Chi-Squared Analysis of Measurements of Two Cosmological Parameters Over Time

Chi-Squared Analysis of Measurements of Two Cosmological Parameters Over Time

Timothy Faerber Uppsala University

VT19

0.1 Abstract

For this project, a historical statistical analysis of the Amplitude of Mass Fluctu-

ations (σ

) and Hubble’s Constant (H

) parameters in the Standard Cosmological

Model was carried out to determine whether or not the given error bars truly repre-

sent the dispersion of values. It was found through analysis of the Chi-Squared (χ

)

values of the data that for σ

(60 data points and χ

between 183.167 and 189.037)

that the associated probability Q is extremely low, with Q = 1.5597 ∗ 10

for the

weighted average and Q = 1.2107 ∗ 10

for the best fit of the data. This was also

the case for the χ

values (163 data points and χ

between 484.3977 and 575.655) of

H

, where Q = 4.2176 ∗ 10

for the linear fit of the data and Q = 1.0342 ∗ 10

for

the weighted average of the data. Through further analysis, it is shown in question,

a linear fit is a better estimate of the data than the weighted average. The general

conclusion is that the statistical error bars have been underestimated (in around

20% of the measurements), or the systematic errors were not properly taken into

account.

1 Introduction

1.1 The Standard Cosmological Model

The Standard Cosmological model is a model that describes the evolution and struc- ture of the universe that we live in from its beginning in the Big Bang and inflation all the way up to the present-day where our universe is dominated by Dark Energy (∼70%) and into the future, when even the slightest difference in a present-day cos- mological parameter could mean the difference between the life and death of the universe and all of its potential inhabitants. The model used to describe the struc- tures we see in our universe has 12 parameters, as can be seen in Table 1, obtained from Croft & Dailey (2015) [29].

Table 1: Parameters in the Standard Cosmological Model

In Table 1, Ω

is the ratio of the current matter density to the critical density,

Ω

is the cosmological constant as a fraction of the critical density, H

is Hubble’s

Constant, σ

is the Amplitude of Mass Fluctuations, Ω

is the baryon density as

a fraction of the critical density, n is the primordial spectral index, β is the red-

shift distortion, m

is the neutrino mass, Γ is Ω

H

/100 kms

Mpc

, Ω

σ

is a

combination of two other parameters that is useful in some peculiar velocity and

lensing measurements, Ω

is the curvature, and w

is the equation of state for the

dark energy parameter [29]. For this study, the two parameters in question are σ

and H

.

1.2 Amplitude of Mass Fluctuations (σ

)

The Amplitude of Mass Fluctuations (σ

) is a parameter in the Standard Cosmolog- ical Model that is concerned with the respective distribution of mass and light in the universe [36]. This is of interest to cosmologists because if σ

' 1, the implication is an ”unbiased” universe in which mass and light are evenly distributed in a sphere of radius R = 8 h

Mpc, whereas if σ

' 0.5, the result would be a ”biased” universe in which mass is distributed more extensively than light in a sphere of radius R = 8 h

Mpc [36]. It is important for cosmologists to study and understand the distri- bution tendencies of mass and light in the universe through σ

because large-scale differences in distribution of matter and energy in the present-day universe tell us about density fluctuations in the early universe on the cluster mass scale of R = 8 h

Mpc [36].

1.3 Hubble’s Constant (H

)

Hubble’s Constant (H

), like the Amplitude of Mass Fluctuations, is a parameter in the Standard Cosmological Model.

Figure 1: The Hubble-Lemaˆitre Law [34]

H

is the slope of the line in the Hubble-Lemaˆitre Law, relating the recession velocity of a galaxy to the distance that it is from an observer. A representation of this law can be seen in Figure 1, obtained from Georges et al. 2017 [42]. In other words, H

relates to the expansion of the universe on cosmic scales and is named after Edwin Hubble who discovered it in 1929 when he realized that galaxies’ velocity away from an observer are directly proportional to their distance from that observer, except for in cases of peculiar velocities [60]. In recent years however, credit has also been given to Georges Lemaˆitre jointly with Hubble for the discovery of this relationship [34]. The parameter is measured in km s

Mpc

and describes the velocity with which a galaxy of distance d from an observer is moving radially away from that observer. For example, if in a hypothetical universe, the value for H

was 50 km s

Mpc

, at a distance of one Mpc away from the observer would be receding at 50 km s

. Since the universe is so large, these recession velocities in the form of redshift (z) are used to describe the distance to far away galaxies rather than units of length. Knowing the exact value of H

is important to cosmologists because H

can also be used to roughly calculate the age of the universe.

