Determination of the Cosmological Constant based on Type Ia Supernovae Redshift Data. Extended Essay Physics

Transcription

1 Determination of the Cosmological Constant based on Type Ia Supernovae Redshift Data Extended Essay Physics Carlota Corbella Alcántara words March 2014

2 Abstract The cosmological constant, Λ, is one of the key parameters that are hypothesized to determine the evolution and ultimate fate of the Universe. It is largely assumed that it corresponds to the energy density in vacuum. This essay aimed to quantify an accurate value in order to estimate its uncertainty. Redshift data from 580 Type Ia supernovae were analysed using Matlab and LoggerPro software. The research was divided into three main blocks. Firstly an analysis of Hubble s law was carried out in order to obtain an accurate value of the Hubble constant, H 0. If all the data were gathered, H 0 = (43.6 ± 0.4) km s 1 Mpc 1, which is a much lower value than the reference one. This means that the linear version of Hubble s law is not applicable to the range of results which have been considered due to relativistic velocities at high redshifts. For low redshifts, however, H 0 = (71.8 ± 0.7)km s 1 Mpc 1, which deviates by 0.3% from the reference value. These results suggest that the nearest supernovae are experiencing some kind of acceleration. Secondly a model based on the hypothesis of a failure of General Relativity theory at high redshifts was explored, which led to incoherent results (Ω M = 0.16 ± 0.01, which deviates by 40% from the reference value). This does not exclude, however, other breakdowns of General Relativity scenarios as possible explanations for the acceleration of the Universe. Finally a dynamic scalar field model with w 0 = 1.01 ± 0.03 was shown to be consistent with the hypothesis that dark energy might be in the form of a cosmological constant fluid that counteracts gravity. Λ s density parameter, Ω Λ, was calculated to be 0.74 ± Using the H 0 and Ω Λ results, the value obtained for a cosmological constant with w = 1 in an homogeneous, isotropic, four-dimensional space was (1.3 ± 0.4) m 2. This result supports the hypothesis that the Universe is experiencing an accelerated expansion driven by an unknown form of energy that acts as a fluid with vacuum energy density. 297 words

3 Contents 1 Introduction Background Research Question Planification of the Investigation Theoretical Background ΛCDM Universe Supernova Cosmology Project Luminosity Distance Hubble s Law Redshift Flat and Infinite Universe Friedmann Equations Cosmological Constant Theoretical Luminosity Distance Dynamic Scalar Field Modification of General Relativity Theory Loglike Error Propagation Results and Discussion Determination of the Hubble Constant Modification of General Relativity Theory Model Dynamic Scalar Field Model Cosmological Constant Model Conclusion Value of the Cosmological Constant Significance of the Result Limitations of the Research Open Questions

4 5 Acknowledgements 23 6 Appendix 23 A Definition of Parameters and Values of Reference 23 B Formation of Type Ia Supernovae 25 C Density Parameter 26 D Friedmann equations 27 E Theoretical Luminosity Distance 29 F E. Hubble s and M. Humason s Data Analysis 31 G Matlab Codes 32 G.1 Hubble analysis G.2 Modification of General Relativity G.3 Dynamical Dark Energy G.4 Cosmological Constant H Letter from external tutor 36

5 1 Introduction 1.1 Background The nature of the acceleration of the Universe stands out as one of the major unresolved physics phenomena nowadays. Whether it is caused by a constant vacuum energy or by a dynamic scalar field has not yet been determined. It is also possible that Einstein s General Relativity theory fails to be correct and both the acceleration of the Universe and its flatness are not due to some unknown dark energy compound but to a breakdown of this theory. The Hubble constant (H 0 ) and the cosmological constant (Λ) are two parameters which provide evidence of the expansion and acceleration of the Universe, respectively. Their values are systematically updated in scientific papers, and vary slightly depending on the observational methods used. Although the density parameter of Λ, Ω Λ, has been estimated to be 0.74, there is no widespread recognition of the exact value of Λ, which is assumed to correspond to the vacuum energy density. Defining this value with precision may prove useful in further cosmological studies. 1.2 Research Question This essay aims to understand how close to a cosmological constant dark energy is by determining the value of Λ. 1.3 Planification of the Investigation Λ will be calculated by using data from Type Ia supernovae measurements. For such, H 0 will be calculated by analysing the validity of the linear relationship of Hubble s law at both high and low redshifts. A particular case of a breakdown of General Relativity theory will be analysed as well with the aim of excluding it from dark energy constraints. Finally, an analysis of the equation-of-state for Λ will be carried out and the value of Ω Λ, calculated, in order to obtain the final result Λ. 1

6 2 Theoretical Background This section contains an overview of the basic equations in cosmology which will be used throughout the essay for a Λ model of Universe. The notation for parameters was used according to the standard cosmology notation. Their definition and reference values are specified in Appendix A. 2.1 ΛCDM Universe The introduction of the parameter Λ to convert a matter-dominated Universe to a dark energydominated Universe is one of the motivations I had to investigate how it affected different cosmological models. Its value and whether or not it dominates the Universe will be discussed throughout this essay. CDM stand for Cold Dark Matter, that is, particles that move much slower than the speed of light and whose interaction with electromagnetic radiation is negligible. According to WMAP (Wilkinson Microwave Anisotropy Probe) results [1], dark matter comprises 23% of the Universe. Atoms only comprise 4.6% of the Universe. The remaining 71.4% 1 corresponds to dark energy, on which this essay will focus. 2.2 Supernova Cosmology Project At present, data based on Type Ia supernovae are the most powerful tool to study dark energy. The luminosity distance at which these supernovae are moving away from our reference point of view (i.e. the Earth, treated as an object of the major frame of reference of the Milky Way) can be measured with precision since they appear to shine fainter than what has theoretically been inferred (see Section 2.3), thus allowing to deduce either that the Universe is experiencing some kind of acceleration, or that some light is being lost in the way between the supernovae and the Earth. The data used for the analysis have been downloaded from the Supernova Cosmology Project [5]. Data from 580 different Type Ia Supernovae included the name of the supernovae,their redshift (z), their distance modulus (µ) and the error in their distance modulus (δ µ ). These data have been assumed to be reliable and a good depiction of the Universe. 1 WMAP results are based on CMB (Cosmic Microwave Background) data, so slightly different values are expected to the ones considered of reference in this essay. 2

