我們通過跑動真空能模型來研究變化的宇宙學常數。在我們的模型中Λ(H) =3νH^2+Λ0，其中ν是模型參數Λ0是宇宙學常數。從宇宙微波背景輻射，弱引力透鏡和重子聲學振盪，哈勃參數H(z) 和加權線性增長f(z)σ_8(z) 的数据中，我們發現RVM 模型中ν= (1.37^+0.72_-0.95)×10^-4 且卡方分佈χ^2 值比ΛCDM 模型的值更小。我們還研究了形式為Λ=3αH^2+3βH^4_0 H^-2+Λ0的特殊跑動真空模型，其中α和β是模型參數，Λ0是宇宙常數。該跑動真空能模型具有非解析的能量密度背景演化，只能用數值解去研究。為了避免產生負的暗能量的能量密度，我們規定α>0。從CMB 功率譜和重子聲學振蕩的分析，我們發現α<2.83×10^-4 和β=(-0.2^+3.9_-4.5)×10^-4(95 % C.L.)。
我們另外也研究了Λ(α)CDM 模型的宇宙學常數演化，其中Λ(α) 是隨時間變化的精細結構常數α的函數。根據宇宙學觀測到的隨時間變化的313個α數據點，我们仔細研究了分別具有一個和兩個模型參數的特定Λ(α)CDM 模型的宇宙學限制。我們發現模型參數被約束在10^-4 左右，這與前人研究的結果相似，但更為準確。

We examine the running vacuum model with Λ(H) =3νH^2 +Λ0 to study the running cosmological constant term, where ν is the model parameter and Λ0 is the cosmological constant. From the data of the cosmic microwave background radiation, weak lensing and baryon acoustic oscillation along with the time-dependent Hubble parameter H(z) and weighted linear growth f(z)σ_8(z) measurements, we find that ν=(1.37^+0.72_-0.95)×10^-4 with the best fitted χ^2 value slightly smaller than that in the ΛCDM model.
We extend our work to a special running vacuum model with Λ= 3αH^2+3βH^4_0 H^-2+Λ0, where α and β are the model parameters and Λ0 is the cosmological constant. This RVM has non-analytic background solutions for the energy densities of matter and radiation, which can only be evaluated numerically. From the analysis of the CMB power spectrum and baryon acoustic oscillation along with the prior of α> 0 to avoid having a negative dark energy density, we find that α<2.83 × 10^-4 and β= (-0.2^+3.9_-4.5)×10^-4 (95% C.L.).
We also study Λ(α)CDM models with Λ(α) being a function of the time-varying fine structure constant α. We give a close look at the constraints on two specific Λ(α)CDM models with one and two model parameters, respectively, based on the cosmological observational measurements along with 313 data points for the time varying α. We find that the model parameters are constrained to be around 10^-4, which are similar to the results discussed previously but more accurately.

[1] A. G. Riess et al. [Supernova Search Team], Astron. J. 116, 1009 (1998).
[2] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999).
[3] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006).
[4] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
[5] S. Weinberg, Gravitation and Cosmology, (Wiley and Sons, New York, 1972).
[6] J. P. Ostriker and P. J. Steinhardt, astro-ph/9505066.
[7] N. Arkani-Hamed, L. J. Hall, C. F. Kolda and H. Murayama, Phys. Rev. Lett. 85, 4434 (2000).
[8] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri and D. Scolnic, Astrophys. J. 876, no. 1, 85 (2019).
[9] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 594, A13 (2016).
[10] K. C. Wong et al., arXiv:1907.04869 [astro-ph.CO].
[11] M. Ozer and M. O. Taha, Phys. Lett. B 171, 363 (1986).
[12] J. C. Carvalho, J. A. S. Lima and I. Waga, Phys. Rev. D 46, 2404 (1992).
[13] J. A. S. Lima and M. Trodden, Phys. Rev. D 53, 4280 (1996).
[14] S. Basilakos, A. Paliathanasis, J. D. Barrow and G. Papagiannopoulos, Eur. Phys. J. C 78, 684 (2018).
[15] E. L. D. Perico, J. A. S. Lima, S. Basilakos and J. Sola, Phys. Rev. D 88, 063531 (2013).
[16] I. L. Shapiro and J. Sola, Phys. Lett. B 530, 10 (2002).
