure is amplified in Fig. 2. But because we are assuming no observational evidence of the dark energy we would rather have drawn the (incomplete) diagram shown in Fig. 3, that is, we would be able to draw the segment c d with slope −1/2 and EoS ω = 0 corresponding to matter domination and we would be able to draw the segment ef corresponding to the 10.  Table I). Work in progress indicate that this strategy is particularly useful when studying specific models of inflation and provides a simple picture from where to extract qualitative as well as quantitative information in an almost effortless way [22] . We mention that this approach can be improved in several ways. It is clear that the de Sitter assumption (that inflation is perfectly exponential or, equivalently, that the Hubble function is constant during inflation) is in general not satisfied by models of inflation although we expect that realistic corrections do not significantly alter our results [22] . More importantly is the imposition of extra constraints (for a recent example see [29]) not yet present in our approach. In any case we believe the strategy presented here is useful and very simple, can be extended in several interesting directions (at the end of the Section II we give a lower bound to the diameter size of the universe at the beginning of observable inflation) and can be applied to any model of inflation. Appendix: Defining the number of e-folds N ij Below and for easy reference we define the notation for the various number of e-folds N ij discussed in the article, where N ij ≡ ln a j a i measures the expansion of the universe from the time the scale factor was a i to the time when it was a j .
N ke + N ep = ln ap a k , from a k during inflation to the pivot scale factor a p . N ke + N e0 = ln a 0 a k , from a k during inflation to the present scale factor a 0 . N ke ≡ ln ae a k , from a k during inflation to the end of inflation at a e . N ep ≡ ln ap ae , from the end of inflation to the pivot scale factor a p . N e0 ≡ ln a 0 ae , from the end of inflation to the present at a 0 . N er ≡ ln ar ae , from the end of inflation to the beginning of radiation. N rp ≡ ln ap ar , from the beginning of radiation to the pivot scale factor a p . N r0 ≡ ln a 0 ar , from the beginning of radiation to the present scale factor a 0 . N req ≡ ln aeq ar , from the beginning of radiation to the matter-radiation equality.