7 2.3 Luminosity Distance Since the absolute magnitude of Type Ia supernovae is known, the distance which separates them from the Earth can be inferred from their apparent magnitudes [9]. µ m app m abs = 5 log 10 (d L ) + 25 (1) d L = 10 µ 25 5 (2) 2.4 Hubble s Law Hubble s law states that objects found in deep space have a recession velocity relative to the Earth that is proportional to z, measured by the Doppler effect [11, p. 202]. For an expanding Universe a system of comoving coordinates can be defined with a d L0 that remains unchanged despite its expansion, which is the distance that separates the body form the Earth at a reference time t 0. The distance between two points will only be related to the comoving distance by terms of this expansion: the scale factor, a(t) a [17]. d L = a d L0 (3) v = ȧ d L (4) v = H d L (5) The farther a celestial body is from the Earth, the higher the speed at which it is moving away (i.e. the recession velocity). However, the linear version of Hubble s law, (5), is only applicable to low z, as relativistic corrections need to be applied for high z data (see Section 2.5). 2.5 Redshift The velocity can also be given as a function of z, which is the fractional amount by which the wavelength associated with photons has been stretched or compressed between its emission and our reception. waves. It is mathematically equivalent to the Doppler formula for sound On physical terms, however, the cause of the stretch or compression of the light wave is not the wave itself, but an enlargement of the space between the emitting object and ourselves [20, p. 58]. z = λ obs λ em λ em (6) z + 1 = λ obs = a(t 0) λ em a a = z (7) 3

8 For z << 1, that is, v << c, as will be shown later, z can be converted to a recession velocity. This implies non-relativistic particles which move much slower than the speed of light. As celestial bodies are mainly formed by atoms, the relativistic Doppler formula for particles moving at nearly the speed of light has not been considered 2 [17, p. 9]. v = c z (8) 2.6 Flat and Infinite Universe If the Universe is assumed to be flat (k = 0, hence Ω k = 0) and the radiation component is assumed to be negligible (Ω R = 0), then in a four-dimensional ΛCDM Universe, Ω = 1 [17, p. 52]. Ω M + Ω Λ = 1 (9) Detailed steps to obtain (9) are shown in Appendix C. 2.7 Friedmann Equations A. Friedmann proposed a set of equations for an expanding homogeneous and isotropic Universe. (10) [17, p. 20] and (11) [17, p. 23] are the standard and acceleration Friedmann equations for a model considering natural units (c = 1) and no curvature in the Universe (k = 0). (ȧ ) 2 H 2 = G 8πρ a 3 (10) ä a = 4πG (ρ + 3p) (11) 3 Detailed steps to obtain (10) and (11) are shown in Appendix D. 2.8 Cosmological Constant In order to be consistent with a static Universe, Einstein introduced a free parameter in his General Relativity equations, Λ [17, p. 51]. (ȧ ) 2 H 2 = G 8πρ a 3 + Λ 3 (12) ä a = 4πG 3 (ρ + 3p) + Λ 3 (13) The value of Λ is related to its density parameter Ω Λ and H 0. Since matter experiences 2 For relativistic velocities, the formula v = c (1+z)2 1 (1+z) 2 +1 should be considered instead. 4

9 gravitational attraction, Λ term would compensate the first term of (13) to give a value of 0 for acceleration. Its validity lies on the assumptions of a homogeneous and isotropic Universe at large cosmological scales and on the consistency of General Relativity theory 3. Λ = 3 H 0 2 Ω Λ (14) 2.9 Theoretical Luminosity Distance Since the Universe is experiencing acceleration, recession velocities for celestial bodies get closer to the speed of light at high z, so relativistic corrections need to be applied to the linear version of Hubble s law. (5) and (8) are substituted by a more complex formula involving a dz, which are a particular formulation of Friedmann equations [21] in which H dependent on z 4. H(z) = H 0 ΩM (1 + z) 3 + (1 Ω M ) (15) A theoretical value of d L can be inferred from the z. d Lth = c z dz (1 + z) H 0 0 ΩM (1 + z) 3 + (1 Ω M ) (16) Detailed steps to obtain (16) are shown in Appendix E Dynamic Scalar Field Another possible explanation for a flat, infinite and accelerating Universe is dark energy in the form of a homogeneous fluid with accelerating properties, which has a specific value at every point in space. As shown in (8), the velocity at which celestial bodies move away from our galaxy 5 depends on z (or on the scale factor a, as it depends on z). The acceleration Friedmann equation (11) can be rewritten as an expression of this constant fluid with equationof-state w 0 p ρ [20, p. 234] at t 0. ä a = 4πG 3 (1 + 3w 0) ρ (17) H(z) = H 0 Ω M (1 + z) 3 + (1 Ω M )(1 + z) 3(1+w0) (18) 3 Einstein then recognised the introduction of Λ constant as his greatest blunder [13] when, in 1929, the Universe was proven to be expanding. Nonetheless, the discovery of an accelerated Universe reintroduced Einstein s free parameter into Friedmann equations to make them consistent with observations. 4 Recall that Ω M + Ω Λ = 1 and that the radiation parameter has been neglected. 5 This is equivalent to the velocity at which supernovae move away from our point of reference in space, the Earth. 5