[17] J. Sola, J. Phys. Conf. Ser. 453, 012015 (2013).
[18] J. Sola, AIP Conf. Proc. 1606, 19 (2014).
[19] J. Grande, J. Sola, S. Basilakos and M. Plionis, JCAP 1108, 007 (2011).
[20] A. Gomez-Valent, E. Karimkhani and J. Sola, JCAP 1512, 048 (2015).
[21] C. Q. Geng, C. C. Lee and L. Yin, JCAP 1708, 032 (2017).
[22] A. Gomez-Valent and J. Sola, Mon. Not. Roy. Astron. Soc. 448, 2810 (2015).
[23] J. Sola, A. Gomez-Valent and J. de Cruz Perez, Astrophys. J. 811, L14 (2015).
[24] J. Sola, A. Gomez-Valent and J. de Cruz Perez, Astrophys. J. 836, 43 (2017).
[25] J. Sola Peracaula, J. de Cruz Perez and A. Gomez-Valent, EPL 121, 39001 (2018).
[26] S. Basilakos, M. Plionis and J. Solà, Phys. Rev. D 80, 083511 (2009).
[27] C. Q. Geng and C. C. Lee, Mon. Not. Roy. Astron. Soc. 464, no. 2, 2462 (2017).
[28] S. M. Carroll, Living Rev. Rel. 4, 1 (2001).
[29] J. D. Barrow and T. Clifton, Phys. Rev. D 73, 103520 (2006).
[30] I. L. Shapiro and J. Sola, Phys. Lett. B 682, 105 (2009).
[31] I. L. Shapiro, J. Sola and H. Stefancic, JCAP 0501, 012 (2005).
[32] F. E. M. Costa, J. A. S. Lima and F. A. Oliveira, Class. Quant. Grav. 31, 045004 (2014).
[33] A. Gomez-Valent, J. Sola and S. Basilakos, JCAP 1501, 004 (2015).
[34] C. Q. Geng, C. C. Lee and K. Zhang, Phys. Lett. B 760, 422 (2016).
[35] D. A. Tamayo, J. A. S. Lima and D. F. A. Bessada, Int. J. Mod. Phys. D 26, 1750093 (2017).
[36] H. Fritzsch, J. Sola and R. C. Nunes, Eur. Phys. J. C 77, 193 (2017).
[37] S. Basilakos, D. Polarski and J. Sola, Phys. Rev. D 86, 043010 (2012).
[38] R. Schutzhold, Phys. Rev. Lett. 89, 081302 (2002).
[39] C. O’Raifeartaigh, M. O’Keeffe, W. Nahm and S. Mitton, arXiv:1711.06890 [physics.hist-ph].
[40] L. Amendola and S. Tsujikawa, Dark Energy Theory and Observations, (Cambridge University Press, Cambridge, 2010).
[41] C. Beck, Physica A 388, 3384 (2009).
[42] A. I. Khinchin, Mathematical Foundations of Information Theory, Dover Publications, New York (1957).
[43] C. G. Boehmer and T. Harko, Phys. Lett. B 630, 73 (2005).
[44] P. S. Wesson, Mod. Phys. Lett. A 19, 1995 (2004).
[45] C. G. Boehmer and T. Harko, Found. Phys. 38, 216 (2008).
[46] C. Q. Geng, C. C. Lee and L. Yin, “Constraints on a special running vacuum model,” submitted to the Eur. Phys. J. C. (2019).
[47] J. J. Zhang, L. Yin and C. Q. Geng, Annals Phys. 397, 400 (2018).
[48] A. Eddington, Proc. Camb. Philos. Soc. 27, 15 (1931); Relativity Theory of Proton and Electrons, Cambridge University Press (1936).
[49] P. A. M. Dirac, Nature 139, 323 (1937).
[50] H. Weyl, Ann. Physik 59, 129 (1919).
[51] L. Nottale, Mach’s Principle, Dirac’s Large Numbers, and the Cosmological Constant Problem, preprint (1993).
[52] J. J. Zhang, C. C. Lee and C. Q. Geng, Chin. Phys. C 43, 025102 (2019).
[53] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. 538, 473 (2000).
[54] A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002).
[55] C. P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995).
[56] J. C. Fabris, I. L. Shapiro and J. Sola, JCAP 0702, 016 (2007).