1.4 Values and Errors

The first step in the process of determining the best observed values for the Am- plitude of Mass Fluctuations parameter (σ

) and Hubble’s Constant (H

) was to compile a list of several tens of measurements of these parameters. For this spe- cific project, 60 values were compiled for σ

between the years of 1993 and 2019 and 163 values were compiled for H

between the years of 1976 and 2019. In addi- tion to the values themselves, we were interested in a few other details about the measurements, namely the year that those measurements were made and the size of the error bars corresponding to the observed values. A list of all 60 observed measurements for σ

163 observed values for H

can be found in Table 2 and Table 3, respectively, in the appendix. For H

values between 1990 and 2010, values from Croft & Dailey [29] were used All values for H

These tables include the observed values along with their year of observation, size of error, and a reference to the source article. All of the referenced papers were found using the Astrophysics Data System (https://ui.adsabs.harvard.edu/), or from the tables in Croft & Dailey [29].

2 Analysis

2.1 Chi-Squared Test

In order to analyze the trends in our data sets when viewed in scatter plots (see

Figure 2 and Figure 3), a good statistical test is a chi-squared test. The colors

of the data points in the figures included in this analysis indicate the values of the cosmological parameter measurement being made, with the lower measurements being shown in blue and the higher measurements being shown in red. The chi- squared value of a set of data gives the likelihood that the trend observed in the data is occurring due to chance, and is also known as a ”goodness of fit” test [72].

The chi-squared value of a data set is given by the following expression:

χ

=

n

X

i=1

(x

− x

)

σ

(1)

where in the case of our data set x

is the observed value for the parameter, x

is the theoretical value for the parameter (weighted average or linear fit), σ

is the variance of the observed parameter value, and n is the degrees of freedom (the number of observed values minus the number of free parameters of the fit).

−5 0 5 10 15 20

0.4 0.6 0.8 1 1.2 1.4 1.6

Year - 2000 σ

Figure 2: σ

vs Year - 2000

This calculation was carried out twice, first using the weighted average σ

and

H

values as the theoretical values (x

), and then again using the best fit values

from a linear fit designed to minimize the value of χ

as x

. Lines representing both

the weighted average of the data set (blue) and the best fit for the data set (red) that were used to calculate chi-squared can be seen with the data points in Figures 4 and 5.

−25 −20 −15 −10 −5 0 5 10 15 20

20 40 60 80 100 120

Year - 2000 H

Figure 3: H

vs Year - 2000

The weighted average (λ

) of the parameters in question were calculated by weighting each point by the inverse square of the standard deviation (also known as variance) of that value, as shown below, where σ

is the variance of data point n:

λ

=

Pn i=1xn

σ_n²

Pn i=1 1

σ_n²

(2)

For σ

, λ

≈ 0.8038 and for H

, λ

≈ 69.3815. Substituting these weighted

averages in for x

in Equation 1 gives χ

≈ 189.037 for σ

and χ

≈ 575.655 for H

.

In order to find the linear fit of the form:

Y = A + B ∗ X (3)

where Y is the theoretical value for the parameter being analyzed and X is the year of that measurement - 2000. A program was written in Python that

minimizes χ

. For σ

this program returned an A value of 0.7769

and a B value of 1.9999 ∗ 10

. With these values for A and B, the function of the linear fit for σ

becomes

Y = 0.7769 + (1.9999 ∗ 10

) ∗ X (4) For H

this program returned an A value of 65.3793

and a B value of 0.257

, making the function of the linear fit for H

Y = 65.344 + 0.257 ∗ X (5)

as can be seen in Figure 4 and Figure 5, represented by the red line. When

replacing Y from Equation 4 for x

in Equation 1, we find that χ

≈ 183.167 for σ

and χ

≈ 484.3977 for H

.