10 d Lth = c z dz (1 + z) (19) H 0 0 ΩM (1 + z) 3 + (1 Ω M )(1 + z) 3(1+w0) Λ is represented in this model by (1 Ω M )(1 + z) 3(1+w0). Since Ω Λ is unchanged by expansion as it does not depend on z, Λ can only fit with this model as long as w = 1 so that (1 + z) 3(1+w0) = Modification of General Relativity Theory As has been discussed in Section 2.8, homogeneity, isotropy and General Relativity theory must be valid in the standard model of cosmology in order to be consistent with Λ. However, General Relativity may not be valid at large scales in an accelerating Universe. This essay will focus on a particular simplified case described by P. Avelino and C. Martins in A Supernova Brane Scan [8]. Ω r 7 is introduced into Friedmann equations to violate the four-dimensional Einstein s matrix, so that the assumption i Ω i = 1 fails to be correct due to the addition of at least one extra spatial dimension to cause a breakdown of General Relativity theory 8. An additional density parameter, Ω r, is introduced to parametrise extra spatial dimensions which give an alternative explanation to the cause of gravity. H(z) = H 0 Ω r + Ω r + i Ω i (1 + z) 3(1+wi) (20) Ω r = (1 Ω M ) 2 4 (21) The only density parameters in the above defined model of Universe are Ω M and Ω 9 r. The model represented by (20) is thus simplified: ( H(z) = H 0 Ωr + ) Ω r + Ω M (1 + z) 3 (22) H(z) = H 0 ( 1 Ω M 2 ) (1 ΩM ) Ω M (1 + z) 4 3 (23) 6 These equations are restricted to a ΛCDM Universe, where w 0 represents the equation-of-state for Λ. w for the whole Universe evolves with time as several of its components do, and has a value between -1 and 0. 7 Note that Ω R, the density parameter for radiation (assumed to be negligible for the present study), is different from Ω r. 8 This does not imply, however, that Ω k 0, as observational evidence corroborates that k = 0. 9 Since the Universe has been assumed to be flat (Ω k = 0) and the radiation component, negligible (Ω R = 0) 6

11 According to (23) and recalling (16), the theoretical luminosity distance is calculated in this model by: d Lth = c z (1 + z) H 0 0 dz ( ) (24) 1 Ω M (1 Ω 2 + M ) Ω M (1 + z) Loglike The likelihood function L for each data point is given by the parameter χ 2 [12, p. 660]. χ 2 = L(j) = exp 580 { i=1 1 2 (d L (i) d Lth (i, j)) i=1 δ dl (i) 2 (25) } (d L (i) d Lth (i, j)) 2 δ dl (i) 2 Given that the logarithmic function is monotonic, by finding loglike 10 the logarithm of the likelihood is being maximised while χ 2 is being minimised [9]. This happens at the data points for which d L is close in value to d Lth. (26) Loglike(j) ln(l(j)) = 1 2 χ2 (27) Loglike(j) = i=1 (d L (i) d Lth (i, j)) 2 δ dl (i) 2 (28) Since (25) has one degree of freedom, σ errors can be related to this function as 1σ, 2σ and 3σ with χ 2 = 1, χ 2 = 4 and χ 2 = 9 respectively [12, p. 697]. The uncertainty associated to the results obtained using (28) (Figures 3, 6 and 7) will be of 1σ Error Propagation In addition to the 1σ error from the loglike function, other errors are to be propagated throughout the essay. Every compilation will deal with d L calculated from the given data value µ, so δ dl is to be inferred from δ µ. log(10) δ dl = δ µ d L (29) 5 H 0 (Section 3.1) is obtained by calculating the inverse of the slope in Figures 1 and 2. δ H is to be inferred from δ slope. δ H0 = H 0 δ slope slope (30) 10 The more widespread term in cosmological papers, loglike, will be used instead of χ 2 throughout this essay. 7

12 Once having been inferred δ ΩM from 1σ error in Figure 3 (Section 3.2), δ Ωr is calculated from propagation of relative errors. δ Ωr = δ Ω M (1 Ω M ) 2 (31) Finally, propagation of relative errors from H 0 and Ω Λ will give the error for Λ, δ Λ. δ Λ = Λ ( 2 δ H 0 + δ ) Ω Λ H 0 Ω Λ (32) Deviation from reference values is calculated as a percentage error. deviation from accepted value reference value - result obtained reference value 100 (33) 3 Results and Discussion Data were obtained from the Supernova Cosmology Project ([5], see Section 2.2) for 580 different Type Ia supernovae. Firstly an analysis on Hubble s law is carried out, for both high and low z values. Secondly a particular case of a modification of General Relativity theory is analysed as a function of loglike for Ω M. The value which adjusts better to the data according to this model is calculated graphically. Finally a dynamic scalar field model with equation-of-state w 0 is studied as a likelihood function for several Ω M values. Λ is then determined from Ω Λ, calculated graphically. 3.1 Determination of the Hubble Constant The calculation of H 0 is relevant for the present study as it will show evidence for expansion. The new value found for H 0 will be substituted into (14) to give the final Λ result. According to the linear relation of Hubble s law (5), H 0 can be calculated from the regression of a linear plot with variables v and d L in the x-axis and y-axis, respectively. 1 = d L H 0 v (34) 8

13 Figure 1: Compiled with Matlab (see Section G.1 in the Appendix for the code). Analysis of the relationship between velocity and luminosity distance for all supernovae data. For each data point, d L and v have been calculated from the known data µ and z (Equations (1) and (8), respectively). δ slope has been calculated using LoggerPro software automatic linear regression. Matlab linear regression in Figure 1 gives the slope a value of (229 ± 2) km 1 s Mpc. Hence, by calculating the inverse in the slope and its error (30), (34), the value obtained for H 0 is: H 0 = (43.6 ± 0.4) km s 1 Mpc 1 This is nearly 2/3 of the reference value, (72±8) km s 1 Mpc 1. Furthermore, the correlation coefficient is approximately and the y-intercept is at Mpc, far enough from the origin of coordinates to conclude that the data are inaccurate or do not follow a linear tendency. The assumption that z does not have an error, since it was not included in the data, should add imprecision to the result. 9