[57] H. A. Borges, S. Carneiro and J. C. Fabris, Phys. Rev. D 78, 123522 (2008).
[58] A. M. Velasquez-Toribio, Int. J. Mod. Phys. D 21, no. 03, 1250026 (2012).
[59] J. Sola and A. Gómez-Valent, Int. J. Mod. Phys. D 24, 1541003 (2015).
[60] S. Basilakos and J. Sola, Phys. Rev. D 92, no. 12, 123501 (2015).
[61] J. Sola, Int. J. Mod. Phys. A 31, no. 23, 1630035 (2016).
[62] J. Sola, J. d. C. Perez and A. Gomez-Valent, arXiv:1703.08218 [astro-ph.CO].
[63] W. H. Press and P. Schechter, Astrophys. J. 187 (1974) 425.
[64] J. R. Bond, S. Cole, G. Efstathiou and N. Kaiser, Astrophys. J. 379 (1991) 440.
[65] R. G. Bower, Mon. Not. Roy. Astron. Soc. 248 (1991) 332.
[66] G. Kauffmann and S. D. M. White, Mon. Not. Roy. Astron. Soc. 261 (1993) 921.
[67] C. G. Lacey and S. Cole, Mon. Not. Roy. Astron. Soc. 262 (1993) 627.
[68] V. R. Eke, S. Cole and C. S. Frenk, Mon. Not. Roy. Astron. Soc. 282 (1996) 263.
[69] S. Borgani et al., Astrophys. J. 561 (2001) 13.
[70] T. H. Reiprich and H. Boehringer, Astrophys. J. 567 (2002) 716.
[71] E. Pierpaoli, D. Scott and M. J. White, Mon. Not. Roy. Astron. Soc. 325 (2001) 77.
[72] D. J. Bacon, R. J. Massey, A. R. Refregier and R. S. Ellis, Mon. Not. Roy. Astron. Soc. 344 (2003) 673.
[73] M. L. Brown, A. N. Taylor, D. J. Bacon, M. E. Gray, S. Dye, K. Meisenheimer and C. Wolf, Mon. Not. Roy. Astron. Soc. 341 (2003) 100.
[74] J. R. Bond et al., Astrophys. J. 626 (2005) 12.
[75] O. Lahav et al., Mon. Not. Roy. Astron. Soc. 333, 961 (2002).
[76] C. Zhang, H. Zhang, S. Yuan, T. J. Zhang and Y. C. Sun, Res. Astron. Astrophys. 14, no. 10, 1221 (2014).
[77] R. Jimenez, L. Verde, T. Treu and D. Stern, Astrophys. J. 593, 622 (2003).
[78] J. Simon, L. Verde and R. Jimenez, Phys. Rev. D 71, 123001 (2005).
[79] M. Moresco et al., JCAP 1208, 006 (2012).
[80] E. Gaztanaga, A. Cabre and L. Hui, Mon. Not. Roy. Astron. Soc. 399, 1663 (2009).
[81] M. Moresco et al., JCAP 1605, no. 05, 014 (2016).
[82] D. Stern, R. Jimenez, L. Verde, M. Kamionkowski and S. A. Stanford, JCAP 1002, 008 (2010).
[83] B. A. Reid et al., Mon. Not. Roy. Astron. Soc. 426, 2719 (2012).
[84] M. Moresco, Mon. Not. Roy. Astron. Soc. 450, no. 1, L16 (2015).
[85] N. G. Busca et al., Astron. Astrophys. 552, A96 (2013).
[86] Y. Hu, M. Li and Z. Zhang, arXiv:1406.7695 [astro-ph.CO].
[87] A. Font-Ribera et al. [BOSS Collaboration], JCAP 1405, 027 (2014).
[88] G. Bruzual and S. Charlot, Mon. Not. Roy. Astron. Soc. 344, 1000 (2003).
[89] C. Maraston and G. Stromback, Mon. Not. Roy. Astron. Soc. 418, 2785 (2011).
[90] H. Okada et al., Publ. Astron. Soc. Jap. 68, no. 33, id.47, 17 (2016).
[91] S. de la Torre et al., Astron. Astrophys. 557, A54 (2013).
[92] C. Blake et al., Mon. Not. Roy. Astron. Soc. 415, 2876 (2011).