−5 0 5 10 15 20 0.4

0.6 0.8 1 1.2 1.4

Year - 2000 σ

Figure 4: σ

vs Year - 2000 with Weighted Average and Best Fit Line

2.2 Reduced Chi-Squared

In order to account for the degrees of freedom in the data, a reduced chi-squared test was used to test the goodness of fit for both the weighted average and best fit values. Reduced chi-squared is commonly used for several purposes in astronomy, namely model comparison and error estimation [4]. The reduced chi-squared value of a data set is simply the chi-squared value divided by the degrees of freedom (n) of that data set, as shown below:

χ

= χ

n (6)

In the case of this project, for the weighted average calculations there were 59

degrees of freedom for σ

and 162 degrees of freedom for H

(one free parameter). For

the linear fit calculations there were 58 degrees of freedom for σ

and 161 degrees of

freedom for H

(two free parameters). When applying the χ

value calculated using

the weighted average of the data set to Equation 5, we get a reduced chi-squared

(or, chi-squared per degree of freedom) of 3.2040 for σ

and a reduced chi-squared

value of 3.5534 for H

.

−25 −20 −15 −10 −5 0 5 10 15 20 40

50 60 70 80 90

Year - 2000 H

Figure 5: H

vs Year - 2000 with Weighted Average and Best Fit Line

Likewise, the reduced chi-squared value obtained from the best fit function meant to minimize reduced chi-squared is 3.1581 for σ

and 3.0087 for H

, which, in ac- cordance to theory, is less than those calculated using the weighted average (0.0459 difference for σ

and 0.5547 for H

).

2.3 Errors in A and B Parameters of Best Fit

In order to calculate the error bars for the parameters A and B, a program was written in Python to estimate the range of values for σ

and H

with an error of 1σ added. The 1σ error (68% C.L.) is obtained by adding the value of 2.3 to the minimum of χ

values of 183.167 (σ

) and 484.3977 (H

) in accordance to the process followed in Avni 1976 [5]. The program returned the following results:

• For parameter A

≈ 0.7769 the maximum value producing a 1σ error is 0.7961 and the minimum value is 0.77, producing a final result of A

= 0.7769

.

• For parameter B

≈ 1.9999∗10

, the positive error is 1.9∗10

and the neg-

ative error is 1.8∗10

producing a final result of B

= 1.9999∗10

.

• For parameter A

≈ 65.3793 the maximum value producing a 1σ error is 65.9431 and the minimum value is 65.0001, producing a final result of A

= 65.3793

.

• For parameter B

≈ 0.257 the maximum value producing a 1σ error is 0.296 and the minimum value is 0.217, producing a final result of B

= 0.257

2.4 Statistical Significance, Q

The probability that a calculated χ

value for a data set with n degrees of freedom is due to chance is represented by Q and is given by the following expression:

Q

= [2

Γ( d 2 )]

Z ∞ χ²

(t)

e

dt (7) where Γ

is given by:

Γ

=

Z ∞ 0

t

e

dt (8)

and is known as the generalization of the factorial function to real and complex arguments [45]. In order to determine which values should be removed as bad values, all values were ranked based on their contribution to χ

by increasing value of [x - (best fit x)]/(error of x) and then again by [x - (weighted average x)]/(error of x), where x is the observed value for the parameter in question, and those with the largest contribution to χ

(bad values) were removed first.