14 It should be noted, however, that the result is highly precise (0.9%) despite large systematic errors at large velocities, and clear deviations from the linear tendency (the largest relative error for d L corresponds to 10% 11 ). This is due to the large amount of data near the tendency at z << 1, which substantially reduces δ slope and compensates the deviation from the linear regression of data points at large velocities (among 580 data points, approximately 30 can be counted on the plot that lie far from the tendency and with large systematic errors; this corresponds to just 5% of the points). Thus, the linear version of Hubble s law, (5), is not applicable to the range of results which have been considered. As explained in Section 2.5, this is due to an approach to relativistic velocities at high z, ultimately caused by the expansion of the space which separates the supernovae from the observer. For a more accurate determination of H 0, only data points at low z should be considered; those which are still consistent with the linear relationship of Hubble s law. Several analysis were carried out, from z < to z < 0.030, all of them yielding to the same H 0 value (accordingly with the significant figures convention). Figure 2 shows the analysis with z < This corresponds to the supernova with z = 1.215, whose 1% relative error for µ is propagated as a 10% relative error for d L. 10

15 Figure 2: Compiled with Matlab (see Section G.1 in the Appendix for the code). Analysis of the relationship between velocity and luminosity distance for for supernovae at z < For each data point, d L and v have been calculated from the known data µ and z (Equations (1) and (8), respectively). δ slope has been calculated using LoggerPro software automatic linear regression. Matlab linear regression in Figure 2 gives the slope a value of (139 ± 1)km 1 s Mpc. Hence, by calculating the inverse in the slope and its error (30), (34), the value obtained for H 0 is: H 0 = (71.8 ± 0.7) km s 1 Mpc 1 This is a much better approximation to the reference value considered in this essay, from which it only deviates by 0.3% according to (33). Furthermore, R 2 = , which is far more accurate than in the previous result (Figure 1). This inaccuracy might be corroborated by large errors and several points outside the tendency. 11

16 The fact that H 0 is larger at low z than at high z suggests that the nearest supernovae are experiencing some kind of acceleration that makes z not proportional to d L, and that the Universe is dominated by some unknown component that produces a repulsive gravity more significant at large scales 12. Expansion has been confirmed in many other studies, and is part of Friedmann equations. This section confirms that the data and the method used to analyse them do fit with an expanding Universe. 3.2 Modification of General Relativity Theory Model The analysis carried out in this section is a faithful reproduction of the model explained in Section 2.11, which considers a Universe with extra-dimensions where Ω 1 as an alternative explanation to a Universe represented by Λ. The analysis consisted of finding the maximum likelihood for the controlled variable Ω M. Ω M was treated as a free variable, which was allowed to adopt any value from 0 to 1 (in steps of 10 3 ), according to which a d Lth was calculated (24). The difference between d Lth and d L gave the value of the likelihood for each Ω M (see Section 2.12), which is represented in Figure The possibility of light being absorbed by some as yet unidentified factor has not been considered, since it would not modify z but the flux, and Figure 1 and Figure 2 only consider z for the analysis. 12

17 Figure 3: Compiled with Matlab (see Section G.2 in the Appendix for the code). Curve of likelihood with the confidence levels given by χ 2 for the free variable Ω M in a particular case of breakdown of General Relativity theory. Hence, as depicted by Figure 3, a Universe considering a failure of Einstein s General Relativity theory implies (according to 1σ uncertainty): Ω M = 0.16 ± 0.01 which deviates by 40% from the reference value, 0.27 ± 0.02 [16]. The model considered can therefore be excluded since it is incompatible with the known matter density by many σ and it is beyond dark energy constraints 13, despite the result being precise with the data (6%). This 13 Note that the values of Ω and Ω r are of no relevance to the present study, as they do not provide any significant result. 13

18 does not exclude other breakdowns of General Relativity scenarios or an extension of the model considered in this section as possible explanations for the acceleration of the Universe. 3.3 Dynamic Scalar Field Model Only a model with w 0 = 1 is consistent with a cosmological constant so that (1+z) 3(1+w0) = 1 (see Section 2.10). Thus, the likelihood function should have a maximum at w 0 = 1, where w 0 is treated as an independent variable. Figure 4 shows a three-dimensional plot in which Ω M has been treated as an independent variable as well The analysis consisted of finding the maximum likelihood for the free variables w 0 -which was allowed to adopt any value from -1.5 to -0.5 (in steps of 10 3 )- and Ω M -from 0 to 1 in steps of 10 3 as well-, according to which a d Lth was calculated (19). The difference between d Lth and d L gave the likelihood a value for each w 0 (see Section 2.12), which is represented in Figure 4. 14

19 Figure 4: Compiled with Matlab (see Section G.3 in the Appendix for the code). Three-dimensional plot with the confidence levels given by χ 2 for the controlled variables w 0 and Ω M. The maximum likelihood corresponds to loglike = 0, which is where χ 2 is maximised (see Section 2.12). Figure 4 does not give a clear result for w 0, as loglike approaches 0 for a wide range of w 0 values. This could be due to a mathematical degeneracy of the data or an unclearness of the plot, as its three-dimensional shape does not show neither a precise nor exact result. Figure 5, in contrast, corresponds to the two-dimensionalisation of Figure 4, in which the loglike function has been expressed as σ contours for the free parameters w 0 and Ω M. 15