[93] Y. S. Song and W. J. Percival, JCAP 0910, 004 (2009).
[94] L. Guzzo et al., Nature 451, 541 (2008).
[95] C. Blake et al., Mon. Not. Roy. Astron. Soc. 425, 405 (2012).
[96] S. Satpathy et al. [BOSS Collaboration], [arXiv:1607.03148 [astro-ph.CO]].
[97] R. Tojeiro et al., Mon. Not. Roy. Astron. Soc. 424, 2339 (2012).
[98] C. H. Chuang et al., Mon. Not. Roy. Astron. Soc. 461, no. 4, 3781 (2016).
[99] H. Gil-Marín et al., Mon. Not. Roy. Astron. Soc. 460, no. 4, 4188 (2016).
[100] L. Samushia, W. J. Percival and A. Raccanelli, Mon. Not. Roy. Astron. Soc. 420, 2102 (2012).
[101] M. Tegmark et al. [SDSS Collaboration], Phys. Rev. D 74, 123507 (2006).
[102] W. J. Percival et al. [2dFGRS Collaboration], Mon. Not. Roy. Astron. Soc. 353, 1201 (2004).
[103] C. Howlett, A. Ross, L. Samushia, W. Percival and M. Manera, Mon. Not. Roy. Astron. Soc. 449, no. 1, 848 (2015).
[104] F. Beutler et al., Mon. Not. Roy. Astron. Soc. 423, 3430 (2012).
[105] M. J. Hudson and S. J. Turnbull, Astrophys. J. 751, L30 (2013).
[106] R. Adam et al. [Planck Collaboration], arXiv:1502.01588 [astro-ph.CO].
[107] N. Aghanim et al. [Planck Collaboration], arXiv:1507.02704 [astro-ph.CO].
[108] P. A. R. Ade et al. [Planck Collaboration], arXiv:1502.01591 [astro-ph.CO].
[109] C. Heymans et al., Mon. Not. Roy. Astron. Soc. 432, 2433 (2013).
[110] F. Beutler et al., Mon. Not. Roy. Astron. Soc. 416, 3017 (2011).
[111] L. Anderson et al. [BOSS Collaboration], Mon. Not. Roy. Astron. Soc. 441, no. 1, 24 (2014).
[112] J. Grande, A. Pelinson and J. Sola, Phys. Rev. D 79, 043006 (2009).
[113] A. Gomez-Valent and J. Sola Peracaula, Mon. Not. Roy. Astron. Soc. 478, 126 (2018).
[114] P. Carter, F. Beutler, W. J. Percival, C. Blake, J. Koda and A. J. Ross, Mon. Not. Roy. Astron. Soc. 481 (2018) no.2, 2371.
[115] E. A. Kazin et al., Mon. Not. Roy. Astron. Soc. 441, no. 4, 3524 (2014).
[116] H. Gil-Marín et al., Mon. Not. Roy. Astron. Soc. 460, no. 4, 4210 (2016).
[117] H. Gil-Marín et al., Mon. Not. Roy. Astron. Soc. 477, no. 2, 1604 (2018).
[118] H. Gil-Marín, W. J. Percival, L. Verde, J. R. Brownstein, C. H. Chuang, F. S. Kitaura, S. A. Rodríguez-Torres and M. D. Olmstead, Mon. Not. Roy. Astron. Soc. 465, no. 2, 1757 (2017).
[119] W. J. Percival et al., Mon. Not. Roy. Astron. Soc. 401, 2148 (2010), arXiv: 0907.1660 [astro-ph.CO].
[120] D. J. Eisenstein et al., Astrophys. J. 633, 560 (2005).
[121] J. Stadler and C. Boehm, arXiv:1807.10034 [astro-ph.CO].
[122] R. A. Battye, T. Charnock and A. Moss, Phys. Rev. D 91, 103508 (2015).
[123] I. G. Mccarthy, S. Bird, J. Schaye, J. Harnois-Deraps, A. S. Font and L. Van Waerbeke, Mon. Not. Roy. Astron. Soc. 476, 2999 (2018).
[124] T. M. C. Abbott et al. [DES Collaboration], Phys. Rev. D 98, 043526 (2018).
[125] N. Aghanim et al. [Planck Collaboration], arXiv:1807.06209 [astro-ph.CO].