2.4.1 Amplitude of Mass Fluctuations

For the value of χ

calculated using the weighted average of σ

(n = 59, χ

≈

189.037), the probability that the observed trend is due to chance is Q = 1.5597 ∗

10

as can be seen in Figure 6. In order to reach a value for Q that is statistically

significant (Q ≥ 0.05), 14 bad values must be removed from the data (n = 45, χ

≈

58.1548), producing a value for Q of 0.0902 as can be seen in Figure 7. For the

value of χ

calculated using the best fit function designed to minimize χ

(n =

59, χ

≈ 183.167), Q = 1.2107 ∗ 10

as can be seen in Figure 8. In order to reach

a statistically significant value for Q, 9 bad values must be removed from the data

(n = 50, χ

≈ 66.698), producing a value for Q of 0.0572 as can be seen in Figure 9.

2.4.2 Hubble’s Constant

For the value of χ

calculated using the weighted average of H

(n = 162, χ

≈ 575.655), the probability that the observed trend is due to chance is Q = 1.0342 ∗ 10

, as shown in Figure 10. In order to reach a value for Q that is statistically significant (Q ≥ 0.05), 36 bad values must be removed from the data (n = 125, χ

≈ 152.5541), producing a value for Q of 0.0538, shown in Figure 11.

Figure 6: Chi-Squared Distribution of Weighted Average σ

Figure 7: Chi-Squared Distribution of Weighted Average σ

Minus 14 Bad Values

Figure 8: Chi-Squared Distribution of Best Fit σ

Figure 9: Chi-Squared Distribution of Best Fit σ

- 9 Bad Values

Figure 10: Chi-Squared Distribution of Weighted Average H

Figure 11: Chi-Squared Distribution of Weighted Average H

- 36 Bad Values

Figure 12: Chi-Squared Distribution of Linear Fit H

Figure 13: Chi-Squared Distribution of Linear Fit H

Minus 24 Bad Values

For the value of χ

(n = 161, χ

≈ 484.3977) calculated using the best fit function designed to minimize χ

, Q = 4.2176 ∗ 10

, shown in Figure 12. In order to reach a statistically significant value for Q, 24 bad values must be removed (n = 136, χ

≈ 160.5772), producing a value for Q of 0.0737, shown in Figure 13.

3 Conclusion

The original Q values for both the weighted average and best fit calculations of the probability of the data for both parameters are extremely low before the removal of bad values. Even though this is the case, a rather large discrepancy can be seen in how many bad values need removing to reach a statistically significant data set (Q ≥ 0.05). For the σ

values, to attain statistical significance, the weighted average calculation needs 14 bad values removed, whereas the best fit calculation needs only 9 bad values removed, making the linear fit of the form y = 0.7769 + (1.999∗10

)x a better estimation of measurements of σ

over time than the weighted average. For the H

values, to attain statistical significance, the weighted average calculation requires 13 bad values be removed, whereas the best fit calculation only needs 10 bad values removed, making the linear fit of the form 65.344 + 0.257x a better estimation of measurements of H

over time than the weighted average.

With the studies of both parameters ending in the aforementioned conclusions, it

is reasonable to conclude that the linear fit with time (Year - 2000) on the x−axis

and measurements of the parameters in question (σ

and H

) on the y−axis is a better estimation of the data than the weighted averaged of the data weighted with the inverse square proportion of the error of each value in question, a linear fit is a better estimate of the data than the weighted average. In particular for H

, we observe a trend to increase the value of the measurements in the last 43 years, which is significant at 6.5-level (B

= 0.257 ± 0.040; zero slope excluded at 6.5-sigma).

Improved research and observations have lead to greater precision in measurements of parameters in the standard cosmological model, and research in this field should not only be continued, but encouraged. In addition to the increasing precision of measurements, it is concluded from this analysis that the error bars of the observed parameters have been largely underestimated in around 20% of the measurements, or the systematic errors of the observation techniques were not fully considered.

In addition, it should also be mentioned that the analysis carried out in this paper can be used to help explain the 4.4σ discrepancy seen between the best estimate in Riess et al. [77] of H

= 74.03 ± 1.42 km s

Mpc

and the predicted value of H

= 67.4 ± 0.5 km s

Mpc

by showing that it is likely that the underestimation of

error bars for H

in around 20% of measurements contributes to the discrepancy

seen between measured values of H

and the predicted value for H

.