20 Figure 5: Compiled with Matlab (see Section G.3 in the Appendix for the code). Two-dimensional plot with the confidence levels given by χ 2 for the controlled variables w 0 and Ω M. Figures 4 and 5 show an inaccurate and imprecise w 0 result to which an Ω M can be associated. 1σ uncertainty errors cover a wide interval of w 0 values together with a limited difference in Ω M results. This narrowed band shape can be associated with mathematical degeneracy, for which a certain combination of values has almost no effect on χ 2. Although w 0 = 1 (corresponding to an accurate Ω M 0.28) for a 2σ uncertainty, the fact that a large interval of values have the same χ 2 does not allow to distinguish any specific value, and hence Figures 4 and 5 do not give a precise result for the data. Moreover, the most likely value for w 0 seems to be around 1.5, which is physically implausible. Values of w < 1 are usually associated to wrong data, to a breakdown of General Relativity theory or to a failure of energy conservation on cosmological scales. No solid conclu- 16

21 sions can therefore be extracted from Figures 4 and 5. It could either be that the data for the study are incomplete (as only Type Ia supernovae data have been gathered) or that the model of the Universe might not be consistent with Λ. The degeneracy might be rearranged either by fixing one of the free parameters or by combining supernovae data with additional data sets, such as CMB (Cosmic Microwave Background) or BAO (Baryon Acoustic Oscillations). In this essay the former case has been analysed by fixing Ω M at 0.26 (Figure 6) and leaving w 0 as the independent variable on which loglike depends. The procedure is similar to the one detailed for Section Figure 6: Compiled with Matlab (see Section G.3 in the Appendix for the code). Curve of likelihood with the confidence levels given by χ 2 for the free variable w 0 and with a fixed parameter Ω M = The analysis consisted of finding the maximum likelihood for the controlled variable w 0. w 0 was treated as a free variable, which was allowed to adopt any value from -1.5 to -0.5 (in steps of 10 3 ), according to which a d Lth was calculated (19). The difference between d Lth and d L gave the likelihood a value for each w 0 (see Section 2.12), which is represented in Figure 6. 17

22 As depicted by Figure 6, the value obtained for the equation-of-state for Λ is: w 0 = 1.01 ± 0.03 which only deviates by 4% from the reference value, 0.97 ± 0.06 [16]. The small inaccuracy in the result obtained may be corroborated by a combination of other observational data (CMB and BAO) for the reference value 16 as well as a small deviation from the reference value for Ω M. At any rate, the uncertainty in the reference value covers the results obtained in Figure 6. It is, moreover, a precise result (3%). The result w 0 1 implies that 1 Ω M Ω 17 Λ, which means that the equation-of-state is constant for Λ. The fact of it being constant (i.e. it does not depend on z) immediately breaks up with the definition of dynamic and scalar field, which states its change with space and time (hence with z, as it depends on the scale factor a, which depends on time). This is, however, a name called upon convenience, as it permits the field represented by Λ to change with time. This change appears to be inappreciable for the data gathered, so it turns out to be constant. Another implication of the result is p ρ, which is consistent with acceleration caused by a cosmological constant in the form of a fluid that somehow counteracts gravity. 3.4 Cosmological Constant Model Once having shown that the data gathered show evidence of the expansion of the Universe (Section 3.1), that they are incompatible with a breakdown of General Relativity theory (Section 3.2) and that they are consistent with w 0 = 1 (i.e. with a Λ model), the value of Λ can be obtained by calculating the value of Ω Λ. The analysis presented in Figure 7 aims to find the value of Ω M by considering the maximum likelihood (the minimum loglike) using a similar procedure to those in Figures 3 and 6 (Sections 3.2 and 3.3) 18. The theoretical model of the Universe found with specific Ω M and Ω Λ values has been included in Figure 1 to give a better approximation of the data (Figure 8). 16 In fact, if only supernovae and BAO data are taken into account, Kowalski et al. s result is w = 1.01±0.08 and Ω M = 0.28 ± 0.02 for a fixed Ω k = 0. If, in contrast, only supernovae and CMB data are considered, w = 0.96 ± 0.06 and Ω M = 0.26 ± 0.02 for a fixed Ω k = Since the redshift term (1 + z) from (1 Ω M)(1 + z) 3(1+w0) tends to The analysis consisted of finding the maximum likelihood for the controlled variable Ω M. Ω M was treated as a free variable, which was allowed to adopt any value from 0 to 1 (in steps of 10 3 ), according to which a d Lth was calculated (16). The difference between d Lth and d L gave the likelihood a value for each Ω M (see Section 2.12), which is represented in Figure 7. 18

23 Figure 7: Compiled with Matlab (see Section G.4 in the Appendix for the code). Curve of likelihood with the confidence levels given by χ 2 for the free variable Ω M. Hence, as depicted by Figure 7, a Universe with Λ implies: Ω M = 0.26 ± 0.01 Ω Λ = 0.74 ± 0.01 which only deviates by 1% from the reference value, Ω Λ = 0.73 ± 0.01 [22]. Nonetheless, other analysis combining data from different observational methods include the value obtained above 19. Figure 8 shows a curve (in green) with fixed Ω M = 0.26 and Ω Λ = A theoretical luminosity distance d Lth was calculated for each z value (according to (16)). 19 A 2008 study combining supernovae and CMB data provided a result of Ω M = 0.26 ± 0.02 [16] for a fixed Ω k = 0, which is the same value as the one obtained. 19

24 Figure 8: Compiled with Matlab (see Section G.4 in the Appendix for the code). Analysis of the relationship between velocity and luminosity distance for all supernovae data. For each data point, d L and v have been calculated from the known data µ and z (Equations (1) and (8), respectively). A model with Ω M = 0.26 and Ω Λ = 0.74 (in green) has been included to the data in order to compare it with the linear model (in purple), in which Ω M = 1. As depicted by Figure 8, a ΛCDM Universe with Ω M = 0.26 and Ω Λ = 0.74 is much more consistent with the data than a matter-dominated Universe with Ω M = 1, since each d Lth is much closer to the actual d L. 20