[126] S. Joudaki et al., arXiv:1906.09262 [astro-ph.CO].
[127] D. Huterer, D. Shafer, D. Scolnic and F. Schmidt, JCAP 1705, 015 (2017).
[128] S. Alam et al. [BOSS Collaboration], Mon. Not. Roy. Astron. Soc. 470, no. 3, 2617 (2017).
[129] A. Pezzotta et al., Astron. Astrophys. 604, A33 (2017).
[130] T. Okumura et al., Publ. Astron. Soc. Jap. 68, no. 3, 38 (2016).
[131] P. Zarrouk et al., Mon. Not. Roy. Astron. Soc. 477, no. 2, 1639 (2018).
[132] A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1917, 142 (1917).
[133] H. Wei, X. B. Zou, H. Y. Li and D. Z. Xue, Eur. Phys. J. C77, 14 (2017).
[134] J. K. Webb, V. V. Flambaum, C. W. Churchill, M. J. Drinkwater and J. D. Barrow, Phys. Rev. Lett. 82, 884 (1999).
[135] J. K. Webb, M. T. Murphy, V. V. Flambaum, V. A. Dzuba, J. D. Barrow, C. W. Churchill, J. X. Prochaska and A. M. Wolfe, Phys. Rev. Lett. 87, 091301 (2001).
[136] J. A. King, J. K. Webb, M. T. Murphy, V. V. Flambaum, R. F. Carswell, M. B. Bainbridge, M. R. Wilczynska and F. E. Koch, Mon. Not. Roy. Astron. Soc. 422, 3370 (2012).
[137] J. K. Webb, J. A. King, M. T. Murphy, V. V. Flambaum, R. F. Carswell and M. B. Bainbridge, Phys. Rev. Lett. 107, 191101 (2011).
[138] J. P. Uzan, Living Rev. Rel. 14, 2 (2011).
[139] H. Wei, Phys. Lett. B 682, 98 (2009).
[140] H. Wei, X. P. Ma and H. Y. Qi, Phys. Lett. B 703, 74 (2011).
[141] H. Terazawa, Nonlin. Phenom. Complex Syst. 20, 241 (2014).
[142] H. Terazawa, Phys. Lett. B 101, 43 (1981).
[143] M. T. Murphy, J. K. Webb, V. V. Flambaum, V. A. Dzuba, C. W. Churchill, J. X. Prochaska, J. D. Barrow and A. M. Wolfe, Mon. Not. Roy. Astron. Soc. 327, 1208 (2001). [144]http://astronomy.swin.edu.au/mmurphy/files/KingJ.12a.VLT+Keck.dat
[145] http://mnras.oxfordjournals.org/content/422/4/3370/suppl/DC1
[146] http:// mnras.oxfordjournals.org/ content/ suppl/ 2013/01/17/ j. 1365-2966.2012.20852.x.DC1/mnras0422- 3370-SD1.txt
[147] M. R. Wilczynska, J. K. Webb, J. A. King, M. T. Murphy, M. B. Bainbridge and V. V. Flambaum, Mon. Not. Roy. Astron. Soc. 454, no. 3, 3082 (2015).
[148] R. G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 81, 061501 (2010).
[149] N. Dalal, K. Abazajian, E. E. Jenkins and A. V. Manohar, Phys. Rev. Lett. 87, 141302 (2001).
[150] Z. K. Guo, N. Ohta and S. Tsujikawa, Phys. Rev. D 76, 023508 (2007).
[151] H. Wei, Phys. Lett. B 691, 173 (2010).
[152] M. Chevallier and D. Polarski, Int. J. Mod. Phys. D 10, 213 (2001); E. V. Linder, Phys. Rev. Lett. 90, 091301 (2003).
[153] Y. Wang, D. Wands, G. B. Zhao and L. Xu, Phys. Rev. D 90, no. 2, 023502 (2014).
[154] Y. Wang, D. Wands, L. Xu, J. De-Santiago and A. Hojjati, Phys. Rev. D 87, no. 8, 083503 (2013).
[155] G. Bazin et al., Astron. Astrophys. 534, A43 (2011).
[156] C. Q. Geng, C. C. Lee and J. L. Shen, Phys. Lett. B 740, 285 (2015).