4 Appendix

Table 2: σ

Data

Date σ

± Reference

1993 0.57 0.05 [99]

1993 1.415 0.165 [99]

1996 0.7 0.05 [94]

1997 0.75 0.1 [21]

1997 0.95 0.1 [21]

1997 0.8 0.15 [89]

1997 0.66 +0.22 −0.08 [48]

1997 0.66 +0.34 −0.17 [48]

1997 0.83 0.15 [36]

1998 1.2 +0.5 −0.4 [9]

1998 0.49 +0.08 −0.07 [81]

1999 0.68 0.09 [32]

1999 0.74 0.05 [18]

2000 0.72 0.1 [49]

2000 0.77 0.15 [49]

2000 0.79 0.08 [66]

2000 0.68 0.04 [67]

2001 1.17 0.1 [19]

2001 0.66 +0.06 −0.05 [15]

2002 0.94 0.17 [74]

2002 1.04 0.104 [35]

2002 1.04 0.078 [58]

2002 0.9 +0.3 −0.2 [7]

2003 0.76 0.09 [68]

2003 0.98 0.1 [8]

2003 0.73 +0.06 −0.03 [20]

2003 1.17 +0.25 −0.2 [90]

2003 0.77 +0.05 −0.04 [70]

2003 0.695 0.042 [3]

2003 0.84 0.04 [92]

2003 0.97 0.13 [6]

2003 0.97 0.35 [46]

2004 0.966 0.048 [73]

Table 2: σ

Data (continued)

Date σ

± Reference

2004 0.71 0.11 [51]

2004 0.72 0.04 [98]

2004 0.85 +0.38 −0.12 [62]

2004 0.94 0.08 [62]

2004 1.0 0.2 [22]

2005 0.90 0.03 [88]

2005 0.88 0.06 [87]

2005 0.68 0.13 [52]

2005 0.85 0.05 [71]

2005 0.88 +0.12 −0.10 [41]

2006 0.89 0.2 [33]

2006 0.77 0.05 [85]

2006 0.91 0.07 [96]

2006 0.67 + − .04 −0.05 [30]

2007 0.761 +0.049 −0.048 [91]

2007 0.84 0.05 [10]

2007 0.97 0.06 [47]

2008 0.79 0.05 [82]

2009 0.85 +0.04 −0.02 [50]

2009 0.812 0.026 [57]

2010 0.79 0.03 [64]

2010 0.811 0.089 [53]

2014 0.83 0.04 [65]

2015 0.710 0.086 [43]

2018 0.811 0.006 [2]

2018 0.76 0.03 [84]

2018 0.80 0.31 [28]

2019 0.786 0.02 [61]

Table 3: H

Data

Date H

± Reference

1976 78 8 [55]

1976 50.3 4.3 [86]

1979 59 8 [97]

1980 75 15 [93]

1983 80 25 [83]

1984 45 7 [56]

1986 67 8 [44]

1988 89 10 [69]

1990 90 10 [29]

1990 75 25 [29]

1990 52 2 [29]

1991 90 17 [29]

1991 87.5 12.5 [29]

1991 40 12 [29]

1992 86 12 [29]

1992 60 10 [29]

1993 51 12 [29]

1993 47 5 [29]

1993 45 12 [29]

1994 85 5 [29]

1994 52 9 [29]

1995 93 1 [29]

1995 90 17 [29]

1995 78 11 [29]

1995 75 12.5 [29]

1995 71 27.5 [29]

1996 84 4 [29]

1996 76 34 [29]

1996 74 11 [29]

1996 72 12 [29]

1996 67 4.5 [29]

1996 64 6 [29]

1996 62 9 [29]

1996 57 4 [29]

1996 56 4 [29]

1996 56 9 [29]

Table 3: H

Data (continued)

Date H

± Reference

1997 78 50 [29]

1997 69 5 [29]

1997 69 8 [29]

1997 66 10 [29]

1997 64 13 [29]

1997 62 7 [29]

1997 58 7.5 [29]

1997 54 14 [29]

1997 51 13.5 [29]

1998 65 1 [29]

1998 62 6 [29]