25 At this stage the value of the Λ can be calculated by substituting the H 0 and Ω Λ results found into (14) and propagating the error as explained in Section 2.13 (32). Thus: Λ = (1.34 ± 0.02) m 2 4 Conclusion 4.1 Value of the Cosmological Constant Λ has been calculated to be (1.34 ± 0.02) m 2 according to the most likely values of Ω Λ and H 0 for the available data. Although is is largely assumed that Λ corresponds to the energy density of vacuum [10], g cm 3 [23], the result obtained based on Type Ia supernovae data is both slightly more accurate and precise (relative uncertainty of 3%) than a value in the range of m 2, which is largely assumed in several scientific projects. The density parameter for the unknown component represented by Λ has been found to be Ω Λ = 0.74 ± 0.01, which represents an accurate deviation of 1% according to the reference value [22] Significance of the Result The consistency of the result with the accepted value implies that w 0 1 for Λ, hence p ρ, which reinforces the idea that dark energy in terms of a cosmological constant exerts a negative pressure (ρ > 0) and has an effect opposed to gravity which increases at large distances. Some work is exerted on the dark energy fluid to make it move inappreciably slowly and change very little with time despite the expansion of the Universe and its increase in volume. 20 However, different analysis provide slightly different values for the dark energy density Ω Λ depending on the observational techniques used to analyse the data. As a result Ω Λ is constantly being updated. Suzuki et al. s sample, the most recent one, from which the reference value has been extracted, included data from 20 more Type Ia supernovae within high z values (0.623 < z < 1.415) than previous samples. This could explain the deviation obtained from the reference value. Other studies, however, provide results whose error margin include the result obtained in this essay, as has been discussed at the end of Section 3.4 (see Footnote 19). 21

26 This fluid has vacuum energy density. In fact, thawing scalar field models can act as vacuum energy at early times, and evolve away from w = 1 at some point in the expansion [13]. Since the fluid is expressed by a constant Λ, it is unaffected by expansion (space or time), so in principle it cannot be described by a dynamic scalar field. Nevertheless, the data available might not describe a small change in the fluid represented by Λ at large cosmological scales, in which case it could indeed be dynamic and scalar. What type of scalar field dark energy could be remains an open question, although transcendental discoveries in other fields of physics may yield valuable insights to understand the nature of dark energy, such as the discovery of the Higgs field, which is known to be scalar. Dark energy might as well be a mixture of several fluids with no interactions. Further studies might show that dark energy could be dynamical instead of a cosmological constant with vacuum energy. 4.3 Limitations of the Research The results obtained from the theoretical models are a computational idealisation of the Universe in several aspects. First of all, data obtained from Type Ia supernovae are concentrated on low z values (as has been discussed in Section 3.1, see Figure 1) and excludes other observational evidence 21, so the results could be slightly modified with additional data. In fact, one of the main findings in Albrecht et al.s Report of the Dark Energy Task Force is that combinations of at least two techniques must be used to extract solid conclusions on this physics field [7]. In addition to this, the periodic update of H 0 may make the value considered here obsolete when new data become available. This might result in a larger deviation for the value obtained in this essay, added to the fact that no Doppler formula has been considered but instead a linear relation of Hubble s law (see footnote 2), which adds an extra inaccuracy to the result. Finally, the restriction of Universe models to those with k = 0 and Ω M + Ω Λ = 1 (such that Ω R = 0 and Ω k = 0) excludes other possible models. Further studies could include models with Ω k 0 and data from other observational techniques. The introduction of a non-zero density parameter for radiation (Ω R ) could help reduce the deviation from the reference value obtained here. It would be relevant as well to analyse other models that assume a failure of General Relativity theory, as well as the study of the equation-of-state for the different components of the Universe. 21 Such as CMB, BAO, Galaxy Clusters, Weak Lensing and H 0 analysis. 22

27 4.4 Open Questions There is strong evidence that the accelerated expansion of the Universe is due either to an unknown component referred to as dark energy or to a failure of General Relativity theory. The analysis presented here shows that dark energy is consistent with Λ, but the reason why it corresponds to a uniform vacuum energy still remains an open question. To sum up, either a failure of General Relativity theory, a cosmological constant that varies with time or a field with w 1 or other yet unformulated hypothesis could describe the expansion of the fragment of the Universe which we are able to analyse, and by extension the rest of the Universe as it is assumed to be homogeneous and isotropic. The explanation for dark energy will shed light on one of the major unresolved cosmological questions. 5 Acknowledgements I would like to acknowledge Carlos Martins, from Porto University, in the frame of the grant from Programa Joves i Ciència from Catalunya-La Pedrera Foundation for introducing me to the basics of cosmology. I would also like to acknowledge Professor Josep Grané from Universitat Politècnica de Catalunya for giving me access to the necessary software to develop this project, and Mrs. Isabel Corominas for having tutored this essay. Finally, I would like to emphasize and particularly acknowledge the CiMs+CELLEX Foundation for the grant that has given me the opportunity to study the International Baccalaureate in Aula Escola Europea. 6 Appendix This section complements the study carried out. The parameters used in the essay are defined and the reference value is specified. A brief explanation on Type Ia supernovae formation is presented. Detailed explanations to obtain several equations which have been used, mainly based on A. Liddle s book An introduction to modern cosmology [17] are also included. In addition to this, further research on the topic has been carried out by analysing E. Hubble s and M. Humason s data in 1929 to obtain H 0. A Definition of Parameters and Values of Reference In this section the symbols used throughout the essay are presented. The accepted value of several parameters is also specified, as well as the year of their update. 23

28 Parameter a c d L Reference Scale Factor a(t 0 ) = 1 by definition. Speed of light km s 1 [19, p. 1586, 2010] Luminosity Distance (Mpc) d Lth Theoretical Prediction of Luminosity Distance δ E E k E m E p F G H (Mpc) Uncertainty or Error Energy (J) Kinetic Energy (J) Mechanical Energy (J) Gravitational Potential Energy (J) Force (N) Newton s Gravitational Constant. ȧ a Hubble constant H 0 = (72 ± 8)km s 1 Mpc 1 [15, 2011] k Curvature Parameter. Assumed to be 0 L λ Likelihood Wavelength (km) Λ Cosmological Constant (m 2 ) m app m abs m M M pc Apparent Magnitude Absolute Magnitude Mass (kg) Mass of a Sphere with Radius d L (kg) Megaparsec km µ Distance Modulus p Pressure (kg m 1 s 2 ) Q r R 2 Heat (J) Characteristic radius (km) Coefficient of Determination ρ Density (kg m 3 ) ρ crit Critical Density (kg m 3 ) t Time (s) v Velocity of Recession (km s 1 ) V w Volume of a Sphere with Radius a Equation-of-State 0.97±0.06 [16, 2008] It includes all supernovae, CMB and BAO data available at the time and is restricted to Ω k = 0. 24

29 W z Ω Ω M Ω Λ Ω r Ω R x 0 x em x obs ẋ ẍ Work (J) Redshift Density Parameter Density Parameter for Matter 0.27±0.02 [16, 2008] It includes all supernovae, CMB and BAO data available at the time and is restricted to Ω k = 0. Density Parameter for Λ 0.73 ± 0.01 [22, 2012] It is calculated upon data sets from supernovae, CMB, BAO and H 0, and is restricted to Ω k = 0 and w = 1. Density Parameter in a modified General Relativity model Density Parameter for Radiation Assumed to be negligible. Subscript to denote at the present time Subscript to denote at the time of emission Subscript to denote at the time of observation dx dt One dot will be used to represent the first derivative with respect to time. d 2 x dt 2 Two dots will be used to represent the second derivative with respect to time. B Formation of Type Ia Supernovae The reason why Type Ia supernovae have been chosen for the study of the expanding Universe is that they release so much energy that they can outshine their whole galaxy for about 45 days [13]. However, the average observed supernovae in every galaxy are just per year, (Sloan Sky Digital Survey (SDSS) monitored about 70,000 galaxies and detected about 500 Type Ia supernovae during 9 months of observation [18]). Also, the usefulness of Type Ia supernovae is that they are standard candles, that is, their absolute luminosity is known and by extension their luminosity distance. 25

30 Type Ia supernovae most commonly happen in binary systems one of whose stars is a white dwarf. A white dwarf is a star which has recollapsed after having run out of its nuclear fuel. Its ions and radiation pressure are a lot less than the degenerate electron gas (formed by electrons which totally fill the lower energy levels of their atoms and become inactive), caused by the increase in the inner density of the star and the constant temperature. The density of a white dwarf is extremely high compared to conventional densities on the Earth; a teaspoon of white dwarf would weight two tons ( g cm 3 ) [3]. Through the waist of the binary system, the white dwarf can start to grow by accreting mass from its companion, and if it gets close to the Chandrasekhar limit, that is, when its mass is approximately 1.38 Solar masses, or kg, it can explode as a Type Ia supernova, causing the destruction of the white dwarf and the rejection of the other star. This happens because the degenerate electron pressure exerted is no longer capable of counteracting the force of gravity, and the result is a collapse. Since the Chandrasekhar limit is a universal quantity and all binary systems as the one described are thought to follow this limit, the intrinsic luminosities of all Type Ia supernovae are nearly the same and so are their absolute magnitudes ( 19.3 ± 0.3) [4]. This is why they can be referred to as standard candles and their luminosity distance can be easily known, as will be explained below. Another case of Type Ia supernovae explosions is produced when one of the stars of the binary system is a helium star which burns to carbon. This occurs when a massive star transfers most of its material to its companion, resulting in a helium star. C Density Parameter The steps developed in this Section aim to show mathematically why Ω M + Ω Λ = 1, which has been a major assumption throughout the essay. The assumption that k = 0 implies a flat and infinite Universe which will never recollapse and it can be described with Euclidean geometry. From the Friedmann equation, for any value of H there is a critical value of the energy density, ρ crit such that the spatial geometry is flat. ρ crit = 3H2 8πG (35) The total energy density can be measured in terms of the critical density by defining the density parameter Ω. Λ has its own density parameter 22, which is similarly dependent on time, as H is (and despite Λ being constant in space and time). Ω ρ ρ crit = 8πG 3H 2 (36) 22 The steps to obtain (37) will not be analysed in this essay. 26

31 Ω Λ = Λ 3H 2 (37) Substituting (36) into the standard form of the Friedmann equation (12) and rearranging it: H 2 = G 8πG 3 Ω ρ crit k a 2 = H2 (Ω + Ω Λ ) k a 2 H 2 (Ω + Ω Λ 1) = k a 2 Ω + Ω Λ = 1 + k H 2 a 2 which, again, since observational evidence leads to the assumption that k = 0: Ω + Ω Λ = 1 (38) D Friedmann equations The steps developed in this Section aim to give a mathematical explanation for the standard form and the acceleration Friedmann equations. For the Universe to be accelerated, it should counteract somehow the gravitational force, of modulus: F = G M m d L 2 (39) Since the Universe is currently accepted to be homogeneous and isotropic on sufficiently large scales, any point can be considered to be its centre, and is only attracted by the bodies at a small distance. In a four-dimensional space, the point at the centre is attracted by the bodies at a distance d L from it, which is continuously changing given that the Universe is accelerating. This could be understood as an spherical perception of the Universe, with radius d L and volume: V = 4 3 π d L 3 (40) A body placed at this radius will exert a gravitational force on the central point (according to (39) and (40)) of modulus: F = G 4 π ρ d L m 3 For which the gravitational potential energy is: (41) E p = G 4π ρ d L 2 m 3 (42) 27