1998 62 6 [29]

1998 55 8 [29]

1998 53 9.5 [29]

1998 51.5 12.5 [29]

1998 47 19 [29]

1998 47 14 [29]

1998 44 4 [29]

1999 87 11 [29]

1999 76 14 [29]

1999 74 8 [29]

1999 72 9 [29]

1999 69 15 [29]

1999 64 3.75 [29]

1999 62 5 [29]

1999 61 7 [29]

1999 60 2 [29]

1999 59 17 [29]

1999 55 3 [29]

1999 54 5 [29]

1999 53 33 [29]

1999 42 9 [29]

2000 77 8 [29]

2000 77 4 [29]

2000 71 6 [29]

2000 68 5.4 [29]

2000 65 1 [29]

Table 3: H

Data (continued)

Date H

± Reference

2000 63 10.5 [29]

2000 63 12 [29]

2000 59 33 [29]

2000 58.5 6.3 [29]

2000 52.2 11.65 [29] height

2000 52 5.5 [29]

2001 75 15 [29]

2001 74 5 [29]

2001 66 12.5 [29]

2001 65 6 [29]

2002 84 19 [29]

2002 78 7 [29]

2002 71 4 [29]

2002 66.5 4.7 [29]

2002 63 15 [29]

2002 60 15.5 [29]

2002 44 9 [29]

2003 85 18.5 [29]

2003 84 26 [29]

2003 75 6.5 [29]

2003 72 14 [29]

2003 72 8 [29]

2003 71 3.5 [29]

2003 70 3 [29]

2003 69 12 [29]

2003 69 4 [29]

2003 68.4 1.7 [29]

2003 66 5.5 [29]

2003 65 31 [29]

2003 59 11 [29]

2004 78 3 [29]

2004 73 4.025 [29]

2004 71 8 [29]

2004 71 7.1 [29]

2004 69 8 [29]

2004 67 24 [29]

2004 56 23 [29]

Table 3: H

Data (continued)

Date H

± Reference

2005 73 6.4 [29]

2005 70 5 [29]

2005 66 12.5 [29]

2006 76.9 3.65 [29]

2006 74.92 2.28 [29]

2006 74 2 [29]

2006 74 6.3 [29]

2006 62.3 5.2 [29]

2007 76 8 [29]

2007 74 3.75 [29]

2007 68 10 [29]

2008 61.7 1.15 [29]

2009 74.2 3.6 [29]

2009 71 4 [29]

2009 70.5 1.3 [29]

2010 79.3 7.6 [29]

2010 69 11 [29]

2010 68.2 2.2 [29]

2010 66 5 [29]

2011 73.8 2.4 [78]

2011 74.8 3.1 [78]

2011 74.4 6.25 [78]

2011 68 5.5 [24]

2012 74.3 2.9 [23]

2012 67 3.2 [13]

2012 74.3 2.1 [40]

2013 68 4.8 [17]

2013 68.9 7.1 [75]

2013 76 1.9 [38]

2014 69.6 0.7 [11]

2015 70.6 2.6 [80]

2015 68.11 0.86 [27]

2016 73.24 1.74 [79]

2017 68.3 +2.7 -2.6 [26]

2017 68.4 +2.9 -3.3 [26]

2017 65 +6.5 -6.6 [26]

2017 67.9 2.4 [26]

Table 3: H

Data (continued)

Date H

± Reference

2017 72.5 +2.5 -8 [12]

2017 69.3 4.2 [16]

2018 66.98 1.18 [1]

2018 64 +9 -11 [95]

2018 73.48 1.66 [76]

2018 67 4 [100]

2018 72.72 1.67 [37]

2018 73.15 1.78 [37]

2018 68.9 +4.7 -4.6 [54]

2018 73.3 1.7 [39]

2018 67.4 0.5 [25]

2018 73.24 1.74 [25]

2019 67 3 [59]

2019 72.5 +2.1 -2.3 [14]

2019 67.5 +1.4 - 1.5 [31]

2019 74.03 1.42 [77]

2019 67.8 1.3 [63]

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