32 And the mechanical energy: E m = E k + E p = 1 2 m v2 G 4π ρ d L 2 m 3 (43) Substituting into (43) the linear version of Hubble s law (4), v = ȧ d L : E m = 1 2 m ȧ2 d 0 2 G 4π ρ a2 d 0 2 m 3 Rearranging the equation: (ȧ a ) 2 = G 8π ρ 3 2 E m m d 0 2 (44) For bodies to escape the gravitational field of other bodies, E m 0. It is unchanged by time or by the real distance d L 23. Hence, the standard form of the Friendmann equation: (ȧ a ) 2 = G 8π ρ 3 k a 2 c2 (45) The acceleration equation is obtained by deriving with respect to time the standard form of the Friedmann equation: 2 ȧ ä a ȧ 2 a a 2 = G 8 π ρ + 2 k ȧ 3 a 3 c2 (46) The expression of ρ in terms of ρ is the so-called fluid equation, (52). It is derived form the first law of thermodynamics under the assumption of an adiabatic expansion (i.e. there is no heat transfer as all energy is transferred as work) in a a spherical region of space with the scale factor as comoving radius. E = Q + W (47) Ė = p V (48) V = 4π 3 a3 (49) V = 4π a 2 ȧ (50) Recalling Einstein s General Relativity theory where E = m c 2 : E = 4π 3 a3 ρ c 2 Ė = 4π a 2 ρ c 2 ȧ + 4π 3 a3 ρ c 2 (51) 23 The evaluation of 2 Em = k c2 m d 2 a 0 2 goes beyond the scope studied in this essay 28

33 By substituting (48) and (51) into (50) and rearranging: 4π a 2 ρ c 2 da + 4π 3 a3 ρ c 2 = p 4π a 2 a ρ = 3 p ȧ ȧ ρ c2 a c 2 ρ = 3 ȧ a ( p c 2 + ρ ) (52) which is known as the fluid equation. By substituting the fluid equation into the standard form of the derived Friedmann equation, (46), and rearranging: 2 ȧ ä a ȧ 2 a a 2 = G 8π 3 3 ȧ a ȧ a (ä a ( p c 2 + ρ ) + 2 k ȧ a 3 c2 ) 2 = G 4π ( pc 2 + ρ ) + k a 2 c2 ȧ ( p ) a = G 4π c 2 + ρ + G 8πρ 3 + k a 2 c2 k a 2 c2 ( ä a = 4πG ρ + 3p ) 3 c 2 (53) which is the Friedmann equation for acceleration. In order to redimensionate the sample and work with natural units c has been set to 1. Equations (45) and (53) are reduced to: (ȧ ) 2 H 2 = G 8πρ a 3 k a 2 (54) ä a = 4πG (ρ + 3p) (55) 3 which are the form of the Friedmann equations used in the basis of this essay. E Theoretical Luminosity Distance This Section presents detailed steps to give Equation (16), which gives a value of d Lth for each dz in order to mend the relativistic corrections that need to be applied to the linear version of Hubble s law. 29

34 An integral considering dz should be evaluated. Thus, assuming an spheric perception of a homogeneous an isotropic Universe (as has been assumed in (41)), the matrix implies 24 [2]. c dt = a dd Lth (56) d Lth = t t 0 dd Lth = t t 0 c dt a (57) By deriving (7) (a = 1 ȧ 1+z ) and recalling the definition of the Hubble constant (H = a ): da = dz (1 + z) 2 = a2 dz d Lth = a a 0 dt = da H a c da z a 2 H = c dz 0 H(z) = c H 0 z 0 H 0 dz H(z) (58) The standard form of the Friedmann equation [21] can be formulated with the parameter H dependent on z: H(z) = H 0 ΩM (1 + z) 3 + Ω R (1 + z) 4 + Ω k (1 + z) 2 + Ω Λ (59) where Ω R represents the density parameter of the CMB, the thermal radiation left in the Universe after the Big Bang. Its density parameter is very small in comparison with those of matter and Λ; it will be neglected from hereafter. Ω k represents the density parameter for curvature, which is assumed to be 0 in order to be consistent with an accelerating Universe. By substituting (58) into (59) under the assumptions of Ω R = 0 and Ω k = 0: d Lth = c z dz (1 + z) (60) H 0 0 ΩM (1 + z) 3 + Ω Λ Hence, for a ΛCDM Universe, (36) can be expressed as: Ω M + Ω Λ = 1 (61) which turns (60) into: d Lth = c z dz (1 + z) H 0 0 ΩM (1 + z) 3 + (1 Ω M ) (62) 24 Please note the use of d Lth instead of d L to represent a calculated luminosity distance instead of the measured data provided by Supernova Cosmology Project. The difference between d Lth and d L is a key point to obtain the maximum likelihood value proposed by (28). 30

35 F E. Hubble s and M. Humason s Data Analysis In this section H 0 is calculated using the data available in 1929 for E. Hubble and M. Humason [14]. The data included the object name, the velocity of recession v, the apparent magnitude m app and the observed magnitude m obs for 47 different objects, both galaxy clusters and nebulae. The objective of this analysis is to compare the value obtained with the reference one in order to understand the relevance of the observational methods used and data available to reconstruct a reliable model of Universe. As previously done in Section 3.1, the Hubble constant can be calculated from the regression of a linear plot with variables v and d L in the axis. In this case, as a simplification for the calculations, d L has been plotted on the x-axis and v, on the y-axis. Figure 9: Compiled with Matlab. Reproduction of the model described by E. Hubble and M. Humason in For each data point, d L has been calculated from the known data m app and m obs (Equations (1) and (2)). δ slope has been calculated using LoggerPro software automatic linear regression. The value of the slope in Figure 9 corresponds to H 0 for E. Hubble and M. Humason s model: H 0 = (500 ± 40) 10 2 km s 1 Mpc 1 31